ramesh k t nanomaterials mechanics and mechanisms

Nanomaterials
Mechanics and Mechanisms
K.T. Ramesh
Nanomaterials
Mechanics and Mechanisms
123
K.T. Ramesh
The Johns Hopkins University
Baltimore, MD 21218
USA
[email protected]
ISBN 978-0-387-09782-4
e-ISBN 978-0-387-09783-1
DOI 10.1007/978-0-387-09783-1
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2008942797
c Springer Science+Business Media, LLC 2009
All rights reserved. This work may not be translated or copied in whole or in part without the written
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Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
To Arjun, Rohan and Priti, for your love and
patience.
Preface
This book grew out of my desire to understand the mechanics of nanomaterials, and
to be able to rationalize in my own mind the variety of topics on which the people
around me were doing research at the time.
The field of nanomaterials has been growing rapidly since the early 1990s. Initially, the field was populated mostly by researchers working in the fields of synthesis and processing. These scientists were able to make new materials much faster
than the rest of us could develop ways of looking at them (or understanding them).
However, a confluence of interests and capabilities in the 1990s led to the explosive growth of papers in the characterization and modeling parts of the field. That
confluence came from three primary directions: the rapid growth in our ability to
make nanomaterials, a relatively newfound ability to characterize the nanomaterials at the appropriate length and time scales, and the rapid growth in our ability to
model nanomaterials at atomistic and molecular scales.
Simultaneously, the commercial potential of nanotechnology has become apparent to most high-technology industries, as well as to some industries that are traditionally not viewed as high-technology (such as textiles). Much of the rapid growth
came through the inventions of physicists and chemists who were able to develop
nanotechnology products (nanomaterials) through a dizzying array of routes, and
who began to interface directly with biological entities at the nanometer scale. That
growth continues unabated.
What has also become apparent is that much of the engineering community
continues to view nanomaterials as curiosities rather than as the potentially gamechanging products that they can be. This book seeks to provide an entré into the
field for mechanical engineers, material scientists, chemical and biomedical engineers and physicists. The objective is to provide the reader with the connections
needed to understand the intense activity in the area of the mechanics of nanomaterials, and to develop ways of thinking about these new materials that could be useful
to both research and application.
Note that the book does not cover the areas associated with soft nanomaterials
(polymer-based or biologically-derived), simply because I am not knowledgeable
about such systems (and the mechanics can be quite different). This is not intended
to minimize the importance of soft nanomaterials or the potential of soft nanotechnology; the reader will simply have to go elsewhere to encounter those areas.
This book is intended to be read by senior undergraduates and first-year graduate students who have some background in mathematics, mechanics and materials
science. It should also be of interest to scientists from outside the traditional fields
of mechanical engineering and materials science who wish to develop core expertise in this area. Although senior undergraduates should be able to read this book
vii
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Nanomaterials
from cover to cover, they may find it somewhat heavy going as a textbook. First and
second year graduate students should find this book challenging but accessible.
My intent has not been to provide a review of the field but rather to provide a
basic understanding. If the reader puts the book down with an appreciation of the
excitement of the field of nanomaterials and the potential applications in mechanical
engineering, materials science and physics, this book will have achieved its objectives. My personal objective with this book is to provide a means for integrating the
discussions that currently go on in the mechanics and materials communities, and at
the same time to provide an accessible path to those from outside these disciplines
who wish to get involved in one of the most exciting fields of our time.
While this book is intended to be read cover to cover, it can also be used as a reference work. The book should serve effectively as a textbook for a course in nanomaterials or nanomechanics as it relates to materials. It is possible that mechanical
engineers using this book will also want to have a materials science reference available as they move through this volume, and conversely, material scientists using this
book may want to have a basic mechanics of solids book as a companion volume.
An effective scenario would be for graduate students who have taken courses in the
mechanics of solids or the mechanical properties of materials to then take a course
that uses this book as a text. The suggested reading at the ends of the chapter will
be useful for those who wish to pursue a particular topic in greater depth than can
be covered in a book such as this.
I am aware that the book in its current form does not have enough problems at
the end of each chapter to make it ideal as a textbook. However, this is a field at the
cutting edge of nanotechnology, rather than an established area with standard problems that can be given to students without considering their specific backgrounds.
The problems that are provided at the ends of the chapters are intended to provoke discussion and further study, rather than to provide training in specific solution
methods.
We are in the midst of a veritable explosion in nanotechnology, and nanomaterials
are at the heart of it. I hope you put this book down as excited about the field as
I have been in the writing of it.
The Johns Hopkins University
Baltimore, MD
K.T. Ramesh
Acknowledgements
This book developed as a result of conversations with many students and colleagues
working in the area of the mechanics of nanomaterials. What began as a simple
attempt to rationalize my own thinking in this area became a concept for a book after
discussions with Elaine Tham, my editor at Springer. Both she and her assistant,
Lauren Denahy, deserve my thanks for their patience with me as I learned how to
write a large document.
My thanks go also to my assistant, Libby Starnes, who has given me a great deal
of help in organizing my materials for this volume, and in rearranging my schedule
to make time for this book. Her resourcefulness and ability to handle details have
been terrific.
I would like to give special thanks to the several anonymous reviewers of this
book. Their suggestions have greatly improved the manuscript, and I appreciate all
the effort involved.
My research group has been bearing the weight of my absentmindedness as
I worked on this book for nearly a year now, and I would like to express my
gratitude to all of them. They have helped me immeasurably by doing the research that I get to talk about, by being willing to critique ideas, and by helping
me make up for the times that I was unavailable. During the writing of this book,
the research group included Dr. Shailendra Joshi, Reuben Kraft, Bhasker Paliwal,
Dr. Qiuming Wei, Dr. Fenghua Zhou, Jessica Meulbroek, Emily Huskins, Sarthak
Misra, Dr. Jamie Kimberley, Dr. Krishna Jonnalagadda, Dr. Buyang Cao, Cindy
Byer, Rika Wright, Guangli Hu, Cyril Williams, Brian Schuster, Dr. Bin Li, Dr. Nitin
Daphalapurkar and Dr. George Zhang. My thanks to all! Several students have been
involved in reviewing chapters of this book, and to them my special thanks: Guangli
Hu, Cindy Byer, Justin Jones; Christian Murphy made up some of the illustrations.
My colleagues at Hopkins have been a big help, explaining (patiently) to me the
physical concepts in nanomaterials and nanomaterial behavior. This is an exploding
field, and my knowledge of it comes almost entirely through my interactions with
the faculty and students around me at Johns Hopkins. Special thanks to Evan Ma,
Kevin Hemker, Bill Sharpe, Jean-Francois Molinari and Todd Hufnagel.
Finally, this book would not have been possible without the love and support of
my family, who had to put up with my occasional pre-occupation, my prolonged
absences, and a lack of social interaction as I wrote. They kept me sane during the
difficult times, and cheerfully stepped in to help when they could. Rohan reviewed
some of the chapters, and everyone has been involved with the proofreading. Arjun
has been a source of good cheer throughout the process, and Priti, I could not do
this without you. To my family, my love and my thanks for all that you do!
ix
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1
Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Length Scales and Nanotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 What are Nanomaterials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Classes of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Making Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Making dn Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Health Risks Associated with Nanoparticles . . . . . . . . . . . . . .
1.4.3 Making Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
5
6
6
7
8
17
18
18
19
2
Fundamentals of Mechanics of Materials . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Review of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Vector and Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Kinematics of Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Forces, Tractions and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Field Equations of Mechanics of Materials . . . . . . . . . . . . . . . . . . . . .
2.4 Constitutive Relations, or Mathematical Descriptions of Material
Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Plastic Deformation of Materials . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
21
21
25
29
34
35
Nanoscale Mechanics and Materials: Experimental Techniques . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 NanoMechanics Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Characterizing Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
61
62
64
3
35
36
43
53
57
57
59
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Nanomaterials
3.3.1 Scanning Electron Microscopy or SEM . . . . . . . . . . . . . . . . . .
3.3.2 Transmission Electron Microscopy or TEM . . . . . . . . . . . . . .
3.3.3 X-Ray Diffraction or XRD . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Scanning Probe Microscopy Techniques . . . . . . . . . . . . . . . . .
3.3.5 Atomic Force Microscopy or AFM . . . . . . . . . . . . . . . . . . . . .
3.3.6 In situ Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Nanoscale Mechanical Characterization . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Sample and Specimen Fabrication . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Nanoindentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Microcompression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Microtensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Fracture Toughness Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.6 Measurement of Rate-Dependent Properties . . . . . . . . . . . . . .
3.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
65
66
66
68
68
71
71
72
74
82
86
86
91
91
91
4
Mechanical Properties: Density and Elasticity . . . . . . . . . . . . . . . . . . . . . 95
4.1 Density Considered as an Example Property . . . . . . . . . . . . . . . . . . . . 95
4.1.1 The Rule of Mixtures Applied to Density . . . . . . . . . . . . . . . . 96
4.1.2 The Importance of Grain Morphology . . . . . . . . . . . . . . . . . . . 101
4.1.3 Density as a Function of Grain Size . . . . . . . . . . . . . . . . . . . . . 103
4.1.4 Summary: Density as an Example Property . . . . . . . . . . . . . . 105
4.2 The Elasticity of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.1 The Physical Basis of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Elasticity of Discrete Nanomaterials . . . . . . . . . . . . . . . . . . . . 107
4.2.3 Elasticity of NanoDevice Materials . . . . . . . . . . . . . . . . . . . . . 110
4.3 Composites and Homogenization Theory . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Simple Bounds for Composites, Applied to Thin Films . . . . . 113
4.3.2 Summary of Composite Concepts . . . . . . . . . . . . . . . . . . . . . . 116
4.4 Elasticity of Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5
Plastic Deformation of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1 Continuum Descriptions of Plastic Behavior . . . . . . . . . . . . . . . . . . . . 121
5.2 The Physical Basis of Yield Strength . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Crystals and Crystal Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Strengthening Mechanisms in Single Crystal Metals . . . . . . . . . . . . . 132
5.4.1 Baseline Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.2 Solute Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.4.3 Dispersoid Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4.4 Precipitate Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4.5 Forest Dislocation Strengthening . . . . . . . . . . . . . . . . . . . . . . . 135
Contents
xiii
5.5 From Crystal Plasticity to Polycrystal Plasticity . . . . . . . . . . . . . . . . . 136
5.5.1 Grain Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.2 Models for Hall-Petch Behavior . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.3 Other Effects of Grain Structure . . . . . . . . . . . . . . . . . . . . . . . . 150
5.6 Summary: The Yield Strength of Nanomaterials . . . . . . . . . . . . . . . . . 154
5.7 Plastic Strain and Dislocation Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.8 The Physical Basis of Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . 156
5.8.1 Strain Hardening in Nanomaterials . . . . . . . . . . . . . . . . . . . . . 158
5.9 The Physical Basis of Rate-Dependent Plasticity . . . . . . . . . . . . . . . . 160
5.9.1 Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.9.2 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.9.3 Dislocation Substructure Evolution . . . . . . . . . . . . . . . . . . . . . 166
5.9.4 The Rate-Dependence of Nanomaterials . . . . . . . . . . . . . . . . . 167
5.10 Case Study: Behavior of Nanocrystalline Iron . . . . . . . . . . . . . . . . . . . 172
5.11 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.12 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.13 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 176
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6
Mechanical Failure Processes in Nanomaterials . . . . . . . . . . . . . . . . . . . . 179
6.1 Defining the Failure of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.2 Failure in the Tension Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.2.1 Effect of Strain Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.2.2 Effect of Rate-Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.2.3 Multiaxial Stresses and Microscale Processes
Within the Neck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.2.4 Summary: Failure in the Simple Tension Test . . . . . . . . . . . . . 189
6.3 The Ductility of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.4 Failure Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.4.1 Nucleation of Failure Processes . . . . . . . . . . . . . . . . . . . . . . . . 194
6.4.2 The Growth of Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.4.3 The Coalescence of Cracks and Voids . . . . . . . . . . . . . . . . . . . 196
6.4.4 Implications of Failure Processes in Nanomaterials . . . . . . . . 196
6.5 The Fracture of Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6.6 Shear Bands in Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.6.1 Types of Shear Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.6.2 Shear Bands in Nanocrystalline bcc Metals . . . . . . . . . . . . . . . 203
6.6.3 Microstructure Within Shear Bands . . . . . . . . . . . . . . . . . . . . . 207
6.6.4 Effect of Strain Rate on the Shear Band Mechanism . . . . . . . 210
6.6.5 Effect of Specimen Geometry on the Shear
Band Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.6.6 Shear Bands in Other Nanocrystalline Metals . . . . . . . . . . . . . 211
6.7 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.8 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 211
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
xiv
Nanomaterials
7
Scale-Dominant Mechanisms in Nanomaterials . . . . . . . . . . . . . . . . . . . . 215
7.1 Discrete Nanomaterials and Nanodevice Materials . . . . . . . . . . . . . . . 215
7.1.1 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7.1.2 Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7.1.3 Nanofibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.1.4 Functionalized Nanotubes, Nanofibers, and Nanowires . . . . . 226
7.1.5 Nanoporous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.1.6 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.1.7 Surfaces and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.2 Bulk Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.2.1 Dislocation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.2.2 Deformation Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.2.3 Grain Boundary Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7.2.4 Grain Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.2.5 Stability Maps Based on Grain Rotation . . . . . . . . . . . . . . . . . 251
7.3 Multiaxial Stresses and Constraint Effects . . . . . . . . . . . . . . . . . . . . . . 256
7.4 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.5 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.6 Problems and Directions for Research . . . . . . . . . . . . . . . . . . . . . . . . . 257
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8
Modeling Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.1 Modeling and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.2 Scaling and Physics Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.3 Scaling Up from Sub-Atomic Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.3.1 The Enriched Continuum Approach . . . . . . . . . . . . . . . . . . . . . 269
8.3.2 The Molecular Mechanics Approach . . . . . . . . . . . . . . . . . . . . 269
8.4 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8.5 Discrete Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.6 Continuum Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.6.1 Crystal Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.6.2 Polycrystalline Fracture Models . . . . . . . . . . . . . . . . . . . . . . . . 279
8.7 Theoretically Based Enriched Continuum Modeling . . . . . . . . . . . . . . 280
8.8 Strain Gradient Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
8.9 Multiscale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.10 Constitutive Functions for Bulk Nanomaterials . . . . . . . . . . . . . . . . . . 292
8.10.1 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.10.2 Yield Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.11 Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
8.12 Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
8.13 Problems and Directions for Future Research . . . . . . . . . . . . . . . . . . . 295
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Acronyms
There are several acronyms that are commonly used in the field of nanomaterials.
Some of the primary acronyms are listed here for use throughout the book.
AFM
cg
CNT
DDD
EAM
FEM
LEFM
MD
MFM
MWCNT
nc
ns
OIM
QC
SEM
STEM
STM
SWCNT
TEM
ufg
XRD
Atomic Force Microscopy
coarse-grained
Carbon NanoTubes
Discrete Dislocation Dynamics
Embedded Atom Method
Finite Element Method
Linear Elastic Fracture Mechanics
Molecular Dynamics
Magnetic Force Microscopy
Multi-Walled Carbon NanoTube
NanoCrystalline
NanoStructured
Orientation Imaging Microscopy
Quasi Continuum
Scanning Electron Microscopy
Scanning Transmission Electron Microscopy
Scanning Tunneling Microscopy
Single-Walled Carbon NanoTube
Transmission Electron Microscopy
ultra fine grained
X-Ray Diffraction
xv
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
Length scales in mechanics and materials and in nature. The topics
of interest to this book cover a large part of this scale domain, but
are controlled by features and phenomena at the nm scale. Note the
sophistication of natural materials and systems at very small length
scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transmission electron micrographs (see Chapter 3) of
nanocrystalline nickel material produced by electrodeposition
(Integran). (a) Conventional TEM micrograph showing the grain
structure. The average grain size is about 20 nm. (b) High resolution
electron microscopy image of the same material. Note that there
is no additional or amorphous phase at the grain boundaries,
although many models in the literature postulate the existence
of such a phase. Micrographs by Qiuming Wei. Reprinted from
Applied Physics Letters, 81(7): 1240–1242, 2002. Q. Wei, D. Jia,
K.T. Ramesh and E. Ma, Evolution and microstructure of shear
bands in nanostructured fe. With permission from American
Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transmission electron micrograph of nanocrystalline iron produced
by consolidation of a nanocrystalline precursor made by ball
milling. There is a range of grain sizes, and many of the grains show
evidence of the prior plastic work produced by the ball-milling
process. The width of the photograph represents 850 nm. . . . . . . . . . .
Schematic of the die used in Equal Channel Angular Pressing
(ECAP). The included angle φ in this case is about 120◦ , but can
be as small as 90◦ . The angle Φ subtended by the external radius
is also an important parameter. Various obtuse die angles and
dimensions may be used. Very large forces are required to move
the workpiece through such dies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Microstructure of tantalum produced by Equal Channel Angular
Extrusion (ECAE), four passes at room temperature through a
90◦ die. (b) Selected Area Diffraction (SAD) pattern for this
sample, showing the presence of many low angle grain boundaries.
The fraction of grain boundaries that are low-angle rather than
high-angle can be an important feature of the microstructure. . . . . . . .
Schematic of High Pressure Torsion (HPT) process. A thin
specimen is compressed and then subjected to large twist within a
constraining die. The typical sample size is about 1 cm in diameter
and about 1 mm thick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
5
11
13
14
15
xvii
xviii
Nanomaterials
1.7
Microstructure of tungsten produced by high-pressure torsion
(HPT). (a) A bright field TEM micrograph. (b) A dark field TEM
micrograph. (c) A selected area electron diffraction pattern from
this field. The grains contain a high density of defects (because of
the plastic work associated with the HPT process). The grains are
also elongated, with widths on the order of 80 nm and lengths of
about 400 nm. Grain orientation appears to be along the shearing
direction. The selected area electron diffraction shows nearly
continuous rings, with no obvious intensity concentration along
the rings, indicating large angle type grain boundaries in the
sample. Reprinted from Acta Materialia, Vol. 52, Issue 7, P. 11,
Q. Wei, L. Kecskes, T. Jiao, K.T. Hartwig, K.T. Ramesh and E. Ma,
Adiabatic shear banding in ultrafine-grained Fe processed by severe
plastic deformation. April 2004, with permission from Elsevier. . . . . . 16
2.1
The stress tensor σ maps the normal vector n at any point on a
surface to the traction vector t at that point. . . . . . . . . . . . . . . . . . . . . . .
Kinematics of deformation: a body initially in the reference
configuration B0 , with material particles occupying locations
X (used as particle identifiers) is deformed into the current
configuration Bt with each particle occupying spatial positions x. . . .
Internal forces on surfaces S and S’ within a body, and the traction
t at a point on a surface with normal n. The entire body must
be in equilibrium, as must be all subparts of the body. (a) The
entire body, in equilibrium with the external forces. (b) Free body
diagram corresponding to part B1 of the whole body, showing
the traction developed at a point on the surface S because of the
internal forces generated on B1 by part B2 . . . . . . . . . . . . . . . . . . . . . . .
Schematic of a simple tension test. The loading direction is the x1
direction, and the gauge length is L. Note the specimen shape,
designed to minimize the influence of the gripping conditions on
the ends. Specimen shapes may be specified by testing standards. . . .
(a) Schematic of tensile stress-strain curve for ductile metal,
showing yield, an ultimate tensile strength, and subsequent failure.
(b) Experimentally obtained curve for nanocrystalline nickel, with
an average grain size 20 nm. Note the unloading line after initial
yield, with the unloading following an elastic slope. Data provided
by Brian Schuster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Illustration of the biaxial tension of a sheet. (b) 2D stress space
corresponding to the sheet, showing the stress path and possible
yield surface bounding the elastic region. . . . . . . . . . . . . . . . . . . . . . . . .
2.2
2.3
2.4
2.5
2.6
23
26
30
44
45
48
List of Figures
2.7
2.8
3.1
3.2
3.3
3.4
3.5
xix
A sharp planar crack in a very large block of linear elastic material.
The coordinate axes are defined so that the crack front (the line
drawn by the crack tip) is the x3 or z axis. Note that all constant
z planes look identical, so that this is essentially a plane strain
problem in the x1 − x2 plane. The mechanics of this problem is
dominated by the behavior as one approaches the crack tip. . . . . . . . . 54
Two views of the crack in Figure 2.7, with (a) being the side view
and (b) the top view looking down on the crack plane. The crack
front is typically curved, but this can be neglected in the limit as
one approaches the crack tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Schematic of a Scanning Probe Microscopy system (not showing
the feedback loop, tunneling amplifiers and other electronics). A
wide variety of such systems exist, some of which contain only a
subset of the features shown here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single crystal copper specimen and tension gripper for in situ
tensile testing in the SEM, from the work of the group of
Dehm, Kiener et al. (this group has also performed compression
experiments using a similar apparatus) (a) SEM image showing
a single-crystal copper tension sample and the corresponding
tungsten sample gripper before the test at a low magnification.
(b) Sample and gripper aligned prior to loading. Image taken from
(Kiener et al., 2008). Reprinted from Acta Materialia, Vol 56, Issue
3, P. 13, D. Kiener, W. Grosinger, G. Dehm, R. Pippan, A Further
step towards an understanding of size-dependent crystal plasticity:
In situ tension experiments of miniaturized single-crystal copper
samples. Feb. 2008, with permission from Elsevier. . . . . . . . . . . . . . . .
Schematic of the NanoIndenter, showing both actuation and force
and position sensing. Such a device may fit comfortably on a large
table top, although vibration isolation may be desirable. . . . . . . . . . . .
Example of force-displacement curve obtained during the
nanoindentation of a nanocrystalline nickel material with an
average grain size of 20 nm. Data provided by Brian Schuster. . . . . . .
(a) Schematic of a microcompression experiment, showing
micropillar and flat-bottomed indenter tip. (b) SEM micrograph
showing a micropillar of a PdNiP metallic glass produced by
focused ion beam machining. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
71
73
74
75
xx
Nanomaterials
3.6
(a) Finite element mesh used for 2D axisymmetric model of the
micropillar in a microcompression experiment. (b) Computed
effective (von Mises) stress distribution in a sample after plastic
compression. Note the nonuniformity at the root of the pillar,
indicating the importance of the root radius (also called the fillet
radius). Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6,
H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design
of accurate microcompression experiments, January 2006, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Input and simulated stress-strain curves for various assumed root
radii (detailed differences observable in the inset) as computed
from the finite element model presented in the last figure. The
“input” curve is the input material behavior used in the finite
element simulations, and the “output” stress-strain curves are
obtained from the forces and displacements computed from the
simulations and then processed in the same way as the force
and displacement data is processed in the experiments. If the
experimental design were to be perfect, the output curves would
be identical to the input curve. Note that the simulated curves
are all above the input data, demonstrating the effect of the end
condition. The outermost curve has the largest root radius, equal
to the radius of the cylinder. As the root radius is decreased, the
simulated curves approach the input curve but are always above
it. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6,
H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design
of accurate microcompression experiments, January 2006, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Computed variation of apparent elastic modulus (normalized by
true elastic modulus) with fillet radius size at fixed aspect ratio,
using the finite element simulations discussed in the text. The
effects of the Sneddon and modified Sneddon corrections are also
shown. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6,
H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design
of accurate microcompression experiments, January 2006, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Schematic of a microtension testing apparatus, showing the
actuation and sensing systems. The air bearing is an important
component. The strain is computed from displacement fields
measured using Digital Image Correlation (DIC) software analysis
of images from the camera. Illustration due to Chris Eberl. . . . . . . . . .
3.10 Example of stress strain curve obtained on nanocrystalline nickel
(20 nm grain size) in uniaxial tension using a microtension setup.
The sample was a thin film. Data due to Shailendra Joshi. . . . . . . . . . .
76
77
79
84
85
List of Figures
3.11 Experimental techniques appropriate for various ranges of strain
rate. The range of rates identified as “specialized machine” is very
difficult to reach. Very few laboratories in the world are able to
achieve strain rates higher than 104 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Schematic of the compression Kolsky bar (also known, incorrectly,
as the split-Hopkinson pressure bar). The projectile is usually
launched towards the input bar using a gas gun. Specimen surfaces
must be carefully prepared for valid experiments. . . . . . . . . . . . . . . . . .
3.13 Stress strain curves obtained on 5083 aluminum using the
compression Kolsky bar at high strain rates (2500 per second).
The lowest curve represents quasistatic behavior. In general, the
strength appears to increase with increasing strain rate, but note
the anomalous softening at the highest strain rate (perhaps due to
thermal softening). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14 The Desktop Kolsky Bar – a miniaturized compression Kolsky bar
arrangement developed by Jia and Ramesh (2004). This device is
capable of achieving strain rates above 104 per second, and can
fit on a standard desktop. Specimen sizes can be on the order of a
cubic millimeter. The large circular object behind the bar is part of
a lighting system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
xxi
87
88
90
90
The linear rule of mixtures. Calculated variation of density in a
nanocrystalline material with volume fraction of grain boundary,
based on Equation (4.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Schematic of (a) a cubical grain of size d with a grain boundary
domain of thickness t, and (b) the packing of such grains to form a
material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Variation of grain volume fraction, grain boundary volume fraction,
triple junction volume fraction, and corner junction volume fraction
with normalized grain size β = dt for the cubic grain morphology.
The junction volume fractions become major contributors
when d ≈ t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Scanning electron micrograph showing the presence of pores
(dark regions) on the surface of a nanocrystalline nickel material
produced by an electroplating technique. . . . . . . . . . . . . . . . . . . . . . . . . 101
Another possible space-filling morphology, using hexagonal tiles
of side s and height h = α s, where α is the aspect ratio of the tile.
The grain boundary domain thickness remains t. . . . . . . . . . . . . . . . . . 102
Variation of grain volume fraction, grain boundary volume fraction,
triple junction volume fraction, and corner junction volume fraction
with normalized grain size β = dt for the hexagonal prism grain
morphology of Figure 4.5 (with an aspect ratio α = 1). The
junction volume fractions become major contributors when d ≈ t. . . . 103
xxii
Nanomaterials
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
5.1
Variation of grain volume fraction, grain boundary volume fraction,
triple junction volume fraction, and corner junction volume fraction
with normalized grain size β = dt for the hexagonal prism grain
morphology of Figure 4.5, with two aspect ratios (10 and 0.1)
corresponding to rods (top) and plates (bottom). . . . . . . . . . . . . . . . . . . 104
Predicted variation of overall density of polycrystalline material
(normalized by ρsc ) with grain size for the cube and hexagonal
prism morphologies (with an aspect ratio α = 10). The assumed
parameters are ρgb = 0.95ρsc , ρt j = 0.9ρsc , and ρc j = 0.81ρsc . . . . . . . 105
Typical interatomic pair potential U(r), showing the equilibrium
position r0 of the atoms. The curvature at the bottom of the
potential well (at the equilibrium position) corresponds to the
effective elastic stiffness of the bond. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Surface effects on a spherical nanoparticle (e.g., surface tension).
Such effects can have a significant impact on the behavior of the
nanoparticle, particularly with respect to its interaction with the
environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Schematic of a thin film on a substrate. The film thicknesses of
interest to industry are typically submicron. . . . . . . . . . . . . . . . . . . . . . . 110
Schematic of a polycrystalline thin film on a substrate. Note the
typical columnar microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Process of homogenization of a composite material. (a) Original
heterogeneous material. (b) Equivalent homogenized material. . . . . . . 112
(a) Schematic of a columnar microstructure for a thin film.
(b) Schematic of a layered microstructure, which can be viewed as
the columnar microstructure loaded in the orthogonal direction to
that shown in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Variation of effective modulus (for fictitious material) with grain
boundary volume fraction, based on Equations (4.30) and (4.34). . . . . 115
Variation of effective Young’s modulus with normalized grain size
in a bulk nanocrystalline material, where the modulus in the grain
boundary domain is defined to be ζ Eg = ζ Esc with ζ = 0.7 in this
figure. The grain size is normalized by the effective thickness t of
the grain boundary domain, typically assumed to be about 1 nm. If
the latter thickness is assumed, the horizontal axis corresponds to
grain size in nm. E1 and E2 correspond to Equations 4.30 and 4.34. . . 117
Schematic of slip in a crystalline solid under shear. (a) Original
crystal subjected to shear stress. (b) Crystal after deformation
(slip) along the shearing plane shown in (a). The final atomic
arrangement within the deformed crystal remains that of the perfect
crystal, except near the free surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
List of Figures
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
xxiii
Dislocations are visible in the transmission electron microscope.
The picture shows dislocations (the lines marked by the black
arrow) in a magnesium alloy (ZK60). Note the length scale in the
picture. This micrograph was taken by Bin Li. . . . . . . . . . . . . . . . . . . . 124
Expanded version of Figure 5.1 showing the primary deformation
mechanism that leads to plasticity in metals: the motion of
line defects called dislocations. Edge dislocations are shown
gliding along the slip plane. The inset shows an expanded
view of an edge dislocation, amounting to an extra plane
of atoms in the lattice (with the trace of the extra plane
on the slip plane defined as the line defect). The motion
of many dislocations results in the macroscopic slip step
shown in (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Schematic of slip with an edge dislocation (top figure) and screw
dislocation (bottom figure), showing the Burgers and line vectors in
each case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Examples of basic crystal structures that are common in metals:
body centered cubic (one atom in the center of the cube), face
centered cubic (atoms at the center of each cube face), and
hexagonal close packed structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Typical planes in a cubic crystal defined using Miller indices. The
(111) plane and similar {111} planes are close-packed planes in a
face-centered-cubic (fcc) crystal and therefore are part of typical
operating slip systems (together with < 110 > type directions). . . . . . 129
Schematic of gliding dislocation bowing around dispersoids that
are periodically spaced a distance L apart along a line. . . . . . . . . . . . . . 134
Schematic of a polycrystalline microstructure with a variety of
plastic strains (represented by color or shading) within each
individual grain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Schematic of grain boundary ledges in a polycrystalline material.
In the grain boundary ledge model, these ledges are believed to act
as dislocation sources and generate a dislocation network. . . . . . . . . . 140
xxiv
Nanomaterials
5.10 A summary of the three basic models for the observed Hall-Petch
behavior. (a) A dislocation pileup at a grain boundary. (b) Grain
boundary ledges in a polycrystalline material. (c) A schematic of
the geometrically-necessary dislocation model. On the left side of
this subpart of the figure, the individual crystals that make up this
polycrystalline material are assumed to have slipped along their
respective slip systems, resulting in misfits between the grains.
However, the polycrystalline material must remain compatible
at the grain boundaries, assuming that voids do not open up at
the boundaries. These misfits can be accommodated by creating
a new set of dislocations (the so-called geometrically-necessary
dislocations) that then result in a dislocation distribution near the
individual grain boundaries. Only one such GND distribution is
shown in one grain on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.11 The variation of the flow stress (measured at 4% strain) with grain
size for consolidated iron, taken from the doctoral dissertation of
Dexin Jia at Johns Hopkins. The variation of hardness (defined here
as H3v , where Hv is the Vickers hardness) is also shown. . . . . . . . . . . . . 144
5.12 Evidence of reduction of hardness with decreasing grain size in
a Ni-P material, from Zhou, Erb et al., Scripta Materialia, 2003
(Zhou et al., 2003). The right-hand axis represents the Vickers
Hardness number HV. Reprinted from Scripta Materialia, Vol. 48,
Issue 6, page 6, Y. Zhou, U. Erb, K.T. Aust and G. Palumbo,
The effects of triple junctions and grain boundaries on hardness
and Young’s modulus in nanostructured Ni-P. March 2003, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.13 Compendium of data on copper in a Hall-Petch-style plot by
Meyers et al. (2006), showing the scatter in data at small grain
sizes but the general deviation from classical Hall-Petch behavior.
Reprinted from Progress in Materials Science, Vol. 51, Issue 4,
Page 130, M.A. Myers, A. Mishra, D.J. Benson, Mechanical
properties of nanocrystalline materials. May 2006, with permission
from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.14 Hall-Petch plots for copper, iron, nickel and titanium compiled
by Meyers et al. (2006), showing the deviation from classical
Hall-Petch at small grain sizes. Note the strength appears to
plateau but not decrease with decreasing grain size. Reprinted
from Progress in Materials Science, Vol. 51, Issue 4, Page 130,
M.A. Myers, A. Mishra, D.J. Benson, Mechanical properties
of nanocrystalline materials. May 2006, with permission
from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
List of Figures
xxv
5.15 Measured grain size distribution in a consolidated iron sample,
based on 392 grains measured from TEM images such as that
shown above (Jia et al., 2003). While some of the grains are clearly
in the nanocrystalline range, others are more than 100 nm in size.
Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia,
K.T. Ramesh, E. Ma, Effects of nanocrystalline and ultrafine grain
sizes on constitutive behavior and shear bands in iron. July 2003,
with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.16 TEM micrographs showing microstructure of a cryomilled 5083
aluminum alloy in (a) extruded and (b) transverse directions
(transverse to the extrusion axis). Note that the two images were
taken at slightly different magnifications. However, elongated
grains are evident in the direction of extrusion (Cao and Ramesh,
2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.17 Stress strain curve for a material, showing the elastic unloading
response and the increased yield strength upon reloading (this
is called strain hardening, since the material has become harder
because of the increased plastic strain). . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.18 Variation of normalized strain hardening with the square root of
grain size for a variety of nanocrystalline and ultra-fine-grain
materials (from Jia’s doctoral dissertation, Jia et al., 2003). The
reference numbers correspond to those in Jia’s dissertation. . . . . . . . . 159
5.19 The dependence of dislocation velocity on shear stress for a variety
of materials, from Clifton (1983). Note that a limiting velocity
is expected, corresponding to the shear wave speed. There is a
domain in the figure where the dislocation velocity is linear with
the applied shear stress, and this is called the phonon drag regime.
Reprinted from Journal of Applied Mechanics, Vol. 50, p. 941–952,
R.J. Clifton, Dynamic Plasticity. 1983, with permission from
original publisher, ASME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.20 Rate-dependence of the flow stress of A359 aluminum alloy over
a wide range of strain rates. All of the flow stresses are plotted
at a fixed strain of 4% (because of the nature of high strain rate
experiments, it is not generally possible to measure accurately the
yield strength of materials at high strain rates). This result is due
to the work of Yulong Li, and includes compression, tension and
torsion data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
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Nanomaterials
5.21 Results of strain rate jump tests on severely plastically deformed
copper (both cold-worked and ECAPed). Note the jump in the
stress associated with the step increase in the strain rate on a
specimen. One of the curves has been shifted to the right for
ease of discrimination. Step increases in strain rate (jump from
a lower strain rate to a higher strain rate) are used because
a step decrease will exacerbate the effect from the machine
compliance, and the interpretation of experimental data becomes
more involved. In this case the strain rate was increased by a factor
of 2 between consecutive rates. Reprinted from Materials Science
and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect
of nanocrystalline and ultrafine grain sizes on the strain rate
sensitivity and activation volume: fcc versus bcc metals. Sep. 2004,
with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.22 The rate-sensitivity of copper as a function of the mean grain size,
including the nanocrystalline, ultra-fine-grain and coarse-grain
domains. Data is presented for materials made through a
variety of processing routes, some of which involve severe plastic
deformation. The figure is taken from the work of Wei et al. (2004a).
Reprinted from Materials Science and Engineering, Q. Wei,
S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and
ultrafine grain sizes on the strain rate sensitivity and activation
volume: fcc versus bcc metals. Sep. 2004, with permission from
Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.23 The rate-sensitivity of bcc metals as a function of grain size from
a variety of sources (the figure is a variant of one published by
Wei et al. (2004a). The decrease of rate sensitivity with decreasing
grain size is the opposite behavior to that observed in fcc metals.
Reprinted from Materials Science and Engineering, Q. Wei,
S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and
ultrafine grain sizes on the strain rate sensitivity and activation
volume: fcc versus bcc metals. Sep. 2004, with permission from
Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.24 Schematic of motion of screw dislocations by kink pair nucleation
and propagation. The dislocation is visualized as a line that needs
to go over the energy barrier. Rather than move the entire line over
the barrier, the dislocation nucleates a kink, which jumps over the
barrier. The sides of the kink pair have an edge orientation, and so
fly across the crystal because of their high mobility, resulting in an
effective motion of the screw dislocation. . . . . . . . . . . . . . . . . . . . . . . . 173
6.1
Stress strain curve for a material obtained from a standard tensile
specimen tested in uniaxial tension, showing the final fracture of
the specimen in uniaxial tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
List of Figures
6.2
6.3
6.4
6.5
6.6
6.7
xxvii
The development of a neck during plastic deformation of a
specimen within a simple tension experiment. The position and
length of the neck are determined by geometric and material
imperfections in the specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
The influence of rate-sensitivity on the total elongation to failure
of materials, from the work of Woodford (1969). Note the strong
effect of the rate-dependence, and the similarity in behavior of a
wide variety of materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Form of the fracture developed in a simple tension test. (a) Original
specimen configuration before loading. (b) Tensile failure of a
brittle material, showing that the fracture surface is essentially
perpendicular to the loading axis. (c) Tensile failure of a ductile
material, showing the cup and cone failure morphology (in section). . 189
The total elongation to failure of copper materials of varying grain
sizes, as presented by Ma et al. (Wang et al., 2002). It is apparent
that the ductility decreases dramatically as the yield strength is
increased. The point labeled E in the figure corresponds to a special
copper material produced by Ma and coworkers that included both
nanocrystalline and microcrystalline grain sizes, with the larger
grains providing an effective strain hardening in the material.
Reprinted by permission from Macmillan Publishers Ltd: Nature,
Vol. 419, Issue 6910, pages 912–915, High tensile ductility in a
nanostructured metal, Yinmin Wang, Mingwei Chen, Fenghua
Zhou, En Ma. 2002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
The total elongation to failure of fcc materials of nanocrystalline
and microcrystalline grain sizes, as presented by Dao et al. (2007).
Reprinted from Acta Materialia, Vol. 55, Issue 12, page 25, M. Dao,
L. Lu, R.J. Asaro, J.T.M. DeHosson, E. Ma, Toward a quantitative
understanding of mechanical behavior of nanocrystalline metals.
July 2007, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 192
A standard compact tension (CT) specimen used for fracture
toughness measurements. The specimens are several centimeters in
size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
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6.8
Nanomaterials
A sequence of in situ TEM micrographs obtained by Kumar
et al. (2003) during the loading of nanocrystalline nickel using
multiple displacement pulses. Images a–d show the microstructural
evolution and progression of damage with an increase in the
applied displacement pulses. The presence of grain boundary
cracks and triple-junction voids (indicated by white arrows in (a)),
their growth, and dislocation emission from crack tip B in (b–d) in
an attempt to relax the stress at the crack tip as a consequence of
the applied displacement, can all be seen. The magnified inset in (d)
highlights the dislocation activity (Kumar et al., 2003). Reprinted
from Acta Materialia, Vol. 51, Issue 2, page 19, K.S. Kumar,
S. Suresh, M.F. Chisholm, J.A. Horton, P. Wang, Deformation of
electrodeposited nanocrystalline nickel. Jan. 2002, with permission
from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.9 Postmortem microscale fracture morphology observed by Kumar
et al. (2003) after the loading of nanocrystalline nickel in tension.
There is no direct evidence of the presence of dislocations in this
image, although some of the grains appear to have necked before
separation. Reprinted from Acta Materialia, Vol. 51, Issue 2,
page 19, K.S. Kumar, S. Suresh, M.F. Chisholm, J.A. Horton,
P. Wang, Deformation of electrodeposited nanocrystalline nickel.
Jan. 2002, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 200
6.10 Postmortem fracture surface morphology
of nanocrystalline (average grain size of
38 nm) gold thin film tested in tension till failure by
Jonnalagadda and Chasiotis (2008). Dimples are seen on the
fracture surface, with an average size of more than 100 nm. The
void size just before coalescence is therefore much larger than
the grain-size, and the growth of the voids must involve plastic
deformation in many many grains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.11 Progressive localization of a block (a), with initially uniform
shearing deformations (b) developing into a shear band (c). The
final band thickness depends on the material behavior. . . . . . . . . . . . . . 202
6.12 Stress-strain curve for a material showing softening after a peak
stress. This is a curve corresponding to a material undergoing
thermal softening, but similar behaviors can arise from other causes
such as grain reorientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
List of Figures
xxix
6.13 Gross deformation features of coarse-grain and ultra-fine-grain iron
samples after compressive deformations to nearly identical strains
(Jia et al., 2003). (a–b) represent quasistatic compression, while
(c) represents high-strain-rate compression. (a) Homogeneous
deformation of coarse-grain (20 µm grain size) Fe. (b) Shear band
pattern development in ultra-fine-grain (270 nm grain size) Fe.
(c) Shear band pattern development in ultra-fine-grain Fe after
dynamic compression at a strain rate of ≈ 103 s−1 . Reprinted from
Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh,
E. Ma, Effects of nanocrystalline and ultrafine grain sizes on
constitutive behavior and shear bands in iron. July 2003, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.14 Shear band patterns evolve with strain in compressed ultra-finegrain Fe (Wei et al., 2002). Note the propagation of existing shear
bands, the nucleation of new shear bands, and the thickening of
existing shear bands. The development of families of conjugate
shear bands is also observed. Reprinted from Applied Physics
Letters, Vol. 81, Issue 7, pages 1240–1242, Q. Wei, D. Jia,
K.T. Ramesh, E. Ma, Evolution and microstructure of shear
bands in nanostructured fe. 2002, with permission from American
Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
6.15 (a) Shearing deformation across one band and (b) the stable
intersection of multiple shear bands in 270-nm Fe after quasistatic
compression. The shear offset is clearly visible across the first
band. Note that no failure (in terms of void growth) is evident at
the intersection of the shear bands. Reprinted from Acta Materialia,
Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E. Ma, Effects of
nanocrystalline and ultrafine grain sizes on constitutive behavior
and shear bands in iron. July 2003, with permission from Elsevier. . . 206
6.16 TEM micrographs (a) within and (b) outside a shear band in
quasistatically compressed Fe with an average grain size of 138 nm
(Wei et al., 2002). The shearing direction is shown by the arrow.
Note the preferred orientation of the grains within the shear
band, while the grains outside the band are essentially equiaxed.
Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages
1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and
microstructure of shear bands in nanostructured fe. 2002, with
permission from American Institute of Physics. . . . . . . . . . . . . . . . . . . 208
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Nanomaterials
6.17 TEM micrograph showing the microstructure near a shear band
boundary in nanocrystalline iron (Wei et al., 2002). The boundary
between the material within the band and that outside the band
is shown by the solid line. Note that the transition occurs over
a transition width that is about one grain diameter. Reprinted
from Applied Physics Letters, Vol. 81, Issue 7, pages 1240–1242,
Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure
of shear bands in nanostructured fe. 2002, with permission from
American Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.1
7.2
7.3
7.4
7.5
7.6
Schematic of a cubic nanocluster consisting of 64 atoms,
approximately 1 nm on a side. The majority of the atoms are on the
surface of this nanocluster, and so will have a different equilibrium
spacing than atoms in a bulk sample of the same material. . . . . . . . . . . 218
Volume fraction of atoms on the surface of a cuboidal nanoparticle
as a function of nanoparticle size, based on Equation (7.1) and
an assumed interatomic spacing of 0.3 nm. Significant surface
fractions are present below about 20 nm. Note that particle sizes
below 0.6 nm are poorly defined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Core-shell model of a nanoparticle viewed as a composite, with a
surface layer that has different properties as a result of the surface
energy and its effects on binding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A spherical nanoparticle carrying several organic molecules
on its surface. The molecules are chosen to perform a specific
function, such as recognizing a molecule in the environment,
and so are called functionalizing molecules. The spacing of the
functionalizing molecules defines the functionalization density.
Since the molecules modify the surface stress state when they
attach to the surface, the functionalization density modifies the
surface stresses and can even modify the net conformation of the
nanoparticle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Schematic of the structure of carbon nanotubes, showing the
armchair, zigzag and chiral conformations. This beautiful
illustration was created by Michael Stróck on February 1, 2006 and
released under the GFDL onto Wikipedia. . . . . . . . . . . . . . . . . . . . . . . . 223
Schematic of two possible modes of deformation and failure in
nanotubes. (a) Buckling of a thin column. (b) Telescoping of a
multi-walled nanotube. The fracture of nanotubes is a third mode,
but is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
List of Figures
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
xxxi
The development of a twinned region in a material. The figures
show the results of a molecular dynamics calculation of the simple
shear of a single crystal of pure aluminum. (a) Atomic arrangement
in undeformed crystal before shearing. The straight lines are drawn
to guide the eye to the atomic arrangement. (b) After shearing, a
region of the crystal has been reoriented (this region is called the
twin). The twin boundaries can be viewed as mirror planes, and the
lines show the new arrangement of the atoms. . . . . . . . . . . . . . . . . . . . . 231
Example of deformation twins in a metal (the hcp metal hafnium).
(a) Initial microstructure before deformation. (b) Twinned
microstructure after compressive deformation at low strain rates
and at 298K. The twinned regions are the lenticular shapes within
the original grain structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Evolution of twin number density in titanium with applied stress,
over a variety of strain rates and temperatures. The twin density is
not correlated with strain rate or temperature (or strain), but only
with applied stress (Chichili et al., 1998). Reprinted from Acta
Materialia, Vol. 46, Issue 3, page 19, D.R. Chichili, K.T. Ramesh,
K.J. Hemker, The high-strain-rate response of alpha-titanium:
experiments, deformation mechanisms and modeling. Jan. 1998
with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
High resolution electron micrograph of deformation twins
developed in nanocrystalline aluminum subjected to large shearing
deformations (Cao et al., 2008). The diffraction pattern on the right
demonstrates the twinned character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
The possible grain boundary motions: displacements δn in the
direction normal to the GB (grain growth or shrinkage), and δt in a
direction tangential to the plane of the boundary (grain boundary
sliding). The unit normal vector is n and the unit tangent vector is
t, as in Equation (7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
(a) Schematic of a grain showing its soft and hard orientations
(with respect to plastic deformation, not elastic stiffness). (b) The
grains are initially randomly oriented in the material, but begin to
orient themselves so that the soft direction is in the direction of
shearing, and the process of grain rotation into the soft orientation
results in the localization of the deformation into a shear band. . . . . . 238
Schematic of simple shearing of an infinite slab, showing the terms
used in examining the shear localization process in simple shear
(Joshi and Ramesh, 2008b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Schematic of the rotation of ensemble of nano-grains occupying
region ℜ embedded in a visco-plastic sea S subjected to shear. The
background image shows the undeformed configuration. . . . . . . . . . . . 240
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Nanomaterials
7.15 A hierarchical approach to homogenization of grain rotation due
to interface traction. (a) Material continuum. (b) Collection of bins
in sample space. (c) Grains within a RVE. The colored shading
represents the average grain orientation in that bin. (d) Interaction
at the grain level. Grains with individual orientations are described
by different colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7.16 Enlarged view of ℜ (Figure 7.14) showing intergranular interaction
in the region. Rotation of the central grain is accomodated by
rotation of the surrounding grains over a length L. . . . . . . . . . . . . . . . . 243
7.17 Evolution of grain orientation fraction (φ ) around the band center.
Note the rapid early growth, the spatial localization, and the
saturation of φ . Also note the spreading of the band, i.e., the
increasing band thickness - this is also observed in experiments. . . . . 248
7.18 Evolution of plastic shear strain rate (γ̇ p ) around the band center.
The greatest activity in this variable is at the band boundaries,
where the grains are reorienting into the soft orientation for shear. . . . 248
7.19 Evolution of the plastic shear strain γ p around the band center.
After localization, γ p evolves slowly indicating that the plastic flow
inside the band develops at the rate of strain hardening at higher
nominal strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.20 Overall stress-strain response for a defect-free sample (curve
A) and a sample with an initial defect in φ (curve B). Curve
B’ indicates the development of the shear band thickness
corresponding to curve B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.21 Evolution of shear band thickness for different grain sizes,
assuming grain rotation mechanism. The material hardening
parameters are held constant for all the grain sizes. The applied
strain rate is 10−3 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.22 The critical wavelength for instability as a function of grain size
for three different metals (note that this is a log-log plot), with the
critical wavelength computed using Equation (7.60). This figure
has been obtained assuming that j = 10 below d = 100 nm, while
for 100 nm < d ≤ 1 µm, we have L = 1 µm and j = Ld . The dashed
straight line represents the condition that λcritical = d, which we
call the inherent instability line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
7.23 Stability map, showing the domains of inherent instability in
materials as a consequence of the rotational accommodation
mechanism. The map is constructed in terms of the strength index
and grain size, so that every material of a given grain size represents
one point on the map, and a horizontal line represents all grain
sizes of a given material. Reprinted figure with permission from
S.P. Joshi and K.T. Ramesh, Physical Review Letters, Stability
map for nanocrystalline and amorphous materials, 101(2), 025501.
Copyright 2008 by the American Physical Society. . . . . . . . . . . . . . . . . 255
List of Figures
8.1
8.2
8.3
8.4
8.5
8.6
8.7
xxxiii
Typical sequence of steps involved in modeling a mechanics of
nanomaterials problem. Note the many layers of approximations
involved, pointing out the danger of taking the results of
simulations at face value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
The typical modeling approaches of interest to the mechanics of
nanomaterials, and the approximate length scales over which each
approach is reasonable. Note the significant overlap in length scales
for several of the modeling approaches, leading to the possibility
of consistency checks and true multiscale modeling. An example
of the observations that can be made at each length scale is also
presented, from a materials characterization perspective. . . . . . . . . . . . 263
The length scales associated with various microstructural features
in metallic materials. For most crystalline materials, the behavior at
larger length scales is the convolution of the collective behavior of
features at smaller length scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Schematic of the Lennard-Jones interatomic pair potential V (d). . . . . 272
The pair potential corresponding to the EAM potential for
aluminum, using the parameters provided by Mishin et al. (1999). . . . 273
Nanocrystalline material constructed using molecular dynamics.
The material is nickel, and the atoms are interacting using an EAM
potential due to Jacobsen. About 30 grains are shown. The color or
shade of each atom represents its coordination number (number
of nearest neighbors), with the atoms in the crystals having the
standard face-centered-cubic coordination number. The change in
coordination number at the grain boundaries is clearly visible. This
collection of atoms can now be subjected to mechanical loading
(deformations) and the motions of the individual atoms can be
tracked to understand deformation mechanisms. . . . . . . . . . . . . . . . . . . 275
High resolution micrograph showing the boundary between two
tungsten grains in a nanocrystalline tungsten material produced
by high-pressure torsion. There is no evidence of any other phase
at the grain boundary, contrary to pervasive assumptions about
the existence of an amorphous phase at the grain boundary in
nanomaterials in crystal plasticity and composite simulations at
the continuum level. Note the edge dislocations inside the grain
on the right. Such internal dislocations are rarely accounted for in
molecular dynamics simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
xxxiv
8.8
Nanomaterials
Finite element model of a polycrystalline mass of alumina, with
the crystals modeled as elastic solids (Kraft et al., 2008). The
model seeks to examine the failure of the polycrystalline mass by
examining cohesive failure along grain boundaries. Reprinted from
Journal of the Mechanics and Physics of Solids, Vol. 56, Issue 8,
page 24, R.H. Kraft, J.F. Molinari, K.T. Ramesh, D.H. Warner,
Computational micromechanics of dynamic compressive loading
of a brittle polycrystalline material using a distribution of grain
boundary properties. Aug. 2008 with permission from Elsevier. . . . . . 280
8.9 Schematic of the structure of a carbon nanotube, viewed as a sheet
of carbon atoms wrapped around a cylinder. The sheet can be
arranged with various helix angles around the tube axis, which
are most easily defined in terms of the number of steps in two
directions along the hexagonal array required to repeat a position
on the helix. Illustration by Volokh and Ramesh (2006). Reprinted
from International Journal of Solids and Structures, Vol. 43,
Issue 25–26, Page 19, K.Y. Volokh, K.T. Ramesh, An approach
to multi-body interactions in a continuum-atomistic context:
Application to analysis of tension instability in carbon nanotubes.
Dec. 2006, with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . 281
8.10 Connection between continuum deformations and atomic positions
as defined by Equation (8.21). We associate atomic positions in the
two configurations with material and spatial vectors. . . . . . . . . . . . . . . 284
8.11 Range of experimental measurements of the elastic modulus of
carbon nanotubes, as summarized by Zhang et al. (2004). Note
the theoretical predictions discussed here were 705 GPa (Zhang
et al., 2002) and 1385 GPa (Volokh and Ramesh, 2006). Reprinted
from Journal of the Mechanics and Physics of Solids, P. Zhang,
H. Jiang, Y. Huang, P.H. Geubelle, K.C. Hwang, An atomistic-based
continuum theory for carbon nanotubes: analysis of fracture
nucleation. May 2004, with permission from Elsevier. . . . . . . . . . . . . . 285
8.12 Range of modeling estimates of the elastic modulus of carbon
nanotubes, as summarized by Zhang et al. (2004), including a wide
variety of first principles and MD simulations. Note the theoretical
predictions discussed here were 705 GPa (Zhang et al., 2002) and
1385 GPa (Volokh and Ramesh, 2006). Reprinted from Journal of
the Mechanics and Physics of Solids, P. Zhang, H. Jiang, Y. Huang,
P.H. Geubelle, K.C. Hwang, An atomistic-based continuum theory
for carbon nanotubes: analysis of fracture nucleation. May 2004,
with permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
List of Figures
xxxv
8.13 Simulation of a copper grain boundary by Warner et al.
(2006) using the quasicontinuum method. The arrows
correspond to the displacement of each atom between
two loading steps. Reprinted from International Journal
of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner,
F. Sansoz, J.F. Molinari, Atomistic based continuum investigation
of plastic deformation in nanocrystalline copper. April 2006, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.14 Computational approach for simulations of nanocrystalline copper
by Warner et al. (2006). The grayscale represents different
orientations of the crystals. Reprinted from International Journal
of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner, F. Sansoz,
J.F. Molinari, Atomistic based continuum investigation of plastic
deformation in nanocrystalline copper. April 2006, with permission
from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
8.15 Comparison of the predictions of the multiscale simulations
of Warner et al. (2006) (identified with the FEM symbol) with
molecular dynamics (MD) simulation results and experimental
data on nanocrystalline copper. Note that the experimental
results are derived from hardness measurements, while the FEM
simulation results correspond to a 0.2% proof strength. Both
MD and FEM simulations predict much higher strengths than
are observed in the experiments. Reprinted from International
Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner,
F. Sansoz, J.F. Molinari, Atomistic based continuum investigation
of plastic deformation in nanocrystalline copper. April 2006, with
permission from Elsevier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
List of Tables
1.1
A broad classification of nanomaterials on the basis of
dimensionality and morphology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
Young’s modulus and Poisson’s ratio for some conventional grain
sized materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1
Some measured moduli of nanotubes, nanofibers, and nanowires,
viewed as 1D nanomaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1
5.2
Slip systems with typical numbering scheme in typical fcc metals. . . 130
Known slip systems in a variety of metals, from a table put together
by Argon (Argon, 2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Hall-Petch coefficients for a variety of materials. . . . . . . . . . . . . . . . . . 143
Hall-Petch coefficients for pure aluminum and select aluminum
alloys. Data taken from the work of Witkin and Lavernia (2006). . . . . 145
5.3
5.4
7.1
7.2
7.3
Scale-dominant mechanisms in nanomaterials, categorized in terms
of materials classification, morphology, and length scale. . . . . . . . . . . 216
Typical intrinsic length scales that arise from dislocation
mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Basic parameters for grain rotation model in polycrystalline iron,
as developed by Joshi and Ramesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
xxxvii
1
There’s plenty of room at the bottom.
Richard Feynman
Nanomaterials
1.1 Length Scales and Nanotechnology
Nanotechnology demands the ability to control features at the nanoscale (10−9 m),
and a variety of techniques have been developed recently that give humanity this
ability. From a fundamental science perspective, issues of physics and chemistry
must be addressed at these scales. Surface and boundary effects can dominate the response. Many of the classical distinctions between mechanics, materials and physics
disappear in this range of length scales, and a new kind of thinking emerges that is
commonly called nanoscience (sometimes humorously interpreted as “very little
science”). The recent rapid development of nanoscience is the result of a new-found
ability to observe and control structure at small length and time scales, coupled with
the development of computational capabilities that are most effective at small scales.
It is useful to develop a physical idea of length scale, and Figure 1.1 shows the
range of length scales of common interest in mechanics and biology (the latter is included because it allows one to develop a human sense of scale). Beginning at small
scales, features associated with atomic radii are on the order of 1 Å (one angstrom,
10−10 m) in size. The atomic lattice spacing in most crystals is of the order of 3 Å.
The diameter of a carbon nanotube is about 2 nm or 20 Å, and this correlates well
with the diameter of a double helix of DNA (which indicates, incidentally, that the
nanotube is a good approach to handling DNA). A tobacco mosaic virus is about
50 nm across (this corresponds approximately in scale with the typical radii of
curvature of the tips of nanomanipulators such as AFM probes). Grains in most
polycrystalline metals have sizes that range from about 1 µm to about 20 µm (grain
boundary thicknesses, to the extent that they can be defined, are typically <1 nm). A
number of bacteria (living organisms) are also about 1 µm in size, a reminder of the
remarkable sophistication of nature. Small-scale failure processes, such as the voids
developed in spallation, are typically of the order of 10 µm in separation. Many cells
in eukaryotic organisms are of this size-scale. Some of the most sophisticated small
devices in engineering, integrated circuit chips, are of the order of 1 cm in size (the
corresponding natural “device” might be a beetle). The author arrives on the scale
at about 1.8 m, while an M1A1 Abrams tank is about four times bigger. Some of
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 1,
c Springer Science+Business Media, LLC 2009
1
2
Nanomaterials
Fig. 1.1 Length scales in mechanics and materials and in nature. The topics of interest to this book
cover a large part of this scale domain, but are controlled by features and phenomena at the nm
scale. Note the sophistication of natural materials and systems at very small length scales.
1 Nanomaterials
3
the largest animals are blue whales, coming in at about 100 m, comparable in size
to some aircraft. Phenomena and features at the nanoscale and microscale can dominate the behavior and performance of devices and structures in the centimeter to
100 m range. Most of the behaviors of interest to this book arise from length scales
of 0.1–100 nm, and this will be our definition of “the nanoscale.”
One way to distinguish between nanotechnology and nanoscience is by distinguishing between what we can control and what we can understand. Most of
what we interact with as humans has structure at the nanoscale, that is, there
is a nanoscale substructure to most materials. Understanding what the nanoscale
structure does (in terms of behavior or phenomena) is the core of nanoscience.
Controlling nanoscale structure so as to achieve a desired end is the essence of nanotechnology. Nanotechnology cannot succeed without nanoscience, and the most
efficient growth of nanotechnology (and growth with the smallest risk) occurs when
the necessary nanoscience is already available.
The enabling science in much of nanotechnology today is the science of nanomaterials (indeed in the broadest sense, nanotechnology would not be possible without
nanomaterials). From a disciplinary viewpoint, most nanoscale phenomena are either controlled or modulated by the mechanics at the nanoscale, and so mechanics
plays a critical role in nanoscience. Nanomechanics controls phenomena as immediately obvious as the interaction of interfaces between nanosize crystals and as subtle
as the folding of proteins (controlling and organizing the living cell). The mechanics
of nanomaterials is therefore the focus of this book.
1.2 What are Nanomaterials?
A nanomaterial is a material where some controllable relevant dimension is of the
order of 100 nm or less. The simple presence of nanoscale structure alone is not
sufficient to define a nanomaterial, since most if not all materials have structure in
this range. The ability to control the structure at this scale is essential. One could
argue, in this sense, that many of the classical alloys and structural materials that
contained nanoscale components by design (e.g., Oxide-Dispersion-Strengthened or
ODS alloys) could be called nanomaterials. Conventionally, however, the modern
usage of the term does not include the classical structural materials. In modern
usage, nanomaterials are newly developed materials where the nanoscale structure
that is being controlled has a dominant effect on the desired behavior of the material
or device.
There are three different classes of nanomaterials: discrete nanomaterials,
nanoscale device materials, and bulk nanomaterials. Discrete nanomaterials or dn
materials are material elements that are freestanding and 1–10 nm in scale in at least
one dimension (examples include nanoparticles and nanofibers such as carbon nanotubes). Nanoscale device materials or nd materials are nanoscale material elements
that are contained within devices, usually as thin films (an example of an nd material
would be the thin film of metal oxide used within some semiconductor fabrication).
4
Table 1.1 A broad classification of nanomaterials on the basis of dimensionality and morphology.
Type of nanomaterial
Dimensionality
Morphology
Characteristics
Remarks
Discrete nano (dn)
materials
0D or 1D
Particles, fibers
Large surface
functionalization
Potential health hazard
Nanoscale device (nd)
materials
Usually 2D, occasionally
1D
Thin films, sometimes
wires
Functionalization,
electrical/thermal
characteristics
Semiconductor fabrication
Bulk (nc or ns)
nanomaterials
3D
Minimum mm3
Mechanical and structural
applications
May be built up from dn
and nd materials
Nanomaterials
1 Nanomaterials
5
Fig. 1.2 Transmission electron micrographs (see Chapter 3) of nanocrystalline nickel material
produced by electrodeposition (Integran). (a) Conventional TEM micrograph showing the grain
structure. The average grain size is about 20 nm. (b) High resolution electron microscopy image
of the same material. Note that there is no additional or amorphous phase at the grain boundaries,
although many models in the literature postulate the existence of such a phase. Micrographs by
Qiuming Wei. Reprinted from Applied Physics Letters, 81(7): 1240–1242, 2002. Q. Wei, D. Jia,
K.T. Ramesh and E. Ma, Evolution and microstructure of shear bands in nanostructured fe. With
permission from American Institute of Physics.
Bulk nanomaterials are materials that are available in bulk quantities (defined here
as at least mm3 volumes) and yet have structure controlled at the nanoscale. Bulk
nanomaterials may be built up of discrete nanomaterials or nanoscale device materials (for example one may construct a bulk nanomaterial that contains a large number of nanofibers). It is possible to view this classification approach (Table 1.1) in a
mathematical light: discrete nanomaterials are either zero-dimensional (particles) or
one-dimensional (fibers), nanoscale device materials are typically two-dimensional
(thin films), and bulk nanomaterials are three-dimensional.
The vast majority of materials are polycrystalline, that is, they are made up of
many crystals (which are also called grains within the materials community). Most
conventional engineering materials have grain sizes of 10–100 µm. In this regard, we
distinguish between two subclasses of bulk nanomaterials: nanocrystalline materials (nc materials) with crystal or grain sizes that are <100 nm, and nanostructured
materials (ns materials) with mixtures of nanoscale and conventional crystal sizes.
An example of a nanocrystalline material is shown in Figure 1.2. Polycrystalline
materials with grain sizes between 100 nm and 1 µm are conventionally called ultrafine-grained or ufg materials. Note that some of the literature includes the range
between 100 nm and 500 nm in the “nanocrystalline” domain.
1.3 Classes of Materials
There are three basic classes of materials categorized on the basis of atomic bond
type and molecular structure: metals, ceramics and polymers. The character of polymers is typically determined by the interactions of large numbers of long-chain
6
Nanomaterials
molecules (typically more than 100 nm long). There are several discrete nanomaterials and nanostructured materials based on polymers, including microspheres
and microballoons (hollow spheres), and some two-dimensional structures (such
as lamellae) that can be used to build three-dimensional bulk nanostructured materials. Some composites consisting of a polymer matrix with clay nanoparticles or
carbon nanotubes as reinforcements are also considered to be bulk nanomaterials.
This book does not examine soft nanomaterials in any detail, simply because the author is not knowledgeable about the field. The book focuses on nanomaterials that
are either metals or ceramics or composites of these two.
1.4 Making Nanomaterials
The ways in which nano materials are made vary widely, and there is not enough
room in this book to discuss all of them. On the other hand, it is important to understand some features of the processes used in the synthesis of nanomaterials,
because the processing route often dominates the behavior of any given material.
The focus here is on processing issues in nanomaterials, rather than on the broader
issues associated with processing nanotechnology – in particular, issues related to
nanobiotechnology are ignored in this discussion.
The fundamental issue associated with the making of nanomaterials is that one
is trying to control the structure of a material at a very fine scale. Stated differently,
one is trying to introduce order into the material at a very fine scale (or equivalently,
trying to reduce the entropy density all over this material). This attempt to reduce the
local entropy density has two direct consequences. First, such processes will require
a significant amount of energy and therefore may incur significant expense. Second,
the material that is generated is often thermodynamically unstable, and may attempt
to revert to a higher entropy state (for example through growth of features that were
initially nanoscale). Some of the more creative processing techniques use this fact
to their advantage: they begin by generating extremely disordered states, which on
equilibration generate the desired nanostructured state.
In a broad sense, the approaches used to make materials can be put into two
categories: top-down approaches, in which one begins with a bulk material that is
then processed to make a nanomaterial, and bottom-up approaches, in which the
nanomaterial is built up from finer scales (in the limit, building the nanomaterial
up one atom at a time). It is evident that bottom-up approaches require control of
processes at very fine scales, but this is not as difficult as it sounds, since chemical
reactions essentially occur molecule by molecule (indeed, the nanomaterials made
by nature are grown through bottom-up approaches).
1.4.1 Making dn Materials
The processes required to generate discrete nanomaterials are usually quite distinct
from the processes required to generate bulk nanomaterials. Many bulk nanoma-
1 Nanomaterials
7
terials are created from dn materials using a second processing step (such bulk
nanomaterials are said to have been created using a two-step process). Discrete
nanomaterials themselves are often made using bottom-up approaches, sometimes
relying on self-assembly to generate the desired nanoparticles or nanofibers. Under
some conditions, a properly tuned bottom-up process (such as condensation from a
vapor) can be used to generate significant quantities of discrete nanomaterials such
as nanopowders. The majority of processing routes for dn materials rely on controlling the nucleation and growth process, since nucleation dominated processes
tend to generate small sizes. Thus, one might nucleate nanoparticles by condensation from a vapor or precipitation from solution, and then intervene to control the
growth of the particles, resulting in nanoparticles. Some of the typical approaches
to creating discrete nanomaterials include:
•
•
•
•
•
•
•
Condensation from a vapor phase (Birringer et al., 1984)
Precipitation from solution (Meulenkamp, 1998)
Chemical vapor deposition (Ren et al., 1998)
Chemical reactions, particularly reduction or oxidation (Brust et al., 1994)
Processes used to generate colloidal phases (Ahmadi et al., 1996)
Self-assembly using surfaces (Li et al., 1999)
Mechanical attrition (Nicoara et al., 1997)
A very large number of processes are used to generate discrete nanomaterials, using
technologies borrowed from a variety of fields. Sometimes nanoparticles and nanotubes are byproducts of reactions that are already industrially relevant, and the usefulness of these byproducts has only recently been recognized. The characterization
and transport of these dn materials are major issues in the development of nanotechnology. The definitive characterization of nanoparticle size and size distribution may
require expensive high-resolution electron microscopy, and optical and X-ray techniques that obtain a reasonable measurement of the average size of nanoparticles
over reasonably large volumes are of industrial interest.
1.4.2 Health Risks Associated with Nanoparticles
There is very little data on the interaction of nanoparticles with biological systems
and the environment. Particles with nanoscale dimensions can easily pass through
biological systems, and may accumulate in undesirable locations within cells, and
so there are potential health risks associated with nanoparticles. Scientific study of
these effects is only just beginning in the early twenty-first century, on two fronts:
the potential health risks, and the potential use of nanoparticles for clinical treatments. Given the lack of data and the ease with which nanoparticles can be redistributed in the environment given their small sizes, it is wise to take precautions
in handling nanoparticles. Industrial operations with nanoparticles should consider
such particles to be a potential hazard, and develop handling protocols that are consistent with potential hazards. Bulk nanomaterials (fully dense nanomaterials) are
8
Nanomaterials
not likely to be a danger in themselves, but note that some nanoparticles may be
generated during the fracture or failure of such materials.
1.4.3 Making Bulk Nanomaterials
There are two basic approaches to the production of bulk nanomaterials, commonly
referred to as the bottom-up and top-down approaches. In bottom-up approaches,
the material is built up one atom, molecule, particle or layer at a time, with the focus on precise control at this scale. While materials made in this way can have very
clean microstructures, the approach is inherently expensive and is generally not suitable for making large quantities of material. In top-down approaches, on the other
hand, the nanomaterial is made by the application of macroscale processes, with the
nanoscale structure caused to develop afterwards. Such approaches generally do not
generate a high-quality material, but are relatively inexpensive. There are also some
approaches that defy such broad categorization.
1.4.3.1 Deposition Techniques
Deposition techniques used to generate nanocrystalline materials include electrodeposition (Elsherik and Erb, 1995), chemical (Bhattacharyya et al., 2001) and physical (Musil and Vlcek, 1998) vapor deposition (CVD and PVD), and a variety of
techniques associated with semiconductor fabrication. These are typically slow processes used for the development of relatively thin films of material, with the best
control of microstructure usually obtained if the film thickness is less than 1 µm
(although some commercial electrodeposited nanocrystalline materials are now
available with thicknesses up to 1 mm). These techniques are well adapted to the
development of coatings, e.g., with mechanical applications in wear resistance
and friction control. The major advantage of these techniques is that very precise
control of the environment allows one to develop very precisely controlled microstructures, and so bulk materials produced in this way can be made in relatively
reproducible fashion. Further, it is possible to generate microstructures with specific
contaminants and specific doping (approaches that are particularly useful in sensor
applications). Figure 1.2 shows the microstructure of a commercially developed
electrodeposited nanocrystalline nickel film from Integran.
The substantial process control that is available does not, however, imply that the
materials that are generated will be free of defects. The development of controlled
fully dense nanocrystalline structures is not an easy task. Process optimization is
expensive, and the classic business problem remains of defining a sufficiently large
market to make process optimization economically viable. For most nanocrystalline
materials, therefore, it remains difficult to obtain reproducible films through deposition techniques. Indeed, much of the early confusion about the properties of
1 Nanomaterials
9
nanocrystalline materials arose from the variability in material microstructures that
are generated through these various processes (Agnew et al., 2000). Specific issues of concern include porosity, inhomogeneous microstructures and particularly
columnar grains, and variable grain size distributions. This makes it difficult to compare the results obtained by one laboratory on a material that they have made with
the results obtained by another laboratory on another nominally identical material
that was made differently. As a consequence, both intensive and extensive characterization of a nanomaterial is essential if one is to understand the behavior. Intensive
characterization procedures attempt to determine structure at a very fine scale, e.g.,
using TEM to determine local grain size. Such intensive procedures are incapable
of providing information over large volumes. Extensive characterization procedures
seek to determine the degree of variation of a structural feature (such as the grain
size) with location over a larger length scale. Such procedures are usually incapable
of resolving detail at the same fine scales as intensive procedures. We discuss some
possible characterization procedures in chapter 3.
Deposition techniques such as those discussed in this section are typically only
capable of making films of material. Thin films are the most common products,
although the use of multiple layers may allow one to build up thicker twodimensional structures, and occasionally up to the “bulk” mm thickness. These
techniques, particularly those that have been optimized for semiconductor fabrication, have tremendous potential for making nanomaterials for applications within
devices, i.e., nd materials. Since such devices represent a large fraction of the current commercial value of nanotechnology, these techniques have much to offer the
nanotechnologist. The optimization of semiconductor fabrication processes for volume and cost allows one to develop relatively large films in terms of surface area,
and thus the potential for volume production should not be minimized.
1.4.3.2 Consolidation of dn Materials
The second major approach to making bulk nanomaterials relies on the already well
developed capacity for generating large volumes of nanoparticles or nanopowders.
The idea is to take a large volume of nanoparticles and then consolidate this to generate a bulk nanomaterial. Many of the techniques used in the powder metallurgy
and ceramics industries can be applied to these nanopowders – these industries have
generated bulk materials from powders for decades. The primary difficulty faced in
the direct application of these classical powder techniques results from the difference in powder size.
The typical steps in the consolidation of bulk materials from powders are:
1. Mixing of the powders, if necessary (it is sometimes advantageous to start with a
variety of powder sizes, and different material powders might be used if a composite material is desired).
2. An initial consolidation step, typically cold isostatic pressing (CIPing) to generate a solid that can be handled, called a green body.
10
Nanomaterials
3. Further consolidation of the green body, either through sintering or hot isostatic
pressing (HIPing) to obtain the desired density. This step usually involves a high
temperature because it is usually controlled by diffusion mechanisms.
4. A finish machining process to remove regions of the material that have not been
completely consolidated.
A variety of other steps are possible, including plastic deformation through rolling
and forging, and the precise recipes used are often jealously guarded as proprietary information by laboratories and companies. The specific temperature histories and pressure histories used in a process have a major influence on the
final microstructure. The relative instability of nanomaterials implies that grain
growth is a major issue at high temperatures, and a compromise that must often be considered is between the desired final density and the desired final grain
size.
Every group making materials by powder consolidation also appears to have a
different definition of “full density,” with some industries defining full density as
98% of the theoretical density (which is calculated using the known densities of the
elements, compounds or phases involved and assuming no porosity). This is a major
problem because the mechanical properties of the nanomaterial can be very strongly
dependent on the presence of even small amounts of porosity. The properties of a
material with 98% density can be very different from that of the same material with
99.5% density (particularly with regard to fracture and failure).
Part of the difficulty with density arises because the more traditional powder
metals produced by the powder metallurgy industry are designed for applications
where small amounts of porosity would be removed by subsequent processes such
as forging, or the small porosity might even be beneficial (as in lubrication systems).
Such finishing processes are often not used in the development of the nanomaterial,
resulting in residual porosity. A related difficulty is that it can be very difficult to
see the porosity within a nanomaterial, because unlike traditional powder processed
materials the porosity might be very finely distributed as very small voids. The consequence is that lower final densities may be considered acceptable by a materials
processing vendor in nanomaterials because the quality control step is not able to
find significant pores.
An example of a bulk nanomaterial (nanocrystalline iron) made by the consolidation of powders is presented in Figure 1.3. First, nanostructured iron particles were
generated by a mechanical attrition process: commercial iron powder with a purity
of 99.9% and a range of powder sizes averaging around 5 µm was used as a starting
material for ball milling. Ball milling (Malow and Koch, 1998) was conducted using
a SPEX 8000 mill with a vial and balls made of stainless steel (SS440). The weight
ratio of balls/powder was 4:1 and the milling time was 18 h. The vial was sealed in
an argon atmosphere and cooled using a fan during milling. The powder particles
after milling had internal grain sizes on the order of 10 nm (Malow and Koch, 1998)
as revealed by X-ray diffraction line broadening. A two-step consolidation procedure was used to form bulk nanocrystalline iron with the desired grain size. In the
first step, the powder was compacted at a pressure of 1.4 GPa at room temperature
for about 10 h using a tungsten carbide (WC) die set. The resulting compact had a
1 Nanomaterials
11
Fig. 1.3 Transmission electron micrograph of nanocrystalline iron produced by consolidation of
a nanocrystalline precursor made by ball milling. There is a range of grain sizes, and many of the
grains show evidence of the prior plastic work produced by the ball-milling process. The width of
the photograph represents 850 nm.
green density of 72–75% of the theoretical density. In the second step, the compact
was transferred into a larger WC die set for hot consolidation in a hot press. A temperature of 753 K and a consolidation time of 30 min at a pressure of 850 MPa was
used to obtain an average grain size of just less than 100 nm and a final density of
99.2% of the theoretical density.
One interesting feature of the nanocrystalline iron shown in this figure is that
there are no visible voids in the material observable within the transmission electron microscope (TEM observations represent very small volumes), even though
there must be 0.8% of porosity somewhere within the material. It is possible that
the voids are very small and so not easily seen, or perhaps the porosity is eliminated by the TEM specimen preparation procedure. Another interesting feature of
the nanocrystalline material shown in this figure is one that is to be expected from
a consolidated material: many of the grains show evidence of the substantial prior
plastic deformation involved in the consolidation process (there is extensive dislocation activity within the grains). This can make it difficult to decouple the effects of
plastic deformation and the effects of the small grain size on the mechanical properties of the nanomaterial, and from a science viewpoint this can be viewed as one
of the disadvantages of the consolidation process. From an engineering viewpoint,
however, the consolidation process allows one to generate substantial quantities of
bulk nanomaterials, and to take advantage of existing technologies in the powder
metallurgy and ceramics industry. At the date of publication of this book, reproducibility of the quality of bulk nanomaterials made by the consolidation process
remains a contentious issue.
12
Nanomaterials
1.4.3.3 Severe Plastic Deformation Processes
A variety of processing techniques for bulk nanomaterials fall under the rubric of
severe plastic deformation (SPD) processes (Valiev et al., 2000; Valiev and Langdon, 2006). The primary idea in all of these is to develop very large plastic deformations within metals, with associated dislocation substructures, and then to use
the processes of recovery, recrystallization and rearrangement of microstructure to
develop the crystalline microstructures of interest. Recovery involves the thermallyenhanced motion of dislocations to reduce internal stresses through rearrangement
and annihilation, while recrystallization involves the growth of new nearly strainfree crystals out of a collection of heavily-worked prior crystals. Large plastic deformations of metals have been known to develop patterned dislocation substructures for many years (Mughrabi, 1983), and the metalworking industry has used this
extensively to their advantage. Large plastic deformations can result in dislocation
structures that reorganize themselves to form cells, subgrains (regions with small
angular misalignment, usually separated by dislocation walls), and eventually large
angle grain boundaries (Hughes and Hansen, 1997), resulting in progressive grain
refinement. The final grain size that is obtained in these processes depends strongly
on the details of the thermomechanical processes involved – in general, larger strains
develop finer grains, but the minimum grain size that can be obtained varies from
material to material. Truly nanocrystalline grains (≤ 100 nm) have been observed
within the centers of adiabatic shear bands developed within some materials, and
have also been developed during high-pressure torsion processes. However, most
SPD processes result in ultra-fine-grain sizes in most materials.
1.4.3.4 Equal Channel Angular Processing
Equal channel angular processing or Equal Channel Angular Pressing, commonly
known as ECAP, is a technique in which very large plastic deformations are developed by forcing a metal rod (the sample) through a die, with a die axis that changes
direction suddenly (resulting in substantial shear deformation). Some variants of
this technique are known as equal channel angular extrusion (ECAE) processes. A
schematic diagram representing such a process is shown in Figure 1.4, with a hard
plunger forcing a metal sample between two channels of equal size connected by
an angular section (thus the name Equal Channel Angular). The magnitude of the
shear strain that is developed in a single pass of the specimen through the die is
determined by the die geometry, and particularly the included angle φ shown in the
figure. The angle Φ subtended by the external radius is also an important parameter.
Various obtuse die angles and dimensions may be used, and die design is a critical
part of an effective ECAP process. Angles closer to 90◦ give larger shear strains,
but require greater workability of the material. The metal specimen that is forced
through the die is typically about 1 inch in diameter and several inches long and is
made of conventional grain size materials. Substantial forces are required to develop
these deformations, and so large presses must be used and particular care must be
taken with die design for high-strength metals.
1 Nanomaterials
13
φ
Φ
Fig. 1.4 Schematic of the die used in Equal Channel Angular Pressing (ECAP). The included angle
φ in this case is about 120◦ , but can be as small as 90◦ . The angle Φ subtended by the external
radius is also an important parameter. Various obtuse die angles and dimensions may be used. Very
large forces are required to move the workpiece through such dies.
Since the angle change renders the process inherently anisotropic, it is common
practice to run the specimen through the die multiple times (these are called multiple
passes), rotating the specimen between consecutive passes. Multiple passes may
also be used to increase the fraction of the sample that has been subjected to the large
deformations (portions of the specimen ends are typically much less deformed). The
various protocols used for these multiple passes and rotations are called processing
routes (commonly referred to as A, B, C . . . routes). Each route has a different yield
in terms of grain size and the likely development of texture (relative orientation
of grains), and also in the fraction of the specimen that achieves significant grain
refinement. The use of multiple passes typically yields ultra-fine-grain (ufg) size
materials relatively easily, but nanocrystalline grain sizes are very difficult to obtain.
These techniques have been used on a variety of metals successfully. In particular,
metals with body-centered cubic (bcc), face-centered cubic (fcc) and hexagonalclose packed (hcp) crystal structures have been processed in this way.
An example of the microstructure of a material made by equal channel angular
pressing is shown in Figure 1.5. The average grain size of this material is 200 nm.
A selected area diffraction (SAD) pattern is also shown, and indicates the presence
14
Nanomaterials
a
b
Fig. 1.5 (a) Microstructure of tantalum produced by Equal Channel Angular Extrusion (ECAE),
four passes at room temperature through a 90◦ die. (b) Selected Area Diffraction (SAD) pattern for this sample, showing the presence of many low angle grain boundaries. The fraction of
grain boundaries that are low-angle rather than high-angle can be an important feature of the
microstructure.
of low-angle grain boundaries (typical of this SPD approach). The high dislocation
density in many of the grains is a result of the large plastic work that must be introduced in order to generate this microstructural grain refinement. As in the consolidated case, materials produced by this SPD process contain large internal dislocation
densities because the processes of grain refinement result in significant numbers of
retained dislocations. There is often a significant texture produced by this process,
although the degree of texture does depend on the specific route used and varies
strongly from one material to another. Elongated grains are another characteristic
feature of such processes, and as a result these materials may have anisotropy in the
yield strength (that is, the yield strength – defined in Chapter 2 – may be different
in different directions). The nature of the grain boundaries produced by equal channel angular pressing is a matter of some controversy. It appears that the structure
of the grain boundary may be strongly dependent on the temperatures at which the
processing is conducted for a given material. In a related issue, the degree to which
rotational recrystallization plays a part in the development of the fine grains also
remains uncertain.
