Àdiv J q ¼ ρT
∂s
∂T
_
T À σ
: _
ε 2 _
ε
e
ð
ÞþA k V k À ρr À T
∂σ
∂T
: _
ε
e
þ
∂A k
∂T
_
V k
ð5:105Þ
By introducing the specific heat defined by
C ¼ T
∂s
∂T
ð5:106Þ
and taking into account Fourier’s Law for isotropic materials
div J q ¼ Àk div grad T
ð
Þ¼Àk∇
2 T
ð5:107Þ
where ∇
2 denotes the Laplacian operator. Using _
ε
p
¼ _
ε 2 _
ε
e , we obtain
k∇
2 T ¼ ρC _
T À σ : _
ε
p
þ A k V k À ρr À T
∂σ
∂T
: _
ε
e
þ
∂A k
∂T
_
V k
ð5:108Þ
This is the fully coupled thermo-mechanical equation, which can simulate the
evolution of temperature influenced by the mechanical work with properly imposed
boundary conditions. A k _
V k represents the non-recoverable energy in the materials
corresponding to other dissipation phenomena. If we simplify the problem to
thermo-mechanical loading at small strain rates on pure metals, we may be able to
ignore other dissipation terms, and then we can write
A k _
V k % 0
ð5:109Þ
which results in the fully coupled elastoplastic thermo-mechanical equation
k∇
2 T ¼ ρC _
T À σ : _
ε
p
2 ρr À T
∂σ
∂T
: _
ε
e
ð5:110Þ
Equation (5.110) also allows us to calculate heat flux J q generated due to elastic
and/or inelastic work in a solid body.
For the isotropic linear thermoelastic materials, the stress-strain relationship in
Newtonian mechanics is given by
σ ij ¼ λδ ij ε kk þ 2με ij À 3λ þ 2μ
ð
Þ δ ij α T À T 0
ð
Þ
ð5:111Þ
where T 0 is the reference temperature, α is the isotropic thermal expansion coefficient, and λ and μ are the Lame’s coefficients
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
231
∂s
∂T
_
T À σ
: _
ε 2 _
ε
e
ð
ÞþA k V k À ρr À T
∂σ
∂T
: _
ε
e
þ
∂A k
∂T
_
V k
ð5:105Þ
By introducing the specific heat defined by
C ¼ T
∂s
∂T
ð5:106Þ
and taking into account Fourier’s Law for isotropic materials
div J q ¼ Àk div grad T
ð
Þ¼Àk∇
2 T
ð5:107Þ
where ∇
2 denotes the Laplacian operator. Using _
ε
p
¼ _
ε 2 _
ε
e , we obtain
k∇
2 T ¼ ρC _
T À σ : _
ε
p
þ A k V k À ρr À T
∂σ
∂T
: _
ε
e
þ
∂A k
∂T
_
V k
ð5:108Þ
This is the fully coupled thermo-mechanical equation, which can simulate the
evolution of temperature influenced by the mechanical work with properly imposed
boundary conditions. A k _
V k represents the non-recoverable energy in the materials
corresponding to other dissipation phenomena. If we simplify the problem to
thermo-mechanical loading at small strain rates on pure metals, we may be able to
ignore other dissipation terms, and then we can write
A k _
V k % 0
ð5:109Þ
which results in the fully coupled elastoplastic thermo-mechanical equation
k∇
2 T ¼ ρC _
T À σ : _
ε
p
2 ρr À T
∂σ
∂T
: _
ε
e
ð5:110Þ
Equation (5.110) also allows us to calculate heat flux J q generated due to elastic
and/or inelastic work in a solid body.
For the isotropic linear thermoelastic materials, the stress-strain relationship in
Newtonian mechanics is given by
σ ij ¼ λδ ij ε kk þ 2με ij À 3λ þ 2μ
ð
Þ δ ij α T À T 0
ð
Þ
ð5:111Þ
where T 0 is the reference temperature, α is the isotropic thermal expansion coefficient, and λ and μ are the Lame’s coefficients
5.4 Thermodynamic Fundamental Equation in Thermo-mechanical Problems
231