Biomedical Engineering Reference
In-Depth Information
Table 3.5 Summary of sufficient restrictions on the material parameters for the O
GDEN
-H
ILL
foam model
Ogden-Hill foam
l
j
a
j
b
j
Polyconvexity
l
j
[ 0
a
j
1
b
j
[ 0
b
j
[
1
Baker-Ericksen
l
j
[ 0
a
j
[ 1
b
j
[
1
=
3
Linear theory
l
j
[ 0
None
Table 3.6 Summary of sufficient restrictions on the material parameters for the O
GDEN
model for
slightly compressible materials
Ogden Slightly Compr.
l
j
a
j
D
j
Baker-Ericksen
l
j
[ 0
a
j
[ 3
=
2
D
j
[ 0
l
j
\0
a
j
\
3
D
j
[ 0
Linear theory
l
j
[ 0
None
D
1
[ 0
The strain-energy function corresponding to the H
OLZAPFEL
-G
ASSER
-O
GDEN
-
model reads (see Sect. 3.5.6.3, Eq. (
3.229
))
X
N
w
¼
C
10
ð
C
I
3
Þþ
1
2D
ð
J
el
2 ln J
el
1
Þþ
k
1
E
hi
2
1
Þ ð
3
:
458
Þ
ð
e
k
2
2k
2
j
¼
1
with
h
E
j
i¼
j
ð
C
I
3
Þþð
1
3j
Þð
C
IV
j
1
Þ
(
where
h
E
j
i¼
0
;
E
j
\ 0
where the first term in (
3.458
) represents the
E
j
;
E
j
[ 0
(isotropic) N
EO
-H
OOKEAN
form describing the deviatoric material response with
the first deviatoric strain invariant C
I
¼
k
2
1
þ
k
2
2
þ
k
2
3
where the k
i
are the devia-
toric principal stretches k
i
¼
J
1
=
3
k
i
. C
IV
j
is a pseudo-invariant given by C
IV
j
¼
k
1
cos
2
#
j
þ
k
2
sin
2
#
j
where
#
represents the mean fibre orientation. The first term
in (
3.458
) is augmented by a second volumetric term to introduce compressibility.
The third term accounts for material anisotropy and is formulated in terms of
exponential function to account for stiffening effects. C
10
;
D
;
k
1
;
k
2
;
j are material
parameters to be determined by experiment. In the isotropic case j is set to
j
¼
1
=
3 and thus
(
0
;
E
j
\ 0
E
j
;
iso
¼
1
=
3 C
I
1
with
E
j
¼
ð
3
:
459
Þ
E
j
;
E
j
0
:
From (
3.458
) and following the outlines given previously, the initial bulk
modulus derives to
