D
p
¼ D
p
ð7:79Þ
Therefore, plastic stretch rate tensor D
p is not objective but invariant to changes in
current reference frame. To summarize, it has been shown that
F, R, _
F, L, W, F
e , R
e , R
p , L
e , W
e are not objective, C,U, F
p , U
e , C
e , U
p , C
p , L
p , D
p
are not objective but invariant to changes in current reference frame, and B,V,D, V
e ,
B
e , V
p , B
p , D
e are objective.
7.3 Unified Mechanics Theory Formulation for Finite
Strain
A thermodynamic potential is concave with respect to temperature (θ) and convex
with respect to all other internal state variables. It provides a basis for satisfying
conditions of thermodynamic stability imposed by Clausius-Duhem inequality.
Specific Helmholtz free energy (ψ) forms such a basis. It is defined as the difference
between specific internal energy (u) and product of absolute temperature (θ) and
specific entropy (s)
ψ ¼ u À θs
ð7:80Þ
First law of thermodynamics (also known as conservation of energy law) states
that energy can be transported, or converted from one form to another, but cannot be
destroyed or created. Accordingly, internal energy of a system can be increased by
heat flow into the system (δq), heat generated within the system due to internal
scattering (r), mechanical work done on the system by external load (δw), or all other
kinds of work done on the system (δw
0
) during any process. Considering only
thermo-mechanical effects, rate of change in internal energy, _
u can be stated as
_
u ¼ Q þ l þ r
ð7:81Þ
where Q is the rate of net heat flow into the system and l is rate of net work done on
the system and r is the heat generated in the system, due to internal friction
(scattering) chemical reactions etc. Second law of thermodynamic states that there
exists a state function, entropy, which increases in universe for all type of processes
due to entropy production. Unlike energy, entropy (s) is not only transferred across
boundaries of system, but it may also be created in the system which is called as
entropy production (S). Clausius-Duhem inequality describes second law of thermodynamics in terms of a nonnegative entropy production rate (γ) per unit volume
for any kind of irreversible process which is defined as
7.3 Unified Mechanics Theory Formulation for Finite Strain
351
p
¼ D
p
ð7:79Þ
Therefore, plastic stretch rate tensor D
p is not objective but invariant to changes in
current reference frame. To summarize, it has been shown that
F, R, _
F, L, W, F
e , R
e , R
p , L
e , W
e are not objective, C,U, F
p , U
e , C
e , U
p , C
p , L
p , D
p
are not objective but invariant to changes in current reference frame, and B,V,D, V
e ,
B
e , V
p , B
p , D
e are objective.
7.3 Unified Mechanics Theory Formulation for Finite
Strain
A thermodynamic potential is concave with respect to temperature (θ) and convex
with respect to all other internal state variables. It provides a basis for satisfying
conditions of thermodynamic stability imposed by Clausius-Duhem inequality.
Specific Helmholtz free energy (ψ) forms such a basis. It is defined as the difference
between specific internal energy (u) and product of absolute temperature (θ) and
specific entropy (s)
ψ ¼ u À θs
ð7:80Þ
First law of thermodynamics (also known as conservation of energy law) states
that energy can be transported, or converted from one form to another, but cannot be
destroyed or created. Accordingly, internal energy of a system can be increased by
heat flow into the system (δq), heat generated within the system due to internal
scattering (r), mechanical work done on the system by external load (δw), or all other
kinds of work done on the system (δw
0
) during any process. Considering only
thermo-mechanical effects, rate of change in internal energy, _
u can be stated as
_
u ¼ Q þ l þ r
ð7:81Þ
where Q is the rate of net heat flow into the system and l is rate of net work done on
the system and r is the heat generated in the system, due to internal friction
(scattering) chemical reactions etc. Second law of thermodynamic states that there
exists a state function, entropy, which increases in universe for all type of processes
due to entropy production. Unlike energy, entropy (s) is not only transferred across
boundaries of system, but it may also be created in the system which is called as
entropy production (S). Clausius-Duhem inequality describes second law of thermodynamics in terms of a nonnegative entropy production rate (γ) per unit volume
for any kind of irreversible process which is defined as
7.3 Unified Mechanics Theory Formulation for Finite Strain
351