Biomedical Engineering Reference
In-Depth Information
f
ð
J
Þ¼
1
f
ð
J
Þ¼
l
b
¼
2m
1
2m
2
k
ð
ln J
Þ
2
l ln J
;
b
ð
J
b
1
Þ
mit
f
ð
J
Þ¼
X
N
l
i
ln J
þ
k
ð
3
:
202
Þ
b
2
ð
J
b
1
þ
b ln J
Þ
f
ð
J
Þ¼
c
ð
J
1
Þ
2
b ln J
;
i
¼
1
f
ð
J
Þ¼
a
ð
J
4
1
Þþ
b
ð
J
2
1
Þ
2
A detailed discussion on various functions f(J) as well as respective convex
and polyconvexity requirements can be found in Doll (1998) and Doll and
Schweizerhof (2000).
Representation as a Function of Principal Stretch. Often, sef are formulated
in terms of the principal stretches k
i
and their modified variants k
i
of the right
stretch tensor U. In Ogden (1972), the following strain energy function for
incompressible materials based on the first invariants of generalized strain tensors
(tensor functions of powers in C and B) is given
w
ð
k
1
;
k
2
;
k
3
Þ¼
X
ð
k
a
1
þ
k
a
2
þ
k
a
k
3
3
Þ
X
X
N
N
3
l
k
a
k
l
k
a
k
ð
k
a
k
i
1
Þ ð
3
:
203
Þ
k
¼
1
k
¼
1
i
¼
1
with k
3
¼
k
1
1
k
2
. A variation of (
3.203
) valid for compressible materials was
proposed by Saleb et al. (1992) and reads
w
ð
k
1
;
k
2
;
k
3
Þ¼
w
ð
k
1
;
k
2
;
k
3
Þþ
f
ð
J
Þ
with w
ð
k
1
;
k
2
;
k
3
Þ
:
¼
X
ð
k
a
1
þ
k
a
k
2
þ
k
a
k
3
3
Þ
X
X
N
N
3
l
k
a
k
l
k
a
k
ð
k
a
k
i
1
Þ
k
¼
1
k
¼
1
i
¼
1
ð
3
:
204
Þ
with decoupled deviatoric (isochoric) and volumetric parts w and f(J) and the
modified principal stretches k
i
defined in (
3.191
)
1
.
A
general
form
based
on
(
3.203
)
(which
satisfies
certain
polyconvexity
requirements) was proposed in Ciarlet (1988)
w
ð
k
1
;
k
2
;
k
3
Þ¼
X
N
a
i
ð
k
a
1
þ
k
a
i
2
þ
k
a
3
Þ
i
¼
1
ð
3
:
205
Þ
þ
X
M
b
j
½ð
k
1
k
2
Þ
b
j
þð
k
1
k
3
Þ
b
j
þð
k
2
k
3
Þ
b
j
þ
f
ð
J
Þ:
j
¼
1
The term trI = 3, however, needed to satisfy condition (
3.195
), is missing in
both summands in (
3.205
). These terms, however, are added with reference to the
literature in the following form equivalent to (
3.205
)
w
ðÞ¼
w F
;
CofF
;
J
ð
Þ¼
w
1
ðÞþ
w
2
Cof
ð Þ
w
3
ðÞ
ð
3
:
206
Þ
