Γ
e
I : L
e
I ¼ T I : L
e
I ¼ T I : D
e
I ¼ J
À1 ∂Ψ I E
e
I , θ
À
Á
∂C
e
I
: _
C
e
I
ð7:142Þ
T I : D
e
I ¼
1
2
F
e
À1
I T I F
e
ÀT
I
: _
C
e
I ¼ J
À1 ∂Ψ I E
e
I , θ
À
Á
∂C
e
I
: _
C
e
I
ð7:143Þ
J F
e
À1
I T I F
e
ÀT
I
¼ S
e
I ¼ 2
∂Ψ I E
e
I , θ
À
Á
∂C
e
I
ð7:144Þ
1
θ
Γ
e
M : L
e
M À J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
¼
ð 7:145Þ
Γ
e
M : L
e
M ¼ T M : L
e
M ¼ T M : D
e
M ¼ J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
ð7:146Þ
T M : D
e
M ¼
1
2
F
e
M T M F
e
M
À
Á : _
C
e
M ¼ J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
ð7:147Þ
J F
e
À1
M T M F
e
ÀT
M
¼ S
e
M ¼ 2
∂Ψ M C
e
M , θ
À
Á
∂C
e
M
ð7:148Þ
γ mech ¼
1
θ
J
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M À J
À1 ∂Ψ D A, θ
ð
Þ
∂A
: _
A
> 0
ð7:149Þ
Equation (7.136) relates entropy flux to heat flux, and Eq. (7.137) defines
irreversible entropy production per unit volume in deformed configuration associated with heat conduction which is always positive according to Fourier’s law
(Eq. (7.150)). Equation (7.140) provides the relation between specific entropy and
specific Helmholtz energy, while Eqs. (7.144) and (7.148) describe stress measures
derived from Helmholtz free energy functions. Equation (7.149) defines irreversible
entropy production due to mechanical dissipation per unit volume in deformed
configuration. The reason for appearance of J
À1 term in front of Eq. (7.149) is that
all stress measures and their conjugate rate measures refer to intermediate (relaxed
configuration), while irreversible entropy production rate density refers to deformed
configuration. Heat flux is given by
J q ¼ Àk∇ x θ
ð Þ
ð7:150Þ
where k is temperature-dependent thermal conductivity of material. Specific heat can
be expressed in terms of specific entropy (Eq. 7.151) and in terms of Helmholtz free
energy (Eq. 7.153) by exploiting the relation in Eq. (7.139). Using linear decomposition assumption in Eqs. (7.85) and (7.86), specific heat can be rewritten in terms of
separate component of specific entropy and Helmholtz free energy Eq. (7.154):
7.3 Unified Mechanics Theory Formulation for Finite Strain
359
e
I : L
e
I ¼ T I : L
e
I ¼ T I : D
e
I ¼ J
À1 ∂Ψ I E
e
I , θ
À
Á
∂C
e
I
: _
C
e
I
ð7:142Þ
T I : D
e
I ¼
1
2
F
e
À1
I T I F
e
ÀT
I
: _
C
e
I ¼ J
À1 ∂Ψ I E
e
I , θ
À
Á
∂C
e
I
: _
C
e
I
ð7:143Þ
J F
e
À1
I T I F
e
ÀT
I
¼ S
e
I ¼ 2
∂Ψ I E
e
I , θ
À
Á
∂C
e
I
ð7:144Þ
1
θ
Γ
e
M : L
e
M À J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
¼
ð 7:145Þ
Γ
e
M : L
e
M ¼ T M : L
e
M ¼ T M : D
e
M ¼ J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
ð7:146Þ
T M : D
e
M ¼
1
2
F
e
M T M F
e
M
À
Á : _
C
e
M ¼ J
À1 ∂Ψ M C
e
M , θ
À
Á
∂C
e
M
: _
C
e
M
ð7:147Þ
J F
e
À1
M T M F
e
ÀT
M
¼ S
e
M ¼ 2
∂Ψ M C
e
M , θ
À
Á
∂C
e
M
ð7:148Þ
γ mech ¼
1
θ
J
À1
Γ
p
I : L
p
I þ J
À1
Γ
p
M : L
p
M À J
À1 ∂Ψ D A, θ
ð
Þ
∂A
: _
A
> 0
ð7:149Þ
Equation (7.136) relates entropy flux to heat flux, and Eq. (7.137) defines
irreversible entropy production per unit volume in deformed configuration associated with heat conduction which is always positive according to Fourier’s law
(Eq. (7.150)). Equation (7.140) provides the relation between specific entropy and
specific Helmholtz energy, while Eqs. (7.144) and (7.148) describe stress measures
derived from Helmholtz free energy functions. Equation (7.149) defines irreversible
entropy production due to mechanical dissipation per unit volume in deformed
configuration. The reason for appearance of J
À1 term in front of Eq. (7.149) is that
all stress measures and their conjugate rate measures refer to intermediate (relaxed
configuration), while irreversible entropy production rate density refers to deformed
configuration. Heat flux is given by
J q ¼ Àk∇ x θ
ð Þ
ð7:150Þ
where k is temperature-dependent thermal conductivity of material. Specific heat can
be expressed in terms of specific entropy (Eq. 7.151) and in terms of Helmholtz free
energy (Eq. 7.153) by exploiting the relation in Eq. (7.139). Using linear decomposition assumption in Eqs. (7.85) and (7.86), specific heat can be rewritten in terms of
separate component of specific entropy and Helmholtz free energy Eq. (7.154):
7.3 Unified Mechanics Theory Formulation for Finite Strain
359