Biomedical Engineering Reference
In-Depth Information
@
I
C
;
@
II
C
@
;
@
III
C
@
III
C
C
1
C
¼
1
C
¼
I
C
1
C
C
¼
:
(11.81)
@
it follows that
T
2
PK
has the representation
1
@
W
I
C
@
W
@
W
þ
@
W
T
2
PK
III
C
III
C
C
1
¼
2
I
C
þ
II
C
C
:
(11.82)
@
@
II
C
@
@
This constitutive relation may also be written in a form which contains
C
2
rather
than
C
1
,
1
C
@
W
I
C
@
II
C
þ
@
W
W
@
W
I
C
@
W
þ
@
W
T
2
PK
III
C
C
2
¼
2
I
C
þ
III
C
II
C
II
C
þ
@
@
@
@
@
III
C
@
(11.83)
by use of Cayley Hamilton theorem that states that a matrix satisfies its own
characteristic equation, thus
C
may replace
2
in (
11.65
),
C
3
I
C
C
2
l
þ
II
C
C
II
I
C
¼
0 . A term-by-term comparison of (
11.78
) with (
11.69
) shows that
the
coefficients
a
o
,
a
1
, and
a
2
in (
11.69
) are given in the case of hyperelasticity by
2
@
W
I
C
@
II
C
þ
@
W
W
2
@
W
I
C
@
W
a
o
¼
I
C
þ
III
C
II
C
;
a
1
¼
II
C
þ
;
@
@
@
@
@
III
C
2
@
W
a
2
¼
(11.84)
@
III
C
Also, in the case of an isotropic hyperelastic material, the coefficients
h
o
,h
1
, and
h
2
in the constitutive relation between the Cauchy stress
T
and the left
Cauchy-Green tensor
B
,(
11.73
), may be expressed in terms of the strain energy
function by the following formulas:
2
J
II
C
@
II
C
þ
@
W
W
2
J
@
W
2
J
III
C
@
W
h
o
¼
III
C
III
C
;
h
1
¼
I
C
;
h
1
¼
(11.85)
@
@
@
@
II
C
Problems
11.9.1. Derive the result
ð@
W
=@
C
Þ¼ð@
W
=@
E
Þ : ð@
E
=@
C
Þ¼ð
1
=
2
Þð@
W
=@
E
Þ
,
(
11.78
), using the indicial notation and the formula 2
E
¼
C
1
. Hint: It
is useful to first derive the formula
ð@
E
gd
=@
C
ab
Þ¼ð
1
=
2
Þd
bd
d
ag
from 2
E
gd
C
gd
d
gd
.
11.9.2. Derive the result (
11.76
) using the indicial notation. Hint: It is useful to first
derive the formula
¼
ð@
C
ab
=@
F
ig
Þ¼ð@=@
F
ig
Þð
F
ka
F
kb
Þ¼d
ik
d
ag
F
kb
þ
F
ka
d
ik
F
T
·
F
in the indicial notation
C
ab
d
bg
beginning from the definition
C
¼
¼
F
ka
F
kb
.
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