In small strain theory, the strain can be written in the form of their additive
decomposition:
ε Total
À ε
pl
¼ ε
e
ð8:87Þ
so that
Ψ ¼ Ψ ε Total À ε
pl
À
Á
, T, V k
À
Á ¼ Ψ ε
e , T, V k
ð
Þ
ð 8:88Þ
Time derivative of the free energy is given by
dΨ
dt
¼
∂Ψ
∂ε e :
dε
e
dt
þ
∂Ψ
∂T
:
dT
dt
þ
∂Ψ
∂V k
:
dV k
dt
ð8:89Þ
From Eq. (8.85) we obtain
T
ds
dt
¼
du
dt
À
dΨ
dt
À s
dT
dt
ð8:90Þ
Using the first principle of thermodynamics, the conservation of energy equation
can be given by
ρ
du
dt
¼ ÀdivJ q þ σ : Grad v
ð Þ þ
X
k
J k Á F k
ð8:91Þ
where u is the total internal energy, J q is the heat flux, σ is the stress tensor, v is the
rate of deformation, J k is the diffusional flux of component k, and F k is the body
force acting on the mass of component k.
Using the conservation of energy equation in the form given by Eq. (8.90) and
Eqs. (8.82) and (8.83), we can write the rate of change of entropy density as follows:
ρ
ds
dt
¼
1
T
ÀdivJ q
À
Á þ σ : Grad v
ð Þþ
X
k
J k • F k À ρ
∂Ψ
∂ε e :
dε
e
dt
þ
∂Ψ
∂T
:
dT
dt
þ
∂Ψ
∂V k
:
dV k
dt
À s
dT
dt
ð8:92Þ
where
σ : Grad v
ð Þ ¼ σ : D þ W
ð
Þ
ð8:93Þ
8.7 Fundamental Equation for Electromigration and Thermomigration
419