Biomedical Engineering Reference
In-Depth Information
T
T
1
PK
F
1
þ
p
ð
Þ
¼
hF
ðÞ;
(11.86)
and
T
2PK
¼
t
(
C
), (
11.56
), becomes
T
2
PK
pC
1
þ
¼
t
ð
C
Þ;
(11.87)
where the functions
h
(
F
) and
t
(
C
) are defined only for deformations or motions that
satisfy the incompressibility condition Det
F
1.
When an elastic model material is both isotropic and incompressible there are
further simplifications in the constitutive relations. For example (
11.69
) and (
11.73
)
are now written in the simpler forms
¼
T
2
PK
pC
1
a
2
C
2
þ
¼
a
1
C
þ
;
(11.88)
and
h
1
B
1
T
þ
p1
¼
h
1
B
þ
;
(11.89)
where the functions of the isotropic invariants in these representations also
simplify,
a
1
¼
a
1
ð
I
C
;
II
C
Þ;
a
2
¼
a
2
ð
I
C
;
II
C
Þ
(11.90)
and
h
1
¼
h
1
ð
I
C
;
II
C
Þ¼
h
1
ð
I
B
;
II
B
Þ;
h
1
¼
h
1
ð
I
C
;
II
C
Þ¼
h
1
ð
I
B
;
II
B
Þ:
(11.91)
When an elastic model material is isotropic, incompressible and hyperelastic
there are even further simplifications in the constitutive relations. In this case the
strain energy per unit volume
W
depends only on
I
C
¼
I
B
and
II
C
¼
II
B
and (
11.93
)
reduces to
h
1
B
1
T
¼
p1
þ
h
1
B
þ
;
(11.92)
where
2
@
W
2
@
W
h
1
¼
I
C
;
h
1
¼
II
C
:
(11.93)
@
@
Even with all these restrictive assumptions (hyperelasticity, isotropy, and
incompressibility) a complete solution of many interesting problems is not possible.
Simpler models based on specialized assumptions but which retain the basic
characteristics of the nonlinear elastic response have been proposed for polymeric
materials and for biological tissues. An example that stems from research on the
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