Biomedical Engineering Reference

In-Depth Information

2G
I
G
I
þ
I
G
I
¼
2
X

3

X

3

M
i
G
M
j
þ
M
j
G
M
i

i
¼
1

j
¼
1

Þ
I
¼
X

X

Þ
I
¼
X

3

3

3

tr
ð
I
¼
trI
G

ð

ð

trM
i
G

Þ
M
j
;

ð

trM
i
G

ð

trM
i
G

Þ
M
j

i
¼
1

j
¼
1

j
¼
1

Þ
M
i
¼
X

3

M
i
;
M
i
G
M
i
¼
trM
i
G

tr
ð
M
i
¼
trI
G

ð

trM
j
G

ð

Þ
M
i

j
¼
1

M
i
G
¼
M
i
G
I
¼
X

M
i
G
M
j
;
G
M
i
¼
I
G
M
i
¼
X

3

3

M
j
G
M
i

j
¼
1

j
¼
1

ð
3
:
280
Þ

whereby (
3.279
) can be transformed in the following form with nine independent

material coefficients u
i

P
II
¼
u
1
trM
1
G
þ
u
2
trM
2
G
þ
u
3
trM
3
ð Þ
M
1

þ
u
2
trM
1
G
þ
u
4
trM
2
G
þ
u
5
trM
3
G

ð

Þ
M
2

þ
u
3
trM
1
G
þ
u
5
trM
2
G
þ
u
6
trM
3
G

ð

Þ
M
3

þ
u
7
M
1
G
M
2
þ
M
2
G
M
1

ð

Þþ
u
8
M
1
G
M
3
þ
M
3
G
M
1

ð

Þ

þ
u
9
M
2
G
M
3
þ
M
3
G
M
2

ð

Þ:

ð
3
:
281
Þ

The material equation (
3.281
) differs from the form presented in Boehler (1975,

1979) by the latter three terms in parentheses, which are identified by M
i
G
þ

G
M
i
(i = 1, 2, 3). They can, however, be transformed into each other employing

(
3.280
)
4
-(
3.280
)
7
.

For transversal isotropic materials, the material tensor (
3.277
) converts to the

following fourth order tensor function of the single direction tensor M using

(
3.224
)
2

þ
l
3
IM
þ
MI

ð
4
Þ

ð
4
Þ

ð
4
Þ

C

:
¼
l
1
II
þ
l
2

I
1

þ
I
2

ð

Þþ
l
4
MM

ð
3
:
282
Þ

M
:

ð
6
Þ

ð
6
Þ

ð
6
Þ

ð
6
Þ

þ
l
5

I
2

þ
I
3

þ
I
11

þ
I
12

In (
3.282
), the underlined term represents the particular case of isotropy.

Substituting (
3.282
)in(
3.276
) leads to the second P
IOLA
-K
IRCHHOFF
stress tensor

with five independent material coefficients u
i
(the isotropic part is underlined)

I

P
II
¼
u
1
trG
þ
u
2
trM
G

ð
3
:
283
Þ

þ
u
2
trG
þ
u
3
trM
G

ð

Þ
M
þ
u
4
G
þ
u
5
M
G
þ
G
M

ð

Þ: