Biomedical Engineering Reference
In-Depth Information
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