φ ε 2 ε
p
ð
Þ, T, V k
ð
Þ¼φ ε
e , T, V k
ð
Þ
ð 5:96Þ
which shows that (Lemaitre and Chaboche 1990)
∂φ
∂ε e ¼
∂φ
∂ε
¼ À
∂φ
∂ε p
ð5:97Þ
And the following expressions define the relation between Helmholtz free energy
and thermodynamic state variables:
σ ¼ ρ
∂φ
∂ε e
ð5:98Þ
s ¼ À
∂φ
∂T
ð5:99Þ
A k ¼ ρ
∂φ
∂V k
ð5:100Þ
where A k is a thermodynamic force associated with the internal variables V k . The
vector formed by the variables is the gradient of the function φ in the space of the
variables T, ε
e , and V k . This vector is normal to the surface φ ¼ constant.
Let us return to the equation of the conservation of energy for small strains (from
Eqs. (5.57) and (5.68)):
ρ _
u ¼ Àdiv J q þ σ : _
ε þ ρr
ð5:101Þ
and replace ρ _
u by the expression derived from Eqs. (5.83a), (5.83b), (5.83c), (5.83d)
and (5.83e):
ρ _
u ¼ ρ _
φ þ ρ_ sT þ ρs _
T
ð5:102Þ
And utilizing _
φ and _
s by their expression as a function of the state variables with
the help of Eqs. (5.98)–(5.100)
_
φ ¼
∂φ
∂ε e : _
ε
e
þ
∂φ
∂T
_
T þ
∂φ
∂V k
_
V k ¼
1
ρ
σ : _
ε
e
À s _
T þ A k _
V k
ð5:103Þ
_
s ¼ À
∂
2 φ
∂ε e ∂T
: _
ε
e
À
∂
2 φ
∂T
2
_
T À
∂
2 φ
∂V k ∂T
_
V k ¼ À
1
ρ
∂σ
∂T
: _
ε
e
þ
∂s
∂T
_
T À
1
ρ
∂A k
∂T
_
V k
ð5:104Þ
We obtain
230
5 Unified Mechanics of Thermo-mechanical Analysis