α t
ð Þ ¼
Z t
0
ffiffi ffi
2
3
r
j _
Ε
pl τ
ð Þjdτ
ð5:169Þ
This is the generalized version of equivalent plastic strain (trajectory) but with the
addition of the gradients of plastic strain. The evolution equations are complemented
by the loading/unloading conditions which allow the determination of the consistency parameter. In terms of the yield function, the following loading/unloading
condition must be satisfied:
γ ! 0 F σ, ℓ
À1 m, α
À
Á
0
ð5:170aÞ
γ ! 0 γF σ, ℓ
À1 m, α
À
Á ¼ 0
ð5:170bÞ
And the consistency condition
γ _
F σ, ℓ
À1 m, α
À
Á ¼ 0
ð5:171Þ
5.8.1.1 Consistency Parameter Determination
Using the definition of the yield function, we have
_
F ¼
∂F
∂σ
: _
σ þ
∂F
∂ℓ
2 1 m
: ℓ
À1
_
m þ
∂F
∂ε p : _
ε
pl
þ
∂F
∂ℓχ pl : _
χ
pl
ð5:172Þ
Using Hooke’s law together with the flow rule yields
_
F ¼ b n : C : _
ε þ b v : D : ℓ _
χ
½
Š
À γ b n : C : b n þ b v : D : b v
½
Š À
∂F
∂ε pl : b
n þ
∂F
∂ℓχ pl : b v
&
'
ð5:173Þ
And imposing the consistency condition yields the consistency parameter γ in the
following form:
γ ¼
b n : C : _
ε þ b v : D : ℓ _
χ
b n : C : b n þ b v : D : b v À
∂F
∂ε pl : b
n þ
∂F
∂ℓχ pl : b v
ð5:174Þ
Using
5.8 Cosserat Continuum Implementation in Unified Mechanics Theory
259