6.3 Perzyna Hardening Model
361
π
∗
(σ vp , σ hk ) = max
˙
vp ,˙ hk
σ vp ˙
vp + σ hk ˙
hk
(6.186)
−
1
2
η |˙ vp |
2
− σ y |˙ vp | − K hk [˙ vp − ˙
hk ]
then reads
5
π
∗
(σ vp , σ hk ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y
for
1
2
|σ vp + σ hk | − σ y
2
η
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.187)
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 hk ) =
1
2
|σ vp + σ hk | − σ y
2
η
.
(6.188)
Identifying ˙
hk with ˙
vp and setting 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. 6.184, 6.185 for the evolution of the visco-plastic and the
kinematic-hardening strains thus follows directly from the reverse Legendre trans5 The expressions for the evolution of the visco-plastic and the kinematic-hardening strains in
Eqs. 6.184, 6.185 result in
σ vp ˙
vp (σ vp , σ hk ) + σ hk ˙
hk (σ vp , σ hk ) =
|σ vp + σ hk |
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp + σ hk | < σ y
for
|σ vp + σ hk | − σ y
η
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
and
1
2
η |˙ vp (σ vp , σ hk )|
2 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp + σ hk | < σ y
for
1
2
|σ vp + σ hk | − σ y
2
η
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
and
σ y |˙ vp (σ vp , σ hk )| + K hk (σ hk )
˙
vp (σ vp , σ hk ) − ˙
hk (σ vp , σ hk )
=
σ y
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp + σ hk | < σ y
for
|σ vp + σ hk | − σ y
η
|σ vp + σ hk | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
.
Taken together, the (total) dual dissipation potential π ∗ (σ vp , σ hk ) follows.