Biomedical Engineering Reference
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ð 4 Þ
where the fourth order material tensor (material tetrad) C
ð 3 : 277 Þ
ð 4 Þ
ð 4 Þ
ð 8 Þ
ð 8 Þ
: ¼ 1
4
ð 8 Þ
ð 8 Þ
ð 8 Þ
ð 8 Þ
ð 8 Þ
ð 8 Þ
ð 8 Þ
ð 8 Þ
C
C
: ¼ S
mit
S
I 1
þ I 2
þ I 10
þ I 53
þ I 54
þ I 68
þ I 69
þ I 89
ð 4 Þ
results from the material tensor C
defined in ( 3.216 ) 2 by a pre-operation with the
ð 8 Þ
. The latter ensures the symmetry of both
eighth order symmetry operator
S
G and P II .
For orthotropic materials, considering ( 3.221 ) 2 , the material tensor ( 3.277 )
transforms to the following fourth order tensor function of the three direction
tensors M i (i = 1, 2, 3):
þ X
3
ð 4 Þ
ð 4 Þ
ð 4 Þ
C
: ¼ l 1 II þ l 2
I 1
þ I 2
l i þ 2 IM i þ M i I
ð
Þ
i ¼ 1
M i þ X
þ X
3
3
X
3
ð 6 Þ
ð 6 Þ
ð 6 Þ
ð 6 Þ
l i þ 5
I 2
þ I 3
þ I 11
þ I 12
l ij M i M j þ M j M i
i ¼ 1
i ¼ 1
j ¼ 1
M i M j
M i M j
þ X
3
X
3
þ I 56
ð 6 Þ
ð 6 Þ
ð 8 Þ
ð 8 Þ
m ij
I 2
þ I 14
þ I 83
i ¼ 1
j ¼ 1
ð 3 : 278 Þ
where the underlined term in ( 3.278 ) indicates the particular case of isotropy.
Substituting ( 3.278 )in( 3.276 ) leads to the equivalent representations of the
second P IOLA -K IRCHHOFF stress tensor as follows (the isotropic part is underlined)
P II ¼ l 1 tr ð I þ 2l 2 G þ X
3
l i þ 2 tr M i G
½
ð
Þ I þ tr ð M i
i ¼ 1
þ X
X
3
3
M i
l ij tr M i G
ð
Þ M j þ tr M j G
i ¼ 1
j ¼ 1
ð 3 : 279 Þ
þ 2 X
3
l i þ 5 M i G þ G M i
ð
Þ
i ¼ 1
þ 2 X
3
m ij M i G M j þ M j G M i
i ¼ 1
Together with ( 3.220 ) 2 , the following identities hold
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