Biomedical Engineering Reference
In-Depth Information
Anisotropic H
OOKE
's Law. In identifying material parameters of anisotropic
materials is shown to be applicable and is thus treated as a special case. Equations
(
3.98
) and (
3.276
) provide a relation between the C
AUCHY
stress tensor and the
second P
IOLA
-K
IRCHHOFF
stress tensor in the following form
F
T
:
ð
4
Þ
S
¼
J
1
F
P
II
F
T
¼
J
1
F
C
G
ð
3
:
284
Þ
=
2 according to (
3.284
) leads
Considering (
3.57
)
1
and G
¼
H
þ
H
T
þ
H
T
H
to
I
þ
H
S
¼
1
2
1
det I
þ
H
ð
4
Þ
Þ
T
: ð
3
:
285
Þ
H
þ
H
T
þ
H
T
H
Þ
I
þ
H
ð
Þ
C
ð
ð
Factoring out the terms in (
3.285
) yields
þ
O H
T
H
ð
4
Þ
S
¼
1
2
1
det I
þ
H
H
þ
H
T
þ
H
T
H
Þ
C
ð
3
:
286
Þ
ð
where terms of higher order in the displacement gradient H are summed up by the
term O H
T
H
.
A consequent geometrical linearization (also shown at the end of
Sect. 3.2.3.6
)
includes the elimination of the quadratic term H
T
H in square brackets in (
3.286
)
as well as reducing the denominator to det I
¼
1 (this can be realized by writing
det I
þ ð Þ
in coordinate notation). Considering (
3.77
), the most general strictly-
linear relation between the C
AUCHY
stress tensor S and the deformator E for
arbitrary anisotropic material behaviour results in:
ð
4
Þ
S
¼
C
E
:
ð
3
:
287
Þ
Based on the approach outlined following Eq. (
3.276
), this constitutive equation
structure for orthotropy between S and E results from (
3.287
) (analogue to (
3.281
))
S
¼
u
1
trM
1
E
þ
u
2
trM
2
E
þ
u
3
trM
3
ð Þ
M
1
þ
u
2
trM
1
E
þ
u
4
trM
2
E
þ
u
5
trM
3
E
ð
Þ
M
2
þ
u
3
trM
1
E
þ
u
5
trM
2
E
þ
u
6
trM
3
E
ð
Þ
M
3
þ
u
7
M
1
E
M
2
þ
M
2
E
M
1
ð
Þþ
u
8
M
1
E
M
3
þ
M
3
E
M
1
ð
Þ
þ
u
9
M
2
E
M
3
þ
M
3
E
M
2
ð
Þ:
ð
3
:
288
Þ
Considering (
3.78
), (
3.219
) and (
3.220
) as well as trM
a
E
¼
e
aa
;
M
a
E
M
b
¼
e
ab
e
a
e
b
(a and b not summed!), r
ij
¼
r
ji
and e
ij
¼
e
ji
,Eq.(
3.288
) leads to
