Biomedical Engineering Reference
In-Depth Information
(
3.245
)
2
and (
3.245
)
3
(regarding the transformation in (
3.245
)
3
, the following
ð
4
Þ
relation based on (
3.69
) is noticeable: o
=
oC
¼
1
=ð
I
1
o
=
o
ð Þ
1
=ðÞ
o
=
o
ð Þ
)
S
¼
2J
1
F
o
w
ðÞ
oC
F
T
;
P
I
¼
JF
1
S
¼
2
o
w
ðÞ
oC
F
T
ð
3
:
245
Þ
P
II
¼
JF
1
S
F
T
¼
2
o
w
ðÞ
oC
o
w
ðÞ
:
oG
Isotropic Materials - Representations in the Form of Invariants. The partial
derivative qw/qC occurring (
3.245
) follows, using (
3.181
) together with the chain
rule from
o
w
ðÞ
oC
¼
o
wC
I
;
C
II
;
C
III
ð
Þ
¼
o
w
oC
I
oC
I
oC
þ
o
w
o
C
II
oC
þ
o
w
o
C
III
oC
: ð
3
:
246
Þ
oC
oC
II
oC
III
Furthermore, the three partial derivations qC
i
/qC (i = I, II, III) occurring in
(
3.246
) are obtained using (
3.180
) based on the G
ÂTEAUX
variation as follows
o
C
I
oC
¼
I
;
o
C
II
oC
¼
C
I
I
C
ð
3
:
247
Þ
o
C
III
oC
¼
C
II
I
C
I
C
þ
C
2
¼
C
III
C
1
:
Substituting (
3.247
)in(
3.246
) finally leads to
I
ow
o
w
ðÞ
oC
¼
C
III
oC
III
C
1
þ
o
w
ow
ow
oC
II
þ
C
I
oC
II
C
ð
3
:
248
Þ
oC
I
establishing the structures of the three stress tensors (
3.245
).
Specifically, substituting (
3.248
)in(
3.245
)
1
and noting that F
I
F
T
¼
F
F
T
¼
B
;
F
C
F
T
¼
F
F
T
F
F
T
¼
B
2
and F
C
1
F
T
¼
F
F
T
F
1
F
T
¼
I as well as using the equality of the invariants C and B, namely C
i
¼
B
i
i
¼
1
;
2
; ð Þ
(this can be inferred from C
I
¼
trC
¼
trF
F
T
¼
trF
T
F
¼
trB
¼
B
I
etc.), the dependence of S on the left C
AUCHY
strain tensor can be derived
to
B
ow
:
oB
III
I
þ
o
w
ow
ow
oB
II
S
¼
g
ð
B
Þ¼
2J
1
oB
II
B
2
B
III
þ
B
I
ð
3
:
249
Þ
oB
I
Equations (
3.245
)-(
3.248
) and (
3.249
) provide the relevant constitutive stress-
strain equations of non-linear hyperelastic isotropic materials, where the specific
strain energy functions w can readily be substituted.
Isotropic Materials - Representations in the Form of Principal Stretches.
The partial derivative qw/qC occurring in (
3.245
) follows using (
3.185
) together
with the chain rule from
