Biomedical Engineering Reference
In-Depth Information
s
ðÞ¼
s
H
ðÞþ
s
D
ðÞ
ðÞþ
R
t
s
H
ðÞ
:
¼
s
0
j t
ðÞ
s
0
t
t
ð Þ
dt
0
with
t
0
¼
0
s
D
ðÞ
:
¼
s
0
ðÞ
<
:
2
4
3
5
=
;
ðÞ
Z
t
t
0
¼
0
ð
4
Þ
þ
F
ðÞ
P
c t
ðÞ
F
1
t
t
ð Þ
s
0
t
t
ð Þ
F
T
F
T
t
t
ð Þ
dt
0
ðÞ
ð
3
:
334
Þ
2
4
3
5
s
0
ðÞþ
ð
4
Þ
Z
t
c t
ðÞ
F
1
t
t
t
ð Þ
s
0
t
t
ð Þ
F
T
t
t
ð Þ
dt
0
t
t
0
¼
0
c
ðÞ
:
¼
_
G
ðÞ
G
0
¼
N
G
i
¼
1
¼
X
j
ðÞ
:
¼
_
K
ðÞ
K
0
N
K
g
i
s
i
e
t
k
i
s
i
e
t
s
G
i
;
s
K
i
and
i
¼
1
In (
3.332
), K(t) is the time-dependent small-strain bulk modulus, K
0
is the
instantaneous bulk modulus, k
?
is the long-term bulk modulus, the k
i
are relative
moduli, the s
i
are relaxation times and the N
K
are model parameters.
In (
3.334
), s
0
according to (
3.329
)
1
, is substituted by s
0
according to (
3.264
)
2
and, s
0
according to (
3.329
)
2
, is substituted by s
J
according to (
3.264
)
1
and, the
respective entities are indexed with ''0''!
The volumetric part s
H
ð
t
Þ
in (
3.334
) is proportional to the identity tensor such
that the following holds for arbitrary tensors A
A
1
s
H
A
¼
s
H
:
ð
3
:
335
Þ
In addition, in Abaqus (2010), the relaxation times and the model parameter
N for shear and compression are assumed equal to each other: s
i
:
¼
s
i
¼
s
i
and
N
G
= N
K
. In the case of highly compressible materials, the coefficients of the
P
RONY
series are assumed to be the same: g
i
= k
i
.
Since material characterization of human soft tissue employing the Ogden
model as well as the H
OLZAPFEL
-G
ASSER
-O
GDEN
model plays an eminent role, the
constitutive equation (
3.334
) is specialized for both models.
O
GDEN
Model for Slightly Compressible Materials. Substitution of (
3.271
)
1
in (
3.334
)
2
leads to the volumetric part (where, in addition, the property of (
3.335
)
becomes apparent)
2
4
3
5
I
k X
k
ðÞþ
Z
t
ðÞ¼
2
X
N
s
H
j t
ðÞ
X
k
t
t
ð Þ
dt
0
with
k
¼
1
t
0
¼
0
"
#
:
D
k
ðÞ¼
D
k
=
1
X
M
2k
1
=
D
k
ðÞ;
1
e
t
=
s
i
X
k
ðÞ
:
¼
J
ðÞ
J
ðÞ
1
½
k
i
i
¼
1
ð
3
:
336
Þ
