336
6 Visco-Plasticity
whereby ˙
vp denotes any admissible visco-plastic strain rate. Then at ˙
vp = 0 it holds
for any admissible ˙
vp that σ p ˙
vp ≤ [σ y + H hi ] |˙
vp | and, with max ˙
vp
{σ p ˙
vp /|˙
vp |} =
|σ p |, the admissible domain follows as |σ p | ≤ σ y + H hi . Moreover, the subdifferential d ˙
hi π p reduces to the partial derivative ∂ ˙
hi π p and renders σ hi = −H hi .
Thus the admissible domain is eventually expressed as |σ p | ≤ σ y − σ hi .
The elastic domain is defined as the interior of the admissible domain, i.e.
int A :=
{σ p , σ hi } | |σ p | − [σ y − σ hi ] < 0
,
(6.129)
whereas the yield surface, which in the present one-dimensional case collapses to
the two lines σ p = ±[σ y − σ hi ], is defined as the boundary of the admissible domain,
i.e.
∂ A :=
{σ p , σ hi } | |σ p | − [σ y − σ hi ] = 0
.
(6.130)
Collectively, the admissible domain in the {σ p , σ hi }-space is characterized by the
yield condition
|σ p | − [σ y − σ hi ] ≤ 0.
(6.131)
States in the interior int A of the admissible domain with |σ p | < σ y − σ hi are
elastic, whereas states on the boundary ∂ A of the admissible domain with |σ p | =
σ y − σ hi are visco-plastic.
The corresponding dual viscous and plastic dissipation potentials π
∗
v and π
∗
p , as
determined from the Legendre transformations
π
∗
v (σ v
) = max
˙
vp
σ v ˙
vp −
1
2
η |˙ vp |
2
(6.132a)
π
∗
p (σ p , σ hi ) = max
˙
vp ,˙ hi
{σ p ˙
vp + σ hi ˙
hi − [σ y + H hi ] |˙ vp | + H hi ˙
hi }
(6.132b)
then read with the stationarity condition σ hi = −H hi (note the minus sign)
π
∗
v (σ v
) =
1
2
1
η
|σ v |
2
(6.133a)
π
∗
p (σ p , σ hi ) = I A (σ p , σ hi ) :=
⎧
⎨
⎩
0
|σ p | ≤ σ y − σ hi
for
∞
|σ p | > σ y − σ hi
⎫
⎬
⎭
,
(6.133b)
where I A denotes the indicator function of the admissible domain A in the {σ p , σ hi }space. The evolution laws (the associated flow rules) for the visco-plastic and the
isotropic-hardening strains then follow either as the partial derivative of the dual
viscous dissipation potential or likewise as some sub-derivatives of the dual plastic
dissipation potential, in either case with respect to their conjugated variables
6 Visco-Plasticity
whereby ˙
vp denotes any admissible visco-plastic strain rate. Then at ˙
vp = 0 it holds
for any admissible ˙
vp that σ p ˙
vp ≤ [σ y + H hi ] |˙
vp | and, with max ˙
vp
{σ p ˙
vp /|˙
vp |} =
|σ p |, the admissible domain follows as |σ p | ≤ σ y + H hi . Moreover, the subdifferential d ˙
hi π p reduces to the partial derivative ∂ ˙
hi π p and renders σ hi = −H hi .
Thus the admissible domain is eventually expressed as |σ p | ≤ σ y − σ hi .
The elastic domain is defined as the interior of the admissible domain, i.e.
int A :=
{σ p , σ hi } | |σ p | − [σ y − σ hi ] < 0
,
(6.129)
whereas the yield surface, which in the present one-dimensional case collapses to
the two lines σ p = ±[σ y − σ hi ], is defined as the boundary of the admissible domain,
i.e.
∂ A :=
{σ p , σ hi } | |σ p | − [σ y − σ hi ] = 0
.
(6.130)
Collectively, the admissible domain in the {σ p , σ hi }-space is characterized by the
yield condition
|σ p | − [σ y − σ hi ] ≤ 0.
(6.131)
States in the interior int A of the admissible domain with |σ p | < σ y − σ hi are
elastic, whereas states on the boundary ∂ A of the admissible domain with |σ p | =
σ y − σ hi are visco-plastic.
The corresponding dual viscous and plastic dissipation potentials π
∗
v and π
∗
p , as
determined from the Legendre transformations
π
∗
v (σ v
) = max
˙
vp
σ v ˙
vp −
1
2
η |˙ vp |
2
(6.132a)
π
∗
p (σ p , σ hi ) = max
˙
vp ,˙ hi
{σ p ˙
vp + σ hi ˙
hi − [σ y + H hi ] |˙ vp | + H hi ˙
hi }
(6.132b)
then read with the stationarity condition σ hi = −H hi (note the minus sign)
π
∗
v (σ v
) =
1
2
1
η
|σ v |
2
(6.133a)
π
∗
p (σ p , σ hi ) = I A (σ p , σ hi ) :=
⎧
⎨
⎩
0
|σ p | ≤ σ y − σ hi
for
∞
|σ p | > σ y − σ hi
⎫
⎬
⎭
,
(6.133b)
where I A denotes the indicator function of the admissible domain A in the {σ p , σ hi }space. The evolution laws (the associated flow rules) for the visco-plastic and the
isotropic-hardening strains then follow either as the partial derivative of the dual
viscous dissipation potential or likewise as some sub-derivatives of the dual plastic
dissipation potential, in either case with respect to their conjugated variables