46
D. Dudin and I. Keller
Considering σ :˙ ε = σ m ˙
ε m + s : ˙
e, where the Cauchy stress tensor is decomposed
into the spherical and deviatoric parts σ = σ m I + s, and using relations (3.5), (3.6),
we can rewrite the thermodynamic inequality (3.13) as
μ A − F A
V A
+ σ m
V A ˙
C A +
μ B − F B
V B
+ σ m
V B ˙
C B − J A · ∇μ A − J B · ∇μ B
+ s : ˙
e
v
+ s : ˙
e
e
+ σ m ˙
ε
e
m −
∂ F e
∂ε e : ˙
ε
e
≥ 0.
(3.14)
Then, following (Brassart et al. 2018), we introduce in terms of the current configuration the volume concentrations of atoms c A , c B , C A = c A , C B = c B , the fluxes
and gradients J A ·∇μ A = j A ·∇
μ A , J B ·∇μ B = j B ·∇
μ B , and denote the rates
of volumetric insertion as i A = V A ˙
C A //, i B = V B ˙
C B //. As the result, relation
(3.14) can be finally written as
μ A − F A
V A
+ σ m
i A +
μ B − F B
V B
+ σ m
i B + s : ˙
e
v
− j A · ∇
μ A − j B · ∇
μ B
+ s : ˙
e
e
+ σ m ˙
ε
e
m −
1
∂ F e
∂ε e : ˙
ε
e
≥ 0.
(3.15)
3.2.4 Elastic Relations and Functions of State
In the absence of irreversible processes, (3.15) degenerates into the equality
s : ˙
e
e
+ σ m ˙
ε
e
m −
1
∂ F e
∂ε e : ˙
ε
e
= 0.
(3.16)
Taking into account the smallness of strains, the expression for the energy of
elastic strains can be written as
F e = Ge
e
: e
e
+
1
2
K ε
e2
m ,
(3.17)
where G is the shear modulus, K is the bulk modulus of elasticity.
Substituting (3.17) into Eq. (3.16) leads to the relation
s − 2Ge
e
: ˙
e
e
+
σ m − K ε
e
m
˙
ε
e
m = 0.
(3.18)
The independence of the bulk and shear elastic strain rates suggests the validity
of the generalized Hooke law
s = 2Ge
e
, σ m = K ε
e
m .
(3.19)