1.4.3.5 High-Pressure Torsion
Another kind of SPD process is shown in Figure 1.6, and is known as the highpressure torsion or HPT process (Liao et al., 2006). Relatively small samples are
subjected to combined high-pressure and severe torsional deformations using large
1 Nanomaterials
15
Applied Pressure
Press
Sample
Die
Applied Torque
Fig. 1.6 Schematic of High Pressure Torsion (HPT) process. A thin specimen is compressed and
then subjected to large twist within a constraining die. The typical sample size is about 1 cm in
diameter and about 1 mm thick.
multi-axial presses. Pressures of several GPa are used, and the torsional strains can
be very large (depending on the total angle of twist). Since thin specimens are being
considered, the strains generated by the compression are relatively small. For any
given increment in the angle of twist θ , the local increment in shear strain is given by
δγ =
rδ θ
,
h
(1.1)
where r is the radial distance from the center of the disc and h is the disc thickness. Very large strains can thus be generated by using small thickness samples. For
example, a full rotation of 360◦ will lead to a shear strain of approximately 30 in
a sample that is 10 mm in diameter and 1 mm thick. These very large strains can
lead to much more refined microstructures than can be obtained through ECAP processes. The superimposed pressure is necessary to keep the sample from breaking
apart, and issues of workability typically still require performing this HPT operation at elevated temperature. The major difficulty with this technique, however, is
the small size of the specimen that is subjected to the very large strain. Further, the
strain is quite inhomogeneous, with the center of the specimen seeing no strain at all.
At this point, therefore, the technique is a good way to understand what the effects
of severe plastic deformation will do to materials, but this is not a good approach
for generating significant quantities of nanomaterials.
16
Nanomaterials
(a)
(b)
(c)
Fig. 1.7 Microstructure of tungsten produced by high-pressure torsion (HPT). (a) A bright field
TEM micrograph. (b) A dark field TEM micrograph. (c) A selected area electron diffraction pattern
from this field. The grains contain a high density of defects (because of the plastic work associated
with the HPT process). The grains are also elongated, with widths on the order of 80 nm and
lengths of about 400 nm. Grain orientation appears to be along the shearing direction. The selected
area electron diffraction shows nearly continuous rings, with no obvious intensity concentration
along the rings, indicating large angle type grain boundaries in the sample. Reprinted from Acta
Materialia, Vol. 52, Issue 7, P. 11, Q. Wei, L. Kecskes, T. Jiao, K.T. Hartwig, K.T. Ramesh and E.
Ma, Adiabatic shear banding in ultrafine-grained Fe processed by severe plastic deformation. April
2004, with permission from Elsevier.
An example of a microstructure generated by HPT is shown in Figure 1.7. The
material shown is pure tungsten, which is a notoriously difficult material to deform plastically. The final grain size has a minimum dimension of about 80 nm, and
some elongation of the grains is observed. An SAD pattern is also shown, demonstrating the nanocrystalline nature of the structure. Note again the large number of
dislocations within the grains, a characteristic of severe plastic deformation processes. Elongation along the shearing direction is evident, and a range of grain
sizes is present. The combination of finer grains but greater heterogeneity of the microstructure is characteristic of HPT processes as compared to ECAP processes. An
excellent discussion of the controllable process parameters is provided by Zhilyaev
(Zhilyaev et al., 2003). There appears to be a limit to the smallest achievable grain
size via HPT, with some indications that the limitation is due to dynamic recrystallization.
1.4.3.6 Surface Mechanical Attrition Treatment
Large plastic deformations can also be obtained on the surface by impacting the surface repeatedly with hardened materials (as in shot peening). The typical approach
involves seating the sample within a chamber that is loaded with a large number of
hardened steel balls, and then shaking the entire chamber using a vibration generator
(Tao et al., 2002). By proper tuning of the frequencies, the steel balls can be caused
to impact the specimen at relatively high velocities and generate fairly high strains
1 Nanomaterials
17
in a surface layer on the sample. Continued vibration can lead to microstructural
refinement of the surface layer. This process is very similar to the mechanical attrition process used in ball milling to generate nanoparticles; since it is constrained
to a surface, the process is called surface mechanical attrition treatment or SMAT.
The distinction is the presence of a surface layer and related strain gradients and
microstructural gradients. Control of the atmosphere in the chamber, the frequency
of vibration, and temperature allows one to control the microstructures that are
generated.
1.4.3.7 Rolling and Accumulative Roll Bonding
Very large plastic deformations can also be generated by repeated rolling. Indeed,
the processes of grain refinement during severe plastic deformation were first examined in repeated rolling processes, because of the applications in the metalworking
industry. While repeated rolling does requires substantial workability, the development of protocols involving rolling followed by annealing and then subsequent
secondary rolling allows for the development of very fine microstructures. Rolling
can also be performed at cryogenic temperatures, which has been shown to allow the
development of finer grains. A version of rolling called Accumulative Roll Bonding
(ARB) involves rolling a material to a thin-sheet, then cutting the sheet in two,
stacking the sheets and rolling again to further reduce the thickness, and then repeating this process (Saito et al., 1999). Accumulative roll bonding has been used
to develop bulk ultra-fine-grained and nanostructured metals, both from alloys and
pure metals, and such an ARB process can be adapted for continuous production of
substantial quantities of material (this is a key difficulty with a number of the other
SPD techniques).
1.4.3.8 Crystallization from the Amorphous State
Another approach to making bulk nanomaterials involves the crystallization of the
material from an amorphous state or from the molten state. Control of the nucleation
and growth process to emphasize nucleation can lead to very fine crystal sizes (Hono
et al., 1995). Such control is, however, quite difficult to maintain, and so very few
bulk nanomaterials are made in this way at this time.
1.5 Closing
Now that we have some sense of how to make nanomaterials, we develop the mechanics of materials in the next chapter. We can then use that understanding to focus
on the mechanics of nanomaterials in subsequent chapters.
18
Nanomaterials
1.6 Suggestions for Further Reading
1. L. Foster, Nanotechnology: Science, Innovation, and Opportunity. Prentice Hall,
Upper Saddle River, NY, 2005.
2. G. Cao, Nanostructures & Nanomaterials: Synthesis, Properties & Applications.
Imperial College Press, London, 2004.
3. Y. Gogotsi, Nanomaterials Handbook. New York, CRC Press, 2006.
4. B. Bhushan, Springer Handbook of Nanotechnology. Springer, New York, 2007.
5. P.C. Hiemenz and R. Rajagopalan, Principles of Colloidal and Surface Chemistry.
CRC Press, New York 3rd ed., 1997.
6. D.D. Brandon and W.D. Kaplan, Microstructural Characterization of Materials.
Wiley, New York, 2nd ed. 2008.
1.7 Problems and Directions for Research
1. Identify a source of silver nanoparticles, and find out what particle size distribution is available. How does the company determine this distribution? What is the
cost of the nanoparticles, and what are the sample masses (or volumes) that are
available?
2. Identify the risks associated with handling silver nanoparticles. What literature is
available on their toxicity, if any? How should they be stored? Is there a danger
of fire if the particles are stored in air?
3. Repeat the problem above, but this time consider aluminum nanoparticles. As
always with respect to safety and risk management, the key is to (a) know and
document the risks, (b) know and document safe handling procedures and (c)
know and document emergency response procedures in case of an accident or
mishandling. Write a three page report that includes (a), (b) and (c) above (one
page each) for aluminum nanoparticles.
4. Consider the ECAP process shown in Figure 1.4. The included angle is φ ≥ 90◦ ,
and often the outside corner is not sharp but rounded, subtending an angle Φ with
the sharp inner corner. Then the total shear strain developed by the ECAP process
is given by γ = √N3 [2 cot ( φ2 + Φ2 ) + Φ csc ( φ2 + Φ2 )], where N is the number of
passes (the number of times the workpiece is reinserted). Using the literature,
identify an ECAP route and the number of passes needed to achieve a shear
strain of 4.
5. Consider the problem of the industrial production of bulk nanomaterials. In your
opinion, which of the processes described in this chapter is best suited to scaleup? Why? What are the potential limitations of this process? What do you think
would be the most expensive part of the facility required for scale-up of this
process? You will need to define an expected output capacity.
1 Nanomaterials
19
References
Agnew, S. R., B. R. Elliott, C. J. Youngdahl, K. J. Hemker, and J. R. Weertman (2000). Microstructure and mechanical behavior of nanocrystalline metals. Materials Science and Engineering
a-Structural Materials Properties Microstructure and Processing 285(1–2), 391–396.
Ahmadi, T. S., Z. L. Wang, T. C. Green, A. Henglein, and M. A. ElSayed (1996). Shape-controlled
synthesis of colloidal platinum nanoparticles. Science 272(5270), 1924–1926.
Bhattacharyya, S., O. Auciello, J. Birrell, J. A. Carlisle, L. A. Curtiss, A. N. Goyette, D. M. Gruen,
A. R. Krauss, J. Schlueter, A. Sumant, and P. Zapol (2001). Synthesis and characterization
of highly-conducting nitrogen-doped ultrananocrystalline diamond films. Applied Physics Letters 79(10), 1441–1443.
Birringer, R., H. Gleiter, H. P. Klein, and P. Marquardt (1984). Nanocrystalline materials an approach to a novel solid structure with gas-like disorder. Physics Letters A 102(8), 365–369.
Brust, M., M. Walker, D. Bethell, D. J. Schiffrin, and R. Whyman (1994). Synthesis of thiolderivatized gold nanoparticles in a 2-phase liquid-liquid system. Journal of the Chemical
Society-Chemical Communications 7, 801–802.
Elsherik, A. M. and U. Erb (1995). Synthesis of bulk nanocrystalline nickel by pulsed electrodeposition. Journal of Materials Science 30(22), 5743–5749.
Hono, K., Y. Zhang, A. Inoue, and T. Sakurai (1995). Atom-probe studies of nanocrystalline
microstructural evolution in some amorphous alloys. Materials Transactions of JIM 36(7),
909–917.
Hughes, D. and N. Hansen (1997). High angle boundaries formed by grain subdivision mechanisms. Acta Mater 45(9), 3871–3886.
Li, M., H. Schnablegger, and S. Mann (1999). Coupled synthesis and self-assembly of nanoparticles to give structures with controlled organization. Nature 402(6760), 393–395.
Liao, X. Z., A. R. Kil’mametov, R. Z. Valiev, H. S. Gao, X. D. Li, A. Mukherjee, J. F. Bingert, and
Y. T. Zhu (2006). High-pressure torsion-induced grain growth in electrodeposited nanocrystalline ni. Applied Physics Letters 88, 021909.
Malow, T. R. and C. C. Koch (1998). Mechanical properties, ductility, and grain size of nanocrystalline iron produced by mechanical attrition. Metallurgical and Materials Transactions aPhysical Metallurgy and Materials Science 29(9), 2285–2295.
Meulenkamp, E. A. (1998). Synthesis and growth of zno nanoparticles. Journal of Physical
Chemistry B 102(29), 5566–5572.
Mughrabi, H. (1983). Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metallurgica 31(9), 1367–1379.
Musil, J. and J. Vlcek (1998). Magnetron sputtering of films with controlled texture and grain size.
Materials Chemistry and Physics 54(1–3), 116–122.
Nicoara, G., D. Fratiloiu, M. Nogues, J. L. Dormann, and F. Vasiliu (1997). Ni-zn ferrite nanoparticles prepared by ball milling. In Synthesis and Properties of Mechanically Alloyed and
Nanocrystalline Materials, Pts 1 and 2 - Ismanam-96, Volume 235- of Materials Science Forum,
pp. 145–150. Transtec.
Ren, Z. F., Z. P. Huang, J. W. Xu, J. H. Wang, P. Bush, M. P. Siegal, and P. N. Provencio (1998).
Synthesis of large arrays of well-aligned carbon nanotubes on glass. Science 282(5391), 1105–
1107.
Saito, Y., H. Utsunomiya, N. Tsuji, and T. Sakai (1999). Novel ultra-high straining process for bulk
materials - development of the accumulative roll-bonding (arb) process. Acta Materialia 47(2),
579–583.
Tao, N. R., Z. B. Wang, W. P. Tong, M. L. Sui, J. Lu, and K. Lu (2002). An investigation of surface
nanocrystallization mechanism in fe induced by surface mechanical attrition treatment. Acta
Materialia 50, 4603–4616.
Valiev, R. and T. Langdon (2006). Principles of equal-channel angular pressing as a processing
tool for grain refinement. Progress in Materials Science 51, 881–981.
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Valiev, R. Z., R. K. Islamgaliev, and I. V. Alexandrov (2000). Bulk nanostructured materials from
severe plastic deformation. Progress in Materials Science 45, 103–189.
Zhilyaev, A. P., G. V. Nurislamova, B.-K. Kim, M. D. Baro, J. A. Szpunar, and T. G. Langdon
(2003). Experimental parameters influencing grain refinement and microstructural evolution
during high-pressure torsion. Acta Materialia 51, 753–765.
...there is no philosophy which is not founded
upon a knowledge of the phenomena, but to get
any profit from this knowledge it is absolutely
necessary to be a mathematician.
Daniel Bernoulli
2
Fundamentals of Mechanics
of Materials
We begin with a brief introduction to the mechanics of materials for the nonspecialist, and then develop a description of elastic and inelastic constitutive behavior. The objective is to develop a common mathematical language that can be
used to describe the mechanics of nanomaterials. A basic familiarity with the concepts of undergraduate strength of materials is assumed, particularly the concepts of
stress and strain.
2.1 Review of Continuum Mechanics
2.1.1 Vector and Tensor Algebra
The equations of interest are written in terms of the independent variables (x,t)
representing spatial position and time, respectively. In our notation, boldface letters
will represent vectorial or tensorial quantities. A quick description of basic vector
and tensor analysis is presented below. An excellent exposition of vector and tensor
algebra as used here is presented by Asaro and Lubarda (2006).
In three dimensions, any vector x can be represented by a linear combination of
three unit vectors {e1 , e2 , e3 } (this collection of “base vectors” is sometimes called
a basis). The most commonly used basis is the Cartesian basis, in which the unit
base vectors are perpendicular to each other and are typically written in the x, y and
z directions in order, so that {e1 , e2 , e3 } = {ex , ey , ez }. In a Cartesian basis the base
vectors do not change from point to point, whereas in a basis such as that used with a
cylindrical coordinate system the base vectors actually change direction from point
to point (consider for example the unit vectors at two points with different polar
angles θ in an r, θ , z coordinate system). Thus, x is represented by
x = x1 e1 + x2 e2 + x3 e3 = xi ei
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 2,
c Springer Science+Business Media, LLC 2009
(2.1)
21
22
Nanomaterials
where the scalars xi are called the components of the vector x in the basis {ei }. The
choice of a different basis (e.g., a basis representing a rotated set of base vectors
{ẽx , ẽy , ẽz }) will result in a different set of components representing the same vector:
x = x̃1 ẽ1 + x̃2 ẽ2 + x̃3 ẽ3 = x̃i ẽi .
(2.2)
Therefore, the components of a vector alone do not represent the vector unless the
base vectors are defined. The second half of either Equation (2.1) or (2.2) demonstrates a condensed way of writing this linear combination through the use of the
summation convention: the use of a repeated index (in this case i) implies summation of similar terms over all values of that index, i.e.,
xi ei = Σ xi ei = x1 e1 + x2 e2 + x3 e3 .
(2.3)
If one chooses to retain the same base vectors throughout the discussion, as we
do in this book, then each vector can be represented simply as an ordered set of
the components, for example, as a row or column matrix. Thus a vector may be
represented by
v = v1 e1 + v2 e2 + v3 e3
(2.4)
v1 v2 v3 .
(2.5)
u(x,t) = u1 (x1 , x2 , x3 ,t) u2 (x1 , x2 , x3 ,t) u3 (x1 , x2 , x3 ,t)
(2.6)
or as a row matrix by
Examples of vectorial dependent variables that will be used in this book include the displacement vector u(x,t), the velocity vector v(x,t), and the force vector
f(x,t), all of which are “field” variables in that they are generally functions of position x. Thus the components of the displacement vector u(x,t) can be represented
by a column matrix:
where the square brackets around the vector in Equation (2.6) indicate that the matrix of components of the vector is specifically being considered. Note that the matrix represents the vector only when the choice of base vectors is explicitly agreed
upon and does not change throughout the course of the discussion. Unless otherwise
stated, throughout this book a Cartesian basis is used with unit vectors defined along
the x, y and z directions in order, so that {e1 , e2 , e3 } = {ex , ey , ez }.
The scalar product of two vectors a and b is defined by
a.b = (ai ei ).(b j e j )
= ai b j (ei .e j )
= ai b j δi j
= ai bi
= a1 b1 + a2 b2 + a3 b3 ,
(2.7)
2 Fundamentals of Mechanics of Materials
23
where the fact that the base vectors are orthonormal (i.e., unit base vectors that
are perpendicular to each other) has been used, and the Kronecker delta δi j is
defined by
δi j = 1 if i = j
= 0 if i = j.
(2.8)
The definition of the scalar product can be used to define the magnitude (or
length) of a vector through
√
u = |u| = u.u
(2.9)
√
= ui ui = u1 2 + u2 2 + u3 2 .
It is assumed that the reader is familiar with the “vector product” commonly used
to define areas.
For our purposes, a tensor is defined as a linear operator that maps one vector
into another vector. The classical example of a tensor in continuum mechanics is
the stress tensor σ , which maps the normal vector n normal to a surface at any point
to the corresponding force density (or traction) vector t at that point (Figure 2.1).
This operation is written as
(2.10)
t = σ n
which is typically read as t equals σ operating on n, where the dot between the
σ and the n represents the operation. The stress tensor can be written as a 3×3
matrix, and has the property of being symmetric (because of the balance of angular
momentum), so that the matrix corresponding to the tensor σ can be written as
⎡
⎤
σ11 σ12 σ13
σ = ⎣σ12 σ22 σ23 ⎦ .
(2.11)
σ13 σ23 σ33
As with vectors, the matrix can be used to represent the tensor only when the
choice of base vectors is explicitly agreed upon and does not change throughout the
course of the discussion (Equation 2.2). Similarly, the components of a tensor will
change when the base vectors used to describe the vector are changed; that is, the
matrix of components of a tensor will change when the basis changes, even though
the tensor itself (defined as always by its operation) does not change. Thus, in terms
n
t = σ.n
Fig. 2.1 The stress tensor σ maps the normal vector n at any point on a surface to the traction
vector t at that point.
24
Nanomaterials
of a new set of base vectors {ẽx , ẽy , ẽz }), the identical tensor σ results in a different
matrix of components:
⎤
⎡
σ̃11 σ̃12 σ̃13
σ |ẽi = ⎣σ̃12 σ̃22 σ̃23 ⎦ .
(2.12)
σ̃13 σ̃23 σ̃33
The matrix notation is particularly useful to show the effect of operating with the
tensor: e.g., Equation (2.10) can be written (in matrix form) as
⎡ ⎤ ⎡
⎤⎡ ⎤
t1
σ11 σ12 σ13 n1
⎣t2 ⎦ = ⎣σ12 σ22 σ23 ⎦ ⎣n2 ⎦
t3
σ13 σ23 σ33 n3
(2.13)
where the ti and the ni are the components of the vectors t and n respectively and
the operation of matrix multiplication is used. This equation provides us with the
components of the resulting vector t (after the mapping) in terms of the components
of the original vector n, since the first component of t in this basis is now
t1 = σ11 n1 + σ12 n2 + σ13 n3
(2.14)
ti = σi j n j .
(2.15)
and more generally
Here the summation convention has been used. Note that the repeated index is j,
so that we are summing over the possible values of j, while the index i is the free
index and takes on each of its three values separately, i.e., Equation (2.15) represents
three scalar equations of the type of Equation (2.14). Equation (2.15) is essentially
the component form of Equation (2.10).
Several simple tensors are now defined in terms of their operation. The identity
tensor is that tensor that operates on every vector a to return the same vector:
a = I a for all possible arguments a,
(2.16)
and the Zero tensor operates on all vectors to give the zero vector:
0 = 0 a for all possible arguments a.
(2.17)
Tensors are always defined in terms of what they do, i.e., in terms of their operation. Two tensors are defined to be equal to each other if they always operate in the
same way. i.e., L ≡ G if and only if
L v = G v for all possible arguments v.
(2.18)
For reasons of brevity, from this point on the operation of a tensor on the vector
immediately following it is no longer explicitly written in terms of a centerdot (),
but is understood from the relative placement of the vector following the tensor.
2 Fundamentals of Mechanics of Materials
25
Thus Equation (2.18) will be written as Lv = Gv, and so forth. It is understood that
the result of a tensor operating on a vector is always a vector, so that the quantity
Lv is known to be a vector, generally different from the operand v.
The transpose LT of a tensor L is defined as a specific operator that provides the
following identity:
a.(Lv) = v.(LT a) for all possible arguments a and v.
(2.19)
The matrix of the transpose LT of a tensor can be shown to be the transpose of
the matrix of the original tensor L:
[LT ] = [L]T , or LiTj = L ji .
(2.20)
The product of two tensors amounts to a sequence of operations, so that we can
write ABv and understand that this means that the tensor A operates on the vector
that results from the operation of the tensor B on the operand v:
ABv = A(Bv) = Cv where by definition C = AB.
(2.21)
One product that is of particular interest is that of the tensor with itself, e.g.,
S2 = SS. This quantity and the transpose of a tensor from Equation (2.19) will be
used extensively in this book.
From a physical point of view, note that all a tensor is doing is mapping one
vector into another vector. Since each vector is defined by a magnitude and a direction, all that a tensor does is change the magnitude and the direction, i.e., the tensor
stretches and rotates the vector on which it operates. The action of the tensor can
thus be decomposed into a sequence of two operations: the first a pure stretch, where
magnitude changes but not direction, and the second a pure rotation, where direction
changes but not magnitude.
2.1.2 Kinematics of Deformations
This section develops an approach to describe the deformation of a body, i.e., the
kinematics of deformation. Consider a body B0 consisting of a collection of material
particles (the subscript 0 indicates that this is the initial configuration or reference
configuration of the body). It is conceptually useful to imagine that each material
particle can be identified by a distinct color (so that their motions through space can
be tracked by watching the motion of the red particle, the green particle, etc.), but for
mathematical purposes it is assumed that each particle can be identified individually
through its initial location X.
As a result of deformation, the various material particles that comprise the body
will occupy new locations x at any given time t, resulting in a new or current configuration of the body that is called Bt (see Figure 2.2). Since each material particle
X will occupy different spatial positions x as a function of time, we have
26
Nanomaterials
χ
Fig. 2.2 Kinematics of deformation: a body initially in the reference configuration B0 , with material particles occupying locations X (used as particle identifiers) is deformed into the current
configuration Bt with each particle occupying spatial positions x.
x = χ (X,t),
(2.22)
where χ (X,t) is called the motion of the material particle X. The function χ (X,t) is
assumed to be invertible, i.e., two material particles cannot occupy the same spatial
position at any given time. Thus it is possible in principle, to find the inverse map
(the inverse of the motion)
X = χ −1 (x,t).
(2.23)
Equation (2.23) can be interpreted as allowing us to compute which material
particle will be at a given spatial position at any given time, whereas Equation (2.22)
tells us which spatial positions a given material particle will occupy at any time.
The latter two equations allow us to take any quantity that is described in terms of
material particles (this is called a material description) and convert it to a quantity
described in terms of spatial positions (this is called a spatial description), and vice
versa. Such conversions are very useful in mechanics problems, particularly those
that involve large deformations (such as fluid flow).
The displacement u(X, t) of the material particle is defined as the difference between the current spatial position and the initial position of the particle:
u(X, t) = x − X = χ (X, t) − X.
(2.24)
Note that possible displacements include rigid translations (e.g., where u is
independent of X) and rigid rotations, but neither of these is of interest to us in
this book. Much of this book considers only situations involving small displacements. Given that the displacement field is a vector field in three dimensions, there
are three independent components of the displacement vector, written in matrix
form as
u(X,t) = u1 (X1 , X2 , X3 ,t) u2 (X1 , X2 , X3 ,t) u3 (X1 , X2 , X3 ,t) .
(2.25)
Note that displacements in general can include rigid translations, rigid rotations,
and deformations. This book considers primarily problems involving small displacements, and rigid motions (rigid translations and rigid rotations) are not of interest to
2 Fundamentals of Mechanics of Materials
27
us. The only displacement fields that are of interest in this book are those representing deformations, with nonzero gradients of the displacement field. The displacement gradient Grad u(X,t) is a tensor, with the matrix of components given by
⎡ ∂u ∂u ∂u ⎤
1
∂ X1
⎢
⎢ ∂ u2
Grad u(X,t) = ⎢ ∂ X
⎣ 1
1
1
∂ X2 ∂ X3
⎥
∂ u2 ∂ u2 ⎥
.
∂ X2 ∂ X3 ⎥
⎦
∂ u3 ∂ u3 ∂ u3
∂ X1 ∂ X2 ∂ X3
(2.26)
This displacement gradient tensor is denoted by ∇u (read as Grad u) for short.
The Grad notation, with a capital G, emphasizes that this is a derivative with respect
to capital X (the material particle descriptor). This tensor is the same as the zero
tensor for rigid-body motions, with all the components of the corresponding matrix
being identically zero. Now, every tensor can be decomposed into symmetric and
antisymmetric parts (like matrices). For example, the displacement gradient tensor
∇u can be decomposed as follows:
∇u = ε + ω ,
(2.27)
where the symmetric part of the displacement gradient tensor is called the infinitesimal strain tensor ε , and is given by
ε=
∇u + (∇u)T
2
(2.28)
while the anti-symmetric part of the displacement gradient tensor is called the infinitesimal rotation tensor ω and is given by
ω=
∇u − (∇u)T
.
2
(2.29)
In component form, we can write Equation (2.27) as
∂ ui
= ui, j = εi j + ωi j ,
∂xj
(2.30)
where the notation ui, j has been used to show the differentiation of ui with respect to
x j . The infinitesimal rotation tensor ω will not be used in this book. The infinitesimal
strain tensor ε in Equation (2.28) has the components
εi j =
1
2
∂ ui ∂ u j
+
∂ X j ∂ Xi
(2.31)
where we have used the fact that [(∇u)T ]i j = u j,i . Thus, for example, we have
ε11 =
1
2
∂ u1 ∂ u1
+
∂ X1 ∂ X1
=
∂ u1
∂ X1
(2.32)
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Nanomaterials
while
ε12 =
1
2
∂ u1 ∂ u2
+
∂ X2 ∂ X1
.
(2.33)
The first of these quantities (Equation 2.32) represents what is called a “normal
strain,” while the second of these (Equation 2.33) represents a “shear strain.” Note
that ε is the infinitesimal strain tensor, rather than a measure of finite strain (i.e., this
ε is a reasonable measure only for small strains). Indeed, for most of this book, the
displacement gradient itself (Equation 2.25) is assumed to be small. Finite deformations are not considered here, but excellent discussions of such deformations can be
found in books on continuum mechanics and mechanics of solids, such as the books
by Lai, Rubin and Krempl (Lai et al., 1999) or Asaro and Lubarda (2006).
In Equation (2.26), the displacement vector was differentiated with respect to one
of its independent variables, X, to obtain the displacement gradient tensor. Differentiation of the displacement vector with respect to the other independent variable
(time) gives us the velocity of each material particle:
v(X,t) =
∂ u(X,t)
∂t
(2.34)
where it is understood that the differentiation occurs with the value of X (identifying
the material particle) held fixed. The resulting velocity vector (like the displacement
vector) is a vector in the material description (i.e., it is a function of X). However, it
can be easily converted into a function of spatial position instead, i.e., into the spatial
description, using Equation (2.23), resulting in a vector field that we call ṽ(x,t):
ṽ(x,t) = v(χ −1 (x, t),t)
(2.35)
The “tilde” symbol above the v is used to remind us that this function has a
different form than v(X,t). The “spatial velocity field” obtained in Equation (2.35)
is the “flow field” that one typically sees in fluid flow applications, where arrows at
each spatial position represent the magnitude and direction of the velocity vector at
each position (with different material particles occupying a given spatial position at
different times).
The gradient of the spatial velocity vector field is a tensor called the velocity gradient or grad ṽ(x,t) and is typically denoted by the symbol L. That is, the velocity
gradient tensor L is given by
L(x,t) = grad ṽ(x,t) =
∂ ṽ
.
∂t
(2.36)
The symmetric part of the velocity gradient tensor is called the rate of deformation tensor D:
L + LT
D=
.
(2.37)
2
The rate of deformation tensor D corresponds to what is traditionally called the
strain rate, and has the following components:
2 Fundamentals of Mechanics of Materials
Di j =
1
2
29
∂ ṽi ∂ ṽ j
+
∂ x j ∂ xi
.
(2.38)
For the particular case of small deformations and small velocity gradients of
interest to this book, this corresponds identically to the time derivative of the infinitesimal strain tensor, i.e.,
Di j =
∂ εi j
= ε̇i j .
∂t
(2.39)
These latter strain rate components provided in Equation (2.39) will be used in
the rest of this text. It is useful to remember that the strain rate arises from the symmetric part of the velocity gradient; the antisymmetric part of the velocity gradient
leads to the concept of spin, and is related to the rate of rotation in the kinematics.
The key quantities that we will take forward to our discussion of the mechanics of
nanomaterials are the infinitesimal strain ε and the strain rate ε̇ .
2.1.3 Forces, Tractions and Stresses
The reader is assumed to be familiar with the concepts of statics and strength of
materials, including the concept of force equilibrium and the calculation of resultant
forces. The physical quantity that we know as a force is mathematically described as
a vector F, and the statement of force equilibrium (which is for the static problem,
with no accelerations) is Σ F = 0 (which can of course be written as three scalar force
equations along the three axes of a Cartesian coordinate system, Σ F1 = Σ Fx = 0,
Σ F2 = Σ Fy = 0, Σ F3 = Σ Fz = 0). The two major types of forces of interest are
surface forces (such as contact forces) and body forces (such as the force due to
gravity).
Our interest here is in relating the relatively intuitive concept of a force applied
on a body to the somewhat more complex concept of internal forces exerted by one
part of a body upon another, and this will lead us to the related concept of the stress
at a point in the body. The key physical concept is presented in Figure 2.3. Consider
the current configuration of a body deforming under the action of external forces.
The body shown is subjected to a number of external forces acting on its boundary.
Consider an imaginary surface S that separates the body into two parts, denoted in
the figure by B1 and B2 . Clearly part B2 of the body must exert a force on part B1 of
the body (and vice versa), across surface S, in order to keep the entire body together.
Now imagine that one makes a cut along the surface S, everywhere except over
a small element of area around the point A on the surface. Since parts B1 and B2
remain connected through the small area around the point A along the surface S,
there must be forces acting on this small area. Since both subparts of the body
remain in equilibrium, part B2 must exert a resultant force on part B1 at the point
A. The point A on the surface S was chosen arbitrarily, and the choice of a different
element of area on the surface would generally produce a different resultant force
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Nanomaterials
t
n
A
B2
B1
S
(a)
t
B1
(b)
Fig. 2.3 Internal forces on surfaces S and S’ within a body, and the traction t at a point on a surface
with normal n. The entire body must be in equilibrium, as must be all subparts of the body. (a) The
entire body, in equilibrium with the external forces. (b) Free body diagram corresponding to part
B1 of the whole body, showing the traction developed at a point on the surface S because of the
internal forces generated on B1 by part B2 .
on the part B1 acting on the new elemental area. It is clear, therefore, that there is a
distribution of internal forces on the surface S.
Consider again the point A shown in Figure 2.3. The normal to the surface S at
the point A is the vector n. The resultant force acting on the point A will in general
have a direction and magnitude that is different from that of the normal vector (the
direction of this normal vector has been chosen so that it is the outward normal from
the part B1 ). Remembering that there is a distribution of internal forces along the
surface S in the uncut body, it is useful to consider the force density on the surface
S at the point A, given by the resultant force acting on part B1 at point A divided
by the area of the element of the surface that has remained uncut. Since the force is
a vector, and the area is a scalar, the force density remains a vector; this vector is
denoted by the symbol t, and is called the traction vector. This is the traction vector
first discussed in Equation (2.10). The direction of the traction vector is identical
to the direction of the resultant force acting at point A, and the magnitude of the
traction vector is equal to the magnitude of the force divided by the elemental area
at A.
Note that the resultant force on the point A in the body is not by itself a welldefined quantity, although the resultant force on the point A on the surface S is a
2 Fundamentals of Mechanics of Materials
31
well-defined quantity. To see this, consider a different surface S’ (shown with the
long-dashed line in Figure 2.3) which also passes through the point A. Clearly the
element of area at the point A defined on the surface S’ is different from the original
element of area on the surface S that was considered in this discussion (although
the elements of area may have the same size, they have different normal vectors).
The body is divided into two slightly different parts by this new imaginary surface,
and the resultant force acting at the point A on the surface S’ will in general be a
different resultant force than that acting at the point A on the surface S. Therefore,
the traction vector acting at the point A on the new element of area (belonging to
the surface S’) will be a different vector than that acting on the point A on the old
element of area (belonging to S). Since the point A is defined by the spatial position
x and the element of area is defined through the normal vector n, the traction vector
depends not only on x but also on the normal n to the element of area passing
through x:
t = t(x, n).
(2.40)
If it is assumed that the dependence of the traction vector on the normal vector
is linear, then the traction t is a linear (vector-valued) function of the vector variable
n, and we can define a linear operator (called a tensor) that maps n into t, as in
Equation (2.10):
(2.41)
t = σ (x,t) n.
The tensor σ (x,t) defined by Equation (2.41) is called the Cauchy stress tensor, and is the most commonly used stress tensor in the mechanics of solids. There
are other stress tensors, defined for example with reference to the initial or reference configuration rather than the current configuration (This is like the engineering
stress versus the true stress in strength of materials). In this book we will confine
ourselves to the Cauchy stress, and refer to this simply as the stress tensor. The components of the stress tensor (which is defined at a point, note) were first presented in
Equation (2.11) (as always, assuming a Cartesian basis):
⎤
⎡
σ11 (x,t) σ12 (x,t) σ13 (x,t)
σ (x,t) = ⎣σ12 (x,t) σ22 (x,t) σ23 (x,t)⎦ .
(2.42)
σ13 (x,t) σ23 (x,t) σ33 (x,t)
Thus, for example, the matrix of components of the stress tensor corresponding
to the problem of the biaxial extension of a thin sheet in the x and y directions in
the plane of the sheet, with no applied stresses in the z direction normal to the sheet,
would be
⎤
⎡
σxx 0 0
σ = ⎣ 0 σyy 0⎦ .
(2.43)
0 0 0
In the discussion following Equation (2.2) it was pointed out that the components
of a vector change when the base vectors used to describe the vector are changed.
Similarly, the components of a tensor will change when the base vectors used to
describe the vector are changed; that is, the matrix of components of a tensor will
32
Nanomaterials
change when the basis changes, even though the tensor itself (defined as always by
its operation) does not change. This process of finding a different set of components of the matrix representing the stress tensor at a point by changing the basis or
coordinate system is called the transformation of the stress components.
One particular set of transformations is especially useful. Suppose that we can
find a direction e (that is, a normal vector e to an internal surface) such that when
the stress tensor acts on e the resulting traction only has components along e, that
is, the traction vector only has a normal component:
σ e = σev e,
(2.44)
where σev is the only component of the traction. Such a statement (where the tensor
operates on a vector to give a resultant vector that is in the same direction as the
original vector but scaled by some magnitude) is called an eigenvalue statement. The
corresponding vector (here e) is called an eigenvector, and the scaling factor (here
σev ) is called the eigenvalue corresponding to that eigenvector. Second-order tensors
such as the stress tensor have three eigenvalues and at least three eigenvectors (it is
possible for all vectors to be eigenvectors, as when the tensor in question is the
identity tensor).
With this in mind, consider again the general matrix of components of the stress
presented in Equation (2.42). The eigenvalues σ1 , σ2 , σ3 of this matrix are called
the principal stresses, and the corresponding eigenvectors are called the principal
directions. It can be shown that for a symmetric tensor like σ , the eigenvectors can
be chosen to be orthonormal, and can be used to provide another set of base vectors.
The stress tensor σ can now be written as a matrix of components in the basis of
the eigenvectors, and in this basis the matrix of the (components of the same stress
tensor) will now be diagonal:
⎤
⎡
σ1 (x,t)
0
0
σ (x,t) |eigenvector basis = ⎣ 0
σ2 (x,t)
0 ⎦.
(2.45)
σ3 (x,t)
0
0
Transformation of the stress components so that the matrix has only diagonal
components (Equation 2.45) is the result of solving the eigenvalue problem for the
matrix and then representing the tensor in terms of a basis made up of orthonormal
eigenvectors. The principal stresses σ1 , σ2 , σ3 can be very useful in descriptions of
the response of materials, but it is important to remember that they are associated
with the directions of the eigenvectors, which can also vary from point to point.
Since one tensor can be represented by many different matrices, these matrices
must be connected in some way (i.e., there must be some way to identify the different matrices as representing the same tensor). For example, it can be shown that all
of the matrices representing a single tensor must have the same determinant, and all
of the matrices representing a tensor must have the sum of the diagonal components
be the same. Such properties are said to be invariant, in that they do not change even
when the base vectors used to describe the tensor change (and therefore when the
matrix representing the tensor changes). The invariants are in fact properties of the
2 Fundamentals of Mechanics of Materials
33
underlying tensor, independent of the basis in which that tensor is described. The
two specific (scalar) invariants that were just discussed are known as the determinant
(det T) of the tensor T and the trace (trace T) of the tensor:
det T = det Ti j = det T̃i j ,
(2.46)
trace T = Tii = T̃ j j = T11 + T22 + T33 = T̃11 + T̃22 + T̃33 .
(2.47)
and
Here Ti j and T̃i j are the components of the matrices representing the tensor T
using two different sets of base vectors ei and ẽi respectively (consider, for example,
a standard Cartesian coordinate system and a rotated coordinate system). Note that
we have made use of the summation convention in Equation (2.47), so that the repeated index i in the equation implies summation: Tii = T11 +T22 +T33 , and similarly
with T̃ j j .
The two scalar invariants of the tensor T defined in Equations (2.46) and (2.47)
are often referred to as TIII = det T and TI = trace T respectively. There is a third
scalar invariant that one can also define for every tensor T, given by
TII =
1
(trace T)2 − trace T2 .
2
(2.48)
Together, the set of scalars TI , TII , TIII are called the set of scalar invariants of the
tensor T, and these identical invariants must be obtained from every matrix that represents T. It turns out that if the set (TI , TII , TIII ) is known, then the set (T1 , T2 , T3 )
of eigenvalues of the tensor T can be completely determined, because (considering
the corresponding diagonal matrix obtained after solving the eigenvalue problem
and using the eigenvectors as base vectors)
TI = T1 + T2 + T3
(2.49)
TII = T1 T2 + T2 T3 + T3 T1
and TIII = T1 T2 T3 .
(2.50)
(2.51)
It is possible to decompose any tensor T into a hydrostatic part, − 13 (trace T)I,
and a deviatoric part T̂, with the latter defined by
1
T̂ = T − (trace T)I.
3
(2.52)
The deviatoric part of a tensor is particularly useful when we are considering
shearing deformations and shearing stresses.
The trace of a tensor has a special meaning for the Cauchy stress tensor σ . The
mean stress at a point, σm , is defined as the scalar measure:
1
1
σm = trace σ = (σ11 + σ22 + σ33 ),
3
3
(2.53)
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Nanomaterials
and physically represents the average hydrostatic tensile stress at the point. The
negative of this quantity is called the hydrostatic pressure p:
1
1
p = −σm = − (σ11 + σ22 + σ33 ) = − (σ̃11 + σ̃22 + σ̃33 )
3
3
(2.54)
which has the conventional physical meaning of the pressure.
The deviatoric part of the Cauchy stress tensor is a particularly useful quantity,
since it represents the part of the Cauchy stress tensor that is left when the hydrostatic (or volume changing) part has been removed, and thus is a measure of the
degree of shear. It is normally given a special name, S, and is known as the deviatoric stress:
1
(2.55)
S = σ̂ = σ − (trace σ )I = σ + pI,
3
and is particularly useful in discussions of plasticity.
Now that both the kinematics of deformations and the relationships between
stresses and forces can be described mathematically, it is possible to construct rigorous ways of approaching problems in the mechanics of materials by using the
conservation laws of physics. The next section presents the governing equations of
mechanics of materials using the terminology developed above.
2.2 Work and Energy
From a thermodynamics perspective, the deformation of a body caused by external
forces acting on the body must involve some work (that is, the external forces do
work on the body in causing it to deform). In basic physics terms, a work or energy
can be described through terms of the type Fdx, where F is the force and dx is
the displacement that occurs through the applied force. A similar concept can be
developed for deformable solids, considering elements of the solid, by defining a
strain energy density W defined by
W = σi j d εi j ,
(2.56)
and the total energy within the body due to the deformation
is then the integration of
the strain energy density over the volume of the body, W dV . This total energy must
of course be related to the work done on the body by the external forces. Similarly,
one can define a kinetic energy density in the body ( 12 ρ v v). Other thermodynamic
quantities can also be defined for application in continuum mechanics through such
“density” approaches. Although we do not pursue the thermodynamics of continua
further in this book, there is a great deal that can be learned through thermodynamic
approaches, and an excellent theoretical description of the thermodynamics of solids
is provided by Ericksen (1998).
2 Fundamentals of Mechanics of Materials
35
2.3 Field Equations of Mechanics of Materials
There is a core set of field equations, derived from continuum mechanics, which
represent most problems of interest in the mechanics of materials. These equations
are:
(i) The conservation of linear momentum
∂ σi j
∂ vi
=ρ
∂xj
∂t
(2.57)
(ii) The conservation of angular momentum
σi j = σ ji
(2.58)
(iii) The conservation of energy, which in one form is
ρ cp
∂T
=κ
∂t
∂ 2T
∂ 2T
∂ 2T
+
+
2
2
∂ x1
∂ x2
∂ x3 2
+β
∂W p
∂t
(2.59)
Here ρ is the density of the material, T is the temperature field, κ is the thermal
conductivity, β is the ratio of plastic work converted to heat, and W p is the plastic
work (this book deals principally with elastic-plastic solids). Note that quantities
such as the stress σ and the strain ε are defined at every point, and so the list of
conservation laws of physics presented above are in a form that must be satisfied at
every point in the continuum. The second law of thermodynamics is not explicitly
considered here, although it does place some restrictions on the kinds of material
behaviors that are possible.
When a mathematical description of the behavior of the material (known as a
constitutive relation or constitutive law, and described in the next section) is added
to this list of the conservation laws of physics, a complete set of governing equations
is obtained for describing the mechanics of materials. Since the governing equations
presented above are partial differential equations in space and time, the solution of
a problem in the mechanics of materials also involves the definition of boundary
conditions and initial conditions. The solution procedures relevant to many such
problems can be found in standard textbooks in continuum mechanics or the mechanics of solids. Examples of useful references of this type include the books by
Lai, Rubin and Krempl (Lai et al., 1999) or Asaro and Lubarda (2006).
2.4 Constitutive Relations, or Mathematical Descriptions
of Material Behavior
An equation or equations representing the behavior of the materials of interest (e.g.,
stress-strain relationships) must be added to this list (Equations 2.57–2.59) of the
conservation laws of physics. Such equations are commonly referred to as constitu-
36
Nanomaterials
tive laws or constitutive relations. Note that constitutive laws are generally expected
to be satisfied pointwise, that is, they are valid at every point in the solid (in a
composite material, for example, different constitutive laws may apply at different
material points). Several examples of constitutive laws that may be of interest to the
mechanics of nanomaterials are discussed in subsequent sections.
2.4.1 Elasticity
A material is defined as being elastic if the stress in the material depends only on
the current deformation (rather than the history of the deformation) and if all of the
work done by the stresses on the deformations is fully recoverable upon unloading.
During the deformation, this work is stored in the material in a form that is called
the strain energy. Since all of the information of interest about the deformation is
contained in the displacement gradient tensor ∇u, an elastic material is one in which
the stress tensor is
σ = σ (∇u).
(2.60)
i.e. the stress is a function of ∇u
The focus here is on linear elastic materials, in which the stress depends linearly
on the displacement gradient. Now the simplest way to evaluate a function is in
terms of a Taylor-series expansion: consider a function y = f (x), and let us suppose
that you know the value of the function (and all of its derivatives) when x = x0 , but
you do not know how the function behaves anywhere else. You want to evaluate the
value of this function at a different value of the argument, i.e., you want to evaluate
the function at x = x0 , but where x is close to x0 (so that the distance x − x0 is
small). The easiest way to get an estimate of the value of the function at x is to use
the information that you have at x = x0 , through a series expansion called a Taylor
series:
∂ f ∂ 2 f f (x) = f (x0 )+
(x−x0 )+
(x − x0 )2 +. . . higher order terms.
∂ x x=x0
∂ x2 x=x0
(2.61)
If you keep only the second term in the expansion (the term that is linear in
x − x0 ), you have a linear approximation of the function (if all of the higher order
terms are actually zero, then the function is in fact a linear function, so that a curve
y = f (x) will be a straight line). Similarly, considering a Taylor-series expansion of
the function σ (∇u) about the initial value ∇u = 0 corresponding to zero strain, and
retaining only the first order terms, we have
∂ σ σ = σ |∇u=0 +
: ∇u + . . . higher order terms.
(2.62)
∂ ∇u ∇u=0
Here the first term on the right hand side corresponds to the stress in the material under zero displacement gradient (i.e., no deformation, so that σ |∇u=0 is like
a residual stress). The second term represents the linear term in the Taylor-series
σ
expansion. The second term on the right hand side involves the derivative ∂∂∇u
,
2 Fundamentals of Mechanics of Materials
37
which is the derivative of the second-order tensor σ with respect to another secondorder tensor u, and is thus a fourth-order tensor (i.e., it will map a second-order tensor into another second-order tensor, just like a second-order tensor like σ maps a
σ
when ∇u = 0
vector into another vector). The result of evaluating the derivative ∂∂∇u
is usually denoted C and referred to as the elastic modulus tensor. The colon in the
second term on the right hand side of equation (2.62) is used to show that the fourthorder tensor is operating on the second-order tensor ∇u, the result of the operation
being another second-order tensor which is the stress tensor (just as a second-order
tensor maps a vector into another vector, a fourth-order tensor maps a second-order
tensor into another second-order tensor). If we assume the material has zero stress
at zero displacement gradient (so that the initial state is stress free), we can write the
elastic constitutive relationship in component form as
σi j = Cijkl
∂ uk
∂ xl
(2.63)
where Cijkl are the components of the elastic modulus tensor and ∂∂ uxk are the compol
nents of the displacement gradient tensor (note the use of the summation convention
on the dummy indices k and l in the equation (2.63)). The components of the elastic
modulus tensor are constants that represent material properties (known as the moduli), and must be measured independently. Using Equation (2.30), we can rewrite
the linear elastic constitutive relationship in terms of the infinitesimal strain and
infinitesimal rotation tensors:
σi j = Cijkl {εkl + ωkl }.
(2.64)
Assuming that pure rotations give zero stress, the linear elastic constitutive relationship reduces to
σi j = Cijkl εkl .
(2.65)
Since the stress and strain tensors each have nine components (remember the
3x3 matrices), the elastic modulus tensor will in general have 81 components. The
constitutive law of Equation (2.65) is appropriate to anisotropic linear elastic solids
(i.e., linear elastic solids in which the properties vary with direction in the material,
such as a fiber-reinforced material or a crystal of quartz). However, the symmetry
of the stress and strain tensors results in some simplifications, since it then requires
that
(2.66)
Cijkl = Cjikl
and
Cijkl = Cijlk
(2.67)
respectively. This reduces the number of independent components of the elastic
modulus tensor to 36. There is a further expectation that every linear elastic material is associated with the existence of a strain energy function W (εi j ) from which
the stress can be derived:
∂W
σi j =
.
(2.68)
∂ εi j
38
Nanomaterials
It follows that the linear elastic moduli can be derived as (using Equation 2.65)
Cijkl =
∂ σi j
∂ 2W
∂ 2W
∂ σkl
=
=
=
= Cklij .
∂ εkl
∂ εkl ∂ εi j
∂ εi j ∂ εkl
∂ εi j
(2.69)
Thus the existence of the strain energy function results in the requirement that
Cijkl = Cklij
(2.70)
and this reduces the number of independent components of the elastic modulus tensor to 21. The restriction represented by Equation (2.70) is observed to be a reasonable restriction for all metals and ceramics deformed to small strains. Thus, the most
general anisotropic linear elastic material would have 21 independent constants that
need to be experimentally determined for complete specification of the material behavior in an elastic domain.
Note that so far no material symmetry has been invoked in our description of the
elastic constants. Invoking such material symmetries (for example, crystal symmetries) further reduces the number of elastic constants that are needed. Thus, while
a general anisotropic linear elastic material requires 21 constants, a monoclinic linear elastic material requires only 13 independent elastic constants, an orthotropic
material requires 9, and a transversely isotropic material requires 5. The limiting
condition of a completely isotropic linear elastic material requires only two independent elastic constants (e.g., the Young’s modulus and the Poisson’s ratio). Excellent derivations of the symmetries are presented from the mechanics perspective
by Lai, Rubin and Krempl (Lai et al., 1999) and from a materials perspective by
Nye (1950).
The fact that the stress and infinitesimal strain tensors are symmetric implies that
each of them has six independent components, and this leads to another way of
describing the anisotropic linear elastic relationship of Equation (2.65), using Voigt
notation, which is useful when the base vectors are Cartesian and unchanging. In
this approach, we first construct column matrices representing the stress and strain.
Now, we use the row index in this column matrix to define a new index called the
Voigt index that takes values 1, 2, . . . 6. For example, in the Voigt notation the matrix
of stress components would be
⎡ ⎤ ⎡ ⎤
σ11
σ1
⎢σ22 ⎥ ⎢σ2 ⎥
⎢ ⎥ ⎢ ⎥
⎢σ33 ⎥ ⎢σ3 ⎥
⎥ ⎢ ⎥
σi j |Voigt = ⎢
(2.71)
⎢σ23 ⎥ ≡ ⎢σ4 ⎥ .
⎢ ⎥ ⎢ ⎥
⎣σ31 ⎦ ⎣σ5 ⎦
σ12
σ6
Similarly, a matrix of strain components can be constructed:
2 Fundamentals of Mechanics of Materials
⎡ ⎤ ⎡ ⎤
ε11
ε1
⎢ε22 ⎥ ⎢ε2 ⎥
⎢ ⎥ ⎢ ⎥
⎢ε33 ⎥ ⎢ε3 ⎥
⎥ ⎢ ⎥
εi j |Voigt = ⎢
⎢ε23 ⎥ ≡ ⎢ε4 ⎥ .
⎢ ⎥ ⎢ ⎥
⎣ε31 ⎦ ⎣ε5 ⎦
ε12
ε6
39
(2.72)
These two matrices for the stress and the strain are then related by (for general
anisotropic linear elasticity, following Equation 2.65) a new matrix of Voigt constants CIK , each of which can be explicitly derived from Cijkl :
σI = CIK εK , with I, K taking values 1, 2, . . . , 6
(2.73)
where the components σI and εK are the components of the column matrices in
Equations (2.71) and (2.72) respectively, and the Voigt constants CIK are obtained
from the moduli Cijkl by replacing i j with a value for I following the pattern in
Equation (2.71). Thus, if i j = 11, then I = 1; if i j = 22, then I = 2; if i j = 23, then
I = 4, and so forth (similarly with the kl being matched to the K index). Thus the
matrix of Voigt constants is a 6x6 matrix, and there are 36 constants. Note, however,
that CIK are NOT the components of a second-order tensor, just as σI are not the
components of a vector—these are simply convenient matrices to use to describe
linear elasticity. The existence of the strain energy function makes (following Equation 2.70) the CIK = CKI , and this symmetry provides the 21 independent constants
in the most general anisotropic linear elastic material. The corresponding matrix of
Voigt constants is
⎤
⎡
C11 C12 C13 C14 C15 C16
⎢C12 C22 C23 C24 C25 C26 ⎥
⎥
⎢
⎢C13 C23 C33 C34 C35 C36 ⎥
⎥
CIK = ⎢
(2.74)
⎢C14 C24 C34 C44 C45 C46 ⎥ .
⎥
⎢
⎣C15 C25 C35 C45 C55 C56 ⎦
C16 C26 C36 C46 C56 C66
The various material symmetries can now be easily seen in the Voigt matrix
(sometimes called the stiffness matrix in materials science). For example, consider
a transversely isotropic material (such as might be produced by a severe extrusion
process). In such a material, every plane perpendicular to a single specific plane is a
plane of symmetry for the material. Consider, for example, a material that has been
extruded along the e3 direction, so that every plane that contains the e3 axis (and
therefore is perpendicular to the plane of isotropy that has normal e3 ) is a plane of
symmetry. For such a transversely isotropic material the Voigt constants would be
⎤
⎡
C11 C12 C13 0 0
0
⎥
⎢C12 C11 C13 0 0
0
⎥
⎢
⎥
⎢C13 C13 C33 0 0
0
⎥,
⎢
CIK |transversely isotropic = ⎢
(2.75)
⎥
0
0
0
C
0
0
44
⎥
⎢
⎦
⎣ 0 0 0 0 C44
0
0 0 0 0 0 12 (C11 −C12 )
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Nanomaterials
and there are only five independent constants. Another case that is of common interest in materials science is that of cubic materials (as in the face-centered-cubic and
body-centered-cubic systems), for which case we have
⎡
⎤
C11 C12 C12 0 0 0
⎢C12 C11 C12 0 0 0 ⎥
⎥
⎢
⎢C12 C12 C11 0 0 0 ⎥
⎥,
⎢
CIK |cubic = ⎢
(2.76)
⎥
⎢ 0 0 0 C44 0 0 ⎥
⎣ 0 0 0 0 C44 0 ⎦
0 0 0 0 0 C44
so that there are three independent constants in this case. In the limiting case of the
fully isotropic material, the Voigt matrix reduces to
⎡
⎤
C11 C12 C12
0
0
0
⎢C12 C11 C12
⎥
0
0
0
⎥
⎢
⎥
⎢C12 C12 C11
0
0
0
⎥ , (2.77)
CIK |isotropic = ⎢
⎥
⎢ 0 0 0 1 (C11 −C12 )
0
0
2
⎥
⎢
1
⎣ 0 0 0
⎦
0
0
2 (C11 −C12 )
1
0 0 0
0
0
2 (C11 −C12 )
and now there are only two independent constants. In the more conventional notation
for the elastic moduli used in Equation (2.65), this fully isotropic material has the
moduli
Ci jkl = λ δi j δkl + µ {δik δ jl + δil δ jk },
(2.78)
and the two independent constants, λ and µ , are called the Lamé moduli. These are
related to the Voigt constants C11 and C12 in Equation (2.77) by C11 = λ + 2µ and
12
C12 = λ , so that C11 −C
= µ.
2
The Lamé moduli λ and µ are related to the conventional Young’s modulus and
Poisson’s ratio by
Eν
λ=
(2.79)
(1 + ν )(1 − 2ν )
and
µ=
E
.
2(1 + ν )
(2.80)
Since only two constants are required to define an isotropic linear elastic material,
we can use either the pair λ , µ or the pair E, ν . Returning to Equation (2.65), we
now see that for an isotropic linear elastic material the stress is related to the strain
by
σi j = λ εkk δi j + 2µεi j .
(2.81)
Here εkk uses the summation convention and therefore is ε11 + ε22 + ε33 ; this quantity
is the total volumetric strain at a point in the material. Thus, for example, we have
σ11 = λ (ε11 + ε22 + ε33 ) + 2µε11
(2.82)
2 Fundamentals of Mechanics of Materials
41
whereas
σ12 = σ21 = 2µε12
(2.83)
and so forth. In terms of the Young’s modulus and Poisson’s ratio, Equation (2.81)
can be rewritten as
σi j =
E
1+ν
εi j +
ν
εkk δi j .
1 − 2ν
(2.84)
Although many mechanical engineers and most materials scientists are most
comfortable with linear elastic materials described through relations of the form
of Equation (2.84), many problems in mechanics of materials are best thought of
in terms of changes in volume (corresponding to the volumetric or hydrostatic
response) and changes in shape (corresponding to the distortional or shearing response). Recalling Equation (2.55), the stress tensor can also be broken up into
hydrostatic and deviatoric parts, so that σ = S − pI, and this decomposition allows
one to break down the constitutive behavior into volumetric and shearing behaviors:
1
p = − σkk = −K εkk ,
3
(2.85)
and Si j = 2µ ε̂i j ,
(2.86)
where ε̂i j are the components of the deviatoric strain tensor given, as in
Equation (2.52), by
1
ε̂i j = εi j − εkk δi j .
(2.87)
3
The coefficient K in Equation (2.85) is called the bulk modulus, and µ is called
the shear modulus.
Thus three different pairs of material properties are commonly used to describe
isotropic linear elasticity: (λ , µ ), (E, ν ), and (K, µ ). The first of these is often convenient while writing governing equations, the second of these is convenient when
the primary data available is obtained in uniaxial tension experiments, and the third
is useful when separating out volumetric and distortional effects. Any of these pairs
can be used, or other combinations involving just two of these properties. Since there
are only two independent properties for an isotropic linear elastic material, there are
relationships between all of these quantities, as in Equations (2.79) and (2.80). Some
additional relationships are particularly useful:
2µ
3
9K µ
E=
3K + µ
1 2µ − 3K
and ν = −
.
2 3K + µ
K=λ+
(2.88)
(2.89)
(2.90)
The linear elastic relationships given above are all presented in terms of calculating the stress components given the strain components. It is possible to invert
42
Nanomaterials
all of these relationships to calculate the strain components when given the stress
components, e.g., inverting Equation (2.84) to solve for the strains gives
εi j =
ν
1+ν
σi j − σkk δi j .
E
E
(2.91)
Similar equations can be written in terms of the other moduli.
It is possible to show, on physical grounds, that E > 0, µ > 0, K > 0 and
−1 < ν ≤ 0.5 (although in most engineering materials, 0 ≤ ν ≤ 0.5). From Equation (2.90) above, we see that ν = 0 only when µ = 3k
2 , but the bulk modulus is
usually greater than the shear modulus, i.e., usually K > µ , so that it is difficult to
find materials with zero Poisson’s ratio (the closest approximation is cork, which is
why this material is used to stop up wine bottles – it does not expand as you push it
into the bottle). Materials with negative Poisson’s ratio are unknown in nature, but
efforts are underway to construct metamaterials that have this property.
For a single crystal, the linear elastic response represents the stiffness of the interatomic bonds within the material near the equilibrium positions of the atoms in
the lattice, and the elastic moduli for a single crystal are not generally isotropic.
Thus one must use the full set of linear elastic constants (making use of available
symmetries) to describe the mechanical behavior of a single crystal in the linear
elastic range, and one must find values for the coefficients Ci jkl corresponding to
Equation (2.74) for each material. For example, for single crystal α -iron, the cubic symmetry allows us to use Equation (2.76) and one only needs to find values
for C1111 , C1122 , and C2323 corresponding to the Voigt constants C11 , C12 , and C44 .
Experimental determination of these constants provides C11 = 230.37 GPa, C12 =
134.07 GPa, and C44 = 115.87 GPa at a temperature of 300 K (the elastic moduli
are functions of temperature). A scalar measure of the elastic anisotropy of cubic
crystals that is often used is
2C44
A=
.
(2.92)
(C11 −C12 )
This measure is usually much greater than 1 for most metals (e.g., A = 2.41 for
α -iron at 300K), with the exception being tungsten, where A ≈ 1. However, most
engineering materials consist of large polycrystalline assemblies, with the crystals
oriented randomly, and the effective behavior of this polycrystalline mass in the
linear elastic range might be nearly isotropic (assuming that there is no significant
texture). For polycrystalline pure iron, therefore, an isotropic linear elastic behavior
is often a reasonable approximation, and the experimental values are E ≈ 210 GPa
and ν ≈ 0.29. Note that the averaging performed over a large number of randomly
oriented crystals has effectively reduced the number of independent constants. A
table of typical values of the moduli for a number of materials (for conventional
grain sizes, in the randomly oriented polycrystalline case) is presented in Table 2.1.
Note the high variability in the elastic data on some polycrystalline ceramics such as
B4 C, which arises because of differences in processing approaches and the resulting
flaw distributions, as well as variations with measurement processes.
Up to this point, the discussion has assumed that the material behavior is linear
elastic. However, most materials demonstrate linear elastic behavior only in a finite
2 Fundamentals of Mechanics of Materials
43
Table 2.1 Young’s modulus and Poisson’s ratio for some conventional grain sized materials.
Material
Young’s modulus E, GPa Poisson’s ratio ␯
Iron
210
0.29
Copper
130
0.34
Aluminum
70
0.33
Vanadium
128
0.37
Tantalum
186
0.34
Nickel
200
0.31
Titanium
116
0.32
Alumina, Al2 O3
396
0.25
Silicon carbide, SiC
410
0.14
Boron carbide, B4 C
362–460
0.18–0.21
range of stresses (or strains), beyond which the deformations are not reversible and
some of the work done by the stresses is associated with permanent changes in the
material. This non-elastic behavior is discussed in the next section.
2.4.2 Plastic Deformation of Materials
Most materials undergo permanent, non-reversible deformations if they are stressed
beyond some critical point, so that the basic assumptions made in the section on
elasticity above are no longer valid. In particular, the behavior of the material may
become such that the stress in the material depends not just on the current state of
deformation, but on the history of the state of deformation. For example, the material behavior can be different during a second loading cycle than it was during the
first loading cycle, a behavior sometimes called hardening. This history-dependent
constitutive behavior is called plasticity – a material that obeys such a constitutive
law is said to be plastic. Most metals in our everyday experience show plastic behavior at some point, and such a response is used heavily in industry in the forming
of materials, components and structures. Understanding plasticity is important both
from a design viewpoint (if permanent deformations are to be avoided) and from a
manufacturing viewpoint (if the material is to be formed into a desired shape).
The classical example of plastic behavior is observed when a rod or strip of a
ductile metal is subjected to uniaxial tensile loading (Figure 2.4) . This is called
the simple tension test, although—as an experiment—there is little that is simple
about it. The shape of the sample is typically the dogbone shown in the figure, and
it is loaded along the x1 direction with a force P. The central section of the sample has a gauge length L and a cross-section area A0 , and sees a nearly uniform
stress state with only one nonzero stress component, σ11 . Note that σ11 is a component of the Cauchy stress, also sometimes called the true stress, which is defined in
the deformed configuration as σ11 = σtrue = PA and so requires a knowledge of the
current cross-section area A (usually very difficult to measure). The corresponding
engineering stress σeng is easier to measure and is defined as the ratio of the applied
44
Nanomaterials
x
P
x
P
Fig. 2.4 Schematic of a simple tension test. The loading direction is the x1 direction, and the gauge
length is L. Note the specimen shape, designed to minimize the influence of the gripping conditions
on the ends. Specimen shapes may be specified by testing standards.
force P to the initial cross-section area A0 :
σeng =
P
.
A0
(2.93)
As the sample is loaded in tension, it elongates, and the change in the length Δ L
can be measured using a variety of devices. The ratio of this change in length to the
original length is a strain measure called the engineering strain:
εeng =
ΔL
.
L
(2.94)
This is a strain measure that uses the initial configuration of the specimen as a
reference. The strain measure that corresponds to the current configuration of the
specimen is called the true strain, and considers the change δ l in the current length
l of the specimen through ε = δll . The true strain should be used whenever the true
stress is used, since they both represent variables in the current configuration of the
specimen. The engineering stress is plotted against the engineering strain, providing
a stress-strain curve that is characteristic of the material rather than of the specimen
(at least until necking of the specimen begins, when the deformation localizes to a
small region that thins rapidly and results in sample failure). A schematic of such
a tensile stress-strain curve is presented in Figure 2.5a. The early part of the stressstrain curve is linear, and represents the linear elastic response of the material (the
slope of this line corresponds to the Young’s modulus). In this linear domain, unloading the sample by decreasing the force P will result in retracing the stress-strain
curve along the straight line, as elastic behavior demands.
As the engineering stress continues to increase, the stress-strain curve deviates
from linearity. The point at which the deviation from linearity begins is called the
proportional limit and the corresponding stress is called the proportional limit stress
σ pl . With a further increase in the stress, the point is reached at which permanent
deformation of the sample begins, and the corresponding stress is called the yield
strength (σy ). In the regime between the proportional limit σ pl and yield σy , unloading and reloading of the sample results in retracing the stress-strain curve along
this nonlinear portion, which is a nonlinear elastic response. For most metallic materials this nonlinear elastic regime is negligible, and so one often identifies the
proportional limit with yield. Note that by convention, a stress level that represents
material behavior is referred to as a strength, as in the yield strength.
Engineering Stress
2 Fundamentals of Mechanics of Materials
45
σUT
σY
σ PL
E
Engineering Strain
True Strain
2000
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
Stress, MPa
Nanocrystalline Nickel
1500
1000
500
Engineering Stress
True Stress
0
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
Engineering Strain
Fig. 2.5 (a) Schematic of tensile stress-strain curve for ductile metal, showing yield, an ultimate
tensile strength, and subsequent failure. (b) Experimentally obtained curve for nanocrystalline
nickel, with an average grain size 20 nm. Note the unloading line after initial yield, with the unloading following an elastic slope. Data provided by Brian Schuster.
Once the applied stress has been taken beyond σy , permanent (plastic) deformation occurs, and unloading of the sample results in an unloading stress-strain
response which does not follow the original stress-strain curve, but instead typically
unloads along a straight line parallel to the linear elastic domain (this is called linear elastic unloading, as shown by the unloading line in Figure 2.5). The plastic
deformation process after yield is called plastic flow. In practice, measuring the true
onset of permanent deformation is not possible, and one instead identifies the yield
strength by determining the point at which a permanent offset strain of 0.2% would
be achieved.
If loading is continued beyond the yield point, the stress-strain curve is nonlinear
in two senses: first, the loading behavior does not follow a straight line response,
and second, the unloading path is different from the loading path. If the stress in
the sample increases after yield, the material is said to show hardening, while if the
stress remains constant after yield the material is said to be perfectly plastic. Since
the unloading is always elastic with a slope corresponding to the original Young’s
modulus E, the permanent plastic strain left over after elastic unloading evolves
46
Nanomaterials
with increasing load. Even in a hardening material, continued loading results in a
maximum in the engineering stress called the ultimate strength σut , where the subscript ut signifies that the strength referred to is the ultimate strength in tension.
While there is a maximum in the engineering stress, the true stress (if it could be
calculated) generally shows no maximum – what is happening is that the sample begins to develop a neck, with a region of the specimen becoming rapidly thinner and
consequently with a rapidly decreasing current cross-section area A. Even though
the load carried by the specimen decreases once necking begins, the current crosssection area A carrying the load is typically decreasing faster, so that the true stress
continues to increase.
The tensile stress-strain curve for the material thus provides two new material
properties that describe the plastic response: the yield strength σy and some measure
of the hardening of the material with increasing plastic strain. For example, a very
common description of the tensile stress-strain curve has the power law form
n
ε
σ = σy
(2.95)
εy
σ
where εy = Ey is the yield strain and the parameter n is called the strain hardening
index and is a new material property. For the purposes of constitutive modeling the
hardening index of the material should be defined in terms of the true stress vs true
strain response, rather than the engineering stress vs engineering strain response.
An example of real experimental data is presented in Figure 2.5b for a nanocrystalline nickel sample with an average grain size of 20 nm. The sample was tested
using a microtension apparatus at a low strain rate of 10−4 s−1 . Both the engineering stress vs engineering strain curve and the true stress vs true strain curve are
shown in the figure. Several points are worthy of note. First, the onset of yield is
not immediately obvious – what should be chosen as the yield strength? The departure from linear elastic response is so smooth that yield is not clearly defined
(the 0.2% onset condition would suggest that σY ≈ 1000 MPa). Second, the elastic
modulus can be obtained directly by allowing a small amount of elastic unloading,
as shown in the figure, and measuring the slope of the unloading line. In this case,
a Young’s modulus of 160 GPa is obtained, quite close to the known bulk response
for coarse-grained nickel. Third, as expected, the true stress is always greater than
the engineering stress (in this tension experiment). Fourth, the material is hardening
for substantial strains, up to about 6%. The necking regime and the ultimate tensile
strength are not shown in this dataset.
2.4.2.1 Plastic Constitutive Relations
Although Equation (2.95) is commonly used by materials scientists to describe the
plastic behavior of metals, it does not provide a true constitutive relation in that it
does not allow one to relate the general stress in the material (and the stress is a tensor) to the strain in the material at that point (the strain is also a tensor). The classical theory of plasticity provides a mechanism for translating the concepts presented
2 Fundamentals of Mechanics of Materials
47
in Equation (2.95) into a generalized constitutive equation that can be used to describe the behavior of materials under arbitrary loading conditions. The first step
is to generalize the concept of a limiting stress (beyond which the material develops
permanent or plastic deformations) to arbitrary stress states rather than the uniaxial
stress state discussed above. In other words, one asks the question: when will a material deform permanently if it is loaded under a stress state that involves multiple
stress components?
The second step is to describe how the stresses and strains are related once plasticity begins. Here one immediately sees the need to distinguish between two parts
of the total strain tensor: the elastic strain εiej (t) and the plastic strain εipj (t), since
the former is recoverable on unloading while the latter accumulates with the deformation. The elastic strain components are always related to the stress components through the elastic moduli (e.g., through Equation 2.91). The generic problem
in plasticity therefore reduces to the following: given σi j (t), determine the plastic
strain εipj (t) that is developed, i.e., given the stress history, find the strain history.
The simplest way to think about the first step is to visualize the problem in
stress space. Recall that there are six independent stress components in the general stress tensor, but that we can solve the eigenvalue problem to find the three
principal stresses and the three principal directions. If we assume that the material
is isotropic, so that the behavior of the material is the same in all directions, then
all we need to consider are the three principal stresses in order to decide whether
or not the material has reached a limiting condition where permanent deformations
begin. Imagine, therefore, a three-dimensional space in which the three principal
stresses are plotted along the three axes of the space. This three-dimensional space
is called the stress space (for the anisotropic material we would have to use a sixdimensional space corresponding to the six independent components of the stress
tensor). Note that the stress tensor is defined at every point in the body, so that the
three principal stresses are defined at every point in the current configuration of the
body, and so there is a three-dimensional stress space associated with every point
in physical space currently occupied by the body. The stress state at a point in the
material is represented by the local principal stresses (σ1 , σ2 , σ3 ) which represent a
point in the stress space. The stress history is the locus of stress states in stress-space
(followed by a material point in the physical body).
Now consider the origin in the stress space, corresponding to the case where
σ1 = 0, σ2 = 0, σ3 = 0. This corresponds to an unstressed material. As we load the
material, the various stress components in the stress tensor begin to increase, and
so do the principal stresses. For example, the uniaxial tension experiment that we
described earlier corresponds to the case where the only nonzero (and increasing)
principal stress is (say) σ1 . For small stresses, our expectation is that the material
will remain linear elastic, even though we have a multiaxial stress state, and Equation (2.65) will hold. Thus, there must be a region around the origin in stress-space
where the material behavior is always linear elastic. What is the boundary of this
region? This question is identical to the question of when permanent deformations
will begin under multiaxial loading, i.e., when will yield begin, and we see that
the answer to this question must represent a surface in this three-dimensional stress
48
Nanomaterials
space. This surface, bounding the elastic region in stress-space, is called the yield
surface, and is analogous to the definition of σy in the preceding subsection. Indeed, by definition the intersection of this yield surface with the σ1 axis will be
σy . Now consider the stress state at a point. If a single principal stress is increased
sufficiently in magnitude, plastic deformation will occur. If more than one principal
stress is changing, the onset of plasticity will depend on the relative magnitude of
the principal stresses.
To see this a little more physically, consider a thin sheet of material under biaxial
tension (see Figure 2.6a), so that both σ1 = 0 and σ2 = 0 in the plane of the sheet,
but σ3 = 0. The corresponding two-dimensional stress space is shown in Figure 2.6b.
Consider first the case where σ2 = 0, which amounts to the simple tension problem;
this case corresponds to moving out along the horizontal (σ1 ) axis in Figure 2.6b,
and clearly yield will occur when σ1 = σy . If instead we consider the case where
σ1 = 0, corresponding to uniaxial tension along the vertical direction, the material
behavior should remain similar to that in the first case since the material is isotropic,
and so yield will occur when σ2 = σy along the vertical (σ2 ) axis in Figure 2.6b. This
much follows just from the simple tension test.
σ
σ
σ
σ
σ
σ
σ
σ
Fig. 2.6 (a) Illustration of the biaxial tension of a sheet. (b) 2D stress space corresponding to the
sheet, showing the stress path and possible yield surface bounding the elastic region.
What happens if both σ1 = 0 and σ2 = 0? In this case the stress state will follow
some arbitrary path in the σ1 − σ2 space of Figure 2.6b, for example the curved
path shown. When will the onset of permanent deformations occur along this path?
In general, the answer to this depends on the material and can only be determined
by experiment; one possible solution is the arc shown in Figure 2.6b and defined
as the yield surface (this is in fact the intersection of the full yield surface with the
plane corresponding to σ3 = 0). The area within this arc and containing the origin
is the elastic region in this stress space; the yield surface is the boundary of this
elastic region. There are many possible shapes for this yield surface, and some of
the common shapes are discussed later in this book. It is already clear that plasticity
is a far more complex constitutive behavior than linear elasticity. However, some of
the basic properties of yield surfaces can be determined simply by considering the
physics of plasticity, considered next.
2 Fundamentals of Mechanics of Materials
49
Mathematically, the yield surface is defined by those stress states that lie on it,
that is, the yield surface is defined by
f (σ ,Yi ) = 0
(2.96)
where the Yi represents a set of material parameters (such as the yield strength in
uniaxial tension). It is expected that a finite stress magnitude is required to produce
plastic deformation, so that
(2.97)
f (0,Yi ) = 0,
that is, the yield criterion is not satisfied when the stress tensor is zero. Further, for
isotropic materials, the yield surface can only depend on the principal stresses, so
that
(2.98)
f (σ ,Yi ) = 0 =⇒ f (σ1 , σ2 , σ3 ,Yi ) = 0,
and this relationship describes the yield surface in our 3D stress space.
So far we have used the concepts of the existence of an elastic region, and the
isotropy of the material. Further determination of the form of this function f (σ ,Yi )
requires the use of the physics of plastic deformation in metals. Plastic deformation
in crystalline metals occurs through the motion of line defects known as dislocations, and dislocation slip occurs largely due to shear stresses. Indeed, it is generally
observed that crystalline metals do not yield under purely hydrostatic stress states.
We thus conclude that we can replace the function f (σ ,Yi ) by a function f (S, Yi ),
remembering (Equation 2.55) that the Cauchy stress tensor σ can be broken down
into deviatoric and hydrostatic parts: σ = S − pI. Since p cannot cause yielding, the
yield surface must be represented by
f (S,Yi ) = 0,
(2.99)
and the concept of material isotropy then leads to
f (S,Yi ) = 0 =⇒ f (S1 , S2 , S3 ,Yi ) = 0 =⇒ f (SI , SII , SIII ,Yi ) = 0,
(2.100)
where we have used the fact that the set of eigenvalues of S can be directly obtained from the three invariants SI , SII , SIII of the deviatoric stress tensor S.
The three invariants SI , SII , SIII of the deviatoric stress tensor are conventionally
called J1 , −J2 , and J3 respectively (note that J2 = −SII ). Since S is deviatoric,
J1 = trace S = 0, and the yield surface can only be described by the yield function
f (J2 , J3 , Yi ) = 0.
(2.101)
Thus, the assumption of dislocation driven plasticity and isotropy leads to Equation (2.101) describing the yield surface (appropriate for crystalline metals). However, what is the form of this function? Further reduction of the form of the
yield surface requires experimental measurements, since a specific functional form
depends on the specific deformation mechanism, and even crystalline metals show a
wide variety of such mechanisms. The most common additional assumption is that
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Nanomaterials
the dependence of the function on J3 = det S is negligible, and that there is only one
material strength parameter Y that is needed to describe the behavior so that
f (J2 , Y ) = 0,
(2.102)
where Y would be related, for example, to the yield strength observed in the simple
uniaxial tension test. The simplest possible form of this function is
f (J2 , Y ) ≡ J2 −Y = 0 =⇒ yield will occur when J2 = Y.
(2.103)
An equation such as Equation (2.103) describing the condition that must be satisfied for yield to occur (that is, for the onset of plastic deformation for an arbitrary stress path) is called a yield criterion. All yield criteria have a similar form, in
that on the left-hand side of the criterion will be a function of the applied stresses,
while on the right-hand side of the yield condition is a material property such as
the yield strength (there are more complex forms of the yield criterion, but they are
not discussed in this book). Yield occurs when the function of the applied stresses
is equal to the material property. In this case, for example, we know that (using
Equation 2.48)
J2 = −SII = −
1
(trace S)2 − trace S2
2
1
= trace S2 , because trace S = 0
2
1
= Si j Si j
2
1 2
2
2
2
2
2
= {S11
+ S22
+ S33
+ 2(S12
+ S23
+ S31
)}
2
1
= (S12 + S22 + S32 )
2
(2.104)
(2.105)
(2.106)
(2.107)
(2.108)
where we have used the summation convention. In terms of the principal stresses
(the eigenvalues of the Cauchy stress tensor), we have
1
1
J2 = {σ12 + σ22 + σ32 − (σ1 + σ2 + σ3 )2 }
2
3
1
2
= {(σ1 − σ2 ) + (σ2 − σ3 )2 + (σ3 − σ1 )2 }.
6
(2.109)
(2.110)
The yield condition at Equation (2.103) can thus be written as
1
Si j Si j = Y,
2
(2.111)
1
{(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 } = Y.
6
(2.112)
or in terms of the principal stresses,
2 Fundamentals of Mechanics of Materials
51
What is Y ? Consider the onset of yield in the simple tension test, where only
σ1 = 0. It follows from Equation (2.112) that at yield we have Y = 13 σy2 . It follows
that for general loading as in Equation (2.112) we must still have
1
Y = σy2 ,
3
(2.113)
where we remember that σy represents the yield strength in tension. Thus the full
yield criterion for multiaxial loading of a crystalline metal would be
1
{(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 } = σy2 ,
2
(2.114)
which essentially says that yield will occur when the combination of applied stress
components on the left side reaches the value defined by the material property on
the right side. This represents a surface called the yield surface in our 3D space of
the principal stresses. This yield criterion is sometimes called the von Mises yield
criterion, and appears to work very well for a large variety of metals. The quantity
on the left-hand side of Equation (2.114) is often used to define a scalar measure of
the applied stress, usually called the von Mises or effective stress σ̄ :
1
σ̄ = √
{(σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 },
(2.115)
2
and in terms of this von Mises stress, the yield criterion can be very simply written
as yield occurs when
σ̄ = σy .
(2.116)
Note that in the two-dimensional case corresponding to the biaxial tension problem of Figure 2.6a, Equation (2.114) reduces to
σ12 − σ1 σ2 + σ22 = σy2 ,
(2.117)
which is the equation for the curve in Figure 2.6b. As the stress increases along any
arbitrary stress path in Figure 2.6a, yield will occur when the path reaches the yield
curve in Figure 2.6b.
Equation (2.116) represents a surface called the yield surface in our 3D space of
the principal stresses. It predicts, for example, that yield will occur in a torsion test
when the shear stress in torsion τ satisfies
1
τ 2 = σy2 ,
3
(2.118)
so that the yield strength τy observed in a simple shear or torsion experiment should
be related to the yield strength in simple tension σy by
1
τy = √ σy .
3
(2.119)
52
Nanomaterials
This is experimentally observed in a very large number of engineering metals.
Let us consider an example of plastic behavior in nanocrystalline metals. Suppose the yield strength in uniaxial tension of a nanocrystalline nickel material with
a grain size of 20 nm (made by electrodeposition) has been measured to be 1.1 GPa
(Figure 2.5). Then it follows that σy = 1.1 GPa. Let us assume that von Mises
yield (Equation 2.116) describes the onset of plastic deformation in nanocrystalline
nickel. This yield criterion would then predict that if we were to take a tube of
nanocrystalline nickel and subject it to torsion, it would yield plastically when the
applied shear stress reaches √13 σy which is 635 MPa. Unfortunately the yield stress
of nanocrystalline nickel has not been measured in torsion as of this writing, so we
do not know if von Mises yield is a good description of the behavior (even though
it is known that conventional grain sized nickel follows von Mises yield behavior).
We now have a relationship (Equation 2.114) that predicts the onset of plastic deformation, but we still do not have what we set out to obtain: a relationship between
the total strain history and the stress history for a plastically deforming material.
Remember that the yield surface is the boundary of the elastic region. Within the
elastic region, we know how the stresses and strains are related (note that in the
elastic region the history of the strain is irrelevant – the current strain determines
the current stress). How are these quantities related once we reach the yield surface?
