F σ, ℓ
À1 m, α
À
Á ¼ jξ
Φ
j À
ffiffi ffi
2
3
r
K α
ð Þ
Flow rule
_
ε
pl
ij ¼ γ
∂F
∂σ ij
γ
f ij
jξ ij j
γb n and ℓ _
χ
pl
ij ¼ γ
∂F
∂ℓ
À1 m ij
γℓ
À1 C ij
jξ ij j
γb ν
Hardening laws
_
α ¼
ffiffi ffi
2
3
r
γ
_
β
Φ
ij ¼ 1 À Φ
ð
Þ
2
3
H
0
γb n ij and ℓ _
η
Φ
ij ¼ 1 À Φ
ð
Þ
2
3
H
0
γb ν ij
Consistency parameter
γ ¼
φ F
ð Þ
h
i
η
5.8.3.1 Viscoplastic Creep Law
The evolution of the generalized viscoplastic strain is specified by the flow rule
which in the enhanced theory specifies creep laws for the strains and the strain
gradients. The following viscoplastic strain rate function is one of the generalized
models. It accounts for multiple creep mechanisms in the polycrystalline materials as
well as coarsening of the crystals:
_
ε
vp
ij ¼
AD 0 Eb
kθ
F
h i
E
n b
d
p
e
ÀQ=Rθ ∂F
∂σ ij
ð5:204Þ
ℓ _
χ
vp
ij ¼
AD 0 Eb
kθ
F
h i
E
n b
d
p
e
ÀQ=Rθ
∂F
∂ℓ
À1 m ij
ð5:205Þ
_
Ε
vp ¼
AD 0 Eb
kθ
F
h i
E
n b
d
p
e
ÀQ=Rθ ∂F
∂Σ
ð5:206Þ
The fluidity parameter η can be summarized as
5.8 Cosserat Continuum Implementation in Unified Mechanics Theory
265