classical form dealing exclusively with the comparison of constrained equilibrium
states are given by
Z t
2
t 1
Z
A
T ∙ _
z dA þ
Z
A
qdA
&
'
dt ¼ V
0 u
2
À u
1
Â
Ã
ð4:17Þ
Z t
2
t 1
Z
A
q=θ
ð
ÞdA
&
'
dt V
0
η
2
À η
1
Â
Ã
ð4:18Þ
where T is the surface stress vector (traction), q is the heat supplied to the surface per
unit area (positive inward), _
z is the deformation rate, u is the internal energy, and η is
the entropy. Time integral from t
1 to t
2 follow the irreversible process. It is important
to point out that elastic deformation here is considered a reversible process; hence,
fatigue under elastic stresses is not possible. Alternatively, during a molecular
dynamics simulations under elastic loading atoms return to their original lattice
site. Based on earlier work by Kestin and Rice (1970), following the classical
approach to irreversible processes for a macroscopically homogeneous system
with internal variables, Rice (1971) views all actual processes as a sequence of
constrained equilibrium states. The work of surface stresses can be written in terms
of internal stresses and strains as V
0 S : _
E for contrained equilibrium states and the
temperature is assumed uniform in the material. Then the first and second laws of
equilibrium thermodynamics can be generalized for any actual homogenous process
as follows
S :
_
E þ Q ¼ _
u
Q ¼
1
V
0
Z
A
q dA θ _
η
ð4:19Þ
where Q is the total heat supply rate per unit reference volume. Rice (1971) further
defines entropy generation rate due to internal variables and conjugate thermodynamic forces as
σ ¼
1
θV
0
f α _
ξ α ! 0
ð4:20Þ
Then, Rice (1971) defines the total change in entropy as summation of Q, the total
heat supply rate per unit reference volume plus the entropy generation due to internal
variable and thermodynamic forces as
Q þ θσ ¼ θ _
η
ð4:21Þ
4.1 Literature Review of Use of Thermodynamics in Continuum Mechanics
123
states are given by
Z t
2
t 1
Z
A
T ∙ _
z dA þ
Z
A
qdA
&
'
dt ¼ V
0 u
2
À u
1
Â
Ã
ð4:17Þ
Z t
2
t 1
Z
A
q=θ
ð
ÞdA
&
'
dt V
0
η
2
À η
1
Â
Ã
ð4:18Þ
where T is the surface stress vector (traction), q is the heat supplied to the surface per
unit area (positive inward), _
z is the deformation rate, u is the internal energy, and η is
the entropy. Time integral from t
1 to t
2 follow the irreversible process. It is important
to point out that elastic deformation here is considered a reversible process; hence,
fatigue under elastic stresses is not possible. Alternatively, during a molecular
dynamics simulations under elastic loading atoms return to their original lattice
site. Based on earlier work by Kestin and Rice (1970), following the classical
approach to irreversible processes for a macroscopically homogeneous system
with internal variables, Rice (1971) views all actual processes as a sequence of
constrained equilibrium states. The work of surface stresses can be written in terms
of internal stresses and strains as V
0 S : _
E for contrained equilibrium states and the
temperature is assumed uniform in the material. Then the first and second laws of
equilibrium thermodynamics can be generalized for any actual homogenous process
as follows
S :
_
E þ Q ¼ _
u
Q ¼
1
V
0
Z
A
q dA θ _
η
ð4:19Þ
where Q is the total heat supply rate per unit reference volume. Rice (1971) further
defines entropy generation rate due to internal variables and conjugate thermodynamic forces as
σ ¼
1
θV
0
f α _
ξ α ! 0
ð4:20Þ
Then, Rice (1971) defines the total change in entropy as summation of Q, the total
heat supply rate per unit reference volume plus the entropy generation due to internal
variable and thermodynamic forces as
Q þ θσ ¼ θ _
η
ð4:21Þ
4.1 Literature Review of Use of Thermodynamics in Continuum Mechanics
123