This is a far more complicated problem, and we merely summarize the results here.
The reader should refer to textbooks on plasticity for a detailed development of the
theory.
Since plasticity is a history dependent behavior, it is useful to think about changes
in the stress and the strain rather than the current stress and strain. Consider the situation when the stress path has just reached the yield surface and we now have an
additional increment δ σi j of the stress along the path. Our goal is to find the increment of the strain δ εi j that results from this increment in the stress. Assume that the
stress increment is a loading increment, that is, the stress path is trying to move out
of the elastic region. By definition, this stress increment will result in a permanent
deformation of the material, that is, in a permanent plastic strain increment δ εipj . On
the other hand, if the additional stress increment were to be removed, the material
would recover a certain amount of elastic strain, so that there must be an elastic
strain increment εiej also associated with the loading stress increment. It is therefore
appropriate to view the total strain increment as having an elastic part and a plastic
part, so that
δ εi j = δ εiej + δ εipj ,
(2.120)
with the additional proviso that the plastic strain increment δ εipj would only be developed if the increment in the stress is the loading increment rather than an unloading increment (all unloading is assumed to be linear elastic). The elastic strain
increment can be related to the stress increment using the equations of linear elasticity (e.g., Equation 2.91):
δ εiej =
ν
1+ν
δ σi j − δ σkk δi j .
E
E
(2.121)
2 Fundamentals of Mechanics of Materials
53
If we can now find a relationship between the plastic strain increments and the
stresses, our objective of relating the stresses and strains will have been achieved,
at least in an incremental sense. First, it is important to recognize that the material
behavior may or may not show hardening (usually called strain hardening). If the
plastic behavior of the material is perfectly plastic, without any strain hardening,
then it is not possible to apply a loading stress increment, and the plastic strain increment is indeterminate (in terms of material response – any number of deformed
states can be attained after yield has occurred, limited only by the boundary conditions and the available energy). If the material behavior shows hardening after
initial yield, then it is possible to determine the plastic strain increment uniquely for
a loading increment in the stress. A textbook on plasticity will demonstrate that the
plastic strain increment is related not so much to the stress increment as it is to the
current deviatoric stress:
δ εipj =
1 δ σ̄ Si j
, if and only if J2 = Y and δ σi j is a loading increment (2.122)
2 h(ε p ) σ̄
where σ̄ is the effective stress measure defined in Equation (2.115) and h(ε p ) is a
function that describes how the material hardens with increasing plastic strain (this
function is itself a material property, relating for example to the strain hardening
index n). Note that Equation (2.122) also contains δ σ̄ , which is the scalar increment
in σ̄ and which depends on δ σi j , but that individual components of the plastic strain
increment depend only on the deviatoric stress components and not on the Cauchy
stress components. Such a model is referred to as a J2 f low model, since it is derived
from a J2 -based plasticity model and is written in an incremental strain form (the
flow). It is important to recognize that δ εipj = 0 if the stress state is in the elastic
region, or if δ σi j is an unloading increment.
Putting Equations (2.120) and (2.122) together, we have
δ εi j =
ν
1+ν
1 δ σ̄ Si j
δ σi j − δ σkk δi j +
.
E
E
2 h(ε p ) σ̄
(2.123)
This is the relation between the strain increments and the stress increments (and
the stresses themselves, because of the history dependence). Together with the yield
criterion (Equation 2.114), this incremental relationship defines the plastic constitutive response. This can be used to describe material behavior under multiaxial stress
states along arbitrary stress paths, but must be solved incrementally, with a process of continually updating the stresses and strains as the nonlinear deformation
develops.
2.4.3 Fracture Mechanics
The concept of a stress concentration is a nearly intuitive concept in elasticity –
it is apparent to most people, based on daily experience, that sudden changes in
cross-section, notches and holes make it easier to cause the failure of materials
54
Nanomaterials
(imagine, for example, how easy it is to tear a sheet of paper that has already had
a hole made in the middle). The concepts of fracture mechanics have similar roots,
and are key concepts in structural design and failure analysis.
The fundamental question of fracture mechanics is this: if a structure has a crack
within it, and the structure is placed under load, will the crack grow? A secondary
question is: if the crack does grow, how much will it grow, and how fast? We will
address these questions in this section.
The simplest form of fracture mechanics involves a crack in a linear elastic material (this is called Linear Elastic Fracture Mechanics or LEFM). Consider a sharp
planar crack within a large block of material (Figure 2.7), loaded in tension in the
x2 -direction at some distance. Define the coordinate system as in the figure. In real
plates, the crack front (the locus of crack tips) is a curve rather than a straight line,
and is sometimes called a thumbnail crack (Figure 2.8). This is because the surface
of the block is essentially in a different stress state (close to plane stress) while the
center of the block is closer to plane strain for a sufficiently large block, and cracks
appear to grow more easily in plane strain. The crack front is the curve separating
the cracked plane from the rest of the material (Figure 2.8).
Fig. 2.7 A sharp planar crack in a very large block of linear elastic material. The coordinate axes
are defined so that the crack front (the line drawn by the crack tip) is the x3 or z axis. Note that
all constant z planes look identical, so that this is essentially a plane strain problem in the x1 − x2
plane. The mechanics of this problem is dominated by the behavior as one approaches the crack tip.
2 Fundamentals of Mechanics of Materials
55
x
x
x
x
Fig. 2.8 Two views of the crack in Figure 2.7, with (a) being the side view and (b) the top view
looking down on the crack plane. The crack front is typically curved, but this can be neglected in
the limit as one approaches the crack tip.
From a mechanics viewpoint, we would like to know all of the stresses, strains
and displacements everywhere in the problem. However, this is more information
than we perhaps need to answer the primary question in fracture mechanics. Far
away from the crack tip, the material and structure do not feel the influence of the
crack, and the stresses and so forth are identical to what one would have from a
simple uniaxial tension problem. In fact, therefore, we only need to investigate what
happens as we get close to the crack tip. This is a recurring theme in linear elastic
fracture mechanics – the stress, strain and displacement fields as one approaches
the crack tip are those of interest, and are indeed expected to dominate the problem
(because the crack tip can be viewed as a stress concentration, and so the stresses
should be highest there). Most of the results in LEFM are only valid in the limit as
one approaches the crack tip (such solutions are called asymptotic solutions).
Thinking of the crack tip as a stress concentration, note that the radius of the
crack tip in the x1 − x2 plane is much smaller (essentially zero) than the radius of
the crack front in the x1 − x3 plane (Figure 2.8b). Thus the dominant behavior as one
approaches the crack tip is the behavior in the x1 − x2 plane. Another way to think of
this is as follows: as one approaches the crack tip, all variations in the x3 direction
are small in comparison to variations in the x1 and x2 directions, that is, none of the
quantities of interest depend on x3 . Note that this implies that (as one approaches
the crack tip) ε33 = 0, that is, the problem looks like a plane strain problem as one
approaches the crack tip. For this reason, most measurements of the toughness of a
material (the resistance to fracture) are performed in plane strain.
As we approach the crack tip, therefore, we have (in terms of functional dependence) σi j = σi j (x1 , x2 ,t), εi j = εi j (x1 , x2 ,t) and ui = ui (x1 , x2 ,t). In quasistatic
problems the dependence on time can also be dropped. Note that u3 = 0 in general.
The problems in LEFM are typically separated into problems with u1 , u2 = 0, u3 = 0
and u1 , u2 = 0, u3 = 0, using the principle that for linear problems we can add the
solutions for the two problems to obtain solutions to more general problems. The
56
Nanomaterials
case where u1 = 0, u2 = 0, u3 = 0 is called the plane problem, and the case where
u1 = 0, u2 = 0, u3 = 0 is called the antiplane problem (this corresponds to the
tearing mode in fracture problems).
Let us define the crack front line as the x3 = z-axis. Note that symmetry requires
that every x3 = constant plane will look the same in terms of stress and strain fields.
Consider the plane problem (Figure 2.8a), so that the problem reduces to considering the stresses in the x1 − x2 or x − y plane. Thus the only stresses of interest
are σxx , σxy , or σyy , because the σ33 = σzz can be computed directly from the linear
elastic constitutive law, given that ε33 = 0. The crack tip can be defined as the origin
in this 2-D space. A transformation to polar coordinates (r, θ ) from the Cartesian
coordinates (x, y) has the crack tip at r = 0, and the crack faces at θ = ±π , and in
this coordinate system the stresses of interest are
σrr , σrθ , or σθ θ . In either case,
distance from the crack tip can be defined as r = x2 + y2 .
Far away from the crack tip, the influence of the crack is negligible, and the stress
state will just be σyy = σ∞ , assuming that the far field tension σ∞ is in the x2 = y
direction. As we approach the crack tip, the influence of the crack is increasingly
felt, and the stresses at the crack tip itself are infinite (for a sharp crack – like a
stress concentration of infinity). How rapidly do the stresses grow as one approaches
the crack tip? A mathematical analysis of this problem, found in any number of
books on mechanics (e.g., Asaro and Lubarda, 2006), shows that all of the stress
components increase in the same way:
KI
σi j = √ fi j (θ ),
r
(2.124)
where the angle-dependent function fi j (θ ) is the only difference between the three
stresses in the plane. Thus all of the stresses increase as √1r , going to ∞ as r → 0.
The derivation of Equation (2.124) uses a specific physical concept: that although
the stresses at the crack tip can be infinite, the displacements and energies at the
crack tip must be finite. The fact that the stresses are infinite at the crack tip is called
a singularity, and the crack tip in the elastic material is said to have a √1r singularity
(other singularities are possible, for example in viscous materials). The coefficient
KI in Equation (2.124) is called the stress intensity factor, and scales the magnitude
of the stress: it is a function of the applied loading and the geometry of the particular
problem. Since the stress intensity factor scales all of the stresses, it is a measure of
the effect of the applied load in the crack problem. The stress intensity factor KI is
tabulated for a very large number of problems, so often one only needs to look it up
for a specific geometry and loading condition (e.g., three-pt bending of a cracked
beam). As the applied load increases, KI increases.
The answer to the first question that was raised at the beginning of this section is
that the crack will grow when the applied stress intensity factor KI reaches a critical
value that is a material property called the plane strain fracture toughness KIc , i.e.,
the fracture criterion is
KI = KIc .
(2.125)
2 Fundamentals of Mechanics of Materials
57
As with all criteria of this type, the left-hand side defines something related to
the applied load and geometry, and the right-hand side defines a material property
(recall the yield criterion σ = σY ). The plane strain fracture toughness is a material
property that must be measured in specific experiments called fracture toughness
experiments, in which specimens containing a precrack are loaded until the initi√
ation of crack growth is perceived. The units of fracture toughness are MPa m.
The fracture toughness is a fundamental material property in terms of the application of materials to problems involving mechanical loading, since fracture is one
of the most common failure modes for materials and structures, and tables of fracture toughness can be found in most materials and mechanical
√ design textbooks.
15
and
60
MPa
m, while for most
In general, for most metals KIc lies between
√
ceramics it lies between 1 and 6 MPa m. Materials with low values of fracture
toughness are said to be brittle; most ceramics are brittle. Many nanocrystalline
metals are believed to be more brittle than the conventional grain size metal, but
there is little real experimental data on the fracture toughness of nanomaterials
(largely because of the difficulty of doing valid fracture experiments on very small
scale samples).
The second question in fracture mechanics, that of how the crack will grow once
it begins to grow, must be answered on an energetic basis. The primary concept here
is that it takes energy to grow a crack, for instance in terms of the addition of energy
required for the free surface corresponding to the crack faces. The amount of crack
growth that would be observed depends on the amount of energy available to grow
the crack. The energetic approach to fracture mechanics is a very powerful approach,
resulting in the ability to solve fairly complex problems and also suggesting ways
in which we can increase the toughness of materials.
2.5 Suggestions for Further Reading
1. A. Bedford and K.M. Liechti, Mechanics of Materials. Good if you have little or
no background in mechanics. Prentice Hall, Upper Saddle River, NJ, 2000.
2. Y.C. Fung and P. Tong, Classical and Computational Solid Mechanics. World
Scientific Publishing, Singapore, 2001.
3. J.R. Barber, Elasticity (Solid Mechanics and its Applications). Springer, New
York, 2003.
4. R. Hill, The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1986.
5. M.F. Kanninen and C. Popelaar, Fundamentals of Fracture Mechanics. Butterworths, London, 1973.
2.6 Problems and Directions for Research
1. Which of the following expressions are meaningful in our notation? Write out
each of the meaningful ones completely for a 3D space: aii , ai j b j , aii bi , ai j bi .
58
2.
3.
4.
5.
6.
7.
Nanomaterials
⎤
1 0 −1
Given the matrix A = ⎣−1 2 0 ⎦, find the quantities Akk , Ai j Ai j , Ai j A ji .
3 0 −1
Consider the simple tension of a circular cylindrical bar of elastic/plastic material (6061-T6 aluminum alloy). Assume that the tension test is performed very
rapidly at a nearly constant strain rate ε̇0 = 10−3 s−1 . Assume further that the
material flows with no strain hardening at a stress σy = 500 MPa. Solve the 1D
energy equation (Equation 2.59) to obtain the temperature distribution in the bar
500 s after the deformation commences, assuming no heat conduction through
the grips, and ignoring convective and radiative losses through the
circumfer
ential surface. Note that the plastic work W p is given by W p = σy ε̇ dt. Use
handbook data as necessary. Provide the temperature distribution in the form of
a plot.
Consider the plastic deformations of a thin sheet of metal deformed in biaxial
extension. The problem is one in which the forces in the two orthogonal directions in the plane are varied independently. Apply the von Mises yield criterion,
and sketch the yield surface in the 2D stress space. If the metal were to be made
of a sheet of nanocrystalline aluminum 1 cm square and 100 µm thick with a
yield strength in tension of 1 GPa, at what load would it yield under equibiaxial
stretching?
(Simple torsion) Equilibrium in torsion of a circular bar gives τ = Tr
J , where τ is
the shear stress in the bar, T is the torque, r is the radius, and J is the polar moment of inertia of the cross-section. The constitutive law then gives γ = rLθ µτ ,
where γ is the shear strain, θ is the angle of twist in radians, and L is the length
of the bar. Now consider a specific problem. A 1 m long steel tube with a diameter of 3 cm and a wall thickness of 1 mm is subjected to torsional loading until a
total angle of twist of 5◦ is achieved. Compute the torque required to accomplish
this, and then calculate the shear stress in the tube. Suppose the tube is simultaneously subjected to a tensile deformation, so that an axial elongation of 1 mm
is obtained. Compute the axial force required to accomplish this elongation, and
compute the resulting tensile stress in the tube along the axial direction. Sketch
the 2D stress state on the surface of the tube after the combined tensile and torsional deformations. Now compute the principal stresses and principal directions
(the directions in which the principal stresses act) for this stress state.
The yield strength of a nanocrystalline aluminum sample is 700 MPa in tension.
If the material’s behavior were to obey the Mises yield surface, at what shear
stress would it yield in torsion? How about in compression?
The surface of a 6061-T6 aluminum component of a machine tool is subjected
to principal stresses of 200 MPa and 100 MPa. The third (out-of-plane) principal
stress is zero. What tensile yield strength is required to provide a safety factor of
2 with respect to initial yielding?
⎡
2 Fundamentals of Mechanics of Materials
59
References
Asaro, R. and V. Lubarda (2006). Mechanics of Solids and Materials. New York: Cambridge
University Press.
Ericksen, J. (1998). Introduction to the Thermodynamics of Solids. New York: Springer.
Lai, W., D. Rubin, and E. Krempl (1999). Introduction to Continuum Mechanics. Burlington, MA:
Butterworth Heinemann.
Nye (1950). Physical Properties of Crystals. Oxford: Oxford Science.
Experimental science does not receive truth
from superior science: she is the mistress and
the other sciences are her servants.
Roger Bacon
3
Nanoscale Mechanics and
Materials: Experimental
Techniques
3.1 Introduction
One of the fundamental difficulties associated with the measurement of the mechanical properties of nanomaterials is that nanomaterial samples generally cannot be obtained in sufficiently large sizes to do traditional mechanical testing. From another
perspective, one must use experimental techniques that can operate at scales that
are relevant to the nanomaterial problem. For both of these reasons, there has been
a drive towards the development of small-scale experimental techniques that can
provide mechanical information on materials or structures. This chapter discusses
a number of these techniques. Note that this is not a comprehensive discussion because these techniques are in rapid evolution, and the reader would be well advised
to review the current literature in this area.
We classify these techniques into two types, based on the objectives:
1. Techniques that attempt to identify or quantify a phenomenon at the nanoscale,
which we refer to as nano mechanics techniques (NMT). Examples of such phenomena include grain boundary failure, dislocation motion, and adhesion of interfaces.
2. Techniques that attempt to measure the mechanical properties of materials using
small sample sizes, which we refer to as nanoscale mechanical characterization
techniques (NMCT). Examples of mechanical properties of interest include elastic modulus, yield strength, and fracture toughness.
Both NMT and NMCT approaches have been developing rapidly since the 1990s.
In practice, some of these techniques involve specimens that are microscale (10−7 to
10−4 m) in dimension, rather than nanoscale, because the observational resolution
may well be nanoscale even when the overall specimen size is in the microscale
regime.
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 3,
c Springer Science+Business Media, LLC 2009
61
62
Nanomaterials
3.2 NanoMechanics Techniques
Understanding the behavior of nanomaterials involves understanding the phenomena that are active at the nanoscale, and this requires two steps: (i) identifying the
phenomena and (ii) being able to model them. The identification of phenomena is
the core objective of nanomechanics techniques. The phenomena themselves may
be mechanics (physics)-dominated or chemistry-dominated, although often it is difficult to determine which parameters are the most relevant ones to measure. Examples of such phenomena include the mechanical behavior of grain boundaries in
terms of the forces required to cause separation or sliding, the forces across interfaces (for example, the interfaces between two elements of a nanoelectromechanical
systems or NEMS device), and the development of instabilities at the nanoscale
such as the buckling of a carbon nanotube. Many of the advances in this area
are fueled by the rapid growth in techniques that can examine atomic scale phenomena, particularly using scanning probe microscopy and transmission electron
microscopy.
What are the typical nanoscale phenomena that are relevant to the mechanical
behavior of nanomaterials and the functioning of nanomaterials or nanostructures?
Since the nature of nanomaterials and nanostructures implies the presence of boundaries at the nanoscale, the dominant question is the influence of such boundaries,
whether these are interfaces between entities or simply surface features. For example, the behavior of grain boundaries is critical to understanding the mechanical
behavior of nanomaterials, while the behavior of the surface of a cantilever beam
can dominate the sensitivity of a cantilever sensor. Thus most nanomechanics techniques focus on the observation of boundaries and interfaces.
Measuring such boundary phenomena requires that one be able to observe at the
appropriate scale as well as be able to quantify at that scale. The classic questions of
precision, accuracy and uncertainty are important for such measurements. It is often
the case that we can observe phenomena at smaller scales than the scales at which
we can accurately quantify those phenomena (the nanoscience version of “one’s
reach is often beyond one’s grasp”). A more fundamental question is whether one
can be certain that one is measuring the phenomenon under the same conditions
each time the measurement is made. For example, we cannot obtain two identical
grain boundaries for examination, because it is difficult to prepare two identical
grain boundaries. Because of the difficulty of preparing samples, we observe what
we can get as specimens, even though there is a very large variety of boundaries that
should be characterized. However, the number of phenomena at the nanoscale may
be very large, and our technology is just beginning to grasp the variety of features
present here, even as foretold by Feynman, when he said that “there’s plenty of room
at the bottom” (Feynman, 1959).
In this regard, our difficulties in identifying and quantifying nanoscale phenomena have strong similarities to the travails of astronomers who are trying to identify
the behavior of stars and understand the stellar lifecycle. Since we cannot make
stars, we have to build our understanding on the basis of those that we can observe.
3 Nanoscale Mechanics and Materials: Experimental Techniques
63
Just so, in the nanomaterials world, we have to build understanding on the basis of
the boundaries that we can observe. Note that the richness of phenomena, and the
difficulty of control, arise for exactly the same reasons in the two cases of astronomy
and nanotechnology: a vast difference of scale from the human condition. We must
turn the difficulty that we are presented with (the variety of structures and phenomena available) into an opportunity, even as the astronomers do, and use these riches
to our advantage.
An analogy might be useful. Imagine an alien spacecraft arrives at the Earth,
and the spaceship commander wishes to obtain an understanding of the human
lifecycle. She could indeed choose to observe a human evolve from an infant to
a full-grown adult and then to old age, but this would take more than 70 years.
On the other hand, if she chose to examine a very large cross-section of the population, she would observe every stage of the human lifecycle in a very short time
and would be able to obtain a fairly good understanding of the process (although
certainly she would see variations based on race, economic status, and environment). This is what astronomers do with stars, and what nanotechnologists need
to do with phenomena in nanostructures. We cannot control all of the nanostructures, but we can observe a large variety of them and try to extract an understanding of the phenomenon of interest. The difficulty would be identifying the
persistent features across a spectrum of nanostructures (comparable to identifying the common aging processes across a spectrum of races and environmental
conditions).
As an example of this approach, consider understanding the mechanical behavior of grain boundaries in a nanocrystalline material. Since the grain boundary
is the boundary between two crystals, there is a very large variety of boundaries
that are possible (depending upon the relative orientations of the two crystals).
It is not physically feasible to prepare a sample of every boundary with a given
relative orientation of the two crystals. However, in any nanocrystalline material, it is likely that we will find representative examples of many of these possible grain boundaries. Thus by examining the population of grain boundaries in a
given nanocrystalline material, and learning what their mechanical behaviors have
in common, we may obtain an understanding of the likely mechanical behavior
of all grain boundaries. This approach inherently requires us to use both experimental and modeling techniques simultaneously to study these phenomena, so that
we can extract the persistent behavior across a variety of examples (as in grain
boundaries).
With that insight, let us consider NanoMechanics Techniques. The two major experimental techniques that comprise NMT are scanning probe microscopy and in
situ deformation. The latter may be performed within an AFM (atomic force microscope), a TEM (transmission electron microscope) or an SEM (scanning electron
microscope). Each of these techniques is described in the following sections. Note
that these are complex techniques that are comprehensively discussed in the literature, so the descriptions here are brief. Other techniques such as optical tweezers
and magnetic tweezer arrangements are more appropriate for biological problems,
and are not discussed here.
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Nanomaterials
3.3 Characterizing Nanomaterials
The characterization of nanomaterials is no easy task. By definition, the feature sizes
that must be observed are very small, being less than half the wavelength of visible
light. This makes it impossible to see these features using traditional optical microscopy. The primary characterization techniques that are used are therefore scanning electron microscopy, X-ray diffraction, and transmission electron microscopy.
Of these three, the highest spatial resolution is obtained with transmission electron
microscopy or TEM, and this is the final arbiter of whether a specific dimension has
been achieved and is necessary for the determination of specific deformation mechanisms. However, transmission electron microscopy is a difficult thing to do, and
specimen preparation in particular is a laborious task requiring substantial skill. As
a consequence, many researchers in the field use X-ray diffraction to obtain a quick
estimate of grain size. Unfortunately X-ray diffraction is unable to distinguish between low angle grain boundaries and high angle grain boundaries, leading to some
confusion in the literature. The scanning electron microscope is sufficient for the observation of many device nanomaterials, and to examine failure mechanisms (often
a question when reliability of devices is an issue).
When surface examination is sufficient, various forms of scanning probe microscopy may be appropriate. The atomic force microscope is often used to provide
images of surfaces with sufficient resolution. In any case, the use of nanomaterials
implies the ability to characterize at the nanoscale, and a substantial investment has
to be made in the infrastructure required to observe at these scales. Each of these
techniques is described very briefly below; the reader should examine the suggested
reading to obtain more detail.
3.3.1 Scanning Electron Microscopy or SEM
A scanning electron microscope generates an image by scanning a surface with an
electron beam. The electron beam is usually generated from a tungsten filament,
and the energy associated with the electrons can be controlled. The beam itself can
be manipulated using electron optics, analogous to traditional optics (lenses and so
forth). The interaction of the electron beam with the surface results in the interrogation of a certain depth of material below the surface. Electrons are scattered off
the material being investigated through both elastic and inelastic scattering, and by
collecting the scattered electrons one can obtain images of the surface (in practice,
the scattered electrons themselves interact with some kind of display, generating
intensity variations that provide the image shown in typical scanning electron micrographs). In the most common mode of operation, the scattered electrons that
are collected are those that have been inelastically scattered, and this interrogates
a very small depth below the surface (several nanometers). Most SEM images thus
provide surface topography information, and resolutions down to a nanometer are
possible. In comparison to optical microscopy, scanning electron microscopy is capable of much greater magnification and resolution, and at the same time retains
3 Nanoscale Mechanics and Materials: Experimental Techniques
65
great depth of field (in light microscopy, high magnification usually results in low
depth of field). Scanning electron microscopy is therefore a great way of looking at
surface information on nanomaterials (the depth of field is particularly useful when
looking at fracture surfaces).
If the scattered electrons that are collected are the elastically scattered electrons
(so-called back-scattered electrons), then the image that results contains a significant
amount of information on the chemical composition of the surface, because different
materials provide different degrees of scattering in this mode (higher atomic numbers – that is, heavier atoms – scatter more). When used in combination with some
form of spectral analysis, a great deal of information on chemical composition can
be obtained.
3.3.2 Transmission Electron Microscopy or TEM
Unlike the SEM, the transmission electron microscope uses the transmission of an
electron beam through a sample to characterize the sample. The sample must be
thin enough for transmission to occur, and so a good deal of specimen preparation is typically necessary. This technique is capable of much higher resolution,
and in some versions is capable of examining individual atomic layers. Since the
beam goes through the sample, the interaction of the beam with the sample is much
more complex, and this can be used to advantage in generating various imaging
modes. Some of these modes include the ability to examine the diffraction of the
beam through the crystal lattice, and to select specific diffraction spots for imaging
purposes. This combined with the ability to tilt the sample in the beam makes it
possible to identify individual crystal structures, to see and identify the character of
dislocations in the crystals, and to examine twin structures. TEM micrographs are
presented throughout this book, together with selected area diffraction (SAD) patterns that provide information on the crystal structures interacting with the beam.
SAD patterns display the diffraction pattern that results from the interaction of the
electron beam with some volume of the sample. In the specific case of the beam
interacting with only one crystal, the SAD pattern is a pattern of dots that represents
the crystal structure (actually the Fourier transform of the crystal structure). When
the beam interacts with a nanocrystalline material, very many grains are averaged
over, resulting in patterns that are more ring-like than spot-like. The technique of
electron energy loss spectroscopy or EELS can be used to provide composition information, and the combination of scanning with transmission modes in a scanning
transmission electron microscope or STEM can provide very useful information on
the location of specific elements at specific regions in the material. In high resolution electron microscopy or HREM, individual atomic layers can be examined,
the locations of atoms can be rigorously interrogated, and grain boundaries can be
examined individually.
The TEM is without question the definitive instrument for use with nanomaterials, since it can examine the interior of materials with the appropriate resolution.
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Nanomaterials
The difficulty with the use of the TEM is twofold: first, specimen preparation is
tedious and exacting and can change the structure being observed if not done properly, and second, the use of the TEM and interpretation of the TEM images requires
a very high degree of expertise. As instruments go, TEMs are expensive to maintain
and operate (much more so than SEMs).
3.3.3 X-Ray Diffraction or XRD
X-rays interact with crystalline structures through diffraction, and X-rays are relatively easily generated, so X-ray diffraction is the easiest way to examine crystal
structures. A large variety of specialized techniques have been developed based on
X-ray diffraction, and careful analysis can reveal information even at the level of the
locations of nearest neighbor atoms in a material. Resolution can thus be very high,
particularly using high-energy beamlines such as those obtained at synchrotrons.
Imaging is not usually the objective, but information can be obtained on grain size
(although low-angle and high-angle grain boundaries are hard to separate), elastic
strain in the crystal, and composition.
3.3.4 Scanning Probe Microscopy Techniques
Scanning Probe Microscopy (SPM) is a generic description for a class of techniques
where a needle tip (called the probe) is scanned across the surface of the sample, and
the interactions between the probe tip and the sample surface are used to generate
an image that represents some characteristic of the surface. In many SPM modes,
the probe (Figure 3.1) never actually touches the surface of the sample, instead being held a fixed distance (∼1 nm) away from the sample through a feedback circuit.
The interactions between the tip and the surface are extremely small in scale, including atomic level interactions. Indeed, it is possible to scan across the surface
of the sample and image every atom on the surface. It is also possible to image
at the micrometer scale. These techniques are thus extremely powerful, and have
revolutionized the study of surface physics and surface chemistry; Gert Binnig and
Heinrich Rohrer won the Nobel Prize in Physics in 1986 for inventing the technique in 1981 (Binnig et al., 1983). Such systems can be used both to image and to
manipulate atoms on the surface.
A very large variety of interactions are possible between the probe and the surface. These include electron tunneling interactions–leading to scanning tunneling
microscopy or STM (Binnig and Rohrer, 1985); contact forces–leading to atomic
force microscopy or AFM (Binnig et al., 1986); and magnetic interactions–leading
to magnetic force microscopy or MFM (Martin and Wickramasinghe, 1987). A variety of technologies have developed that utilize the extreme sensitivity of the measurements to these interactions. The particular interaction that is measured using
3 Nanoscale Mechanics and Materials: Experimental Techniques
67
Laser Diode
Source
Piezoelectric
Scanning
Actuators (x,y,z)
To Position
Sensitive
Photodetector
Cantilever
Sample Surface
Fig. 3.1 Schematic of a Scanning Probe Microscopy system (not showing the feedback loop, tunneling amplifiers and other electronics). A wide variety of such systems exist, some of which
contain only a subset of the features shown here.
the scanning probe system determines the characteristic of the surface that will
be imaged and the name of the kind of microscopy that is being used. There is a
large variety of commercial versions of scanning probe microscopy instruments on
the market, and many online resources that provide detailed descriptions of these
instruments.
There are four critical components to scanning probe microscopy techniques:
• First, the nature of the probe tip determines the resolution of interaction.
• Second, the ability to scan in a controllable way over very small distances with
high-resolution determines the spatial resolution of the technique (this is usually
done with a piezoelectric system).
• Third, a feedback system is usually used to maintain the probe a fixed distance
away from the surface being scanned, and there is a very important relationship
(which can be designed to some degree) between the feedback control and the
specific quantity being measured.
• Fourth, the ability to develop images that visualize the distribution of these interactions across the surface of the sample through appropriate software is critical
to the practical use of these techniques. The interpretation of these images often requires substantial expertise, and may require a real understanding of the
physics of the interactions.
These are very powerful techniques, but come with a strong caveat. The ease
with which an image is obtained using a commercial SPM instrument may hide very
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Nanomaterials
complex physics that might be very important to the issue of interest, because the
software that provides the image may make assumptions that are hidden from the
viewer. It is critical to understand (a) what the hardware is capable of doing and
(b) what the software assumes in generating the image (for example, what are the
defaults in the parameters used by the software).
Examples of applications of scanning probe microscopy that are of interest in
this book include the examination of grain boundary character at the atomic scale,
the determination of the development and motion of ledges on crystal surfaces, the
examination of compositional variation close to boundaries, and the characterization
and fabrication of nanomaterial surfaces.
3.3.5 Atomic Force Microscopy or AFM
Atomic force microscopy (AFM) is a very high-resolution SPM technique for examining the surfaces of materials. The image is formed by scanning a probe over
the surface and using the interaction of the probe with the atoms on the surface to
generate an image. Many different kinds of interactions can be examined, and images can be generated of surface topography, surface magnetic character, and so on.
This technique is often used with discrete nanomaterials and thin films, and less so
with bulk nanocrystalline materials.
3.3.6 In situ Deformation
In situ deformations are defined as deformations that are developed while the sample
is within an imaging environment. The ability to image during the deformation allows us to determine the mechanisms that are active and to examine the deformation
processes at the microscale. The development of an in situ deformation technique
requires two components:
• First, an arrangement that allows one to develop controlled deformations within
the microscope
• Second, an imaging capability through microscopy (in order to be able to examine the length scales of interest)
From a mechanics viewpoint, it is useful to be able to measure the forces within
the system at the same time as the deformations are being imaged through the microscope. This can be very hard to do at nanometer scales, because the forces that
are developed can be in the nano-Newton range. The primary in situ deformation
techniques that are available are accomplished within AFM, TEM, and SEM systems. The major difficulty in in situ deformation techniques is that it is difficult if not
impossible to obtain samples of identical structure; one has to look at what one has
3 Nanoscale Mechanics and Materials: Experimental Techniques
69
available, rather than being able to examine model structures for a given material
(such as identical grain boundaries).
3.3.6.1 In Situ Deformation Within an Atomic Force Microscope
Atomic force microscopy involves the measurement of the forces involved in the
scanning of a probe along the surface of a material (often including contact). The
probe tip is often made from single crystal silicon. Probe tip radii of curvature can
be of the order of 1 nm. Typically, the probe is carried on a cantilever beam called
a microcantilever. The force of interaction between the probe tip and the surface results in the bending of the microcantilever, and the forces are measured by recording
the bending of the beam using an optical arrangement. The forces that are measured
might be contact forces, van der Waals forces, or electrostatic forces. In terms of
spatial resolution, this technique often does not have the spatial resolution of STM
but is much more robust in operation. A common approach to in situ deformation
within the atomic force microscope involves the use of thin films of material that are
deformed using a mechanical arrangement (Nishino et al., 2000). The large majority
of such experiments do not record the forces within the sample during the deformation. However, recent approaches to mechanical loading using a MEMS type device
show significant promise (Chasiotis et al., 2004). Images of the deformed sample
can then be correlated to measures of the current load to estimate the relationship
between the stress state and the deformation. Note that the observation of deformation processes using imaging may be worthwhile even without load measures.
For example, the evolution of surface features in a nanocomposite can be imaged
effectively using this technique, as can grain boundary migration.
3.3.6.2 In Situ Deformation Within a Transmission Electron Microscope
The Transmission Electron Microscope (TEM), as the name implies, is an instrument that uses the transmission of electrons through a material to determine the
structure of the material. An electron beam is accelerated using a high voltage and
is passed through a sample; an image is constructed on the basis of the electrons
that have been transmitted through the sample and that strike a detector placed on
the opposite side of the sample. The most common application of the TEM is to provide microstructural characterization of crystalline materials. The interaction of the
electron beam with the crystal can be studied by using the principles of diffraction,
and highly sophisticated techniques have been developed to infer the structure of
the material on the basis of the distributions of the transmitted electrons. Among the
characteristics of the material that can be inferred are the local grain size, the crystal
structure, and the defect density, in addition to the characterization of boundaries
and identification of composition. At nanometer length scales, the TEM is often
believed to provide “ground truth” in terms of materials characterization. The primary difficulty with the use of the TEM is that the samples must be made electron
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Nanomaterials
transparent, and the production of good TEM specimens is essentially an art that
requires a great deal of experience. Most TEM specimens are disks about 3 mm in
diameter with a central dimple containing a region of material that has been thinned
to about 100 nm in thickness. The process of generating a good TEM specimen may
itself (if done incorrectly) result in substantial modification of the structure that is
being examined. With these caveats, there is no substitute for high-quality transmission electron microscopy work in the nanocrystalline materials area for determining
the material characteristics.
In situ deformation experiments within a TEM are also complicated by the need
to develop an electron transparent region within the sample. This essentially limits
the in situ samples to thin films. There are a number of commercially available testing stages that can be mounted within TEM vacuum chambers for such experiments,
but the development of an effective technique requires substantial effort to ensure
that the technique is optimized to the specific microscopy facility. A common way
to develop local strains in the sample is to use a thermal misfit rather than to apply
controlled mechanical loads. Again, recent developments in MEMS based mechanical testing techniques show strong promise in this area (Haque and Saif, 2004).
Some instruments have been developed that accomplish compression and tension
within a TEM using a combination of focused ion beam machining to generate the
specimens and in situ nanoindentor based loading techniques (Minor et al., 2001;
Stach et al., 2001), and these are just beginning to have a significant impact on our
understanding of deformation mechanisms at the nanoscale.
3.3.6.3 In Situ Deformation Within a Scanning Electron Microscope
The scanning electron microscope (SEM) is an imaging instrument that also uses an
electron beam, but generates an image by looking at the electrons that are scattered
off a surface. The electron beam is scanned across the surface, generating images of
the surface over a very wide range of magnifications. This is the premier technique
for examining material surfaces in materials science and engineering. One of the
major advantages of this technique is that the depth of field is considerable, so that
it is possible to look at very rough surfaces such as fracture surfaces. Unlike the
TEM, the SEM can be used by a relatively unskilled operator and specimen preparation is not a major problem except for some special cases. The spatial resolution
of SEM systems has improved dramatically over the last two decades, and nanometer resolution is now possible. Further, it is possible to identify crystal orientations
using SEM systems in Orientation Imaging Microscopy mode (this is very useful
for examining polycrystalline systems).
In situ deformation experiments within an SEM have a particular attraction because of the ease of use of the instrument, the wide range of magnifications that are
possible, and the substantial depth of field. The coupling of these capabilities with
the recently developed digital image correlation (DIC) technique for determining
displacements and strains from multiple images has made this a very useful technique for in situ deformation studies. Commercial in situ testing stages for the SEM
3 Nanoscale Mechanics and Materials: Experimental Techniques
71
Fig. 3.2 Single crystal copper specimen and tension gripper for in situ tensile testing in the SEM,
from the work of the group of Dehm, Kiener et al. (this group has also performed compression
experiments using a similar apparatus) (a) SEM image showing a single-crystal copper tension
sample and the corresponding tungsten sample gripper before the test at a low magnification.
(b) Sample and gripper aligned prior to loading. Image taken from (Kiener et al., 2008). Reprinted
from Acta Materialia, Vol 56, Issue 3, P. 13, D. Kiener, W. Grosinger, G. Dehm, R. Pippan, A Further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments
of miniaturized single-crystal copper samples. Feb. 2008, with permission from Elsevier.
are now available, and are easily fitted to specific facilities. These techniques have
been used, for example, to understand the size-dependence of plastic behavior in
metal single crystals (Kiener et al., 2008). An example of the specimen size and
shape is presented in Figure 3.2 from the work of Kiener et al. (2008).
3.4 Nanoscale Mechanical Characterization
3.4.1 Sample and Specimen Fabrication
The objective of a mechanical characterization technique is to measure the behavior of the material. However, what we in fact measure is the behavior of a sample
of the material, and we then infer material behavior from sample behavior. This
works very well if we (a) understand the relation between sample behavior and material behavior and (b) have not changed the material response while trying to make
the samples that we will test. An example might be useful. Suppose one wished to
measure the tensile strength of a relatively brittle material like pure tungsten, and
so one performs a standard tensile test on dogbone-shaped samples (Figure 2.4) of
the tungsten. Suppose the dogbone-shaped samples are made by machining from a
larger tungsten block using a milling machine, but one set of samples is tested immediately after milling, while the other set of samples is subject to a finish grinding
operation (after milling) before the tensile tests are performed. A very interesting
behavior is observed. The strength of the material measured from the finish-ground
samples is found to be much higher than the strength of the material measured
from the as-milled samples (a similar behavior has been shown for compression;
Lennon and Ramesh, 2000). This is because the as-milled samples have small surface cracks generated during the milling process, and these cause premature failure
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Nanomaterials
of the sample. The finish-ground samples do not have these surface cracks, because
the top layer of damaged material has been ground away, and these samples sustain
higher strengths (and higher strains to failure).
Thus sample fabrication processes can be critical. Note that this is an issue that
goes above and beyond the issue of the generation of high-quality material, with few
defects and reproducible microstructures. Sample preparation can be a particular
problem for nanocrystalline materials, which are typically high-strength and less
tolerant of defects introduced by fabrication processes. Sample fabrication processes
may include various kinds of machining, extrusion, and thermal cycling, and the
potential effects of all of these processes must be considered.
3.4.2 Nanoindentation
The mechanics of the indentation of one solid into another have been known for
many years. The first technical discussions of such matters examined the scratch
testing of materials, asking the question whether one material would scratch another.
This led to the famous Moh’s scale of hardness (diamond was ranked as the hardest material). The mathematical solution of the problem of the elastic indentation
of one material by another was provided by Hertz, who calculated the stress state
beneath an indentor. This mathematical solution is one of the fundamental solutions
in contact mechanics (Johnson, 1987), and describes the relationship between the
applied force on the indentor P and the depth δ of the indentation:
P=
16
9
1
(
1−νi2
Ei
+
2
1−νs2
Es )
Ri Rs 3/2
δ
Ri + Rs
(3.1)
where the subscripts i and s refer to the indenter and the sample respectively (e.g., Ri
and Rs are the radii of curvature of the indenter and sample). Note that the applied
force depends on the displacement to the 3/2 power, and so that relationship would
be expected for elastic behavior. If one performs an indentation experiment using
a relatively sharp indentor against a flat sample surface (so that Rs is very large),
measuring both the force and the displacement of the indentor tip, and then plots the
force against δ 3/2 , a straight line should be obtained, and the slope of that line can
be used to compute Es using Equation (3.1) if one knows the indentor radius and
indentor modulus Ei .
If the indented material can develop permanent deformations (i.e., has elasticplastic behavior), then a permanent indentation will remain after the removal of
the load from the indentor tip. Extending the Hertz contact analysis to the elastic
plastic case, the development of these permanent indentations within materials can
be described mathematically (although the description can be quite complicated if
the plastic behavior of the material deviates from simple classical plasticity such as
described in the chapter on mechanics). For the simple classical plasticity case, the
size of the remaining indent depends on the strength of the material. This has led
3 Nanoscale Mechanics and Materials: Experimental Techniques
73
to a very common testing technique in materials science called Vickers indentation
where the indentor has a specific pyramidal geometry, and Vickers hardness is a
common way of estimating the strength of the material. Other indentor shapes can
also be used, resulting in different hardness measures such as the Knoop hardness
and Brinell hardness.
Oliver and Pharr (1992) extended this approach to the problem of very smallscale indentation, calling this technique nanoindentation. The primary characteristic
of nanoindentation is that both the force on the indentor and the displacement of the
indentor tip are measured simultaneously with very great accuracy, and very small
forces and very small displacements can be measured. A force-displacement curve
is obtained for each indent. If the indentor is made out of an extremely stiff material such as diamond, the indentation force-displacement curve is dominated by the
response of the material being indented. An example of such a curve is presented
in Figure 3.4, which shows the result of a nanoindentation experiment on nanocrystalline nickel. Commercial nanoindentation instruments are now widely available.
The coupling of this technique to software for automated indentation makes it possible to develop hardness maps that contain substantial information about the mechanical characteristics of materials and structures at the small scales of interest to
nanomechanics and nanomaterials.
The device that provides a controlled indentation while simultaneously measuring both indentor force and indentor displacement for very small indents (submicron) is called a nanoindenter. A schematic of such a device is shown in
Figure 3.3, indicating the critical components. Current commercial nanoindenters
have a displacement resolution better than 0.1 A, and a load resolution of the order
of 1 nano-Newton. This high precision gives the technique popular application in
Load Frame
Actuator
Force
Sensor
Position
Sensor
Indenter Tip
Sample
Fig. 3.3 Schematic of the NanoIndenter, showing both actuation and force and position sensing. Such a device may fit comfortably on a large table top, although vibration isolation may be
desirable.
74
Nanomaterials
many areas of materials science and engineering, such as the measurement of mechanical properties of very thin films (thickness < 100 nm), and tribological measurements of coatings. In a conventional nanoindentation experiment, a Berkovich
(tetrahedral) tip is usually used to probe the nano-scale mechanical behavior (such
as hardness and elastic modulus) as a function of indentation depth. For example,
the indentation size has a strong effect on the hardness of single crystal silver (Pharr
and Oliver, 1989).
The typical output is a force-displacement curve that may deviate from the form
presented in Equation (3.1), as shown in Figure 3.4 for nanocrystalline nickel. The
primary mechanical properties extracted from a nanoindentation test are the hardness and the elastic modulus. This is extremely useful information, particularly for
nanocrystalline materials. However, the nanoindentation test cannot provide a full
description of the mechanical behavior of the material in terms of the stress-strain
curve, and such a description is needed in order to be able to use a material within
a specific application. The microcompression and microtension tests described next
can provide such a constitutive description.
3.4.3 Microcompression
Microcompression is an experimental technique where a modified indenter tip geometry is utilized within a nanoindenter to probe the nano-scale mechanical behavior of micro-scale or sub-micron pillars (Uchic et al., 2004). In this modification,
300
Nanocrystalline Ni, 20 nm
250
Load, mN
200
150
100
50
0
0
500
1000
1500
2000
Displacement Into Surface, nm
Fig. 3.4 Example of force-displacement curve obtained during the nanoindentation of a nanocrystalline nickel material with an average grain size of 20 nm. Data provided by Brian Schuster.
3 Nanoscale Mechanics and Materials: Experimental Techniques
75
the sharp Berkovich tip is truncated, resulting in a flat end tip which enables one to
convert the indentation system into a compression system. Since both the force on
the indentor tip and the displacement of the indentor tip can be measured, using this
flat tip to compress the top of a cylindrical sample allows one to estimate the stress
strain behavior of the material of the sample. This technique is commonly called
“micro-compression” or µ -compression. This technique is of great use for probing
the mechanical behavior of nanomaterials which are difficult to obtain in bulk form,
and whose tensile ductility is limited (which makes micro-tension tests difficult).
It is currently difficult to produce bulk forms of many metals and ceramics with
nanocrystalline microstructure, but they can be readily produced in thin foils and
small cubes or cylinders with dimensions less than 1 mm. Conventional mechanical
testing techniques are not useful due to the small size of specimens.
Micro-compression makes use of a conventional nano-indenter and a flat-end tip
to measure the stress-strain curves of materials using posts with a dimension as
small as 250 nm in diameter (Schuster et al., 2007). A schematic of a cylindrical
post sitting on a base of the same material is shown in Figure 3.5a. Technologies
such as focused ion beam (FIB) enable researchers to fabricate such microposts
(Figure 3.5b) with ease. These posts usually have one end (the bottom) fixed on
the matrix (or the base material), while the top end is pressed by the flat-end tip.
In a conventional uniaxial compression test, both ends of the specimen are free to
deform in a nearly frictionless manner. When can the data from microcompression
measurements be used to represent the bulk materials behavior? How sensitive is
the test to variations in material behavior?
Finite element simulations can be used to determine the errors resulting from
the compliant base condition, and to provide guidelines for the proper design of
microcompression experiments (Zhang et al., 2006). Consider first an axisymmetric
finite element model (Figure 3.6a) of the micropost. A velocity boundary condition
Fig. 3.5 (a) Schematic of a microcompression experiment, showing micropillar and flat-bottomed
indenter tip. (b) SEM micrograph showing a micropillar of a PdNiP metallic glass produced by
focused ion beam machining.
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Nanomaterials
(a)
(b)
Fig. 3.6 (a) Finite element mesh used for 2D axisymmetric model of the micropillar in a microcompression experiment. (b) Computed effective (von Mises) stress distribution in a sample after
plastic compression. Note the nonuniformity at the root of the pillar, indicating the importance of
the root radius (also called the fillet radius). Reprinted from Scripta Materialia, Vol 54, Issue 2,
page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments, January 2006, with permission from Elsevier.
corresponding to the imposed strain rate is applied to the post top, and the bottom
of the base is not allowed to move in the x2 -direction as shown in the figure. The
true stress-strain behavior of the post can be obtained from the reaction force P
generated on the post top and the displacement u of the post top using the following
equations:
σ̄22 =
1
π r2
r
0
ε̄22 = ln(1 +
P2 |x2 =h dx1
(3.2)
u2
)
h
(3.3)
where the subscripts 2 indicate the x2 components of the corresponding vector, r is
the radius of the post, h is the height of the post with respect to the base, and σ22
and ε22 are the stress and strain components that would be appropriate for a uniaxial
compression experiment (note that in the microcompression experiment σ22 will not
be the only non-zero stress component). These equations are essentially those used
in reducing experimental data.
In the finite element simulations, the material of the post is assumed to have the
behavior described by the Ramberg-Osgood equation
ε̄ =
σ
σ0 σ
+ β ( )( )N
E
E σ0
(3.4)
where E is the Young’s modulus, N the inverse of the traditional strain hardening
exponent (see Equation 2.95), β is a material constant and σ0 is a reference stress
(the Poisson’s ratio ν is also needed to complete the description of the linear elastic behavior). For the calculations shown here, the following parameter set is used,
3 Nanoscale Mechanics and Materials: Experimental Techniques
77
corresponding to an aluminum (Al) alloy: E = 70 GPa, ν = 0.33, σ0 = 65 MPa,
β = 2.69, and N = 4.18. The question of interest is: under what conditions would
the calculated σ22 and ε22 have the same relationship as the input material behavior
given by Equation (3.4)? Those conditions would represent the design of an appropriate microcompression experiment, since the computed effective behavior would
be identical to the assumed material behavior.
3.4.3.1 Post Geometry Effects
h
) on the output
The effects of the fillet radius rc and the post aspect ratio (α = 2r
stress-strain curves are considered first. We begin by fixing α = 2 and varying rc
(expressed in terms of the post radius r). Figure 3.7 displays the input stress-strain
curve from Equation (3.5), along with the output stress-strain curves from the simulations with different fillet radii rc = 0.1r, 0.2r, 0.5r, and r. For clarity, a portion of
the curves has been enlarged in the inset of Figure 3.7. Due to the fixed end effect,
all the simulated output curves are above the input curve, indicating an overestimate
Fig. 3.7 Input and simulated stress-strain curves for various assumed root radii (detailed differences observable in the inset) as computed from the finite element model presented in the last
figure. The “input” curve is the input material behavior used in the finite element simulations, and
the “output” stress-strain curves are obtained from the forces and displacements computed from
the simulations and then processed in the same way as the force and displacement data is processed in the experiments. If the experimental design were to be perfect, the output curves would
be identical to the input curve. Note that the simulated curves are all above the input data, demonstrating the effect of the end condition. The outermost curve has the largest root radius, equal to
the radius of the cylinder. As the root radius is decreased, the simulated curves approach the input
curve but are always above it. Reprinted from Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang,
B.E. Schuster, Q. Wei and K.T. Ramesh, The design of accurate microcompression experiments,
January 2006, with permission from Elsevier.
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Nanomaterials
of the flow stress of the material in the plastic range, and the strength of this effect
depends on the fillet radius/post radius ratio. The larger the fillet size (with respect
to the post radius), the greater the error. When the fillet radius ratio rrc is less than
0.5, the output curves are within 3% of the input data. This suggests that if the fillet
size is well controlled the fixed-end compression measurements can still be used to
represent the bulk material’s behavior in terms of flow stress. For the aspect ratio
α = 2, even with the largest fillet radius/post radius ratio ( rrc = 1) the error in flow
stress from the microcompression test is less than 6%.
The distribution of the von Mises stresses in the post and the base (Figure 3.6)
for various fillet radius/post radius ratios shows that with increased fillet size, the
stress concentration at the fillet is alleviated. For rrc ≥ 0.5 there is no significant
stress concentration at the fillet, the maximum von Mises stress being along the post
axis. Excessive stress concentrations may result in localized specimen failure prior
to general yielding. Thus, the choice of fillet radius for an experiment represents a
compromise between the need to accurately recover material behavior in the plastic
range (requiring small rrc ) and the desire to avoid localized failure at the post root
(which requires large rrc ). The simulations suggest that a choice of rrc in the range of
0.2 ∼ 0.5 represents a good compromise. The fillet size can be controlled to some
degree by the FIB or micro-machining parameters.
The second important geometric factor that may affect the accuracy of the meah
). The aspect ratio of the posts has a relasurement is the post aspect ratio (α = 2r
tively small effect on the output flow stress curves if the aspect ratio is larger than 2.
Aspect ratios of less than 2 are not appropriate, since the output strain hardening of
the post material then deviates from the input value due to the constraints from the
post base. However, aspect ratios much larger than 2 can cause buckling (discussed
later in this chapter and in Chapter 7).
3.4.3.2 Accuracy of Measurement of Elastic Response
In a standardized mechanical experiment the machine compliance is eliminated by
using extensometers or strain gages and the elastic modulus of the specimen material
can then be calculated. For micro-compression, this type of compliance correction
is not possible. Therefore, it is important to examine the factors that will affect the
accuracy of elastic modulus measurement. Figure 3.8 displays the error of elastic
modulus measurements as a function of the fillet size (the apparent elastic modulus is normalized by the known input modulus). The post aspect ratio is fixed at 2.
It is apparent that increased fillet size improves the accuracy with which the elastic modulus is measured, but the apparent modulus still underestimates the input
modulus. The reason is that the base will always deform along with the post, and
this deformation contributes to the system compliance, which is not accounted for
when collecting the displacement data. However, this error may be alleviated by accounting for the base compliance using a result from Sneddon. Sneddon computed
the depth of penetration D of an elastic half-space by a flat-ended rigid cylinder of
radius a under a load P as
3 Nanoscale Mechanics and Materials: Experimental Techniques
79
Normalized Modulus, E/Einput
1.4
Effect of fillet size ( = 2)
Sneddon Correction
Modified Sneddon Correction
1.3
1.2
1.1
1
0.9
0.8
0.7
(b)
0
0.3
0.6
0.9
1.2
Normalized Fillet Radius, rc /r
Fig. 3.8 Computed variation of apparent elastic modulus (normalized by true elastic modulus)
with fillet radius size at fixed aspect ratio, using the finite element simulations discussed in the
text. The effects of the Sneddon and modified Sneddon corrections are also shown. Reprinted from
Scripta Materialia, Vol 54, Issue 2, page 6, H. Zhang, B.E. Schuster, Q. Wei and K.T. Ramesh, The
design of accurate microcompression experiments, January 2006, with permission from Elsevier.
D=
P(1 − ν )
4µ a
(3.5)
where µ and ν are respectively the shear modulus and Poisson’s ratio of the halfspace. We then obtain the base compliance (Zhang et al., 2006) as
Cb =
1−ν
1 − ν2
=
4µ a
2Eac
(3.6)
where ac = η (r + rc ) is the effective contact radius, with η as a proportionality constant to be determined. The choice of a in Equation (3.6) is difficult because of the
geometry of the post-base connection and we have set a = ac . By subtracting this
base compliance from the total compliance, a corrected post stiffness (and hence
modulus) is obtained. If η = 1 (the Sneddon correction), the modulus is overestimated. Setting η = 1.42 provides a significantly improved correction (the modified
Sneddon correction). Figure 3.8 also includes both the Sneddon and modified Sneddon corrections. Similar results on the error in the modulus measurement are found
with other aspect ratios.
3.4.3.3 Accuracy of Measurement of Plastic Response
The accuracy of micro-compression measurements may be affected by the plastic flow properties (strain hardening and strain rate hardening) of the post material. Finite element simulations have shown that the microcompression technique is
most accurate for materials that exhibit low strain hardening. Such materials include
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Nanomaterials
nano-structured metals, metals and alloys with ultrafine microstructure, metallic
glasses, and ceramics. Strain rate effects are also important in many applications,
especially when plastic instability is of serious concern. This is because the susceptibility to plastic instability of metals and alloys is strongly dependent on the
strain rate sensitivity of the material. There are various ways to define the strain
rate sensitivity. An engineering definition assumes a power-law strain rate hardenσ
ing such that the strain rate sensitivity is written as m = ∂∂ln
lnε̇ , where ε̇ is the imposed
strain rate. Finite element simulations assuming different values of m indicate that
microcompression can capture the strain rate sensitivity of the material. Note that the
practically useful strain rate achievable for microcompression with a nano-indenter
is probably below 1 s−1 , i.e., in the quasistatic regime.
3.4.3.4 Buckling, Friction and Taper Effects
Buckling, either elastic or plastic, is a common concern for a compression test.
Elastic Euler buckling of a column occurs when the applied compressive load
reaches a critical value given by
Pcr =
π 2 EI
Le 2
(3.7)
where I is the moment of inertia of the column (here the micropost) and Le is the
effective length of the column (Le = 2h for the end conditions developed in microcompression). The applied load in the microcompression experiment must be at
least large enough to cause yielding of the sample, so P > σY A where A is the crosssection area of the post. The minimum aspect ratio at which Euler buckling is likely
to occur in microcompression can then be computed, and is found to be greater than
10 given the typical ratio of E/σY for most materials. Thus Euler buckling is not an
issue in microcompression for aspect ratios less than 10.
The plastic buckling problem must be addressed with 3D finite element analysis.
Effects of friction between the indenter (usually made of diamond) and the post of
top must also be considered, as must the effect of “taper” of the post and the system misalignment on the simulation output. In these 3D analyses the smallest fillet
radius ratio used is rrc = 0.1 for α = 2. Both 3D cylindrical and cuboidal posts are
considered. The compression is exerted by a rigid surface representing the flat-end
nano-indenter tip. Misalignment of the experimental setup is considered by tilting
the rigid surface with respect to the axis of the post.
3D simulations show that when there is no friction, aspect ratios of approximately
2 provide good measures of behavior and avoid buckling, while aspect ratios of 5
cause buckling after small overall strains.
However, micro-compression experiments are usually performed without applying lubrication to the tip-post interface. A range of values for Coulomb friction coefficients exist in the open literature for a number of diamond-metal surface pairs:
static coefficients of friction in the range of 0.1–0.15, and kinetic values of 0.03 ∼
0.06. Friction between the indenter tip and the post top delays the plastic buckling
3 Nanoscale Mechanics and Materials: Experimental Techniques
81
of cylindrical posts; however, the friction does increase the estimated flow stress.
The ease of development of plastic buckling is strongly dependent on the strain
hardening of the material. Thus large aspect ratio posts should also be avoided due
to the possible occurrence of plastic buckling when one tests low strain hardening
materials: aspect ratios of 2 ∼ 3 are recommended.
Another important geometric factor that may have a significant effect on the accuracy of microcompression experiments is the “taper” of the post. Taper is typically
a result of fabrication inaccuracies. Taper can cause errors in the estimates of the
modulus and yield stress and result in a spurious strain hardening response (due
to propagation of the plastically deforming zone down the post axis). Taper should
therefore be avoided by careful manipulation of the micro-post fabrication process.
3.4.3.5 Effect of Misalignment
Misalignment of the system is manifested primarily in the deviation of the elastic
part of the stress-strain curve, giving rise to an underestimate of the elastic modulus
of the material. A decrease of the measured elastic modulus is observed with increase of misalignment (note that a lower elastic modulus will be observed even for
perfect alignment is due to the compliance of the post base). In addition, excessive
misalignment will result in buckling of the post.
3.4.3.6 Summary Recommendations for Microcompression Experiments
1. The post fillet radius affects both the accuracy of the microcompression test (in
terms of the flow stress) and the stress concentration at the post root. Fillet radius/post radius ratios of 0.2 ∼ 0.5 are recommended to provide sufficient test
accuracy. The fillet radius also has a profound effect on the accuracy of the elastic modulus due to the compliance of the base. This effect can be alleviated using
a modified Sneddon correction to account for the base compliance.
2. Strain hardening of the material significantly affects the microcompression test
accuracy, particularly when the fillet radius is large (close to the post radius). The
results suggest that microcompression experiments are better suited for materials
with low strain hardening.
3. For aspect ratios ≥ 5, first-mode plastic buckling is suppressed by the friction
between the indenter tip and the post top, but higher-mode buckling may still
occur, particularly for high aspect ratios and low strain hardening materials. Post
aspect ratios of 2 ∼ 3 are therefore recommended.
4. Taper of the post results in overestimated elastic modulus, spurious strain hardening and increased apparent yield and thus should be minimized during micro-post
fabrication.
5. Misalignment of the microcompression system significantly affects the test accuracy by underestimating the elastic modulus, and may result in buckling of
the post. This should be minimized through careful post sample preparation, tip
machining, and experimental alignment.
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Nanomaterials
3.4.4 Microtensile Testing
Conventional tension testing is the standard approach to obtaining the mechanical response of materials (indeed, there are very well-developed ASTM standards
for such testing). These standards involve the use of specially shaped specimens,
designed to develop a uniaxial tensile stress along a specific gauge length and to
obtain accurate measures of strain. The result is a stress strain curve in uniaxial tension (Figure 3.10) that is the basis of most constitutive modeling in the plastic range.
However, conventional tension tests require specimens that are at least several centimeters long. As of 2008, there are very few nanomaterials which can be obtained
in sizes sufficiently large to permit conventional tension tests. A number of research
groups have therefore developed the ability to perform tension tests on very small
samples (typically obtained from thin films). Sample sizes of interest are typically
lengths of 10–1000 µm, widths of 10–100 µm, and thicknesses of 10 nm to 100 µm.
There are three critical requirements for a good tension test. The first is the development of a significant region of the specimen (called the gauge length) within
which a uniaxial tensile stress state exists; the second is the ability to measure that
tensile stress accurately; and the third is the ability to measure the tensile strain along
the direction of the stress within the gauge section. Finally, there is the question of
actuation: how is the tension generated?
3.4.4.1 Stress State and Specimen Design
The first requirement determines the design of the specimen. The primary complexity in tension tests arises from the need to apply the tensile force to the specimen,
which requires the use of some type of grip. Many, many grip types are used in conventional tension testing, ranging from hydraulic and pneumatic grips to threaded
ends and adhesively bonded ends. The stress state at the grip is very complicated,
and so careful specimen design is necessary to obtain a gauge length that is primarily under uniaxial tension. An important issue is the degree of stress concentration
near the grips, which is particularly important for high-strength and relatively brittle
materials like nanomaterials. For the case of materials with a significant degree of
plastic deformation, a very common specimen design that provides a gauge length
containing nearly uniaxial tensile stresses is obtained with the “dogbone” shape of
conventional tension testing.
Very good computational design of tensile specimens can be performed using
the finite element method, accounting for the expected material behavior, and so it
is no longer necessary to rely on a standard to ensure that reasonable results are
being obtained. When it comes to testing very small nanomaterials samples, the
total forces that are developed are small and so unique grip designs are often used
that are mated to the specimen shapes that can be easily fabricated. There is also
a strong correlation between the fabrication or machining techniques used to make
the nanomaterial and the tension test sample design. Many micro-tension specimens
end up being essentially thin films, and so many of the techniques associated with
thin-film applications are relevant to such experiments.
3 Nanoscale Mechanics and Materials: Experimental Techniques
83
3.4.4.2 Accurate Measurement of Stress
The second requirement (that of determining the tensile stress within the gauge
length of the specimen) is relatively easy to satisfy, because the conditions of force
equilibrium demand that the force along the force chain in the testing technique
be constant. Thus the simple insertion of a load cell into the force chain provides a
measure of the force in the specimen, and then knowing the cross-section area of the
specimen allows one to calculate the tensile stress in the sample gauge length. The
only difficulty involves ensuring that the load cell is sufficiently accurate and has
the resolution to measure the small forces typically developed in small samples. A
rough estimate of the force that would be generated in the specimen can be obtained
as σy A where σy is the yield stress of the material and A is the cross-section area of
the specimen, which scales as the square of the specimen width (A ∝ b2 ). Thus for
a nanocrystalline material with a yield strength of 1 GPa and a specimen width of
10 µm, the force that must be measured is of the order of 0.1N. In the elastic range,
the force scales as E ε A, and so to have a strain resolution of ε = 10−4 and a Young’s
modulus of 100 GPa one needs a force resolution of 1 mN.
3.4.4.3 Accurate Measurement of Strain
The biggest difficulty in microscale tension experiments is accurate measurement of
the strain in the specimen. In traditional tension testing, strain is measured in one of
two ways. The simplest and least accurate is to measure the displacement δ l of the
crosshead (or actuator) and divide this by the specimen gauge length l0 :
ε=
δl
.
l0
(3.8)
A much more accurate approach involves measuring the strain directly on the
specimen using some kind of in situ strain measurement technique such as a strain
gauge. However, it is difficult to find strain gauges that are of the appropriate size
for most micro sample tests. Approaches that are currently used in microsample
testing include digital image correlation or DIC (Chu et al., 1985) together with in
situ imaging using an optical microscope, SEM, or TEM; the measurement of the
displacement of fiducial markers placed on the specimen, and specialized techniques
such as the interferometric strain/displacement gauge (ISDG). A simple measure of
the difficulty involved can be obtained by recognizing that the displacements that
must be measured are of the order of ε l0 , and so for a specimen length l0 of the order
of 100 µm and a strain resolution of ε = 10−4 one needs to measure a displacement
of δ u = 10 nm. Such displacement resolutions are difficult to achieve.
3.4.4.4 Actuation Systems for Micro-Tensile Testing
Uniaxial tension experiments are typically performed either under load control or
under displacement control (that is, one controls the load while the displacement is
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Nanomaterials
measured as the output of the system, or vice versa). A simple (and idealized) example of a load controlled test is a tension test performed using dead weights to provide
the load. The actuator system used to generate tension in conventional tension tests
is typically either a servohydraulic system or a screw driven system. These systems
are capable of carrying very high loads, and control systems for such devices are
highly developed. However, such systems are not appropriate for the small loads
and small displacements that must be used to study microscale specimens: the excursions in force or displacement that are developed in these conventional systems
before full control is established are typically larger than the forces or displacements
σ
required to cause failure of the microscale specimens. Initial displacements of Ey l0
σ
are sufficient to cause yield of the specimen, and since for most materials Ey ∼ 10−3 ,
it takes only 100 nm of displacement to cause yield in a specimen that is 100 µm
long. An actuator with extreme control of displacement is therefore necessary for
micro-tensile testing. Examples of actuators with such control include piezoelectric actuators and MEMS-based actuator systems (Figure 3.9). The advantage of
piezoelectric actuator systems is that they can be purchased rather than having to be
fabricated.
Fig. 3.9 Schematic of a microtension testing apparatus, showing the actuation and sensing systems. The air bearing is an important component. The strain is computed from displacement fields
measured using Digital Image Correlation (DIC) software analysis of images from the camera.
Illustration due to Chris Eberl.
In most tensile tests, the compliance of the testing machine is a significant factor in the design of the experiment. Under the load, some of the displacement that
is observed in the system is a result of the deformation of the machine because of
machine compliance. There are standard approaches to correcting for machine compliance so that accurate measurements of specimen strain can be obtained, at least
for conventional tension testing. This can be particularly important for micro-tensile
testing.
3 Nanoscale Mechanics and Materials: Experimental Techniques
85
The result of a microtensile experiment on a nanocrystalline nickel film is shown
in Figure 3.10. The specimen was a thin film made by electrodeposition (an Integran
material), with a film thickness of 100 nm and a mean grain size of 20 nm. Note the
significant flow stress that is obtained, and the unload-reload cycle used to estimate
the modulus (to minimize machine compliance issues). The apparent strain hardening that is observed early in the stress-strain curve will be discussed later in the
mechanical properties chapter.
True Stress, MPa
2000
1500
1000
500
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
True Strain
Fig. 3.10 Example of stress strain curve obtained on nanocrystalline nickel (20 nm grain size) in
uniaxial tension using a microtension setup. The sample was a thin film. Data due to Shailendra
Joshi.
A remarkable variety of nanomechanical and micromechanical tensile testing
systems have been developed over the last decade (Sharpe et al., 1997; Espinosa
et al., 2003; Legros et al., 2000; Yi et al., 2000; Haque and Saif, 2001; Srikar and
Spearing, 2003; Jonnalagadda and Chasiotis, 2008). Several of them have been developed with an eye towards thin-film applications rather than testing nanomaterials,
but some of them can be used as nanoscale mechanics characterization techniques
with application towards nanomaterials. Since these techniques involve multiple
innovations and some uncertainty about accuracy and repeatability, it is important
to compare the results of these experiments with the results of other experimental
techniques on similar materials. Such comparisons are difficult because the research
groups that operate such techniques typically do not work on identical materials, and
so it is unclear whether the differences in behavior that are observed are the result
of experimental technique variations or the material variations. Note, in particular,
that comparisons of nanomaterial behavior measured using these systems with the
result of macroscale experiments on large-sample coarse-grain materials may not be
meaningful, because of the potential differences in materials processing routes and
specimen fabrication.
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Nanomaterials
3.4.4.5 Summary Recommendations for Micro-Tensile Experiments
1. Finite element methods should be used to accomplish specimen design, with the
objective being to optimize the specimen design with respect to the mechanical
property that must be measured (for example, optimal design for elastic modulus
measurement would be distinct from optimal design for strength measurement).
2. For minimum specimen dimensions larger than several microns, piezoelectric
actuators systems provide a good compromise between accuracy and controllability.
3. In situ strain measurement techniques such as digital image correlation are recommended, and are essentially required if computational design of the specimen
has not been performed.
4. Ideally, experimental results with any new technique should be compared with
the results of other experimental techniques on a standard material in order to
obtain confidence in the particular experimental technique that is being used.
Unfortunately, at this point (2008) it is not possible to recommend a specific
standard material.
3.4.5 Fracture Toughness Testing
The fracture toughness of nanomaterials is extremely difficult to measure. Standard
fracture toughness measurement techniques require standard specimen sizes, all of
which are too large to be used with most nanomaterials (as of 2008). Indeed, it is
not clear that the concept of fracture toughness as it is normally defined is relevant
to most small-scale systems. Thus, most nanomaterials are most easily obtained in
thin-film form, and so plain strain fracture toughness is not a sensible concept. Further, it is not clear that there is a significant K-dominant (see Chapter 2) domain
within a nanomaterial containing a crack. That said, estimates of fracture toughness
are needed in order for materials to be used within the design of complex components, and it is important to have at least a theoretical understanding of how fracture
toughness should change with grain size. It is often stated that nanomaterials are
brittle, but what is typically meant is that the tensile ductility of a specific nanomaterial specimen is low.
The two existing approaches to fracture toughness testing with nanomaterials are
(a) examining the failure of a thin film containing a pre-crack and (b) bending techniques developed for understanding thin film behavior. We explore these techniques
further in the chapter on the failure of nanomaterials, because these techniques show
material behavior that is intimately coupled to test design.
3.4.6 Measurement of Rate-Dependent Properties
The primary experimental techniques associated with the measurement of the rate
dependent properties of materials are described in Figure 3.11 (note that the stress
10–6
10–4
10–2
100
102
Strain Rate,
104
87
Pressure-shear plate impact
Miniaturized Kolsky bar
Conventional Kolsky bars
Specialized machines
Servohydraulic machines
3 Nanoscale Mechanics and Materials: Experimental Techniques
106
108
s–1
Fig. 3.11 Experimental techniques appropriate for various ranges of strain rate. The range of rates
identified as “specialized machine” is very difficult to reach. Very few laboratories in the world are
able to achieve strain rates higher than 104 s−1 .
states developed within the various techniques are not necessarily identical). An excellent and relatively recent review of these methods is presented by Field et al.
(2004), and a summary of the key experimental issues is presented by Ramesh in a
chapter in the Handbook of Experimental Mechanics (see suggested reading). For
the purposes of this discussion, strain rates above 102 s−1 are classified as high
strain rates, strain rates above 104 s−1 are called very high strain rates, and strain
rates above 106 s−1 are called ultra-high-strain-rates. Conventionally, strain rates
at or below 10−3 s−1 are considered to represent “quasistatic” deformations, and
strain rates below 10−6 s−1 are considered to be in the “creep” domain. This is an
important range for some nanomaterials. Creep experiments are typically performed
at relatively high temperatures, and a variety of specialized machines exist for these
kinds of loading; dead loads are often of particular interest. Quasi static experiments are typically accomplished through a variety of servohydraulic machines,
and ASTM standards exist for most of these experiments. Most servohydraulic machines are unable to develop strain rates larger than 100 s−1 repeatably, but some
specialized servohydraulic machines can achieve strain rates of 101 s−1 . We focus
here on the higher strain rates (greater than 102 s−1 ).
3.4.6.1 Split-Hopkinson or Kolsky Bars
The now-classical experimental technique in the high-strain-rate domain is the
Kolsky bar or split-Hopkinson pressure bar (SHPB) experiment for determining
the mechanical properties of various materials (e.g., Nicholas and Rajendran, 1990;
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Nanomaterials
Ravichandran and Subhash, 1994; Walley and Field, 1994 in metals, ceramics, and
polymers respectively) in the strain rate range of 102 s−1 to 8 × 103 s−1 . The terms
“split-Hopkinson pressure bar” and Kolsky bar are often used interchangeably, although the latter is more appropriate (the technique was invented by Kolsky, and
neither John nor Bertram Hopkinson used a split-bar to study material behavior).
One should note also that the term “split-Hopkinson pressure bar” implies the performance of compression experiments, whereas the term “Kolsky bar” is more general and may include compression, tension, torsion or combinations of all of these.
Since the fundamental concept involved in this technique was developed by Kolsky
(1949), we will use the term Kolsky bar.
A schematic of the typical compression Kolsky bar experimental apparatus is
shown in Figure 3.12. The device consists of two long bars (called the input and
output bars) that are designed to remain elastic throughout the test. These bars sandwich a small specimen (usually cylindrical), which is expected to develop inelastic
deformations. The bars are typically made of high-strength steels such as maraging
steel, with a very high yield strength and substantial toughness. One end of the input
bar is impacted by a projectile made of a material identical to that of the bars; the
resulting compressive pulse propagates down the input bar to the specimen. Several
reverberations of the loading wave occur within the specimen; a transmitted pulse is
sent into the output bar and a reflected pulse is sent back into the input bar. Typically,
resistance strain gages are placed on the input and output bars for measuring (i) the
incident pulse generated by the impacting projectile, (ii) the reflected pulse from the
input bar/specimen interface, and (iii) the transmitted pulse through the specimen
to the output bar. The strain gage signals are typically measured using high-speed
digital oscilloscopes with at least 10-bit accuracy and preferably with differential
inputs to reduce noise.
Fig. 3.12 Schematic of the compression Kolsky bar (also known, incorrectly, as the splitHopkinson pressure bar). The projectile is usually launched towards the input bar using a gas
gun. Specimen surfaces must be carefully prepared for valid experiments.
Let the strain in the incident pulse be denoted by εI , that in the reflected pulse
by εR , and that in the transmitted pulse by εT (these are bar strains as measured by
the strain gages). By assuming stress equilibrium, uniaxial stress conditions in the
specimen and 1D elastic stress wave propagation without dispersion in the bars, the
nominal strain rate ės , nominal strain es , and nominal stress ss (all in the specimen)
can be estimated using the following equations:
3 Nanoscale Mechanics and Materials: Experimental Techniques
2cb
εR (t)
l0
(3.9)
ės (τ )d τ
(3.10)
Eb Ab
εT (t)
As
(3.11)
ės (t) = −
es (t) =
89
t
0
and ss (t) =
where cb = Eρb is the wavespeed in the bar, l0 is the initial length of the cylinb
drical specimen, Eb is the Young’s modulus of the bar material, ρb is the density
of the bar material, Ab is the bar cross-section area, and As is the specimen initial
cross-section area. Note the negative sign in Equation (3.9). This arises because
the strain in the reflected pulse has the opposite sign of the strain in the incident
pulse; the latter is compressive and is conventionally considered positive in these
experiments. Thus the specimen strain rate expressed by Equation (3.9) is a positive
strain-rate, i.e., in this convention, a compressive strain-rate. The true strain and true
strain rate in the specimen are then given by Ramesh and Narasimhan (1996):
εs (t) = − log (1 − es (t))
and
ε̇s (t) =
ės (t)
.
1 − es (t)
(3.12)
(3.13)
Note that compression is defined as positive. Assuming plastic incompressibility,
the true stress σs in the specimen is obtained as
σs (t) = ss (t)(1 − es (t)).
(3.14)
From Equations (3.12) and (3.14), a stress-strain curve is obtained at a strain rate
defined by an average taken over the strain rate history obtained in Equation (3.13);
an example of such a stress-strain curve is presented in Figure 3.13.
Several fully miniaturized versions of the compression Kolsky bar have been
developed (e.g., see Jia and Ramesh 2004). In the Jia and Ramesh version (see
Figure 3.14), the bars are 3 mm in diameter and 300 mm long, and may be made of
maraging steel or tungsten carbide. Sample sizes are cubes or cylinders on the order
of 600 µm to 1 mm on a side; cube specimens are used when the failure mode must
be imaged using a high-speed camera, or when the amount of material is so small
that only cuboidal specimens can be cut. The technique is simple, and the entire system can be designed to fit on a desktop, and so the technique is sometimes referred
to as the “desktop Kolsky bar.” The miniaturized Kolsky bar or “desktop Kolsky
bar” experimental technique is particularly effective when it comes to understanding
the properties of nanostructured materials, or of materials with very fine grain
sizes.
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Nanomaterials
Fig. 3.13 Stress strain curves obtained on 5083 aluminum using the compression Kolsky bar at
high strain rates (2500 per second). The lowest curve represents quasistatic behavior. In general,
the strength appears to increase with increasing strain rate, but note the anomalous softening at the
highest strain rate (perhaps due to thermal softening).
Fig. 3.14 The Desktop Kolsky Bar – a miniaturized compression Kolsky bar arrangement
developed by Jia and Ramesh (2004). This device is capable of achieving strain rates above 104 per
second, and can fit on a standard desktop. Specimen sizes can be on the order of a cubic millimeter.
The large circular object behind the bar is part of a lighting system.
3 Nanoscale Mechanics and Materials: Experimental Techniques
91
3.5 Suggestions for Further Reading
1. K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge,
1987.
2. W.N. Sharpe, Jr., Handbook of Experimental Mechanics. Springer, New York,
2008. See, in particular, the chapter on High Strain Rate and Impact Testing by
K.T. Ramesh.
3. B. Bhushan, Springer Handbook of Nanotechnology. Springer, New York, 2007.
3.6 Problems and Directions for Research
1. Determine the resolution limits of the transmission electron microscope and
scanning electron microscopes in your institution. Will they be able to resolve
a 10-nm grain of nanocrystalline aluminum? Can you see a carbon nanotube in
your SEM?
2. Look up the ASTM standard for testing hard materials (determining the yield
strength). Determine the size of specimen that would be required to meet the
standards if the specimen were to be made of a nanocrystalline aluminum that
has a yield strength of 1 GPa.
3. What is the smallest force that can be measured in one of the atomic force microscopes in your institution (or the nearest AFM, if your institution does not
possess one)? Can you measure the force required to develop a strain of 1% in a
single-walled carbon nanotube of Young’s modulus 1 TPa, assuming elasticity?
4. Perform a tensile stress-strain test on any aluminum alloy in a servohydraulic
testing machine, and then determine how to correct for the machine compliance.
Provide a corrected tensile stress-strain curve.
References
Binnig, G. and H. Rohrer (1985). Scanning tunneling microscopy. Surface Science 152, 17–26.
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From the many, one.
– A liberal translation of E Pluribus Unum
4
Mechanical Properties: Density
and Elasticity
Should nanomaterials have properties that are any different from conventional materials? Since we have defined nanomaterials as materials where some controllable
relevant dimension is of the order of 100 nm or less, there should not be a difference
in properties unless the property of interest is affected by some size scale. A big
part of understanding the properties of nanomaterials will therefore be developing
an understanding of the length scales associated with these properties. We will approach this discussion in two parts. First, in this chapter, we will describe the simple
mechanical properties of nanomaterials: the density and the linear elastic behavior.
Second, in the next chapter, we will discuss the plastic behavior of nanomaterials,
where the couplings of deformation mechanisms with size scale are much more
complex. The objective of the current chapter is to introduce the concepts behind
the coupling of the grain size and other size scales with the mechanical properties
through the simple scalar property of density and through the simple linear property
of elasticity.
4.1 Density Considered as an Example Property
We can learn a great deal about size effects in nanocrystalline materials by considering a simple scalar property like the density. Our intuitive sense is that the density
of a metal like iron is a fixed number. Elementary physics tells us that the density
must depend on both the atomic weight and the relative spacing of the atoms; the
spacing of the atoms depends on interatomic interactions, and therefore – at a fundamental level – on position within the periodic table of the elements. Looking up
iron in the periodic table of the elements, one would find that pure iron has a density of 7874 kg m−3 . This is the density one would obtain if one were to measure
the density of a single crystal of pure iron (which is a body-centered-cubic solid at
room temperature), whatever the size of that single crystal. However, pure iron is
generally obtained as a polycrystalline material. If the grain size (i.e., crystal size)
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 4,
c Springer Science+Business Media, LLC 2009
95
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Nanomaterials
were to be 10 µm, a cubic millimeter of iron would contain about a million grains.
The density of this polycrystalline pure iron will be very slightly different than the
density of single crystal pure iron, because some fraction of the atoms in the polycrystalline material are at grain boundaries, where their interatomic spacing is not
the same as that in the single crystal. Since the spacing of atoms at grain boundaries is usually somewhat larger than the spacing of atoms within the single crystal
(i.e., the atoms are more loosely packed), the polycrystal is usually slightly lower in
density.
4.1.1 The Rule of Mixtures Applied to Density
If the density of the material in the single crystal is ρsc , and the density of the
material corresponding to atoms that are at grain boundaries is ρgb , then the overall
density ρ of the iron in the polycrystalline material can be calculated by using the
balance of mass:
V ρ = Vsc ρsc +Vgb ρgb
(4.1)
where V is the total volume of the material, and Vsc and Vgb are the volumes of the
atoms in the single crystal part and grain boundary part of the polycrystal respectively. Dividing Equation (4.1) throughout by V , we obtain
ρ=
Vgb
Vsc
ρsc +
ρgb .
