energy of the process. The elastic energy is the maximum amount of work that could
be produced by a device between any given two states. If the device is work
absorbing, the elastic energy work of the process is the minimum amount of work
that must be supplied (Li 1989).
In order to find the explicit form of the entropy balance equation (5.81), we insert
the expressions (5.72) for
du
dt into Eq. (5.85) with the time derivatives given by
Eq. (5.39):
ρ
ds
dt
¼ À
div J q
T
þ
1
T
σ : D À
ρ
T
dw
e
dt
þ
ρr
T
ð5:87aÞ
Noting that
div J q
T
¼ div
J q
T
þ
1
T
2
J q Á grad T
ð5:87bÞ
It is easy to cast Eq. (5.87) into the form of a balance equation (5.81):
ρ
ds
dt
¼ Àdiv
J q
T
À
1
T
2
J q Á grad T þ
1
T
σ : D À
ρ
T
dw
e
dt
þ
ρr
T
ð5:88Þ
From comparison with Eq. (5.81), it follows that the expressions for the entropy
flux and the entropy production rate are given by
J S ¼
J q
T
ð5:89Þ
γ ¼
1
T
σ : D À
ρ
T
dw
e
dt
À
1
T
2
J q Á grad T þ
ρr
T
ð5:90Þ
Equation (5.90) represents the entropy generation rate by the internal dissipations.
The sum of the first two terms is called the intrinsic dissipation or mechanical
dissipation. It consists of plastic dissipation plus the dissipation associated with the
evolution of other internal variables. The last two terms are the thermal dissipation
due to the conduction of heat and the internal heat source. The structure of the
expression for γ is that of a bilinear form: it consists of a sum of products of two
factors. One of these factors in each term is a flux quantity (heat flow J q , σ stress
tensor) already introduced in the conservation laws. The other factor in each term is
related to a gradient of an intensive state variable (gradients of temperature and
velocity). These quantities which multiply the fluxes in the expression for the
entropy production are called thermodynamic forces. As we discussed earlier,
actually the assignment of flux and thermodynamic force is rather arbitrary.
However, their multiplication must yield the entropy generation rate.
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
227
be produced by a device between any given two states. If the device is work
absorbing, the elastic energy work of the process is the minimum amount of work
that must be supplied (Li 1989).
In order to find the explicit form of the entropy balance equation (5.81), we insert
the expressions (5.72) for
du
dt into Eq. (5.85) with the time derivatives given by
Eq. (5.39):
ρ
ds
dt
¼ À
div J q
T
þ
1
T
σ : D À
ρ
T
dw
e
dt
þ
ρr
T
ð5:87aÞ
Noting that
div J q
T
¼ div
J q
T
þ
1
T
2
J q Á grad T
ð5:87bÞ
It is easy to cast Eq. (5.87) into the form of a balance equation (5.81):
ρ
ds
dt
¼ Àdiv
J q
T
À
1
T
2
J q Á grad T þ
1
T
σ : D À
ρ
T
dw
e
dt
þ
ρr
T
ð5:88Þ
From comparison with Eq. (5.81), it follows that the expressions for the entropy
flux and the entropy production rate are given by
J S ¼
J q
T
ð5:89Þ
γ ¼
1
T
σ : D À
ρ
T
dw
e
dt
À
1
T
2
J q Á grad T þ
ρr
T
ð5:90Þ
Equation (5.90) represents the entropy generation rate by the internal dissipations.
The sum of the first two terms is called the intrinsic dissipation or mechanical
dissipation. It consists of plastic dissipation plus the dissipation associated with the
evolution of other internal variables. The last two terms are the thermal dissipation
due to the conduction of heat and the internal heat source. The structure of the
expression for γ is that of a bilinear form: it consists of a sum of products of two
factors. One of these factors in each term is a flux quantity (heat flow J q , σ stress
tensor) already introduced in the conservation laws. The other factor in each term is
related to a gradient of an intensive state variable (gradients of temperature and
velocity). These quantities which multiply the fluxes in the expression for the
entropy production are called thermodynamic forces. As we discussed earlier,
actually the assignment of flux and thermodynamic force is rather arbitrary.
However, their multiplication must yield the entropy generation rate.
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
227