Biomedical Engineering Reference
In-Depth Information
that in the following strain energy functions w, in the isotropic case J = 1, the
modified measures C
i
and k
i
;
degenerate to the original variants C
i
and k
i
,
respectively. Some sef taken from the literature are listed below. The issue of
material stability restrictions resulting from convexity requirements, with respect
to their respective material parameters, is treated in
Sect. 3.4
in the context of
material identification. N and M may be considered as model or order parameters.
The volumetric and deviatoric parts within the representations of w generally
follow from the postulate given in (
3.194
).
Polynomial Form as a Function of Invariants. The most general power series
expansion of (
3.181
)ofw in the three invariants of C yields
w
¼
w
ð
C
I
;
C
II
;
C
III
Þ¼
X
X
X
1
1
1
c
ijk
ð
C
I
3
Þ
i
ð
C
II
3
Þ
j
ð
C
III
3
Þ
k
ð
3
:
198
Þ
i
¼
0
j
¼
0
k
¼
0
where the c
ijk
are the expansion coefficients and, c
000
must be equal to zero. The
powers of the invariants C
i
(summation to ?) may further be reduced using the
C
AYLEY
-H
AMILTON
-theorem. In (Abaqus 2010), instead of (
3.198
) and according to
(
3.193
), a polynomial representation of w is used
w
¼
w
ð
C
I
;
C
II
Þþ
f
ð
J
Þ
with w
ð
C
I
;
C
II
Þ
:
¼
X
N
X
M
ð
3
:
199
Þ
c
ij
ð
C
I
3
Þ
i
ð
C
II
3
Þ
j
i
¼
0
j
¼
0
with the deviatoric (isochoric) and the volumetric parts w
ð
C
I
;
C
II
Þ
and f(J), the
expansion coefficients (material parameters) c
ij
and, c
00
= 0. For N = 1, M = 0
and
in C
I
c
10
= l/2
and
based
on
(
3.199
),
the
special
case
of
the
linear
N
EO
-H
OOKE
model follows
w
ð
C
I
Þ
:
¼
l
w
¼
w
ð
C
I
Þþ
f
ð
J
Þ
with
2
ð
C
I
3
Þ:
ð
3
:
200
Þ
In (
3.200
), the first part is based on Treolar (1943) and f(J) according to
(
3.202
)
2
on Blatz (1971). For N = M = 1 and c
10
= l
1
/2 and c
01
= l
2
/2, the
special case of the M
OONEY
-R
IVLIN
model (Mooney 1940; Wriggers 2008) which is
linear in C
I
and C
II
yields
w
¼
w
ð
C
I
;
C
II
Þþ
f
ð
J
Þ
with w
ð
C
I
;
C
II
Þ
:
¼
l
1
ð
3
:
201
Þ
2
ð
C
I
3
Þþ
l
2
2
ð
C
II
3
Þ:
Several proposals have been made regarding the ''volumetric function''
f(J) and, some are listed subsequently (Blatz 1971; Ogden 1984; Ciarlet 1988;
Abaqus 2010)