V
V
(4.2)
This can be rewritten as
ρ = fsc ρsc + fgb ρgb
where fsc =
fgb =
Vgb
V
Vsc
V
(4.3)
is the volume fraction of atoms with the single crystal packing, and
is the volume fraction of atoms with the grain boundary packing. Note that
Vsc +Vgb = V,
(4.4)
fsc + fgb = 1.
(4.5)
so that
It follows that Equation (4.3) can be rewritten as
ρ = (1 − fgb )ρsc + fgb ρgb .
(4.6)
An equation of the form of Equation (4.6) is called a rule of mixtures: it provides
an overall property of the material (from the mechanics perspective, viewed as a
composite) in terms of the corresponding properties of its constituents, with a linear
dependence on the volume fraction of each constituent. This is the most common
first approximation for the overall properties of any material made out of multiple
4 Mechanical Properties: Density and Elasticity
97
Overall Density
ρSC
ρgb
0
0.2
0.4
0.6
0.8
Volume fraction of grain boundary, fgb
1
Fig. 4.1 The linear rule of mixtures. Calculated variation of density in a nanocrystalline material
with volume fraction of grain boundary, based on Equation (4.6).
subcomponents (in this case, crystal atoms and grain boundary atoms). The result of
using such a rule of mixtures to predict the overall density of the material is shown
in Figure 4.1.
Note that Equation (4.6) establishes that the density of the polycrystalline material may be different from the density of the single crystal, but this does not by
itself ensure that there will be a grain size effect. What is the volume fraction fgb
of the grain boundary atoms? This depends, of course, on the shapes of the grains
and the nature of the grain boundaries. Consider the highly idealized case of a material made up entirely of grains that are cubes (the simplest space filling shape).
Assume that the grain size, given by the side of the cube, is d, and that the thickness
of the grain boundary is t (Figure 4.2). Then it follows that the volume fraction of
the material that is inside the grain (rather than at the grain boundary) is
fsc =
d3
(d + t)3
,
(4.7)
and that the volume fraction of the material at the grain boundary is
fgb = 1 − fsc = 1 −
d3
(d + t)3
.
(4.8)
The key question is whether the grain boundary thickness changes as the grain
size changes. Both experimental observations and atomistic simulations suggest otherwise (Latapie and Farkas, 2003). A simple intuitive way to understand this is to
ask how many layers of atoms are affected by the disordered conditions at the grain
boundary: it is apparent that the number of layers of atoms should remain the same
whether the grain is 1 µm or 100 µm in size. Thus, for instance, if we assume that
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Nanomaterials
a
t
d
b
d
Fig. 4.2 Schematic of (a) a cubical grain of size d with a grain boundary domain of thickness t,
and (b) the packing of such grains to form a material.
no more than three layers of atoms are affected by the grain boundary disorder, the
typical grain boundary thickness will be on the order of 1 nm (in metals, the interatomic spacing is of the order of 3 Å). It is apparent that the influence of the grain
boundary increases dramatically at small grain sizes. This size effect arises because
the grain boundary thickness provides a length scale within the problem.
A rapid estimate of the effect of the grain boundary thickness can be shown by
recasting Equation (4.7) in terms of the grain size ratio β , defined by β = dt . We
then have
β3
,
(4.9)
fsc =
(β + 1)3
and we see that when the grain boundary thickness is about the same as the grain
size, i.e., t = d so that β = 1, we have fsc = 18 and so fgb = 1− fsc = 78 ! Thus the fraction of the material that is in the grain boundary increases dramatically as the grain
size decreases to the approximate size of the fixed grain boundary thickness. We
refer to this limit (β = 1) as the grain boundary limit, and the corresponding grain
gbl
boundary volume fraction as fgb
. Note that the material would be about half grain
boundary ( fgb = 0.5 or 50%) and half single crystal when d50 = 3.85t, and grain
boundary mechanical behaviors can be expected to begin to dominate the overall
behavior at least at this point (which in this case amounts to a grain size of about
4 nm). Thus the characteristic length scale t combined with the assumed morphology (space filling cubes) gives us a characteristic grain size scale at which we should
4 Mechanical Properties: Density and Elasticity
99
expect a significant change (or transition) in behavior. Changing the assumed crystal
morphology would certainly change this d50 size scale.
However, at small grain sizes, other complexities arise. Examining the space filling cubic morphology, three different domains can be defined along grain boundaries. The domains that we have been discussing at this point in terms of the lower
density at the grain boundaries represent planar regions where two crystal facets
meet. The regions where these planes themselves meet are either nearly-linear regions of finite size that we will call triple junctions, or cuboidal regions at cube
corners that are corner junctions. (Note that triple junctions get their name because
in most microstructures they are formed by the intersection of three planes.) It is
likely that the densities ρt j and ρc j of the material at triple junctions and corner
junctions respectively are even lower than that of the grain boundaries, because of
the increased lack of compatibility between the atoms in the crystal structure. The
volume of each triple junction is t 2 d, and so the volume fraction ft j of triple junctions in the polycrystalline material of cubic morphology is
ft j =
6t 2 d
3
4(d + t)
=
3β
(β + 1)3
.
(4.10)
Similarly, the volume of each corner junction is t 3 , and the volume fraction fc j
of corner junctions in the polycrystalline material of cubic morphology is
fc j =
6t 3
8(d + t)3
=
3
4(β + 1)3
.
(4.11)
The variation of each of these volume fractions (for the grains, the grain boundaries, the triple junctions, and the corner junctions) with the normalized grain size
(normalized by the assumed grain boundary thickness t, so that this axis represents
the grain size ratio) is shown in Figure 4.3 for the cubic morphology. The single
crystal grain volume fraction decreases continuously as the grain size is decreased.
The grain boundary volume fraction increases rapidly as the grain size approaches
t, but has a maximum at a value of d just greater than t. This is because for smaller
grain sizes the contributions of first the triple junctions and then the corner junctions increase dramatically, while the sum of all the volume fractions must be equal
to unity. Thus any mechanisms that are triggered at such junctions are likely to
become very important at small grain sizes. Note that the triple junction volume
fraction is significant in the important domain of 1–10 nm (assuming that t ≈ 1 nm).
When one now considers the grain boundary, triple junction, and corner junction
parts of the polycrystalline material together, the effective density of the polycrystalline material is given by
ρ = (1 − fgb − ft j − fc j )ρsc + fgb ρgb + ft j ρt j + fc j ρc j ,
(4.12)
because we must have
fsc + fgb + ft j + fc j = 1.
(4.13)
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Nanomaterials
1
Cube
Morphology
fsc
0.8
fgb
ftj
Volume fraction
Grains
0.6
fcj
Corner
junctions
0.4
Grain
boundaries
0.2
Triple
junctions
0
0.1
1
10
100
1000
Normalized grain size d/t
Fig. 4.3 Variation of grain volume fraction, grain boundary volume fraction, triple junction volume
fraction, and corner junction volume fraction with normalized grain size β = dt for the cubic grain
morphology. The junction volume fractions become major contributors when d ≈ t.
The lower limit of the density would now be ρc j , where the atoms are typically
the most disordered. How different is this from ρsc ? This is not known a priori. The
density ρgb is usually assumed not to be dramatically smaller than ρsc , which would
imply that the absolute density change with grain size should not normally be an
issue until small grain sizes (say of the order of 50 nm) are achieved. This may of
course not be true for other properties, where the property of the grain boundary
may be substantially different from the property of the crystal.
In fact, however, the density of many nanocrystalline materials is measurably
less than the density of the corresponding conventional grain size material even for
relatively large grain sizes of order 100 nm. Why is this so? The primary reason is
because the processes that are used to make nanocrystalline materials typically result in the generation of significant quantities of defects in the material, particularly
porosity. An example of the porosity that might be observed in a nanocrystalline material is presented in Figure 4.4, which shows the surface of a nanocrystalline nickel
material made by electroplating. A significant area fraction of pores is observed on
the surface.
The effect of the pores on the density can be very easily understood by considering the pores to represent a third phase of the material of density ρdefect , where
ρdefect = 0. The porosity φ of the material is essentially the volume fraction of the
V
pores, φ = fdefect = Vpores . With this fourth constituent (the defects) included, the
total
overall density of the polycrystalline material is now
ρ = (1− fgb − ft j − fc j − fdefect )ρsc + fgb ρgb + ft j ρt j + fc j ρc j + fdefect ρdefect , (4.14)
because we must have
fsc + fgb + ft j + fc j + fdefect = 1.
(4.15)
4 Mechanical Properties: Density and Elasticity
101
Fig. 4.4 Scanning electron micrograph showing the presence of pores (dark regions) on the surface
of a nanocrystalline nickel material produced by an electroplating technique.
Since ρdefect = 0 when the defects are pores, it follows that
ρ = (1 − fgb − ft j − fc j − φ )ρsc + fgb ρgb + ft j ρt j + fc j ρc j .
(4.16)
The effect of the porosity can be quite dramatic, because the porosity φ itself
can be substantial, depending on the processing route used to make the nanocrystalline material. Much of the literature considers material with a porosity of 3–5%
to be “fully dense” even though this level of porosity has a substantial impact on
mechanical properties.
4.1.2 The Importance of Grain Morphology
The effects of grain morphology on the behavior of a material can be very substantial
at small grain sizes. For example, consider the alternative space-filling hexagonal
prism morphology shown in Figure 4.5, with side s and height h. Suppose we define
the aspect ratio parameter α = hs . The effective grain size in the plane of the hexagon
is d = 2s, but note that if the aspect ratio is either much less than 1 or much greater
than 1, a single grain size is no longer a reasonable description of the polycrystal
structure.
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Nanomaterials
t
s
h
Fig. 4.5 Another possible space-filling morphology, using hexagonal tiles of side s and height
h = α s, where α is the aspect ratio of the tile. The grain boundary domain thickness remains t.
√
The volume of the hexagonal prism is area × h, which is ( 3 2 3 s2 h). For this
hexagonal morphology, the volume fraction of the material in the crystal phase is
fsc |hex =
1
,
2α
1
1+(√
+
)g
+
( α3 + √23 )g2 + 13 g3
α
3
(4.17)
where the boundary-crystal size ratio is g = 2st . For the special case where α = 1 so
that h = s, we find that fgb = 0.5 at a transition grain size d50 = 2.7 nm, somewhat
less than in the cubic morphology case. However, if α = 0.1, corresponding to a
plate morphology (e.g., from a rolling process), d50 = 12 nm; if α = 10, corresponding to a rod morphology (e.g., from an extrusion process), we obtain d50 = 1.6 nm.
An even stronger effect of the morphology of the polycrystal is seen on the volume
fraction of the material that is in the grain boundary when the grain boundary limit
g = 1 is reached. For the space filling hexagonal prism morphology with α = 1, we
gbl
gbl
have fgb
= 0.8; when α = 10, we have fgb
= 0.64; and when α = 0.1, we have
gbl
fgb
= 0.96! We should thus expect the rod morphology and the plate morphology
to produce very different rates of change of behavior as the grain size is decreased.
The sensitivity to morphology is of course related to another specific issue, which
is that the grain size is not a well-defined quantity for highly asymmetric structures.
What single grain size number should be used for a long rod shaped crystal? Should
it be the diameter of the rod or the length of the rod? One possibility, sometimes used
in the literature, is to define the grain size as the size of the cube of identical volume
to the grain in the actual polycrystal. In this way of thinking, the effective grain size
d to be used in considering the space filling hexagonal prism morphology shown in
√
1
3
Figure 4.5 would be given by d = ( 3 2 3 α ) s. This approach is useful if one needs to
use a single number to define the grain size in the material. For many applications,
however, it is more useful to simply remember that perhaps two effective grain sizes
need to be used.
For the space filling hexagonal morphology, the volume of each triple junction is t 2 h, and so the volume fraction ft j of triple junctions in the polycrystalline
material is
4 Mechanical Properties: Density and Elasticity
103
1
Hexagonal Prism
Morphology
fsc
fgb
Volume fraction
0.8
ftj
Grain
boundaries
Grains
fcj
0.6
0.4
0.2
Triple
junctions
Corner
junctions
0
0.1
1
10
100
1000
Normalized grain size d/t
Fig. 4.6 Variation of grain volume fraction, grain boundary volume fraction, triple junction volume
fraction, and corner junction volume fraction with normalized grain size β = dt for the hexagonal
prism grain morphology of Figure 4.5 (with an aspect ratio α = 1). The junction volume fractions
become major contributors when d ≈ t.
ft j |hex =
2t 2 h
√
.
2
3 3
√t
2 (s + 3 ) (h + t)
(4.18)
The volume fraction of the corner junctions is given by
fc j |hex =
1
36(s +
√t
2
3
) (h + t)
.
(4.19)
The variation of these volume fractions with grain size for the hexagonal morphology is presented in Figure 4.6. The difference between this variation and that
for the cube morphology in Figure 4.3 is considerable, particularly in terms of the
importance of the triple and corner junctions (these become important at smaller
grain sizes in this hexagonal case).
The hexagonal prism morphology affords us the opportunity to also study the
effects of grain aspect ratio (α ). The variation of the several volume fractions with
grain size are presented in Figure 4.7 for aspect ratios of 10 (rods) and 0.1 (plates).
The aspect ratio clearly has a strong influence, particularly in determining the grain
boundary limit and the relative importance of the triple junctions.
4.1.3 Density as a Function of Grain Size
What is the net effect of these variations in volume fraction of grains, grain boundaries and junctions on the variation of the density of a material as a function of grain
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Nanomaterials
1
Hexagonal, α = 10
fsc
fgb
ftj
fcj
Volume Fraction
0.8
0.6
0.4
0.2
0
0.1
1
10
100
1000
Normalized Grain Size, d/t
1
Hexagonal, α = 0.1
fsc
fgb
ftj
fcj
Volume Fraction
0.8
0.6
0.4
0.2
0
0.1
1
10
100
1000
Normalized Grain Size, d/t
Fig. 4.7 Variation of grain volume fraction, grain boundary volume fraction, triple junction volume
fraction, and corner junction volume fraction with normalized grain size β = dt for the hexagonal
prism grain morphology of Figure 4.5, with two aspect ratios (10 and 0.1) corresponding to rods
(top) and plates (bottom).
size? The density of the polycrystalline material is given by Equation (4.16). The
density of the material near the grain boundaries and at the triple junctions is generally not known, because there is a very wide range of atomic arrangements possible
at the boundaries (some of these are discussed later in this chapter). To visualize
the possibilities, let us assume that the density near the grain boundaries is 95% of
the density ρsc within the single crystal, that at the triple junctions is 90% ρsc , and
that at the corner junctions is 81% ρsc . The resulting variation of the density with
grain size is presented in Figure 4.8 for the cube and hexagonal prism morphologies
(with an aspect ratio α = 10). The overall density of the material is then predicted
(using Equation (4.16)) to decrease substantially as the grain size of the polycrys-
4 Mechanical Properties: Density and Elasticity
105
1
Overall Density
0.95
0.9
0.85
Cube Morphology
Hex Prism Morphology
0.8
0.1
1
10
100
1000
Normalized Grain Size, d/t
Fig. 4.8 Predicted variation of overall density of polycrystalline material (normalized by ρsc ) with
grain size for the cube and hexagonal prism morphologies (with an aspect ratio α = 10). The
assumed parameters are ρgb = 0.95ρsc , ρt j = 0.9ρsc , and ρc j = 0.81ρsc .
tals decreases below 50 nm or so for these assumed parameters. There is negligible
change once the polycrystal size increases beyond perhaps 100t. Thus the apparent
density of nanocrystalline materials should be expected to be somewhat lower than
that of conventional large-grain polycrystalline versions, and this may account for
some of the situations where lower densities are observed but no pores are visible
even in the TEM (e.g., Jia et al., 2003).
4.1.4 Summary: Density as an Example Property
In summary, this discussion of the scalar property which is the density of the material establishes several issues that are important with respect to a change in the
properties of nanocrystalline materials:
• There must be an internal length scale in the material (in this case the grain
boundary thickness).
• The properties of one part of the material (here the grain boundaries) should be
substantially different from the properties of another part.
• A first order estimate of the change in a property can be obtained using the rule
of mixtures.
• The morphology of the polycrystalline material, and in general the microstructure, can have a strong influence on the effective property.
• The presence of defects can dominate the behavior, and so characterization of defects is important. Processing issues often enter into defining the effective properties through the defects that the process generates.
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Nanomaterials
4.2 The Elasticity of Nanomaterials
One of the simplest mechanical behaviors of materials is elasticity (see Chapter 2).
The simplest case of elasticity is that for a homogeneous isotropic material; that is,
for a material in which the properties do not change either with position (homogeneous) or with direction (isotropic). As demonstrated in the chapter on mechanics,
this results in the definition of two material properties, which could be chosen to
be the Lamé moduli, or the bulk modulus and the shear modulus, or the Young’s
modulus and the Poisson’s ratio.
4.2.1 The Physical Basis of Elasticity
It is necessary here to understand the fundamental physics associated with elasticity. The elastic response of materials essentially arises from the interatomic forces
within the material. Consider the interatomic potential U(r) that defines the interaction of two atoms separated by a distance r, usually called a pair potential. The
shape of this interatomic potential for most solids is similar to that presented in
Figure 4.9. The energy has a minimum at a position r0 , which corresponds to the
equilibrium spacing of the atoms (e.g., in a crystal). By definition, the force between
the two atoms is given by
∂U
F(r) =
,
(4.20)
∂r
Energy, U(r)
and so of course is zero at the minimum energy. The stiffness of the interatomic
bond represents the elasticity of the system, and is given by
r
Distance, r
Fig. 4.9 Typical interatomic pair potential U(r), showing the equilibrium position r0 of the atoms.
The curvature at the bottom of the potential well (at the equilibrium position) corresponds to the
effective elastic stiffness of the bond.
4 Mechanical Properties: Density and Elasticity
107
∂F
∂ 2U
;
(4.21)
=
∂r
∂ r2
i.e., the elastic stiffness is the curvature of the interatomic potential (Ashby and
Jones, 1996). Different kinds of interatomic interactions lead to different elastic
stiffnesses. For example, the interatomic potential that represents a covalent bond
is significantly stiffer than the interatomic potential that represents a metallic bond
(this is why ceramics are stiffer than metals).
Generating a strain in the material amounts to moving the atoms either closer together or further apart, that is, moving the atoms away from the equilibrium position
(which can be maintained with no applied force). Thus, when no force is applied,
all of the atoms in the material are sitting in their equilibrium positions (although
thermal vibrations about the equilibrium position will occur). In a crystalline material, these equilibrium positions define the lattice spacing. The lattice spacing may
be different in different directions, corresponding to the various symmetries of crystals. In elastic deformations, when a force is applied, all of the atoms move away
from their equilibrium positions into non-equilibrium positions corresponding to the
magnitude of the applied force (resulting in a strain). These nonequilibrium strained
positions correspond to a higher energy state (see Figure 4.9), and we say that the
material has stored a certain amount of elastic strain energy. When the applied force
is released, the stored energy is recovered as the atoms move back to their equilibrium locations. The idea that the elastic modulus is a constant (i.e., does not depend
on strain – this defines linear elasticity) amounts to assuming constant curvature of
the interatomic potential, which is typically only reasonable near the minimum energy (in effect, this amounts to assuming that the potential can be approximated by
a quadratic function at the equilibrium spacing of the atoms). Nonlinear elastic approximations are also possible, although rarely used for metals (except in molecular
dynamics).
E(r) =
4.2.2 Elasticity of Discrete Nanomaterials
Are these elastic moduli different for nanomaterials than for conventional materials?
Remembering our three classes of nanomaterials (discrete nanomaterials, nano device materials, and bulk nanomaterials), we see that this question is not well posed
in the case of discrete nano materials, because there is no “conventional” version
of discrete nanomaterials. In this case, a different question must be asked: do the
elastic properties change with the size of the particle (the 0D case) or fiber (the 1D
case)?
4.2.2.1 Nanoparticles
Most nanoparticles are essentially very small single crystals. On the basis of the
discussion above, there is no reason for the moduli of the crystal to depend on
the size of the crystal, provided the interatomic potential does not change and the
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Nanomaterials
equilibrium positions of the atoms do not change with crystal size. However, there
is reason to expect that the equilibrium positions of the atoms would be different in
free crystal particles of different sizes. The argument goes as follows. The atoms at
the surface of a nanoparticle (indeed at any free surface) perceive an environment
that is distinctly different from that perceived by atoms within the bulk of the crystal (since there are no atoms to interact with on the outside of the free surface). The
result is an effective surface tension on the surface of these particles, which implies
that the atoms within the particles are under an effective force (Figure 4.10). As
a consequence, the atoms within the nanoparticles should have a different equilibrium spacing than the atoms within bulk single crystals. We discuss this further in
Section 7.1.1 in the chapter on scale-dominant mechanisms.
Fig. 4.10 Surface effects on a spherical nanoparticle (e.g., surface tension). Such effects can have
a significant impact on the behavior of the nanoparticle, particularly with respect to its interaction
with the environment.
A continuum mechanics calculation (see Section 7.1.1) based on spherical nanoparticles indicates that there should be a significant change in the lattice spacing for
nanoparticles that are 5 nm or less in size (Huang et al., 2007). This change in the
lattice spacing is indeed observed experimentally in the case of palladium (Lamber
et al., 1995). The measured reduction in lattice spacing is of the order of 3%. This
transition size of course depends on the magnitude of the surface tension for a given
material. Particles larger than the transition size are not significantly affected by the
surface tension.
We note here that these continuum calculations essentially assumed a constant
modulus and computed a change in the equilibrium lattice spacing with particle size,
rather than computing a change in the modulus with particle size. It is extremely
difficult to measure the modulus of a given nanoparticle while it is relatively easy to
measure the lattice spacing. The effect of the decreased size in terms of the increased
importance of the surface tension essentially amounts to computing the solution
of an equilibrium elasticity problem on each particle, resulting in a change in the
4 Mechanical Properties: Density and Elasticity
109
particle configuration and a change in the material properties (discussed in a later
chapter).
There is a very different calculation involved with the effect of the insertion of
nanoparticles into a polymer matrix to develop a nanocomposite of increased stiffness, and we will address this calculation in a later chapter. Another important topic
is that of the properties of functionalized nanoparticles, and we will discuss this as
well in Chapter 7.
4.2.2.2 Nanotubes, Nanofibers, and Nanowires
The elastic response of nanotubes, nanofibers, and nanowires is generally dominated
by the stiffness of a specific structure rather than by a specific interatomic interaction. For example, the apparent elastic modulus of a nanotube depends not only on
the specific arrangement of atoms within the nanotube, which depends on the interatomic potential, but also on the geometry of the structure (e.g., the diameter of the
nanotube, whether it is a singlewall or multiwall nanotube, and the distribution of
defects within the nanotube wall).
Unlike the case of nanoparticles, the elastic moduli associated with onedimensional discrete nanomaterials is of significant technical interest. Such tubes,
fibers, and wires are often used as reinforcing elements within composite structures,
and their high elastic stiffness is critical for successful reinforcement. The measured
effective Young’s moduli of a variety of one-dimensional discrete nanomaterials
is presented in Table 4.1. Size dependence is often observed, which is sometimes
attributed to defects and sometimes to surface effects (Ma and Xua, 2006). Note
that these one-dimensional materials have essentially zero stiffness in compression,
since they generally buckle under compressive loading. This is another sense in
which these discrete nanomaterials should be viewed as structures rather than as
materials alone. The stiffness of carbon nanotubes is a topic of great interest, and is
discussed in detail in Chapter 8.
Table 4.1 Some measured moduli of nanotubes, nanofibers, and nanowires, viewed as 1D nanomaterials.
Material
Single-wall carbon nanotube
Multi-wall carbon nanotube
ZnO nanowires
Silica nanowires
Modulus, GPa
1002
11–63
140–200, depending on size
20–100
Reference
(Yu et al., 2000a)
(Yu et al., 2000a)
(Chen et al., 2006)
(Silva et al., 2006)
4.2.2.3 Nanobelts
Nanobelts are thin strips of material, usually a few nanometers thick and a few tens
of nanometers in width, typically made of oxides or other compounds. The belts
have both a tensile Young’s modulus, and a flexural modulus, but as with nanowires
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Nanomaterials
and nanotubes, they have essentially zero stiffness in compression. The elastic response of nanobelts is also a structural response rather than a material response
alone. There is very little data at present on these materials. The oxide nanobelts
typically involve covalent atomic bonds, and so the intrinsic stiffness of the material is very high. A good review of the properties of nanobelts is provided by Wang
(2004).
4.2.3 Elasticity of NanoDevice Materials
The largest fraction of nanodevice materials are thin films; a few are nanowires
or nanotubes. A schematic of a thin film structure is shown in Figure 4.11. The
elastic moduli associated with thin films can be substantially different from that of
the bulk material, for a variety of reasons discussed in Chapter 8. Thin films can
be either single crystal or polycrystalline, depending on how they are grown. The
polycrystalline thin films will be affected by the behavior of grain boundaries, but
this will not be an issue for single crystal films, so we shall consider single crystal
films first.
Substrate
Fig. 4.11 Schematic of a thin film on a substrate. The film thicknesses of interest to industry are
typically submicron.
Consider a single crystal thin film of thickness h f . If h f is sufficiently large, the
elastic behavior of the thin film material will simply be that of the bulk material.
As the thickness decreases, the fraction of atoms that feel the free surface and interface conditions increases, and the overall behavior of the thin film will become
perceptibly different. When the behavior of interest is elasticity, at what thickness
do the surface and interface effects become significant? A good model for the elasticity of thin films should predict the transition thickness. The concepts of surface
stress and surface energy have both been used to analyze the elasticity of thin-film
structures. The thin film essentially has one free surface (two free surfaces if it is
a freestanding film), and the atoms on the free surface have different equilibrium
positions than in the bulk. The atoms at the interface with the substrate also have
positions that are different from the equilibrium positions in the bulk, because they
4 Mechanical Properties: Density and Elasticity
111
Fig. 4.12 Schematic of a polycrystalline thin film on a substrate. Note the typical columnar microstructure.
must interact with the different atoms of the substrate. An excellent description of
the effects of surfaces is presented by Dingreville et al. (Dingreville et al., 2005).
The stiffness of a thin film is often difficult to measure, because freestanding thin
films are difficult to generate, so the quality of the dataset on thin films is sometimes
contested. Nanoindentation can be used to measure elastic response for thin films on
a substrate, but even this technique fails when h becomes of the order of 2–3 times
the indentation depth. Microtension tests on freestanding thin films have recently
been performed for a small number of films, almost all of them polycrystalline rather
than single crystal, e.g., Jonnalagadda et al. (2008).
Polycrystalline thin films provide us with an excellent opportunity to understand
the effects of size scale and microstructure on elastic behavior. In terms of elastic response, single crystals are anisotropic, with up to 21 elastic moduli Cijkl (or
CIJ in the Voigt notation) as described in Chapter 2. For the purposes of modeling, we assume that the elastic behavior of the grain boundary is isotropic, and
therefore described by two elastic moduli, e.g., the Young’s modulus Egb and the
Poisson’s ratio νgb . A polycrystalline thin film is thus a composite (Figure 4.12)
made up of an anisotropic elastic phase (the single crystals) and an isotropic elastic phase (the grain boundaries). In general, therefore, the polycrystalline thin film
will be elastically anisotropic, with specific effective moduli that depend on the arrangement of the grains as well as the volume fractions of the two phases. Calculating the effective anisotropic moduli of the thin film can involve a fairly complex
computation.
4.3 Composites and Homogenization Theory
How does one define the effective moduli of a material like that shown in Figure 4.12? The elastic moduli reflect a linear relationship between the stress and the
strain in the material, as in Equation (2.65), and the constitutive law (material behavior) of linear elasticity is defined at every point. This is a problem, however, in
composite materials, because any given point in the material might be in one phase
112
Nanomaterials
(a)
(b)
Fig. 4.13 Process of homogenization of a composite material. (a) Original heterogeneous material.
(b) Equivalent homogenized material.
or the other (in this case either in the crystal or within a grain boundary region).
Thus, even to define a constitutive law like linear elasticity for a composite material
like this polycrystal involves replacing the real heterogeneous material (with grains
and grain boundaries) with an equivalent homogenized material which has material
properties that in some specified way are related to the properties of the original
heterogeneous material. This process of homogenization is shown schematically in
Figure 4.13. The behavior of the homogenized material is now defined at every point
as a relationship between the overall stress and overall strain in the homogenized
material. The overall stress σ i j in the homogenized material is defined as a volume
average stress over a representative volume element or RVE (in the heterogeneous
material), which is a volume element that is sufficiently large that it captures the
effective behavior of the composite. Mathematically, the definition of the overall
stress is
1
σij =
σi j dv,
(4.22)
VR
where VR is the volume of the RVE. Similarly, the overall strain is defined by the
volume average of the strain over a representative volume element:
1
εi j =
VR
εi j dv.
(4.23)
Note that the stress and strain components in the two phases within the RVE
will vary from point to point, but the overall stress and strain computed using Equations (4.22) and (4.23) are defined at a point corresponding to the center of the RVE.
If the RVE is defined by an effective dimension (say a radius) rR , we cannot in such
a homogenized approach usefully examine behavior at length scales smaller than
rR . Thus the process of homogenization introduces a modeling length scale into the
problem, and this should ideally be associated with a microstructural length scale in
the heterogeneous material. Now that the stress and strain in the homogenized material are known by computation from those in the composite, the effective elastic
properties Cijkl of the homogenized composite can be defined through the relation
σ i j = Cijkl ε kl .
(4.24)
4 Mechanical Properties: Density and Elasticity
113
Using Equation (4.22) and the definition of the elastic modulus σi j = Cijmn εmn at
each point in the heterogeneous material, we have
Cijkl ε kl =
1
VR
Cijmn εmn dv.
(4.25)
Because the elastic moduli Cijmn of each phase vary from point to point, it is
in general not possible to pull these moduli out of the integral. Equation (4.25)
therefore provides a nice formal way to define the effective elastic properties but
it is not easy to compute Cijkl . In some special cases, however, this can be done
explicitly, and we demonstrate this next.
4.3.1 Simple Bounds for Composites, Applied to Thin Films
Consider the simpler idealized case of a polycrystalline thin film consisting of
isotropic elastic grains (with elastic modulus Eg and Poisson’s Ratio νg ) with
isotropic elastic grain boundaries (with elastic modulus Egb and Poisson’s Ratio
νgb ). Even though both of the constituent phases are isotropic, it is possible for
the composite material to be anisotropic. For example, consider one possible microstructure (Figure 4.14a), where the grains are all grown perpendicular to the
plane of the film (this is called a columnar microstructure, and is fairly common
in polycrystalline thin films). The effective elastic properties of this thin film are
different in the directions perpendicular to the film plane and parallel to the film
plane, so that this film is effectively anisotropic. A simple calculation of these effective properties follows.
Let us idealize the thin film as the two-dimensional microstructure shown in
Figure 4.14a, corresponding to grains that are infinitely thick perpendicular to the
plane of the paper. Imagine that the thin film is being pulled along the film plane by
a tensile force P. The stress in the film plane is then uniform, and is simply given by
σ = PA , where A is the cross-section area of the film. That is
Fig. 4.14 (a) Schematic of a columnar microstructure for a thin film. (b) Schematic of a layered
microstructure, which can be viewed as the columnar microstructure loaded in the orthogonal
direction to that shown in (a).
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Nanomaterials
σtotal = σg = σgb =
P
,
A
(4.26)
while the total strain in the film is the sum of the strains in the grains and the grain
boundaries in the material. Given the volume fractions in the material, we have
εtotal =
1
V
ε dv = fg εg + fgb εgb
(4.27)
using an argument identical to that in Equation (4.3). Using the definitions of the
Young’s modulus of the grains and the grain boundaries, we have
εtotal = fg
But by definition we have εtotal =
σgb
σg
+ fgb
.
Eg
Egb
σtotal
,
E inplane
(4.28)
so that we have
σgb
σg
σtotal
= fg + fgb
,
Eg
Egb
E inplane
(4.29)
from which it follows that (using Equation 4.26)
1
1
1
= fg + fgb
.
Eg
Egb
E inplane
(4.30)
This last equation represents the rule of mixtures for elastic compliance. The
compliance is the inverse (and in this isotropic case, the reciprocal) of the elastic
modulus. Solution of this equation provides the effective modulus of the thin film in
the in-plane direction.
If, on the other hand, we consider the elastic properties of the thin film in the
direction perpendicular to the thickness of the film (putting aside for the moment
the physical difficulty of carrying out such an experiment), the mechanical loading
situation is essentially that corresponding to Figure 4.14b. Now the displacements
of the grain and grain boundary phases are the same, and since they have the same
initial lengths, the strains in the grain and grain boundary phases are the same. It
follows that now we have
εtotal = εg = εgb =
δh
,
h
(4.31)
while the total stress in the film is the sum of the stresses in the grains and the grain
boundaries in the material. Given the volume fractions in the material, we have
σtotal =
1
V
σ dv = fg σg + fgb σgb .
(4.32)
Again using the definitions of the elastic moduli, we have
E outofplane εtotal = fg Eg εg + fgb Egb εgb ,
(4.33)
4 Mechanical Properties: Density and Elasticity
115
from which it follows that (using Equation 4.31)
E outofplane = fg Eg + fgb Egb .
(4.34)
This last equation represents the rule of mixtures for the elastic modulus directly,
and provides the effective modulus of the thin film in the out-of-plane direction.
This out-of-plane effective Young’s modulus is clearly different from the in-plane
effective Young’s modulus, and thus the thin film behaves in an anisotropic manner
even though it is composed of two isotropic components (the grains and the grain
boundaries).
Real thin film microstructures may not be well-represented by the idealized case
in Figure 4.14. However, it is possible to show that any other microstructure for the
material will result in effective moduli that lie between the two bounds represented
by Equations (4.30) and (4.34) for linear elastic materials. Thus, all polycrystalline
linear elastic solids (that do not contain pores) will have moduli between these two
bounds, which are represented in Figure 4.15 in terms of variation with volume
fraction of grain boundary.
However, there are two big caveats. First, in a composite model such as that
shown in Figure 4.14, one makes an assumption about the behavior of the interface
between the two materials (typically, the assumption is that the interface is perfect).
This assumption seems trivial here, because we do not have a conception of an interface between a grain and a grain boundary. In general composite mechanics terms,
however, the interface behavior may be a dominant issue. For example, in the simple case that the interface between the two materials sustains no stress, the material
in Figure 4.14a will carry no load, while the material in Figure 4.14b will carry
load, so that the behaviors are dramatically different from the behaviors shown in
Figure 4.15. Second, one typically assumes that the properties of the individual
phases are independent of size – only the volume fraction shows up in Equation (4.34), without a length scale.
75
75
Overall Modulus, GPa
ESC
70
70
65
65
E1 = fgbEgb + (1–fgb)Esc
60
60
1/E2 = fgb /Egb + (1–fgb)/Esc
55
55
50
Egb
50
45
0
0.2
0.4
0.6
0.8
1
45
Volume fraction of grain boundary, fgb
Fig. 4.15 Variation of effective modulus (for fictitious material) with grain boundary volume fraction, based on Equations (4.30) and (4.34).
116
Nanomaterials
4.3.1.1 Size Effects in Thin Films
The primary cause of the size effects that we have been discussing in this polycrystalline material is the lengthscale associated with the thickness t of the grain
boundary domain. This lengthscale will exist in ANY polycrystalline material and
is not specific to thin films. The size scale that is specific to thin films is the film
thickness h f . As discussed earlier, even single crystal thin films have elastic properties that are a function of film thickness. In the case of polycrystalline thin films,
both film thickness effects and grain size effects will exist. As the grain size decreases from ∞ (corresponding to the single crystal), the fraction fgb of the thin film
material that is in the grain boundary domain will increase, while the fraction fsurf
of the material that is in the surface-affected domain will remain constant. At some
specific grain size, these two volume fractions will become comparable.
4.3.2 Summary of Composite Concepts
There are several important concepts that should be extracted from the previous
discussion on polycrystalline materials:
• First, even when we are talking about a scalar parameter (the Young’s modulus),
the issue of interest is that of the mechanical behavior in tension, which has a
principal direction (the tensile axis). The film with the columnar microstructure
of Figure 4.14 has two very different behaviors in the two tensile directions (inplane and out-of-plane). This is because the material microstructure itself has a
directionality (e.g., the direction perpendicular to the grain boundary planes in
Figure 4.14a, b), and the anisotropy (directionality) of the mechanical response
follows from the relationship between the tensile axis and the normal to the grain
boundary planes. The directionality of the arrangement of two individual phases
(the grain boundary and the grains) results in this anisotropy, even though each
of the phases is itself isotropic in mechanical response. This is one of the fundamental concepts of composite mechanics.
• Second, most nanomaterials will not have the nice regular microstructure shown
in Figure 4.11. In general, the arrangement of the phases is not periodic in any
particular direction. For the general microstructure, we cannot easily extract effective moduli. However, composite mechanics theory places bounds on the moduli, as exemplified by Equations (4.30) and (4.34). Any given sample of material
will have moduli that lie within those bounds. This concept of bounds on the
properties is also a fundamental concept of composite mechanics.
• Third, specific assumptions (related to the microstructures) had to be made to
obtain the effective response of the nanomaterial. In one case it was assumed that
the strains were uniform (i.e., the same in both grain and grain boundary), and
in the other that the stresses were uniform. These assumptions are idealizations
that lead to specific models, corresponding to Equations (4.30) and (4.34). The
4 Mechanical Properties: Density and Elasticity
117
development of models is always based on the assumption of specific idealizations, and later, specific mechanisms.
• Fourth, we are NOT assuming that there is a different phase of the material at the
grain boundary (as does exist, for example, in some sintered ceramics). Rather,
the assumption is that the behavior of material in a finite thickness domain at the
grain boundary is different from that of the bulk material. This is a point that
causes some confusion in the literature. The fact that a different crystallographic
or amorphous phase cannot be identified at the grain boundary does NOT imply
that the models described above cannot be used – all that is needed is a different
response at the grain boundary (in this case a lower apparent stiffness), which
may arise merely from the increased disorder or looser packing at a boundary.
4.4 Elasticity of Bulk Nanomaterials
Bulk nanomaterials are either nanocrystalline or nanostructured materials, as described in Chapter 1. Much of our discussion in the previous section applies directly to bulk nanocrystalline materials, and in particular, Equations (4.30) and
(4.34) describe the bounds on the elastic properties that will be obtained (see also
Figure 4.15). The predicted variation of the effective modulus of the material with
grain size is presented in Figure 4.16.
Normalized Modulus E /Esc
1
0.8
0.6
0.4
0.2
0
0.1
E1
Cube
Morphology
1
E2
10
100
1000
Normalized grain size d/t
Fig. 4.16 Variation of effective Young’s modulus with normalized grain size in a bulk nanocrystalline material, where the modulus in the grain boundary domain is defined to be ζ Eg = ζ Esc with
ζ = 0.7 in this figure. The grain size is normalized by the effective thickness t of the grain boundary domain, typically assumed to be about 1 nm. If the latter thickness is assumed, the horizontal
axis corresponds to grain size in nm. E1 and E2 correspond to Equations 4.30 and 4.34.
118
Nanomaterials
Note that in either case (for either bound), a 5% drop in the effective modulus is
predicted to occur only when the grain size falls below about 20t, and for t ≈ 1 nm
this implies grain sizes less than about 20 nm. The behavior is essentially that of
the grain boundary once d ≈ t. There is very little reduction in modulus expected
for grain sizes in the range d > 50 nm (although, for the reasons pointed out in the
discussion on density effects, this depends on the assumed grain morphology).
However, experimental measurements of the moduli of nanocrystalline materials
often show values substantially smaller than those of their conventional grain size
counterparts (Shen et al., 1995). This suggests that some component of these materials has not been included in the modeling. Examination of the literature shows that
the measured moduli are relatively insensitive to grain size in the nanocrystalline
range, and change considerably with the route of synthesis of the nanomaterial. It
is likely, therefore, that the observed reduction in modulus is due to defects such as
porosity generated during the synthesis of the material (Shen et al., 1995). We will
return to this issue of the effects of defects and porosity on the mechanical properties
of materials in detail in a later section.
4.5 Suggestions for Further Reading
1. R.M. Christensen, Mechanics of Composite Materials, John Wiley, New York,
1979.
2. J.-M. Berthelot, Composite Materials: Mechanical Behavior and Structural
Analysis, Springer, New York, 1999.
3. R. Dingreville, J.M. Qu, and M. Cherkaoui, Surface free energy and its effect
on the elastic behavior of nano-sized particles, wires and films, Journal of the
Mechanics and Physics of Solids, 53, 8, 1827–1854, 2005.
4. R. Cammarata, Surface and interface stress effects in thin films. Progress in Surface Science, 46, 1, 1994.
4.6 Problems and Directions for Research
1. Consider a nanocrystalline nickel material. Assuming that the grain boundary
region has 90% of the Young’s modulus of the interior of the single crystal, and
assuming a 1 nm thickness for the affected layer, compute the apparent Young’s
modulus of nanocrystalline gold with a grain size of 20 nm and cube-morphology
grains. How will the modulus change if the material has 2% porosity?
2. Solve the identical problem, but this time assume a hex-prism morphology with
an aspect ratio of 5.
3. Put together a table of the elastic moduli of bulk nanocrystalline materials, using
only experimental data from the literature. How do these numbers compare with
the bulk response?
4 Mechanical Properties: Density and Elasticity
119
References
Ashby, M. F. and D. Jones (1996). Engineering Materials I (Second ed.), Volume 1. ButterworthHeinemann.
Chen, C. Q., Y. Shi, Y. S. Zhang, J. Zhu, and Y. J. Yan (2006). Size dependence of young’s modulus
in zno nanowires. Physical Review Letters 96.
Dingreville, R., J. Qu, and M. Cherkaoui (2005). Surface free energy and its effect on the elastic
behavior of nano-sized particles, wires and films. Journal of the Mechanics and Physics of
Solids 53(8), 1827–1854.
Huang, Z. P., P. Thomson, and S. Di (2007). Lattice contractions of a nanoparticle due to the
surface tension: A model of elasticity. Journal of Physics and Chemistry of Solids 68, 530–535.
Jonnalagadda, K., N. Karanjgaokar, I. Chasiotis, A. Mahmood, and D. Peroulis (2008). Strain rate,
thickness, and processing effects on the mechanical behavior of nanocyrstalline au films. Acta
Materialia to be submitted.
Lamber, R., S. Wetjen, and N. Jaeger (1995). Size dependence of the lattice parameter of small
palladium particles. Physical Review B 51, 10968–10971.
Latapie, A. and D. Farkas (2003). Effect of grain size on the elastic properties of nanocrystalline
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Ma, F. and K. Xua (2006). Size-dependent theoretical tensile strength and other mechanical properties of [001] oriented au, ag, and cu nanowires. Journal of Materials Research 21(11), 2810–
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Shen, T. D., C. C. Koch, T. Y. Tsui, and G. M. Pharr (1995). On the elastic moduli of nanocrystalline fe, cu, ni and cu-ni alloys prepared by mechanical milling/alloying. Journal of Materials
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Silva, E. C. C. M., L. Tong, S. Yip, and K. J. V. Vliet (2006). Size effects on the stiffness of silica
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Yu, M. F., B. S. Files, S. Arepalli, and R. S. Ruoff (2000a). Tensile loading of ropes of single wall
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breaking mechanism of multiwalled carbon nanotubes under tensile load. Science 287(5453),
637–640.
5
Be faithful in small things, because it is in them
that your strength lies.
Mother Theresa
Plastic Deformation of
Nanomaterials
This chapter discusses the much more complex mechanical behaviors associated
with inelastic deformations. Although the grain size, volume fraction and composite
approaches discussed in the last chapter are still relevant, the interactions between
the size scales and the deformation mechanisms involved in plasticity do not always
allow the simple use of continuum approximations to predict the overall behavior.
The chapter will first use a materials approach to discuss mechanisms, and then
attempt to link the mechanics to these mechanisms and thus define the associated
size scales.
5.1 Continuum Descriptions of Plastic Behavior
Once the stress (or strain) in a material has increased to the elastic limit, plastic
behavior or fracture occurs. Plasticity is a behavior that is generally only observed in
metals rather than in ceramics or polymers (except under exceptional circumstances
such as very high temperatures or pressures), and so we focus on bulk nanometals
in this chapter. The plastic behavior of nanomaterials is not generally relevant to the
applications of discrete nanomaterials and nanodevice materials. We focus therefore
on the plasticity of bulk nanomaterials.
There are essentially four parameters that describe the simple plastic behavior of
a material (assumed isotropic, and following the typical von Mises type of behavior,
see Chapter 2.):
• The yield strength σy
ε ∂σ
σ ∂ε
σ
ε̇ ∂ σ
strain rate hardening index m, where m = ∂∂ ln
ln ε̇ = σ ∂ ε̇
σ
thermal softening coefficient ν , where ν = ∂∂ ln
ln θ , with
• The strain hardening index n, where n =
• The
∂ ln σ
∂ ln ε
=
• The
temperature
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 5,
c Springer Science+Business Media, LLC 2009
θ being the absolute
121
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Nanomaterials
All of these are generally known for conventional grain size metals. It is known
that all four parameters are strong functions of the homologous temperature Th = TTm
(where Tm is the melting temperature), but for the purposes of this discussion let us
limit ourselves to considering low homologous temperatures (Th ≤ 0.3). The early
work on nanomaterials was confined to evaluating the yield strength, and indeed,
much of the early apparent promise of structural nanomaterials was in the development of very high strengths at small grain sizes. The other three parameters are
more difficult to measure, since they relate to the derivatives of the strength with
respect to the strain (in the case of n), the strain rate (m), and the temperature (ν ). It
is only since 2004 or so that attention has become focused on the strain hardening
and strain rate hardening behaviors (in part because synthesis and characterization
techniques have achieved sufficient reliability to make such evaluations possible).
There is as yet little data on the thermal softening of nanomaterials, largely because
of the competition with grain growth mechanisms. In this chapter, we will address
each of these parameters in turn.
5.2 The Physical Basis of Yield Strength
By definition, the onset of plastic deformation in metals is associated with the permanent rearrangement of atoms and atomic bonds (because irreversible deformations must result). The physical basis of the yield strength is thus completely different from the physical basis of the elastic modulus. What is the stress required to
cause a permanent rearrangement of the atoms of a given crystalline metal?
As with most questions involving irreversible deformations, there are many ways
of thinking about this question from a physical viewpoint. First, of course, we must
define what “permanent rearrangement of the atoms” means. Consider the perfect
crystal shown in Figure 5.1a and subjected to a shearing stress as shown. If we imag-
Fig. 5.1 Schematic of slip in a crystalline solid under shear. (a) Original crystal subjected to shear
stress. (b) Crystal after deformation (slip) along the shearing plane shown in (a). The final atomic
arrangement within the deformed crystal remains that of the perfect crystal, except near the free
surface.
5 Plastic Deformation of Nanomaterials
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ine the specific permanent rearrangement consisting of the sliding of one layer of
atoms over another along the dashed line in the figure, we see that this rearrangement involves the motion of the top half of the crystal over the bottom half. At the
end of the sliding process, the deformed crystal (Figure 5.1b) has two ledges on
the free surfaces corresponding to the sliding. Within the crystal however, even after deformation, the atomic arrangement looks identical to that in a perfect crystal.
Thus merely examining the interior of a crystal after deformation does not allow
one to determine whether a permanent rearrangement of the atoms has occurred. In
this case the signature of the permanent rearrangement is actually at the surfaces of
the crystal. Another signature of the permanent rearrangement is that a significant
quantity of non-recoverable work must have been done in order to accomplish the
sliding.
We see here the two key signatures of plastic deformation. First, the observable
permanent deformations are macroscopic, not at the atomic scale; the net result is a
continuum permanent strain. Second, plastic deformation involves the expenditure
of energy, known as the work of plastic dissipation. The two classical ways of examining plastic deformation therefore involve the examination of the development
of plastic strain, and the examination of the development of plastic dissipation. The
former involves a close examination of the kinematics of the deformation process,
and the latter involves a close examination of the energetics of the process.
How does the crystal actually go from configuration (a) to configuration (b) in
Figure 5.1? It turns out to be energetically unfavorable to move an entire half of
a crystal over the other half (put another way, the stress required to perform such
µ
where µ is the shear
an operation would be prohibitively large, of the order of 10
modulus of the material (Asaro and Lubarda, 2006), whereas the measured yield
strength of most metals is of the order of 10−3 µ ). It is far cheaper, in terms of energy,
to cause the overall sliding of the top over the bottom (Figure 5.1) by moving small
defects along the slip plane until the entire block has slid across (Figure 5.3). The
nearest everyday analogy is that it is easier to move a large carpet a short distance by
generating a ripple in the carpet and then moving the ripple across the carpet, rather
than moving the entire carpet at one time. The corresponding ‘ripples’ in the crystal
are line defects (i.e., defects in which the atomic arrangement deviates from the
perfect crystal along a line) called dislocations, and were first hypothesized in the
1920s by G.I. Taylor. The existence of such line defects has since been established
through direct visualization in a transmission electron microscope (Figure 5.2).
Figure 5.3 shows a schematic of one such line defect, called an edge dislocation,
which amounts to an extra plane of atoms in the otherwise perfect lattice (the line at
the bottom of the extra plane is essentially the line defect). The motion of this line
defect is energetically favorable to the motion of the entire block of atoms above
the slip plane at one time. This turns out to be a general principle: the inelastic
deformation of crystalline metals is controlled by the motion of defects, because
this motion is energetically much cheaper than large scale simultaneous breaking
and rearranging of atomic bonds.
Almost every crystal contains defects, many of which are developed during the
process of nucleation and growth of the crystal. The defects in crystals occur at four
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Fig. 5.2 Dislocations are visible in the transmission electron microscope. The picture shows dislocations (the lines marked by the black arrow) in a magnesium alloy (ZK60). Note the length scale
in the picture. This micrograph was taken by Bin Li.
levels of dimensionality, all of which are of interest to the mechanics of nanomaterials:
• 0D or point defects are either atoms missing from the lattice (therefore called
vacancies), atoms of the wrong type in a given lattice position (substitutional
defects) or atoms that are in located in the interstices between normal lattice
positions (interstitial defects). Such point defects play a very important role in
the high temperature deformation of materials, and in particular in the inelastic
deformation process known as creep.
• 1D or line defects are the dislocations previously discussed, and are of two broad
types called edge dislocations (shown in Figure 5.3b) and screw dislocations (as
the name suggests, a helical line defect). Dislocation motion dominates the plastic deformation of metals.
• 2D or planar defects are primarily defects in the stacking sequence in a lattice,
and so are called stacking faults. The boundaries of a stacking fault may be described in terms of line defects (dislocations). One particular consequence of
stacking faults can be the development of regions of a crystal of different orientation called twins.
• 3D or volume defects are essentially either pores or inclusions in a crystal. Such
defects are typically immobile, but may be the sites for the nucleation of dislocations, as well as damage mechanisms such as cracks and voids, which can then
lead to macroscopic failure.
5 Plastic Deformation of Nanomaterials
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Fig. 5.3 Expanded version of Figure 5.1 showing the primary deformation mechanism that leads
to plasticity in metals: the motion of line defects called dislocations. Edge dislocations are shown
gliding along the slip plane. The inset shows an expanded view of an edge dislocation, amounting to an extra plane of atoms in the lattice (with the trace of the extra plane on the slip plane
defined as the line defect). The motion of many dislocations results in the macroscopic slip step
shown in (c).
Dislocations are the defects that control the plastic deformation of metals, and the
onset of the motion of these line defects determines the yield strength. A thorough
treatment of the theory of dislocations is beyond the scope of this book; excellent
references include the classical work of Hirth and Lothe (1992). For our purposes, it
is sufficient to identify dislocations in terms of two vectors: the line vector (a vector
tangent to the dislocation line) which we shall call ξ , and a unit Burgers vector b
which represents the disregistry in the crystal lattice when a circuit around the line
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defect (Figure 5.3c) is completed (this is, in effect, one measure of the size of the
defect in the lattice).
In the case of edge dislocations, the line vector and the Burgers vector are perpendicular to each other, while in the case of screw dislocations, the line and Burgers
vectors are parallel to each other (Figure 5.4). Thus for an edge dislocation we have
ξ · b = 0, while for a screw dislocation we have ξ · b = ±b, where b is the magnitude
of the Burgers vector (the line vector ξ is a unit vector). The slip plane or glide plane
is that plane along which slip occurs and is defined as the plane with normal b × ξ .
For a screw dislocation, we have b × ξ = 0, which implies that the glide plane is
nonunique or indeterminate for a screw dislocation, so that a screw dislocation can
move on a variety of planes at any time. The combination of a slip plane normal and
a slip direction defines a slip system. In general, ξ (x,t), that is the line vector, varies
with position in the crystal, but the Burgers vector for the dislocation is determined
by the particular slip system that is active.
ξ
b
bξ
Fig. 5.4 Schematic of slip with an edge dislocation (top figure) and screw dislocation (bottom
figure), showing the Burgers and line vectors in each case.
5 Plastic Deformation of Nanomaterials
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By definition, as a line defect, a dislocation line cannot begin or end inside the
crystal (but it can begin or end at a free surface). Thus dislocation lines generally
exist within crystals as dislocation loops or as line segments ending in free surfaces.
A dislocation loop within the crystal will have, in general, both edge and screw configurations, as well as regions where the dislocation is of mixed type (both edge and
screw). In such regions, the Burgers vector can be decomposed into two parts in
terms of its relationship with the line vector: b = be + bs , where be · ξ = 0 (corresponding to the edge part of the mixed dislocation) and bs × ξ = 0 (corresponding
to the screw part of the mixed dislocation). Note that for the mixed dislocation the
pair be and ξ define the glide plane, since that plane must contain both the Burgers
vector and the line vector.
Since the dislocation represents a defect in the crystal, there is a stress field within
the crystal in the region surrounding the dislocation (because the atoms are not in
their ideal positions with respect to the interatomic potential). The stress field around
a dislocation can be calculated using the theory of linear elasticity (note, however,
that linear elasticity theory breaks down as one approaches within one or two atoms
of the center of the defect, a region which is defined as the dislocation core). The
stress field around a dislocation (both edge and screw) is observed to have a finite
range (that is, the stress decreases rapidly with distance r, as 1r , as one moves away
from the dislocation core). This finite range of the stress fields means that dislocations do not interact with each other until they get within some critical distance.
Similarly, a dislocation that gets sufficiently close to a free surface effectively feels
a force (called an image force) due to the existence of the free surface. The stresses
that we have discussed in this paragraph are all self stresses, in that they are developed simply because of the existence of the defect, not because of the application
of an external force. The consequence of the existence of these self stresses is that
assemblies of dislocation lines can rearrange themselves to minimize the internal
stresses, a process known as recovery that is particularly important at high homologous temperatures.
What does all of this have to do with the yield strength? Recall, from the chapter
on mechanics, that the plastic deformation of materials is primarily the result of
shear stresses, and that the plastic deformations do not involve volume change. The
movement of dislocations in metals results in the development of deviatoric (distortional) strains, but not of volumetric strains (e.g., look at Figure 5.3d). In crystalline
metals, the plastic deformation turns out to be developed by the collective motion of
large numbers of dislocations. When an external stress field is applied to a crystal
containing dislocations, each dislocation feels a corresponding local force per unit
length that results in motion of the dislocation if a critical condition is reached, corresponding to a critical value of the applied stress. The yield strength of the material
is defined by the critical applied stress at which the dislocation motion begins. The
direction of subsequent motion of the dislocation depends on the applied stress tensor and the character of the dislocation (as defined by the Burgers vector and the
line vector).
It has been experimentally observed that the movement of the dislocations in
glide occurs only along specific combinations of slip planes and slip directions that
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are defined for each crystal type. This specificity of the slip planes and slip directions
arises from such features as the spacing of the planes and the packing of the atoms
within a plane; close-packed planes are preferred for energetic reasons.
5.3 Crystals and Crystal Plasticity
A discussion of crystal structure and crystallography is beyond the scope of this
book, but the reader will benefit greatly from a knowledge of basic crystallography
and in particular of Miller indices for describing planes and directions inside crystals; an excellent reference is Courtney (2005). The majority of metals of interest
have crystal structures that fall into three categories (Figure 5.5):
1. Body centered cubic or bcc structures, example of which are iron (Fe), vanadium
(V), tantalum (Ta), tungsten (W) and niobium (Nb)
2. Face centered cubic or fcc structures, example of which are aluminum (Al), copper (Cu), gold (Au), and silver (Ag)
3. Hexagonal close-packed or hcp structures, examples of which are titanium (Ti),
zinc (Zn), zirconium (Zr) and magnesium (Mg)
Fig. 5.5 Examples of basic crystal structures that are common in metals: body centered cubic (one
atom in the center of the cube), face centered cubic (atoms at the center of each cube face), and
hexagonal close packed structures.
Directions in a cubic crystal are typically defined through Miller indices, which
are sets of integers which can be viewed in terms of an orthonormal basis consisting of unit vectors e1 , e2 , e3 which are aligned with the axes of the cube.
The notation [h, k, l] then defines a direction parallel to a unit vector with com√
ponents ( hs e1 + ks e2 + sl e3 ), where s = h2 + k2 + l 2 essentially normalizes the
vector to unit length. Thus the set of Miller indices [1, 1, 2] represents the direction ( √16 e1 + √16 e2 + √26 e3 ). Negative directions are denoted by an overbar, e.g.,
[1, 1, 2̄] represents the direction ( √16 e1 + √16 e2 − √26 e3 ). Families of similar directions are conventionally denoted by < h, k, l >. Now, planes are essentially
5 Plastic Deformation of Nanomaterials
129
defined when the normal to the plane is defined, and the normal is just another direction, so a plane in the crystal can be defined in similar terms using Miller indices.
In order to distinguish between planes and directions, the notation (p, q, r) using
parentheses rather than square brackets
is used to define a plane with unit normal
n = ( ps e1 + qs e2 + rs e3 ), where s = p2 + q2 + r2 . Thus the notation (1, 1, 1) represents a plane with unit normal n = ( √13 e1 + √13 e2 + √13 e3 ). Examples of such planes
are shown in Figure 5.6. Families of similar planes (similar in the sense of crystal
symmetry) are conventionally denoted by {p, q, r}.
(0,0,1)
(0,1,0)
x
(1,0,0)
x
Z
x
Z
Z
Y
Y
(1,1,0)
Y
(2,0,0)
(1,1,1)
x
x
x
Z
Z
Y
Z
Y
Y
Fig. 5.6 Typical planes in a cubic crystal defined using Miller indices. The (111) plane and similar
{111} planes are close-packed planes in a face-centered-cubic (fcc) crystal and therefore are part
of typical operating slip systems (together with < 110 > type directions).
Hexagonal close packed or hcp metals do not have cubic symmetry, and therefore cannot be described using just the three integers defined in the last paragraph;
instead, sets of four numbers are used, e.g., [h, k, l, m]. The typical approach here
is to use three vectors defining directions in the basal plane of the crystal, and one
vector defining the normal to the basal plane. Clearly only two vectors are needed
to define any direction in the plane, so only two of the first three vectors are independent. The reader should find a discussion of hcp notation in any textbook for
materials scientists that discusses crystal structure, e.g., Courtney (2005).
Now that we can define planes and directions in a crystal, we can list the various
slip planes and slip directions in typical crystal structures. The combination of a
slip plane (defined by the normal n) and a slip direction (defined by the vector s) is
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Table 5.1 Slip systems with typical numbering scheme in typical fcc metals.
System
Primary system
Number Designation Normal Slip direction
1
B4
(111̄)
[101]
2
B5
(111̄)
[011]
3
B2
(111̄)
[11̄0]
Conjugate system
4
C1
(11̄1)
[110]
5
C5
(11̄1)
[011]
6
C3
(11̄1)
[101̄]
Cross-glide system
7
D4
(11̄1̄)
[101]
8
D1
(11̄1̄)
[110]
9
D6
(11̄1̄)
[101̄]
Critical System
10
A3
(111)
[01̄1]
11
A6
(111)
[01̄1]
12
A2
(111)
[11̄0]
called a slip system. The slip systems that are commonly observed in fcc structures
are presented in Table 5.1. Since dislocation motion only occurs within specific
slip systems, the stress associated with the onset of plastic deformation (i.e., yield)
is related to the stress on those slip systems. The various slip systems in a given
crystal structure are typically presented in some specific order, and then numbered
(thus one speaks of the second slip system in fcc metals, and so forth). There are 12
slip systems in fcc metals, and as many as 48 in bcc metals (Table 5.2).
Let us consider using these concepts together with the mechanics concepts that
we learned in Chapter 2. When an external stress σ (x,t) is applied to the crystal,
specific tractions (Equation 2.41) will be developed on every slip plane within the
crystal. Consider a slip plane with normal n; the traction vector (Figure 2.1) at a
point on the plane is given by the operation of the applied stress tensor on the normal
vector, t = σ (x,t) n. The resulting traction vector is essentially a vector made up
of stress components, see Chapter 2, and will in general have components normal
to the plane (defining the normal stress on the plane) as well as in the plane. The
component of the traction vector in the slip direction s will be the shear stress in the
slip direction, and is given by
τ = t · s = (σ n) · s = σi j n j si ,
(5.1)
where the last term represents the component form of the expression (see Chapter 2).
Equation (5.1) provides the resolved shear stress on a given slip system; changing
the slip plane normal and the slip direction changes the slip system, and so will result
in a different resolved shear stress. That is, for a given applied stress σ , the α th slip
system (which has slip plane normal nα and slip direction sα ) will have a resolved
stress τ α . For a given crystal orientation and given applied stress, some particular
slip system will have the maximum resolved shear stress in comparison to all of the
available slip systems – we will refer to this as the maximal slip system. It has been
observed experimentally that in most metals dislocation motion begins when the
resolved shear stress on the maximal slip system reaches a critical value, known as
the Critical Resolved Shear Stress (CRSS) C: that is, the onset of dislocation motion
corresponds to
5 Plastic Deformation of Nanomaterials
131
τ α ≥ Cα .
(5.2)
This is sometimes called the Schmid law. The critical resolved shear stress Cα in
Equation (5.2) is the slip system equivalent of yield strength, and defines the onset
of plastic deformation. Although the convention is to call Cα the critical resolved
shear stress, we shall refer to it as the critical resolved shear strength, because we
need to distinguish applied stresses from the material property that is the strength.
When a crystal is subjected to an externally applied stress, the slip system with
the largest resolved shear stress is the one on which dislocation motion is likely to
start, and the external stress at which this motion begins essentially defines a yield
strength.
It is worth noting that bcc metals seem to show significant non-Schmid effects,
i.e., they do not necessarily follow equation 5.2, but these concepts continue to be
used for bcc metals as a simplifying assumption.
Note that, by the definition above, the yield strength is a function of crystal orientation, reflecting the fundamental anisotropy of plastic deformation at the single
crystal level. Simultaneously, note that the plastic deformation occurs along specific
slip planes in specific slip directions, reflecting the fundamental heterogeneity of
plastic deformation at the single crystal level. Single crystal plasticity is inherently
anisotropic and heterogeneous, and in that sense the “strength” of a single crystal is
not a well-defined single quantity. The plasticity concept of a single yield strength
number for a material is a result of the averaging of a variety of local orientationdependent yield strengths for the very large number of crystals within a polycrystalline metal, and the plasticity concept of a plastic strain at a point is a result of
the averaging of a very large number of heterogeneous dislocation motions over a
number of slip systems. Experiments on single crystals are difficult to do, not least
because the development of slip on a given slip system results in an effective rotation (because shear involves a rotation), and consequently the resolved shear stress
on a given slip system changes with continuing deformation if the loading axis remains fixed. Thus even in single crystals, the strength values that are obtained from
experiment are usually averages over a number of slip systems.
To summarize, the physical basis of yield strength is the stress required for the
onset of the motion of dislocations in crystals along specific slip planes and in specific slip directions, combined with averaging over a large number of crystals within
a polycrystalline material. Two consequences of this physical basis are of interest:
• First, the apparent yield strength of a material is a convolution of the fundamental behavior of single crystals of the material and the arrangement of the grains
within the material. For example, the yield strength of a material where all of the
grains are oriented in one direction (these oriented polycrystalline structures are
said to be textured) may well be different from the yield strength of the material
when the grains are randomly oriented.
• Second, the yield strength of a material can be controlled by making it difficult to move the dislocations, for example by placing obstacles in the path of
dislocations. Obstacles of interest can include dispersoids, particles, and grain
boundaries.
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Table 5.2 Known slip systems in a variety of metals, from a table put together by Argon (Argon,
2008).
Crystal Structure
Cu, Au, Ag, Ni,
CuAu, α -CuZn,
AlCu, AlZn
Al
Lattice Type
FCC
α -Fe
BCC
BCC
BCC
BCC
BCC
HCP c/a>1.85
Mo, Nb, Tz, W, Cr, V
Mo, Nb, Tz, W, Cr, V
Cd, Zn, ZnCd
FCC
Mg
HCP c/a = 1.623
Be
HCP c/a = 1.568
Ti
HCP c/a = 1.587
Ge, Si, ZnS
As, Sb, Bi
Diamond cubic
Rhombohedral
Slip plane Slip direction
{111}
110
Reference
Seeger (1958)
111
100
110
112
123 111
110
112
(0001)
(001̄0)
(112̄2)
(0001)
101̄1
101̄0
(0001)
101̄0
101̄0
101̄1
(0001)
111
(111)
(111̄)
110
110
111
111
Seeger (1958)
111
111
(21̄1̄0)
[112̄0]
[1̄1̄23]
21̄1̄0
21̄1̄0
2110
21̄1̄0
21̄1̄0
21̄1̄0
21̄1̄0
21̄1̄0
101
[101̄]
[101]
Seeger (1958)
Seeger (1958)
Seeger (1958)
Seeger (1958)
Seeger (1958)
Seeger (1958)
Seeger (1958)
Schmid and Boas (1935)
In the next two sections, we first discuss the strengthening mechanisms in single
crystal metals, and then examine the strength of polycrystalline metals. We will then
describe ways of determining the overall response of engineering materials.
5.4 Strengthening Mechanisms in Single Crystal Metals
There are multiple ways of increasing the strength of metals, not the least of which
is decreasing the grain size. We discuss the various mechanisms within single crystals here, leaving aside the interaction of multiple crystals for a later section. There
remains the very interesting question of whether it is appropriate to assume that
the final yield strength of the material is obtained by simply adding the strength increments provided by each individual mechanism, or whether it makes more sense
to add up the inelastic strain increments provided by each mechanism for a fixed
applied stress. From a conceptual viewpoint, this question is identical to the one of
whether we should assume that the strains are uniform or the stresses are uniform
in a polycrystalline material. For the moment, we will follow the popular approach
of adding the strength increments from each mechanism, and consider other options
later. An excellent and very detailed discussion of strengthening mechanisms is provided in the recent book by Argon (Argon, 2008).
5 Plastic Deformation of Nanomaterials
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5.4.1 Baseline Strengths
The most fundamental resistance to dislocation motion through a crystal lattice
comes from the lattice itself, and is called the Peierls stress or Peierls-Nabarro
stress, which we shall call σPN , the Peierls-Nabarro strength. The magnitude of
the Peierls-Nabarro strength varies from one material to another, but also varies
substantially on the basis of crystal structure. The Peierls-Nabarro strength is quite
high for bcc crystals and quite low for fcc crystals, and so the strength of pure
bcc metals is typically higher than that of pure fcc metals. In many ways the
Peierls-Nabarro strength (also called the friction stress) of the pure metal represents a baseline strength for a material, and so this quantity is sometimes viewed
as a fundamental strength. However, this stress can be very low in fcc metals
(≈ 10−6 µ , where µ is the shear modulus), so much so that it is essentially ignored from an applications perspective. Modifications to the material, such as the
addition of alloying elements, will increase the strength above this fundamental
strength.
5.4.2 Solute Strengthening
The addition of solute atoms (by alloying) can make it more difficult to move dislocations and thereby increase the strength of a material. It is typically found that
a slightly impure metal has a much higher strength than the very pure metal, and
that different solute atoms provide different levels of strengthening. For most solute
atoms, the additional strengthening Δ σss provided by a solute can be described in
terms of a power law function:
Δ σss = Kcn̄ ,
(5.3)
where c is the concentration of the solute atom and K and n̄ are solute-dependent
parameters. Thus, for example, the addition of small amounts of magnesium can
greatly strengthen aluminum, leading to a class of Al-Mg alloys (including, with
some additional compositional variations, the structural aluminum alloy Al 5083).
The addition of 4.8 atomic % of magnesium to aluminum raises the strength by
67 MPa. This strengthening occurs largely because the solute atoms diffuse towards the core of dislocations because of the local stress state there, and this
essentially pins the dislocations, making it difficult for them to begin to move.
There is no length scale associated with this strengthening mechanism except in
that the spacing of the solute atoms decreases as the concentration increases. Since
the operation of the strengthening mechanism is essentially through a cloud of
solute atoms, this length scale is essentially irrelevant, and this strengthening mechanism should be observed over nearly all specimen sizes down to the 100 nm
scale.
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Nanomaterials
5.4.3 Dispersoid Strengthening
Another way to strengthen materials is to put obstacles in the path of the dislocations, such as small particles called dispersoids (typically 10–100 nm in diameter).
When a dislocation line moves through the crystal and bumps up against such a particle, it has three options: (a) stop, (b) cut through the particle, or (c) move around
the particle through the rest of the crystal (e.g., climb over the particle). If the particle is too strong to cut through, and if the dislocation does not have the character
(combination of b and ξ ) to move around the particle, then that segment of dislocation line behind the particle comes to a stop. However, parts of the dislocation line
farther away from the particle continue to move, resulting in the dislocation taking
on a curved shape. If the particles have some spacing L, then the dislocation takes
on the bowed shape shown in Figure 5.7.
Fig. 5.7 Schematic of gliding dislocation bowing around dispersoids that are periodically spaced
a distance L apart along a line.
This is called the Orowan bowing mechanism, and the corresponding strengthening is called Orowan strengthening. The additional strengthening obtained through
the Orowan mechanism is (Arzt, 1998)
Δ σds = A1
µb
,
L − 2a
(5.4)
where A1 is a prefactor of order 1, L is the particle spacing (which depends on
the dispersoid volume fraction), and a is the particle radius. Note that there is a
relationship between the particle spacing, the particle size, and the particle volume fraction (for the same volume fraction, smaller particles must be closer together). This strengthening mechanism is the dominant one in many engineering materials, and can lead to strength increases of the order of 102 MPa. Note
that a length scale defined by the particle spacing now exists in the material;
this particle spacing is typically much less than the grain size in conventional
materials.
5 Plastic Deformation of Nanomaterials
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5.4.4 Precipitate Strengthening
A variety of particles can be generated inside materials, often by precipitating out
another phase within the primary crystal. Some (typically larger) particles can be
sheared through by the dislocation, in which case the strengthening that is attained
through the Orowan mechanism is limited by the particle strength. The effective
precipitate strengthening obtained through a distribution of these particles becomes
(Arzt, 1998)
3
Fm 2
Δ σ ps = A2 √
,
(5.5)
µ Lb2
where A2 is a prefactor of order 1, and Fm is the force required to shear the precipitate
particle. If the particle has a shear strength τ p , then this force is Fm = τ p π a2 , and so
we have
3
3
π 2 τ p 2 a3
Δ σ ps = A2 √
.
(5.6)
µ Lb2
This mode of strengthening has an optimal particle size for fixed particle volume fraction, as described by Arzt (1998). Strengthening increments on the order of
102 MPa can be generated in this way.
5.4.5 Forest Dislocation Strengthening
Most crystals contain large numbers of dislocations. A measure of this is the dislocation density, ρd , defined as the total dislocation line length in a crystal divided
by the crystal volume. The units of dislocation density are m12 , and therefore √1ρ
d
represents a length scale in the crystal – it is the average spacing between dislocations. One viewpoint that we can take is to imagine that a dislocation line sweeping
through a crystal perceives a forest of other dislocations around it, and these other
forest dislocations act as obstacles. Thus other dislocations can cause strengthening
of the material. The strength increment Δ σ f d provided by forest dislocations acting
as obstacles is given by
√
Δ σ f d = AT µ b ρd ,
(5.7)
where AT is a constant of order 0.1 − 1. Equation (5.7) is often called the Taylor
equation, and provides a measure of the increase in strength of the single crystal
because of the presence of some non-zero dislocation density. Now the dislocation
density itself evolves during deformation, because there are a number of sources
of dislocations inside most crystals – e.g., Frank-Read sources (Courtney, 2005).
Indeed, in modeling material behavior, one must often write an evolution equation
for the dislocation density:
∂ ρd
= Ac fc (ρd ) − Aa fa (ρd ),
∂t
(5.8)
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where fc (ρd ) and fa (ρd ) are functions that represent the rate of creation and annihilation of dislocations respectively, and Ac and Aa represent scaling parameters.
For most materials under most conditions, ρd increases with strain, and so Equation (5.7) is often said to represent a strain-hardening behavior (but note that the
dislocation density must evolve as in Equation (5.8) for this truly to represent such a
hardening behavior). A much more detailed understanding of this process can be obtained by considering multiple slip systems that are simultaneously active within a
crystal: one expects that dislocations sliding along one slip plane will intersect with
the traces of dislocations sliding along another slip plane, resulting in obstacles that
cause hardening on the first slip plane. This process is sometimes called latent hardening (Wu et al., 1991). Equation (5.7) represents a generalization of such ideas over
all slip systems in the crystal.
Adding all of these strength increments due to all of these mechanisms together,
we obtain the strength of the crystal, averaged in some way over all slip systems, as
3
σsc = σPN + Kcn + A1
√
µb
Fm 2
+ AT µ b ρd .
+ A2 √
2
L − 2a
µ Lb
(5.9)
Other strengthening mechanisms may be considered in a similar way. Later, when
we consider dislocation dynamics, we will return to this issue and suggest another
way to add these mechanisms together. Note that the third, fourth, and fifth terms on
the right hand side all contain length scales (particle size, particle spacing, and dislocation spacing) that may generate size effects (for example, behavior that depends
on sample size).
5.5 From Crystal Plasticity to Polycrystal Plasticity
The development of models that explain the transition from single crystal plasticity to polycrystalline plasticity involves the implementation of a number of averaging concepts that are similar to those we have discussed earlier in this chapter.
One issue is that of translating an external stress applied to an assemblage of many
crystals to the local stress experienced by each crystal in the polycrystalline mass.
Once we know the local stress in each crystal, we can use the crystal plasticity approach described above to determine when yield would occur in any given crystal.
But does yield in one crystal out of a large polycrystalline mass amount to yield
of the polycrystalline material? In principle, of course, the answer is yes. In practice, we define yield strength by when a measurable amount of permanent deformation has taken place (a common number is 0.2% of permanent strain, leading
to what is sometimes called a 0.2% yield strength), where this measure is an average measure over the entire polycrystal. That is, by the time that a measurable
amount of plastic strain has developed in the polycrystalline material (Figure 5.8),
some of the crystals may well have been deformed a great deal. The behavior of
5 Plastic Deformation of Nanomaterials
137
Fig. 5.8 Schematic of a polycrystalline microstructure with a variety of plastic strains (represented
by color or shading) within each individual grain.
the overall material is defined by the volume average stresses and the volume average strains, rather than by the stresses and deformations within each individual
crystal.
This is another version of the problem encountered with multiple crystals in the
elasticity case: we do not typically know the local stresses and strains in the polycrystal without doing full blown computations of the crystal plasticity. A number
of assumptions are commonly made to make analysis feasible. The most common
is known as Taylor averaging, where it is assumed that the strains are uniform in
the polycrystal, that is, that all of the crystals see the same strain, which is equal
to the average strain computed from the boundaries of the polycrystalline mass. In
this case, the stress will of course vary from one crystal (grain) to the next. An
alternative but less common assumption is sometimes called static averaging, and
states that the stresses are the same everywhere, while the strains vary from crystal
to crystal. These two assumptions are analogs to the two bounding cases that we discussed with respect to elastic moduli earlier in this chapter. Clearly neither of these
assumptions represents reality, and real behavior will be somewhere in between the
two extremes.
A number of crystal plasticity codes have been developed that allow for the analysis of the deformation of a polycrystal while simultaneously resolving deformations in individual crystals. This is particularly important for textured materials or
low-symmetry materials (such as some of the hcp metals). For our purposes, we
will simply assume that the averaging can be done, and ask what some of the consequences of such an averaging will be on the macroscopic response.
If we know the behavior of the single crystal, what is the simplest way to modify
this to obtain the polycrystal behavior? This raises a different question: how do
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Nanomaterials
we quantify the polycrystalline character of the material? The primary variables
defining the polycrystalline structure are:
1.
2.
3.
4.
The mean grain size, d
The grain size distribution, g(d)
The grain orientation distribution function, gθ (θ )
The grain shape (e.g., the grain aspect ratio α )
One should expect the properties of the polycrystal to depend on all of these variables. However, most of these variables are not reported by most experimentalists
who present data in the literature on the yield strength.
5.5.1 Grain Size Effects
The one variable that is routinely reported in the literature is the mean grain size,
d, and thus the most common approach in terms of yield strength is to add to the
strength in Equation (5.9) an additional term Δ σgs that captures the effects of polycrystallinity through the mean grain size. Experimental measurements of grain size
effects on the yield strength of metals has shown that it is often well described by
an inverse square-root law:
1
Δ σgs = ky d − 2 .
(5.10)
This behavior is called the Hall-Petch behavior, and ky is a material parameter
called the Hall-Petch coefficient or sometimes the Hall-Petch slope. For most materials, the “Hall-Petch equation” is written as
1
σy = σi + ky d − 2 ,
(5.11)
where the left-hand side of the equation represents the material’s strength, and the
term σi is called the intercept stress and represents the net strength of the material
from all of the other strengthening mechanisms put together. This Hall-Petch behavior can be rationalized in a number of ways. The most commonly cited model is that
of dislocations piling up at grain boundaries, with the grain boundary thus acting as
an obstacle to dislocation motion. This behavior may break down at very small values of d in the range of tens of nm, although such breakdowns remain controversial
because of possible variations in materials created by different processing routes. Of
course, it is expected that there will be a breakdown of such behavior: after all, if the
grain size becomes sufficiently small, it is no longer possible to retain a dislocation
loop within the grain (Arzt, 1998).
5.5.2 Models for Hall-Petch Behavior
The dependence of yield strength on grain size described by Equation (5.10) was
first developed simply to explain experimental observations in steel (Hall, 1951;
5 Plastic Deformation of Nanomaterials
139
Petch, 1953). A wide range of experimental measurements since then have demonstrated similar behavior in a wide range of metals. Efforts to describe Hall-Petch
behavior on a theoretical basis have evolved over the last five decades. Since the
early 1990s, modeling efforts have also focused on identifying possible reasons for
the breakdown of Hall-Petch behavior at very small grain sizes. We discuss some,
but not all, of these models in this section. The majority of these models fall into
three classes.
5.5.2.1 The Dislocation Pile-up Model
This was the first (and for many years the most popular) way of rationalizing the
observed √1d behavior. In its essence, the model states that a dislocation gliding
on a slip system stops when it arrives at a grain boundary, because the boundary
represents an obstacle and the next grain is not oriented for easy slip along the
same direction for the current applied stress. The dislocation gliding behind the first
dislocation, on the same slip system, bumps up against the stalled dislocation, and
this process continues, creating a dislocation pile-up at the grain boundary. The
result of this pileup is the generation of a force per unit length on the leading (first)
dislocation:
π Nbτ
π (1 − ν )τ 2 d
F
=
=
,
(5.12)
L
4
4µ
where F is the force, N is the number of dislocations involved in the pileup, τ is the
resolved shear stress on the system, ν is the Poisson’s ratio for the material (assumed
isotropic), µ is the shear modulus, and d is the grain size. Note that this result
(Equation 5.12) was originally derived for a symmetric system, that is dislocation
loops gliding through the crystal, so that there is an equal and opposite array of
dislocations on the other side of the crystal.
Because of this force, the dislocation pile-up generates an increased stress in the
adjacent grain, across the boundary. As the pile-up continues to grow, the stress in
the adjacent grain also grows, until it reaches some critical value τ ∗ at which dislocation motion can begin on another slip system in the adjacent grain. This critical
∗
value corresponds to the development of a critical force per unit length FL on the
lead dislocation. The external or applied stress at which this critical force is reached
can be obtained by solving Equation (5.12) for the critical resolved stress and then
scaling by a factor A that relates the external stress to the critical resolved shear
stress in the adjacent grain:
4µ
1
F∗ 1
∗
√ = K1 √ ,
(5.13)
σ =
π (1 − ν ) L d
d
4µ F ∗
where K1 = π (1−
ν ) L is a scaling factor that clearly depends on the elastic properties but also depends on the critical force F ∗ . Further plastic deformation is now
accommodated by slip in the adjacent grain. In real materials, it appears that the
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Nanomaterials
total external stress cannot be described through Equation (5.13), but rather, this
equation describes a strengthening increment due to the grain boundaries (like the
second term in Equation 5.11). A very complete discussion of the effects of pileups is found in Hirth and Lothe (1992). In simplified terms, when the grain size is
decreased, the number of dislocations within a pileup that can be accommodated
within a given grain decreases. Consequently a greater applied stress is needed to
cause slip to begin in the adjacent grain. In this model, this is the source of increased
strengthening at smaller grain sizes. The major drawback of this model is that dislocation pileups are not always (or even commonly) seen in materials, and thus there
is not a great deal of experimental evidence to support the primary mechanism postulated by the model. Further, similar grain size strengthening is observed of the
flow stress of a material at finite plastic strain (i.e., well beyond initial yield), and
the dislocation pileup argument does not make much sense at large strains when a
significant network dislocation density already exists inside each grain. Finally, it
is not clear that such a pile-up model is relevant to nanocrystalline materials where
the grain size may be too small to accommodate full dislocation loops (Arzt, 1998),
the number of dislocations in a pileup may be too low, or the number of dislocation
sources may be too small.
