6.3 Perzyna Hardening Model
337
˙
vp (σ v
) = ∂ σ v π
∗
v (σ v
),
˙
vp (σ p , σ hi ) ∈ d σ p π
∗
p (σ p , σ hi ) = d σ p I A (σ p , σ hi ),
˙
hi (σ p , σ hi ) ∈ d σ hi π
∗
p (σ p , σ hi ) = d σ hi I A (σ p , σ hi ),
(6.134)
with
∂ σ v π
∗
v (σ v ) =
1
η
σ v
(6.135a)
and
d σ p π
∗
p (σ p , σ hi )
= d σ p I A (σ p , σ hi ) =
⎧
⎪ ⎨
⎪ ⎩
0
|σ p | < σ y − σ hi
for
λ
σ p
|σ p |
|σ p | = σ y − σ hi
⎫
⎪ ⎬
⎪ ⎭
(6.135b)
and
d σ hi π
∗
p (σ p , σ hi )
= d σ hi I A (σ p , σ hi ) =
⎧
⎨
⎩
0
|σ p | < σ y − σ hi
for
λ
|σ p | = σ y − σ hi
⎫
⎬
⎭
,
(6.135c)
whereby d σ p π
∗
p and d σ hi π
∗
p denote the sets of sub-derivatives, i.e. the sub-differentials
of π
∗
p with respect to σ p and σ hi , respectively, and λ is a positive Lagrange (or rather
plastic) multiplier.
Obviously, the expressions in Eqs. 6.124, 6.125 and 6.134 are inverse relations.
The smooth viscous dissipation and dual viscous dissipation potentials π v = π v (˙ vp )
and π
∗
v = π
∗
v (σ v ) together with the resulting smooth constitutive relations σ v =
σ v (˙ vp ) and ˙
vp = ˙
vp (σ v ) are similar to those displayed in Fig. 4.2. Identifying ˙
hi
with |˙ vp | and setting σ hi = 0, the remaining non-smooth plastic dissipation and dual
plastic dissipation potentials π p = π p (˙ vp ) and π
∗
p = π
∗
p (σ p ) together with the resulting non-smooth constitutive relations σ p = σ p (˙ vp ) and ˙
vp = ˙
vp (σ p ) are similar to
those displayed in Fig. 5.3.
Visco-Plastic Stress
Alternatively, the Perzyna isotropic hardening model may be formulated further by
considering the visco-plastic stress. To this end the viscous and the plastic stress need
to be related to the visco-plastic stress.
Remarkably, since at yield the plastic stress satisfies |σ p | = σ y − σ hi , the viscous
overstress σ v = σ vp − σ p allows representation in terms of the yield condition that
is, however, evaluated in terms of the visco-plastic stress σ vp
σ v =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
+|σ vp | − σ y + σ hi
0
−|σ vp | + σ y − σ hi
if
σ vp > +σ y − σ hi
else
σ vp < −σ y + σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(6.136)