384
6 Visco-Plasticity
π
∗
(σ vp , σ hi , σ hk ) =
(6.233)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
1
2
|σ vp + σ hk | − [σ y − σ hi ]
2
η
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
The above relations may conveniently be condensed by the help of the Macaulay
bracket :=
1
2
[• + | • |], e.g. the dual dissipation potential is expressed as
π
∗
(σ vp , σ hi , σ hk ) =
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
.
(6.234)
Identifying ˙
hi with |˙ vp | and ˙
hk with ˙
vp and setting σ hi = 0 and σ hk = 0, the
remaining non-smooth (total) dissipation and dual (total) dissipation potentials π =
π(˙ vp ) and π
∗
= π
∗
(σ vp ) together with the resulting non-smooth constitutive relations
σ vp = σ vp (˙ vp ) and ˙
vp = ˙
vp (σ vp ) are similar to those displayed in Fig. 6.3.
The result in Eqs. 6.229, 6.230, 6.231 for the evolution of the visco-plastic and
the isotropic- and kinematic-hardening strains thus follows directly from the reverse
Legendre transformation
π(˙ vp , ˙
hi , ˙
hk ) = max
σ vp ,σ hi ,σ hk
(6.235)
d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) −
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
,
whereby d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) := σ vp ˙
vp + σ hi ˙
hi + σ hk ˙
hk denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.235
embodies the unconstrained optimization problem
˜
1/η (σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) :=
(6.236)
−d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) +
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
→ min
σ vp ,σ hi ,σ hk
,
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility
constraint |σ vp + σ hk | ≤ σ y − σ hi penalized by the penalty parameter 1/η. In accorand
σ y + H hi (σ hi )
|˙ vp (σ vp , σ hi , σ hk )| − H hi (σ hi ) ˙
hi (σ vp , σ hi , σ hk )+
K hk (σ hk )
˙
vp (σ vp , σ hi , σ hk ) − ˙
hk (σ vp , σ hi , σ hk )
=
σ y
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
η
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎬
⎪ ⎭
.
Taken together, the (total) dual dissipation potential π ∗ (σ vp , σ hi , σ hk ) follows.
6 Visco-Plasticity
π
∗
(σ vp , σ hi , σ hk ) =
(6.233)
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
1
2
|σ vp + σ hk | − [σ y − σ hi ]
2
η
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
The above relations may conveniently be condensed by the help of the Macaulay
bracket :=
1
2
[• + | • |], e.g. the dual dissipation potential is expressed as
π
∗
(σ vp , σ hi , σ hk ) =
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
.
(6.234)
Identifying ˙
hi with |˙ vp | and ˙
hk with ˙
vp and setting σ hi = 0 and σ hk = 0, the
remaining non-smooth (total) dissipation and dual (total) dissipation potentials π =
π(˙ vp ) and π
∗
= π
∗
(σ vp ) together with the resulting non-smooth constitutive relations
σ vp = σ vp (˙ vp ) and ˙
vp = ˙
vp (σ vp ) are similar to those displayed in Fig. 6.3.
The result in Eqs. 6.229, 6.230, 6.231 for the evolution of the visco-plastic and
the isotropic- and kinematic-hardening strains thus follows directly from the reverse
Legendre transformation
π(˙ vp , ˙
hi , ˙
hk ) = max
σ vp ,σ hi ,σ hk
(6.235)
d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) −
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
,
whereby d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) := σ vp ˙
vp + σ hi ˙
hi + σ hk ˙
hk denotes the dissipation power density. Interestingly, the reverse Legendre transformation in Eq. 6.235
embodies the unconstrained optimization problem
˜
1/η (σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) :=
(6.236)
−d(σ vp , σ hi , σ hk ; ˙
vp , ˙
hi , ˙
hk ) +
1
2
|σ vp + σ hk | − [σ y − σ hi ]]
2
η
→ min
σ vp ,σ hi ,σ hk
,
whereby ˜
1/η is a penalized Lagrange functional incorporating the admissibility
constraint |σ vp + σ hk | ≤ σ y − σ hi penalized by the penalty parameter 1/η. In accorand
σ y + H hi (σ hi )
|˙ vp (σ vp , σ hi , σ hk )| − H hi (σ hi ) ˙
hi (σ vp , σ hi , σ hk )+
K hk (σ hk )
˙
vp (σ vp , σ hi , σ hk ) − ˙
hk (σ vp , σ hi , σ hk )
=
σ y
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp + σ hk | < σ y − σ hi
for
|σ vp + σ hk | − [σ y − σ hi ]
η
|σ vp + σ hk | ≥ σ y − σ hi
⎫
⎪ ⎬
⎪ ⎭
.
Taken together, the (total) dual dissipation potential π ∗ (σ vp , σ hi , σ hk ) follows.