5.5.2.2 The Grain Boundary Ledge or Network Dislocation Model
This approach was first introduced by Li (1963), and invokes the grain boundaries
as sources of dislocations rather than merely as barriers. The concept is that grain
boundary ledges (Figure 5.9) act as dislocation sources, and if the ledge density
(i.e., number of ledges per unit grain boundary area) is essentially constant (i.e.,
independent of grain size), then the density ρntwk of the dislocation network that is
generated is inversely proportional to grain size (because of the area/volume ratio).
Fig. 5.9 Schematic of grain boundary ledges in a polycrystalline material. In the grain boundary
ledge model, these ledges are believed to act as dislocation sources and generate a dislocation
network.
5 Plastic Deformation of Nanomaterials
141
Consequently, using a strengthening increment of the form of Equation (5.7), we
have
√
1
(5.14)
Δ σgs = AT µ b ρntwk = K2 √ ,
d
where K2 is proportional to AT µ b, and AT represents the polycrystalline averaging
and is of order 1. This has a form identical to that of Equation (5.10), and does not
require the existence of dislocation pile-ups. A refinement of this model is provided
by Scattergood and Koch (1992). Note the dependence of the coefficient K2 on the
shear modulus, as in the case of K1 from Equation (5.13).
5.5.2.3 Geometrically Necessary Dislocation and Strain Gradient Models
The mismatch between grains that occurs at grain boundaries will lead to the development of high local dislocation densities. These dislocations are generated to
accommodate the mismatch (rather than to accommodate externally driven plastic
flow), and are therefore said to be geometrically necessary; the corresponding dislocation density is called the geometrically necessary dislocation density or GND
density (Ashby, 1970). Associated with the development of this GND density is
the development of a strong strain gradient at the grain boundaries (Shu and Fleck,
1999). The GND density is also inversely proportional to grain size, so a relationship exactly like that in Equation (5.14) follows. Strain gradient models are somewhat more complex (Shu and Fleck, 1999), in that it is not immediately clear that
the length scale of interest is the grain size, but the essential behavior remains the
same as with the GND case. Related models are of the core-boundary type, such as
the Meyers-Ashworth (Meyers and Ashworth, 1982) and Fu et al. (Fu et al., 2001)
models, postulating a hardened grain boundary layer. Many of these models differ
in terms of their predicted behaviors at very small grain sizes; we will return to this
issue later.
A summary of the three basic models for the observed Hall-Petch behavior is
presented in Figure 5.10 in schematic form. The schematic in Figure 5.10a shows a
dislocation pileup at a grain boundary, while the schematic in Figure 5.10b shows
grain-boundary ledges at one of the boundaries in a polycrystalline material. The
schematic in Figure 5.10c shows (on the left), the misfits between many grains as
a result of deformation along their individual slip systems, while on the right the
grains have been put back together in a compatible form with the misfits now accounted for by a (geometrically necessary) dislocation distribution near the grain
boundaries.
5.5.2.4 Observations of Hall-Petch Behavior
A very wide range of materials has been studied in terms of the grain size dependence of the yield strength, and there is a very substantial literature on the Hall-Petch
strengthening of materials. Most of the literature presents Hall-Petch
√ coefficients in
terms of Equation (5.11), where the units of ky would be MPa m. However, this
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a
Fig. 5.10 A summary of the three basic models for the observed Hall-Petch behavior. (a) A dislocation pileup at a grain boundary. (b) Grain boundary ledges in a polycrystalline material. (c) A
schematic of the geometrically-necessary dislocation model. On the left side of this subpart of the
figure, the individual crystals that make up this polycrystalline material are assumed to have slipped
along their respective slip systems, resulting in misfits between the grains. However, the polycrystalline material must remain compatible at the grain boundaries, assuming that voids do not open
up at the boundaries. These misfits can be accommodated by creating a new set of dislocations
(the so-called geometrically-necessary dislocations) that then result in a dislocation distribution
near the individual grain boundaries. Only one such GND distribution is shown in one grain on the
right.
5 Plastic Deformation of Nanomaterials
143
dimensional form can lead to some confusion, so in this book we choose to examine
material behavior in terms of the normalized Hall-Petch coefficient K̃, where
K̃ =
ky
√ ,
µ b
(5.15)
so that the traditional Hall-Petch coefficient is normalized by the shear modulus
and the square root of the Burgers vector. Other normalizations are also possible, of
course. For most materials, the normalized coefficient K̃ turns out to be in the range
of 0.1–0.6.
Hall-Petch coefficients (both traditional and normalized) for a number of materials are presented in Table 5.3, with the caveat that the reliability of some of
these numbers may be low because of variations in materials processing and in the
accuracy of the mechanical tests. A particular concern arises when Hall-Petch coefficients are extracted from measures of hardness variation with grain size rather
than yield strength variation with grain size, since hardness measures do not correlate well with yield strength for all specimen geometries (consider, for example, the errors introduced by a substrate interaction just below an indent in a
thin film). Note that most fcc metals (Cu, Al, Au, Ni) have small K̃, while most
bcc metals (Fe, Mo, W, Ta) have high K̃.
Table 5.3 Hall-Petch coefficients for a variety of materials.
Material
1. Copper
2. Aluminum
3. Gold
4. Nickel
5. Zinc
6. Magnesium
7. Titanium
8. Cobalt
9. Armco Iron
10. Molybdenum
11. Tantalum
12. Tungsten
ky, MPa
0.14
0.04
0.25
0.16
0.22
0.28
0.40
0.29
0.58
1.77
0.31
3.1
√
m Shear modulus, MPa Burgers
vector, m
45,000
2.56E-10
26,000
2.86E-10
27,000
2.88E-10
79,000
2.49E-10
36,285
2.60E-10
17,300
3.25E-10
45,000
2.95E-10
82,000
2.51E-10
81,400
1.43E-10
126,000
2.73E-10
69,200
2.83E-10
160,600
2.70E-10
K̃ Reference
0.19
0.09
0.55
0.13
0.38
0.89
0.52
0.22
0.60
0.85
0.27
1.17
Hansen (2005)
Hansen (2005)
Chew et al. (2007)
Hansen (2005)
Armstrong (1970)
Armstrong (1970)
Armstrong (1970)
Karimpoor et al. (2007)
Courtney (2005)
Armstrong (1970)
Koo (1962)
Wei and Kecskes (2008)
An example of Hall-Petch behavior is presented in Figure 5.11, which shows the
variation of the flow stress (the strength of a material at fixed strain) as a function
of grain size for consolidated ball-milled iron. The flow stress is used because these
particular experiments were unable to accurately capture the yield strength of the
iron. This material was produced by ball-milling a nanopowder, followed by coldpressing and then hot-pressing to √achieve nearly full density. The corresponding
Hall-Petch slope is ky = 0.69 MPa m, which is significantly higher than observed
in some pure irons (Table 5.3), perhaps as a result of the ball-milling that was used.
Notice that if the Hall-Petch plot is made in terms of hardness, defined here as a
third of the Vickers hardness, the slope is slightly different.
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3500
Consolidated Iron (Jia et al., 2003)
Quasistatic Flow Stress (MPa)
3000
y = 139.38 + 0.69478x R = 0.99685
2500
2000
1500
1000
Flow Stress (MPa)
Hardness (MPa)
500
0
0
500
1000
1500
2000
2500
3000
3500
4000
1/Sqrt (grain size (m))
Fig. 5.11 The variation of the flow stress (measured at 4% strain) with grain size for consolidated
iron, taken from the doctoral dissertation of Dexin Jia at Johns Hopkins. The variation of hardness
(defined here as H3v , where Hv is the Vickers hardness) is also shown.
In considering the use of an equation such as Equation (5.9) in conjunction with
a supposed strengthening increment such as in Equations (5.10) or (5.11), i.e., when
multiple mechanisms are active simultaneously, the question of coupling immediately arises. That is, should one expect, for example, that the parameters associated
with Orowan strengthening are independent of the solute content of the material?
This question must be addressed for each strengthening mechanism. In the case of
grain size strengthening, the question amounts to asking whether ky (or K̃) is independent of solute content, precipitate content, and so forth. That is, would the K̃
measured (or computed from a model) for a pure polycrystalline metal also be appropriate for use with an alloy of the metal? It is known that the elastic moduli of
alloys do not change dramatically with respect to the pure metal, so Equations (5.13)
and (5.14) suggest that K̃ for the polycrystal should be about the same as K̃ for the
alloy. Unfortunately, this turns out not to be the case: the ky and K̃ are functions of
substructure:
ky = f (solutes, precipitates, dispersoids, dislocation substructure...).
(5.16)
Some of this is of course to be expected, because it is known that the CRSS is
itself a function of solute concentration; for a discussion, see Argon (2008). However, the degree of coupling at the the level of ky is difficult to predict, as we see now
for the specific case of aluminum. Table 5.4 presents Hall-Petch coefficient data for
a variety of aluminum-based materials (Han et al., 2003), including pure aluminum.
5 Plastic Deformation of Nanomaterials
145
Table 5.4 Hall-Petch coefficients for pure aluminum and select aluminum alloys. Data taken from
the work of Witkin and Lavernia (2006).
√
Material
ky , MPa m Reference
Pure Aluminum
Al (99.999%)
0.13
Al (99.5%)
0.08
Al and Al-Al2 O3 composites Cryomilled Al
Ball-milled Al
ECAP Al
SAP Al
0.09
0.012
0.12
0.03-0.07
Witkin and Lavernia (2006)
Perez (1998)
Robinson (1953)
Perez (1997); Phillips
(1953); Rodriguez (2003)
Al-Mg alloys
Cryomilled Al 5083
MA IN905XL (Al-4Mg-1.5Li-0.4O)
Al-3.5Mg
Al 5052 (Al-2.6Mg)
Al 5754 (Al-3Mg)
Al-5Mg
Al-5Mg-0.2Mn
ECAP Al-1Mg
ECAP Al-3Mg
Al-6Ni
0.28
0.32
0.12–0.25
0.21
0.27
0.15
0.22
0.13
0.15
0.14
Witkin and Lavernia (2006)
Rosler (1992)
Semiatin (2001)
Siegel (1994)
Siegel (1994)
Siegel (1994)
Siegel (1994)
Robinson (1995)
Robinson (1995)
Mechanically alloyed Al composites
Al plus C, O (14.1 vol.%)
Al plus C, O, Cu (12.8 vol.%)
Al plus C, O, Y (5.9 vol.%)
Al plus C, O, Nb (16.1 vol.%)
Al plus C, O, Y, Nb (15.9 vol.%)
Al plus C, O, Ti (31.6 vol.%)
0.23
Sun (2000)
0.31
0.20–0.27
0.16
0.22
0.25
One notes from Table 5.4 that the Hall-Petch coefficients of aluminum-based materials vary over a wide range. This is because strengthening mechanisms such as
solid solution strengthening and precipitate strengthening appear to have an effect
not just on the strengthening increments presented in Equations (5.3)–(5.8), but also
on the effective Hall-Petch coefficient in these materials. A significant part of the
data on aluminum in Table 5.4 is derived from the work of Lloyd and co-workers,
who conducted a comparative study of several aluminum alloys of controlled composition, and we can discuss these coupling effects in the context of their work.
For example, the Hall-Petch coefficient for cryomilled Al 5083 is apparently much
larger (Witkin
aluminum alloy
√ and Lavernia, 2006) than that for the coarse-grained√
(0.068 MPa m) and that for cryomilled pure aluminum (0.09 MPa m) (Han et al.,
2003). By comparing the ky values of different aluminum and Al-Mg based aluminum alloys (Han et al., 2003), it is observed that the effective Hall-Patch coefficient is related to the solid solution, the distribution of dispersoids, and the oxygen
and nitrogen contents in these alloys. Parts of this behavior have been predicted in
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Nanomaterials
terms of the grain boundary ledge argument by J.C.M. Li for the specific case of
solute concentrations.
Lloyd and Court (2003) discussed the effects of Mg and Mn content on the intercept stress and slope of the Hall-Petch equation for these Al alloys. Their work
shows that Mg and Mn contribute to both the intercept stress and the Hall-Petch
coefficient as solute atoms and through precipitates. Examining this data, it appears
(on a heuristic basis) that the effective k¯y for an Al-Mg-Mn alloy can be obtained
from
k¯y = kyPure Al + kyMg addition + kyMn addition ,
(5.17)
where kyPure Al is the Hall-Petch coefficient for pure Al, and kyMg addition and kyMn addition
are the contributions of Mg and Mn to the effective coefficient. From experimental data, √
the term kyMg addition in a cryomilled Al 5083 can be approximated as
Al-5Mg (0.16) and cryomilled pure
0.07 MPa m (the difference of ky between
√
Al (0.09)), and kyMn addition is 0.09 MPa m (the difference of ky between Al-5Mg0.5Mn(0.25) and Al-5Mg (0.16)). Equation (5.17) allows us to compute
the effective
√
Hall-Petch coefficient k¯y for the√cryomilled Al 5083 as 0.25 MPa m. This value is
close to the value of 0.28 MPa m estimated from the experimental data of Lavernia et al. (Witkin and Lavernia, 2006). This approach explains why the Hall-Petch
coefficient for cryomilled Al 5083 is so much larger than that of the conventional
grain size version of the same Al alloy, and demonstrates that when the Hall-Petch
equation is used to estimate grain size strengthening in an alloy, the solution and
precipitation mechanisms may couple into the grain size strengthening mechanism.
Since the moduli of alloys generally do not vary dramatically from the moduli of
the pure polycrystalline metal, models such as Equations (5.13) and (5.14) suggest
that the critical force F ∗ or the network dislocation density ρntwk must be affected
by these alloying additions (e.g., through precipitation at grain boundaries, or solute
segregation to grain boundaries).
5.5.2.5 Limits of Hall-Petch Behavior
The grain-size dependence of yield strength that is described by the Hall-Petch behavior appears to suit experimental data very well for metallic materials with grain
sizes that are larger than about 500 nm. The approach breaks down in two limits.
first, at very large grain sizes, the observed yield stress does not necessarily approximate that of the single crystal. This is of course to be expected, given the averaging
issues involved. Second, and of greater interest to us in this book, the Hall-Petch
dependence of yield strength on grain size appears to be inaccurate for small grain
sizes. This is sometimes known as Hall-Petch breakdown or deviations from HallPetch behavior. Indeed, at very small grain sizes, it has been claimed that the yield
strength of materials decreases with grain size, sometimes called the inverse HallPetch effect. This type of behavior was first observed by Chokshi et al. (1989), when
they examined the hardness of copper and palladium at grain sizes ≈ 10 nm. Most
if not all of the deviations from Hall-Petch that are observed at small grain sizes
5 Plastic Deformation of Nanomaterials
147
are in the direction of lower than expected strength (although most measurements
in this grain size range are of hardness, (e.g., Zhou et al. (2003), not strength, see,
e.g., Figure 5.12). These deviations are important because, from a technological perspective, they imply that the strength of a nanocrystalline solid cannot be increased
indefinitely by decreasing the grain size.
d (nm)
9
28.9
17.1
7.1
4.1
650
550
7
450
6
5
0.1
HV
H(GPa)
8
0.2
0.3
0.4
350
0.5
1/d0.5
Fig. 5.12 Evidence of reduction of hardness with decreasing grain size in a Ni-P material, from
Zhou, Erb et al., Scripta Materialia, 2003 (Zhou et al., 2003). The right-hand axis represents the
Vickers Hardness number HV. Reprinted from Scripta Materialia, Vol. 48, Issue 6, page 6, Y. Zhou,
U. Erb, K.T. Aust and G. Palumbo, The effects of triple junctions and grain boundaries on hardness
and Young’s modulus in nanostructured Ni-P. March 2003, with permission from Elsevier.
A very large number of experiments have demonstrated a deviation of the
strengths of nanocrystalline materials from the classical Hall-Petch behavior at very
small grain sizes. However, the data remains somewhat inconsistent. This is partly
because of the difficulty of making materials with very small grain sizes, and partly
because the differences between materials may be larger than just the grain size (for
instance the material with the grain size of 50 nm made by electrodeposition may
be very different in character from a material of identical grain size made by severe
plastic deformation). An example of the degree of variability in measured material
properties can be seen in Figure 5.13, which is taken from the excellent review article by Meyers et al. (2006) and which examines only the properties that have been
measured for copper. Even in this one material, very different yield strengths have
been measured at similar grain sizes in the range of grain size below 25 nm. In some
cases the yield strength is perceived to become independent of grain size, while in
other cases the yield strength is perceived to decrease with decreasing grain-size.
However, what is certainly apparent is that the behavior deviates from the classical
Hall-Petch behavior at grain sizes of less than 25 nm. Similar deviations from the
classical Hall-Petch behavior have been observed in other materials, including iron,
nickel and titanium (Meyers et al., 2006). In all of these cases, however, the scatter
in the behavior at these very small grain sizes remains substantial, so much so that it
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Fig. 5.13 Compendium of data on copper in a Hall-Petch-style plot by Meyers et al. (2006), showing the scatter in data at small grain sizes but the general deviation from classical Hall-Petch
behavior. Reprinted from Progress in Materials Science, Vol. 51, Issue 4, Page 130, M.A. Myers,
A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials. May 2006, with permission from Elsevier.
is difficult to use the data to distinguish between different possible mechanisms for
the deviations from Hall-Petch.
An examination of the literature on nanocrystalline materials indicates that the
majority of experiments demonstrate not so much an inverse Hall-Petch effect
(a decrease of the yield strength of decreasing grain size), but rather a reduction
in the degree of strengthening that is obtained as the grain size is decreased (e.g.,
Figure 5.14). It is difficult to draw conclusions about material behavior given the
scatter evident in the current dataset (as of 2008), with most of the variability related
to either materials synthesis or measurement difficulties. The difficulties associated
with making measurements of material behavior at very small grain sizes arise primarily from the difficulty of obtaining samples of sufficient size for traditional testing, as discussed in the chapter on experimental techniques. The rapid evolution of
these techniques suggest that this controversy will be resolved in the near future.
What are the possible causes of Hall-Petch breakdown? A number of theories
have been advanced to explain this behavior. These “Hall-Petch breakdown” models
can generally be separated into two categories.
• First, models which attempt to show that the primary mechanism for Hall-Petch
strengthening (whether viewed as dislocation pileup or GND localization at the
boundaries) will break down at small grain sizes. An example of such a model
would be one that states that a minimum grain size is required to develop a dislocation loop (Nieh and Wadsworth, 1991).
5 Plastic Deformation of Nanomaterials
149
(a)
(b)
(c)
(d)
Fig. 5.14 Hall-Petch plots for copper, iron, nickel and titanium compiled by Meyers et al. (2006),
showing the deviation from classical Hall-Petch at small grain sizes. Note the strength appears
to plateau but not decrease with decreasing grain size. Reprinted from Progress in Materials Science, Vol. 51, Issue 4, Page 130, M.A. Myers, A. Mishra, D.J. Benson, Mechanical properties of
nanocrystalline materials. May 2006, with permission from Elsevier.
• Second, models which attempt to account for the increasing contributions of the
grain boundary and triple junctions as the grain size is decreased e.g., Zhou et al.
(2003). Such models often introduce new mechanisms such as grain boundary
diffusion (Chokshi et al., 1989), grain boundary sliding (Hahn and Padmanabhan,
1997) and grain rotation. Of critical importance here are the grain boundary and
triple junction volume fractions, which we have discussed in the text following
Figures 4.3 and 4.6.
Such breakdown models will not be examined in detail in this chapter, in part
because the experimental dataset remains too sparse to provide solid distinctions
between possible models (this is an excellent example of an area in which it is far
easier to develop a plausible model than it is to develop a good experiment). A
number of these mechanisms will, however, be discussed in other contexts later in
the book.
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5.5.2.6 Summary of Grain Size Effects on Strength
We have now spent a great deal of time discussing the effects of grain size on the
yield strength of materials, with specific reference to pure nanocrystalline materials.
This is perhaps appropriate, because it is the strength of nanomaterials that has been
best studied (in terms of mechanical properties) with respect to grain size. However,
this does not imply that other features of the polycrystalline materials do not have an
impact on nanocrystalline material behavior. In the next few paragraphs, we explore
briefly the consequences of grain size distributions, grain orientation distributions
and grain shape on the plastic behavior of nanocrystalline materials.
5.5.3 Other Effects of Grain Structure
5.5.3.1 Effects of Grain Size Distributions
All of the preceding discussions on the effects of grain size on yield strength have
essentially assumed that the polycrystalline metal had a single grain size. This is,
of course, a fallacy. It is nearly impossible to make materials with a single grain
size; instead, there is always a grain size distribution. An example of the grain size
distribution in a near-nanocrystalline metal is provided in Figure 5.15, which shows
a size distribution derived from TEM analysis of a consolidated iron sample. From
a mathematical viewpoint, the grain size distribution can be described in terms of
a probability density function, using for example, a Gaussian distribution, a lognormal distribution, or a Weibull distribution. For any given distribution function,
we can define the mean grain size and the variance of the grain size (a measure of the
spread of the distribution – the square root of the variance is the standard deviation).
Other distribution parameters such as the skewness may also be of interest. Our
discussion up to this point has simply examined the effects of the mean grain size.
How important is the spread of the grain size distribution?
Consider two materials that have identical mean grain sizes but very different
standard deviations in the grain size distribution. Material A is essentially monodisperse, that is, all of its grains are essentially the same size. Material B has a large
standard deviation in its grain size distribution, so that some of the grains are much
larger than the mean grain size and some are much smaller. In both cases, we assume
that all of the grains are randomly oriented and have essentially the same shape.
Imagine that both materials are subjected to a slowly increasing external load. Since
the grains are randomly oriented, there is an equal probability that deformation will
begin in any given grain at any given load. Thus the onset of plastic deformation
(that is the yield) should be essentially the same for the two materials, assuming
that yield can be perceived at very small strains.
There is an important subtlety here. It is certainly the case that the strength of
polycrystalline metals is a function of the grain size. However, the strength of any
given crystal is not a function of the size of the crystal (aside from pathological
5 Plastic Deformation of Nanomaterials
151
200nm
0.16
Number Fraction
0.14
d =138 nm
392 grains
0.12
0.10
0.08
0.06
0.04
0.02
0.00
40
440
840
1240
Grain Size (nm)
Fig. 5.15 Measured grain size distribution in a consolidated iron sample, based on 392 grains
measured from TEM images such as that shown above (Jia et al., 2003). While some of the grains
are clearly in the nanocrystalline range, others are more than 100 nm in size. Reprinted from Acta
Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E. Ma, Effects of nanocrystalline and
ultrafine grain sizes on constitutive behavior and shear bands in iron. July 2003, with permission
from Elsevier.
cases where free surfaces dominate the behavior of the crystal, so that dislocations
can easily run out to the free surface). In spite of this, a number of polycrystalline
plasticity simulations essentially assume that the strength of each crystal is proportional to the inverse square root of the crystal size (this is done in an effort to account
for the Hall-Petch behavior). The results of such simulations should be viewed with
caution for initial yield of the polycrystalline metal, since they essentially enforce
a grain boundary constraint that has not been established for single crystals. The
approximation is less of a concern for materials that have been subjected to a prior
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plastic deformation, because the development of geometrically necessary dislocations will provide an effective strengthening mechanism, resulting in a distribution
of effective yield strengths in crystals of various sizes.
The big effect of grain size distributions is subsequent to yield. The interaction
of the dislocations with the grain boundaries (or equivalently, the development of
geometrically necessary dislocation densities) is certainly a function of the grain
size. Thus the effective hardening of the material will be a function of the grain size
distribution, and a greater hardening rate should be expected for the material with
the widest spread in the grain size distribution. This approach is used to some benefit
in the development of nanomaterials with greater ductility than is typically obtained
in materials with narrow grain size distributions. The addition of the larger grains in
the wider grain-size distribution does not substantially decrease the yield strength,
for the reasons discussed previously, but does improve the overall hardening and
therefore the ductility. Indeed, bimodal grain size distributions appear to hold great
promise for nanostructured materials systems (Wang et al., 2002).
5.5.3.2 Effects of Grain Orientation Distributions
We have essentially assumed that the material has randomly oriented grains within
the polycrystalline structure. If the grains are not randomly oriented, then the overall
material behavior will reflect the anisotropy of the crystals to some degree. The effective orientation distribution of the grains leads to what is called the texture of the
polycrystalline material. The degree to which this is important obviously depends
on the anisotropy of the single crystal. The influence of texture is dramatic in low
symmetry crystal structures such as hexagonal-close-packed structures, and must
be accounted for in the modeling of such materials. It is not clear at this point that
the texture effect is significantly different in nanocrystalline materials as compared
to conventional grain size materials, but it is not immediately apparent that there
should be a coupling between grain size and texture. Of perhaps greater influence
are issues such as the development of specific kinds of grain boundaries in materials
in which texture is important after severe plastic deformation processes.
5.5.3.3 Effects of Grain Shape
The effect of grain shape on the yield strength can be very simply viewed in the
following manner. Imagine a material with the microstructure consisting of hexagonal grains of the morphology shown in Figure 4.5, and consider three very different
aspect ratios for the grains: α = 10, corresponding to rods, α = 1 corresponding
to nearly equisized structures, and α = 0.1 corresponding to plates. As we have
demonstrated earlier, the grain boundary and triple junction volume fractions are
very strong functions of these shapes. Further, it is not clear what mean grain size
one should use in estimating the response of the rod and plate morphologies. If one
5 Plastic Deformation of Nanomaterials
153
looks along and perpendicular to the axis of the hexagonal prism, one would estimate two different sizes (or equivalently, different distances between grain boundaries). If we assume that Hall-Petch behavior is operative in both directions, it
should be expected that the material behaves like an anisotropic plastic material,
with different yield strengths in the two directions. To avoid the complexity of
anisotropy, some authors use as an effective grain size the size of a cube of equivalent volume.
For example, consider the microstructure of the cryomilled Al 5083 alloy shown
in Figure 5.16. This material was first cryomilled, then HIPped (Hot Isostatic
Pressed, i.e., subject to high pressure while at high temperature) and extruded, resulting in an ultra-fine-grain (but not quite nanocrystalline) material. The resulting
microstructure shows grain anisotropy between the extrusion and transverse directions (Figure 5.16). Using multiple TEM observations, the linear dimensions of
the ultra-fine grains can be measured in each direction. The grains are elongated
along the extrusion direction with a mean length of 390 nm (Figure 5.16a). In the
transverse direction, equiaxed grains are observed with a mean diameter of 150 nm
(Figure 5.16b). If each of these independent grain sizes were to be used with a
Hall-Petch strengthening increment, the transverse direction would give a strengthening increment that is 60% larger than the extrusion direction. However, clearly the
entire grain participates in the deformation even if the loading is only in the transverse direction, so it is not obvious that the grain diameter in Figure 5.16b should
be used for loading in the transverse direction. Thus using the grain size in each
cross-section to compute the anisotropic strengthening is not necessarily correct
(indeed, in this particular material the strength in the transverse direction was found
to be smaller than the strength in the extrusion direction). What effective grain size
Fig. 5.16 TEM micrographs showing microstructure of a cryomilled 5083 aluminum alloy in
(a) extruded and (b) transverse directions (transverse to the extrusion axis). Note that the two
images were taken at slightly different magnifications. However, elongated grains are evident in
the direction of extrusion (Cao and Ramesh, 2009).
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Nanomaterials
d¯ should be used for this material within the Hall-Petch relation (Equation 5.11)?
An effective grain size can be calculated by assuming that the grains are cylin2
ders with a volume given by V = π d4 h (d being the diameter of the equiaxed grain
in the transverse direction, and h being the length of each elongated grain in the
extruded direction). The side of a cube of equivalent volume can be used to de1
1
2
3
fine the effective grain size d¯ through d¯ = V 3 = ( π d4 h ) , which works out to
d¯ = 190 nm.
5.6 Summary: The Yield Strength of Nanomaterials
Like conventional materials, the yield strength of nanomaterials is determined by the
ease with which dislocations can be moved through the material to develop plastic
strain. All of the strengthening mechanisms that are active in conventional grain
size materials are also active in nanocrystalline materials. The key characteristics
associated with the yield strength of nanomaterials, and specifically nanometals, are
the following:
• The primary additional strengthening mechanism that arises in nanocrystalline
materials is a result of the interaction of grain boundaries with dislocation generation and motion.
• The grain morphology of a nanocrystalline material can be usefully characterized
in terms of its grain size, grain size distribution, grain orientation distribution and
grain shape.
• The grain size has the dominant influence on the yield strength, determining
the strengthening through a Hall-Petch type response (Equation 5.11). Very high
strains can be obtained in nanocrystalline metals.
• At very small grain sizes (of the order of 25 nm or less in most materials) the
strength increment deviates from the Hall-Petch behavior, typically resulting in
less strengthening than would be expected through Hall-Petch.
• The so-called inverse Hall-Petch effect, where the strength of the material actually decreases with decreasing grain size, remains controversial and is not commonly seen.
• The breakdown of Hall-Petch behavior at small grain sizes can be described in
terms of grain boundary volume fractions, triple junction volume fractions, grain
boundary sliding, and grain boundary diffusion.
• The grain-size distribution appears to have at most a weak effect on yield strength
but may have a significant role to play in the hardening and the stability of
nanocrystalline systems.
• The grain orientation distribution leads to texture effects, and is particularly important in low-symmetry systems such as hexagonal-close-packed systems.
• Grain morphology effects can include variations in grain boundary volume fractions and the development of plastic anisotropy of the yield strength.
5 Plastic Deformation of Nanomaterials
155
5.7 Plastic Strain and Dislocation Motion
Now that we have an understanding of yield strength, let us consider the relationship
between the development of plastic strain and the motion of dislocations. Remember
that dislocations are line defects that move along crystallographic slip planes with
normal n in specific slip directions s. Consider the special case of slip along only one
slip system in a crystal, and let us suppose that the dislocation line of length l moves
the distance δ x. The corresponding increment of shear strain that is developed in
the crystal will be
bl δ x
,
(5.18)
γ=
V
where V is the volume of the crystal and b is the Burgers vector. Of course, this
is the shear strain increment along the direction of slip. Since the dislocation does
not move back, this is a permanent (plastic) strain increment. How does this plastic
shear strain increment relate to the tensorial plastic strain increment δ εipj that we
have discussed in the mechanics chapter? This relationship must depend on the slip
plane normal n and the slip direction s, and can be shown to be given by
δ εipj = γ µi j =
bl δ x
bl δ x
(si n j + ni s j ),
µi j =
V
2V
(5.19)
where
1
µi j = (si n j + ni s j )
(5.20)
2
is called the transformation tensor. In tensorial notation, the transformation tensor
is
1
(5.21)
µ = (s ⊗ n + n ⊗ s).
2
In practice, there will be many dislocations on many different slip systems. Given
that multiple slip systems may be operative at one time, it follows that the net plastic
strain increment δ ε˜ipj will be
α
blm (δ x) α
δ ε˜ipj = ∑
µ i j,
V
(5.22)
where the summation is carried out over the α separate slip systems. The dislocation
line length involved here is lm , which is the total length of the mobile dislocation
line segments (note that not all dislocations will be mobile, some being pinned by
various mechanisms). We define the mobile dislocation density as
ρm =
lm
,
V
(5.23)
so that the net plastic strain increment is
δ εipj = bρm (δ x)α µ α i j ,
(5.24)
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Nanomaterials
where the summation symbol has been dropped for clarity of the dependence of
the plastic strain increment on the dislocation slip. The distinction between the mobile dislocation density and the total dislocation density can be very important in
some materials under some conditions. In terms of the scalar shear strain, the analog to Equation (5.24) would be γ = bρm (δ x), a notation in common use in materials
science but somewhat confusing in that the mobile dislocation density is not welldefined over a single slip system.
5.8 The Physical Basis of Strain Hardening
Engineering Stress
Strain hardening in a material is, by definition, the increase of the strength with
the strain (when this is thought of as increase of strength with plastic work, the
process is called work hardening). Recall from Chapter 2 that after a material has
been loaded plastically to a given strain, unloading is elastic, and reloading is along
the elastic line (Figure 5.17) until the former flow stress has been achieved (which
is thus a new yield strength). Why would the yield strength increase with increasing strain? Examining the list of strengthening mechanisms that we have discussed
(solutes, dispersoids, precipitates, other dislocations, grain boundaries), we see that
only one of these changes with increasing strain, and that is the dislocation density.
All of the other strengthening obstacles are predetermined and do not themselves
change with strain (e.g., the solute concentration does not evolve with strain). Thus,
the primary mechanism for strain hardening is the increase of dislocation density
with increasing plastic strain, and the subsequent interaction of a moving dislocation with this increased array of dislocation-derived obstacles.
Strain hardening in single crystals is normally thought of as occurring in three
stages (called Stages I, II, and III) as slip continues beyond yield; some authors
σ
σ
σ
Engineering Strain
Fig. 5.17 Stress strain curve for a material, showing the elastic unloading response and the increased yield strength upon reloading (this is called strain hardening, since the material has become
harder because of the increased plastic strain).
5 Plastic Deformation of Nanomaterials
157
also define Stages IV and V. Stage I covers very little strain and shows very little
hardening (corresponding to single slip at the CRSS). The resolved shear strength
of a single crystal typically increases nearly linearly with strain in Stage II, and
this can occur for significant strains. For large strains the strength increases less
than linearly with strain, with a decreasing rate of strengthening – this is Stage III.
For the polycrystalline and in particular nanocrystalline systems of interest to this
book, the early stages of strain hardening are difficult to observe (except in some
pathological highly textured cases), and so we will ignore this nomenclature. Most
of what is observed in nanocrystalline materials is already effectively Stage III. Very
sophisticated theories of strain hardening have been developed for fcc metals, and
can be found in a number of reference works, e.g., Chapter 7 in Argon (2008). Strain
hardening in bcc and hcp materials is less well understood.
The emphasis on strain hardening in this book is through the evolution of the
dislocation density (Equation 5.8) and its consequences on the strengthening of the
material (Equation 5.7). Whether or not the dislocation density grows depends on
the relative rates of creation and annihilation of dislocations, which is a strong function of temperature. Indeed, it is possible to dramatically reduce the overall dislocation density by holding the material at a fixed high temperature for some time (a
process often referred to as recovery). Conversely, it is possible to increase greatly
the dislocation density by severely deforming the material (for example by rolling).
An additional point to recognize is that although Equation (5.7) uses an average
dislocation density, at high dislocation densities, it is energetically favorable for the
dislocations to pattern themselves into a dislocation substructure that may involve
dislocation cells or subgrains, and this may require special treatment (Mughrabi,
1983). It is known that the dislocation density in the material cannot increase indefinitely, and instead tends to saturate at some level corresponding to the development
of specific dislocation substructures (this correlates well with the effect of saturation
of strain hardening observed in many materials).
How does strain hardening evolve in nanocrystalline metals? Viewing nanocrystalline metals as simply special cases of polycrystalline metals in which the grain
size is very small, the contributions to strain hardening that are specific to nanocrystalline metals will arise from Equations (5.7) and (5.10), with the latter being involved in that the normalized Hall-Petch coefficient K̃ may well be a function of the
averaged dislocation density (Equation 5.16). Briefly, still assuming that strengthening mechanisms can be added in terms of strength increments, the overall strength
of the material may be written as
√
1
√
σnet = σi + AT µ b ρd + K̃ µ bd − 2 ,
(5.25)
where the evolution of the dislocation density is given by Equation (5.8). Note
that in this way of thinking the magnitude of the strength increment due to the
dislocation substructure itself does not change as a result of grain size. It is certainly possible that the normalized Hall-Petch coefficient K̃ does change with increasing dislocation density, although this appears to be a small effect (e.g., see
Table 5.4 for aluminum). The consequences of this in terms of overall constitutive
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function are considerable. Since a considerable degree of strengthening has already
been obtained in a nanocrystalline material using the reduction in grain-size, the
additional increment of strengthening generated through strain-hardening is relatively small, and as a consequence the apparent strain hardening of nanocrystalline
materials tends to be relatively small. In addition, many nanocrystalline materials
are made by processes that involve severe plastic deformation (see Chapter 1),
and so the dislocation substructure may have evolved into an equilibrium state,
where as many new dislocations are being generated as are being annihilated
within the substructure so that the substructure no longer evolves. This implies
that the strengthening increment provided by the dislocation substructure no longer
changes, resulting in the near-vanishing of the strain hardening. This has important implications for the stability of deformation in nanomaterials, as we shall see
later.
Consider, for example, the commonly used power law expression (Equation 2.95)
for describing the tensile stress-strain curve for a material:
n
ε
σnet = σ0
,
(5.26)
ε0
where ε0 = σE0 is a reference strain and the material parameter n is called the strain
hardening index (the stresses and strains in this equation are the true stresses and
strains (see Chapter 2). Let us define a new variable q, called the normalized hardening, which is a normalized measure of the strain hardening:
q=
1 ∂ σnet
.
σnet ∂ ε
(5.27)
This measure has the advantage that at any given strain it is proportional to the
strain hardening index n, and in addition it is normalized by the strength of the
material (so that it is possible to compare two materials with very different strengths
in terms of their relative hardening with strain). The term ∂ ∂σεnet is simply the tangent
to the stress-strain curve at any given strain (the slope of the stress-strain curve, a
more intuitive measure of the hardening).
5.8.1 Strain Hardening in Nanomaterials
When one considers nanocrystalline materials, the normalizing factor σnet becomes
very large because of the grain size strengthening (Equation 5.25, assuming small
grain sizes) and so the normalized hardening becomes small. That is, the apparent
strain hardening of the nanocrystalline material is small even though quantitatively
the additional strengthening provided due to the dislocation density may be the same
as in the conventional grain sized version of the material. Indeed, because σnet is
dominated by the grain size term for nanocrystalline materials, for such materials
we will have
5 Plastic Deformation of Nanomaterials
159
q∝
√
(5.28)
d.
This is in fact observed in many nanocrystalline materials, as demonstrated in
Figure 5.18, which presents the q (at a plastic strain of 3%) plotted against the square
root of grain size for various materials reported in the literature. Reduced values of
q can be seen with decreasing grain size in the figure, where all the data is from
uniaxial compression (thus avoiding the problems associated with the apparent reduction of hardening during necking in tension). The reduction of q with decreasing
grain size is similar to the reduction of observed ductility with decreasing grain size
shown in Koch et al. (1999).
The influence of strain hardening on the ductility of materials is dramatic. A
fairly simple way to think about this is to consider the simple tension test discussed
in Chapter 2. Simple tension of a perfectly plastic material leads to instabilities at
very small strains, because any perturbation in the geometry of the specimen should
result in a local increase in the true stress because of the reduction in the area, but the
inability of the material to sustain a higher stress implies that the strain in that area
has to increase dramatically (and is essentially bounded only by the finite amount
of energy available for the deformation). Thus the necking instability begins very
Fe - Current Study
Pd [10]
Fe - Khan et al [2]
Cu [10]
Fe - da Silva & Ramesh [3]
Cu [11]
Alpha-Fe - Ostwaldt et al. [4]
Cu [12]
Fe-10%Cu - Carsley [5]
Cu [13]
Fe28Al2Cr - Jain & Christman [6]
Ni [13]
Ti - Current Study
Cu [14]
Ti - Chichili et al. [8]
Cu [15]
Ti - Nemat-Nasser et al. [9]
Normalized Strain Hardening, q
20
15
10
5
0
–5
2
4
6
Grain Size, d 1/2 (d in microns)
8
10
Fig. 5.18 Variation of normalized strain hardening with the square root of grain size for a variety
of nanocrystalline and ultra-fine-grain materials (from Jia’s doctoral dissertation, Jia et al., 2003).
The reference numbers correspond to those in Jia’s dissertation.
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rapidly in a perfectly plastic material, leading to an effective loss of ductility. The
presence of some strain hardening allows one to delay the onset of the necking instability, effectively providing some additional ductility. As a consequence, unless
a strain-hardening mechanism is incorporated in the material, nanocrystalline materials tend to have very little ductility in tension. This is discussed further in the
section on failure mechanisms.
5.9 The Physical Basis of Rate-Dependent Plasticity
The material behaviors that have been discussed up to this point have essentially
been relationships between the stress, the strain, and the microstructure. However,
the behavior of most materials is also a function of the rate of deformation, rather
than simply of the deformation alone (that is, the behavior is also a function of
the strain rate rather than just of the strain). This rate-dependence arises from three
sources:
• The force required to move a dislocation depends, in a fundamental way, on the
velocity of the dislocation. Moving the dislocation at a higher velocity requires
a higher force, or a higher stress. Since the velocity of dislocation motion is
directly related to the strain rate, a higher stress is required to achieve a higher
strain rate in the material. This is essentially a result of dislocation dynamics.
• Dislocations in materials are constantly in motion as a result of available thermal
energy. At higher temperatures, they oscillate to a greater degree around a mean
state. This results in a relationship between the motion of the dislocation and the
temperature, resulting in both easier deformation at higher temperatures but fixed
rate of deformation, and more resistance to deformation at higher velocities but
fixed temperature. This is essentially a result of thermal activation.
• The rate of increase of the dislocation density is a function of the rate of deformation because the mechanisms associated with the annihilation and creation of
dislocations depend on the strain rate. Since the dislocation density affects the
strength of the material, the strengthening of the material depends implicitly on
the strain-rate. This is essentially a result of dislocation substructure evolution.
5.9.1 Dislocation Dynamics
Many experimental observations on single crystals have demonstrated that the dislocation velocity is a strong function of the applied stress. Many of these experimental
results are collected in Figure 5.19, taken from the review article by Clifton (1983).
For small stresses at small dislocation velocities, the relationship between the stress
and the velocity is not clear and may vary substantially from material to material.
However, at sufficiently high stresses and sufficiently high velocities, the velocity first becomes linearly dependent on the applied shear stress (a domain known
5 Plastic Deformation of Nanomaterials
161
Fig. 5.19 The dependence of dislocation velocity on shear stress for a variety of materials, from
Clifton (1983). Note that a limiting velocity is expected, corresponding to the shear wave speed.
There is a domain in the figure where the dislocation velocity is linear with the applied shear
stress, and this is called the phonon drag regime. Reprinted from Journal of Applied Mechanics,
Vol. 50, p. 941–952, R.J. Clifton, Dynamic Plasticity. 1983, with permission from original publisher, ASME.
as dislocation drag) and then becomes highly nonlinear, with the stress becoming
very large
as the dislocation velocity approaches the limiting speed of a shear wave
(cs = µρ , where µ is the shear modulus and ρ is the mass density) in the material
(this is called the relativistic domain).
To connect this behavior with the rate-dependence of the stress, we use the rate
form of the equation defining plastic strain in terms of dislocation motion (Equation 5.24), so that we can relate the plastic strain rate to the dislocation velocity:
ε̇ipj = bρm ẋµi j = bρm vd µi j ,
(5.29)
where vd is the dislocation velocity and the tensor µi j relates to the slip system and
was defined in Equation (5.20). A scalar strain rate measure can be obtained from
this equation by defining a scalar called the effective plastic strain rate (analogous
to our definition of the von Mises stress in Chapter 2) as
ε̇¯ p = 2ε̇ipj ε̇ipj ,
(5.30)
and using Equation (5.29) we obtain
ε̇¯ p = bρm vd .
(5.31)
This equation is sometimes called the Orowan equation, although in materials
science textbooks it is usually written in terms of a shear strain rate γ̇ which is
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Nanomaterials
identically equal to our effective plastic strain rate ε̇¯ p , i.e, γ̇ = ε̇¯ p . For a tensorial
stress state σi j , we can also extract a scalar shear stress measure, which is the von
Mises stress σ̄ (Chapter 2). Now Figure 5.19 gives us a relationship between the
shear stress and the dislocation velocity for a given material: vd = f (σ̄ ). Using this
in Equation (5.31) we obtain
ε̇¯ p = bρm f (σ̄ ).
(5.32)
This can be used to obtain a measure of the strengthening due to dislocation dynamics (note that in obtaining this equation we have not considered any of the other
strengthening mechanisms). For example, if we assume that the applied stress is
such as to put us in the linear portion of Figure 5.19, we have vd = f (σ̄ ) = Bσ̄ ,
where B is a constant called the drag coefficient, and it follows that in this case
ε̇¯ p = bρm Bσ̄ ,
(5.33)
which corresponds to a strengthening increment due to dislocation drag that shows
up as a rate-dependent strengthening that is linear in the strain rate:
Δ σ̄ =
1 ¯p
ε̇ .
bρm B
(5.34)
The rate-dependent strengthening of materials due to dislocation drag is generally observed only at very high strain rates (conventionally assumed to be rates
larger than 106 s−1 ). At lower rates, the thermal activation and substructure evolution are traditionally considered to dominate the behavior.
5.9.2 Thermal Activation
The mechanisms discussed up to this point have ignored the effects of temperature.
In practice, atoms do not stay in fixed positions within the lattice, but rather vibrate
around a mean position with the degree of vibration determined by the current temperature (eventually the temperature becomes high enough that the atoms are able
to move freely, corresponding to melting). Similarly, defects within the crystal are
in constant motion at nonzero temperature, oscillating gently around the mean position that is defined by a minimum energy state. One way to visualize deformation
is to imagine that the units of deformation (in this case dislocations) are moving
from one minimum energy state to another, and that between these two minima is
an energy barrier that represents the difficulty of getting around a specific obstacle.
Work is required to move the dislocation over the energy barrier, and this work is
typically supplied by the applied stress. However, in any given state, the energy of
the dislocation is actually oscillating around the minimum because of the available
thermal energy, and so the effective height of the energy barrier must be compared
with the available thermal energy.
5 Plastic Deformation of Nanomaterials
163
Consider the dislocation that is attempting to cross an energy barrier. The frequency with which it attempts to cross this barrier is designated ν0 . Using the
Arrhenius approximation, the frequency ν with which it is successful in crossing
the barrier is
−G
,
(5.35)
ν = ν0 exp
kB T
where G is the height of the barrier (sometimes referred to as Δ G in the literature to
emphasize the fact that it is a difference of energies), kB is the Boltzmann constant,
and T is the absolute temperature. In general, the dislocation perceives the identical
energy barrier moving in either direction, i.e., the barrier height is G whether the
dislocation is moving forwards or backwards. Since the probability of moving forwards or backwards is identical, the average motion of the dislocation is zero, and
this corresponds to the expected thermal vibration around the mean position. However, when a shear stress τ is applied in a specific direction, the stress essentially
provides additional work that can be used to overcome the energy barrier. In effect,
the dislocation perceives a lowered energy barrier in the direction of the applied
stress.
How does the barrier height depend on the applied stress τ ? In general, this will
be some function G(τ ) that will change depending on the specific obstacle that
defines the barrier (an example of such an obstacle might be a precipitate particle).
The standard approach to handling functions that are poorly known is to use a Taylor
series expansion of the function about the point at which the value of the function is
known (in this case, that point is τ = 0, and G = G0 when τ = 0):
G(τ ) = G0 +
∂G
| τ + higher order terms,
∂ τ τ =0
(5.36)
where only the first term has been written down because it is assumed
that τ is small
(this is called linearizing the function). Identifying ∂∂Gτ |τ =0 as a constant (−Ω ),
the barrier height becomes G0 − τΩ in the direction of the applied stress, where Ω
is a parameter (with the dimensions of volume) that is called the activation volume.:
Ω =−
∂G
| .
∂ τ τ =0
(5.37)
For the moment, let us assume that the activation volume is a constant. Note that
the dislocation perceives an increased activation barrier G0 + τΩ in the direction
working against the applied stress. Thus because of the applied stress, the frequency
ν f with which the dislocation is successful in crossing the barrier in the forward
direction, i.e., the direction of the applied stress, is
ν f = ν0 exp
−G0 + τΩ
,
kB T
(5.38)
while the frequency νb with which the dislocation is successful in crossing the barrier in the backward direction is
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Nanomaterials
νb = ν0 exp
−G0 − τΩ
.
kB T
(5.39)
Given the arguments of the exponentials, the forward frequency will be greater
than the backward frequency, so the net motion of the dislocation is in the forward
direction and has a success frequency
νnet = ν0 sinh
−(G0 − τΩ )
.
kB T
(5.40)
If the dislocation moves the distance λ (that is, the slip distance) as a result of
crossing the barrier, the corresponding dislocation velocity is
vd = λ νnet = λ ν0 exp
τΩ
−G0
sinh
.
kB T
kB T
(5.41)
This is called a thermally activated motion. The corresponding effective plastic
strain rate is then (using Equation 5.31) given by
τΩ
σ̄ Ω
−G0
−G0
ε̇¯ p = bρm vd = bρm λ ν0 exp
sinh
= bρm λ ν0 exp
sinh
, (5.42)
kB T
kB T
kB T
kB T
where the shear stress τ has been replaced by the effective stress σ̄ for consistency
of the definitions (otherwise, using τ , we would have to have γ̇ on the left hand side
of Equation 5.42). This provides a relationship between the applied stress and the
strain rate that results from the thermally activated motion of the dislocations. Now,
the definition of the hyperbolic sine function is
sinh x =
ex − e−x
,
2
(5.43)
and when the argument x of the function is large, we have ex ≫ e−x , so that sinh x →
ex
σ̄ Ω
2 . In our case we have x = kB T , so that for reasonably large stresses the forward
rate is much higher than the backward rate (which can therefore be ignored), and
Equation (5.42) then reduces to
σ̄ Ω
σ̄ Ω
−G0
ε̇¯ p = bρm λ ν0 exp
exp
= ε̇0 exp
,
kB T
kB T
kB T
(5.44)
0
where we have defined ε̇0 = bρm λ ν0 exp −G
kB T . This equation can be solved for the
stress in terms of the strain rate, providing a relation for the strengthening increment
due to thermally activated motion of the dislocations (remembering that only this
mechanism is being considered):
Δ σ̄ =
kB T ε̇¯ p
ln ,
Ω
ε̇0
(5.45)
5 Plastic Deformation of Nanomaterials
165
where it should be noted that ε̇0 is typically much greater than ε̇¯ p . Note that this
dependence of the stress on the strain rate is logarithmic, and much weaker than
that obtained from the dislocation drag argument of Equation (5.34).
Although this is a very common approach to estimate the effects of thermal activation on rate-dependent response, note that we have made two possibly contradictory approximations to obtain this relationship. First, we assumed that τ was
sufficiently small in order to linearize the barrier height G(τ ) (Equation 5.36);
second, we assumed that τ was sufficiently large in order to reduce the sinh function
to an exponential function (Equation 5.44). Is there a range of τ where both of these
conditions are satisfied? It is not obvious that there is, because the acceptable range
depends on the specific obstacle being considered. This caveat is worth recalling as
specific models are built for any given thermally activated deformation mechanism,
but is not a concern if it is true that (for that material) G(τ ) = G0 − τΩ for all τ .
An example of the rate-dependent behavior of a material is shown in Figure 5.20,
which presents a range of strength data for A359 aluminum (a cast aluminum alloy).
All of the strengths were measured at a fixed strain but varying strain rates; because
of the nature of high strain rate experiments, it is not generally possible to measure accurately the yield strength of materials at high strain rates. A logarithmic
dependence like that in Equation (5.44) appears to be a reasonable description of
the rate-dependent behavior for strain rates less than 104 s−1 .
3 102
A359
Flow Stress (MPa)
2.5 102
2 102
1.5 102
1 102
5 101
0 –4
10
10–2
100
102
Strain Rate (s–1)
104
Fig. 5.20 Rate-dependence of the flow stress of A359 aluminum alloy over a wide range of strain
rates. All of the flow stresses are plotted at a fixed strain of 4% (because of the nature of high strain
rate experiments, it is not generally possible to measure accurately the yield strength of materials
at high strain rates). This result is due to the work of Yulong Li, and includes compression, tension
and torsion data.
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Nanomaterials
5.9.3 Dislocation Substructure Evolution
There is one significant concern that arises when rate-dependence of the material
behavior is presented in the form of a figure like Figure 5.20: is it reasonable to
compare flow stresses at fixed strain? The difficulty is that some materials, particularly fcc metals, appear to have strain hardening behavior that is itself a function of
strain rate (higher strain rates leading to higher strain hardening). Thus the increase
in strength (at fixed strain) with strain rate can be viewed as a strain hardening behavior rather than a “truly rate-dependent” behavior.
Strain hardening is essentially a consequence of the evolution in dislocation density. It turns out that in some materials the rate of change of the dislocation density is
a function of the rate of deformation: the dislocation density appears to grow faster
at higher rates of deformation. The evolution of the dislocation density ρd is given
by Equation (5.8):
∂ ρd
= Ac fc (ρd , ε̇ ) − Aa fa (ρd ),
(5.46)
∂t
and it appears that (for fcc metals) the rate of creation of dislocations fc is a function not just of ρd but also of the strain rate ε̇ . As a consequence, ρd grows faster
at higher strain rates, and then because of Equation (5.7) the strength of the material also increases with strain rate (because the strain hardening has increased).
This kind of rate-dependence is said to be the result of dislocation substructure
evolution.
This particular situation brings out one of the classic differences between the mechanics and materials perspectives in this area: when is a material considered to be
rate-dependent? From a mechanics perspective, the material is described in terms of
a constitutive relation such as σ = f (ε , ε̇ , T ) where ε and ε̇ are the current strain
and current strain rate, and T is the current temperature. The constitutive function
f () has a specific form and specific values of parameters for each material, but the
function typically does not contain microstructural parameters such as the dislocation density. If the strength of a material changes with strain rate at fixed strain and
temperature, the material is rate-dependent from the mechanics perspective. From
a materials perspective, the material behavior is typically described in terms of microstructural parameters such as the dislocation density, which are viewed as evolving with the deformation, and the material is typically said to be rate-dependent if
the behavior is sensitive to strain rate at fixed (dislocation) structure. Thus from a
materials perspective the increase of strength with strain rate as a consequence of the
increase of dislocation density with strain rate may not be considered to be a ratedependent behavior. The identical behaviors, therefore, are viewed differently by
individuals with mechanics and materials backgrounds. Materials scientists would
prefer to look at figures like Figure 5.20 with the flow stress measured at constant
structure. This can be accomplished, for example, through strain rate jump tests,
where the strain rate is changed suddenly and the immediate response of the stress
is measured, assuming that the dislocation substructure has not had time to evolve
(Figure 5.21).
5 Plastic Deformation of Nanomaterials
167
490
480
Stress (MPa)
470
460
450
440
430
420
Cold Deformed Cu
ECAP+Cold Deformed Cu
0.02
0.024
0.028
0.032
0.036
Strain
Fig. 5.21 Results of strain rate jump tests on severely plastically deformed copper (both coldworked and ECAPed). Note the jump in the stress associated with the step increase in the strain
rate on a specimen. One of the curves has been shifted to the right for ease of discrimination. Step
increases in strain rate (jump from a lower strain rate to a higher strain rate) are used because
a step decrease will exacerbate the effect from the machine compliance, and the interpretation of
experimental data becomes more involved. In this case the strain rate was increased by a factor of 2
between consecutive rates. Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng,
K.T. Ramesh, E. Ma, Effect of nanocrystalline and ultrafine grain sizes on the strain rate sensitivity
and activation volume: fcc versus bcc metals. Sep. 2004, with permission from Elsevier.
This effect of the evolution of substructure would be captured from a mechanics perspective by considering the effects not just of the current strain rate but of
the strain rate history. It is clear that if the dislocation substructure evolves differently at different strain rates, then the strength of the material is determined not
just by the current strain rate but also by the history of the strain rate, because
this controls the dislocation density and thus the strength through Equation (5.7).
In this viewpoint, fcc metals are not just rate-dependent but also strain rate history
dependent.
5.9.4 The Rate-Dependence of Nanomaterials
Although experiments employing strain rate changes are very useful in revealing
deformation mechanisms, there is very limited experimental data on the strain-rate
sensitive mechanical properties of nc/UFG metals. The primary material parameter
of interest is the strain rate sensitivity, which is related to the activation volume
defined earlier (Equation 5.37). A common definition of the strain rate sensitivity
(common in engineering) is in terms of a rate sensitivity parameter m defined as
m=
∂ ln σ
ε̇ ∂ σ
=
.
∂ ln ε̇
σ ∂ ε̇
(5.47)
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Nanomaterials
This measure of rate sensitivity is valid for any constitutive function (for examm
ple, it is directly the strain rate exponent in the power law function: σ = σ0 ε̇ε̇0 ), and
is an indicator of the strain-rate response that is useful for technological purposes.
The “engineering” strain rate sensitivity m and the activation volume Ω are related
as follows (in the particular case of thermal activation):
m=
kB T
.
σΩ
(5.48)
Experiments of the type shown in Figure 5.21 are typically used to estimate m,
using the slopes of linear fits on a log-log plot of stress vs strain rate. The activation volume can then be computed using Equation (5.48). The dependence of
rate-sensitivity on grain size can be obtained by performing such experiments on
materials with a range of grain sizes.
5.9.4.1 FCC Nanometals
Carreker and Hibbard (1953) first discussed the grain size dependence of the rate
sensitivity for tensile deformation of high-purity Cu. They observed that m increased
slightly from 0.004 (large d) to 0.0072 (small d), but the grain size range studied was
very small (12 − 90 µm). Early work on Ag and Cu also showed that polycrystalline
samples had larger values of m than single crystals (Conrad, 2003). Coarse-grained
Cu (d = 40 µm) has a rate sensitivity m of the order of 0.006 according to Follansbee
and Kocks (1988). More recent studies of ultrafine-grained Cu also showed elevated
m (Gray et al., 1997). A very large m (0.14) for Cu was reported by Valiev et al.
(2002) on a material obtained using 16 passes of ECAP. The rate-sensitivity (m =
0.03) of a UFG Cu processed by ECAP+CR has been measured at large plastic
strains (Wang et al., 2002b) using load-reload compression tests. The tensile jump
tests shown in Figure 5.21 give m = 0.02 for a similar ECAPed Cu (d = 200 nm).
Thus the available data indicate a substantial increase in m when the grain size is
reduced into the UFG and nc regime, regardless of the technique used to process the
material. This behavior is summarized in Figure 5.22, showing the change in the rate
sensitivity of copper over a wide range of grain sizes. Similar behavior is observed in
other fcc metals, such as nickel and gold. Note that all of these rate-sensitivities are
for deformations that are at nominal strain rates from 10−4 to 103 s−1 . The apparent
rate-sensitivity is much higher (m 0.5) for all grain sizes of fcc metals in the creep
domain, because of diffusional processes.
What causes the increase in rate-sensitivity with decreasing grain size in fcc metals? There is no clear explanation for this at present, but one plausible explanation
is that the evolution of dislocation substructure (which is known to be important in
fcc metals) is controlling the behavior. For coarse-grained fcc metals, the primary
obstacles to the motion of glissile dislocations are known to be forest dislocations,
leading to the rate dependence of flow stress through thermal activation. According
to the work of Conrad (1964), the activation volume can be written as
5 Plastic Deformation of Nanomaterials
169
Rate-sensitivity, m
0.1
0.08
0.06
[47,48]
[41]
[37]
[49]
[21]
Cu, this work
[22]
Cu, this work
[25]
Cu, this work
0.04
0.02
0
101
102
103
Grain size, nm
104
105
Fig. 5.22 The rate-sensitivity of copper as a function of the mean grain size, including the
nanocrystalline, ultra-fine-grain and coarse-grain domains. Data is presented for materials made
through a variety of processing routes, some of which involve severe plastic deformation. The
figure is taken from the work of Wei et al. (2004a). Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline and ultrafine grain
sizes on the strain rate sensitivity and activation volume: fcc versus bcc metals. Sep. 2004, with
permission from Elsevier.
Ω = bxl ∗ ,
(5.49)
where x is the distance swept out by a gliding dislocation during one activation
event and l ∗ is the length of the dislocation line segment involved in the thermal
activation (this is also the Friedel sampling length, which scales with the average
contact distance between two obstacles, so it is the obstacle spacing). For large-grain
fcc metals with forest cutting as the dominant mechanism for plastic deformation,
l ∗ is roughly the average forest spacing which scales with the forest density √1ρ f ,
or following Taylor’s relation (Equation 5.7), l ∗ with µτb . The sliding distance x
may also decrease slightly with increased applied stress (Kobrinsky and Thompson,
2000). It thus follows that the activation volume would decrease with applied stress.
As an example, for heavily deformed Cu, Ω decreases to about 100 b3 (Dalla Torre
et al., 2004).
Combining Equations (5.48) and (5.49) we have
m=
kB T
kB T
=
,
σΩ
σ bxl ∗
(5.50)
in which x is roughly approximated to be constant. Therefore, the variation of m has
to be understood from the combined variation of σ and l ∗ through the product σ l ∗ .
We expect that the flow stress has contributions arising from both grain boundaries
and dislocations, and it can be written as
1
√
σ = σ0 + AT µ b ρ f + ky d − 2 ,
(5.51)
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Nanomaterials
where the first term accounts for all of the other strengthening components, the
second arises from the Taylor contribution (Equation 5.7), and the third arises from
the Hall-Petch relation. The obstacle spacing or “sampling length” l ∗ is assumed to
have two possible limits, l ∗ 1 and l ∗ 2 :
1
l ∗ 1 = α1 √
ρf
(5.52)
l ∗ 2 = α2 d,
(5.53)
where α1 and α2 are proportionality factors. These limits are chosen because they
represent, respectively, the two classes of obstacles of interest: forest dislocations
(l ∗ 1 ) and grain boundaries (l ∗ 2 ). Of the two possible limits for the obstacle spacing,
l ∗ 1 is likely to be the controlling length scale when the grain sizes and dislocation
densities are large, and l ∗ 2 is likely to be the controlling length scale at very small
grain sizes (and small dislocation densities). In this model, as the grain size is decreased there will be a transition grain size when the controlling sampling length
changes from l ∗ 1 to l ∗ 2 . A crude estimate of the transition size can be obtained by
asking when l ∗ 1 and l ∗ 2 are of the same order of magnitude, and assuming that α1
and α2 are of the same order. This gives
1
dtransition ≈ √ .
ρf
(5.54)
Using typical dislocation densities in metals of 1012 − 1017 m−2 , one estimates that
dtransition is of the order of 10–500 nm.
What are the perceived material behaviors in the two limits? At conventional
grain sizes we know that the forest cutting is dominant, so that we have (using
Equations 5.51 and 5.52)
1
1
1
√
σ l ∗ = σ l ∗ 1 = (σ0 + AT µ b ρ f + ky d − 2 )α1 √ = AT µ bα1 + ky α1 , (5.55)
ρf
dρ f
where the initial strengthening term has been dropped since it is usually small for
pure fcc metals (note that this is not a good approximation for alloys). Using Equation (5.55) in Equation (5.50), we obtain
m=
1
kB T
bx AT µ bα1 + ky α1 √1
,
(5.56)
dρ f
Equation (5.56) indicates that when the dislocation density is increased (for example by large deformations) in conventional grain-sized metals, the strain rate sensitivity should increase. This explains the observations reported by Zehetbauer and
Seumer (1993).
Consider now the case of nanocrystalline fcc materials, where the grain size is
refined into the UFG/nc regime, the controlling sampling length becomes l ∗ 2 , and
we now have
5 Plastic Deformation of Nanomaterials
171
√
1
√
√
σ l ∗ = σ l ∗ 2 = (σ0 + AT µ b ρ f + ky d − 2 )α2 d = AT µ bα2 d ρ f + ky α2 d, (5.57)
and now
m=
1
kB T
√ .
√
bx AT µ bα2 d ρ f + ky α2 d
(5.58)
Equation (5.58) indicates that when the grain size is refined into the UFG/nc
regime, the rate sensitivity parameter m should increase with reduced grain size, as
was observed in Figure 5.22. Further, as d is reduced to below the transition size,
the forest dislocation density in the grain interior may become very low, whereas
the obstacle density associated with grain boundaries becomes very high. It is thus
plausible that the controlling obstacles are the grain (or subgrain) boundaries.
5.9.4.2 BCC Nanometals
Very different rate-dependent behavior is observed in nanostructured bcc metals.
In these materials it is observed that the effective rate-sensitivity decreases with
decreasing grain size. Such a decrease has been observed in iron, tantalum, tungsten
and vanadium, and a summary of these data is presented in Figure 5.23.
0.07
Rate Sensitivity, m
0.06
0.05
0.04
Consol. Fe [16]
ECAP Fe, C [34]
nc Fe, [51]
nc Fe, [53]
ECAP Ta, C [35]
Consol. V [52]
ECAP Fe
ECAP+Roll Fe
ECAP+Roll Ta
JMR model
0.03
0.02
0.01
0 1
10
102
103
Grain size (nm)
104
105
Fig. 5.23 The rate-sensitivity of bcc metals as a function of grain size from a variety of sources
(the figure is a variant of one published by Wei et al. (2004a). The decrease of rate sensitivity with
decreasing grain size is the opposite behavior to that observed in fcc metals. Reprinted from Materials Science and Engineering, Q. Wei, S. Cheng, K.T. Ramesh, E. Ma, Effect of nanocrystalline
and ultrafine grain sizes on the strain rate sensitivity and activation volume: fcc versus bcc metals.
Sep. 2004, with permission from Elsevier.
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Nanomaterials
The source of this behavior is the deformation mechanism that dominates bcc
metal plasticity, that of kink nucleation and propagation, and is discussed in detail in
the next section. In brief, the explanation is that the plastic deformation of nanocrystalline bcc metals is controlled by the Peierls-Nabarro stress and the temperatureand rate-insensitive grain boundary controlled Hall-Petch mechanism, and for the
very high-strength nanocrystalline bcc metals these terms are much larger than the
rate-dependent terms, resulting in a reduction in the effective rate-sensitivity.
5.10 Case Study: Behavior of Nanocrystalline Iron
As a case study, let us work through understanding the plastic deformation of
nanocrystalline iron (we discussed the elastic behavior of single crystal iron in
Chapter 2). Iron is a bcc metal and has been studied extensively both for fundamental science reasons and as the primary constituent of steel. Plasticity in metals is
a consequence of dislocation motion, and the nature of dislocation slip in bcc metals is somewhat different than that in fcc systems. One of the major differences is
that in bcc metals the edge and screw dislocations (or edge and screw components
of a dislocation loop) have vastly different mobilities. Screw dislocation cores are
dissociated into a non-planar configuration in bcc structures, which makes it difficult for them to move (Duesbery and Vitek, 1998). In contrast, edge dislocations
move freely, and the mobility of the edge components of dislocation loops is much
higher than the mobility of the screw components. As a consequence TEM images
of dislocations in bcc metals tend to show lots of screw dislocations and few edge
dislocations, because the edges have all run out to the free surfaces. Since the edges
are so mobile, plastic deformation in bcc metals is generally controlled by the motion of screw dislocations, which means the plasticity is controlled by the barriers
to the motion of the screws.
The primary barriers to the motion of screw dislocations in bcc metals turn out
to be lattice-related, rather than forest dislocations: the Peierls-Nabarro stress is relatively high. This high friction stress derives in part from the fact that the screw
dislocation core tends to be nonplanar. These lattice barriers essentially define the
energy landscape in the thermal activation theory discussed previously. Moving the
entire dislocation line across the lattice costs a significant amount in energy terms
and, nature being as resourceful as she is, the dislocation finds an energetically
cheaper way to move. Essentially, rather than move the entire screw dislocation
at one time, one small segment of the screw collapses back into a linear configuration and jumps across the barrier, resulting in a kinked configuration (Figure 5.24).
The sides of the kink pair (which are called the kinks) turn out to have an edge orientation and have high mobility, so they fly out to the sides, causing effectively a
propagation of the screw across the barrier. This is the mechanism of motion of the
screws in bcc metals, and the kink mechanism determines the mobility of the screw
dislocations.
5 Plastic Deformation of Nanomaterials
173
Fig. 5.24 Schematic of motion of screw dislocations by kink pair nucleation and propagation. The
dislocation is visualized as a line that needs to go over the energy barrier. Rather than move the
entire line over the barrier, the dislocation nucleates a kink, which jumps over the barrier. The
sides of the kink pair have an edge orientation, and so fly across the crystal because of their high
mobility, resulting in an effective motion of the screw dislocation.
The kink mechanism involves two steps: kink nucleation and then the propagation of kink pairs. Kink propagation is rapid, so (at low homologous temperatures)
the screw dislocation mobility is controlled by kink nucleation. The kink pair nucleation rate νk is given by
−Gk
νk = νk0 exp
,
(5.59)
kB T
where νk0 is the attempt frequency and Gk is the activation energy for kink-pair
nucleation. The corresponding average dislocation velocity is
vd = λ νk = λ νk0 exp
−Gk
,
kB T
(5.60)
where λ is the distance between the barriers (of the same order as the Burgers vector
b in bcc metals). The plastic shear rate corresponding to this mechanism is then,
using the Orowan equation,
ε̇ p = ρm bλ νk0 exp
−Gk
,
kB T
(5.61)
where a Taylor factor of the order of 3 has been absorbed into the assumed value for
the mobile dislocation density in Equation (5.61), since ρm is usually only known
to within an order of magnitude in any case. The key remaining question is the
dependence of the activation energy on the stress (e.g., we had assumed a linear
dependence in Equation (5.36). A common approach to this definition of the barrier
shape is to use an empirical function first proposed by Kocks et al. (1975):
e a b
σ̂
Gk = Gk0 1 −
,
σ0
(5.62)
where a and b are parameters that define the barrier shape, σ̂ e is the effective stress
(which we shall call a Peierls stress) for this thermally activated mechanism, and σ̂0
is a material parameter. Various values for a and b are used in the literature, with particularly common values (for bcc metals) being a = 12 , b = 32 . Using Equation (5.62)
in (5.61) and solving for the stress, the effective thermally assisted “Peierls stress”
σ̂ e is then given by
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Nanomaterials
σ̂ e = σ̂0
1 1a
b
kB T
ρm bλ νk0
1−
.
ln
p
Gk0
ε̇
(5.63)
The collection of microscopic terms in the numerator of the logarithmic function
is sometimes referred to as a “reference” strain rate ε̇0 = ρm bλ νk0 . Considering
typical values for these parameters for an annealed metal, we see that ε̇0 ≥ 105 s−1
(although usually many orders of magnitude larger).
The effective Peierls stress σ̂ e is the component of the flow stress σ related to
thermally activated motions over short-range barriers (corresponding to the Peierls
barriers in this case). To develop the constitutive function for the material we need
to add to this the “athermal” stress corresponding to that needed to overcome the
long-range barriers perceived by the moving dislocations, i.e., σ = σathermal + σ̂ e ,
so that we obtain the flow stress as
e
σ = σathermal + σ̂ = σathermal + σ̂0
1 a1
b
kB T
ρm bλ νk0
ln
1−
.
p
Gk0
ε̇
(5.64)
The primary part of the athermal stress (i.e., the most important long-range barriers) for the materials that we are considering is that arising from the grain boundaries, and this brings in a grain size dependence that we assume to be of the HallPetch form:
√
σathermal = σa0 + ky d + g(ε p ),
(5.65)
where we have also included a strain hardening function g(ε p ) in an additive
fashion, since experimental evidence for bcc metals generally indicates that the
strain hardening function is not affected by strain rate or temperature in the lowtemperature regime, e.g., Ono (1968). We now have an effective response function
that should be applicable to our bcc iron:
2 2
3
√
kB T
ε̇0
p
σ = σa0 + ky d + g(ε ) + σ̂0 1 −
ln
,
(5.66)
p
Gk0
ε̇
using the common bcc values for a and b in Equation (5.66). We have thus obtained
a constitutive formalism that we can use to compare with experimental data. The
strain hardening of BCC metals is typically described fairly well by a power law,
p
and so we may write g(ε p ) = σg0 ( εε0 )n , where σg0 and ε0 are a reference stress and
a reference strain used to fit a quasi-static stress-strain curve and n is the material
parameter known as the strain hardening index. Using this function for the strain
dependence, we have
p n
2 2
3
√
kB T
ε
ε̇0
σ = σa0 + ky d + σg0
+ σ̂0 1 −
ln
.
(5.67)
ε0
Gk0
ε̇ p
The key feature of Equation (5.66) for our purposes arises from the assumed
mechanism of plastic deformation in bcc metals – we should expect that the change
5 Plastic Deformation of Nanomaterials
175
of grain size will not affect the additive stress term corresponding to the ratesensitive response of the material. In terms of experimental data, Equation (5.66)
tells us that it is useful to examine the variation with strain rate of the rate-dependent
term alone as in Equation (5.63) (i.e., we examine the flow stress difference σ − σ f 0 ,
where σ f 0 is the average flow stress for each grain size measured at the low rates)
since the effect of grain size and strain represent additive terms for the assumed
deformation mechanism. The model is able to capture the primary features of the observed behavior using physically reasonable parameters for this strain rate and temperature regime. Although the absolute increases in strength as a result of changes
in strain rate are consistent across these grain sizes in accord with the model predictions, the relative rate-dependent strengthening of the nc-Fe is much smaller than
that observed at conventional grain sizes, so that the total flow stress is less sensitive
to strain rate in ultra-fine-grain and nanocrystalline iron (again as predicted by the
model). This is illustrated in Figure 5.23, which plots the rate-sensitivity as a function of grain size for a number of bcc metals. The effective rate-sensitivity of these
materials (as described, for example in the index of a power-law rate-dependence
function) decreases as a function of grain size because the dominant part of the
strength is the grain boundary strengthening at small grain sizes.
5.11 Closing
The elasticity and plasticity of materials discussed in the last two chapters are examples of the constitutive behavior of materials. The corresponding constitutive
equations (Chapter 2, Section 2.4) relate the stresses, strains and strain rates at every point in the material, and each of these constitutive equations contain material
parameters (such as the Young’s modulus, Poisson’s ratio, and the yield strength)
that must be obtained from experimental measurements, or from calculations (such
as atomistic or first principles calculations) that are derived from a lower scale,
as we will discuss in a subsequent chapter. An equation such as Equation (5.67)
describes the dependence of the material’s strength on the strain, the strain rate,
the temperature, as well as an internal variable (the grain size). The application of
Equation (5.67) for the development of a full constitutive equation for a material (in
this case iron) will be shown in a later chapter. First, in the next chapter, we discuss
the processes of mechanical failure in nanomaterials because these processes define
the limits of application of such materials.
5.12 Suggestions for Further Reading
1. R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials.
John Wiley, New York, Fourth Edition, 1995.
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Nanomaterials
2. A.S. Argon, Strengthening Mechanisms in Crystalline Metals. Oxford University
Press, Oxford, 2008.
3. R.J. Asaro and V.A. Lubarda, Mechanics of Solid Materials. Cambridge University Press, Cambridge, 2006.
5.13 Problems and Directions for Research
1. In fcc systems, the Miller indices essentially define the components of the vector.
Consider a face centered cubic single crystal. The slip systems available are listed
in Table 5.1. Define the slip system normal vector m and the slip direction s for
each slip system using that table.
2. Now compute the “transformation tensor” µ , defined as µi j = 12 (si n j + ni s j ) for
each slip system (Equation 5.20). Each µ that you compute corresponds to one
of the µiαj tensors in Equation (5.26), i.e., the transformation tensor associated
with that slip system.
3. Consider now a single crystal of a face centered cubic material (say copper) that
is being pulled in tension along the [100] type direction. Calculate the resolved
shear stress on each slip system in this case.
4. Search the literature to obtain a critical resolved shear stress for copper along
slip system 10 in Table 5.1. Assuming that the crystal is being pulled in simple
tension, calculate the applied stress at which slip will begin in the crystal.
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Stone cracks from a hard enough blow. Steel
shatters. The oak fights the wind and breaks.
The willow bends where it must and survives.
Robert Jordan, The Wheel of Time
6
Mechanical Failure Processes
in Nanomaterials
The definition of the failure of a material is of course determined by the intended application. For example, in the majority of traditional mechanical design problems,
the material is intended to operate in the elastic range. Therefore, in that application, the material may be said to fail when it begins to deform plastically (that is,
when the yield strength is reached). We have discussed the onset of plasticity (yield,
or the elastic limit) in the previous chapter. Beyond yield, the material continues
to carry load in a nonlinear manner, corresponding to the plastic deformation; the
nonlinear response is something that designers often avoid because of the complexity associated with the constitutive behavior in that domain. In this chapter, our
interest is primarily in the applications of nanomaterials beyond the elastic limit,
assuming a modicum of plastic deformations. Situations where there is no nonlinear
deformation after the elastic limit is reached, as in ceramics, will be discussed as
well.
As any good mechanical designer knows, once the material goes plastic, it is
desirable to have the component continue to carry load for at least some time so as
to provide an additional safety margin in the application. How much deformation
can be sustained in the nonlinear range before a complete loss of load-carrying
capacity occurs? This determines how gracefully the component will fail and is
a key part of effective design. From a mechanical perspective, the complete loss of
load-carrying capacity is the ultimate definition of mechanical failure. If the amount
of nonlinear deformation that can be sustained before complete loss of load-carrying
capacity occurs is nearly zero, we say that the material is effectively brittle. This is a
condition that is generally to be avoided in safety critical applications (equivalently,
this is a condition under which the designer must incorporate a much larger factor of
safety in her design calculations). If a significant amount of nonlinear deformation
can be sustained before complete loss of load-carrying capacity, the material is said
to be ductile. We shall develop more precise definitions of brittleness and ductility
within this chapter.
It is important to recognize that the loss of load-carrying capacity of a component
or structure is fundamentally a structural response that is modulated by the material
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 6,
c Springer Science+Business Media, LLC 2009
179
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Nanomaterials
behavior. That is, mechanical failure by this definition occurs because of the failure
of the structure, which depends on the failure of the material but also involves the
geometry and boundary constraints provided by the structure. As an example, the
collapse of a steel-frame building can be triggered by the failure of the steel within
one of the columns, but the collapse process itself is dominated by the structural features of the building – the relative mass distribution, the locations of other columns
and the nature of the joints between the columns. Similar considerations are relevant
to mechanical components and small engineering structures, which are after all the
eventual applications of nanomaterials. With this caveat, in this chapter we focus on
the failure of materials rather than on the failure of structures.
6.1 Defining the Failure of Materials
Engineering Stress
How is the failure of a material defined after yield? This is a complicated question
which has caused much handwringing in the mechanical design and materials literature. Most authors define material failure by the loss of the load-carrying capacity
of a specimen of the material subjected to a specific kind of load (e.g., the failure
of a specimen in a uniaxial tensile test, see Chapter 2). This approach amounts to
defining a canonical structure (such as a tensile test specimen) and defining material failure as corresponding to the failure of the canonical structure made of the
material.
The most common definition of material failure of this type is that of final fracture in the simple tension test (Figure 6.1). In this experiment, a specimen is subjected to a state of nearly uniaxial tensile stress, and the stress is plotted against the
strain along the direction of tension (the axial strain). Beyond yield, the stress-strain
curve is nonlinear and has a maximum when looked at in terms of the engineering
σ
σ
σ
Engineering Strain
Fig. 6.1 Stress strain curve for a material obtained from a standard tensile specimen tested in
uniaxial tension, showing the final fracture of the specimen in uniaxial tension.
6 Mechanical Failure Processes in Nanomaterials
181
stress and engineering strain (the conventional approach). Note that the engineering
stress is defined in terms of the load P per unit initial area A0 , while the engineering strain is defined in terms of the change in length δ l divided by the initial gauge
length l0 (see Chapter 2). This maximum engineering stress is called the ultimate
tensile strength, and is often cited as a material property. After the maximum engineering stress is reached, the engineering strain continues to increase while the
engineering stress is decreasing until specimen fracture occurs and the specimen is
no longer capable of carrying a load. Truly brittle materials such as ceramics have
no apparent plastic deformation after the elastic limit, and the elastic limit in tension
is essentially also the fracture strength.
Since the simple tension test involves the measurement of two quantities (the
uniaxial stress and the axial strain), this definition of material failure produces two
parameters that define the failure: the stress at the time of fracture (called the fracture
stress, σ f = AP0 ) and the strain at the time of fracture (called the fracture strain,
ε f = δl0l ). Most material scientists ignore the fracture stress and use the fracture strain
as the measure of failure. There are several variations on the fracture strain that are
in common currency in the materials science literature, including the elongation to
failure, the reduction in the area, and the so-called ductility. When two materials are
compared, the material with the larger fracture strain is said to be more ductile.
The implicit assumption is that the fracture strain is in fact a material property.
However, the fracture strain is actually a property of the specimen rather than of
the material alone, and changing the specimen dimensions can lead to a change in
the measured fracture strain for the same material. Consider two tension specimens
(of one material) that are identical in gauge length (say 30 mm) but have very different diameters: one has a diameter of 10 mm, and the other has a diameter of
100 µm. The fracture strain that is measured with the small diameter specimens will
be considerably smaller than the fracture strain measured with the large diameter
specimen (we will see why this is so once we have understood the failure process).
Thus the fracture strain should only be used as a material property if it is agreed, as
a matter of convention, that only a standard specimen will be used with a standard
tensile test protocol. This is not a problem for conventional materials, where there
are agreed-upon standards for tensile testing that include a specification of specimen shape and dimensions. Thus it is perhaps reasonable to compare the fracture
strains obtained on 5083-H131 aluminum alloy and 6061-T6 aluminum alloy within
an ASTM standard tensile test.
Specific ASTM standards have been defined for tensile testing of steels (ASTM
A370), aluminum alloys (ASTM B557) and so forth, but these are typically for
conventional materials in terms of grain size and microstructure. There is a specific ASTM standard for high-strength materials, but that involves a different specimen geometry. The use of this geometry can result in a different failure process
within the material, so that different fracture strains will be measured for the same
material using the two different ASTM standards. One immediately sees a problem: nanocrystalline materials are typically very strong, so the measurement of
the elastic modulus and the yield strength would require the use of the ASTM
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standard for high-strength materials. However, the measured fracture strain of the
nanocrystalline material cannot then be compared to that of the conventional-grainsize version of the same material using the other ASTM standard. Further, nanomaterials generally cannot be obtained in quantities large enough to perform a standard
tension test, and so they tend to be subject to a variety of specialized mechanical
testing techniques. Given the dependence of the failure process on the specimen
geometry and testing protocol, it is generally not reasonable to compare the fracture strains of different nanomaterials tested in tension using different specimen geometries. However, the literature is replete with comparisons of the fracture strain
(the so-called ductility) of materials as a function of grain size. These comparisons
should be viewed with a very jaundiced eye.
It is useful to remember why fracture strains are measured in the first place. The
intent is to compare one material to another, and to be able to determine which of
these materials would be best suited for a specific application in which failure is
to be avoided or controlled. If the fracture strain is unable to distinguish between
the usefulness of materials in a specific application, then it is not a useful material
failure parameter for that application.
It is straightforward to find applications for which the fracture strain is not a useful parameter. Consider two materials A and B, for both of which standard tensile
tests have been performed resulting in the measurement of fracture strains ε f A and
ε f B . The two materials are now used to make identical short columns which carry
only an axial compressive load. Let us suppose that ε f A > ε f B . In general, it is not
possible to conclude that the column made of material B will lose its load carrying
capacity before the column made of material A. The failure processes are generally different in the two stress states of tension and compression, even though the
yield strengths are typically the same in tension and compression for conventional
metallic materials. The failure process in a shear-dominated problem may be different again. We must, therefore, develop an understanding of failure processes that
we can apply to a variety of stress states, because a variety of stress states will be
observed in application. This is obviously an issue in conventional materials, but becomes much more important in nanocrystalline materials for reasons that we shall
see in this chapter.
Another important difference between constitutive behaviors (such as elasticity
and plasticity) and the failure process within the material is that failure processes
tend to localize (so that all of the action occurs, for example, at the crack tip, rather
than everywhere within the loaded solid). This is the primary reason why failures
tend to be so sensitive to structural geometry and boundary constraints, since the
latter may have a strong effect on the localization process. A related consequence
is particularly significant: the localization associated with failure means that failure
processes are very defect sensitive, with localization (and the associated failure) occurring much more easily in regions of high defect density. The control of failure
processes is therefore often tied to the control of defect distributions in materials
(this is particularly important in ceramics, but is an issue even in ductile metals).
Two nanomaterials that are otherwise similar can show very different failure behaviors because of differences in defect density.
6 Mechanical Failure Processes in Nanomaterials
183
6.2 Failure in the Tension Test
When a tensile test specimen (Figure 6.2) is loaded beyond yield along the specimen
axis, increasing plastic strains are developed while the stress increases because of
strain hardening (for the moment we only consider the strain hardening case). If
the specimen is properly designed so that the stress concentrations at the grips have
been minimized, the plastic strains are first developed within the gauge section of
the specimen. As loading continues, the plastic strain continues to increase in the
gauge section of the specimen and spreads to some degree into the region of the
grips. A simple finite element calculation of the development of plastic strains inside
the tensile specimen shows that the strains are not homogeneous outside the gauge
section. The inhomogeneities at the ends of the specimen arise due to the changing
geometry near the grips, and these inhomogeneities can propagate into the gauge
section to some degree. This results in several complications. The gauge length is
typically used to calculate the strain in the specimen, on the basis of the assumption
that the deformations within the gauge section are uniform. However, calculations
show that the fraction of the specimen that has uniform plastic strains depends on
material properties such as strain-hardening.
h
P
neck
P
Fig. 6.2 The development of a neck during plastic deformation of a specimen within a simple
tension experiment. The position and length of the neck are determined by geometric and material
imperfections in the specimen.
What is the effective gauge length le of the material in the plastic range? This
length is needed for calculating the strain in the specimen, but the number itself depends on the material properties, which are unknown (indeed that is why the tension
test is being conducted). If the original gauge length (based on elastic deformations
of the specimen) is used to calculate the engineering strain, then the strain measure
will be inaccurate to some degree. This problem is recognized by most experts in
mechanical testing, and so local (true) strains are measured on the specimen using
strain gauges or digital image correlation. However, plastic deformations in the tension test may not be stable, and even this local measure of the strain may not be
useful once the deformations become unstable.
Consider the effect of a small geometric imperfection in a metal tension specimen, so that at some location in the gauge length there is a slightly smaller crosssection area (for example, as a result of machining-induced surface roughness).
Since force equilibrium requires that the force be the same at every cross-section
in the specimen, the local stress at the imperfection will be higher because of the
smaller cross-section area. Because the local stress is higher, the local strain must
increase to keep pace with the true material behavior (the true stress vs true strain
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Nanomaterials
curve). However, since plastic deformation in metals occurs at constant volume, an
increase in the local strain (which implies an increase in length at the imperfection)
results in a decrease in the local cross-section area and therefore now a further increase in the local stress. This process is unstable, and leads to the development
of a local region of much smaller cross-section area called a neck. Eventually the
neck becomes so small that the specimen separates (breaks) into two pieces, and
that results in a complete loss of load-carrying capacity.
In a tension test, the load carried by the specimen has a maximum at the point
corresponding to the ultimate tensile strength. Why is there a maximum in the load?
This is at the heart of the question of the stability of plastic deformations in tension.
Consider the derivative of the relationship P = σ A between the load P, the true stress
σ along the direction of tension, and the local cross-section area A:
dP = Ad σ + σ dA.
(6.1)
The load has a maximum when dP = 0, at which point
dσ
dA
=− .
σ
A
(6.2)
However, the assumption that plastic deformation occurs at constant volume implies that
Al = constant =⇒ Adl + ldA = 0 =⇒
dA
dl
= − = −d ε ,
A
l
(6.3)
so that a maximum in the force that can be carried by the specimen will occur when
dσ
= σ.
dε
(6.4)
The maximum strength of the material will therefore occur when Equation (6.4)
is satisfied, implying that the maximum strength that can be achieved depends on
the material behavior, and we discuss this next.
6.2.1 Effect of Strain Hardening
The constitutive law for the material, that is the stress-strain curve for the material,
provides a relationship between the axial stress and the axial strain: σ = f (ε ), so that
dσ
′
d ε = f (ε ) for the material. Thus a maximum in the load carried by the specimen
will occur when
f ′ (ε ) = σ .
(6.5)
In the particular case of a power-law relationship between stress and strain in the
uniaxial tensile test,
6 Mechanical Failure Processes in Nanomaterials
σ = f (ε ) = σ0
185
ε n
= αε n ,
ε0
(6.6)
which is sometimes called the Hollomon equation. We now have f ′ (ε ) = εn σ , so
that a maximum in the load will occur when (using Equation 6.5)
ε = n.
(6.7)
This equation is called the Considere criterion and essentially states that a maximum in the load-carrying capacity of the specimen will be attained when the local
strain on the specimen reaches a value equal to the strain hardening index. That
is, a specimen made of a material with a low strain hardening index will reach a
maximum in the load-carrying capacity at a small strain (e.g., if the strain hardening
index for a steel is 0.1, the maximum load will occur when the true strain in the specimen is 0.1). There is a strong temptation, therefore, to conclude that low hardening
materials will have small fracture strains, since the specimen usually fails soon after a maximum load is reached. As a consequence, much of the effort to develop
nanocrystalline materials with improved fracture strain has focused on the development of some strain hardening within the material together with the high strengths
produced by the reduced grain size. However, note that the Considere criterion only
addresses the development of the ultimate tensile strength, not the fracture strain
itself: what happens after the ultimate tensile strength is reached?
Experiments show that necking in specimens typically begins just before the development of the peak load, that is, just before the development of the ultimate
tensile strength. Many authors assume that necking begins at the ultimate tensile
strength, but this is generally not so. The precise point at which necking begins depends on the uniformity of the specimen, and on the possible presence of defects
within the material (that is, on geometric defects and material defects). Once necking begins, the deformation is said to have localized: most of the plastic deformation
now begins to occur only in the next region, while the rest of the specimen unloads
elastically. As a consequence, the stress-strain curve is no longer meaningful as a
constitutive behavior for the material once necking begins, since the stress-strain
curve assumes that the stresses and strains are uniform in the gauge section. A correction to the stress-strain curve is still feasible after the necking begins using the
Bridgman analysis developed by Bridgman (1952), but this correction is very rarely
provided in the literature.
The necked region is inherently unstable for the reasons discussed earlier, and
so the question of when the final failure will occur is a question of the stability
of the necking deformations. An important general question when discussing the
stability of deformations is the nature of the loading: is the tensile test being performed at controlled loading rate or controlled displacement rate? This question is
intimately coupled to the compliance of the testing machine, but we shall not discuss
this complexity here. For our purposes, it is sufficient to note that a test performed at
constant loading rate would by definition not be able to maintain a constant loading
rate once a maximum load is reached. Assume, therefore, that the simple tension
test being discussed here involves a controlled displacement rate (that is, the ends of
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the tension specimen are separated from each other at a controlled velocity). Under
this condition, once necking begins, the rate-sensitivity of the material becomes an
important factor in the stability of the deformations.
6.2.2 Effect of Rate-Sensitivity
To see this in a simplified manner, consider the definition of the strain rate given in
Chapter 2 through Equation (2.39), where we see that the strain rate is essentially
a velocity gradient. Equation (2.120) shows that the total strain rate is made up of
elastic and plastic parts, but once yielding has begun the elastic strain rate is usually
relatively small, so that the total strain rate is almost all due to plastic deformation.
Suppose the two ends of the specimen are being separated at a constant velocity
difference δ V . Then the average strain rate in the specimen is given by ε̇uniform =
δV
l0 during uniform plastic deformations, where l0 is as before the gauge length.
However, once necking begins, all of the plastic deformation concentrates within
the region of the neck, which is much smaller than the original gauge length. Let
us suppose the necking begins over a region of length h is shown in Figure 6.2;
since, once necking begins, the rest of the specimen is only elastically deformed,
all of the action is occurring within the necked region and the local strain rate is
approximately given by
δV
.
(6.8)
ε̇neck =
h
Since h is smaller than the gauge length, the strain rate in the neck is larger than
the average strain rate before necking began: ε̇neck > ε̇uniform . Now the material response within the neck depends on the rate-sensitivity, in that for a rate-sensitive
material the stress within the necked region must be increased to account for the increase in the strain rate. Thus maintaining the deformation in the neck now requires
a higher load. This effect competes against the effect of the reduction in area, which
requires a lower load to develop a given stress. Thus the rate-sensitivity of the material acts to stabilize the deformations, and reduces the tendency of the specimen
to further necking.
The influence of rate sensitivity is actually pervasive, affecting more than the
post-necking behavior (Hutchinson and Neale, 1977). The Considere analysis presented previously is only appropriate for a rate-independent material. In the case of
the rate-dependent material, the onset of necking can be much delayed from that
predicted by the Considere criterion, and indeed the total elongation of a material
can be a strong function of the rate-sensitivity (see Figure 6.3). A more rigorous approach to this problem is to consider a stability analysis of the equations describing
plastic deformation in tension. Conceptually, this amounts to asking the question
whether perturbations in some variable that is part of the equations will grow or
shrink with the deformation. The perturbation is prescribed with some amplitude
aN and some wavelength λ , e.g., a = aN cos( 2λπ x ). If the perturbation grows, the
system is unstable while if it shrinks the system is stable; growth may occur for
6 Mechanical Failure Processes in Nanomaterials
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STRAIN RATE SENSITIVITY, m
1
0.1
0.01
Trend line
0.001
1
10
Fe - 1.3% Cr - 1.2% Mo
Fe - 1.2% Cr - 1.2% Mo - 0.2% v
Ni
Mg - 0.5% Zr
Pu
Pb - Sn
Ti - 5%AI - 2.5% Sn
Ti - 6%AI - 4%V
ZIRCALOY-4
100
TOTAL ELONGATION %
1000
10,000
Fig. 6.3 The influence of rate-sensitivity on the total elongation to failure of materials, from the
work of Woodford (1969). Note the strong effect of the rate-dependence, and the similarity in
behavior of a wide variety of materials.
some wavelengths and not for others (e.g., a surface scratch may not cause necking,
while a wide machining groove might). In the uniaxial tension problem, it would
be appropriate to consider perturbations in the geometry (essentially a defect in the
geometry such as machining grooves) or perturbations in the material behavior (a
defect within the material), and to ask whether these perturbations would result in
the growth of necks. There are two versions of such stability analyses, known as
linearized stability analysis and nonlinear analysis. Such stability analyses become
particularly important when both strain hardening and rate-sensitivity are considered in the material behavior: σ = f (ε , ε̇ ). The Considere criterion discussed above
corresponds to the long wavelength instability in strain hardening materials. Hart
(Lin et al., 1981) and others have identified the onset of stability in rate-sensitive
materials using linear stability analysis, resulting in criteria such as
γ = 1 − m,
(6.9)
for onset of instability, where γ = σ10 ∂∂ εf 0 is a work-hardening coefficient and m =
ε̇0 ∂ f
σ0 ∂ ε̇ |0
is the rate-sensitivity (the subscript 0 indicates the uniform behavior far
from a neck, i.e., the behavior without a neck). This is called the Hart criterion for
instability.
Hutchinson and Neale (1977) conducted a nonlinear analysis of the influence of
rate-sensitivity on necking and showed that the linearized stability analyses of the
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type that provide the Hart criterion miss important features of the necking behavior
in rate-dependent materials. However, the key behavior remains that rate-sensitivity
stabilizes the deformation. For this reason, some researchers attempt to increase the
rate-sensitivity of nanomaterials. Note that in the case of fcc materials, going to
smaller grain sizes appears to increase the rate-sensitivity, while the reverse is true
for bcc materials. This suggests that the plastic deformations of fcc nanocrystalline
materials will typically be more stable than those of bcc nanocrystalline materials.
6.2.3 Multiaxial Stresses and Microscale Processes
Within the Neck
The stress state within the necked region can no longer be simple uniaxial tension,
because the axial strains within the necked region result in lateral contractions that
are resisted by the material surrounding the neck; this in effect adds a transverse
stress to the system. The resulting stress state is said to be triaxial in that there are
stresses in three directions (one along the axes and two perpendicular to the axis of
loading). A common measure of the degree of triaxiality of the stress state is called
the triaxiality χ , and is defined as the ratio of the mean stress σm to the effective
stress σ̄ :
σm
χ=
,
(6.10)
σ̄
where the mean stress relates to the hydrostatic pressure, and the effective stress
relates to a shearing stress (see Chapter 2, Equations 2.53 and 2.115). The triaxiality within the necked region increases as necking increases, as shown by full
numerical simulations of the developing neck, through the development of an increasing tensile hydrostatic stress. A consequence of this hydrostatic tension may
be the development of small voids within the material, through nucleation, growth
and subsequent coalescence (which leads to final failure of the specimen).
In the case of brittle materials, the fracture surface in a simple tension experiment is developed after very little necking, is oriented essentially perpendicular to
the tensile axis (Figure 6.4b), and shows a faceted character. The fracture surface
developed after simple tension of a ductile material involves a substantial neck, and
typically, a cup and cone fracture, with one side shaped like a cup and the other side
shaped like a cone (Figure 6.4c); both surfaces show a dimpled character, demonstrating the growth of voids.
Microvoid development is an example of a microscale failure process that is not
externally visible until final failure, but the signatures of the voids (appearing as
dimples) are quite clear on the fracture surface after failure has occurred. The consequences of the microvoid development are twofold. First, the actual failure of the
specimen is determined by the microvoid nucleation, growth and coalescence process, and so this is what determines the fracture strain (as distinct from the onset
of necking, which does not depend on the voiding process). Second, once the voids
have nucleated, the effective constitutive behavior of the material (that is the average
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Fig. 6.4 Form of the fracture developed in a simple tension test. (a) Original specimen configuration before loading. (b) Tensile failure of a brittle material, showing that the fracture surface is
essentially perpendicular to the loading axis. (c) Tensile failure of a ductile material, showing the
cup and cone failure morphology (in section).
material behavior that would be observed) changes dramatically from the classical
plasticity description discussed previously: the deformation is no longer volume
preserving. Thus it is not a priori reasonable to use classical plasticity descriptors to
examine the development of mature necks in materials.
6.2.4 Summary: Failure in the Simple Tension Test
Our considerations of failure in the simple tension test allow us to identify several
key concepts that will be useful throughout the rest of this chapter:
1. Failure in simple tension is an unstable process that involves both the material
response and the structural response of the tensile specimen and the testing machine.
2. The instability is observed as the rapid growth of a neck within the tensile specimen. The necking begins just before the development of the ultimate tensile
strength.
3. The onset of the instability (and associated necking) depends on material properties such as the strain hardening and rate sensitivity, both of which must be
defined in terms of the true stress and true strain.
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4. High strain hardening and high rate sensitivity both delay the development of
the instability. Criteria such as the Considere and Hart criteria (Equations 6.7
and 6.9 respectively) may underestimate the strain developed before the onset of
necking.
5. The fracture strain is larger than the strain associated with the ultimate tensile
strength, but how much larger depends on many factors including microscopic
failure processes within the material.
6. The stress state within the neck is no longer uniaxial but rather triaxial, and void
nucleation and growth may dominate the evolution of the neck.
7. Final failure, which defines the fracture strain, is determined by the interaction
of microvoid processes within the neck.
8. The fracture strain is not itself a material property – it is a property determined
by the combination of material behavior and test conditions such as geometry.
Comparisons of fracture strains made with different specimen geometries should
be viewed with great caution.
6.3 The Ductility of Nanomaterials
The effective ductility of a material is loosely defined as the tensile strain that can be
carried by a material before complete loss of load-carrying capacity occurs in a simple tension test. Initial work on the mechanical behavior of nanomaterials focused
primarily on the effect of increasing strength that could be obtained by changing
the grain size of bulk nanomaterials. Once it was apparent that high strains could
be achieved, it became important to identify the conditions under which these nanomaterials could be used in applications, and in particular to define when mechanical
failure would occur. The first bulk nanomaterials that were developed had significant
strengths, but failed immediately after the yield strength was reached in tension. A
significant amount of effort has been directed in the last decade towards developing nanostructured and nanocrystalline metals that can withstand significant tensile
strains before failure.
It is generally true, even in conventional materials, that an increase in strength is
usually associated with a reduction in ductility. Thus very high strength steels are
usually much less ductile than mild steel. This reduction in ductility has two proximate causes. First, high strengths in conventional materials are typically obtained
at the expense of strain hardening and rate-sensitivity, i.e., most high-strength materials have low strain hardening and low rate sensitivity. Second, given a standard
specimen geometry, the high-strength material in the tension test is more sensitive
to geometric and material defects because the growth rate of microscale failure processes is often related to the applied stress. Both of these issues also result in reduced
ductility in many nanostructured materials as the strength is increased. As a specific
example, consider the total elongation to failure of copper specimens made of copper materials of decreasing grain size (corresponding to increasing yield strength)
as shown in Figure 6.5. Similar behaviors have been observed in a broad range
6 Mechanical Failure Processes in Nanomaterials
191
E
Fig. 6.5 The total elongation to failure of copper materials of varying grain sizes, as presented by
Ma et al. (Wang et al., 2002). It is apparent that the ductility decreases dramatically as the yield
strength is increased. The point labeled E in the figure corresponds to a special copper material produced by Ma and coworkers that included both nanocrystalline and microcrystalline grain sizes,
with the larger grains providing an effective strain hardening in the material. Reprinted by permission from Macmillan Publishers Ltd: Nature, Vol. 419, Issue 6910, pages 912–915, High tensile
ductility in a nanostructured metal, Yinmin Wang, Mingwei Chen, Fenghua Zhou, En Ma. 2002.
of nanocrystalline FCC metals, as shown in Figure 6.6. The elongation to failure
(which is essentially the ductility) of the copper materials drops drastically with increasing yield strength, which corresponds to decreasing grain size. Ma et al. (Wang
et al., 2002) showed that it was possible to obtain increased ductility together with
high strength in a copper material by providing a bimodal grain-size distribution
of both nanocrystalline and microcrystalline sizes, with the larger grains allowing
the development of some effective strain hardening in the overall material. Thus
the importance of strain-hardening in improving ductility of nanostructured metals
is clearly established. Unfortunately, our understanding of strain hardening mechanisms in truly nanocrystalline materials remains weak, and so improving the overall
strain hardening of the fully nanocrystalline structure remains difficult. Special cases
seem to abound, such as apparent hardening due to stress assisted grain growth and
hardening through the incorporation of deformation twinning (discussed in a later
chapter).
The influence of internal defects on the apparent ductility of nanostructured and
nanocrystalline materials is clearly evident if one considers the scatter of the data
that is observed in measurements of the ductility as compared to that observed in
data on the yield strength of materials. A much larger scatter is observed in the ductility, and correlates quite well with the variety of fabrication procedures used for
making both the nanomaterials and the specimens used for the testing. The larger
scatter in ductility as compared to strength is a consequence of the fact that the
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Nanomaterials
80
nc Cu [31. Sanders PG. Acta, 1997;45:4019]
nc Cu [62. Legros M. et al, Phil. Mag. A 2000;A80:1017]
nc Cu [63. Nieman GW. et al, JMR 1991;6:1012]
nc Cu [64. Wu XJ, Nanostrust Mater., 1999;12:221]
nc Cu [65. Cheng S et al, Acta. 2005;53:1521]
nc Cu [66. Wang YM et al, APL, 2003;83:3185]
nc Cu [67. Conrad H. Yang D., Acta Mater. 2002;50:2851]
nc Cu [41. Lu et al, Science, 2004]
nc Cu [68. Youssef KM, et al, APL, 2005;87:091904]
nc Ni [69. Schwaiger R. et al, Acta mater., 2003;51:5159]
nc Ni [70. Torre FD. et al, Acta Mater., 2002;50:3957]
nc Co [71. Karimpoor AA, et al, Scripta mater, 2003;49:651]
nc Pd [72. Nieman GW et al, JMR, 1991;6:1012]
CG Cu [65]
CG Ni [70]
CG Pd [72]
CG Co [71]
70
Elongation (%)
60
50
40
30
20
10
0
0
200
400
600
800
1000 1200 1400 1600 1800
Yield strength (MPa)
Fig. 6.6 The total elongation to failure of fcc materials of nanocrystalline and microcrystalline
grain sizes, as presented by Dao et al. (2007). Reprinted from Acta Materialia, Vol. 55, Issue 12,
page 25, M. Dao, L. Lu, R.J. Asaro, J.T.M. DeHosson, E. Ma, Toward a quantitative understanding
of mechanical behavior of nanocrystalline metals. July 2007, with permission from Elsevier.
ductility is sensitive to defects in the materials, while the strength is not as sensitive
to as wide a range of defects. A fairly simple example of such a defect influence can
be identified in the porosity. A few large pores may have a small effect on the scatter
of the yield strength of a material, particularly if the yield strength is measured using hardness testing. However, those few large pores can have a dramatic impact on
the apparent ductility of the material in tension. The overall ductility of nanocrystalline materials has improved as the procedures for fabricating these materials have
become better defined and more capable of generating high-quality and consistent
material.
The influence of defects on the ductility is nicely illustrated by the work of
Youssef et al. (2005), who were able to make a relatively defect-free nanocrystalline
(mean grain size of 23 nm) copper material by using special fabrication procedures.
The use of the special procedures resulted in the development of a material that had
a total elongation to failure on the order of 15%, which is substantially higher than
had been observed in copper of similar nanocrystalline grain-sizes in the past.
It is important to recognize that some of the apparent lack of ductility of nanomaterials is a consequence of how they are tested. Most nanostructured materials are
typically subjected to tension in experimental configurations that deviate very far
from the ASTM standards even for high-strength materials, because it is difficult to
get sufficiently large quantities of these materials to perform standard tension tests.
The much smaller tensile specimens that are being evaluated are typically much
more sensitive to geometric defects, and therefore generally produce lower elongations to failure. In addition, some of the data in the literature are almost certainly
6 Mechanical Failure Processes in Nanomaterials
193
compromised by experimental artifacts associated with testing machine compliance
and inaccurate measures of strain. Further, new mechanisms may become possible
in some specimen geometries (such as grain growth in thin films) that are not possible in others, and that therefore color the results. Improvements in processing, specimen fabrication and experimental design are all required to develop a clearer sense
of the ductility of nanocrystalline and nanostructured materials. That said, recent
improvements in the apparent tensile failure strains shown by new nanomaterials
are very promising.
A number of specific approaches have been suggested to improve the ductility
of nanomaterials, while retaining their high strengths (Ma, 2006). Most of these involve techniques for improving the strain hardening or rate sensitivity of the material
(for example, through the use of bimodal grain-size distributions or through the addition of alloying phases), while some address the issue of the defect sensitivity by
reducing the number of defects generated by the processing. Other approaches include the incorporation of special boundaries within the material through twinning.
It is likely that each nanomaterials system will require a different cocktail of such
mechanisms in order to achieve the desired ductility.
What is the acceptable ductility within a nanomaterial system? Much of the literature assumes that a total elongation to failure of ≥10% will be acceptable, but
the precise number of course depends on the specific application. Product designers learn to live with the limitations of the materials that are available to them, but
the market for bulk structural nanomaterials will improve substantially if ductilities
comparable to that of low carbon steel (ε f = 0.25) can be obtained.
6.4 Failure Processes
There are several failure processes within materials, each of which tends to occur
under a specific stress state, and some of which are more likely to occur in certain
kinds of materials. This section describes the microscopic failure processes that lead
to macroscopic failure:
1. Crack nucleation, propagation and coalescence. These processes are what most
people think about in terms of mechanical failure.
2. Void nucleation, growth and coalescence, leading to what is typically called ductile failure.
3. Shear band initiation and evolution. These are processes of localization of shearing deformations.
4. Relatively specialized processes such as crazing and kink band formation in polymers and composites respectively. We shall not discuss these in detail.
These failure processes may be coupled: for example, voids are often formed
within shear bands in materials, and cracks may follow along shear band planes. In
the next few sections, each of these failure processes is examined in terms of the
failure of nanomaterials, with an emphasis on identification of phenomena rather
than on the modeling of the failure process.
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There are three fundamental steps in all of these failure processes. First, the microscopic failure process (for example a micro-crack) must be nucleated. Second,
the microscopic failure must propagate or grow. Finally, multiple instances of the
failure must coalesce and cause a macroscopic failure (or a microscopic failure must
grow to become a macroscopic failure).
6.4.1 Nucleation of Failure Processes
The nucleation process usually depends on the achievement of a critical stress, such
as at a stress concentration. Since defects are major stress concentrators, microscopic failure nucleation typically occurs at a defect (whether a geometric defect –
like a surface scratch – or a material defect, like an inclusion). In some steels, for
example, microvoid nucleation typically occurs at manganese sulfide inclusions,
while the initiating defect is a carbonaceous inclusion in many hot-pressed ceramics. A greater density of such defects (inclusions) will lead to a larger number of
nucleation sites, and therefore more nucleation of the failure at a give stress state.
A classical nucleation site is a pre-existing pore. Given this dependence on nucleation site density, we now understand why processing-induced defects play such a
big role in the apparent ductility of nanostructured materials – they provide an array
of nucleation sites. Careful processing that avoids the development of initial porosity and minimizes the number of inclusions therefore improves the tensile behavior
of nanomaterials, delaying macroscopic failure, as demonstrated by Youssef et al.
(2005).
Note that the typical size of defect that is involved in failure process nucleation is
of the order of the grain size or larger in conventional grain-sized materials, because
most of these failure processes do not develop easily within single crystals. Failure
processes in single crystals can be initiated by smaller stress concentrators such as
dislocation jogs, but these require much higher stresses before cleavage facets or
significant microvoid nucleation can occur.
What are the critical stresses associated with nucleation? This is not a question
with a clear-cut answer, because the critical stress and the kind of stress will vary
both with type of failure process (cracks, voids or shear bands) and with the type
of nucleating defect (e.g., initial porosity vs hard inclusion). In a generic sense,
however, cracks nucleate when critical tensile stresses are attained, and voids nucleate when critical hydrostatic tensile stresses are reached. The critical conditions for
shear band initiation are not understood. Since nanocrystalline and nanostructured
materials are generally very strong, the stresses that they carry are relatively high,
and so there is often enough stress available to nucleate failures (this is a general
argument that can be used for high-strength materials).
A comparison between the nucleation of failure processes for nanomaterials and
conventional materials may be useful. In conventional materials, the nucleation sites
are typically defects such as inclusions and pores. Depending on the process used
to make the nanomaterial, these defects may be more common in nanomaterials,
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or may be more widely distributed (it is not clear, for example, whether the inclusions will also change in size scale when the overall material is nanostructured). On
the other hand, it is typically the case that (in nano metals) homogeneous plastic
deformation has become more difficult, and so the overall stresses associated with
deformation are higher, and as a consequence there is more stress available to nucleate the failure process. Thus, both tendencies work towards making the nucleation
of failure processes more common in nanomaterials.
6.4.2 The Growth of Failures
Whereas the nucleation of failure processes is typically stress controlled, the growth
of failure processes is controlled by the available energy. This is most easily seen in
the case of cracks. Propagating cracks require the generation of new surface area,
and the generation of new surface area requires additional energy in the form of
surface energy. Thus crack propagation requires the delivery of energy to the crack
tip (the amount of energy that must be delivered is related to the fracture toughness
of the material). If the energy is not available, the crack will not grow. If the energy
is not provided fast enough, the crack will be forced to grow slowly (this controls
the speed of crack growth, and is an inherent problem in dynamic fracture mechanics). The energy for crack growth (and similarly for other failure processes such
as void growth) typically comes from strain energy locally stored in the structure
that is undergoing the failure process. The magnitude of the energy that is required
for the growth of failure processes arises depends on the processes involved. For
example, in the case of cracks, the energy is required for developing new surface
area; in the case of voids in a metallic material, the energy is required to develop
the plastic work that must occur at the surface of the voids as the voids grow; in the
case of shear bands, the energy is required to develop plastic work within the shear
bands.
The growth of failures can therefore be controlled by limiting the available energy. Conversely, if the available energy is very large, the growth of failures is nearly
impossible to control. An excellent example of this is observed in the testing of
ceramics. Once microcracks have nucleated in ceramics within a tension or compression experiment, it is typically very difficult to control the growth of the cracks
because so much strain energy is stored in the testing machine and is available for
crack growth, while the energy needed for crack growth is relatively small in ceramics. Thus it is very difficult to recover ceramic specimens that contain only small
amounts of microcracking, although metallic specimens containing small amounts
of microcracking can often be recovered after a mechanical test. The availability
of energy is also related to the question of whether the application involves load
control or displacement control, or equivalently whether the application has fixed
available energy or fixed available load. If the application involves fixed available
energy, the growth of the failure processes will be limited to that possible with the
available energy.
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6.4.3 The Coalescence of Cracks and Voids
Final macroscopic failure is the result of the coalescence of microscopic failure
processes. The process of coalescence is extremely difficult to model, since it is essentially a runaway instability. Consider, for example, two voids that are growing
within a material after nucleating at nucleation sites that are a distance l apart. In the
early stages of void growth, both the stress fields and the available energy around
each void is essentially unaffected by the presence of the other void. Thus each
void grows as though it is alone within an infinite medium, and this has been modeled quite carefully in the mechanics literature (Wu et al., 2003). A simple model
of a void growing in a uniform external hydrostatic tensile stress field shows that
most of the deformation occurs right near the void surface in a region that is heavily deformed plastically, and as one moves radially away from the void surface the
deviatoric stresses decrease until plastic deformation can no longer occur, so that
there is an effective plastic zone around each void. Once the voids get to be sufficiently large, however, they begin to interact with each other through their stress
fields (a quick but relatively inaccurate measure of when interactions begin is to
consider the time when the two plastic zones of the voids begin to overlap). The result of the interaction is typically to accelerate the growth of the voids, thus leading
to an instability. As the voids continue to grow, the space between the voids eventually shrinks to a mere ligament. From here on, the interaction between the voids
is occurring through the plastic extension of the ligament, and finally the ligament
breaks, resulting in the coalescence of the two voids. Such a process repeated across
the entire specimen results in the macroscopic ductile failure of the specimen, and
the holes that remain within the specimen after the ligaments are broken give the
fracture surface a dimpled appearance.
Although the coalescence of failure processes is difficult to model, it is also true
that the runaway instability associated with coalescence usually results in macroscopic failure that occurs very soon after coalescence begins (that is, there is very
little global deformation – or strain – accommodated once coalescence begins). For
practical purposes, therefore, it may be sufficient to consider the onset of coalescence and use that to define the onset of microscopic failure.
6.4.4 Implications of Failure Processes in Nanomaterials
The preceding discussion has demonstrated that in order to control failure in nanomaterials one must control the nucleation, growth, and coalescence of the failure
processes that develop within each material. Controlling nucleation is largely a question of controlling the available defects that are the nucleation sites, that is, one requires excellent control of the processing so as to reduce the available defects. It
is useful to recognize here, however, that some microstructures inherently contain
the defects that act as nucleation sites for failure, and that these “defects” may in
fact be important in producing the strength or the strain hardening that is required
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for the application. For example, the incorporation of a weak phase within the microstructure may result in enhanced strain hardening, but the weak phase may also
result in the easier nucleation of voids within the material. More broadly, this is a
recognition of the fact that the microstructural features that control plastic deformation may have significant implications on failure processes, so that a trade-off must
sometimes be considered.
Once the failure has nucleated, the growth of the failure is typically not controllable in microstructural terms, although it may be controllable from an applications
standpoint (for example by controlling the applied loading). Inertial effects ensure
that the growth of failures is typically rate dependent (Wright and Ramesh, 2008).
In a broad sense, however, the relative roles of nucleation and growth dominate the
failure process, and so the control of the failure process can be viewed in terms of
controlling the relative rates of nucleation and rates of growth. For example, if there
are relatively few nucleation sites, the failure process may begin at the same stress as
in the material with many nucleation sites, but the failure process will remain stable
for a longer time because the interaction of the failures (and eventual coalescence)
does not occur until later in the material with few nucleation sites.
6.5 The Fracture of Nanomaterials
As is generally the case in this field, the fracture of nanomaterials has primarily been
studied in FCC materials. Recall that failure processes in materials are highly localized, and are generally sensitive to the presence of geometric and material defects. In
order to be able to compare the resistance to fracture of one material with the resistance to fracture of another, one must perform experiments in which the localization
occurs in a specific place through the use of a controlled defect (either a notch or
a crack). From a mechanics perspective, fracture should be studied through the use
of specialized fracture mechanics experiments which evaluate the likelihood of the
propagation of a pre-existing crack within the material. The result of such an experiment is the evaluation of the fracture toughness KIC of the material (see Chapter 2).
This involves two distinct steps. First, a fracture mechanics experiment needs to be
designed so that the appropriate quantities can be measured. Second, a pre-crack
must be generated in the specimen. Standardized experimental techniques exist for
the measurement of the fracture toughness of materials (for example, a standardized fracture mechanics experiment for plane strain fracture toughness measurement uses ASTM E399), but these generally require fairly large specimens (e.g.,
Figure 6.7), and such specimens typically cannot be made with current quantities
of most nanomaterials. Few if any standard fracture mechanics experiments have
been conducted on nanomaterials, and thus there are no fracture toughness data on
nanometals or nanoceramics, although there are some data on polymer nanocomposites. Such data will require the development of larger quantities of bulk nanomaterials, or the development of specialized experimental techniques for small-scale
fracture.
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Fig. 6.7 A standard compact tension (CT) specimen used for fracture toughness measurements.
The specimens are several centimeters in size.
Recall, from Chapter
√ 2, that materials with low values of fracture toughness (typically KIC ≤ 10 MPa m) are considered
√ to be brittle, but materials with high fracture
toughnesses (typically KIC ≥ 20 MPa m) are said to be tough (sometimes, they are
said to be ductile). Tougher materials have a wider range of application, since they
are less sensitive to minor cracks. It turns out that there is a very strong correlation
between the fracture toughness of a material and the character of the fracture surface after a fracture mechanics experiment. Brittle materials show faceted fracture
surfaces, with the facets representing typically crystalline cleavage (fracture along
crystallographic planes) or fracture along the grain boundaries, while ductile materials show dimpled fracture surfaces, with the dimples being the signatures of void
growth and coalescence leading to separation of the two fracture surfaces. It appears
to take less energy to grow the faceted fracture surfaces as compared to the dimpled
fracture surfaces, so that the fracture toughness of brittle materials is lower than
that of ductile materials. There is also a strong correlation between the brittleness
of the material and the form of the fracture surface developed in a simple tension
experiment. From a materials perspective, therefore, fracture processes are typically
studied through the examination of the fracture surfaces developed after a standard
tensile test.
What kinds of fracture surfaces are typically observed in nanocrystalline materials? Nanocrystalline ceramics behave in a brittle fashion in tension, and their fracture surfaces are faceted, as would be expected. Nanocrystalline metals, on the other
hand, typically show dimpled fracture surfaces. Thus the local failure processes in
6 Mechanical Failure Processes in Nanomaterials
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nanometals are of the ductile variety, indicating that the macroscopic low ductility
observed in many nanometals can perhaps be mitigated, and that it may be possible
to make nanometals reasonably tough.
An interesting example of the development of failures within a nanomaterial is
provided by the in situ observations of Kumar et al. (2003). These authors examined
the propagation of tensile failures in nanocrystalline nickel within a transmission
electron microscope, so that they could observe the microscopic process of failure.
A sequence of images from their in situ TEM work is presented in Figure 6.8. They
Fig. 6.8 A sequence of in situ TEM micrographs obtained by Kumar et al. (2003) during the loading of nanocrystalline nickel using multiple displacement pulses. Images a–d show the microstructural evolution and progression of damage with an increase in the applied displacement pulses.
The presence of grain boundary cracks and triple-junction voids (indicated by white arrows in (a)),
their growth, and dislocation emission from crack tip B in (b–d) in an attempt to relax the stress at
the crack tip as a consequence of the applied displacement, can all be seen. The magnified inset in
(d) highlights the dislocation activity (Kumar et al., 2003). Reprinted from Acta Materialia, Vol.
51, Issue 2, page 19, K.S. Kumar, S. Suresh, M.F. Chisholm, J.A. Horton, P.Wang, Deformation of
electrodeposited nanocrystalline nickel. Jan. 2002, with permission from Elsevier.
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observe the development of grain boundary cracks, void nucleation at triple junctions, and the simultaneous movement of dislocations within the material. Thus at
the microscopic scale, both fracture processes and plastic deformation processes are
active within the nanocrystalline nickel.
If, on the other hand, the fracture surface of the nanocrystalline nickel (typical
grain size of 40 nm) studied by Kumar et al. (2003) is examined after loading, what
is observed is the jagged fracture surface shown in Figure 6.9. Such a post mortem
observation leads one to underestimate the importance of plastic deformation during
the failure process simply because dislocations are not immediately evident. This
work therefore points out the difficulty of inferring processes (and in particular assuming the lack of specific processes such as dislocation motion) from post mortem
observations.
Dimpled fracture surfaces are often observed in nanocrystalline metals after tensile failure (again, most of this work has been in FCC nanometals). Jonnalagadda
and Chasiotis (2008) have examined the tensile behavior of thin films of nanocrystalline fcc metals (gold and platinum) using a microtension set up that is capable of
a range of strain rates. An example of the failure surface morphology observed in
nanocrystalline gold (with an average grain size of 38 nm) after the tensile failure
of thin films is presented in Figure 6.10. Large (≥100 nm) dimples are seen on the
fracture surface, with the dimple size many times larger than the average grain size.
Significant plastic deformation is therefore occurring before macroscopic failure. It
is not clear what the nucleation sites were for the voids that produced the dimples
(note that these are essentially pure nanocrystalline metals produced by electrodeposition, and therefore would appear to contain few processing defects).
Fig. 6.9 Postmortem microscale fracture morphology observed by Kumar et al. (2003) after the
loading of nanocrystalline nickel in tension. There is no direct evidence of the presence of dislocations in this image, although some of the grains appear to have necked before separation.
Reprinted from Acta Materialia, Vol. 51, Issue 2, page 19, K.S. Kumar, S. Suresh, M.F. Chisholm,
J.A. Horton, P. Wang, Deformation of electrodeposited nanocrystalline nickel. Jan. 2002, with permission from Elsevier.
6 Mechanical Failure Processes in Nanomaterials
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Fig. 6.10 Postmortem fracture surface morphology of nanocrystalline (average grain size of
38 nm) gold thin film tested in tension till failure by Jonnalagadda and Chasiotis (2008). Dimples are seen on the fracture surface, with an average size of more than 100 nm. The void size just
before coalescence is therefore much larger than the grain-size, and the growth of the voids must
involve plastic deformation in many many grains.
Dimpled fracture surfaces have been observed in a number of fcc nanometals,
including copper, gold, platinum, and nickel. There are few if any observations of
fracture surfaces in bcc and hcp metals.
6.6 Shear Bands in Nanomaterials
Shear banding is a plastic shear flow instability in which initially uniform shearing
deformations localize into a narrow region called the shear band. This is a common
mode of failure in a variety of materials, particularly under shear-dominated and
compressive loadings (in tension-dominated loadings, the crack and void growth
processes are typically favored from an energetic viewpoint). The shear localization
process is illustrated in Figure 6.11. An initially uniform shearing deformation (in
the plastic range) can begin to become nonuniform, with most of the plastic deformation occurring over a small region that becomes increasingly narrow, until all of
the plastic deformation occurs within this narrow region (called the shear band) with
only elastic deformations occurring outside the shear band.
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Fig. 6.11 Progressive localization of a block (a), with initially uniform shearing deformations (b)
developing into a shear band (c). The final band thickness depends on the material behavior.
Stress
There are many reasons for the localization of the plastic deformation into a band,
but in a broad sense such localizations occur when the material behavior locally
reaches a softening domain where increasing deformations can be accommodated at
decreasing stresses as in Figure 6.12. Consider, for example, a slab of material (with
a constitutive behavior corresponding to Figure 6.12) that is deformed in simple
shear as in Figure 6.11, with slowly increasing shearing stresses prescribed on the
boundaries. Suppose that some location in the slab has a local stress concentration
(e.g., due to the presence of an inclusion). Then the material at that point will reach
the peak stress in Figure 6.12 before the material elsewhere in the slab. At that
material point, therefore, runaway deformation occurs, because increasing deformations can be sustained at decreasing stress. This then causes neighboring points
to reach the peak stress more quickly, and a cascade process begins leading to a
localized zone of deformation as in Figure 6.11. Note that the local softening may
not be immediately observable at the macroscopic scale, because much of the slab
is still in the hardening regime. Once the shear band has run all the way through
the slab, macroscopic softening may also be observed. At intermediate states, the
Strain
Fig. 6.12 Stress-strain curve for a material showing softening after a peak stress. This is a curve
corresponding to a material undergoing thermal softening, but similar behaviors can arise from
other causes such as grain reorientation.
6 Mechanical Failure Processes in Nanomaterials
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macroscopic behavior may be that of decreasing hardening, or even perfect plasticity (when the hardening and softening regions balance each other).
6.6.1 Types of Shear Bands
Two kinds of softening dominate materials problems: softening resulting from local reorientation and texture development (this is often called geometric softening)
and softening resulting from temperature rise (thermal softening). Shear bands arising from geometric softening are often simply called deformation instabilities in
shear, while shear bands arising from thermal softening are said to be adiabatic shear
bands. Adiabatic shear bands are typically observed in dynamic loading conditions,
when the heat generated by plastic work does not have enough time to be conducted
away (hence the term adiabatic).
A very large number of materials will localize into adiabatic shear bands if driven
at sufficiently high strain rates. The likelihood of the development of such adiabatic
shear bands is determined by the combination of the plastic properties of the material and the thermal properties (thermal conductivity and specific heat). An excellent
summary of the mechanics issues associated with such adiabatic shear bands is provided by Wright (2002), and a description with a materials emphasis is provided
by Bai and Dodd (1992). Because nanostructures change the plastic properties of
materials, the onset of adiabatic shear band development will also be influenced by
grain size, but this has not been carefully studied. In this book the discussion is focused on shear bands generated in nanocrystalline materials as result of geometric
softening.
Localized shearing deformations (shear bands) have been reported in many
nanocrystalline metals. Localization of plastic deformation into shear bands has
been observed in nc- Fe-10%Cu (Carsley et al., 1998) and nc-Fe (Jia et al., 2003).
Malow et al. (1998) observed shear bands in nc-Fe samples after microhardness
tests. Shear bands have also been found in nc-Pd (Sanders et al., 1997) under compression, in nc-Cu under fatigue (Vinogradov and Hashimoto, 2001) and in nc-Ni
under high-rate tension (Dalla Torre et al., 2002). We discuss shear band development in nanocrystalline metals through the particular example of shear band development in bcc iron, discussed in a series of papers by the author’s group (particularly
Jia, Wei and others – Jia et al., 2003, 2000; Wei et al., 2004b).
6.6.2 Shear Bands in Nanocrystalline bcc Metals
The observations of Jia et al. (2003) and of others (e.g. Carsley et al., 1998), indicate
that there is a transition from uniform to non-uniform deformation as the grain size
decreases down to the nanoscale. In the case of iron (Jia et al., 2003), when the grain
size is above ≈ 1 µm, the plastic deformation takes place uniformly at least up to a
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plastic strain of about 15% over the entire sample. However, for smaller grain sizes
(≤ 300 nm) shear band development was observed to occur immediately after the
onset of plastic deformation, correlating to an observed change in apparent strain
hardening at those grain sizes. Most of the plastic deformation was localized in the
narrow shear bands, often along planes containing the highest shear stresses. Comparative optical micrographs are presented in Figure 6.13 after low-rate compressive
deformations to similar plastic strains, showing (a) the uniform deformations in Fe
with a mean grain size of 980 nm and (b) the strongly non-uniform deformations in
Fe with a mean grain size of 268 nm. Shear band populations similar to Figure 6.13b
were observed in all specimens with grain sizes. Figure 6.13c shows the strongly
non-uniform deformations observed in the 268 nm Fe after high-rate loading to a
similar strain as in Figure 6.13a, b. Thus the shear bands were observed during both
quasistatic and high-rate deformations for these grain sizes, suggesting that the shear
bands resulted from geometric rather than adiabatic (thermal) softening.
The process of development of these shear bands was studied using “progressive” low-rate compression tests performed on the 268 nm-Fe with repeated loading/unloading/reloading at controlled nominal levels of strain. The evolution of the
shear band population over an entire specimen as a function of strain is described
by Wei et al. (2002) and is presented in Figure 6.14. These authors show that the
onset of plasticity in the nanostructured iron corresponds to the appearance of shear
bands emanating from one of the edges of the specimen, i.e., there is negligible uniform plastic deformation before localized deformations begin. It was demonstrated
in Wei et al. (2002) that additional shear bands appear with increasing strain and
(a(a) )
(b)
(b)
(c)
(c)
Fig. 6.13 Gross deformation features of coarse-grain and ultra-fine-grain iron samples after compressive deformations to nearly identical strains (Jia et al., 2003). (a–b) represent quasistatic
compression, while (c) represents high-strain-rate compression. (a) Homogeneous deformation of
coarse-grain (20 µm grain size) Fe. (b) Shear band pattern development in ultra-fine-grain (270 nm
grain size) Fe. (c) Shear band pattern development in ultra-fine-grain Fe after dynamic compression at a strain rate of ≈ 103 s−1 . Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia,
K.T. Ramesh, E. Ma, Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior
and shear bands in iron. July 2003, with permission from Elsevier.
6 Mechanical Failure Processes in Nanomaterials
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III
III
A
II
II
iv
I
I
(a)
(b)
Fig. 6.14 Shear band patterns evolve with strain in compressed ultra-fine-grain Fe (Wei et al.,
2002). Note the propagation of existing shear bands, the nucleation of new shear bands, and the
thickening of existing shear bands. The development of families of conjugate shear bands is also
observed. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages 1240–1242, Q. Wei,
D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in nanostructured fe.
2002, with permission from American Institute of Physics.
that the newly generated shear bands have similar orientations (in the four possible
shearing planes for these cuboidal specimens). Large numbers of shear bands are
observed, rather than single dominant bands that lead to failure.
The process of shear banding that develops during a quasistatic experiment can
be observed in Figure 6.14, which shows a single area (well away from the edges)
of a 268 nm Fe specimen at nominal plastic strains of (a) 3.7% and (b) 7.8%. The
loading axis is vertical in these figures. The straight lines produced by the directed
polishing described earlier are visible in both figures and may be used to deduce
the amount of shear in each band as the nominal strain develops. Three shear bands
denoted I, II and III are visible in Figure 6.14a, including one band tip denoted by A
(note that band tips are atypical in our data in unconfined regions). By the time the
nominal plastic strain has reached the level of 7.8% in Figure 6.14b, a minimum of
eight shear bands are visible (i.e., five new bands have appeared). Bands I, II, and
III are all considerably broader (Band I, for example, has grown in width by more
than 50%). Band III has propagated, including several branches, one of which has
apparently merged with Band II. One of the new bands (Band IV) is at least as broad
as the current size of Bands II and III, indicating that the deformation within a newly
nucleated band can be substantially larger than that in an existing band. Tracing the
polishing lines, one observes that the total strain in Band I has increased. Observations with optical microscopy and TEM for the deformed UFG-/nc-Fe specimens
show that large plastic deformations are developed within the bands (Wei et al.,
2002). Thus, it appears that the process of shear banding in this material involves
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the nucleation of new bands, propagation along the shear plane, increase in width
(broadening) and increase in strain (flow) within the band.
Figure 6.15a shows a high-magnification view of a single shear band with welldefined boundaries (this band was observed at a nominal plastic strain of 0.3%, just
after yield). The shear deformation within the band can be traced by following a
polishing scratch across the band. The measured width of the shear band is ≈ 16 µm
and the shear strain in the band is 25% (nearly two orders of magnitudes higher than
the nominal specimen strain).
~7°
(a)
(b)
30 µm
Fig. 6.15 (a) Shearing deformation across one band and (b) the stable intersection of multiple
shear bands in 270-nm Fe after quasistatic compression. The shear offset is clearly visible across
the first band. Note that no failure (in terms of void growth) is evident at the intersection of the
shear bands. Reprinted from Acta Materialia, Vol. 51, Issue 12, page 15, D. Jia, K.T. Ramesh, E.
Ma, Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands
in iron. July 2003, with permission from Elsevier.
Although no shear bands are present along the conjugate planes in Figure 6.14,
the bulk of the specimen develops shear bands along more than one of the conjugate
planes. As a result, shear band intersections are common and organized networks of
shear bands are observed. Figure 6.15b shows a network of shear bands at about 45◦
to the loading axis in a 268-nm Fe specimen after low-rate loading to 7.8% nominal
plastic strain. The relative uniformity of the shear band spacing must be described
by models of this failure mode (that is, a model that correctly captures the physics
of multiple shear banding should predict this shear band spacing in compression).
The intersections of shear bands are observed to remain compatible within optical
resolution, which is itself remarkable given the large strains within each shear band.
For most of the shear bands, the two boundaries can be clearly identified, so that
the width can be accurately measured. The band widths (all measured using the
optical microscope) for the 138-nm and 268-nm Fe are the thickest bands observed
at low plastic strains (<0.5 %) while that for the 80-nm Fe is the thickest band
observed at a plastic strain of 2.3% after unloading (tests at a lower plastic strain
were not performed on the 80-nm Fe). Wider bands were observed in the 268-nm Fe
than in 138-nm Fe or 80-nm Fe. In contrast, the band-width/grain-size ratio, R, for
all the three grain sizes remains similar, on the order of 50. In all the three UFG-/ncFe, the orientations of most bands (defined as the angle between the normal to the
6 Mechanical Failure Processes in Nanomaterials
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shear band plane and the loading axis) were 41–46◦ , suggesting that the orientations
are insensitive to grain size.
At sufficiently high strains, numerous bands may group into “shear zones”
(two diagonal shear zones can be seen in the dynamically loaded specimen of
Figure 6.13c). The boundaries of the bands in the zones become more difficult to
identify whereas the boundaries of the zones remain clear. The development of the
zones is controlled by the specimen aspect ratio, because of kinematic constraints
from the specimen ends. The maximum width of the shear band zones measured in
the 268-nm Fe at a plastic strain of 7.8% is 158 µm, which is close to the results
reported by Carsley (1996) on Fe-10%Cu.
6.6.3 Microstructure Within Shear Bands
Observation of microstructure within shear bands is very difficult, because the small
features involved require the use of transmission electron microscopy. The electrontransparent region in the TEM specimen must be made within the shear band, and
the location of the electron transparent region is difficult to control. An effective
approach is to use tripod polishing, which generates a fairly large transparent region
with some control of location (however, the technique requires considerable skill).
TEM observations of the shear bands in bcc iron were made by Wei using a tripodpolishing technique (Wei et al., 2002). After obtaining perforation at the wedge
edge near shear bands, the specimens were cleaned with a dual-gun ion mill at liquid
nitrogen temperature. In addition, both electron energy loss spectroscopy and energy
dispersive X-ray spectroscopy mapping were employed to rule out the segregation of
impurities, using a Philips CM300 microscope equipped with a Gatan Image Filter
system. [One sees here the complexity associated with TEM specimen preparation
and materials characterization.]
An example of the TEM results is presented in Figure 6.16, which shows TEM
micrographs obtained from outside and within shear bands in nearly-nanocrystalline
iron. Figure 6.16a shows a bright-field TEM image taken outside a shear band and
Figure 6.16b an image inside the shear band. The TEM specimen was from a sample
that has an average grain size of 110 nm and was tested to a nominal overall strain of
14% in low rate compression. Figure 6.16b, obtained from a region outside the shear
bands, shows equiaxed grains. Figure 6.16a, obtained from within the shear band,
shows elongated grains (elongated along the shearing direction) with an average
aspect ratio of 10 and often decorated by high densities of dislocations. There is no
evidence of recrystallization within the shear bands, and the shear band boundaries
are relatively sharp. Further, no gradient of microstructure is observed across the
shear bands.
Selected area diffraction (SAD) from outside the band showed a uniform ring
pattern corresponding to bcc Fe, with no evidence of preferential orientation of the
grains. In contrast, the SAD pattern of the crystals inside the shear band (not shown
in Figure 6.16) showed strong maxima on the ring of 110 reflections, suggesting
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(a)
(a)
(b)
(b)
Fig. 6.16 TEM micrographs (a) within and (b) outside a shear band in quasistatically compressed
Fe with an average grain size of 138 nm (Wei et al., 2002). The shearing direction is shown by the
arrow. Note the preferred orientation of the grains within the shear band, while the grains outside
the band are essentially equiaxed. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages
1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in
nanostructured fe. 2002, with permission from American Institute of Physics.
texturing. Non-uniform rings can also be observed for other reflections. The sixfold symmetry observed in the SAD pattern allows us to index the pattern as along
the < 111 > zone axis. We therefore infer that the texturing involves the activation
of the < 111 > /110 slip systems of bcc Fe. Although slip systems of < 111 >
/110, < 111 > /112 and < 111 > /123 are all reported for bcc metals such as
Ta, Mo, Cr, < 111 > /110 appears to be a much preferred slip system for bcc Fe
(Nemat-Nasser et al., 1998). Since the material within the shear band shows this
texture, it follows that the large plastic deformation observed is developed through
conventional dislocation mechanisms. Thus even at this relatively small grain size,
dislocation mechanisms dominate the material behavior.
In summary, within the shear band we have elongated and textured grains, suggesting that the grains have reoriented themselves for easy slip. In other words, the
grains have experienced rotations and alignment in forming the band with texture,
allowing easy slip propagation in certain orientations. This is a geometric softening
mechanism. It is conceivable that such cooperative deformation across the band is
easier when it involves many tiny grains rather than large grains, possibly explaining the enhanced propensity for shear banding when the grain size is in the nc/UFG
regime. An approach to modeling such a softening mechanism is presented in the
next chapter.
These observations of microstructure within the shear bands are interesting from
another perspective. Various investigators (e.g., Meyers et al., 1994; Chichili et al.,
1998) have examined the microstructures of shear bands in some conventional grainsized metals in which the shear bands were developed only under high-rate loading.
When the interiors of adiabatic shear bands are examined in the transmission electron microscope, significant amounts of recrystallization are typically observed (the
6 Mechanical Failure Processes in Nanomaterials
209
recrystallization is thought to be due to either a significant temperature rise or due
to very large plastic deformations). Indeed, one typically observes grain refinement
within the center of an adiabatic shear band (rather than grain elongation), and the
refined grains tend to be equiaxed and nearly dislocation free. Originally microcrystalline microstructures can become refined into nanocrystalline structures inside an
adiabatic shear band; 50-nm grains were obtained in a commercially pure titanium
that was originally 30 µm in grain-size by Chichili et al. (1998). In such adiabatic
shear bands, the TEM also reveals a gradient of microstructure across the shear
band. The shear bands that are observed in nanocrystalline iron are clearly very
different. In this case the microstructures are relatively uniform across the band,
with relatively sharp boundaries, as shown by the montage presented in Figure 6.17.
It appears, therefore, that the shear bands observed in nanocrystalline iron are not
adiabatic shear bands, but rather deformation instabilities developed as a result of
geometric softening.
Fig. 6.17 TEM micrograph showing the microstructure near a shear band boundary in nanocrystalline iron (Wei et al., 2002). The boundary between the material within the band and that outside the band is shown by the solid line. Note that the transition occurs over a transition width
that is about one grain diameter. Reprinted from Applied Physics Letters, Vol. 81, Issue 7, pages
1240–1242, Q. Wei, D. Jia, K.T. Ramesh, E. Ma, Evolution and microstructure of shear bands in
nanostructured fe. 2002, with permission from American Institute of Physics.
We can learn much about the mechanisms associated with shear band development and the deformation in the nanocrystalline iron by examining the processes of
shear band evolution presented in Figure 6.14. First, the fact that existing bands continue to widen indicates that a saturation condition is present in the system. Thus if
the center of the band reaches some saturated condition, so that no further geometric
softening can occur (for example, because the optimum orientation has already been
achieved), then the band must widen because geometric softening can still occur at
the edges of the band. Second, the fact that additional bands are nucleated indicates
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that a hardening mechanism must also exist (for example, local strain hardening
due to dislocation accumulation), because otherwise further deformation could simply accommodated by widening of the existing bands. Examination of Figure 6.17
shows that the length scale associated with the softening mechanism is relatively
small, only about one grain wide, since the transition between the outside of the
band and the saturated orientation condition within the band occurs over this range.
There are thus three distinct length scales in this problem at any given time: (i) the
width of the band, (ii) the width of the transition zone, and (iii) and the spacing of
the shear bands. Presumably the spacing of the shear bands is associated with the
competition between the nucleation and growth of the shear bands, with nucleationdominated problems showing smaller band spacings.
6.6.4 Effect of Strain Rate on the Shear Band Mechanism
High-strain-rate experiments were also performed by Jia et al. (2003) on the nanocrystalline iron, using controlled strains (by choosing the length of the projectile and
the initial velocity). For a similar strain level, the distribution of shear bands was
more uniform in the low-rate specimens (Figure 6.13b) than in the high-rate specimens (Figure 6.13c). This arises primarily because of the differences in specimen
aspect ratio (note the shear zones in Figure 6.13c) because the high-rate specimens had a length:diameter aspect ratio of ≤1 while the low-rate specimens have a
length:diameter aspect ratio of ≥1. Length to diameter ratios of less than one guarantee that a shear band at an angle of 45◦ to the loading direction will intersect one
of the loading boundaries, resulting in constrained deformations. The shear band
orientations and widths, however, remain similar. Wei’s TEM observations at high
rates (Wei et al., 2002) did not differ qualitatively from those at low rates (in the
case of the nanocrystalline iron). No macroscopic shear bands were observed in the
20-m Fe or 980-nm Fe loaded at high strain rates. Thus the basic mechanism of
shear banding does not appear to change between the low-rate and high-rate loading, and the failure mechanism appears to be mediated by grain size rather than
strain rate. This also indicates that the bands observed in nanocrystalline bcc iron
are not adiabatic shear bands.
6.6.5 Effect of Specimen Geometry on the Shear Band Mechanism
Jia et al. (2003) also tested a number of short specimens (with the same aspect ratio
as that used for the high-strain-rate tests) at low strain rates to examine possible effects of the different specimen geometries conventionally used in such tests. In the
20-m Fe and 980-nm Fe, which deformed uniformly like conventional ductile metals, no effects of specimen geometry were observed on the stress-strain behavior.
There was also no significant difference in the observed stress-strain curves for the
long and short specimens of 268-nm Fe, even though the shear band mechanism is
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211
active at this grain size. The specimens (both long and short) were able to sustain
very large deformations without failure. The experiments showed that the measured
yield strengths and the shear band orientations were identical for the long and short
specimens, but the total strain to catastrophic failure was much larger for the short
specimens (as is expected, because of the effective constraint provided by the compression platens in the small aspect ratio case). In summary, the grain sizes at which
the shear band mechanism is observed in iron, the stress levels at which the shear
bands are formed, the orientations of the shear bands, and the morphology (width,
microstructure) of the shear bands all appear to be independent of specimen geometry. However, final failure as a result of this instability is dependent on geometry (as
with the tensile instability called necking).
6.6.6 Shear Bands in Other Nanocrystalline Metals
Shear bands are not commonly seen in other nanocrystalline materials (other than
bcc metals) unless the material is driven to extreme conditions such as through very
high strain rates or within highly deformed ligaments in the last stages of necking.
The development of shear bands in the final stages of the tensile instability is well
known even in conventional grain sized metals, and is also observed in nanocrystalline metals. It is generally also true that conventional grain size hcp metals are
prone to shear band development because of the lack of symmetry in their plastic
deformations, but nanocrystalline hcp metals have not been studied in great detail.
6.7 Suggestions for Further Reading
1. Bai, Y. and B. Dodd. Adiabatic Shear Localization. Pergamon Press, New York
1992.
2. Wright, T.W. The Physics and Mathematics of Adiabatic Shear Bands. Cambridge University Press, Cambridge, 2002.
3. Anderson, T.L. Fracture Mechanics: Fundamentals and Applications. 3rd Edn.
CRC Press, New York 2004.
4. Broberg, K.B. Cracks and Fracture. Academic Press, London 1999.
6.8 Problems and Directions for Research
1. Make a plot of the tensile strain to failure (as quoted in the literature) as a function
of grain size, for example fcc and bcc nanocrystalline metals.
2. Using the Considere and Hart criteria, compute the strain to onset of instability
in tension for conventional grain-size copper, aluminum, and iron. How does this
compare with experimental data on these materials?
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Nanomaterials
3. It appears that grain size distributions can have a profound influence on the total
strain to failure in tension. Identify three reasons why this might be so.
4. The double torsion test has been used to measure the fracture toughness of very
small diamond specimens – see the work of Field et al. (Davies, 2004). Can this
technique be used to study nanomaterial samples (e.g., electrodeposited samples
available in thin sheets)?
5. Relatively thick (<1 mm) sheets of nanomaterials are now commercially available, typically made by electrodeposition. However, these sheets sometimes develop a columnar microstructure. Using some of the concepts presented in Chapter 4, discuss the potential effects on the tensile failure of the material. Note that
other issues, such as porosity, may also play a role in very thick films.
6. Void growth is a common failure process in metals. Typically, void growth is
easier in materials with lower yield strength. However, the available nucleations
sites for voids can also effect the macroscopic failure process. In nanometals, the
yield strength is greatly increased, but it is possible that the number of nucleation
sites available also increases. Discuss the tradeoffs associated with this failure
process as applied to nanomaterials.
References
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Bridgman, P. (1952). Studies in Large Plastic Flow and Fracture, with Special Emphasis on the
Effects of Hydrostatic Pressure. New York: McGraw-Hill.
Carsley, J., A. Fisher, W. Milligan, and E. Aifantis (1998). Mechanical behavior of a bulk nanostructured iron alloy. Metallurgical Materials Transactions A 29A, 2261–2271.
Carsley, J. E. (1996). Mechanical Behavior of Nanostructured Metals. Ph. D. thesis, Michigan
Technological University.
Chichili, D., K. Ramesh, and K. Hemker (1998). The high-strain-rate response of alpha-titanium:
Experiments, deformation mechanisms and modeling. Acta Materialia 46, 1025–43.
Dalla Torre, F., H. V. Swygenhoven, and M. Victoria (2002). Nanocrystalline electrodeposited ni:
microstructure and tensile properties. Acta Materialia 50, 3957–3970.
Dao, M., L. Lu, R. J. Asaro, J. De Hosson, and E. Ma (2007). Toward a quantitative understanding
of mechanical behavior of nanocrystalline metals. Acta Materialia 55, 4041–4065.
Davies, A. R. (2004). The toughness of free-standing cvd diamond. Journal of Materials Science 39(5), 1571–1574.
Hutchinson, J. W. and K. W. Neale (1977). Influence of strain-rate sensitivity on necking under
uniaxial tension. Acta Metallurgica 25(8), 839–846.
Jia, D., K. Ramesh, and E. Ma (2000). Failure mode and dynamic behavior of nanophase iron
under compression. Scripta Mater 42, 73–78.
Jia, D., K. T. Ramesh, and E. Ma (2003). Effects of nanocrystalline and ultrafine grain sizes on
constitutive behavior and shear bands in iron. Acta Materialia 51(12), 3495–3509.
Jonnalagadda, K. and I. Chasiotis (2008). Tensile behavior of thin films of nanocrystalline fcc
metals. Submitted for publication.
Kumar, K. S., S. Suresh, M. F. Chisholm, J. A. Horton, and P. Wang (2003). Deformation of
electrodeposited nanocrystalline nickel. Acta Materialia 51(2), 387–405.
Lin, I. H., J. P. Hirth, and E. W. Hart (1981). Plastic instability in uniaxial tension tests. Acta
Metallurgica 29(5), 819–827.
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Ma, E. (2006). Eight routes to improve the tensile ductility of bulk nanostructured metals and
alloys. Journal of Materials (JOM) April, 49–53.
Malow, T., C. Koch, P. Miraglia, and K. Murty (1998). Compressive mechanical behavior of
nanocrystalline fe investigated with an automated ball indentation technique. Materials Science
& Engineering A A252, 36–43.
Meyers, M., G. Subhash, B. Kad, and L. Prasad (1994). Evolution of microstructure and shear-band
formation in alpha -hcp-titanium. Mechanics of Materials 17, 175–93.
Nemat-Nasser, S., T. Okinaka, and L. Q. Ni (1998). A physically-based constitutive model for bcc
crystals with application to polycrystalline tantalum. Journal of the Mechanics and Physics of
Solids 46(6), 1009–1038.
Sanders, P., C. Youngdahl, and J. R. Weertman (1997). The strength of nanocrystalline metals with
and without flaws. Materials Science and Engineering A 234–236, 77–82.
Vinogradov, A. and S. Hashimoto (2001). Multiscale phenomena in fatigue of ultra fine grain
materials-an overview. Materials Transactions JIM 42, 74–84.
Wang, Y. M., M. W. Chen, F. H. Zhou, and E. Ma (2002). High tensile ductility in a nanostructured
metal. Nature 419(6910), 912–915.
Wei, Q., D. Jia, K. Ramesh, and E. Ma (2002). Evolution and microstructure of shear bands in
nanostructured fe. Applied Physics Letters 81(7), 1240–1242.
Wei, Q., L. Kecskes, T. Jiao, K. T. Hartwig, K. T. Ramesh, and E. Ma (2004b). Adiabatic shear
banding in ultrafine-grained fe processed by severe plastic deformation. Acta Materialia 52(7),
1859–1869.
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Quarterly 62(1), 291.
Wright, T. (2002). The Physics and Mathematics of Adiabatic Shear Bands. Cambridge: Cambridge University Press.
Wright, T. W. and K. T. Ramesh (2008). Dynamic void nucleation and growth in solids: A selfconsistent statistical theory. Journal of the Mechanics and Physics of Solids 56(2), 336–359.
Wu, X., K. Ramesh, and T. Wright (2003). The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading. Journal of the Mechanics and Physics of
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091904.
If you take care of the small things, the big things
take care of themselves.
Emily Dickinson
7
Scale-Dominant Mechanisms
in Nanomaterials
The unusual properties of nanomaterials all arise from mechanisms that are not
intuitive to us at the human length scale. To gain an understanding of these properties – and to control them – we must discover the mechanisms that are operative at
the length scales associated with the specific nanomaterial. From a science perspective, many of these unusual properties arise from the interactions of intrinsic length
scales (associated with underlying deformation or functional mechanisms) with the
extrinsic or external length scales associated with the nanomaterial or device. This
chapter discusses a variety of such interactions, and describes the implications of
some of them in terms of mechanical behavior. The emphasis is on deformation and
failure mechanisms rather than on functional mechanisms, but many of the deformation mechanisms discussed here have direct implications for function.
A broad brush view of the major mechanisms associated with nanomaterials is
presented in Table 7.1. Using the materials classification scheme of Chapter 1, this
table lists the primary mechanisms associated with discrete nano (dn) materials,
nano device (nd) materials, and bulk nanocrystalline (nc) or nanostructured (ns)
materials. In each case, we consider the extrinsic and intrinsic length scales associated with the morphology, define the important interactions, and discuss some of
the implications. For ease of discussion, we consider the discrete nanomaterials and
nano device materials at the same time, because of the great commonality in the
morphology.
7.1 Discrete Nanomaterials and Nanodevice Materials
7.1.1 Nanoparticles
A quick search of the internet will demonstrate that there is a large variety of materials that are colloquially considered to be nanoparticles. The term nanoparticle is
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 7,
c Springer Science+Business Media, LLC 2009
215
Type of nanomaterial
Discrete nano (dn)
materials
Morphology or
application
conditions
Equiaxed
nanoparticles
Nanotubes
Nanofibers
Nanodevice (nd) materials Nanoparticles
Nanowires and
nanotubes
External length
scales
Particle diameter
Diameter, wall
thickness, length
Diameter, length
Diameter
Diameter, length
Nanoporous structures Thickness
Bulk nanocrystalline
nanomaterials
Thickness
Micropillars
Diameter, height
Microtension
specimens
Diameter, length
Nanoindentation
Indent size
Components
Component size
Mechanisms
Surface stress equilibration
Fracture, surface interactions
Remarks
Suggests core-shell
nanostructures
Very high stiffness and
strength
Fiber-matrix interactions
Surface stresses due to
functionalization
Functionalization-induced
deformations, twinning, flexure
Interactions at optical
wavelengths
Highly sensitive through
flexure
Atomic spacing, defect spacing Buckling, telescoping, fracture
Atomic spacing, molecular
weight
Atomic spacing,
functionalization density
Atomic spacing, defect
spacing, functionalization
density
Pore size, pore spacing,
functionalization density
Atomic spacing, dislocation
spacing, ledge spacing,
functionalization density
Interactions with environment
Optical interactions,
molecular sensitivity
Surface stresses, surface mobility, Highly developed
2D mechanisms, passivation
fabrication techniques
layers, substrate interactions,
diffusion
Grain size, dislocation density Dislocation mechanisms,
Specimen size effects,
twinning, grain boundary (gb)
strain gradients
processes, grain rotation, shear
failure, buckling, constraint
effects
Grain size
Dislocation mechanisms,
Specimen size effects,
twinning, gb processes, grain
strain gradients
rotation, constraint effects,
necking, fracture
Grain size, dislocation spacing Dislocation mechanisms,
Indentation size effects,
twinning, gb processes, grain
strain gradients
rotation, release processes,
pressure effects
Grain size
Dislocation mechanisms,
Design of subscale
twinning, gb processes, grain
structure, as in metallic
rotation, failure processes
foams
Nanomaterials
Thin films
Internal length scales
Atomic spacing
216
Table 7.1 Scale-dominant mechanisms in nanomaterials, categorized in terms of materials classification, morphology, and length scale.
7 Scale-Dominant Mechanisms in Nanomaterials
217
generally used to identify any particle that is less than 100 nm in the largest dimension. A variety of morphologies is thus possible within this loose definition:
•
•
•
•
Spherical nanoparticles (nanospheres)
Rod-like nanoparticles (nanorods)
Plate-like nanoparticles (nanoplates)
Some very complex shapes including dendrites
These morphologies can have very different properties, and indeed this accounts
for the vast range of applications that are attributed to nanoparticles. Note that
the surface area-to-volume ratio of nanorods and nanoplates can be very different
from those of nanospheres. Since this ratio dominates many of the applications of
nanoparticles, the morphology has a dramatic impact on the application.
Large fractions of the nanoparticles used in assorted technologies are also heterogeneous, consisting of complex structures made up of more than one material.
Some nanoparticles appear to be nearly amorphous, with no clearly defined crystal
structure, while others are clearly single crystals showing crystalline facets. The variety of mechanisms that are appropriate to the deformation of nanoparticles is as
large as the variety of nanoparticles themselves, and we will not be able to discuss
all of them in this book.
We will focus our discussion on nanoparticles that are nearly equiaxed, that is,
with aspect ratio (ratio of smallest to largest dimension) close to one. The external
length scale associated with a nanoparticle is then the particle diameter (by definition, less than 100 nm). The only internal length scale associated with the mechanical properties of the nanoparticle is the inter-atomic spacing. The magnitude
of this spacing is typically on the order of 3 Åfor most elements, so that a nanometer contains perhaps 3 atomic spacings (Figure 7.1) separating 4 atoms, and a cubic
nanometer would consist of perhaps 64 atoms. Particles consisting of fewer than
100 atoms are often called nanoclusters.
Consider a nanocluster of 64 atoms in a cubic lattice arrangement – if it were
a cube of side 1 nm, most (about 80%) of those atoms would be on the surface
of the nanocluster. Surface atoms are not surrounded by similar atoms on all sides
and so have a different equilibrium spacing, and thus the cubic lattice must be distorted from the bulk equilibrium configuration. Such surface character can lead to
unusual structures. Further, the surface atoms can interact strongly with other atoms
or molecules in the environment, leading to the development of nanoparticles that
are surface-active. Core-shell type nanoparticles can be developed relatively easily
as a consequence, and these have great value in bioengineering.
Let us assume that the equilibrium interatomic spacing is a in the bulk crystal,
while the particle dimension is D. If the particle is a cube of side D and the atoms
are arranged in a cubic lattice, the total volume of the cube is D3 , while the volume
corresponding to the atoms that are inside the cube (not exposed to the surface)
is (D − 2a)3 . The volume fraction fext of the atoms that are on the surface of the
nanoparticle is therefore (note that the volume fraction is of the same order as the
number fraction)
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Nanomaterials
Fig. 7.1 Schematic of a cubic nanocluster consisting of 64 atoms, approximately 1 nm on a side.
The majority of the atoms are on the surface of this nanocluster, and so will have a different
equilibrium spacing than atoms in a bulk sample of the same material.
fext = 1 −
(D − 2a)3
8a3 6a2 6a
= 3 − 2 + .
3
D
D
D
D
(7.1)
A quick solution of the resulting cubic equation shows that about half the atoms
will be on the surface of a nanoparticle with dimension D ≈ 15a. Since interatomic
spacings are typically of the order of 3 Å, cuboidal nanoparticles are surface dominated (in terms of structure and mechanical response) if they are less than 5 nm
in typical dimension. Larger nanoparticles (up to 20 nm, see Figure 7.2) have very
large surface area-to-volume ratios in comparison to conventional micron-size particles (remember that this ratio goes as D1 ), but the surface equilibration effect on
internal structure becomes less of an issue.
The variation of the volume fraction of surface atoms with nanoparticle size is
shown in Figure 7.2 for an assumed interatomic spacing of 0.3 nm. The rapid growth
of the surface fraction as the particle sizes decrease below 20 nm is apparent, and
the surface fraction dominates in the nano cluster domain.
Surface atoms see a different distribution of neighbors than interior atoms, and
so have different energies than interior atoms. Note that it is not just the outermost
layer of atoms that is affected – the influence of the surface will typically be felt by
several layers of atoms (depending on the range of the interatomic potential). The
precise number of layers that must be considered can only be derived from atomistic
analysis considering the interatomic potentials. The influence of the surface fraction
on the mechanical behavior depends on the difference between the energies of the
atoms on the surface and the energies associated with the atoms in the interior of the
7 Scale-Dominant Mechanisms in Nanomaterials
219
Fraction of Atoms on the Surface
1.00
0.80
0.60
0.40
0.20
0.00
0
20
40
60
80
100
Particle Size, nm
Fig. 7.2 Volume fraction of atoms on the surface of a cuboidal nanoparticle as a function of
nanoparticle size, based on Equation (7.1) and an assumed interatomic spacing of 0.3 nm. Significant surface fractions are present below about 20 nm. Note that particle sizes below 0.6 nm are
poorly defined.
particle. The discussion of the surface energy effects that follows is taken largely
from the summary of the mechanics issues provided by Dingreville et al. (2005). If
we define the difference between the energy of an atom at the surface and the energy
of an atom at the interior as esi , then we can define a surface free energy density (first
defined by Gibbs, the grandmaster of thermodynamics) as the areal density Γ :
Γ dA = ∑ esi ,
(7.2)
where the summation is over all atoms in the layers affected by the surface element
dA, the capital A indicating that the surface element is in the undeformed or reference configuration (remember that the atoms rearrange themselves as a result of
these surface interactions). The element da in the deformed configuration is related
to the undeformed element dA through the surface strains:
s
s
+ ε22
),
da = dA(1 + ε11
(7.3)
where the coordinate axes are chosen so that x3 is normal to the surface (that is,
s , ε s , and ε s ). These surface strains are related to the
the in-plane stresses are ε11
22
12
s , using a concept similar to
surface free energy density through surface stresses Tαβ
that presented in Equation (2.68) discussed in Section 2.4.1:
s
Tαβ
=
∂Γ
s ,
∂ εαβ
(7.4)
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Nanomaterials
where the Greek subscripts α and β are used to imply that only surface quantities
are being considered, with the indices taking on values of 1 and 2 only. This is
effectively a definition of the surface stress, and we see that it can be derived from
the surface energy density. How does the surface energy density Γ depend on the
surface strain? As a first approximation, one can expand the surface energy in a
s = 0, and in principle the various terms in this
Taylor series about the point εαβ
series can be computed using atomistic calculations. For our purposes, it is simplest
just to assume that for small strains the surface energy density is quadratic in the
surface strain, with all higher order terms being negligible (the identical assumption
is made in linear elasticity for the strain energy density). It then follows that the
surface stress is given by (using Equation 7.4)
s
s
= Gαβ γδ εγδ
,
Tαβ
(7.5)
where again the Greek subscripts indicate that they range only over 1, 2 and the
summation convention is implied. The coefficients Gαβ γδ are the components of the
surface elasticity tensor, which is a two-dimensional analog of the elastic modulus
tensor discussed in Equation (2.65) of Section 2.4.1, but derived specifically for
surfaces as a result of the fact that the surface energy Γ is different from the strain
energy W in the interior of the solid. Note that like the elastic modulus tensor, we
have
(7.6)
Gαβ γδ = Gβ αγδ = Gαβ δ γ = Gγδ αβ ,
for reasons identical to those discussed in Section 2.4.1. The values of the components of the surface elasticity tensor must be obtained either through atomistic calculations and then using Equation (7.2), or by direct measurement, although such
measurements are extraordinarily difficult.
Equation (7.5) tells us that the effect of the free surface is to cause the top few
layers of atoms to behave effectively as though they are made of a different material
in terms of elastic behavior. One could view the particle as being a composite structure (Figure 7.3), with the inner core having the normal bulk properties and the outer
αβγδ
Fig. 7.3 Core-shell model of a nanoparticle viewed as a composite, with a surface layer that has
different properties as a result of the surface energy and its effects on binding.
7 Scale-Dominant Mechanisms in Nanomaterials
221
shell having different elastic properties, based on the surface energy. Whether the
outer shell is stiffer or more flexible depends on the specific atomic species being
considered (note that many nanoparticles are not single element particles, e.g., ZnO
and CdS particles). From a mechanical viewpoint, the behavior of the surface layers
will depend not just on the surface elasticity tensor but also on the particle shape
(morphology), since a starfish-shaped particle can be expected to have a different
surface response (at the level of Equation 7.2) than a spherical particle of the same
material.
