Biomedical Engineering Reference
In-Depth Information
where D
k
ð
t
Þ
is a time-dependent ''coefficient'' and D
k
is the instantaneous bulk
modulus.
The deviatoric part follows by substitution of (
3.271
)
2
in (
3.334
)
3
(C
t
is the
relative right C
AUCHY
strain tensor):
!
ðÞ¼
2
X
X
X
X
3
N
N
3
l
k
a
k
k
a
i
n
i
n
i
2
3
l
k
a
k
k
a
k
j
s
D
I
i
¼
1
k
¼
1
k
¼
1
j
¼
1
2
4
3
5
Z
t
þ
2
ð
4
Þ
X
3
X
N
l
k
a
k
c t
ðÞ
k
a
k
i
t
t
ð Þ
F
1
t
t
t
ð Þ
n
i
n
i
F
T
t
t
ð Þ
dt
0
t
i
¼
1
k
¼
1
t
0
¼
0
2
4
3
5
with
Z
t
ð
4
Þ
X
X
N
3
2
3
l
k
a
k
c t
ðÞ
k
a
k
j
t
t
ð Þ
C
1
t
C
t
:
¼
F
t
F
t
:
t
t
ð Þ
dt
0
k
¼
1
j
¼
1
t
0
¼
0
ð
3
:
337
Þ
H
OLZAPFEL
-G
ASSER
-O
GDEN
Model. Substitution of (
3.296
)in(
3.334
)
2
leads to
the volumetric part of the material equation
2
4
3
5
I
;
ðÞ
:
¼
Y
ðÞþ
Z
t
s
H
j t
ðÞ
Yt
t
ð Þ
dt
0
Y
ðÞ
:
¼
J
2
½
ðÞ
1
=
D
ðÞ:
t
0
¼
0
ð
3
:
338
Þ
The diviatoric part follows from (
3.329
)
1
by substitution of (
3.299
)in(
3.334
)
3
and considering the transformation of arbitrary tensors A (where
ð
4
Þ
and
ð
4
Þ
are to
be used according to equations (
3.257
) and (
3.261
))
F
T
¼
P
ð
4
Þ
F
1
ð
4
Þ
A
F
1
A
F
T
ð
3
:
339
Þ
to finally lead to the following expression
2
4
3
5
B
ðÞ
þ
Z
t
ð
4
Þ
ð
4
Þ
ðÞ
2
ð
4
Þ
U
1
ðÞ
I
1
c t
ðÞ
U
1
t
t
ð Þ
P
s
D
t
t
ð Þ
dt
0
t
0
¼
0
8
<
9
=
2
4
3
5
þ
Z
t
Þ
k
1
X
N
ð
4
Þ
ð
4
Þ
t
t
ð Þ
dt
0
ð
4
Þ
U
2
i
ðÞ
I
1
c t
ðÞ
U
2
i
t
t
ð Þ
P
þ
2 p
j 1
3j
ð
:
;
i
¼
1
t
0
¼
0
h
i
F
ðÞ
K
0i
F
T
ðÞ
ð
3
:
340
Þ
Isotropy. In the case of isotropic materials, it holds that j
¼
1
=
3 where (
3.340
)
reduces to (note that C
I
¼
B
I
)
