Biomedical Engineering Reference
In-Depth Information
In terms of material symmetry, the sef (
3.171
) must be insensitive to a change
of the reference configuration from ICFG 1 to ICFG 2 such that considering
(
3.174
) the following relation yields
h
i
g C
¼
g F
T
F
¼
g
T
F
K
1
w
¼
g
ðÞ¼
!
F
K
1
¼
g K
T
F
T
F
K
1
ð
3
:
175
Þ
Taking the definition for the right C
AUCHY
strain tensor (
3.64
)
1
into account,
from (
3.175
) the following symmetry condition arises
:
w
¼
g
ðÞ¼
!
g K
T
C
K
1
ð
3
:
176
Þ
In (
3.176
), w must take the same value independent of the deformation process
being controlled by C or by the variation of change of configuration systems
K
T
C
K
1
generated by K.
Based on (
3.176
) various material symmetries can be defined which, in addi-
tion, can be associated with so-called crystal classes (not treated here). Extreme
cases are outlined, whereas in the case of an anisotropic representation of w the
reader is referred to
Sect. 3.2.6.3
.
1. Fully Anisotropic Materials. The tensor of the ''change in reference con-
figurations'' is the identity tensor such that
F
¼
F
K
¼
I
and according to
ð
3
:
174
Þ;
respectively
;
ð
3
:
177
Þ
whereby the material exhibits different material properties in each direction.
Relation (
3.176
) thereby is identically satisfied and can not be further reduced.
2. Isotropic Materials. The tensor of the ''change in reference configurations''
comprises all orthogonal transformations Q (cf. (
3.154
) such that
F
¼
F
Q
1
K
¼
Q
and according to
3
:
17
ð Þ;
respectively
;
ð
3
:
178
Þ
Q
Q
T
¼
Q
T
Q
¼
I
K
1
¼
Q
1
detQ
¼þ
1
respectively
and
Based on (
3.154
) the relations K
1
¼
Q
1
¼
Q
T
and K
T
¼
Q
T
¼
Q hold
such that (
3.176
) transforms to the following condition of isotropy for scalar-
valued tensor functions of a tensor-valued variable
:
w
ðÞ¼
w Q
C
Q
T
ð
3
:
179
Þ
Materials that satisfy condition (
3.179
) exhibit equal material properties in all
directions (directional independence or isotropy) such that w always takes the
same values independent of the initial orientation of the body, no matter if the
process is controlled by C or by the variation Q
C
Q
T
generated by Q.
Based on representation theorems it can be shown (Spencer 1965; Smith 1969;
Wang 1969a, b) that condition (
3.179
) is satisfied if C is replaced by its three basic
invariants (which are insensitive regarding orthogonal transformations Q)
