Biomedical Engineering Reference
In-Depth Information
Observe from (
11.108
)that
W
cannot depend upon
x
3
2
x
5
when either
x
2
¼
0or
x
4
¼
0. To require that
W
be a single valued function of the components
C
ij
that enter
into
x
3
2
x
5
,itisnecessaryfor
W
to be periodic, with period 2
p
,in
x
3
2
x
5
.
Therefore it may be assumed that
W
depends on
. From
these observations it follows that
W
can be expressed as a function of the invariants
x
3
2
x
5
through cos
ðx
3
2
x
5
Þ
2
2
2
2
4
4
C
12
¼
C
13
þ
C
23
¼ x
C
33
;
C
11
þ
C
22
¼
2
x
1
; ð
C
11
C
22
Þ
þ
4
x
;
;
C
13
C
23
Þþ
4
C
12
C
23
C
31
¼
2
x
2
x
2
ð
C
11
C
22
Þð
4
cos
ðx
3
2
x
5
Þ:
(11.111)
An equivalent set of five invariants is given by the isotropic invariants
I
¼
I
B
¼
I
C
;
II
¼
II
B
¼
II
C
;
III
¼
III
B
¼
III
C
;
(11.112)
and complemented by
C
13
þ
C
23
¼ x
2
4
IV
;
V
C
33
:
(11.113)
Thus
W
ð
C
Þ
has the representation
W
ð
C
Þ¼
W
ð
I
;
II
;
III
;
IV
;
V
Þ
and it follows from
(
11.97
) that the stress has the representation
T
2
J
F
@
W
@
Y
@C
@
F
T
T
¼
;
or
T
ij
Y
T
F
mj
X
2
J
F
ik
@
W
@
@
Y
¼
(11.114)
Y
@
C
km
Y¼I;II;III;IV;V
which may be rewritten as
1
II
@
II
III
@
III
@
þ
@
I
III
@
II
þ
@
IV
@
þ
@
V
2
J
C
B
1
T
¼
C
þ
C
B
C
M
C
N
;
(11.115)
@
C
@
@
@
where
X
2
M
ij
¼
C
a
3
ð
F
ia
F
j
3
þ
F
ja
F
i
3
Þ;
N
ij
¼
F
i
3
F
j
3
;
(11.116)
a¼
1
which are related to
IV
and
V
by
T
;
T
F
ik
@
IV
F
ik
@
V
@
V
@F
@
IV
@F
2
M
ij
¼
F
jk
;
2
N
ij
¼
F
jk
;
or 2
N
¼
F
2
M
¼
F
:
@
@
(11.117)
If this transversely isotropic hyperelastic material is also incompressible, then
III
¼
1 and
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