The entropy generation rate γ must be zero if the thermodynamic equilibrium
conditions are satisfied within the system. Another requirement which Eq. (5.90)
must satisfy is that it is invariant under the transformation of different reference
frames, since the notions of reversible and irreversible behavior must be invariant
under such a transformation. It can be seen that Eq. (5.90) satisfies these requirements. Finally it may be noted that Eq. (5.88a) also satisfies the Clausius-Duhem
inequality:
σ : D À ρ
dφ
dt
þ s
dT
dt
À J q Á
grad T
T
! 0
ð5:91Þ
Between two particles which are at different temperatures, heat is transferred only
by conduction, a process which takes place at the molecular and atomic levels. The
law of heat conduction for isotropic bodies may be stated as follows:
J q ¼ Àk grad T
ð5:92Þ
where k, with units of Btu/ft h
F, is the thermal conductivity and where J q is the
heat flux.
This law of heat conduction was stated first by Fourier who based it on experimental observation. Fourier’s law expresses a linear relation between the heat flux
vector J q and its dual variable grad T. Since solid, opaque bodies are of primary
interest here, heat is transferred from point to point within this body solely by
conduction. The field equation of the boundary value problem will, therefore, always
be some form of the Fourier heat conduction equation. Of course, heat may be
transferred to the surface of the body by other modes of heat transfer which
correspond to various thermal boundary conditions. Then the expression for the
internal entropy generation rate for thermo-mechanical problems can be simplified
as
γ ¼
1
T
σ : D À
ρ
T
dw e
dt
þ
k
T
2
grad T
j
j
2 þ
ρ r
T
ð5:93aÞ
Total entropy generation is of course time integration of Eq. (5.93a):
s ¼
1
ρ
Z t 2
t 1
γ dt
ð5:93bÞ
The specific entropy production rate for small strain thermo-mechanical problems
in metals can be simplified as
228
5 Unified Mechanics of Thermo-mechanical Analysis
conditions are satisfied within the system. Another requirement which Eq. (5.90)
must satisfy is that it is invariant under the transformation of different reference
frames, since the notions of reversible and irreversible behavior must be invariant
under such a transformation. It can be seen that Eq. (5.90) satisfies these requirements. Finally it may be noted that Eq. (5.88a) also satisfies the Clausius-Duhem
inequality:
σ : D À ρ
dφ
dt
þ s
dT
dt
À J q Á
grad T
T
! 0
ð5:91Þ
Between two particles which are at different temperatures, heat is transferred only
by conduction, a process which takes place at the molecular and atomic levels. The
law of heat conduction for isotropic bodies may be stated as follows:
J q ¼ Àk grad T
ð5:92Þ
where k, with units of Btu/ft h
F, is the thermal conductivity and where J q is the
heat flux.
This law of heat conduction was stated first by Fourier who based it on experimental observation. Fourier’s law expresses a linear relation between the heat flux
vector J q and its dual variable grad T. Since solid, opaque bodies are of primary
interest here, heat is transferred from point to point within this body solely by
conduction. The field equation of the boundary value problem will, therefore, always
be some form of the Fourier heat conduction equation. Of course, heat may be
transferred to the surface of the body by other modes of heat transfer which
correspond to various thermal boundary conditions. Then the expression for the
internal entropy generation rate for thermo-mechanical problems can be simplified
as
γ ¼
1
T
σ : D À
ρ
T
dw e
dt
þ
k
T
2
grad T
j
j
2 þ
ρ r
T
ð5:93aÞ
Total entropy generation is of course time integration of Eq. (5.93a):
s ¼
1
ρ
Z t 2
t 1
γ dt
ð5:93bÞ
The specific entropy production rate for small strain thermo-mechanical problems
in metals can be simplified as
228
5 Unified Mechanics of Thermo-mechanical Analysis