Biomedical Engineering Reference
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from which it follows
dC
0
d
d
d
dW
dC
0
:
dW
dC
0
: ½
O
C
0
Þ¼
C
0
þ
C
0
T
W
ð
0
¼
f
¼
O
;
(11.103)
f
or
d
d
dW
dC
0
:
C
0
Þ¼
C
0
W
ð
0
¼
2
O
:
(11.104)
f
Substitution of (
11.101
) into (
11.104
) above yields
e
ij
3
O
3
C
im
dW
dC
mj
¼
;
0
(11.105)
which requires that
dW
dC
m
2
¼
C
2
n
dW
C
1
m
dC
n
1
;
(11.106)
f
if (
11.105
) is to be true for all
. Expanding the two sets of summation indexes in
(
11.106
) we obtain
dW
dC
22
dW
dC
11
dW
dC
12
þ
C
13
dW
C
23
dW
C
12
þð
C
11
C
22
Þ
dC
23
dC
13
¼
0
(11.107)
,is
obtained by solving this differential equation (Ericksen and Rivlin
1954
; Januzemis
1967
). Simplification of this result
The form of the function
W
ð
C
Þ
invariant under the rotation (
11.98
), for all
f
is obtained if the following notation is
introduced,
C
11
¼ x
1
þ x
2
cos
x
3
;
C
22
¼ x
1
x
2
cos
x
3
;
C
12
¼ x
2
sin
x
3
;
C
13
¼ x
4
cos
x
5
;
C
23
¼ x
4
sin
x
5
;
(11.108)
where
x
2
0,
x
4
0 in which case (
11.107
) reduces to
2
dW
d
dW
d
x
3
þ
x
5
¼
0
;
(11.109)
thus
W
¼
W
ð
C
33
; x
1
; x
2
; x
3
; x
3
2
x
5
; x
4
Þ:
(11.110)
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