Biomedical Engineering Reference
In-Depth Information
Similarly, the principle C
AUCHY
-stresses deduced from the O
GDEN
model for
slightly compressible materials (
3.272
)
2
, read
!
X
3k
a
j
i
X
þ
2
X
N
3
N
l
j
S
ii
¼
2
3
j
D
j
a
j
J
a
3
1
k
a
j
k
J
ð Þ
2j
1
ð
3
:
452
Þ
j
¼
1
k
¼
1
j
¼
1
In conjunction with (
3.434
) the first B
AKER
-E
RICKSEN
inequality yields
2
X
N
l
j
a
j
J
3
a
j
1
ð
k
a
j
i
k
a
k
Þð
k
i
k
k
Þ
[ 0
:
ð
3
:
453
Þ
j
¼
1
A sufficient condition regarding the choice of parameter l
j
in (
3.453
)is
l
j
[ 0
for
j
¼
1
;
...
;
N
ð
3
:
454
Þ
with no restrictions on parameter a
j
. Due to the form of the first B
AKER
-E
RICKSEN
inequalities no restriction can be deduced for parameter D
j
.
Based on the second B
AKER
-E
RICKSEN
inequalities (
3.436
), the following
inequalities for the O
GDEN
model for slightly compressible materials (
3.452
) are
obtained which restrict the choice of parameter a
j
and D
j
(
"
#
)
k
a
j
k
k
a
j
i
þ
X
X
þ
4J
X
N
3
N
l
j
a
j
k
1
ð
2j
2
j
Þ
D
j
2
3
J
3
a
j
1
1
3
a
j
þ
1
ð
J
1
Þ
2j
2
a
j
3
[ 0
i
j
¼
1
k
¼
1
j
¼
1
ð
3
:
455
Þ
Sufficient conditions regarding the choice of parameters l
j
, a
j
and D
j
in (
3.451
)
are
a
j
[
3
2
l
j
[ 0
and
and
D
j
[ 0
for
j
¼
1
;
...
;
N
ð
3
:
456
Þ
or
l
j
\0
and
a
j
\
3
and
D
j
[ 0
for
j
¼
1
;
...
;
N
:
ð
3
:
457
Þ
Summary:
As outlined in Tables
3.5
and
3.6
, from linear theory, it is required that
P
j
¼
1
l
j
[ 0 or sufficiently l
j
[ 0 for all j. This disqualifies the sufficient condi-
tions of l
j
\0 and a
j
\
3 for the slightly compressible O
GDEN
model.
In addition to the O
GDEN
model, the anisotropic hyperelastic material model
based on an invariant formulation proposed by Gasser et al. (2004) and Holzapfel
et al. (2000), in the form implemented by Abaqus (2010), is employed in
the parameter optimization process. This provides additional parameter sets for
long-term gluteal fat and transversally loaded (passive) muscle tissue behavior.
The material is assumed to be isotropic by specific choice of parameter j.
