Biomedical Engineering Reference
In-Depth Information
h
i
þ
f
ðÞ:
Þþ
k
1
2k
2
e
k
9
C
I
ð Þ
2
w
C
I
3
ðÞ¼
c
1
ð
1
ð
3
:
233
Þ
Correlation between (
3.226
) and (
3.233
): In the isotropic case, the fourth
ð
4
Þ
order material tensor A
in (
3.227
) is replaced by the group of fourth order iso-
tropic tensors according to
¼
X
3
ð
4
Þ
ð
4
Þ
ð
4
Þ
¼
!
A
J
k
j
I
j
ð
3
:
234
Þ
j
¼
1
(the latter represents the special case for s = 4 and M = 3 according (
3.213
)
1
)
such that the isotropic variant of (
3.227
) reads
ð
4
Þ
ð
4
Þ
G
¼
1
Q :
¼
G
J
G
J
G
2
ð
C
I
Þ:
ðÞ
with
ð
3
:
235
Þ
Multiplication in (
3.235
) leads to arbitrary second order tensors A due to
ð
4
Þ
ð
4
Þ
ð
4
Þ
A
¼
A
T
;
I
1
A
¼
A
;
I
2
I
3
A
¼
tr
ð
I
I
A
I
ð
3
:
236
Þ
and A
¼
A
T
as well as
ð
4
Þ
ð
4
Þ
A
¼
A
A
I
A
ð Þ
trA
2
¼
: II
A
A
I
1
A
¼
A
I
2
ð
3
:
237
Þ
ð
4
Þ
A
¼
tr
ð
A
I
tr
ð
2
tr
2
A
¼
I
A
A
I
A
I
3
where I
A
and II
A
are referred to as the first two fundamental invariants of A.
Using A
G and considering k
1
þ
k
2
l
2
and k
3
l
1
and (
3.237
), the
anisotropic Fung model (
3.226
) transforms to the isotropic case
þ
f
ðÞ
with
e
Q
1
w C
; ð Þ¼
c
2
Q
¼
l
1
tr
2
G
þ
l
2
tr G
2
¼
l
1
I
2
G
þ
l
2
II
G
:
ð
3
:
238
Þ
Due to I
G
G
I
¼
tr G
¼
2
trC
3
C
I
3
ð
Þ
2
ð
Þ
and substituting
2
k
2
;
c
2
k
1
l
1
2
3
2k
2
;
l
2
0
;
ð
3
:
239
Þ
it follows from (
3.238
) that the second term of the isotropic H
OLZAPFEL
-G
ASSER
-
O
GDEN
-model (
3.233
), as a special case of the isotropic F
UNG
-model where the
effect of the second fundamental variant II
G
:
¼
trG
2
;
is neglected!
Veronda and Westman used the following strain energy function (Veronda and
Westman 1970)