Composite mechanics approaches can be used to compute the effective elastic
behavior of particles having such a core-shell structure. For example, Dingreville
et al. (2005) show that the contribution of the surface energy to the effective modulus
of a spherical nanoparticle is inversely proportional to the particle size. For large
particles the effect can be ignored, but the effect can be significant for nanoparticles.
7.1.1.1 Functionalization
The applications of many nanoparticles are based on the fact that the surfaces of
the nanoparticles can be modified through the addition of specific molecules in
order to accomplish specific functions. This process is known as functionalization.
For example, metallic nanoparticles might be coated with a layer of an organic
compound, and then have specific biological molecules attached to the organic
compound (Figure 7.4) so that this functionalized nanoparticle can interact with the
biological environment (Bruchez et al., 1998).
The process of functionalization changes the surface stresses on the nanoparticle
by changing the surface energies and, correspondingly, certain kinds of functionalization are easier for nanoparticles that have specific surface stresses. The surface
Fig. 7.4 A spherical nanoparticle carrying several organic molecules on its surface. The molecules
are chosen to perform a specific function, such as recognizing a molecule in the environment, and
so are called functionalizing molecules. The spacing of the functionalizing molecules defines the
functionalization density. Since the molecules modify the surface stress state when they attach to
the surface, the functionalization density modifies the surface stresses and can even modify the net
conformation of the nanoparticle.
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Nanomaterials
stress modification of the nanoparticle will depend on the density η f of the functionalizing molecule that is applied to the surface (the density is defined as the
number of functionalizing molecules per unit surface area). The reciprocal of the
square root of this functionalization density provides another intrinsic length scale
in the problem. There is an increasing array of biomedical applications of functionalized nanoparticles (Sanvicens and Marco, 2008), including potential treatments
for some kinds of malignant tumors. Other kinds of functionalization can lead to
the interaction of nanoparticles at optical wavelengths, leading to various sensor
applications (a specific example in this area is the application of quantum dots;
Alivisatos, 1996).
7.1.2 Nanotubes
There is a vast literature on carbon nanotubes, which are perhaps the best understood
of the discrete nanomaterials. We do not attempt to reprise the entire literature on the
mechanics of carbon (and other) nanotubes here, but some references are suggested
at the end of this chapter. Here, we present just a brief discussion of the major issues
and mechanisms.
By definition, nanotubes are tubes that have nanoscale diameters and some predefined wall thickness, with length-to-diameter ratios that are usually greater than 10.
Carbon nanotubes consist of single or multiple walled structures that can be thought
of as rolled up graphene sheets (see Figure 7.5). Each sheet is an atom-thick layer of
carbon atoms arranged in a specific structure – depending on exactly how the sheets
are rolled up, one obtains varieties of nanotube known as armchair, chiral or zigzag
nanotubes (the adjectives indicate the apparent atomic arrangement). The hexagonal
two dimensional lattice is mapped on to a cylinder of radius R with helicities characterized by so-called rolling vectors (n, m), and the different types of nanotubes are
characterized by these (n, m) pairs.
The external length scales associated with a nanotube are the diameter, the wall
thickness (different for single-wall and multi-wall nanotubes), and the length. The
internal length scales associated with nanotubes are the interatomic spacing (as in
nanoparticles) and the defect spacing (defects can control the mechanical properties
of nanotubes). There are (Figure 7.6) three associated deformation mechanisms: column and shell buckling of the thin-walled tube, telescoping of the tube, and fracture
of the tube. Telescoping is a deformation mode that is only possible in multi-walled
nanotubes, but buckling and fracture can occur in all nanotubes.
Buckling is a structural instability in which a structure deforms suddenly with
an abrupt drop in load-carrying capacity, and is thus a structural failure mechanism
(as distinct from a material failure mechanism like fracture). There are many different kinds of buckling, the best known being the compressive buckling of slender
columns (called Euler buckling, see the discussion of Equation 3.7). This is a structural instability in which the column ceases to remain straight, adopting a bent shape
when the load exceeds a critical value called the Euler buckling load:
7 Scale-Dominant Mechanisms in Nanomaterials
223
Fig. 7.5 Schematic of the structure of carbon nanotubes, showing the armchair, zigzag and chiral
conformations. This beautiful illustration was created by Michael Stróck on February 1, 2006 and
released under the GFDL onto Wikipedia.
Pcr =
π 2 EI π 2 EA π 2 EA
=
=
,
Le 2
Le 2
SR 2
(7.7)
ρg
where Le is the effective length of the column, and I =
Aρg 2 is the moment of inertia,
with A the cross-section area of the column and ρg AI is a geometric property of
the cross-section called the radius of gyration. For a circular cross-section of
√radius
r(r+t)
r, ρg = 2r , while for a tube with inner radius r and wall thickness t, ρg = √2 .
The ratio of the effective length of the column to the radius of gyration is called the
slenderness ratio SR = ρLge . Large slenderness ratios thus imply low critical buckling
loads in compression.
Since the lengths of nanotubes are usually orders of magnitude larger than the diameters, such compressive buckling occurs very easily, and so nanotubes are rarely
intended for use under compressive loading. The primary use of thin fibers like
224
Nanomaterials
Fig. 7.6 Schematic of two possible modes of deformation and failure in nanotubes. (a) Buckling
of a thin column. (b) Telescoping of a multi-walled nanotube. The fracture of nanotubes is a third
mode, but is not shown.
nanotubes is always in tension. However, even when the intended application is
tension, the nanotubes are typically also subjected to flexural forces (for example,
as a result of off axis load), and in such cases the additional mode of asymmetric
buckling of the tube wall becomes a possibility. An excellent discussion of the mechanics of buckling of carbon nanotubes viewed in a continuum context is presented
by Pantano et al. (2004). Simulations of the buckling of single walled carbon nanotubes are presented from both molecular dynamics and continuum approaches by
a number of authors, for example, Cao and Chen (2006).
A multiwalled carbon nanotube in tension may first fail through a telescoping
process in which the inner walls are pulled out with respect to the outer walls (Cumings and Zettl, 2000, 2004). This mode arises because the shear strength of the interface between the walls is significantly lower than the strength of the multiwalled
nanotube itself. The relative sliding of the walls appears to be nearly frictionless
(Kis et al., 2006). The telescoping mode also suggests applications in sensing of
small displacements and as encoders (Jiang et al., 2007).
If the nanotube is placed under tension, the primary failure mode is that of fracture (Yu et al., 2000b). The mode of fracture depends on the structure and composition of the nanotube, as well as on the number and location of defects within the
nanotube (note that defects in a nanotube are primarily missing atoms or incorrectly
located atoms). Unlike conventional crystalline metallic structures, nanotubes do
not contain dislocations, and so dislocation plasticity mediated necking failures cannot occur. Multi-walled carbon nanotubes can use sliding between the layers in the
walls to generate apparent plastic deformation before final fracture (this is related
7 Scale-Dominant Mechanisms in Nanomaterials
225
to the telescoping mode) but single walled carbon nanotubes do not have this option. Multi-walled carbon nanotubes often fracture through failure first of the outer
shell (this mode is sometimes called the “sword-in-sheath” mode) (Yu et al., 2000).
When bundles of single walled nanotubes are tested in tension, a bundle appears to
neck down to a single nanotube before fracture, but this corresponds to the progressive figure of multiple nanotubes within the bundle (Yu et al., 2000). Simulations
indicate that single walled carbon nanotubes may fail ultimately through the reorganization of the atomic structure until individual bonds carry the load – an atomic
version of necking.We discuss models for nanotube fracture in the next chapter.
7.1.3 Nanofibers
Nanofibers are by definition solid fibers in which the diameter is less than 100 nm,
and typically the fiber lengths are many orders of magnitude greater than the fiber
diameters. The primary external length scales are therefore the fiber diameter and
the fiber length. Although there are a variety of nanofibers available, the vast majority of nanofibers of interest in nanotechnology are made from polymer precursors,
typically by processes such as electrospinning. An excellent review of the latter
topic is provided by Huang et al. (2003).
For polymer nanofibers, an internal length scale that appears is related to the
molecular weight of the fiber, and can be viewed variously as the molecular length
or as the persistence length. The persistence length is the length over which the orientation of one part of a long molecule is correlated with the orientation of another,
so that it essentially measures the stiffness of the molecule, large persistence lengths
implying stiff molecules (think of a really floppy molecule – the orientation of the
end of the molecule has very little correlation with the orientation of the beginning
of the molecule). For example, the persistence length of a rigid rod would be infinite, while the persistence length of DNA is around 50 nm. In conventional polymer
fibers, with fiber diameters in the microns, the polymer persistence length is usually
much smaller than the fiber diameter. The stiffness of the fiber is then determined
not by the molecular stiffness as much as by the variety of configurations of the
multiple molecules that make up the fiber. This is known as an entropic stiffness,
the idea being that highly disordered states – with many possible configurations –
are preferred over ordered states (fewer possible configurations) from an entropic
viewpoint.
When the fiber diameter becomes comparable to the persistence length of the
molecule, one should expect a dramatic change in the behavior of the fiber (because the molecular stiffness is itself now important, rather than simply the entropic
stiffness). This suggests that nanofibers should have very different properties from
conventional-sized fibers. Note that most polymer chains have relatively stiff carbon
chain backbones. The process of making the nanofibers typically involves an effective drawing of the fiber, resulting in oriented molecules and again greater stiffness.
Thus, in general, one should expect that the stiffness of the nanofiber would exceed
226
Nanomaterials
the stiffness of the conventional size fiber. Additional effects arise because of the
increased surface area to volume ratio associated with nano fibers, for the same
energetic reasons that were discussed in the case of nanoparticles.
There is very little experimental data on the mechanical properties of single polymer nanofibers, but there is a wealth of data on the mechanical properties of polymer
nanofiber-reinforced composites (Huang et al., 2003). In such composites, the fiber
matrix interactions become a dominant term in determining the mechanical properties. The flexibility and strength of these fibers also makes them attractive for device
applications, where the large surface area can be used for functionalization. These
fibers are often used as building blocks for larger structures, such as fiber mats
(sometimes called nanomats) and in three-dimensional structures such as polymer
scaffolds for tissue engineering (these provide matrices for cell implantation and the
development of extra-cellular matrix, essential for tissue generation). The mechanical properties of the fiber mats and polymer scaffolds are themselves interesting, but
the great applications of these structures are in the provision of large surface area
for interaction with the biological and chemical environment.
7.1.4 Functionalized Nanotubes, Nanofibers, and Nanowires
Surface functionalization of one-dimensional nanostructures (nanotubes, nanofibers,
and nanowires) has multiple applications, ranging from the modification of a fibermatrix interaction for a composite to the development of sensors and actuators in the
nanoscale domain. As with the nanoparticles, the functionalization of the surface results in a modification of the surface energies and therefore of the surface stresses,
and the reciprocal of the square root of the functionalization density η f (i.e., √1η f )
provides an additional length scale. Such modifications can result in equilibrium
shapes for the nanofibers that are far from straight (referred to as functionalizationinduced deformation). The morphology of the nanowires and the nanotubes naturally lends itself to applications that take advantage of the easy flexural deformations. Indeed, such devices are highly sensitive to the environment through the
combination of functionalization and the flexural mode (i.e., the bending stiffness is
modified by the coupling to the environment, as when the functionalizing molecules
interact with molecules in the environment). One specific approach that is often used
in sensor applications is to examine shifts in the resonant frequency of the device
(which depends on the flexural stiffness) as a result of interactions with the environment.
7.1.5 Nanoporous Structures
Nanoporous structures are typically thin film structures that are deliberately constructed to contain large numbers of nanosized pores. A common way to produce
these materials is to construct a binary alloy made up two highly miscible metals,
such as gold and silver, and then to dissolve out one element using a reactive agent.
7 Scale-Dominant Mechanisms in Nanomaterials
227
The resulting porosity is extremely small, and subsequent thermal processing affords some control of size and distribution. It is also possible to make nanoporous
structures by highly controlled etching procedures. The volume fraction of the pores
can be very large, depending on the application. The primary external length scale
in this case is the film thickness t f , while the important internal length scales are the
pore size l p and the pore spacing λ p . If the pore surface is functionalized, then the
functionalization density also provides an intrinsic length scale through √1η f , which
must by definition be less than l p . The pore size and the pore spacing are related for a
given pore volume fraction: smaller pores will be closer together, because there have
to be a greater number of small pores to achieve a given pore volume fraction. The
pore surface area to pore volume ratio increases as the pore size decreases, and so
nanoporous structures are very efficient in terms of making surface area available for
activation. If l p ≈ t f , columnar structures are likely to be obtained. If λ p ≈ t f , then
the thin film typically contains one pore in its layer thickness. If the pore volume
fraction is large enough, the pores are usually connected (essentially a percolation
limit), and this is usually desirable for sensing applications. The resolution of the
sensor (if functionalized) is related to η f .
The large surface area generated by the pores results in strong interactions
with the environment, and some interesting applications arise in catalysis. Such
nanoporous structures are typically constructed in order to interact strongly with
the environment, particularly in applications involving the sensing of gaseous environments (such as small concentrations of toxic gases or explosive residues). These
devices can also be constructed to provide optical interactions, that is to provide optically perceptible changes on exposure to the target molecule (Chan et al., 2000).
7.1.6 Thin Films
The subject of thin films and the mechanics issues therein has been extensively investigated because of the applications in semiconductor technology, and we will
not try to cover these issues here; the reader is referred to the supplementary reading material for detailed analyses. In terms of length scales of interest, however,
the intrinsic length scales include the defect density, the dislocation spacing (if any
dislocations are present), the surface ledge spacing (atomic-sized ledges are sometimes present on the surfaces), and grain size. The primary extrinsic length scale is
of course the film thickness. The mechanics issues are very different for films with
device and electronic applications as compared to films with mechanical applications (such as wear resistance).
7.1.7 Surfaces and Interfaces
Phenomena that occur at surfaces and interfaces can play a very large role in nanotechnology, and so also in nanomaterials. The mechanics associated with surface
and interface interactions can be quite complex, ranging from issues such as friction
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Nanomaterials
and contact angle (wetting) to residual stresses and delamination. Surface stresses
can be used to control surface phenomena (for example in the self-organization of
structures on surfaces). These are very important topics in nanotechnology, but will
not be dealt with in this book.
7.2 Bulk Nanomaterials
A variety of internal mechanisms lead to scale-dependent or size-dependent behavior of bulk nanomaterials, and these are discussed here. This is an area of significant
research activity as of this writing. In broad terms, the internal mechanisms are:
•
•
•
•
•
Dislocation-mediated plasticity
Deformation twinning
Grain boundary sliding
Grain boundary propagation (grain growth or shrinkage)
Grain rotation
Each of these mechanisms will be discussed in this section. We begin with a
discussion of dislocation mechanisms, relating primarily to issues of strength and
plastic flow.
7.2.1 Dislocation Mechanisms
We have discussed dislocation plasticity in some depth in the sections on the plastic behavior of nanomaterials, but that section focused on the overall behavior of
bulk nanocrystalline materials, and in particular on the effects of grain-size. A more
general discussion of dislocation mechanisms is of interest in this chapter, because
some of these mechanisms are only active under specific conditions. As always, it is
the interaction of length scales that determines what the dominant mechanisms will
be in any given problem (Greer, 2006).
There are a number of intrinsic length scales (Table 7.2) associated with dislocation mechanisms, and these are listed below:
• The dislocation core size Rc defines the smallest length scale that can be identified with a given dislocation.
• If the dislocation dissociates into two partial dislocations (a dissociation that will
only occur if it is energetically favorable to do so), the spacing ws f of the partials
provides another length scale. The spacing of the partials is called the width of
the stacking fault ws f , because a stacking fault exists between the two partials. A
stacking fault is an unexpected stacking, or error in the stacking, of the atomic
planes. The energy associated with the stacking fault is called the stacking fault
energy; if the stacking fault energy is high, the equilibrium spacing of the two
7 Scale-Dominant Mechanisms in Nanomaterials
•
•
•
•
•
229
dissociated partials will be small. One could view the dissociated partials as representing an extended core for the dislocation, although the core is no longer
cylindrical.
The effective range Re of the elastic stress fields around the dislocation core
provides another length scale.
If the dislocation density is ρD , then the average spacing between dislocations is
given by λ = √1ρD . Essentially, the other dislocations are considered to be either
obstacles or sources of energy for re-patterning of the dislocation substructure.
The average spacing LS of dislocation sources (there are many possible dislocation sources, the classic example being the Frank-Read source). This spacing
may relate to other features such as the dislocation density.
The average spacing between obstacles (such as dispersoids) within a given crystal provides another length scale, LO .
The distance between grain boundaries (that is, the grain size d) provides another
length scale. This length scale was the focus of discussion in the chapter on the
mechanical behavior of nanocrystalline materials.
Table 7.2 Typical intrinsic length scales that arise from dislocation mechanisms.
Dislocation mechanism length scale
Symbol
Typical size
Dislocation core size
Rc
1 nm
Stacking fault width
ws f
Several nm
Range of elastic stress field
Re
?
Dislocation spacing
λ = √1ρD Highly variable, from 10 nm to 1 µm
Dislocation source spacing (inside grains)
LS
?
Obstacle spacing (inside grains)
LO
?
Distance between grain boundaries (grain size)
d
10 nm – 10 µm
Scale-dominant behaviors (i.e., behaviors that are the consequence of specific
scaling) can be expected whenever an external length scale approaches one of these
internal length scales. Such scale-dominant behaviors are to be viewed differently
from the grain size dependent mechanical behaviors described in the previous chapter, in that they arise as a consequence of the competition between an internal and
an external length scale, rather than as a consequence of competition between two
internal length scales (which are the cause of most grain size strengthening mechanisms).
Consider, for instance, the mechanical behavior of a micropillar studied using the
micro-compression technique described in Chapter 3. The two external length scales
that define a micropillar are the diameter dm and the height. Most experimentalists
ensure that the height is a fixed multiple of the diameter (usually twice the diameter) in order to obtain accurate measurements of material behavior (Zhang et al.,
2006), and so we can view the diameter as the only external length scale of interest.
Clearly one would not expect the micropillar to have the same behavior as the bulk
polycrystalline material if the grain size is comparable to the pillar diameter (i.e,
dm ≈ d), since the averaging associated with many crystals is no longer possible.
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Nanomaterials
If the micropillar diameter were to be smaller than the typical grain size, then
the typical micropillar specimen would be a single crystal specimen. IF the single
crystal contains an initial dislocation density, then the interaction that develops is
certainly between the pillar size and the dislocation spacing, but there is also now
the question of whether new dislocations can be generated inside the single crystal,
which relates to the dislocation source spacing. If there are other obstacles inside
the single crystal, such a dispersoids or precipitates, these scale interactions will
also be important. In the specific case of a single crystal containing no other obstacles and no additional dislocation sources, the interaction between the specimen
size and the dislocation spacing can evolve in a complex way, since a pre-existing
dislocation density can be modified by the presence of free surfaces (which attract
the dislocations because of the image forces), resulting in a reduction of the effective dislocation density. Such a dislocation starvation process (Greer and Nix, 2006)
could result in the development of apparent enhanced strength. Similar issues arise
with metal nanowires and some thin films (Nix et al., 2007).
7.2.2 Deformation Twinning
Deformation twins are small finite regions of a crystal that become reoriented during deformation (Figure 7.7), and deformation twinning provides another inelastic
deformation mechanism, because collections of deformation twins are able to provide a macroscopic inelastic deformation. This is a deformation mechanism that is
observed very often in some materials (for example, HCP metals like zirconium and
hafnium) and only rarely in others (for example, aluminum). The mechanism is of
interest to us because it is sometimes observed in nanomaterials even when it does
not appear under equivalent conditions in coarse-grained versions of the material; it
is thus a mechanism that is scale-dependent.
Deformation twinning processes have been studied for many years. One of the
seminal papers in this area is that of Christian and Mahajan (Christian and Mahajan, 1995). Several characteristics of deformation twinning have been identified
(although most of the work is in metals):
• First, the onset of deformation twinning appears to be relatively athermal, controlled more by the applied stress than by the current temperature or strain rate
(in contrast, note the dislocation mechanisms are typically both temperature and
strain rate-sensitive)
• Second, it appears that a critical stress is required to develop substantial twinning,
although there is substantial argument over whether this is a threshold stress or
simply a stress range
• Third, materials with high stacking fault energies (like aluminum) appear to have
difficulty twinning
Nanocrystalline materials generally (but not always) twin more easily than their
conventional grain-sized cousins. The reason for this remains somewhat unclear.
7 Scale-Dominant Mechanisms in Nanomaterials
231
Fig. 7.7 The development of a twinned region in a material. The figures show the results of a
molecular dynamics calculation of the simple shear of a single crystal of pure aluminum. (a)
Atomic arrangement in undeformed crystal before shearing. The straight lines are drawn to guide
the eye to the atomic arrangement. (b) After shearing, a region of the crystal has been reoriented
(this region is called the twin). The twin boundaries can be viewed as mirror planes, and the lines
show the new arrangement of the atoms.
Part of the difficulty is fundamental: there is no single accepted model that explains
all of the observations of deformation twinning. A brief and high-level review of the
basic concepts associated with deformation twinning is presented here.
Experimental observations demonstrate that deformation twins are nucleated,
grow in length and width (Figure 7.8), and then interact with other twins. Models
for deformation twinning must therefore account for nucleation, growth, and interaction. There is some understanding of the processes associated with the nucleation
of deformation twins, and we shall discuss these processes shortly. However, there
is little understanding of the dynamics associated with the growth of deformation
twins, and there are few models that handle the processes associated with the interaction of twins. This lack of understanding of the fundamental mechanism places
real limits on the development of models for the scale-dependence of deformation
twinning in conventional and nanocrystalline materials:
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Nanomaterials
Fig. 7.8 Example of deformation twins in a metal (the hcp metal hafnium). (a) Initial microstructure before deformation. (b) Twinned microstructure after compressive deformation at low strain
rates and at 298K. The twinned regions are the lenticular shapes within the original grain structure.
• The classical view of twin nucleation uses the idea that a twinned region in a crystal can be generated by the coordinated development of stacking faults within the
crystal. A stacking fault is generated when a full dislocation dissociates into two
partials, and one of these partial dislocations (called the leading partial) moves
through the crystal ahead of the trailing partial. If a second stacking fault is generated on an adjacent layer of the crystal through the motion of partials on that
layer, the crystal now contains a two layer defect. Conventionally, two stacking
faults are not considered sufficient to be called a twin. If a third set of partials
moves through the crystal on the third adjacent layer, the resulting three layer
rearrangement is often called the twin nucleus. The question of twin nucleation
thus amounts to the question of the generation of a twin nucleus consisting of
three layers through the coordinated motion of partial dislocations. Twin nucleation has thus been addressed by examining the energetics of the generation of
stacking faults and the forces associated with the motion of partial dislocations
(e.g., Kibey et al., 2007).
• The growth of twins is a much thornier problem, because it involves several components: the energetics of growth, the dynamics of growth and the interaction
with potential dislocation mechanisms. Twin growth is relatively fast, and many
materials scientists speak of twin growth as essentially instantaneous. However,
from an energetic viewpoint, the rate of twin growth must be controlled by some
as-yet undefined driving force (the beginnings of such models have been developed for the twins that are observed in shape memory alloys (Rosakis and Tsai,
1995). Such growth models are important in being able to understand completely
the common observation that twins occur more often in materials subjected to
high-rate deformations.
• The interaction of twins with existing dislocation structures, the interaction of
dislocations with twin boundaries, and the interactions of twins with other twins
represent very complex mechanics and materials issues. Multiple twin populations are often observed in metals subjected to large strains at high-strain-rates. It
7 Scale-Dominant Mechanisms in Nanomaterials
233
is apparent that twin nucleation is affected by the presence of pre-existing twins,
and the shapes of twinned boundaries appear to be connected to the character of
the local twin population. Current models are not capable of predicting even the
likely fraction of the crystal that is twinned after a given deformation.
Twin Number Density
A general, if phenomenological, way of thinking about deformation twinning is
to view twin development as a mechanism that competes with dislocation mobility.
The idea here is that twinning will occur if a critical stress is reached, and that the
material chooses either to deform through dislocation motion or through twinning,
based on the available stress. The dislocation mobility is low for low temperatures
or high strain rates, and so high stresses are required in order to develop plastic
deformations. Deformation twins may develop instead of dislocation slip as a consequence of the high stresses. Thus, one would argue that at low temperatures, the
mobility of dislocations is relatively low, and so a material may prefer to twin rather
than deform through dislocation motion. Similarly, under high rates of loading, inertial or strain-rate history effects may be such as to make it difficult to move dislocations, so that twinning becomes the preferred mechanism. For the specific case
of titanium, this was shown by Chichili et al. (1998) in Figure 7.9, which presents
the measured twin number density as a function of the applied stress for a variety of
conditions, including low and high strain rates and low and high temperatures.
400
500
600
700
800
900
True Stress, MPa
Fig. 7.9 Evolution of twin number density in titanium with applied stress, over a variety of strain
rates and temperatures. The twin density is not correlated with strain rate or temperature (or strain),
but only with applied stress (Chichili et al., 1998). Reprinted from Acta Materialia, Vol. 46, Issue
3, page 19, D.R. Chichili, K.T. Ramesh, K.J. Hemker, The high-strain-rate response of alphatitanium: experiments, deformation mechanisms and modeling. Jan. 1998 with permission from
Elsevier.
How does this relate to the question of twinning in nanomaterials? One could
argue (as some do) that as the overall stress level that must be applied to deform
nanomaterials increases, the probability of finding twins increases (i.e., because of
the high applied stresses, it becomes possible to nucleate twins in a nanocrystalline
version of a material that would not twin easily in conventional grain sizes). A more
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Nanomaterials
substantive approach is to consider the stress levels required to nucleate partial dislocations as compared to full dislocations, e.g., (Chen et al., 2003a). The shear stress
τ f required to nucleate a full dislocation from a dislocation source of size d, assuming that the grain boundary is the source, is
2α µ b f
,
(7.8)
d
where α = 0.5 for edge dislocations and α = 1.5 for screw dislocations, µ is the
shear modulus, and b f is the Burgers vector corresponding to the full dislocation. In
contrast, the shear stress τ p required to generate the partial dislocation that results
in twin nucleation must account for the additional work associated with the stacking
fault:
2α µ b p
γ
+ ,
τp =
(7.9)
d
bp
τf =
where b p is the Burgers vector associated with the partial dislocation and γ is the
stacking fault energy. Identifying the grain boundaries as the dislocation sources
allows one to associate these stresses with the grain size. For most materials under
most conditions, τ p > τ f , full dislocations are preferentially generated, and twinning
is unlikely. The stresses required to nucleate full and partial dislocations will be
equal when
2α µ (b f − b p )b p
.
(7.10)
dc =
γ
At grain sizes below this critical grain size dc , the nucleation of partial dislocations becomes easier than the nucleation of full dislocations and this partial dislocation generation implies that twinning may become a preferred deformation mode.
Note that the Burgers vector for the full dislocation is always greater than the Burgers vector for the partial dislocation. The preferential generation of partial dislocations over full dislocations in nanocrystalline materials has also been observed in
molecular dynamics simulations (Yamakov et al., 2002). More sophisticated models of this type have also been developed, which account for the different stresses
that are required to generate leading partials, trailing partials and twinning partials
(Liao et al., 2003). The potential increased incidence of twinning in nanocrystalline
materials will have implications for the multiaxial constitutive functions that must
be used to describe the mechanical properties of these materials.
An example of twinning in a nanocrystalline aluminum (mean grain size of
50 nm) is shown in Figure 7.10 (Cao et al., 2008). Aluminum has a high stacking
fault energy, and does not appear to twin at conventional grain sizes, but develops deformation twins at nanocrystalline grain sizes, as first shown by (Chen et al.,
2003a).
A second way in which deformation twinning has become of interest in nanocrystalline materials is based on the idea that the twin boundaries represent additional
obstacles (akin to grain boundaries) or provide additional degrees of freedom in
terms of deformation. Some highly twinned nanocrystalline or ultrafine grained materials have been observed to have both high strength and high ductility (Dao et al.,
2006; Chen and Lu, 2007). The precise reasons for this behavior remain unclear as
of this writing.
7 Scale-Dominant Mechanisms in Nanomaterials
235
Fig. 7.10 High resolution electron micrograph of deformation twins developed in nanocrystalline
aluminum subjected to large shearing deformations (Cao et al., 2008). The diffraction pattern on
the right demonstrates the twinned character.
7.2.3 Grain Boundary Motion
The fact that the grain boundary volume fraction begins to dominate the behavior of
bulk nanocrystalline materials leads one to examine the possible mechanisms that
are directly associated with the grain boundary. To this point, we have discussed two
different ways in which the grain boundary can affect the behavior of the material.
First, the grain boundary may act as an obstacle to dislocation motion, or in general
as an obstacle to the development of compatible deformations. Second, the grain
boundary may act as a source of dislocations (or twins), and thus may actually
aid the development of plastic deformations. However, the existence of so many
grain boundaries also makes possible deformation mechanisms that directly invoke
movement of (or at) the grain boundary. Idealizing the grain boundary as a plane
separating two grains, one sees (Figure 7.11) that two motions are possible: (a)
normal to the grain boundary and (b) in the plane of the grain boundary. In vector
form, this can be expressed as
δ = δn n + δt t
(7.11)
where δn is the normal displacement of the boundary and δt is the tangential (inplane) displacement of the boundary. The first component results in the growth or
the shrinkage of the grain, while the second results in the relative sliding of two
grains.
In general, thermodynamics does not favor the shrinkage of the grain, unless substantial amounts of energy are provided. In contrast, particularly in nanocrystalline
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Nanomaterials
δ
δ
Fig. 7.11 The possible grain boundary motions: displacements δn in the direction normal to the
GB (grain growth or shrinkage), and δt in a direction tangential to the plane of the boundary (grain
boundary sliding). The unit normal vector is n and the unit tangent vector is t, as in Equation (7.11).
materials, thermodynamic processes favor grain growth. Thermally assisted grain
growth is familiar to most mechanical engineers and materials scientists, since this
is a classical method of grain coarsening and develops routinely during annealing
processes. However, grain growth can also be a stress assisted process (Gianola
et al., 2006), and this may be of importance to many deforming nanocrystalline
solids.
Grain boundary sliding is a mechanism that is well known from studies of creep
deformations of materials. This mechanism is not discussed in any detail here, but
the approach itself is a classical one in materials science (e.g., Ashby and Verrall,
1973), involving diffusive processes.
7.2.4 Grain Rotation
As the grain size decreases, the grains in the bulk nanocrystalline materials become
harder (in the sense that they do not deform easily). Is there a point at which a
nanocrystalline material can be treated like a granular solid? The limiting condition
would appear to be one of elastic grains deforming as a granular assembly. However,
most bulk nanocrystalline materials cannot be viewed in this way at any length scale,
because they generally maintain compatible deformations even at the grain boundaries (except perhaps at triple points or junctions) and because they usually retain
the ability to develop some plastic deformations. Plastic deformations have been
observed in the TEM for grain sizes as small as 5 nm. Still, the increasing strength
suggests that some of the concepts that are relevant to granular materials may
7 Scale-Dominant Mechanisms in Nanomaterials
237
also be important for nanocrystalline materials. We examine one of these concepts
in this section.
There are two motions of grains (rather than grain boundaries) that may be of
interest in nanocrystalline solids: grain translation and grain rotation. The rigid
translation of grains is generally not a significant mechanism, although this may
be important in the case of very heterogeneous grain size distributions. The rotation of grains, however, plays a significant part in accommodating deformations.
For example, grain rotations have been observed in the development of shear
bands in nanocrystalline metals (Wei et al., 2002). Joshi and Ramesh (2008b) have
demonstrated that grain rotations lead directly to the development of localized
shearing deformations in nanocrystalline solids, and suggest that this mechanism
may result in the development of limited stable deformations and specific grain
size regimes (Joshi and Ramesh, 2008c). The core concepts are discussed in this
section.
There are few theoretical investigations of shear banding in bulk nanocrystalline
materials as a function of grain size. The continuum approach cannot predict such
size effects because of the absence of a length-scale in the governing equations.
Strain gradient plasticity theories can be used, but these theories do not explicitly
account for grain size (Carsley et al., 1997; Zhu et al., 1997). Joshi and Ramesh
modeled the growth of shear bands in nanocrystalline and ultrafine-grained materials using an explicit consideration of grain rotation driven geometric softening. The
model considers the development of shearing instabilities through a visco-plastic
constitutive law for the material response coupled with micromechanics-based softening through grain rotation, motivated by the observations of Jia and co-workers
(Jia et al., 2003; Wei et al., 2002). The grain size enters naturally through the grain
rotation mechanism in this model.
The essence of the model is as follows. Assume that each grain in the material
has anisotropic plastic behavior, with a hard orientation and a soft orientation, as
shown in Figure 7.12a. The initial state of the material has all of the grains randomly
oriented. Now, the average response of the polycrystalline material will clearly be
different if the grains are all oriented in a specific direction. For example, a material with all of the grains oriented so that the soft orientation is in the direction of
shearing will need a lower shear stress (than the randomly oriented polycrystalline
material) to sustain further plastic deformation. Because the shear strength of the
material decreases with grain orientation (this is called geometric softening), the
grains will tend to rotate so that the soft plastic direction rotates into the direction of
shearing. The model of Joshi and Ramesh demonstrates that a consequence of this
specific geometric softening behavior is shear localization, that is, the development
of the shear band. This is illustrated in Figure 7.12b, where the random orientation of the grains is shown as having changed into a specific orientation within a
localized region called the shear band.
Consider the one-dimensional shear problem (Figure 7.13) of an infinitely long
slab with finite thickness W subjected to a constant nominal shear strain rate γ˙a .
In the absence of inertia (i.e., for slow loading), the balance of linear momentum
reduces to an equilibrium equation, ∂∂Xτ = 0, where τ is the shear stress and X is the
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Nanomaterials
s
h
Fig. 7.12 (a) Schematic of a grain showing its soft and hard orientations (with respect to plastic
deformation, not elastic stiffness). (b) The grains are initially randomly oriented in the material, but
begin to orient themselves so that the soft direction is in the direction of shearing, and the process
of grain rotation into the soft orientation results in the localization of the deformation into a shear
band.
thickness direction. For simplicity, only isothermal problems are considered here
(note that this eliminates the consideration of adiabatic shear bands). The material
is assumed to be elastic/viscoplastic, so that the total strain rate can be decomposed
into an elastic strain rate and a viscoplastic strain rate: γ̇ = γ˙e + γ˙p . From this we
obtain
∂τ
τ̇ =
(7.12)
= µ γ˙e = µ (γ̇a − γ̇ p ),
∂t
where µ is the elastic shear modulus, γ˙e is the average
elastic strain rate, γ˙p is the
thickness-averaged plastic strain rate defined by W1 γ̇ p dX, and the total applied
strain rate is γ̇a = γ̇e + γ̇ p .
γ
W
Fig. 7.13 Schematic of simple shearing of an infinite slab, showing the terms used in examining
the shear localization process in simple shear (Joshi and Ramesh, 2008b).
7 Scale-Dominant Mechanisms in Nanomaterials
239
A viscoplastic constitutive law is assumed for the material, assuming powerlaw strain hardening and power-law rate hardening, and allowing for the geometric
softening that arises from grain rotation:
γ̇ p m
γp n
τ
= 1+
(1 − c̄φ ),
(7.13)
1+
τ0
γ̇0
γp
where φ is an internal variable defined (in the next paragraph) as the number fraction
of grains in a given region that have the soft orientation aligned to the shearing
direction, and γ p is the plastic strain. The terms γ̇0 , τ0 , γ0 , n, and m in Equation
(7.13) represent the characteristic strain rate, yield strength (of the polycrystalline
material), yield strain (corresponding to τ0 ), strain hardening index, and strain rate
sensitivity, respectively. The factor c̄ is defined as c̄ = 1 − ττs0 where τh0 and τs0 are
h0
the yield strengths of the grain in the hard and soft orientations (corresponding to a
single crystal).
By solving this equation for the strain rate, we obtain the evolution of plastic
strain corresponding to this constitutive law:
1
γ p −n m
τ
1
γ̇ p = γ̇0 {
} −1 ,
1+
τ0 (1 − c̄φ )
γ0
where is defined through f =
(7.14)
( f +| f |)
.
2
7.2.4.1 Definition of the Internal Variable
Consider a representative volume element (hereafter referred to as an RVE or a bin)
that contains a large number of grains Nb (b for bin). We define the internal variable as φ (t) = NNs , where Ns is the number of grains in the bin that have the soft
b
orientation aligned with the shearing direction, and the evolution of φ represents the
evolution of rotation of the grains. Thus, as more and more grains rotate into the soft
shearing orientation, φ increases and (through the (1 − c̄φ ) term) a softening mechanism develops. Joshi and Ramesh demonstrated that this softening mechanism leads
to localization, with a shear band that is essentially the region where φ ≈ 1.
7.2.4.2 Evolution of the Internal Variable
The fraction φ of soft-oriented grains evolves through contributions from two specific mechanisms:
φ̇ = φ̇1 + φ̇2 .
(7.15)
The first component φ̇1 in Equation (7.15) represents the contribution of overall
plasticity (Figure 7.14) to grain rotation, while φ̇2 represents the contribution due
to interaction of a grain with nearest neighboring grains (Figure 7.15d). φ ceases
to grow once it reaches unity (since no further re-orientation is possible), so that
φ̇ = 0 when φ = 1.
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Nanomaterials
Fig. 7.14 Schematic of the rotation of ensemble of nano-grains occupying region ℜ embedded in a
visco-plastic sea S subjected to shear. The background image shows the undeformed configuration.
Fig. 7.15 A hierarchical approach to homogenization of grain rotation due to interface traction.
(a) Material continuum. (b) Collection of bins in sample space. (c) Grains within a RVE. The
colored shading represents the average grain orientation in that bin. (d) Interaction at the grain
level. Grains with individual orientations are described by different colors.
7.2.4.3 Contribution of Overall Plastic Deformation
We focus on the contribution to rigid rotation of an ensemble of grains (ℜ) due to
overall plastic deformations in the viscoplastic sea (S) surrounding the ensemble
(Figure 7.14). Strain compatibility at the interface of ℜ and S gives a rotation rate
that is proportional to the rate of plastic deformation. Thus, the contribution of overall plastic deformation to the evolution of φ is given to first order by
7 Scale-Dominant Mechanisms in Nanomaterials
¯ γ̇ p
φ̇1 = ψ (φ , d)
241
(7.16)
where d¯ is the mean grain size within the ensemble. The function ψ indicates the
ease with which the ensemble ℜ can rotate rigidly in the sea, and is referred to as a
fabric factor (because it represents, in some sense, a measure of the local fabric or
microstructure). An ensemble residing in a region where surrounding grains are soft
(φ > 0) will rotate with relative ease as the soft surrounding medium deforms plastically. On the other hand, in a region where the grains are randomly oriented (φ ≈ 0),
rotation (in the direction consonant with macroscopic shear) is difficult because the
¯ = ψ0 ,
surrounding grains will not deform easily. For simplicity, let us set ψ (φ , d)
where ψ0 is of order 1–10.
In practice, ensembles with very small d¯ will tend to rotate to accommodate the
global plastic deformations, while those with large d¯ will tend to deform plastically
through crystallographic slip. The mean grain size in the ensemble is determined
from the grain size distribution, and in this way the grain size distribution may affect
φ̇1 .
7.2.4.4 Contribution of Inter-Granular Interaction
Grain rotation may also be caused by the local interactions of a single grain with
its neighboring grains of similar size. Grain rotation models based on interfacial
contact resistance have been successfully implemented in simulating granular flow
(Iwashita and Oda, 1998) including elastoplastic contact (Hu and Molinari, 2004).
This latter elastoplastic contact mechanism can be included in this model for grain
rotation in nanocrystalline materials through a hierarchical scheme. Using a threegrain model, the concept of interfacial interaction at the grain level is used to determine a driving force for rotation; subsequently, an evolution law for grain rotation
is prescribed in terms of the driving force.
In granular materials, rolling resistance causes neighboring grains to create interfacial traction. Consider the interaction between three adjacent grains as shown in
Figure 7.15 (for simplicity, we consider a 2D case).
Let the current rotation angles of the three grains (a, b and c) in Figure 7.15d
be denoted as θa , θb and θc , respectively. Focusing on the central grain (b) and
assuming that θa > θb > θc , the interfacial tractions on grain b are then
F|ab = kt |ab δt |ab
(7.17)
due to the interaction with grain a, and F|bc due to the interaction with grain c:
F|bc = kt |bc δt |bc ,
(7.18)
where kt is the effective contact stiffness and the tangential displacements are δt (see
Equation 7.11). These tangential displacements are of the form
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Nanomaterials
δt |ab = δt |a − δt |b + Ra (θa ) − Rb (θb ),
(7.19)
with Ra and Rb denoting the radii of grains a and b. The grain boundary sliding
mechanism is represented by the term δt |a − δt |b . Since this section is only considering grain rotation, we set the grain boundary sliding to zero, so that δt |a − δt |b = 0.
Further, let us assume that all of the grains are the same size, so that Ra = Rb = d2 ,
where d is the grain diameter. Then Equation (7.19) gives us
δt |ab =
d
(θa − θb ),
2
(7.20)
andδt |bc =
d
(θb − θc ).
2
(7.21)
The net interface traction (per unit length) on grain b is
[Δ F]b = Fab − Fbc ,
(7.22)
and using Equations (7.20) it follows that
kt d
[θa − 2θb + θc ].
2
The corresponding net torque on grain b is then given by
[Δ F]b =
(7.23)
d
kt d 2
[Δ F]b =
[θa − 2θb + θc ].
(7.24)
2
4
But what is the tangential contact stiffness kt ? This can be obtained from elasticplastic contact micromechanics (Stronge, 2000). The contact stiffness of grains undergoing elasto-plastic deformation is given in terms of the radius rc of the contact
area by
8rc
kt |ab =
,
(7.25)
Cab
Tb =
where Cab is called the contact compliance for the two grains in the contact and is
given by (Stronge, 2000)
Cab =
1 − νa 1 − νb
2(1 − ν )
+
=
,
µa
µb
µ
(7.26)
using the shear modulus (µ ) and Poisson’s ratio (ν ), and assuming that these moduli
are identical for the two grains.
In general, the contact area will evolve as grains rotate (Conrad and Narayan,
2000; Van Swygenhoven and Derlet, 2001), and will be higher for plastic contacts
than for elastic contacts. Let us, however, assume a fully plastic contact for a polycrystalline material and set the radius of the contact equal to the grain radius, so
7 Scale-Dominant Mechanisms in Nanomaterials
243
that rc = d2 . Using Equations (7.26) in (7.25) and then using the result for kt |ab in
Equation (7.24), we obtain the driving force for grain rotation as the torque:
Tb ==
µ d3
[θa − 2θb + θc ].
2(1 − ν )
(7.27)
How will the grain respond to this driving force? The response will clearly depend on the surrounding grains and their grain boundaries as well. To simplify the
problem, we postulate a kinetic relation between the evolution of grain rotation and
the driving force, with the rate of rotation given by
θ̇b = (mr T )b ,
(7.28)
where mr is a mobility coefficient, representing the resistance provided by the surrounding grains to the rotation of grain b. Equation (7.28) is similar to those adopted
in describing grain rotation coalescence (Moldovan et al., 2001). In the latter work
the driving force is due to anisotropic grain boundary energies and the plastic deformation is accommodated by grain boundary and lattice diffusion mechanisms,
which are believed to be dominant at very small grain sizes (d < 10 nm) and at
elevated temperatures. Similarly, in defining the mobility, we account for the viscoplastic resistance offered by the region surrounding the rotating grain.
Fig. 7.16 Enlarged view of ℜ (Figure 7.14) showing intergranular interaction in the region. Rotation of the central grain is accomodated by rotation of the surrounding grains over a length L.
The key concept with respect to grain rotation is the following. Consider the rotation of a specific grain such as the central grain shown in Figure 7.16. If you were to
be an observer sitting on another grain that is sufficiently far away from the rotating
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grain, you will not even be aware that the central grain has rotated. However, grains
neighboring the central grain will have to rotate to some degree to accommodate
the rotation of the central grain. That is, the rotation of any single grain must be
accommodated by the surrounding grains (and the grain boundaries), but there is
a finite length (L) over which this accommodation occurs (Figure 7.16) and grains
sufficiently far away from the rotating grain are unaffected by the rotation. This finite length (L) will depend on the interactions between the grains, and in particular
on interactions at the grain boundaries (although the degree of plastic deformation
of the grains will also be involved). The length scale for rotational accommodation will be smaller for elastoplastic grains (as in polycrystalline metals) than for
purely elastic grains (a limiting example of which would be sand). This rotational
accommodation mechanism must be present in all granular solids that contain some
dissipative mechanisms (which may be localized to grain boundaries).
Using this rotational accommodation concept, we can (from a dimensional analysis viewpoint) define the mobility due to the viscoplastic resistance in terms of the
accommodation length scale as
(mr )b =
1 1
(
),
L3 ηeff
(7.29)
where ηeff is the effective viscosity of the accommodating region. This can be estimated in some cases using molecular dynamics simulations (Van Swygenhoven and
Caro, 1997). From a mechanistic viewpoint, the effective viscosity itself can be derived (through addition of the corresponding strain rates) in terms of the intragranular and grain boundary mechanisms using the following argument. The effective
strain rate γ̇eff in the accommodating region is given by
γ̇eff = γ̇ p + γ̇gb ,
(7.30)
where γ̇ p is the strain rates due to plastic deformation mechanisms and γ̇gb is the
strain rate due to deformation at the grain boundaries. Assuming that the stress is
uniform across this region, and defining the viscosities in the classical way, we have
from which it follows that
τ
τ
τ
=
+
,
ηeff
η p ηgb
(7.31)
1
1
1
=
+
.
ηeff
η p ηgb
(7.32)
Thus the effective viscosity of the accommodating region has contributions from
dislocation plasticity (η p ) and grain boundary mechanisms (ηgb ). For example, the
effective viscosity corresponding to grain boundary diffusion-type mechanisms can
be computed using
kB T d 3
ηgb =
,
(7.33)
64δgb Ω Dgb
7 Scale-Dominant Mechanisms in Nanomaterials
245
where kB is the Boltzmann constant, T is the absolute temperature, δgb is the grain
boundary thickness, Ω is the atomic volume, and Dgb is the grain boundary diffusion
coefficient (Murayama et al., 2002). The contribution to the effective viscosity (η p )
from dislocation plasticity is represented by the viscoplastic resistance provided by
the grains themselves, and a lower bound for this is
ηp =
τs0
.
γ̇0
(7.34)
What is the magnitude of L in Equation (7.29)? As the grain size increases, the
likelihood of global plastic deformation dominates the likelihood of rotation, and
eventually this grain rotation mechanism is no longer a significant contributor to the
deformation and slip becomes the dominant mode of deformation. For most metals,
where dislocation slip is relatively easy, grain rotation is unlikely to be an important
contributor for grain sizes larger than about 1 µm, and may not be a significant contributor even in the ultrafine grain range. In ceramics, however, grain rotation may
be significant even for much larger average grain sizes, because there is so little
plasticity in ceramics (on the other hand, however, grain boundary mechanisms for
inelastic deformation in ceramics are relatively few). In general, one can describe L
in terms of the number j of other grains involved (in a given linear direction in the
plane) in accommodating the rotation of a given grain, so that L = jd. This length
scale is a natural length-scale for a bin. Since very large grains will deform by slip
rather than rotate, we assume that L is a fixed length scale for a given material (for
example, this might be 1 µm for metals). The number of grains involved in accommodating the rotation is j = Ld . In a nano-grained material, many grains ( j > 1) are
involved in the accommodation distance L. A quantitative measure of L remains to
be established, but L can be viewed in a manner analogous to the microstructural
length scales derived in strain gradient plasticity theories (Gao et al., 1999).
7.2.4.5 Homogenized Representation of Grain Rotation
An expression for the second contribution (φ̇2 ) to the evolution of grain rotation can
be derived by homogenizing the evolution of θ over the RVE (Figure7.15c). The
average grain orientation over the RVE is
θ̄ =
1
|θk |.
Nb ∑
(7.35)
The number of favorably oriented grains in a bin is counted by performing a
weighted averaging with a penalty function ζk = δ (|θk | − θs ), where θs is the favorable orientation (aligned with the band orientation). Note this uses a delta function
δ (|θk | − θs ), which has value unity when the argument (|θk | − θs ) is zero, that is
|θk | = θs , and is zero for all other θk . Further, note that as θ̄ → θs , φ → 1. Thus for
the RVE we have
1
φ2 =
ζk |θk |.
(7.36)
Nb ∑
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Nanomaterials
The average rate of change of φ2 is then (using the kinetic relation, Equation 7.28)
φ̇2 =
1
1
ζk θ˙k =
ζk (mr T )k .
Nb ∑
Nb ∑
Using Equation (7.27), we obtain
1
kt d
φ̇2 =
ζk (θa − 2θb + θc ) .
4 jηe j2 d 2 Nb ∑
(7.37)
(7.38)
Examining Equation (7.38), we identify the terms in the square brackets as ap2
proximating the second derivative ∂∂ Xφ2 in the continuum limit, so that we develop
the rate equation
∂ 2φ
kt d ∂ 2 φ
φ̇2 =
=
D
,
(7.39)
r
4 jηe ∂ X 2
∂ X2
where we have identified a new rotational diffusion coefficient Dr as
Dr =
kt d
1 µ d2
=
(
).
4 jηe
jηe 2 − ν
(7.40)
Equation (7.39) may be viewed as characterizing a rotational diffusion mechanism that tends to diffuse variations in φ through inter-granular interactions.
Putting together Equations (7.15), (7.16), and (7.40), the net evolution of the
internal variable φ is
∂ 2φ
φ̇ = ψ γ̇ p + Dr 2 .
(7.41)
∂X
This evolution equation can be integrated together with the momentum balance
and constitutive laws to solve the problem of a nanocrystalline material in which
grain rotation is a possible mechanism. We now see that the primary consequences
of the grain rotation mechanism are the existence of a diffusive term and the corresponding length scale.
The governing equations for the problem of simple shear of a nanocrystalline
material incorporating grain rotation are summarized below:
∂τ
=0
∂X
∂τ
= µ γ̇a − γ̇¯p
∂t
1
γ p −n m
τ
1
γ̇ p = γ̇0
−1
1+
τ0 (1 − c̄φ )
γ0
φ̇ = ψ γ̇ p + Dr
∂ 2φ
.
∂ X2
(7.42)
(7.43)
(7.44)
(7.45)
Boundary conditions and initial conditions must be added to solve these equations. Joshi and Ramesh considered the simple shear of a slab of nanostructured
7 Scale-Dominant Mechanisms in Nanomaterials
247
iron, with initial conditions that are zero everywhere except for the initially uniform
shear strain rate. In order to account for typical material variability, an initial defect
2
X
− 0.5) ], where φ0 is
distribution φi was assumed in the form φi (X) = φ0 exp[−A( W
the largest initial value of φ in the distribution and A is a dimensionless parameter
which is like a wave-number. Their results on nanostructured iron are of interest
because of the experimental observations of shear bands in this material (Jia et al.,
2003).
The values for the various parameters used by Joshi and Ramesh are listed in
Table 7.3 (corresponding to polycrystalline Fe). Given the heavy plastic work during
severe plastic deformation processes, the parameter c̄ was obtained at large strains
(∼ 1) from the results of Spitzig and Keh (1970) on iron single crystals.
Table 7.3 Basic parameters for grain rotation model in polycrystalline iron, as developed by Joshi
and Ramesh.
Parameter
Value
Average grain size d, in nm
300
Shear modulus µ , in GPa
76
Poisson’s ratio ν
0.3
Normalizing strain rate γ̇0 , per second
5 × 10−4
Normalizing stress τ0 , MPa
900
Perturbation “wave number”
5 × 104
Strain hardening index n
0.01
Rate hardening index m
0.005
Anisotropy factor c̄
0.07
Plastic contribution coefficient ψ
10
Maximum initial value of φ , φ0
0.005
Far-field initial value of φ , φ∞
0
Rotational accommodation length scale L, µm
1.0
For a material with this initial distribution in φ , Figure 7.17 shows the evolution
of φ as a function of nominal strain near the band. The internal variable (representing the degree of rotation of the grains into the soft orientation) increases rapidly
at the center of the initially perturbed region once the strain reaches 0.02, and stops
evolving (φ̇ = 0) once φ reaches the maximum value (φ = 1) in the center. Physically, this represents a stage where following the localization the grain reorientation
process saturates, so that no further reorientation occurs in that region. At the onset of severe localization, the plastic shear strain rate is very high (≈ 103 γ̇a ). With
increasing nominal strain, γ̇ p drops in the band center (Figure 7.18) in this strain
hardening case because φ is constant there but the plastic strain continues to grow.
Following a fully developed band (defined by a finite thickness over which φ = 1),
γ̇ p remains high at the edges of the band where φ has not saturated. Subsequent
band growth is driven by rotational diffusion and strain hardening. The plastic shear
strain in the band center is fully resolved by the simulation (Figure 7.19).
In Figure 7.20, the overall stress-strain response for a defect-free material is presented (curve A) and compared with the response obtained from a sample with
the perturbation (curve B, solid line). With the perturbation, the sample exhibits
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Nanomaterials
Fig. 7.17 Evolution of grain orientation fraction (φ ) around the band center. Note the rapid early
growth, the spatial localization, and the saturation of φ . Also note the spreading of the band, i.e.,
the increasing band thickness - this is also observed in experiments.
Fig. 7.18 Evolution of plastic shear strain rate (γ̇ p ) around the band center. The greatest activity in
this variable is at the band boundaries, where the grains are reorienting into the soft orientation for
shear.
Fig. 7.19 Evolution of the plastic shear strain γ p around the band center. After localization, γ p
evolves slowly indicating that the plastic flow inside the band develops at the rate of strain hardening at higher nominal strains.
7 Scale-Dominant Mechanisms in Nanomaterials
249
a brief period of strain hardening following yield. The strain hardening is followed
by a rapid drop in the stress corresponding to severe localization, with the magnitude
of the drop determined by the material hardening and the anisotropy parameters. The
nominal strain corresponding to the stress collapse is termed the critical strain γcr ,
and should be related to (but less than) experimental measures of the critical strain
for localization. Following the stress drop, a softening response is observed. Such
a macroscopic softening response of a strain hardening material indicates the dominance of geometric softening over material hardening (Yang and Bacroix, 1996).
The softening regime exhibits serrated behavior in the presence of strain hardening
(this can be seen at a higher magnification on the stress-strain curve) due to the
competition between the hardening and softening variables.
Fig. 7.20 Overall stress-strain response for a defect-free sample (curve A) and a sample with
an initial defect in φ (curve B). Curve B’ indicates the development of the shear band thickness
corresponding to curve B.
We define a fully developed shear band as a region where φ reaches the maximum
value and use this definition to also determine the thickness of the shear band. After
localization, the material hardening together with rotational diffusion controls the
broadening of the shear band. The development of the thickness of the shear band
is also shown in Figure 7.20 as a function of overall strain.
7.2.4.6 Effect of Grain Size
The grain size explicitly appears in two quantities in the governing equations: the
polycrystalline yield strength (τ0 ) and the rotational diffusion coefficient Dr . Consider three grain sizes (100 nm, 300 nm, and 1 µm), with n = 0.01 and m = 0.005
held constant for all the grain sizes under consideration. Using the Hall-Petch
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Nanomaterials
σ
relation for Mises yield in iron, the shear strengths τ0 = √y3 are 1300 MPa, 900 MPa
(300 nm), and 290 MPa (1 µm).
Figure 7.21 shows the band thickness evolution for different grain sizes. Whether
a shear band will develop is strongly grain size dependent. If a shear band does
develop, the band thickness is larger for larger grain sizes (at equivalent nominal
strains), because the rotational diffusion term depends on the grain size.
Fig. 7.21 Evolution of shear band thickness for different grain sizes, assuming grain rotation mechanism. The material hardening parameters are held constant for all the grain sizes. The applied
strain rate is 10−3 s−1 .
7.2.4.7 Relative Contributions of Grain Boundary and Grain Mechanisms
to ηe
With this definition of the effective viscosity ηe , a transition from dislocation dominated mobility to grain boundary dominated mobility will occur (comparing the two
terms in Equation 7.32) at a transition grain size given by
dtr =
64δ Ω Dgb τs0
kB T γ̇0
1
3
.
(7.46)
This transition size depends on grain boundary diffusion and grain boundary
thickness variables, and is about 9 nm in the specific case of iron. The grain boundary viscosity ηgb is likely to be dominant for smaller grain sizes (although it
will certainly continue to play a role even at much larger grain sizes). Note that
7 Scale-Dominant Mechanisms in Nanomaterials
251
below the transition grain size we will then have the rotational diffusion coefficient increase as the grain size decreases, because in that domain ηe ∼ ηgb , and
then Dr ∝ d1 .
7.2.4.8 Comparison with Experiments
Joshi and Ramesh considered two quantitative comparisons between the model and
specific experiments: (a) bcc-Fe of Jia et al. (2003), and (b) fcc-Al of Sun et al.
(2006). Both experiments under consideration involved uniaxial loading whereas
the model was developed for the case of simple shear and considers only a single
shear band. However, the model was able to distinguish between the two material
behaviors.
Experiments (Jia et al., 2003; Wei et al., 2002) on nanostructured and ultrafine grain iron show that shear bands occur immediately following yield. Joshi
and Ramesh were able to reproduce this behavior in the large using physically
reasonable parameters. However, in real bcc structures the hardening parameters
may be modified by the grain size due to the variety of processing routes used
for generating bulk nanostructured materials. Consolidated Fe (Jia et al., 2003) exhibits low but finite strain hardening (n ≈ 0.01 − 0.04) in the nanostructured regime
whereas at larger grain sizes n is high (≈ 0.26) (Malow et al., 1998). Thus the strain
hardening increases with grain size in NS-Fe, and the model should account for
such variations (similar changes occur in the rate sensitivity, which decreases with
grain size).
Using the same perturbation parameters as in the bcc-Fe case, the model predictions were compared with the experimental observations of Sun et al. (2006)
on equal channel angular extruded commercially pure aluminum (ECAE-CP-Al).
Unlike in the case of the iron, the model does not predict shear banding for these
three grain sizes in CP-Al. The quasi-static compression experiments (Sun et al.,
2006) also showed no shear bands for 350 and 470 nm cases, although apparently
anomalous shear bands were observed for the 590 nm case. Our model does not
predict shear band development in such rate-sensitive materials as nanostructured
and UFG-fcc metals. Indeed, shear bands have not been reported in NS or UFG
pure fcc metals in general, although very “diffuse” bands have sometimes been
observed in tension. The rate sensitivity effects are considered further by Joshi
and Ramesh (2008a) when applying the grain rotation model to different crystal
structures.
7.2.5 Stability Maps Based on Grain Rotation
Some additional understanding can be obtained by considering a scaled version of
the governing equations (Equation 7.42) for the simple shear problem including
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grain rotation. Let us set up the same governing equations in terms of normalized
variables X̂ for position, τ̂ for shear stress, and tˆ for time, defining these as follows:
X
L
τm
τ̂ =
τs0 c̄
tˆ = γ̇0t
X̂ =
(7.47)
(7.48)
(7.49)
where the scaling uses the parameters defined earlier. Using these scaled variables,
the equilibrium condition is now
∂ τ̂
= 0.
∂ X̂
(7.50)
Similarly, the elastic relation from Equation (7.42) (the second equation in that
set) becomes
1 ˆ
(7.51)
τ̂ =
γ̇a − γ̇ˆp ,
M
where the strength index M has been defined as
M=
τs0 c̄
,
µm
(7.52)
and will be utilized in constructing a stability map. The normalized evolution equation for the internal variable is
Dr ∂ 2 φ̂
φ̂ = ψ γ̇ˆp +
,
γ̇0 L2 ∂ X̂ 2
(7.53)
where γ̇ˆp is the scaled visco-plastic shear strain rate.
The solution of this system of equations will tell us how the stress, strain, strain
rate, and orientation number fraction will evolve for a given material subjected to
simple shear. However, we can learn a great deal about the problem by studying the
stability of the equations – that is, by finding out under what conditions a perturbation to one of the variables will grow (corresponding to instability) rather than decay
(corresponding to stability). The basic approach is as follows: we assume a perturbation in one or more of the variables (e.g., the plastic strain), insert the perturbed
solution into the system of equations, and ask whether the perturbation will grow. If
it will grow, the system is unstable in that it cannot survive a perturbation; if it will
not grow, the system is stable in that it suppresses perturbations and tends towards
homogeneous solutions. In what follows, we analyze the stability of the governing
equations using linear perturbation analysis (a limited kind of perturbation analysis,
which effectively restricts our attention to early times, i.e., the onset of instability
rather than the later stages of unstable growth). With τ̂ required to be homogeneous
(which is the equilibrium condition), the stability of deformation is investigated by
7 Scale-Dominant Mechanisms in Nanomaterials
253
perturbing the plastic strain γ p and the internal variable φ from their homogeneous
solutions, so that
γ¯p = γ p h + δ γ p (t˜) sin kX̂,
(7.54)
and
φ̄ = φ h + δ φ (t˜) sin kX̂,
(7.55)
where, for example, φ h is the homogeneous solution, δ φ (t˜) is the normalized temporal component of the perturbation to the solution (assumed small compared to φ h ),
and k is the non-dimensional wave-number of the perturbation (we are essentially
perturbing the amplitudes of these variables in space with a perturbation wavelength
proportional to the reciprocal of k). Substituting the perturbed solutions in the normalized evolution equation (Equation 7.53) and retaining only the first order terms
(linearized perturbation analysis) we find that the perturbations will grow only if
∂ γ̂ p
k2 Dr
>
,
∂φ
ψ γ̇0 L2
(7.56)
where the rotational diffusion coefficient Dr and the rotational accommodation
length scale L are the quantities defined previously. Equation (7.56) represents the
condition for instability. Letting φ h = 0, the constitutive law gives
∂ γ̂ p
c̄
≈ γ̂ ph .
∂φ
m
(7.57)
Thus the condition for instability is
c̄ h
k2 Dr
γ̂ p >
,
m
ψ γ̇0 L2
(7.58)
from which it follows that the critical wavelength of a perturbation that will lead to
a runaway instability is given by
λcritical =
γ̇0 γ̂ ph ψ c̄
Dr m
12 −1
.
(7.59)
This instability condition can be rewritten using the strength index defined earlier
(Equation 7.52), giving
M
d2 ηp
λcritical =
(2 − ν )ψ γ̂ ph j ηe f f
21
.
(7.60)
Equation (7.60) should be interpreted as follows. It describes the critical wavelength of a perturbation of the system that will grow. That is, if the internal variable
φ has a perturbation (a local variation) over a region with a size greater than λcritical
(amounting to a wavelength greater than the critical wavelength), then that perturbation will grow, defining an instability that will become a shear band as shown in
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Nanomaterials
the last section. Perturbations smaller than the critical wavelength will decay and
will thus not be sufficient to initiate a shear band. The grain size dependence of
λcritical appears through Dr in Equation (7.59). Note that if the material parameters
are such that λcritical ≈ d, the material is inherently unstable in that this perturbation
necessarily exists in the material, given the grain size. The condition λcritical < d is
mathematically possible but is not physically meaningful unless grain size distributions are included.
For most materials the fabric factor ψ will range between 1 and 10. Setting
ψ = 10 and γ̂ ph = 1, with γ̇0 = 5 × 10−4 s−1 , we can examine the dependence of
λcritical on grain size. Figure 7.22 shows the relationship between λcritical and grain
size for three materials (Joshi and Ramesh, 2008c), bcc-Fe, fcc-Cu, and fcc-Al. For
all these materials, there are in general three regimes in this log-log plot as the grain
size is decreased. In the first regime (d ≥ 100 nm), the critical wavelength decreases
with decreasing grain size, the maximum number of participating grains is determined by fixed L, and η p is the rate limiting process. This rate limiting process
continues to dominate the second regime (100 nm ≥ d ≥ 10 nm), which is governed
by the limited number of participating grains, and λcritical decreases with a different slope. In the third regime (d ≤ 10 nm), the critical wavelength λcritical increases
again as the grain size decreases. This is because the dominant rate limiting process
changes from being bulk response to grain boundary driven. The region below the
line λcritical = d in Figure 7.22 is the inherently unstable domain. As an example, for
300 nm grain size Fe, λcritical is approximately equal to the grain size. A similar calculation for 300 nm grain size Cu yields λcritical ≈ 440 nm, which is somewhat higher
than the grain size. Consequently, the Fe is inherently susceptible to instability over
Fig. 7.22 The critical wavelength for instability as a function of grain size for three different metals
(note that this is a log-log plot), with the critical wavelength computed using Equation (7.60). This
figure has been obtained assuming that j = 10 below d = 100 nm, while for 100 nm < d ≤ 1 µm,
we have L = 1 µm and j = Ld . The dashed straight line represents the condition that λcritical = d,
which we call the inherent instability line.
7 Scale-Dominant Mechanisms in Nanomaterials
255
a larger range of grain sizes than the Cu. For some materials the entire curve will
be above the inherent instability line (e.g., the Al-curve in Figure 7.22). The grain
size range over which a material is inherently unstable (if at all) is intimately tied
to its elastic properties, plastic anisotropy, and the strain rate sensitivity. Equation
(7.60) shows that materials with high crystallographic plastic anisotropy and low
rate sensitivity are generally more susceptible to instability.
Fig. 7.23 Stability map, showing the domains of inherent instability in materials as a consequence
of the rotational accommodation mechanism. The map is constructed in terms of the strength index
and grain size, so that every material of a given grain size represents one point on the map, and
a horizontal line represents all grain sizes of a given material. Reprinted figure with permission
from S.P. Joshi and K.T. Ramesh, Physical Review Letters, Stability map for nanocrystalline and
amorphous materials, 101(2), 025501. Copyright 2008 by the American Physical Society.
This stability analysis allows one to probe the likelihood of instability in the
materials and microstructure space. Figure 7.23 presents a stability map where the
ordinate is the previously defined strength index M that enables one to compare different materials while the abscissa is the grain size. The curve is the locus of points
satisfying the inherent instability criterion (λcritical = d, which was the straight line
in Figure 7.22). If the point representing a given material and grain size lies in the
saddle (the blue region), then the material is inherently susceptible to the rotational
mechanism of shear instability. In general, materials with low strength indices show
susceptibility over a smaller range of grain sizes compared to those with higher
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Nanomaterials
strength indices. Thus, the iron is inherently susceptible to rotational instability over
a larger grain size range than the copper, while aluminum does not exhibit any inherent instability. This prediction is qualitatively consistent with the available data
on nc-Fe (Jia et al., 2003), nc-Cu (Cheng et al., 2005), and nc-Al (Sun et al., 2006).
Note that a point in this figure (a combination of a given material and grain size)
lying outside the inherent instability region (the orange region) does not unambiguously guarantee a stable response, as instability can still occur at larger perturbation
wavelengths, but such a material would not be inherently unstable. Such maps provide useful guidelines to determine the probability of a material being unstable with
respect to this softening mechanism for a range of defect wavelengths.
7.3 Multiaxial Stresses and Constraint Effects
It appears that nanomaterials have multiple mechanisms that are sensitive to multiaxial stresses and to constraints (some of which are manifested as pressure effects).
Unfortunately these are difficult to characterize experimentally, and so they are discussed in the next chapter (focusing on some modeling approaches).
7.4 Closing
There are a variety of deformation mechanisms in nanomaterials that arise from
length-scale interactions, with specific mechanisms developing in specific structures
and for specific applications. These mechanisms sometimes result in remarkable
properties, and a full understanding of the mechanisms may suggest applications
that would not be conceived of otherwise. Some of these mechanisms are only discoverable from an experimental viewpoint, unless very large scale simulations are
performed to capture the richness of possible phenomena.
7.5 Suggestions for Further Reading
1. Argon, A. (2008). Strengthening Mechanisms in Crystal Plasticity. Oxford
University Press, Oxford.
2. Van Swygenhoven, H. and J. R. Weertman (2006). Deformation in nanocrystalline metals. Materials Today 9(5): 24–31.
3. Meyers, MA, A. Mishra, DJ. Benson, (2006). Mechanical properties of nanocrystalline materials. Progress in Materials Science 51: 427.
4. Kocks, U.F. A.S. Argon, M.F. Ashby, (1975). Thermodynamics and kinetics of
slip. Progress in Materials Science 19: 1.
7 Scale-Dominant Mechanisms in Nanomaterials
257
7.6 Problems and Directions for Research
1. Examine grain size effects in hard granular materials in terms of macroscopic
inelastic deformation and failure processes. This provides a sense of the behaviors that are possible when grain boundary and contact mechanisms are primarily
responsible for deformation.
2. The development of twins is often related to the ability to emit and move partial
dislocations. What does this say about the maximum speed of twin tips?
3. Let us view a young deciduous tree as a buckling problem (a long thin column
with a weight – corresponding to the foliage – concentrated at the top). Assume
the tree trunk is circular, with a diameter of 3 cm, and assume the tree is about
3 m tall. Given that the Young’s modulus of the wood is about 10 GPa, compute
the maximum allowable weight of the foliage before buckling will occur. Can
you climb to the top of this tree (in the winter) without inducing buckling?
4. What is the critical compressive buckling load for a “tree” made of a singlewalled carbon nanotube? Obtain the cross-section geometry from the literature,
but assume a length of 2 µm. The Young’s modulus of a thin-walled carbon nanotube is about 1 TPa.
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...In that Empire, the craft of Cartography attained such Perfection that
the Map of a Single province covered the space of an entire City, and
the Map of the Empire itself an entire Province. In the course of Time,
these Extensive maps were found somehow wanting, and so the College
of Cartographers evolved a Map of the Empire that was of the same Scale
as the Empire and that coincided with it point for point.
Jorge Luis Borges, On Exactitude in Science
8
Modeling Nanomaterials
8.1 Modeling and Length Scales
The length scales that are of interest to the mechanics of materials were discussed
in Chapter 1. The focus there was on the devices and structures that typified the
various length scales, and the intent was to define the range appropriate for nanomaterials. In subsequent chapters, typical mechanics approaches to characterizing and
describing nanomaterials were considered. As in any field of science, however, we
cannot truly understand nanomaterials unless we can develop mathematical models
for them. This chapter focuses on the modeling approaches that are appropriate for
nanomaterials.
It is worthwhile to consider explicitly the modeling process, in order to understand the various approximations being made. This process is represented schematically in Figure 8.1. The process is as follows:
• One starts with a physical problem of interest – let us say this is the problem of
nanoindentation of a diamond Berkovich tip into a single crystal of copper.
• Immediately, the necessity for some form of idealization presents itself: for example, we do not know what the friction is between the nanoindenter tip and the
sample, so we will typically idealize the problem as a zero friction problem at
the interface. This is a physical idealization made for modeling purposes. A set
of such physical idealizations results in an idealized problem that we still think
is of interest.
• Next, we replace the physical problem with a mathematical problem: we choose
the set of equations, with boundary and initial conditions, that we think best represents the idealized physical problem. Thus we may choose to represent the
problem through Newton’s equations for the motion of the individual atoms,
with the atomic interactions specified through an interatomic potential (e.g.,
the Lennard-Jones potential). This translation into a mathematical problem represents another set of assumptions: we assume the potential is reasonable for
copper (the Lennard-Jones potential is not), and that we can ignore quantumlevel effects.
• Now that we have a mathematical problem that in some sense represents our
physical problem, we choose a numerical approach to solving the mathematical
K.T. Ramesh, Nanomaterials, DOI 10.1007/978-0-387-09783-1 8,
c Springer Science+Business Media, LLC 2009
261
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problem: by choosing, for instance, a specific code with all of the specific algorithms incorporated in the code. There are different levels of accuracy with which
specific numerical schemes will solve the same set of mathematical equations, so
again we have a set of now numerical approximations.
• We then implement the numerical scheme and consider the solutions.
Physical Problem
Idealized Physical Problem
Mathematical Problem
Numerical Problem
Numerical Solution
Examine (e.g. visualize) Solutions
Fig. 8.1 Typical sequence of steps involved in modeling a mechanics of nanomaterials problem.
Note the many layers of approximations involved, pointing out the danger of taking the results of
simulations at face value.
It is important to remember when we look at the solutions that we have made
(a) physical, (b) mathematical, and (c) numerical approximations, and we should
always ask whether we have ended up solving the problem that we set out to solve.
This is sometimes stated as the verification and validation problem for complex
codes: in the words of Rebecca Brannon, are we solving the right equations, and are
we solving the equations right? Eventually, the simulation results must be compared
to experiments or to the results of entirely independent simulations.
Consider again the range of length scales presented in Chapter 1. Some of the
modeling approaches that can be used to describe phenomena over this range of
length scales are presented in Figure 8.2, together with the corresponding suite of
materials (rather than mechanical) characterization techniques. As an example, we
discuss the use of this figure to understand the plastic deformation of a material.
At very fine scales below those shown in Figure 8.2, the quantum mechanics
problem must be solved, leading to calculations that are variously called ab initio
or first principles computations (such calculations account for electronic structure).
Slightly larger scale calculations (Figure 8.2) account for every atom (but not the
electrons) by assuming an interatomic potential between the atoms and then solving
the equations of motion for every atom. Such computations are called molecular dynamics calculations, and are generally limited by length scale and time scale (a mil-
263
Discrete Dislocation
Dynamics
Continuum Modeling
Modeling
Molecular
Dynamics
10–6
10–8
10–10 m
AFM
Observations
Enriched CM
Finite Element Methods
10–2
10–4
SEM
TEM
nanomaterials
Optical Microscopy
100
Radar
10+2
8 Modeling Nanomaterials
Fig. 8.2 The typical modeling approaches of interest to the mechanics of nanomaterials, and the
approximate length scales over which each approach is reasonable. Note the significant overlap
in length scales for several of the modeling approaches, leading to the possibility of consistency
checks and true multiscale modeling. An example of the observations that can be made at each
length scale is also presented, from a materials characterization perspective.
lion atom computation, with 100 atoms on the side of a cube, would be examining
a physical cube that is about 40 nm on a side). Although computations of this size
scale can address a number of important issues, they are incapable of accounting for
a large number of dislocations.
Discrete dislocation dynamics simulations are larger scale computations that no
longer have atomic resolution but that can track many thousands of dislocations.
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Nanomaterials
Such computations are able to capture the development of some dislocation substructures, and to interrogate possible sources and sinks of dislocations. At larger
scales still, continuum modeling computations such as crystal plasticity calculations
do not attempt to resolve individual dislocations but do resolve individual crystals.
Such calculations smear the dislocations into the continuous medium representing
each crystal, and replace the dislocation evolution with hardening laws on given slip
systems.
At even larger scales, the individual crystals are no longer resolved, and instead
the computations examine the deformations of a homogeneous continuum that has
some prescribed plastic behavior (such as those discussed in Chapter 2). This is the
level of classical continuum mechanics, which contains no internal length scales.
Some models incorporate ad hoc microstructural length scales into structured continuum calculations, and we refer to these as enriched continuum models (for example, the strain gradient models in vogue in 2008).
A computational approach derived from classical continuum mechanics that has
been used with some success over a very wide range of length scales is the finite
element method, particularly when coupled to computational techniques that are
better able to resolve very fine scale physics. We will not discuss this numerical
technique in any detail: there are many excellent books on the finite element method
and its applications in mechanics.
An interesting observation from Figure 8.2 is the overlap between the scales
at which each modeling approach is effective. These overlapping regions can be
viewed in two ways. First, these are regions where information on the behavior of
the material can be obtained using two different approaches, and such situations
often greatly enhance our understanding. Second, these are regions in which the
ability to resolve behavior at finer scales with one method while computing over
larger scales with another method can lead to multiscale approaches that are able to
capture phenomena that would otherwise be missed.
The range of typical interest in nanomaterials is shown in the big vertical box
in Figure 8.2. Larger scales may be of interest for applications of nanomaterials,
but the box describes the range where nanomaterials phenomena are determined,
i.e., scale-dominant behaviors are typically observed. The microstructural characterization techniques associated with this range have been discussed in Chapter 3.
All of the modeling approaches in the figure are useful for modeling nanomaterials, although it is clear that part of this size range is accessible only by molecular
dynamics. The range of length scales where a number of modeling approaches overlap demonstrates that there is significant potential for understanding nanomaterials
using multiscale approaches.
Another way to think about these length scales is to think about them in terms
of the features of the material that are important at each length scale. This is illustrated in Figure 8.3, which shows the microscopic features typically associated with
each length scale in metallic systems. At very small length scales, the behavior is
dominated by atomic motions and so the atom is the critical feature. As the length
scales increase, the arrangements of the atoms become important, and so we begin
to see the effects of atomic clusters, point defects such as vacancies, and the unusual
arrangement of atoms around dislocation cores.
8 Modeling Nanomaterials
Features
Heterogeneities: pores, inclusions, particle
Clusters, Point
Stacking Faults
Defects,
Grain
ai boundaries,, grain size,
e texture
Dislocation
Dislocation
substructure:
cells,
tangles,
cores
Boundary conditions, component geometry,
walls
A
constraints, residual stresses
Atoms
Dislocations
Twins
10 –10 m
1
10–8
10–6
10–4
10–2
100
10+2
Fig. 8.3 The length scales associated with various microstructural features in metallic materials. For most crystalline materials, the behavior at larger length
scales is the convolution of the collective behavior of features at smaller length scales.
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Nanomaterials
At slightly greater length scales, dislocations themselves become the features
that dominate the mechanical behavior, and dissociated dislocations lead to stacking
faults. We then begin to see the influence of dislocation substructures, that is, the
collective behavior of large numbers of dislocations: this includes the development
of dislocation tangles, dislocation walls, cells and sub-grains.
At larger scales still, deformation twins begin to become important in some materials. Grain boundary effects, grain size effects, and texture effects can occur over
a very wide range of length scales. At similar lengthscales, one begins to see the
influence of heterogeneities in the material, such as inclusions, pores and precipitates. Most component features such as boundary conditions, geometry, constraints
and residual stresses are at the larger scales in this figure. We have discussed most
of these mechanisms and features in previous chapters.
From a modeling perspective, what becomes clear is that it is not possible to link
directly actions and events at the atomic scale to mechanical behavior at the macro
scale without accounting for the specific microstructural features that may develop
at intermediate scales. The mechanical behavior at larger scales is modulated by the
collective behavior of features and mechanisms at a number of smaller scales – it
does not make sense to try to compute the behavior of a steel structure in terms
of atomistic calculations unless one is certain that one can also capture all of the
features of dislocation motion, dislocation tangling, and dislocation-grain boundary
interactions. Microscale and mesoscale interactions are affected by atomistic behavior, but must be specifically accounted for as well in order to predict the material
behavior.
Now that we have some insight into the physical processes active at each length
scale, let us think again about the process of modeling discussed in Figure 8.1. Once
we have chosen the physical problem that we would like to understand, we typically
need to idealize that physical problem into something that we can actually solve.
This process of idealization tends to be discipline-specific, in that the idealizations
chosen by materials scientists tend to be different from the idealizations chosen
by physicists or mechanical engineers. When we think more clearly about these
idealizations, we realize that very often what we are doing is deciding which features
of the deformation (Figure 8.3) our model is intended to capture. Thus, for example,
we may choose to construct a model that accounts for dislocation density but does
not account for individual dislocations or of the distributions of dislocation density
associated with dislocation cell walls and dislocation tangles.
How do we decide which features we intend to include in the model? Often
these are based on experimental observations that indicate that these particular microstructural features are (or are not) observed in the phenomenon or behavior that
we wish to describe using the model. For example, suppose we are attempting to describe the deformations of an aluminum alloy. The modeler may know that dislocations are important in plastic deformations, and so she builds a model that accounts
for dislocation density. Her model may not be capable of describing the clumping
of dislocation density that characterizes dislocation cell walls, and she may believe
this to be a reasonable idealization of the physics of deformation. However, TEM
observations show that dislocation cells are important in describing the large plastic
8 Modeling Nanomaterials
267
deformations of aluminum, and the model will obviously not account for this feature
of the deformation. The modeler cannot, in general, put into the model microstructural features that she does not know exist.
There is one modeling approach that should in principle account for all possible microstructural features, even those that the modeler does not consciously put
into the model. In this approach, one accounts at a fundamental level for quantum
mechanical interactions (some in the literature refer to this as a “first principles” approach), and then computes everything else explicitly from that basis. In principle,
an explicit computation of this type that accounts for all of the possible interactions
between the atoms should be able to reproduce all the possible behaviors at all possible length and timescales. In practice, of course, one is never likely to have the
computing capacity to actually perform such a calculation and so this might be an
exercise in futility.
However, the advantage of such an approach (even over limited scales) is that
one might discover phenomena that one would otherwise never know about, because the fact is that performing experiments over a complete range of length scales
and time scales is also prohibitively expensive, and so there are certainly phenomena that we have never observed simply because we have not known to look for
them. A computation that handles the true complexity of nature at the fundamental
level may thus truly discover the existence of phenomena, and we could then design
an experiment to examine the veracity of such a discovery. This complexity-aware
approach to modeling is extraordinarily expensive from a computational viewpoint,
but has a certain intellectual appeal, and is likely to be of increasing importance as
computational capabilities escalate.
