ds
dt
þ
1
ρ
div
q
T
À
1
T
du
dt
À σ Á D þ div q
! 0
ð3:91Þ
Since
div
q
T
¼
div q
T
À
q Á grad T
T
2
ð3:92Þ
Multiplying each term with T,
ρ T
ds
dt
À
du
dt
þ σ : D À q Á
grad T
T
! 0
ð3:93Þ
Helmholtz specific free energy is given by
Ψ ¼ u À Ts
ð3:94Þ
Differentiating this, we obtain
dΨ
dt
¼
du
dt
À T
ds
dt
À s
dT
dt
ð3:95Þ
We can write
À
dΨ
dt
þ s
dT
dt
¼ T
ds
dt
À
du
dt
ð3:96Þ
If we substitute this new relationship in fundamental inequality, we get
σ : D À ρ
∂Ψ
dt
þ s
dT
dt
À q Á
grad T
T
! 0
ð3:97Þ
We should point out that in conservation of energy, time derivative of internal
energy is multiplied by ρ density. It was dropped in this derivation for consistency of
units.
Now we can substitute
dΨ
dt in the fundamental equation.
Meanwhile, we can split D, the deformation gradient tensor, into elastic and
plastic components:
D ¼ D
e
þ D
p
ð3:98Þ
104
3 Thermodynamics
dt
þ
1
ρ
div
q
T
À
1
T
du
dt
À σ Á D þ div q
! 0
ð3:91Þ
Since
div
q
T
¼
div q
T
À
q Á grad T
T
2
ð3:92Þ
Multiplying each term with T,
ρ T
ds
dt
À
du
dt
þ σ : D À q Á
grad T
T
! 0
ð3:93Þ
Helmholtz specific free energy is given by
Ψ ¼ u À Ts
ð3:94Þ
Differentiating this, we obtain
dΨ
dt
¼
du
dt
À T
ds
dt
À s
dT
dt
ð3:95Þ
We can write
À
dΨ
dt
þ s
dT
dt
¼ T
ds
dt
À
du
dt
ð3:96Þ
If we substitute this new relationship in fundamental inequality, we get
σ : D À ρ
∂Ψ
dt
þ s
dT
dt
À q Á
grad T
T
! 0
ð3:97Þ
We should point out that in conservation of energy, time derivative of internal
energy is multiplied by ρ density. It was dropped in this derivation for consistency of
units.
Now we can substitute
dΨ
dt in the fundamental equation.
Meanwhile, we can split D, the deformation gradient tensor, into elastic and
plastic components:
D ¼ D
e
þ D
p
ð3:98Þ
104
3 Thermodynamics