Biomedical Engineering Reference
In-Depth Information
C
C
¼
F
T
F
¼
J
2
=
3
F
T
F
J
2
=
3
I
C :
¼
F
T
F
J
2
=
3
C
ð
3
:
192
Þ
and
and, based on (
3.192
) considering (
3.180
), (
3.184
) and (
3.191
)
2
the following
invariants of C yield
C
I
¼
tr C
¼
I
C
¼
k
1
þ
k
2
þ
k
3
C
II
¼
1
2
h
i
¼
k
1
k
2
þ
k
2
k
3
þ
k
1
k
3
¼
1
2
C
I
tr C
2
I
ð Þ
2
C
C
ð
3
:
193
Þ
¼
det F
T
det
ð Þ¼
det
ð Þ
2
J
2
¼
1
:
C
III
¼
det C
¼
det
F
T
F
By means of (
3.190
)to(
3.193
) and split into a deviatoric w
ðÞ
and a volumetric
part f(J), a decoupled representation of the strain energy function w can be pos-
tulated as follows
w
ðÞ¼
w
ðÞþ
f
ðÞ:
ð
3
:
194
Þ
Important Properties of Strain Energy Function. Besides the previously
mentioned representation of a strain energy function satisfying material objectivity
and material symmetry, w in addition, must satisfy the following conditions:
1. Positive Definiteness: According to
w
¼
w
ðÞ¼
[ 0
;
fur
C
6¼
I
ð
3
:
195
Þ
¼
0
;
fur
C
¼
I
and due to w [ 0, there is always strain energy needed to reach a deformed state
(C = I and k
i
= 1, respectively) whereas w in a strain-free state (C = I respec-
tively k
i
= 1) equals zero and thus takes a global minimum (in the ICFG).
2. Growth Condition: According to
lim
J
!
o
w
¼1
and
J
!1
w
¼1;
lim
ð
3
:
196
Þ
An infinite strain energy is needed to compress a volume element (and a
continuum) to a point (J ? 0) or to infinitely stretch it (J ? ?) (Ciarlet 1988).
Due to (
3.195
)
2
, the parts of (
3.194
) must satisfy
w C
¼
I
ð
Þ
0
and
fJ
¼ ð Þ¼
0
:
ð
3
:
197
Þ
3.2.6.2 Isotropic Representations of Strain Energy Functions
Based on
Sect. 3.2.6.1
, particularly (
3.181
), (
3.185
), (
3.195
) and (
3.196
), special
forms of sef may be generated. Compressible and incompressible materials are
distinguished, whereby most sef used here is restricted to (slightly) compressible
materials, so that the incompressible case is not treated in detail. Note, however,
