380
6 Visco-Plasticity
π
∗
v (σ v ) =
1
2
1
η
|σ v |
2
(6.224a)
π
∗
p (σ p , σ hi , σ hk ) = I A (σ p , σ hi , σ hk ) :=
(6.224b)
⎧
⎨
⎩
0
|σ p + σ hk | ≤ σ y − σ hi
for
∞
|σ p + σ hk | > σ y − σ hi
⎫
⎬
⎭
,
where I A denotes the indicator function of the admissible domain A in the
{σ p , σ hi , σ hk }-space. The evolution laws (the associated flow rules) for the viscoplastic and the isotropic- and kinematic-hardening strains then follow either as the
partial derivative of the dual viscous dissipation potential or likewise as some subderivatives of the dual plastic dissipation potential, in either case with respect to their
conjugated variables
˙
vp (σ v
) = ∂ σ v π
∗
v (σ v
),
˙
vp (σ p , σ hi , σ hk ) ∈ d σ p π
∗
p (σ p , σ hi , σ hk ) = d σ p I A (σ p , σ hi , σ hk ),
˙
hi (σ p , σ hi , σ hk ) ∈ d σ hi π
∗
p (σ p , σ hi , σ hk ) = d σ hi I A (σ p , σ hi , σ hk ),
˙
hk (σ p , σ hi , σ hk ) ∈ d σ hk π
∗
p (σ p , σ hk , σ hk ) = d σ hk I A (σ p , σ hk , σ hk ),
(6.225)
with
∂ σ v π
∗
v (σ v ) =
1
η
σ v
(6.226a)
and
d σ p π
∗
p (σ p , σ hi , σ hk ) = d σ p I A (σ p , σ hi , σ hk ) =
(6.226b)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ p + σ hk | < σ y − σ hi
for
λ
σ p + σ hk
|σ p + σ hk |
|σ p + σ hk | = σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
and
d σ hi π
∗
p (σ p , σ hi , σ hk ) = d σ hi I A (σ p , σ hi , σ hk ) =
(6.226c)
⎧
⎨
⎩
0
|σ p + σ hk | < σ y − σ hi
for
λ
|σ p + σ hk | = σ y − σ hi
⎫
⎬
⎭
and