РАЗРАБОТКА МАТЕМАТИЧЕСКОГО ОПИСАНИЯ ВНУТРЕННЕГО

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A.A. Zaborskikh, V.M. Tverier
Perm National Research Polytechnic University
DEVELOPMENT OF MATHEMATICAL DESCRIPTION
OF THE INTENAL STRUCTURE OF CANCELLOUS BONE
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,
.
;
(fabric tensor).
,
.
:
,
(
,
)
,
,
,
.
Consideration of structural features of the dentofacial system units is one of the main problems of
contemporary dental biomechanics. Heterogeneity of spongy structure can be described by methods
of quantitative stereology. At the same time, structural bone tissue features are described by means of
the fabric tensor . The measuring procedure for stereological investigations is analyzed. Properties
of fabric tensor by constructing it for a sample of cancellous bone, carved out of the femoral
head of human was investigated.
Keywords: biomechanical modelling, Wolff’s law, bone tissue structure, cancellous bone tissue,
fabric tensor, ellipse structure.
200
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f 2 (1 − cos ψ1 ) − f1 (1 − cos ψ 2 )
  ψ + ψ2 

+ κ  ctg  1
 (1 − cos ψ1 ) − sin ψ1  , (2)
cos ψ 2 − cos ψ1
  2 

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f1 (1 + cos ψ 2 ) − f 2 (1 + cos ψ1 )
  ψ + ψ2 

− κ  ctg  1
 (1 + cos ψ1 ) + sin ψ1  , (3)
cos ψ 2 − cos ψ1
2
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def
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(4)
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f 3 − k ⋅ f 2 + f1 ( k − 1)
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-
cos ψ 3 − cos ψ1
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cos ψ 2 − cos ψ1
def
k=
sin ( ψ1 − ψ 2 ) + sin ( ψ 2 − ψ 3 ) + sin ( ψ 3 − ψ1 )
≠ 0.
2 f1 f 2 f 3
:
,
1
= 0°,
2 = 120°
3
= 240°.
(2)–(4)
:
mαα = f1 ,
mββ =
1
( 2 ⋅ ( f 2 + f3 ) − f1 ) ,
3
3
( f3 − f 2 ) ,
3
mαβ =
k = 1.
f1 > 0,
(
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2 ( f1 f 2 + f 2 f3 + f1 f3 ) > f12 + f 22 + f32 .
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165
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24.
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.,
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//
18.10.2012
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-11, e-mail: kichenko.alex@yandex.ru.
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168
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, -mail: tverier_55@perm.ru.
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