6.2 Perzyna Model
313
union of the elastic domain and the yield surface, compare the representation in
Fig. 5.8. Thereby, the admissible domain may either be determined directly from the
expression of the sub-differential d ˙
vp π p in Eq. 6.72, or, alternatively, from evaluating
the formal definition of the sub-differential
d ˙
vp π p (˙ vp ) = {σ p | σ p [˙
vp − ˙
vp ] ≤ σ y
|˙
vp | − |˙ vp |
∀˙
vp },
(6.74)
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 |˙
vp | and, with max ˙
vp
{σ p ˙
vp /|˙
vp |} = |σ p |, the
admissible domain follows as |σ p | ≤ σ y .
The elastic domain is defined as the interior of the admissible domain, i.e.
int A := {σ p | |σ p | − σ y < 0},
(6.75)
whereas the yield surface, which in the present one-dimensional case collapses to
the two end points σ p = ±σ y , is defined as the boundary of the admissible domain,
i.e.
∂ A := {σ p | |σ p | − σ y = 0}.
(6.76)
Collectively, the admissible domain in the σ p -space is characterized by the yield
condition
|σ p | − σ y ≤ 0.
(6.77)
States in the interior int A of the admissible domain with |σ p | < σ y are elastic,
whereas states on the boundary ∂ A of the admissible domain with |σ p | = σ y 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
(6.78a)
π
∗
p (σ p ) = max
˙
vp
{σ p ˙
vp − σ y |˙ vp | }
(6.78b)
then read
π
∗
v (σ v ) =
1
2
1
η
|σ v |
2
(6.79a)
π
∗
p (σ p ) = I A (σ p ) :=
⎧
⎨
⎩
0
|σ p | ≤ σ y
for
∞
|σ p | > σ y
⎫
⎬
⎭
,
(6.79b)
where I A denotes the indicator function of the admissible domain A in the σ p -space.
The evolution law (the associated flow rule) for the visco-plastic strain then follows
either as partial derivative of the dual viscous dissipation potential or likewise as some