316
6 Visco-Plasticity
then reads
3
π
∗
(σ vp ) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp | < σ y
for
1
2
|σ vp | − σ y
2
η
|σ vp | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.85)
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 ) =
1
2
|σ vp | − σ y
2
η
.
(6.86)
The 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 Eq. 6.83 for the evolution of the visco-plastic strain thus follows
directly from the reverse Legendre transformation
π(˙ vp ) = max
σ vp
d(σ vp ; ˙
vp ) −
1
2
|σ vp | − σ y
2
η
,
(6.87)
whereby d(σ vp ; ˙
vp ) := σ vp ˙
vp denotes the dissipation power density. Interestingly,
the reverse Legendre transformation in Eq. 6.87 embodies the unconstrained optimization problem
˜
1/η (σ vp ; ˙
vp ) := −d(σ vp ; ˙
vp ) +
1
2
|σ vp | − σ y
2
η
→ min
σ vp
,
(6.88)
3 The expression for the evolution of the visco-plastic strain in Eq. 6.83 results in
σ vp ˙
vp (σ vp ) = |σ vp |
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
η
|σ vp | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
and
1
2
η |˙ vp (σ vp )|
2 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp | < σ y
for
1
2
|σ vp | − σ y
2
η
|σ vp | ≥ σ y
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
and
σ y |˙ vp (σ vp )| = σ y
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp | < σ y
for
|σ vp | − σ y
η
|σ vp | ≥ σ y
⎫
⎪ ⎬
⎪ ⎭
.
Taken together, the (total) dual dissipation potential π ∗ (σ vp ) follows.