free energy per unit volume in reference configuration describes elastic energy stored
in molecular network structure in terms of first invariant of stretch in polymer chains
as
Ψ M C
e
M , θ
À
Á ¼ À
1
2
μ M I M ln 1 À
I 1 À 3
I m
ð7:182Þ
where μ M and I M are temperature-dependent rubbery shear modulus and limit on
extensibility of polymer chains, respectively. Since volume change in material is
considered in elastic deformation gradient associated with intermolecular structure,
it is essential to define Gent free energy in terms of distortional elastic deformation
gradient in network structure, F
e
M
À Á
d
(Eq. (7.183)), which produces no change in
volume. I 1 is first invariant of elastic distortional Cauchy tensor in network structure
C
e
M
À Á
d
F
e
M
À Á
d
¼ J
À1 = 3 F
e
M
ð7:183Þ
det F
e
M
À Á
d
¼ 1
ð7:184Þ
C
e
M
À Á
d
¼ F
e
M
À Á T
d
F
e
M
À Á
d
ð7:185Þ
I 1 ¼ tr C
e
M
À Á
d
h
i
ð7:186Þ
Second Piola-Kirchhoff stress S
e
M
À Á
and Cauchy stress (T M )can be derived from
Gent free energy as follows:
S
e
M ¼ 2
∂Ψ M C
e
M , θ
À
Á
∂C
e
M
ð7:187Þ
T M ¼ J
À1 F
e
M S
e
M F
e
T
M
ð7:188Þ
Using Gent free energy definition in and stress definitions in
S
e
M ¼ J
À2= 3 μ M 1 À
I 1 À 3
I M
À1
I À
1
3
tr C
e
M
À Á
d
À
Á
C
e
M
À Á
d
À1
h
i
ð7:189Þ
T M ¼ J
À1
μ M 1 À
I 1 À 3
I M
À1
dev B
e
M
À Á
d
À
Á
ð7:190Þ
Elastic distortional Almansi tensor, B
e
M
À Á
d
, is defined in terms of distortional
elastic deformation gradient F
e
M
À Á
d
À
Á
as
368
7 Unified Micromechanics of Finite Deformations