The major modeling approaches used to study nanomaterials will be described
in this chapter, with an emphasis on ultimately providing descriptions that can be
viewed in a mechanics context. First, however, we discuss some of the scaling approximations and physical idealizations made by classical continuum mechanics.
8.2 Scaling and Physics Approximations
All models must make approximations in order to be useful. The question, of course,
is how does one decide which approximations to make? In this section, we attempt
to address this question with respect to nanomaterials. Since most of us approach the
nanomaterials problem with a prior training in the macroscale sciences, it is important to remember that some phenomena at the nanoscale may have no macroscale
counterparts, and we will not naturally think about them in terms of scaling down
from the macroscale.
Continuum models (such as the mechanics models that we normally use at the
macroscale) usually make four core approximations:
• It is assumed that classical mechanics (rather than quantum mechanics) represents the behavior. This will clearly be an issue when considering very small
particles, such as quantum dots.
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Nanomaterials
• Material behavior is assumed to hold pointwise in a continuous medium, where
every material particle has an infinitesimal size. Thus the atomic or molecular
structure is averaged out in this representation.
• Thermal effects are averaged out (we do not consider thermal vibrations of atoms
or molecules explicitly).
• The chemistry is usually ignored, although it can be incorporated in some average
sense.
The second and third items above both relate also to the concepts of differentiability and localization of equations (that is, when can we write the governing
equations as field equations valid at every point?).
An interesting example of the influence of such approximations is that of the
scaling of thermal effects. In solid mechanics, temperature shows up primarily in
terms of thermal conduction as a diffusion term (or as a parameter that modifies
material properties, e.g., through thermal softening). Thermal diffusion is represented mathematically in continuum mechanics through a second derivative (as in
Equation 2.59). The idea of differentiability shows up here – is the second derivative
well-defined (i.e., is the first derivative sufficiently continuous for us to be able to
take a second derivative)? Physically, one can only even define the diffusion term
if the temperature is smooth over a sufficiently large length scale: thermal energy
is moved through the system by atomic or molecular collisions. As we consider
smaller size scales, the mean free path of the molecule between collisions can become an issue – if the size scale of interest is comparable to the mean free path, the
continuum definition of thermal diffusion is no longer sensible. Further, situations
with strong gradients (such as interfaces and surfaces) begin to dominate behavior at
small length scales. For these reasons, for example, the definition of thermal conduction along and across a carbon nanotube cannot be the continuum definition. Thus
the assumptions implicit in continuum mechanics models must often be challenged
on a case-by-case basis when dealing with nanomaterials.
The process of integrating information from multiple scales into a model to
understand the behavior of a material (or more generally a system) is called multiscale modeling. Traditionally, all length scales (and timescales) below the current
scale in a simulation are called subscales, and models are frequently said to represent various levels of subscale information in specific ways. We discuss some of
these approaches next.
8.3 Scaling Up from Sub-Atomic Scales
There is a significant scientific effort focused towards the goal of being able to simulate the behavior of human-scale objects using atom-scale information. While there
is an excellent argument to be made for invoking complexity only when one needs
it, there is also the excellent argument that our intuition is so bad at the nanoscale
that we do not know what complexity to invoke, and so we may learn a great deal by
incorporating the complexity of atomic structure as the foundations of a modeling
8 Modeling Nanomaterials
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approach. But how do we feasibly scale up from atomistics? There are many ways
of approaching this issue, but for the purposes of this book we identify two specific approaches: the enriched continuum approach, and the molecular mechanics
approach.
8.3.1 The Enriched Continuum Approach
This approach to the problem is often taken by those scientists coming to the problem from a continuum scale, and essentially views the equations of continuum mechanics as core equations. That is, it is assumed that the equations of continuum
mechanics (such as the governing equations presented in Section 2.3) will be used
for the solution of the problems of interest. This, of course, presupposes that the
problems of interest are at ≥ nm scales, which is certainly the case for this book.
Given this assumption, atomic-level information primarily enters into the system
of equations through the constitutive equation. This approach therefore views the
constitutive equation as “informing” other continuum equations of the microscopic
degrees of freedom in an averaged sense. Note that this has always been done at
the microstructural level (that is our constitutive equations, such as in plasticity,
have incorporated microstructural information such as the grain size). However, we
recognize that the microscopic degrees of freedom that we choose to incorporate
can also come from the quantum (sub-atomic) level.
This approach results in the incorporation of quantum-level information into continuum mechanics problems in a well-defined and consistent way. It has the advantage of being a great approach for the modeling of mechanics of materials, but it
also has the disadvantage that we must do averaging, and depending on the way we
take the average, we may end up averaging out many of the effects of interest to
nanoscale phenomena. In terms of the topics of interest to this book, the informed
continuum approach may be very effective for bulk nanomaterials but may miss important phenomena in discrete nanomaterials (an excellent example being quantum
dots viewed as nanoparticles - it is the quantum effects that are key in this case).
An excellent and detailed description of these issues is provided by Phillips
(2001).
8.3.2 The Molecular Mechanics Approach
This approach gets its name from the idea that the mechanics problem is considered at the molecular level. However, the fundamental concept here is that we must
make relevant approximations at very small scales (specifically, sub-atomic scales
such as electrons and nuclei) to develop the equations appropriate at the next scale
up. We can then calculate interatomic or intermolecular interactions directly, and
use these to define interatomic or intermolecular potentials. In a subsequent step, to
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Nanomaterials
move to the continuum scale, we can use the interatomic potentials to compute the
behavior of many-atom systems. The primary benefit of this approach is that it handles the molecular structure directly, and so leads to efficient molecular dynamics
calculations.
Such an approach has the advantage that it is not tied to continuum dynamics (it
does not assume the standard continuum equations), but has the disadvantage that
the computations are much more complex, and therefore the physical limitations of
the computing system may result in the simulations being scale limited. For example, as of this writing it is difficult to simulate a billion atom system, which amounts
to a cube of side ≈300 nm.
For us, the primary interest is in interatomic bonds and bond strengths (which
arise naturally from electron cloud interactions). It is immediately obvious that
the problem quickly becomes intractable if we were to attempt to handle the
interactions of every atom with every other atom in terms of electron cloud interactions. However, we can recast this problem in terms of short-range and longrange interactions, and handle each of these separately as approximations. That is,
we do not need to consider short range interactions between every atom pair in
the simulation, but only those between nearby atoms (within some specified zone
of influence defined by the range of the short-range interaction). Even the longrange interactions will typically have some cut-off distance, beyond which they are
negligible.
How are these short-range and long-range interactions obtained? Typically, one
starts with Schrodinger’s equation from quantum mechanics, which is something
the reader has probably seen for a single particle interacting with a potential
V (r,t):
−
∂ ψ (r,t)
ℏ2 2
▽ ψ (r,t) +V (r,t)ψ (r,t) = iℏ
,
2m
∂t
(8.1)
where r is the position and m is the mass of the particle, ℏ is Planck’s constant,
and ψ (r,t) is the particle wave function. This latter quantity is most easily understood by noting that in one dimension |ψ |2 dx is the probability that the particle is
between x and x + dx. The quantity |ψ |2 dx is called the probability density function
for the particle. The right-hand-side of this equation reduces to E ψ (r) in the timeindependent case, where E is an energy (really an energy eigenvalue – one obtains
different solutions ψα corresponding to energies Eα ). We can write a similar equation to Equation (8.1) for an N-particle system, although it is somewhat more complex, and we would have to invoke the Pauli exclusion principle. The point, however,
is that this can be done (writing the equation, that is – solving it can quickly become
difficult).
In the case of interacting charged particles, for instance, the potential V (r,t)
would be
qi q j
,
(8.2)
V (r,t) = ∑
4πε0 di j
8 Modeling Nanomaterials
271
where qi is the charge on the ith particle, and di j is the distance between the ith and
jth particles.
For example, in the simple 1D case of a particle oscillating within a quadratic
potential well, we have V (r,t) = V (x,t) = 12 kx2 , and the now time-independent
Schrodinger equation reduces to
−
ℏ2 d 2 ψ (x) 1 2
+ kx ψ (x) = E ψ (x),
2m dx2
2
(8.3)
which is the equation to a harmonic oscillator and is easily shown to have solutions
ψn (x) =
1
1
2n/2 n! 2
mω
πℏ
1
4
mω
mω 2
exp −
x Hn
x ,
2ℏ
ℏ
(8.4)
with the corresponding energies being quantized as En = (n + 12 )ℏω , n being an
integer. In the latter two equations, the frequency ω is defined by ω = mk , exactly as in the classical spring-mass system. The point of presenting this solution
is to demonstrate that the Schrodinger equation behaves a little differently from the
usual continuum spring-mass system, in that the energy states are discrete rather
than continuous (only certain energies are allowed). Further, the probability of finding the particle (after all, this is what is defined by ψn ) is non-zero at all x even
for the discrete states. That is, you may find the particle anywhere, but it can only
have certain discrete energy states. Some of the quantum character of the solution
becomes evident.
The fact is that Schrodinger’s equation is often written but rarely solved in books
on things nano, particularly when it comes to books of this type that attempt to bring
together multiple disciplines. This is because while Schrodinger’s equation is easy
to write, it quickly becomes difficult to solve for a many-body system, such as, for
example, a cluster of 10 aluminum atoms with all of the nuclei and electrons. Thus
various approximations become necessary.
One typical approximation is called the Born-Oppenheimer approximation, and
uses the fact that the nuclei of atoms are much heavier than electrons, so that in most
problems of interest the nuclei move slowly in comparison with the electrons. Thus,
by separating the problem into the rapid motions of electrons and slower motions
of nuclei, one can compute the electron ψ and energy E in a field of initially fixed
nuclei. Then one updates the positions of the nuclei, and recomputes ψ and E.
In most cases, the problem reduces to finding the total energy ET of a collection
of particles as a function of the positions of the nuclei Rn and electrons re :
ET = ET (Rn , re ).
(8.5)
Note that the number of nuclei and number of electrons are usually very different
in the problems relevant to materials. We can make some further approximations. If,
for instance, we decide that we will approximate the total energy of the nuclei and
electrons given by Equation (8.5) as
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Nanomaterials
ET (Rn , re ) ≈ Eion (Rn ),
(8.6)
we are essentially replacing the total energy of the system with an energy that is
calculated from the positions of the nuclei alone, and this is like computing an effective interaction between ions. If, on the other hand, we replace the large number
of electrons instead with an electron density function ρ (r), we obtain
ET (Rn , re ) ≈ Eapprox (Rn , ρ (r)),
(8.7)
and approaches that solve problems using this approximation are called density
functional theories.
A particularly interesting example of the approximation corresponding to
Equation (8.6) is obtained when one sets the energy Eion to be dependent only on the
relative distances between particles di j , that is, ignoring directional influences, as
Eion =
1
V (di j ),
2∑
(8.8)
where V (di j ) is called a pair potential, and di j is the scalar distance between ion i
and ion j. Many such pair potentials are used in the literature, with the canonical
example being the Lennard-Jones potential:
12 6 ξ
ξ
−
,
V (d) = V0
d
d
(8.9)
Energy, U(r)
with V0 defining the depth of the potential well and ξ defining the length scale
corresponding to the equilibrium separation of the atoms. This potential represents
the interatomic interaction presented in Figure 4.9, which is reproduced here as
Figure 8.4 for convenience.
r0
Distance, r
Fig. 8.4 Schematic of the Lennard-Jones interatomic pair potential V (d).
In principle, this pair potential still involves interactions between every pair of
atoms in the simulation. In practice, most simulations assume that the potential dies
8 Modeling Nanomaterials
273
off after the nearest neighbors, so that only a small number of interactions need to
be included for each atom in the simulation.
A very large fraction of the calculations in the literature use such pair potentials. Increasingly sophisticated potentials are also used, incorporating some of the
directional dependences of the bonds.
Another approximation of interest is the embedded atom method (EAM), which
was developed for metals (Daw and Baskes, 1984) and essentially considers each
metal atom to be embedded within an electron density field derived from the rest of
the metal:
Etotal = Einteraction of nuclei + Eembedding of
1
= ∑ φ (di j ) + ∑ F(ρn ),
2
atom in electron gas
(8.10)
(8.11)
√
where the electron density is ρ . For example, in the special case of F(ρ ) = ρ , we
obtain the Finnis-Sinclair potential.
EAM potentials are extensively used in simulations of the deformation of metals,
and much of the literature on the atomistic modeling of nanocrystalline metals is
based on such potentials. The parameters for such potentials are obtained in a variety
of ways, and the usefulness of the simulation is typically determined by the quality
of the potential that is used. Good EAM potentials are available for a small number
of metals, including copper, nickel and aluminum. An example of such a potential is
that developed by Mishin et al. (1999), and the resulting pair potential for aluminum
is presented in Figure 8.5.
0.25
Aluminum EAM
0.20
Potential (eV)
0.15
0.10
0.05
0.00
–0.05
–0.10
2
3
4
5
6
Distance between atoms (angstrom)
7
Fig. 8.5 The pair potential corresponding to the EAM potential for aluminum, using the parameters
provided by Mishin et al. (1999).
In the next section, we consider computations involving multiple atoms that use
such potentials in order to understand material behavior. Such computations that
include the dynamics are called molecular dynamics.
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Nanomaterials
8.4 Molecular Dynamics
In a molecular dynamics (MD) calculation, one seeks to determine the motions of all
atoms in the simulation by solving Newton’s equations of motion (rather than quantum mechanical equations), with the atoms interacting through assumed interatomic
potentials (such as that in Equation 8.8). Aside from length scales, time scales also
become an issue. Very small integration time steps are needed in order to obtain an
accurate solution, and most MD calculations never go beyond a nanosecond in total
time duration. In order to achieve the strain of interest in that time frame, MD computations typically run at equivalent strain rates on the order of 109 s−1 . This section
will not attempt to show the reader how to perform MD simulations, but rather to
survey some of the issues involved as regards such simulations for nanomaterials.
An excellent and relatively concise introduction to molecular dynamics simulations
is provided by Rapaport (2004).
The basic computational process is as follows: one starts with the positions and
velocities of a number of atoms (preferably a large number). The potential energy
of the system is used to compute the force on each atom using an assumed interatomic potential. Since the masses of the atoms are known, these calculated forces
allow us to compute the accelerations of each atom, and these accelerations can be
used to update the velocities of the atoms, and the velocities are used to update the
atomic positions. We then begin the process again. These can be very large simulations, depending on the number of atoms involved, given that every atom interacts
at least with its nearest neighbors, and depending on the total times (total number
of timesteps) that must be computed. There are very sophisticated codes now available in the public domain for solving problems using molecular dynamics. The best
known of these is called LAMMPS, and is able to run effectively on a parallel processing platform (many computers working in parallel in terms of the numerical
algorithms used for solving MD problems). Given the availability of such codes,
molecular dynamics simulations are now in the reach of any scientist willing to
make the effort.
As always with scientific computations, however, the key is to understand (a) precisely what is the scientific objective of the calculation (what are you looking for?),
(b) precisely what is being computed (what equations are being solved numerically?) and (c) when can the computation provide reasonable results (what are the
limits of the computational approach?). There is a significant danger in using canned
computational packages, in that the user tends to assume that the results are always
reasonable, while the developers may not have provided warnings or flags to indicate when the output is inaccurate or even meaningless.
There is a significant level of research activity, as of this writing, that uses molecular dynamics simulations to enhance our understanding of nanomaterials. Some of
the work has been on the special case of carbon nanotubes, which involve carboncarbon interactions. Since such interactions are covalent and quite directional, a
classic pair potential involving only the atomic separation is not appropriate. Modifications of pair potentials that include some directional characteristics (sometimes
called bond-angle character in the literature) include the Tersoff (Tersoff, 1988)
8 Modeling Nanomaterials
275
and Brenner (Brenner et al., 2002) potentials. In the case of bulk nanocrystalline
materials, most of these simulations have been on metallic nanomaterials, using
embedded atom method potentials. EAM potentials have been constructed and validated to various degrees for copper, nickel, and aluminum (e.g., Mishin et al., 1999).
As is generally the case in nanomaterials, fcc metals are the best studied. Reviews
of MD simulations in nanomaterials appear in the literature every two years or so, a
recent example being Van Swygenhoven and Weertman (2006).
An example of a three-dimensional nanostructure developed for solution using
molecular dynamics is presented in Figure 8.6. The figure shows several tens of
grains of nanocrystalline nickel constructed using a Voronoi tessellation approach,
and modeled using the LAMMPS software. The colors represent the coordination
number associated with each atom, and the grain boundaries are identified by the
distinctly different coordination numbers of the atoms compared with the atoms in
the face-centered cubic crystals. This collection of atoms can now be subjected to
mechanical loading (deformations) and the motions of the individual atoms can be
tracked to understand deformation mechanisms.
Fig. 8.6 Nanocrystalline material constructed using molecular dynamics. The material is nickel,
and the atoms are interacting using an EAM potential due to Jacobsen. About 30 grains are shown.
The color or shade of each atom represents its coordination number (number of nearest neighbors),
with the atoms in the crystals having the standard face-centered-cubic coordination number. The
change in coordination number at the grain boundaries is clearly visible. This collection of atoms
can now be subjected to mechanical loading (deformations) and the motions of the individual atoms
can be tracked to understand deformation mechanisms.
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Nanomaterials
What have we learned, and what can we learn, from MD simulations of nanomaterials? In principle, of course, such simulations have tremendous potential for
understanding the physics of these materials, because they give us access to time
scales and length scales that are extremely difficult to investigate through experiments. We will discuss some of the advances in our understanding obtained through
MD computations in the next paragraphs. The limitations of these calculations arise
also from the scales at which they operate, and the continuing limitations of computing capacity. In general, larger simulations (in terms of number of atoms or length
scale) are run for shorter total times because of the available computer cycles. Even
the largest simulations that are currently feasible are unable to sample a cubic micron of volume for a total time of 1 ns. As a consequence, any physical mechanism
that requires significant time to develop or significant space to organize itself cannot be captured using such simulations (although many MD aficionados continue to
extrapolate from short times and small scales).
Some other distortions of the physics can also occur. To generate significant
strains in very short times, extraordinarily high strain rates are applied (almost always higher than 107 s−1 ), and thus the mechanisms that are observed in an MD
simulation will not necessarily be observed in a physical experiment at strain rates
many orders of magnitude smaller. Further, the dislocations that are developed in
MD simulations often have extremely high accelerations, which can generate some
pathological situations in terms of mechanics. Thermal equilibration of systems is
a continuing challenge, since there is insufficient time for typical thermal diffusion
mechanisms; some ingenious approaches have been developed to define a reasonable thermal state (Liu et al., 2004). The small sample sizes that are simulated and
the sometimes two-dimensional nature of the simulations can restrict the number of
available slip systems and deformation mechanisms. Most MD simulations also do
not account for the initial defect structure that exists in real materials, and therefore
miss many of the sources and sinks that are active in many nanomaterials (e.g., most
of the simulated grains are completely clean and contain no initial defect structures,
while most experimentally observed nanocrystals contain large numbers of initial
defects such as stacking faults and pre-existing dislocations). Issues with regard to
the types of grain boundaries considered must also be addressed. Finally, the nature
of the (often periodic) boundary conditions that are applied can couple into the dynamics, causing difficulties with modeling some phenomena that are driven by wave
dynamics.
Even with all of these limitations, however, MD simulations have been very useful for understanding nanomaterial behavior. For example, such simulations (e.g.,
Schiotz and Jacobsen, 2003) have indicated that there may in fact be a maximum in
the strengthening that can be obtained by decreasing the grain size, so that below
a certain critical grain size the strength begins to decrease again as the grain size
decreases. This is called the inverse Hall-Petch effect and is somewhat controversial, in that it has been difficult to obtain incontrovertible experimental evidence that
the effect exists (because of specimen preparation and testing difficulties). The MD
results (such as those of Schiotz and Jacobsen, 2003) demonstrate that there are at
least some physical reasons why such an effect might exist. Further, such simula-
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277
tions have demonstrated that grain boundaries can be significant sources of dislocations (Farkas and Curtin, 2005) and that there are competing dislocation nucleation
mechanisms that may become dominant as the grain size changes (Yamakov et al.,
2001). The likely dominance of partial dislocations rather than full dislocations in
nanocrystalline aluminum has been predicted by MD and subsequently observed
through experiments (Chen et al., 2003a). Grain boundary sliding has been shown
to be a plausible mechanism for effective plastic deformation in very small grain
size metals (Derlet et al., 2003). The simulations also show that dislocations may
not survive in the interior of small grains without externally applied stresses, that
is, they may be emitted and then reabsorbed by grain boundaries (Yamakov et al.,
2001). As a consequence, not seeing dislocations in the interior of nanograins within
the TEM does not guarantee that dislocation-mediated plastic deformation did not
occur. This is an insight that could not have been obtained definitively without the
MD simulation.
It is apparent that such simulations continue to have great potential in this field.
However, the limitations of strain rate, time scale and length scale must be carefully
considered in attempting to relate MD simulations results to experimental measurements and to predictions of engineering behavior and possible applications. In the
next section we discuss another modeling approach that examines larger scales in
space and time.
8.5 Discrete Dislocation Dynamics
We have seen that the length scales that can be simulated with molecular dynamics
are relatively small, <1 µm3 . Even though such simulations are able to consider
nanocrystalline materials (because the grain sizes are less than 100 nm), they are
unable to consider time scales large enough to observe most of the conventional
deformation mechanisms of interest to engineering. In addition, such small length
scales do not permit the development of organized micron-scale structures, or the
examination of materials containing grain size distributions.
A larger scale approach that shows some promise is that of discrete dislocation
dynamics. This modeling approach does not seek to track the motions of individual
atoms, but rather seeks to track the motions of individual dislocations as they move
through a continuous medium. Each dislocation is supposed to interact with the
medium in a specific way (typically derived from elasticity theory, but sometimes
from atomistics), and the interactions of dislocations with each other can be explicitly handled using rules derived from well-developed dislocation theory (Hirth
and Lothe, 1992). Very large numbers of dislocations can be tracked in this way,
with specific rules for their motion through the medium and for their interactions
with each other. The physics associated with the interactions of such dislocations
is fairly well understood, but the equations of motion for dislocations at low velocities are still somewhat empirical. The most common equation of motion involves
a dislocation drag type behavior, which is probably only accurate at relatively high
dislocation velocities.
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Nanomaterials
The advantage of such simulations is (aside from sheer length scale) that they
can follow dislocation structures as they evolve, and can examine the development
of deformation mechanisms that involve multiple dislocations (or dislocation densities). While discrete dislocation dynamics (often called DDD) simulations are now
often used in the simulation of plastic deformations in single crystals and large polycrystals, they are only rarely used with respect to nanocrystalline materials at this
time. Some examples of work of interest include that of Noronha and Farkas (2002,
2004) and that of Kumar and Curtin (2007). A significant effort to marry the length
and time scales is needed to take advantage of this technique in the nanomaterial
domain, but the return is likely to be well worth the effort.
8.6 Continuum Modeling
Much of this book has been focused on the continuum viewpoint with respect to
nanomaterials, as in our discussions of elastic and plastic behavior. Some of the discussion has incorporated some microstructural features in an implicit sense, as when
the grain size was used through the Hall-Petch term to change the yield strength
(which is the continuum parameter).
8.6.1 Crystal Plasticity Models
Explicit continuum computations of the effect of microstructure are also possible,
for example through the explicit computation of the behavior of a polycrystalline
mass using crystal plasticity codes. Such calculations consider the simultaneous deformation of multiple crystals of various grain sizes, where each crystal is modeled
as a continuum made up of a material that has anisotropic elastic properties and that
develops plastic strain along specific slip systems. We can view such a model as
being one level up in length scale from the discrete dislocation dynamics model, in
that the crystal plasticity model does not account for individual dislocations but does
account for the collective effects of dislocations through the assumption of a critical resolved shear stress for the onset of slip (the analog of the onset of motion of
dislocations) and hardening rules for slip along primary and conjugate slip systems
(the consequence of the interaction of dislocations). However, a continuum crystal
plasticity model has no internal length scale, and therefore is unable to distinguish
between a collection of large crystals and a collection of small crystals.
Some researchers have modified crystal plasticity models to incorporate a length
scale by invoking the onset of yield differently for different sized crystals, typically by assuming a Hall-Petch style behavior. Such models do not add any new
physical insight, since the length scale effect is assumed, but may be useful in looking at the behavior of textured polycrystalline systems. Other researchers have formally incorporated another phase at the grain boundary in a polycrystalline mass
8 Modeling Nanomaterials
279
(Wei et al., 2006), giving the new phase properties that correspond to an amorphous
material, and computing the deformation of the crystals using a crystal plasticity
model. While such a model certainly has academic interest, an amorphous phase
is almost never found at grain boundaries in nanomaterials (see, e.g., Figure 8.7.
The mechanics justification of the assumption of an amorphous phase at the grain
boundary is typically that the atoms at the grain boundary are somewhat disordered,
and therefore can be viewed as being part of an amorphous material. However, there
is no evidence that the atoms at the grain boundary behave like an amorphous material in terms of plastic response. Indeed, molecular dynamics simulations suggest
that the material at the grain boundary can be a source of dislocations, rather than
behaving like an independent harder phase.
Fig. 8.7 High resolution micrograph showing the boundary between two tungsten grains in a
nanocrystalline tungsten material produced by high-pressure torsion. There is no evidence of any
other phase at the grain boundary, contrary to pervasive assumptions about the existence of an
amorphous phase at the grain boundary in nanomaterials in crystal plasticity and composite simulations at the continuum level. Note the edge dislocations inside the grain on the right. Such internal
dislocations are rarely accounted for in molecular dynamics simulations.
8.6.2 Polycrystalline Fracture Models
Finite element simulations of two-dimensional microstructures of polycrystalline
ceramic materials are also performed, with a view to identifying the fracture processes in such materials, e.g., Kraft et al. (2008). An example of such a polycrystal
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Nanomaterials
simulation is shown in Figure 8.8, which shows crystals of alumina modeled as
elastic solids, with grain boundaries that can fail (with the failure process modeled through a cohesive function). The identical issues of length scale arise in such
simulations (although dynamic simulations that incorporate wave propagation effectively include a length scale because of the finite wave speed). The current status of
polycrystalline fracture computations is that three-dimensional fracture of the polycrystalline mass is extremely difficult to simulate, particularly with regard to the
lack of mesh convergence when solving combined contact and fragmentation problems. The computational simulation of nanocrystalline ceramics remains a major
challenge.
Fig. 8.8 Finite element model of a polycrystalline mass of alumina, with the crystals modeled as
elastic solids (Kraft et al., 2008). The model seeks to examine the failure of the polycrystalline
mass by examining cohesive failure along grain boundaries. Reprinted from Journal of the Mechanics and Physics of Solids, Vol. 56, Issue 8, page 24, R.H. Kraft, J.F. Molinari, K.T. Ramesh,
D.H. Warner, Computational micromechanics of dynamic compressive loading of a brittle polycrystalline material using a distribution of grain boundary properties. Aug. 2008 with permission
from Elsevier.
8.7 Theoretically Based Enriched Continuum Modeling
The discussions up to this point show the value of developing multiple scale approaches that can rigorously couple atomistic and continuum behaviors. Most of
the methods by which this is done are computational, and indeed, this is one of the
fastest-growing fields within scientific computing. We do not discuss such multi-
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281
scale simulations in any great detail in this book. In this section we focus on some
theoretically based approaches to enriching the continuum response with atomic
level information, focusing primarily on the use of the Cauchy-Born rule.
As a specific example, we will consider the deformations of carbon nanotubes.
Much more detailed discussions of such approaches can be found in the works of
Huang et al. (e.g., Zhang et al., 2002, 2004) and Volokh (Volokh and Ramesh, 2006).
This discussion follows the work of Volokh.
Carbon nanotubes typically have diameters on the order of ≤1 nm and lengths
of 10–1000 nm. They are made entirely of carbon atoms, which are typically arranged in a hexagonal structure (Figure 8.9). The easiest way to think of a carbon
nanotube is in terms of a two-dimensional sheet of carbon atoms arranged in the
hexagonal structure and then wrapped around a cylinder of radius r. Let us consider
a single-walled carbon nanotube, in which case only one sheet of atoms exists in
the nanotube. While we understand how to model tubes of material from solid mechanics, how do we describe the mechanics of a tube such as that in Figure 8.9? The
atomic structure must be accounted for in some fashion, and we show how that can
be directly integrated into a continuum description next.
Fig. 8.9 Schematic of the structure of a carbon nanotube, viewed as a sheet of carbon atoms
wrapped around a cylinder. The sheet can be arranged with various helix angles around the tube
axis, which are most easily defined in terms of the number of steps in two directions along the
hexagonal array required to repeat a position on the helix. Illustration by Volokh and Ramesh
(2006). Reprinted from International Journal of Solids and Structures, Vol. 43, Issue 25–26,
Page 19, K.Y. Volokh, K.T. Ramesh, An approach to multi-body interactions in a continuumatomistic context: Application to analysis of tension instability in carbon nanotubes. Dec. 2006,
with permission from Elsevier.
Since all of the atoms are carbon, the interactions between the atoms is the
carbon-carbon interaction. Since carbon-carbon bonds are covalent and highly di-
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rectional, a simple pair potential does not adequately describe the interaction.
Tersoff (1988) and Brenner et al. (2002) developed models for carbon-carbon interactions that we will use here. Let us suppose that all of the carbon atoms in a
nanotube, before deformation, are at positions Ri . After deformation, we define the
position of the ith carbon atom as ri , and assume that there are n atoms in the nanotube under consideration. Then Tersoff and Brenner identify the total energy of
these atoms as
1
Etotal = ∑ V (di j ),
(8.12)
2
where di j is the distance between the ith and jth atoms, and is itself given by
di j = (ri − r j ) (ri − r j ).
(8.13)
The pair potential V (di j ) itself is given by
√
2
V0
V (di j ) =
fc (di j ) exp[− 2Sβ (di j − Di j )] − SB(i j) exp −
β (di j − Di j ) ,
S−1
S
(8.14)
where Di j = (Ri − R j ) (Ri − R j ) is the initial separation between atoms, V0 is a
scaling measure, usually assumed to be 6 eV , and S = 1.22 where β = 21nm−1 .
The equilibrium bond length is set to Di j = 0.145 nm. Equation (8.14) also contains
a cut-off distance function fc (di j ), which ensures that the potential function only
needs to be exercised over a finite distance, and yet rolls off to zero in a smooth
fashion (thus retaining continuity). A typical cut-off function is
fc = 1, di j < a1
π (di j − a1 )
1 1
= + cos
, a1 < di j < a2
2 2
a2 − a1
= 0, di j < a2 ,
(8.15)
(8.16)
(8.17)
with a1 = 0.17 nm and a2 = 0.2 nm, thus including only the first-neighbor shell for
each carbon atom. Equation (8.13) also contains a term written as B(i j) , which is a
common way of writing the symmetric part of Bi j :
1
B(i j) = (Bi j + B ji ).
2
(8.18)
The term Bi j itself represents the three-body coupling that is needed to define an
angle (note two atoms define a line, while three atoms define an angle). Consider
three atoms at locations ri , r j , and rk . We can define the vector joining the ith and jth
atoms as di j = ri − r j and similarly we have dik = ri − rk . Then the angle between
the three atoms is given by φi jk , where
cos φ(i jk) =
di j dik
.
di j dik
(8.19)
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Note the differences between the vectors and the scalars in the latter equation.
Now that the angle between the three atoms is define, Tersoff-Brenner defines the
three-body coupling term Bi j as
−δ
c20
c0 2
B(i j) = 1 + ∑ a0 1 +
fc (di j )
− 2
,
d0
d0 + 1 + cos φi jk 2
(8.20)
where the various parameters are a0 = 0.00020813, c0 = 330, d0 = 3.5, and δ =
0.5. We can use this Tersoff-Brenner potential to calculate the interactions between
the carbon atoms. Now that we have defined the atomistics, how can we connect
the atomic positions to the continuum behavior? We do this by associating atomic
positions to continuum deformations (Weiner, 1983). The initial (Di j = Ri − R j )
and current (di j = ri − r j ) relative positions of any two atoms are assumed to be
connected to the deformation of the corresponding continuum by
di j = FDi j ,
(8.21)
where F is the continuum deformation gradient tensor. This tensor is defined in
continuum terms as the tensor that maps the vector dX between two nearby material
particles in the reference configuration to the corresponding vector dx between the
corresponding spatial positions of those particles in the current configuration:
dx = FdX.
(8.22)
Here x = χ (X,t) is the spatial position in the current configuration of the material particle X in the reference configuration (see the section on kinematics in
Chapter 2). In effect, we assume that the ith and jth atoms are in the vicinity of the
point X, as shown in Figure 8.10. Equation (8.21) provides a link between the atomistics and the continuum mechanics in terms of the deformations (the Cauchy-Born
approximation), and was originally invoked for elasticity theory.
We have now connected the kinematics at the two scales, but we do not truly
have a linking of scales unless we can link the corresponding energies. The key is
to identify the strain energy in the continuum with the energies associated with the
interatomic interactions (the bond energies) (Born and Huang, 1954). To do this, we
define the potential energy density W of the continuum problem to be
W = W (E) =
1
V (di j ),
V0 ∑
(8.23)
where V0 is the initial volume and E is the Green strain tensor defined by
1
E = [FT F − I],
2
(8.24)
with I the identity tensor. The Green strain tensor is simply the finite deformation
analog of the infinitesimal strain tensor ε defined in Equation (2.28). Note that it
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X
F=
Dijij
x
∂x
∂X
dij
dij
d ij
= FD
F ij
Current Configuration
Reference Configuration
Fig. 8.10 Connection between continuum deformations and atomic positions as defined by
Equation (8.21). We associate atomic positions in the two configurations with material and spatial vectors.
is not actually necessary to sum over all atoms to be able to compute the energy
density function W . The sum can be replaced by an integral:
W (E) =
1
V0
V Bv dV,
(8.25)
where Bv is called the volumetric bond density function (Gao and Klein, 1998).
In terms of the atomic positions, we now have
di j = Di j (I + 2E)Di j ,
(8.26)
and the angles between the atoms are related to the Green strain tensor by
cos φi jk =
Di j (I + 2E)Di j
.
di j dik
(8.27)
The stress tensor can then be computed as the derivative of the potential energy
density through
∂W
.
(8.28)
S=
∂E
Note that this stress tensor is defined in the reference configuration (in mechanics
terms it is called the second Piola-Kirchhoff stress tensor), but this is a refined point
that is not critical to understand for the reader who does not wish to delve into the
depths of the mechanics of finite deformations. The point is that the stress can be
computed from the energy density, which can be computed from the interatomic
potential.
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The material property corresponding to the stiffness is obtained through the stiffness tensor M:
∂ S ∂ 2W
.
(8.29)
=
M=
∂E
∂ E2
This is the analog of the modulus of elasticity tensor (and is identical to that tensor
in the linear elasticity limit). Thus we can compute the stiffness of the material at the
continuum scale from the atomic positions and the interatomic potential. There are
several refinements of this approach that are needed for the specific case of carbon
nanotubes, because the atomic arrangements there are not those of a simple lattice.
The reader is referred to the work of Huang et al. (Zhang et al., 2004) for examples
of these details.
The elastic modulus of carbon nanotubes computed in this way (Zhang et al.,
2002) is in fairly good agreement with that computed using MD simulations, and
lies within the range of moduli measured using experimental methods. The Young’s
modulus predicted by Zhang et al. (2002) is E = 705 GPa, while Volokh and Ramesh
predict E = 1385 GPa. These numbers fall within the range of numerous experimental (Figure 8.11) and modeling predictions (Figure 8.12) reviewed by Zhang et al.
(2004). However, the assessment of the Young’s modulus and stresses in the CNT is
not trivial because the thickness of the CNT wall is difficult to define. The estimates
of the Young’s modulus given above were obtained with the assumption that the
wall thickness is 0.335 nm (Zhang et al., 2002).
Fig. 8.11 Range of experimental measurements of the elastic modulus of carbon nanotubes, as
summarized by Zhang et al. (2004). Note the theoretical predictions discussed here were 705 GPa
(Zhang et al., 2002) and 1385 GPa (Volokh and Ramesh, 2006). Reprinted from Journal of the
Mechanics and Physics of Solids, P. Zhang, H. Jiang, Y. Huang, P.H. Geubelle, K.C. Hwang,
An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. May
2004, with permission from Elsevier.
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Fig. 8.12 Range of modeling estimates of the elastic modulus of carbon nanotubes, as summarized
by Zhang et al. (2004), including a wide variety of first principles and MD simulations. Note the
theoretical predictions discussed here were 705 GPa (Zhang et al., 2002) and 1385 GPa (Volokh
and Ramesh, 2006). Reprinted from Journal of the Mechanics and Physics of Solids, P. Zhang,
H. Jiang, Y. Huang, P.H. Geubelle, K.C. Hwang, An atomistic-based continuum theory for carbon
nanotubes: analysis of fracture nucleation. May 2004, with permission from Elsevier.
It is also possible to use this mixed atomistic and continuum approach to estimate the likely failure strength of a carbon nanotube, using bifurcation analyses
(Zhang et al., 2004; Volokh and Ramesh, 2006. As an example of the utility of such
approaches, consider the results of Volokh and Ramesh (2006). These authors incorporate a physically distinct approach to handling the multi-body interactions needed
to model carbon nanotube and similar systems, accounting for bond stretching and
bond bending separately, but the scaling approach remains essentially that discussed
above. They predict the onset of tensile instability in carbon nanotubes to occur at
21–25% Green strain (the theory predicts the onset of instability, rather than the final failure strain, which is the more easily measured quantity in experiments). Using
observations within a TEM, Troiani et al. (2003) reported that they observed 50%
elongation of the single-walled nanotubes before failure.
Some comparison with the experiments performed by Yu et al., (2000b) on multiwalled CNTs is also possible. These authors measured the critical strain to be between 10% and 14% experimentally on these multiwalled CNTs. The theoretical
analysis of Volokh and Ramesh estimated the critical strains to be 21% for singlewall zigzag CNT and 25% for single-wall armchair CNT (these are two possible
types of carbon nanotubes). Assuming that the addition of the wall layers leads to
the stiffening of the CNT, the results of the theoretical analysis seem reasonable,
because the critical strains would be expected to decrease and the critical stresses
are expected to increase with CNT stiffening.
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There is one additional caveat when comparing the results of such theoretical
predictions with experimental measurements. Idealized and perfect nanostructures
are assumed in the analyses, while real imperfect nanostructures are tested in practice. It is extremely difficult to develop defect-free nanotubes, and this is a particular
problem when the tensile instability will be modulated by the defects (defects should
trigger earlier onset of instabilities). The single SWCNT tests of Troiani et al. (2003)
appear to be close to defect-free structures, and they appear to show the largest failure strains (note that these are failure strains, not the onset of the instability, which
is what is predicted by this theoretical analysis). Other experiments have generally
shown a wide range of failure strains (Yu, 2004), typically significantly smaller, but
these are typically in less-clean structures. The presence of atomic defects can disturb the results of the measurements. Inclusion of the atomic imperfections in the
theoretical models is of interest, but this is difficult to do.
Finally, note that theoretical predictions made using bifurcation analysis can also
predict the bifurcation mode (like the buckling mode). Symmetric bifurcations are
typical of rod necking (necking along the axis) while asymmetric bifurcations are
typical of shell buckling (buckling of the wall of the nanotube). Volokh and Ramesh
predict, for example, that ideally perfect armchair nanotubes are shell-like in terms
of failure mode, while zigzag nanotubes are rod-like in failure mode. Such predictions have not yet been examined in experiments.
8.8 Strain Gradient Plasticity
Another approach to enriching continuum theories to account for subscale phenomena is to include explicitly a length scale in the continuum model. The most common
way in which this is done is through strain gradient theories. Such theories have typically been developed to model size effects in mechanics, rather than specifically for
nanomaterials, but the issues are related.
Many experiments, mainly from microindentation and nanoindentation, e.g., Nix
and Gao (1998), have shown that materials display strong size effects when the
characteristic length scale associated with non-uniform plastic deformation is in the
range of a micron or less, and these effects are believed to result from intrinsic length
scales in the materials studied. Classical plasticity theories do not have an intrinsic
material length scale and thus fail to capture this strong size dependency of material
behaviors. Several strain gradient plasticity theories have been proposed to address
this problem. As an example, a strain gradient plasticity theory based on dislocation
theory was introduced by Gao et al. (1999) and Huang et al. (2000), and is briefly
discussed here.
An intrinsic length scale can be defined in materials through the concept of geometrically necessary dislocations (Ashby, 1970). The idea is that some of the dislocations in a material exist in the structure only to accommodate certain misfits,
i.e., they are geometrically necessary in order to maintain compatibility (consider,
for example, the misfit strains produced by a hard particle in a soft matrix when the
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assembly is deformed plastically). These misfits amount to gradients in the plastic
strain, and hence the term strain gradient. The total dislocation density in a material
is then the sum of two components, called the statistically stored dislocations and
the geometrically necessary dislocations:
ρ = ρS + ρG .
(8.30)
In this light, the Taylor relation between the shear strength τ and the total dislocation density in a material becomes
τ = α µ b ρS + ρG ,
(8.31)
where α ≈ 0.3. From the Taylor relation, Gao et al. (1999) then derive a constitutive
function that contains an internal length scale:
σ = σy f 2 (ε ) + lg η ,
(8.32)
where σy is the yield stress, η is the effective plastic strain gradient, lg is an intrinsic
material length, and σ = M̄ τ is the effective stress, with M̄ the Taylor factor or the
ratio of tensile flow stress to shear flow stress (for polycrystalline FCC materials
M̄ ≈ 3 (Bishop and Hill, 1951a,b). The function f (ε ) is a plastic strain-hardening
function. In the absence of a strain gradient (η = 0 so that ρG = 0), the density of
statistically stored dislocations ρS can be expressed in terms of the strain hardening
function:
σy ( f (ε ) − 1)2
ρS =
.
(8.33)
2
M̄ α µ b
Geometrically necessary dislocations appear in the strain gradient field for compatible deformation. The relation between the density of geometrically necessary
dislocations ρG and the effective plastic strain gradient η is
ρG =
λη
,
b
(8.34)
where λ is Nye’s factor (in polycrystalline FCC materials, λ = 1.85 for torsion, and
λ = 1.93 for bending). Using Equation (8.34) in Equation (8.31), one obtains the
intrinsic material length as
2
M̄ α µ
λ b.
(8.35)
lg =
σy2
For example, for polycrystalline FCC metals in bending this material length scale
2
would be lg = 18α 2 b σµ 2 λ b. For non-uniform plastic deformations, the effects of
y
plastic strain gradient are important when the characteristic length associated with
the deformation becomes comparable to the intrinsic material length lg . For most
structural engineering materials, the intrinsic material length is on the order of mi-
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crons, but for very pure annealed metals, it could be an order of magnitude larger.
Using this as a basis, Gao et al. (1999) and Huang et al. (2000) proposed a multiscale approach to construct the constitutive laws, the equilibrium equations and the
boundary conditions for strain gradient plasticity.
The presence of this length scale in strain gradient theories allows one to describe
size effects in macroscopic behavior, such as the indentation size effect (smaller
indentations appear to make the material seem harder) and particle-size effects in
composites. An example of the application of strain gradient plasticity theory to
nanocrystalline materials is presented by Gurtin and Anand (2008).
8.9 Multiscale Modeling
A number of multiscale modeling approaches have been developed that seek explicitly to couple atomistic and continuum models. Broadly speaking, these multiscale
methods are classified in two ways: as hierarchical approaches or as concurrent
approaches. In hierarchical approaches, measures from simulations at one scale are
passed on to simulations at other scales, and most of the models that we have discussed so far are of this type (these are also called approaches to coarse-graining
or fine-graining). In concurrent approaches, multiple length scales are simultaneously simulated, with different spatial resolutions in different regions of the body,
and with some well-defined approach to hand off the scale-appropriate variables to
each scale in each region. Entire books are devoted to these techniques, e.g., Liu
et al. (2006) and we cannot usefully discuss them here. One particular approach,
however, is mentioned because it has grown rapidly in usage in recent years and
is relatively easily used with traditional finite element methods. This technique is
called the Quasi-Continuum (QC) method and was developed in a series of papers
by Tadmor et al. (1996) and Shenoy et al. (1999). The method links the atomistic
and continuum scales through the finite element method, and therefore is able to
utilize much of the expertise associated with computational mechanics. A review
of the method is presented by Miller and Tadmor (2002) and there is an associated
website www.qcmethod.com that acts as a clearing house for related research.
Examples of the use of this method for nanocrystalline materials are provided by
Sansoz and Molinari (2005) and Warner et al. (2006). In a multiscale simulation of
the deformation of nanocrystalline copper, Warner et al. (2006) use the quasicontinuum method to determine the properties of a suite of grain boundaries of various
types (Figure 8.13). These results are used to develop constitutive equations for
grain boundaries, and then these grain boundary properties are incorporated into a
simulation of a collection of polycrystals (Figure 8.14) with a range of grain boundaries. These are therefore hierarchical multiscale simulations.
Based on these computations, Warner et al. (2006) predict the strength of nanocrystalline copper as a function of grain size, and compare their results (Figure 8.15)
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Fig. 8.13 Simulation of a copper grain boundary by Warner et al. (2006) using the quasicontinuum method. The arrows correspond to the displacement of each atom between two loading
steps. Reprinted from International Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner,
F. Sansoz, J.F. Molinari, Atomistic based continuum investigation of plastic deformation in
nanocrystalline copper. April 2006, with permission from Elsevier.
with the results of MD simulations and the available experimental data. This comparison is fairly typical for nanocrystalline materials.
Note the molecular dynamics simulations predict much higher stresses than are
observed in experiments. The finite element simulations also predict higher stresses
than the experiments, but these are closer in magnitude to the observed behavior.
The higher stresses that are derived in both kinds of simulations are partly because
of the number of mechanisms that exist in the real world but that are not accounted
for in the simulations due to issues regarding length scale and time scale (for example, diffusion and grain boundary migration). In addition, the simulations inevitably
8 Modeling Nanomaterials
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Fig. 8.14 Computational approach for simulations of nanocrystalline copper by Warner et al.
(2006). The grayscale represents different orientations of the crystals. Reprinted from International
Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner, F. Sansoz, J.F. Molinari, Atomistic
based continuum investigation of plastic deformation in nanocrystalline copper. April 2006, with
permission from Elsevier.
Fig. 8.15 Comparison of the predictions of the multiscale simulations of Warner et al. (2006)
(identified with the FEM symbol) with molecular dynamics (MD) simulation results and experimental data on nanocrystalline copper. Note that the experimental results are derived from
hardness measurements, while the FEM simulation results correspond to a 0.2% proof strength.
Both MD and FEM simulations predict much higher strengths than are observed in the experiments. Reprinted from International Journal of Plasticity, Vol. 22, Issue 4, Page 21, D.H. Warner,
F. Sansoz, J.F. Molinari, Atomistic based continuum investigation of plastic deformation in
nanocrystalline copper. April 2006, with permission from Elsevier.
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have limitations that arise from the assumptions made to render the problem computationally tractable. For example, the Warner et al. finite element simulations are
2D plane strain, and these authors point out that the fraction of grain boundary
material is substantially lower in 2D simulations. Although the magnitudes of the
stresses are very different, all of the simulations and the experiments indicate that
the grain-size driven strengthening eventually saturates or even turns into a softening. These modeling results are consistent with the general observation of deviations
from Hall-Petch behavior at small grain sizes that was discussed earlier.
8.10 Constitutive Functions for Bulk Nanomaterials
Now finally returning to the full continuum scale, we recognize that the solutions
of problems in the mechanics of nanomaterials require the specification of full constitutive equations for these materials. These constitutive equations must be defined
with parameters that can be determined either through experiments or through subscale or multiscale simulations (a sophisticated constitutive function that cannot
have its parameters determined in some way will never be used). In some cases, the
traditional definition of a constitutive equation may no longer be appropriate, and
the nanomaterial may be better treated as a structure than as a material (for example,
the behavior of a carbon nanotube used as an AFM tip). In other cases, the constitutive equation may need to be enriched by the addition of an internal length scale, or
through the incorporation of an internal variable that has an independent evolution
equation determined by subscale physics (e.g., the influence of grain rotation on the
behavior of bulk nanocrystalline solids). In still other cases, almost entirely limited
to bulk nanocrystalline materials, the traditional mechanics constitutive equations of
elasticity, plasticity and fracture are all that are needed (with perhaps the additional
considerations of viscoelasticity, creep or superplasticity).
8.10.1 Elasticity
The elasticity of bulk nanocrystalline materials is very similar to that of their coarsegrained cousins, with two caveats. First, the nanocrystalline material may have different anisotropy, as discussed in Chapter 4 on density and the elastic properties,
because of the grain structure and perhaps orientation. Second, the nanocrystalline
solid will typically have lower moduli, again as discussed in that chapter. From
an experimental viewpoint, the stiffnesses of nanomaterials are dominated by the
specifics of the processing route used to create the material, since defect populations, grain sizes, grain orientations, and grain size distributions are all controlled
by the processing.
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8.10.2 Yield Surfaces
Specifying the plastic response of nanomaterials requires first a specification of the
yield behavior. In earlier chapters we have discussed the effects of grain size on the
yield strength. However, most nanocrystalline materials do not behave the same in
tension and compression, and therefore the traditional Mises yield surface cannot be
used to specify the onset of yield.
Consider, for example, the case of nanocrystalline nickel. Nanocrystalline Ni
pillars were studied through the microcompression of 20 µm diameter posts by
Schuster et al. (2006). Since there are about a million grains across the cross section and about a billion grains in each post, this technique provides the effective
response of the bulk material. An MTS Nanoindenter XP with the Continuous
Stiffness Measurement extension was used with a truncated Berkovich indenter to
apply the compressive load at a strain rate of 10−3 s−1 . Following the method proposed by Zhang et al. (2006) , the pillar compliance was corrected for the effect of
the elastic base and the corresponding stress and strain measures were corrected.
The nano-Ni specimens showed very consistent compressive behavior. The compressive yield strength at 0.2% offset was an average of 1210 ± 20 MPa (σC ).
The identical batch of material was also examined by Wang et al. (2004). They
report a flow strength (at 3.5% strain) of 1260 MPa in tension (σT ). The microcompression results show a compressive flow strength (σC ) at 3.5% strain of 1460 MPa,
16% higher than Wang’s tensile values. Therefore, the Mises yield surface is not
appropriate for this material, because that yield surface assumes identical behavior in tension and compression. In this case, the compression-tension asymmetry is
σC
σT = 1.16 (note that we compare the flow strengths at a given strain rather than yield
due to general uncertainties in measuring initial yield). As an example of the differ= 1.5 if
ences between testing techniques for estimating strength, one obtains σσCh
T
the compressive strength (σCh ) is estimated using the approximation that H ≈ 3σCh
and using Wang’s measured microhardness.
Why is there a tension-compression asymmetry in nanomaterials? Cheng et al.
(2003) addressed compression-tension asymmetry using a strength model based on
dislocation emission from grain boundary sources for various regimes of grain size,
and relating the uniaxial strength of materials to the self energy of a dislocation
under pressure. Lund and Schuh (2005) addressed compression-tension asymmetry across a broad range of grain sizes using MD simulations and analysis. Jiang
and Weng (2003) present analytical solutions for Cu using a weighted average of a
crystalline phase (with grain size dependent mechanical properties) and an amorphous grain boundary (with size independent properties). In these analyses, the
volume fraction of the grain boundaries increases with decreasing grain size and
thereby affects the overall response. For example, the analysis applied to Cu shows
increasing compression-tension asymmetry with decreasing grain size. The results
on nanocrystalline nickel obtained from microcompression and comparison with
the data of Wang et al. closely match the predictions of Jiang and Weng of the
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compression-tension asymmetry (≈ 1.2) for Cu at a similar grain size. However,
high-resolution electron microscopy of the nickel does not show the presence of an
amorphous phase at the grain boundary.
Given this compression-tension asymmetry, what would be a reasonable yield
surface? One simple yield surface that can incorporate separate tensile and compressive behaviors is the Mises-Schleicher yield surface, defined by
f (σ ) = σ̄ 2 + (σC − σT )σkk − σC σT = 0,
(8.36)
where σ̄ is the effective stress defined by Equation (2.115) and the yield function
f (σ ) was first discussed in the paragraphs following Equation (2.96). The MisesSchleicher yield criterion takes into account the difference between yield strengths
in tension and compression as well as the effect of the hydrostatic stress σkk . In the
special case that σC = σT , Equation 8.36 reduces to the traditional von Mises yield
criterion (Equation 2.114).
Unfortunately, there is not enough data in other multiaxial states (e.g., in torsion)
to be able to determine a full yield surface at this point, and so we cannot state
that the Mises-Schleicher yield criterion is the appropriate one for nanocrystalline
nickel. This is an area where additional experimentation is sorely needed in other
stress states, but these are difficult to come by (in part because of the paucity of bulk
nanocrystalline materials). Some suggestions for yield surfaces come from simulations, but these are still too heavily constrained to be relied upon for engineering
purposes with respect to yield (remembering that the simulations are typically nonconservative, predicting unrealistically high stresses).
8.11 Closing
This chapter has discussed a wide range of modeling approaches to the problem of
the mechanics of nanomaterials, but the reader should recognize that this is a field
in explosive growth, and much continues to be done to push the boundaries of our
modeling capabilities. Certainly much remains to be done. We are not at the point,
as of this writing, where we can identify specific constitutive laws for nanocrystalline materials, or specify with certainty that one particular material, made in a
particular way, is better than another for a specific engineering application. We do
have excellent measures of strength, however, and these indicate great promise.
It appears clear that dramatic progress in nanomaterials can be made through a
coordinated effort involving processing research (for improving consistency of materials and reducing defect populations), sophisticated modeling (to tell the producers what is worth making and where it is best to commit effort and investment) and
mechanical and materials characterization experiments over a wide range of length
and time scales (to determine true material behavior, remembering that our intuition
is not reliable over this range of scales).
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8.12 Suggestions for Further Reading
1. Phillips, R.W. (2001). Crystals, Defects and Microstructures, Cambridge
University Press, Cambridge.
2. Weiner, J.H. (1983). Statistical Mechanics of Elasticity. Wiley, New York.
3. Rapaport, D.C. (2004). The Art and Science of Molecular Dynamics Simulation.
Cambridge University Press, Cambridge.
4. Liu, W.K., E.G. Karpov, et al. (2004). An introduction to computational nanomechanics and materials. Computational Methods in Applied Mechanics and
Engineering 193: 1529–1578.
5. Curtin, W.A. and R.E. Miller (2003). Atomistic/continuum coupling in computational materials science. Modeling and Simulation in Materials Sciences and
Engineering 11: R33–R68.
6. Van Swygenhoven, H. and J.R. Weertman (2006). Deformation in nanocrystalline metals. Materials Today 9(5): 24–31.
7. Liu, W.K., E.G. Karpov, et al. (2006). Nano Mechanics and Materials: Theory,
Multiscale Methods and Applications. Wiley, New York.
8.13 Problems and Directions for Future Research
1. Consider the development of a molecular dynamics simulation of the simple
tension problem (using, for example, LAMMPS) for a single crystal of an fcc
metal – say nickel – loaded at a constant high strain rate. Find an EAM potential for nickel in the literature, and implement the parameters in your MD
code. Decide what temperature you are going to use for your simulations. Next,
choose which orientation, e.g., [110] to use for the tensile loading axis. Determine what simulation size you can manage with your computing system. How
will you make this simulation sample representative of the larger experimental sample? Can you use periodic boundary conditions? If you do, how will the
traction boundary conditions be implemented (tensile strain rate imposed along
the long axis, traction-free along the sides)? What does a traction-free boundary
mean for a collection of atoms?
Construct the single crystal sample for your simulation, and ensure that you have
relaxed your structure. Determine the meaning of the following terms for your
simulation: thermostat and barostat. How will you impose the constant strain rate
loading that you would like to apply? Which atoms will have imposed velocities?
Compute the stress histories that you obtain from your simulations. How are
these stresses defined? What are virial stresses? What is the volume over which
you average to obtain the stresses? Do you average over a number of timesteps
as well as a volume? If so, how do you decide how many timesteps to average
over to compute the stress state? Did you achieve your desired uniaxial tensile
stress state, and if not, what is the magnitude of the error?
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What are your choices for visualizing the atomic motions? Can you distinguish
atoms that are in irregular lattice positions (such as around dislocation cores)?
2. A common assumption in dislocation dynamics codes is that the force on the
dislocation is proportional to the dislocation velocity (the so-called dislocation
drag regime, see Figure 5.19). Using the Orowan equation (Equation 5.31) and
Figure 5.19, compute the plastic strain rate at which this becomes a reasonable
assumption in a metal such as lead. Note that the typical mobile dislocation density in a metal is in the range of 1013 –1015 m−−2 . 3. Find a downloadable crystal
plasticity code that you can use in conjunction with a finite element program. A
common example is a “user-material subroutine“ or UMAT for use in conjunction with the commercial finite element software known as ABAQUS (Simulia)
and written by Huang & Kysar. Use this code to simulate the deformation of
a microcompression pillar (see Chapter 3) of copper. You will need to obtain
some properties for the slip systems and hardening rates in the copper from the
literature.
References
Ashby, M. (1970). Deformation of plastically non-homogeneous materials. Philosophical Magazine 21(170), 399.
Bishop, J. F. W. and R. Hill (1951a). A theoretical derivation of the plastic properties of a polycrystalline face-centred metal. Philosophical Magazine 42(334), 1298–1307.
Bishop, J. F. W. and R. Hill (1951b). A theory of the plastic distortion of a polycrystalline aggregate
under combined stresses. Philosophical Magazine 42(327), 414–427.
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Index
A
Activation volume, 163, 167–168, 169
Anisotropic, 13, 37, 38, 39, 47, 111, 113, 115,
131, 153, 237, 243, 278
Atomic force microscope (AFM), 1, 63, 64,
68–69, 91, 263, 292
B
Baseline strengths, 133
Basis, 4, 5–6, 21, 23–24, 31–32, 33, 57, 62, 69,
82, 106–107, 122–128, 131, 133, 139,
146, 156–172, 268, 289
Berkovich, 74, 75, 261, 293
Biology, 1
Body centered cubic (bcc), 13, 40, 95, 128
Bottom-up, 6–7, 8
Brittle, 57, 71, 82, 86, 179, 181, 188, 189, 198
Buckling, 62, 78, 80, 81, 216, 222, 223, 224,
257, 287
Bulk modulus, 41, 42, 106
Bulk nanomaterials, 3, 5, 6, 7, 8, 9, 11, 17, 18,
107, 117–118, 121, 190, 197, 228–256,
269, 292
Burgers vector, 125–126, 127–128, 143, 155,
173, 234
C
Carbon nanotube, 1, 3, 6, 62, 91, 109, 222–225,
257, 268, 274, 281, 285–286, 292
Cartesian basis, 21–22, 31
Cauchy stress tensor, 31, 33–34, 49, 50
Cells, 1, 7, 12, 157, 265, 266
Ceramics, 5–6, 9, 11, 38, 42, 57, 75, 80, 88,
107, 117, 121, 179, 181, 182, 194, 195,
197, 198, 245, 280
Characteristic length scale t, 98–99
Chemical vapor deposition (CVD), 7, 8
Coalescence, 188, 193, 196, 197, 198, 201,
243
Cold isostatic pressing, 9
Colloidal phases, 7
Columnar grains, 9
Columnar microstructure, 111, 113, 116, 212
Compact tension, 198
Components, 22, 23–24, 26, 27, 28–29, 31–32,
33, 37, 38–39, 41–42, 43, 47, 51, 53, 56,
67, 68, 73, 76, 86, 97, 112, 128, 130,
170, 172, 176, 180, 216, 220, 232, 288
Composite, 6, 9, 36, 69, 96–97, 109, 111–117,
118, 145, 193, 197, 220, 221, 226, 279,
289
Condensation, 7
Considere criterion, 185, 186–187
Consolidation, 9–11
Constitutive relation, 35–57, 166
Control, 1, 3, 6, 7, 8, 10, 16, 17, 63, 67, 83–84,
125–126, 182, 185, 196, 197, 207, 215,
222, 227, 228
Core-shell, 216, 217, 220, 221
Corner junctions, 99, 100, 103, 104
Crack, 54, 55, 56, 57, 86, 182, 193, 194, 195,
199, 201
Crazing, 193
Creep, 87, 124, 168, 236, 292
Critical resolved shear stress (CRSS), 130,
131, 139, 144, 157, 176, 278
Crystallography, 128
Crystal morphology, 99
Crystal structure, 13, 65, 66, 69–70, 99, 101,
128, 129, 130, 132, 133, 152, 217, 251
Cup and cone, 189
Cup and cone fracture, 188
D
Defects, 8, 16, 49, 72, 100, 101, 105, 109,
118, 123–124, 125–126, 155, 162, 185,
190–191, 192–193, 194, 196, 197, 200,
222, 224, 264, 265, 276, 287
Deposition techniques, 8–9
Desktop kolsky bar, 89, 90
Determinant, 32–33
Deviatoric part, 33, 34, 41
Deviatoric stress, 34, 49, 53, 196
Digital image correlation, 70, 83, 84, 86, 183
Discrete nano materials, 107
Dislocation core, 127, 172, 228, 229, 264
Dislocation drag, 161, 162, 165, 277
Dislocation dynamics, 136, 160–162, 263,
277–278
311
312
Nanomaterials
Dislocation loops, 127, 139, 140, 172
Dislocation pile-up, 139–140, 141
Dislocations, 12, 14, 16, 49, 65, 123, 124,
125–126, 127, 133, 134, 139, 140, 141,
142, 151, 154, 156, 157, 160, 162, 164,
166, 169, 173, 199, 200, 207, 224, 227,
228, 229, 230, 232, 233–234, 235, 257,
263–264, 265, 266, 276, 277, 278, 279,
287, 288
Dislocation velocity, 160, 161, 164, 173
Dispersoids, 131, 144, 145–146, 156, 229, 230
Dispersoid strengthening, 134
Displacement, gradient, 27, 28, 36–37,
DNA, 1, 225
Dogbone, 43, 71, 82
Drag coefficient, 162
Ductile, 179, 181, 188, 189, 190, 193, 196,
198, 210
F
E
G
Ecae, 12, 14, 251
ECAP (Equal Channel Angular Pressing), 12,
13, 15, 16, 18, 145, 167, 168, 171
Edge, 172, 207
Edge dislocation, 123, 124, 125, 126, 172, 234,
279
Effective elastic properties, 112, 113
Effective grain size, 101, 102, 152, 153
Effective modulus, 114, 115, 117, 221
Effective stress, 51, 53, 164, 173, 188, 288,
293
Eigenvalues, 32, 33, 49, 50
Eigenvectors, 32–33
Elasticity, 36–43, 48, 52, 53–54, 57, 91,
95–118, 127, 137, 175, 182, 220, 221,
277, 283, 284–285, 292, 295
Elastic modulus tensor, 37, 38, 220
Elongation to failure, 181, 187, 190–191, 192,
193
Energy, 6, 34, 35, 53, 57, 58, 64, 65, 66,
106–107, 110, 123, 159, 160, 162, 163,
172, 173, 195–196, 198, 207, 219, 220,
221, 228–229, 234, 235–236, 268, 270,
271–272, 274, 281–282, 284, 293
Engineering strain, 44, 45, 46, 156, 168, 180,
181, 183
Engineering stress, 44–45, 46, 156, 180, 181
Equal channel angular processing, 12–14
Equilibrium, 29–30, 42, 58, 83, 88–89, 106,
107, 108–109, 110–111, 157–158,
183–184, 217, 218, 226, 228–229,
237–238, 252–253, 272, 282, 289
Equivalent homogenized material, 112
Geometrically necessary dislocation, 141, 142,
151–152, 287, 288
Glide plane, 126, 127
GND (geometrically necessary dislocation
density), 141, 142, 148
Grain aspect ratio, 103, 138
Grain boundaries, 5, 12, 13, 14, 16, 61, 64, 65,
66, 69, 96, 97, 99, 103–104, 105, 110,
111, 112, 113, 114, 115, 131, 138, 140,
141, 142, 146, 151–152, 169, 170, 174,
198, 229, 234, 235, 236, 237, 243, 244,
275, 276, 277, 279, 280, 289, 293
Grain boundary ledge, 140–141, 146
Grain boundary limit, 98, 102, 103
Grain boundary thickness, 1, 97, 98, 99, 105,
245, 250
Grain
growth, 10, 122, 192–193, 228, 236
morphology, 100, 101–103, 104, 118, 154
rotation, 149, 214, 236–256, 291
shape, 138, 150, 152–153, 154
Grain size, 5, 8–9, 10, 11, 12, 13, 15–16, 42,
43, 45, 46, 52, 57, 64, 66, 69, 74, 85,
86, 89, 95–96, 97, 98–99, 100, 101, 102,
103, 104–105, 117, 118, 119, 120, 132,
134, 138, 141, 143, 144, 146–147, 152,
153, 154, 157, 158, 168, 169–170, 171,
174, 175, 181, 182, 185, 188, 190–191,
192, 193, 194, 200, 201, 203, 204, 207,
208, 210, 211, 216, 227, 228, 229,
230–231, 233–234, 236–237, 241, 243,
245, 247, 249–250, 254, 255–256, 266,
269, 276–277, 278, 289, 292, 293
Face-centered-cubic (fcc), 13, 40, 128, 129,
176, 275
Failure processes, 179–212, 216, 257, 280
Field equations, 35, 268
Finite element model, 75, 77, 280
Force, 12, 13, 22, 23, 29–34, 43, 44, 58, 62,
66–67, 68–69, 72–73, 74, 76, 77, 82,
83, 84, 91, 107, 108, 113–114, 135,
139–140, 146, 151, 160, 183–184, 195,
224, 232, 241, 243, 274
Forest dislocation strengthening, 135–136
Fracture mechanics, 53–57, 195, 197–198
Fracture strain, 181–182, 185, 188–189, 190
Fracture stress, 181
Fracture toughness, 56–57, 61, 86, 195,
197–198
Friction stress, 133, 172
Index
distribution, 8–9, 138, 150–152, 154, 191,
193, 212, 237, 241, 254, 277, 292
ratio, 98, 99, 206
Growth, 3, 6, 7, 12, 17, 57, 60, 123–124,
186–187, 188–189, 191–192, 193,
195–197, 198, 199, 201, 206, 210, 212,
228, 231, 232, 235, 237, 247, 252, 294
313
K
Kinematics, 25–29, 34, 123, 283
Kink band, 193
Kink pair nucleation, 172–173
Kolsky bars, 87–90
L
H
Hall-Petch, 138–150, 151, 153, 154, 157, 170,
172, 249–250, 276, 278, 292
Hall-Petch breakdown, 146, 148
Hardening, 43, 45–46, 136, 152, 154, 193, 202,
249, 250, 251, 278
Hart criterion, 187, 188
Hertz, 72
Heterogeneous material, 112–113
Hexagonal close-packed (HCP), 13, 128, 129,
132, 137, 152, 154, 157, 201, 211, 230,
232
Hexagonal prism morphology, 101, 102, 103
Homogeneous, 106, 183, 195, 204, 252, 253,
264
Homogenization, theory, 111–117, 240
Homologous temperature, 122, 127, 173
Hot isostatic pressing, 10
HPT (High-pressure torsion), 14, 15, 16
Hydrostatic part, 33, 34, 49
Hydrostatic pressure, 34, 188
I
Identity tensor, 24, 32, 283
Image force, 127, 230
Inclusions, 124, 194, 202, 266, 287
Infinitesimal rotation, 27, 37
Infinitesimal strain, 27, 28, 29, 37, 38, 283
In situ deformation, 63, 68–71
Interatomic, 42, 95–96, 98, 109, 217, 218, 219,
222, 269–270, 272, 284
Interatomic potential, 106, 107–108, 109, 127,
218, 261, 262, 270, 274, 284–285
Intermolecular, 269–270
Internal forces, 29, 30
Internal length scale, 105, 216, 217, 222, 225,
227, 229, 264, 278, 288, 292
Interstitial defects, 124
Invariant, 32–33, 49
Isotropic, 38, 40, 41, 42, 47, 48, 49, 106, 111,
113, 114, 115, 116, 121, 139
J
J2 flow, 53
Lamé moduli, 40, 106
Lattice spacing, 1, 107, 108
Layered microstructure, 113
LEFM (Linear Elastic Fracture Mechanics),
54, 55
Lennard-Jones, 261, 272
Line vector, 125–126, 127
Localization, 148, 182, 193, 197, 201–202,
203, 237, 238, 239, 247, 249, 268
M
Materials, 6–7, 9–11, 57, 65–66, 68, 69–70,
74, 75, 79–80, 81, 82, 84, 85, 86, 87, 115
Mean stress, 33, 188
Mechanical attrition, 7, 10, 16
Mechanical design, 57, 179, 180
Mechanical properties of, 10, 11, 61, 74,
87–88, 95, 118, 167, 217, 222, 226, 234
Mechanics, 1, 2, 3, 17, 21–58, 61–91, 96, 106,
108, 115, 116, 118, 124, 127, 130, 155,
166, 196, 197, 203, 219, 221, 222, 224,
227–228, 232–233, 237, 242, 261, 262,
263, 264, 267, 269–273, 276, 279, 281,
283, 284, 287, 289, 292, 295
Metals, 1, 5, 6, 10, 11, 12, 13, 17, 38, 42, 43,
46, 49, 51, 52, 57, 75, 80, 88, 98, 107,
121, 122, 123, 124, 125, 127–132, 133,
138, 146, 151, 154, 157, 166, 167, 169,
170, 171–172, 173, 174, 175, 182, 184,
190, 191, 192, 195, 201, 203–207, 208,
210, 226, 230, 232, 244, 245, 251, 254,
273, 277, 288
Microcompression, 74–81, 293, 296
Microposts, 75
Microscale, 3, 61, 68, 83, 84, 188, 190, 200,
266
Microtensile, 82–86
Microvoid, 188–189, 190, 194
Miller indices, 128, 129, 176
Mixed type, 127
Mobile dislocation density, 155, 173
Moduli, 37, 38, 39, 40, 42, 47, 107, 109, 110,
111–112, 113, 114, 115, 116, 118, 137,
144, 146, 242, 285, 292
314
Molecular dynamics, 107, 224, 231, 234, 244,
262–263, 264, 270, 273, 274–277, 279,
290, 291
Monoclinic, 38
Multiscale modeling, 263, 268, 289–292
N
Nanobelts, 109–110
Nanocrystalline, 5, 8, 10, 11, 12, 13, 16, 46,
52, 57, 58, 63, 65, 68, 70, 72, 73, 74, 75,
83, 85, 91, 95, 97, 100, 101, 105, 117,
118, 140, 147–148, 149–150, 151, 152,
153, 154, 157–160, 169, 170, 172–175,
181–182, 185, 188, 190, 191, 192–193,
194, 198, 199, 200, 201, 203–207, 209,
211, 215, 228, 229, 230–231, 233–234,
235–236, 237, 241, 246, 273, 275, 277,
278, 279, 289, 290, 291, 292, 293, 294
Nanoindentation, 72–74, 111, 261, 287
Nanoindenter, 73, 74–75, 261, 293
Nanomaterials, 1–18, 57, 61–91, 106–111,
116, 117–118, 121–176, 179–212,
215–257, 261–296
Nanoparticles, 3, 6, 7, 9, 17, 107–109,
215–222, 226, 269
Nanopowders, 7, 9, 143
Nanoscale, 1, 3–5, 6, 7, 8, 61–91, 222, 226,
267, 268–269
Nanoscience, 1, 3, 62
Nanostructured, 5–6, 10, 17, 89, 117, 152, 171,
190, 192, 193, 194, 204, 215, 246–247,
251
Nanotechnology, 1, 3, 6, 7, 9, 63, 225, 227–228
Neck, 44, 46, 159, 160, 183–187, 188–189,
190, 200, 211, 216, 224, 225, 287
Network dislocation, 140–141, 146
NMCT (nanoscale mechanical characterization
techniques), 61
NMT (nano mechanics techniques), 61, 63
Normalized Hardening, 158
Nucleation, 7, 17, 123, 124, 172, 173, 188,
190, 193–197, 200, 205, 210, 231, 232,
233, 234, 277
O
Orientation distribution function, 138
Orowan equation, 161, 173
Orowan mechanism, 134, 135
Orowan strengthening, 134, 144
Orthonormal, 23, 32, 128
Orthotropic, 38
Overall density, 96, 97, 100, 104, 105
Overall stress, 112, 195, 233, 247, 249
Nanomaterials
P
Pair potential, 106, 272, 273, 274, 282
Peierls-Nabarro strength, 133, 172
Perfectly plastic, 45, 53, 159
Phonon drag, 161
Physical, 1, 8, 25, 29, 34, 42, 47, 48, 56, 63,
106–107, 122–128, 131, 156–172, 175,
247, 251, 254, 261, 262, 263, 266, 267,
268, 270, 276, 278, 286
Physics, 1, 34, 35, 36, 48, 49, 60, 66, 67,
68, 93, 106, 118, 205, 211, 264, 266,
267–268, 276, 277, 291
Plane strain, 54, 55, 56, 57, 197, 289
Plastic
dissipation, 121
flow, 45, 79, 141, 228, 248
strain, 45, 46, 47, 52, 53, 121, 131, 136, 137,
140, 154–155, 156, 158, 160, 161, 164,
168, 183, 203, 204, 205, 206, 238, 239,
247, 252, 253, 278, 287, 288
Plasticity, 34, 43, 46, 47, 48, 49, 52, 53, 57, 72,
119, 125, 128, 131, 136–153, 160–171,
172, 175, 177, 182, 189, 202, 204, 224,
228, 237, 239, 244, 245, 256, 264, 269,
278–279, 287–288, 291
Plate morphology, 102
Point defects, 124, 264, 265
Poisson’s ratio, 38, 40, 41, 42, 43, 76, 79, 106,
111, 139, 175, 242, 247
Polycrystalline, 1, 5, 42, 70, 95, 96, 99, 100,
102, 104, 105, 110, 111, 113, 115, 116,
131, 132, 136, 137, 138, 140, 141, 142,
144, 146, 150, 151, 152, 157, 168, 229,
237, 239, 242, 244, 247, 249, 278,
279–280, 288
Polycrystal plasticity, 136–154
Polymers, 5, 88, 109, 121, 193, 197, 225, 226
Pores, 10, 100, 101, 105, 115, 124, 192, 194,
214, 226, 227, 265, 266
Porosity, 9, 10, 11, 100, 101, 118, 192, 194,
227
Position, 21, 22, 26, 28, 31, 67, 73, 95, 106,
107, 124, 126, 162, 163, 183, 252, 270,
281, 283
Precipitate strengthening, 135, 145
Precipitation, 7, 146
Principal directions, 32, 47, 58, 116
Principal stresses, 32, 47, 48, 49, 50, 51, 58
Proportional limit, 44
PVD, 8
Q
Quasicontinuum, 289, 290
Index
R
Rate of deformation, 28, 160, 166
Rate-dependent plasticity, 160–172
Rate-sensitivity, 168, 169, 171, 175, 186–188,
190
Recovery, 12, 127, 157
Recrystallization, 12, 14, 16, 207, 208
Reference configuration, 25, 26, 31, 219, 283,
284
Representative volume element (RVE), 112,
239, 240, 245
Resolved shear stress, 130, 139, 278
Rod morphology, 102
Rolling, 10, 17, 102, 157, 222, 241
Rotation, 13, 14, 15, 25, 26, 27, 29, 37, 131,
149, 208, 216, 228, 236–256, 292
Rule of mixtures, 96–101, 105, 114, 115
S
SAD (A selected area diffraction), 13, 14, 16,
65, 207
Scanning electron microscopy, 64–65
Scanning Probe Microscopy (SPM), 62, 63,
64, 66–68
Screw dislocations, 124, 126, 172, 234
Self-assembly, 7
Self stresses, 127
Semiconductor fabrication, 4, 8, 9
Severe plastic deformation, 12, 15, 16, 17, 147,
152, 158, 169, 247
Shear band, 12, 193, 194, 195, 201–211, 237,
238, 239, 247, 249, 250, 251, 253
Shear modulus, 41, 42, 79, 106, 123, 133, 139,
141, 143, 161, 234, 238, 242, 247
Simple tension, 43, 44, 48, 51, 58, 159, 180,
181, 183, 185, 188, 189–190, 198
Sintering, 10
Slip
plane, 123, 125, 126, 127, 129, 130, 131,
132, 136, 155
system, 126, 129, 130, 131, 132, 136, 139,
141, 142, 155, 156, 161, 208, 264, 276,
278
SMAT (surface mechanical attrition treatment),
16–17
Solute strengthening, 133
Spallation, 1
SPD (severe plastic deformation processes),
12, 14, 16, 17, 152, 247
Split-hopkinson pressure bar (SHPB), 87, 88
Stacking faults, 124, 228, 229, 230, 232, 234,
265, 266, 276
315
Strain energy, 34, 36, 107, 195, 220, 283
Strain energy function, 37, 38, 39
Strain gages, 78, 88
Strain gradients, 17, 141, 216, 237, 245, 264,
287–289
Strain hardening, 58, 76, 78, 79, 81, 85, 122,
136, 156, 157, 158–160, 166, 174, 183,
184–186, 187, 189, 190, 191, 193, 196,
204, 210, 247, 248, 249, 251, 288
Strain hardening index, 46, 53, 121, 158, 174,
185, 239, 247
Strain rate, 28, 29, 46, 58, 76, 79, 80, 87, 88,
89, 90, 91, 121, 122, 160, 161, 162, 164,
165, 166, 167, 168, 170, 174, 175, 186,
187, 200, 203, 204, 210, 230, 232, 233,
237, 238, 239, 244, 247, 248, 250, 252,
255, 274, 276, 277, 293
Strain rate hardening index, 121
Stress
history, 47, 52
intensity factor, 56
space, 47, 48, 49, 58
tensor, 23, 31, 32, 33, 34, 36, 37, 41, 47, 49,
50, 127, 130, 284
Subgrains, 12, 157, 171
Substitutional defects, 124
Substrate, 110, 111, 143, 216
Summation convention, 22, 24, 33, 37, 40, 50,
220
Surface mechanical attrition treatment, 16–17
Surface tension, 108
T
Taylor averaging, 137
Telescoping, 216, 222, 224, 225
Tensor, 21–25, 27, 28, 29, 31, 32, 33, 34, 36,
37, 38, 39, 41, 46, 47, 49, 50, 127, 130,
155, 161, 176, 220, 221, 283, 284
Tensor analysis, 21
Textured, 131, 137, 157, 208, 278
Thermal activation, 160, 162–165, 169, 172
Thermal softening, 121, 122, 202, 203, 268
Thermodynamics, 34, 35, 219, 235
Thin films, 3, 4, 5, 8, 9, 68, 69, 70, 74, 82, 85,
86, 110, 111, 113–116, 143, 193, 200,
201, 216, 226, 227, 230
Trace, 33, 34, 49, 50, 125, 136, 206
Traction, 23, 29–34, 70, 130, 188, 240, 241,
242
Transformation, 32, 56, 155
Transformation tensor, 155
Transition thickness, 110
316
Transmission electron microscopy, 62, 64,
65–66, 70, 207
Transversely isotropic, 38, 39
Triaxiality, 188
Triple junctions, 99, 100, 102, 103, 104, 147,
149, 152, 154, 199
True stress, 43, 44, 45, 46, 76, 85, 89, 158,
159, 183, 184, 189, 233
Twins, 65, 124, 230–235, 265, 266
Two-step process, 7
Nanomaterials
156, 157, 162, 163, 165, 167, 168, 169,
171, 187, 190, 192, 194, 195, 199, 200,
201, 202, 211, 230, 234, 237, 243, 247,
274, 278, 281, 285
Workability, 12, 15, 17
X
X-ray diffraction, 10, 64, 66
U
Ultimate strength, 46
Ultra-fine-grained, 17
V
Vacancies, 124, 264
Vapor deposition, 7, 8
Velocity, 22, 28, 75, 160, 161, 164, 173, 186,
210
Velocity gradient, 28, 29, 186
Voids, 193, 195, 196, 198, 199, 200, 201, 206
Voigt notation, 38, 111
Volume fraction, 96, 97, 98, 99, 100, 102, 103,
104, 111, 114, 115, 116, 134, 135, 149,
152, 154, 217, 218, 219, 227, 235, 293
Von mises, 51, 52, 58, 76, 78, 121, 161, 294
Y
Yield criterion, 49, 50, 51, 52, 53, 57, 58, 294
Yield strain, 46, 239
Yield strength, 14, 44, 46, 49, 50, 51, 52, 58,
59, 83, 88, 119, 120–128, 131, 132, 136,
138, 141, 143, 146, 147, 148, 149, 150,
151, 152, 154, 156, 165, 175, 177, 179,
182, 190, 191, 192, 210, 211, 239, 249,
278, 292, 293
Yield surface, 48, 49, 51, 52, 58, 293–294
Young’s modulus, 38, 40, 41, 43, 44, 45, 46,
76, 83, 89, 91, 106, 109, 111, 114, 115,
116, 117, 118, 175, 285
W
Z
Work, 11, 12, 13, 16, 18, 34, 35, 36, 43, 51, 58,
70, 71, 85, 120, 121, 125, 145, 146, 153,
Zero tensor, 24, 27