Γ
p
I ¼ dev M
e
I
À Á
ð7:129Þ
Γ
p
M ¼ dev M
e
M
À
Á
ð7:130Þ
Finally, elastic second Piola-Kirchhoff stress tensor in intermolecular structure
and molecular network structure can be defined as follows:
S
e
I ¼ JF
e
À1
I T I F
e
ÀT
I
ð7:131Þ
S
e
M ¼ JF
e
À1
M T M F
e
ÀT
M
ð7:132Þ
Using definitions of elastic Mandel stress (Eqs. (7.127) and (7.128)) and symmetric second Piola-Kirchhoff stress (Eqs. (7.131) and (7.132))
M
e
I ¼ C
e
I S
e
I
ð7:133Þ
M
e
M ¼ C
e
M S
e
M
ð7:134Þ
Therefore, it has been shown that stress measures Γ
e
I , Γ
e
M , Γ
p
I , Γ
p
M
À
Á
or
Τ I , Τ M , M
e
I , M
e
M
À
Á
with deformation rate conjugates of D
e
I , D
e
M , D
p
I , D
p
M
À
Á
form a
frame indifferent framework for dual-mechanism elastic-viscoplastic constitutive
model. The thermodynamic restrictions on constitutive relations can be obtained
by substituting Eqs. (7.86), (7.89), and (7.93) and into Eq. (7.84), as follows:
div J s
ð Þ À div
J q
θ
¼ 0
ð7:135Þ
J s ¼
J q
θ
ð7:136Þ
γ ther ¼ À
1
θ
2
div J q
À Á • ∇ x θ
ð Þ þ
ρr
θ
> 0
ð7:137Þ
J
À1 ∂Ψ I E
e
I , θ
À
Á
∂θ
þ
∂Ψ M C
e
M , θ
À
Á
∂θ
þ
∂Ψ D A, θ
ð
Þ
∂θ
þ ρs
!
_
θ ¼ 0
ð7:138Þ
ρs E
e
I , C
e
M , A, θ
À
Á ¼ ÀJ
À1 ∂Ψ E
e
I , C
e
M , A, θ
À
Á
∂θ
ð7:139Þ
s E
e
I , C
e
M , A, θ
À
Á ¼ À
∂ψ E
e
I , C
e
M , A, θ
À
Á
∂θ
ð7:140Þ
1
θ
Γ
e
I : L
e
I À J
À1 ∂Ψ I E
e
I , θ
À
Á
∂C
e
I
: _
C
e
I
¼
ð 7:141Þ
358
7 Unified Micromechanics of Finite Deformations