displacement, elastic deformations and ignoring many entropy-generating mechanisms, they do not satisfy laws of thermodynamics, in the strict sense. Therefore, it is
more appropriate to call these forces pseudo-thermodynamic forces or pseudodissipative forces.
It is easier to formulate material model constitutive laws in state variables that can
easily be calculated or measured. Therefore, Legendre-Fenchel transformation can
be used to define the complementary potential corresponding to dissipation potential.
Dissipation variables for thermo-mechanical system are given in Table 3.2.
The corresponding potentialφ
Ã
σ, A k , g
!
is the dual of the dissipation
potential φ _
E, _
V k , q
!
=T
. For details of the Legendre-Fenchel transformation of the
dissipation, potential readers are referred to Lemaitre and Chaboche (1990).
The corresponding potential must be differentiable. The normality rule also applies
to the corresponding potential. Therefore, constitutive relation can be given by
_
E ¼
∂φ
Ã
∂σ
ð3:108Þ
À _
V k ¼
∂φ
Ã
∂A k
ð3:109Þ
À
q
!
T
¼
∂φ
Ã
∂ g
!
ð3:110Þ
The potentials φ and φ
à must be nonnegative, convex, and zero at the origin to
satisfy Clausius-Duhem inequality. According to Lemaitre and Chaboche (1990),
the normality rule is sufficient to ensure the satisfaction of the second principle of
thermodynamics, but it is not a necessary condition. However, this assumption
cannot be substantiated mathematically because it ignores entropy generation due
to other mechanisms. This rule applies to “generalized standard materials” under
thermo-mechanical loads only. Lemaitre and Chaboche (1985) define standard
material as that for which only the first of the above three rules _
E
P
¼
∂φ
Ã
∂σ
applies.
This first relation yields the plasticity or viscoplasticity. The standard material here
would only include metals under thermo-mechanical loading.
Determination of dissipation potential in Newtonian mechanics is an empirical
process. The dissipation potential represents the energy dissipated in the system by
heat, mechanical work at the lattice level, and all other mechanisms. While potential
cannot be directly measured, state variables that define the potential can be measured
or indirectly calculated. Therefore, these potentials are defined in terms of state
variables _
E, _
V k ,
À q
!
T or dual variables σ, A k , g
! .
Table 3.2 Dissipation flux
variables and dual variables
(Lemaitre and Chaboche
1990)
Flux of state variables
Dual variables
_
E
σ
ÀV k
A k
À
q
!
T
g
! ¼ grad
! T
3.3 Second Law of Thermodynamics
107
more appropriate to call these forces pseudo-thermodynamic forces or pseudodissipative forces.
It is easier to formulate material model constitutive laws in state variables that can
easily be calculated or measured. Therefore, Legendre-Fenchel transformation can
be used to define the complementary potential corresponding to dissipation potential.
Dissipation variables for thermo-mechanical system are given in Table 3.2.
The corresponding potentialφ
Ã
σ, A k , g
!
is the dual of the dissipation
potential φ _
E, _
V k , q
!
=T
. For details of the Legendre-Fenchel transformation of the
dissipation, potential readers are referred to Lemaitre and Chaboche (1990).
The corresponding potential must be differentiable. The normality rule also applies
to the corresponding potential. Therefore, constitutive relation can be given by
_
E ¼
∂φ
Ã
∂σ
ð3:108Þ
À _
V k ¼
∂φ
Ã
∂A k
ð3:109Þ
À
q
!
T
¼
∂φ
Ã
∂ g
!
ð3:110Þ
The potentials φ and φ
à must be nonnegative, convex, and zero at the origin to
satisfy Clausius-Duhem inequality. According to Lemaitre and Chaboche (1990),
the normality rule is sufficient to ensure the satisfaction of the second principle of
thermodynamics, but it is not a necessary condition. However, this assumption
cannot be substantiated mathematically because it ignores entropy generation due
to other mechanisms. This rule applies to “generalized standard materials” under
thermo-mechanical loads only. Lemaitre and Chaboche (1985) define standard
material as that for which only the first of the above three rules _
E
P
¼
∂φ
Ã
∂σ
applies.
This first relation yields the plasticity or viscoplasticity. The standard material here
would only include metals under thermo-mechanical loading.
Determination of dissipation potential in Newtonian mechanics is an empirical
process. The dissipation potential represents the energy dissipated in the system by
heat, mechanical work at the lattice level, and all other mechanisms. While potential
cannot be directly measured, state variables that define the potential can be measured
or indirectly calculated. Therefore, these potentials are defined in terms of state
variables _
E, _
V k ,
À q
!
T or dual variables σ, A k , g
! .
Table 3.2 Dissipation flux
variables and dual variables
(Lemaitre and Chaboche
1990)
Flux of state variables
Dual variables
_
E
σ
ÀV k
A k
À
q
!
T
g
! ¼ grad
! T
3.3 Second Law of Thermodynamics
107