Biomedical Engineering Reference
In-Depth Information
small deformation theory is hyperelastic. The strain energy per unit volume
W
is
obtained from the specific strain energy, that is to say the strain energy per unit mass
by multiplying it by
r
R
is the density function in the initial configuration.
In terms of the Cauchy stress and the first and second Piola-Kirchhoff stress tensors
the definition of a hyperelastic material has the following forms:
r
R
, where
T
;
:
¼
r
r
R
@
W
¼
@
W
@
@
W
T
1
PK
T
2
PK
F
1
T
F
F
;
¼
(11.75)
@
F
@
F
On first encounter, the variety of forms for the constitutive relation for
hyperelastic materials is bewildering. Not only are there three different stress
measures, but there are many different strain measures,
C
,
c
,
F
,
E
,
e
, etc. Thus,
for example, if we introduce the right Cauchy-Green deformation tensor
C
, since
C
U
2
FF
T
,
@
C
@
¼
¼
F
¼
2
F
then
@
W
@
@
W
@
F
¼
2
F
C
;
(11.76)
and the constitutive relations (
11.57
) take the form
T
2
r
r
R
@
W
@C
@
W
@C
;
2
@
W
@C
:
F
T
T
1
PK
T
2
PK
T
¼
F
;
¼
2
F
¼
(11.77)
Alternatively, these relations can be expressed in terms of the Lagrangian strain
tensor
E
,2
E
¼
C-1
,by(
11.29
), thus
@
@C
¼
@
W
@E
:
@
W
E
@C
¼
1
2
@
W
@E
;
(11.78)
and
T
¼
r
r
R
@
W
@
W
¼
@
W
@
F
T
T
1
PK
T
2
PK
T
F
;
¼
F
E
;
E
;
(11.79)
@
E
@
for example.
In the special case of an isotropic hyperelastic material the strain energy function
2
I
2
II
2
III
W
¼
W
ð
C
Þ¼
W
ðl
; l
; l
Þ
depends upon
C
only through the (isotropic)
invariants
ð
I
C
;
II
C
;
III
C
Þ
of
C
, thus
W
¼
W
ð
I
C
;
II
C
;
III
C
Þ;
(11.80)
where
are given by (
11.66
). Substituting this isotropic expression for
the strain energy into (
11.72
) and making use of the following expressions for the
derivatives of the invariants
I
C
,
II
C
and
III
C
with respect to
C
,
ð
I
C
;
II
C
;
III
C
Þ
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