332
6 Visco-Plasticity
Table 6.6 Summary of the generic Perzyna model
(1) Strain
= e + vp
(2) Energy ψ = ψ( − vp )
(3) Stress
σ = ∂ ψ ≡ σ ≡ −σ
vp
(4) Potential π = π(˙ vp )
(5) Stress
σ vp ∈ d ˙
vp π σ
vp
or
(4) Potential π ∗ =
1
2 φ(σ vp ) 2 /η
(5) Evolution ˙
vp = λ ∂ σvp φ with λ : = =φ(σ vp )/η
as
π
∗
(σ vp ) =
⎧
⎨
⎩
0
φ(σ vp ) ≤ 0
for
1
2
φ(σ vp )
2
/η
φ(σ vp ) > 0
⎫
⎬
⎭
=
1
2
φ(σ vp )
2
η
.
(6.116)
Finally for an equivalent (visco-plastic) stress that is homogeneous of degree one
in the visco-plastic stress (thus σ vp ∂ σ vp ϕ = ϕ), the dissipation d = σ vp ˙
vp is exclusively given in terms of the overstress function φ (with abbreviation λ := =φ/η ≥ 0
for the visco-plastic multiplier and equivalent stress ϕ = φ + σ y ≥ 0), since then
d = λ σ vp ∂ σ vp ϕ = λ ϕ = =φ [φ + σ y ]/η.
(6.117)
The generic Perzyna model is summarized in Table 6.6.
6.3 Perzyna Hardening Model
The Perzyna model of a hardening elasto-visco-plastic solid (in short the Perzyna
hardening model) consists of a serial arrangement of (1) an elastic spring and (2)
a hardening viscous frictional slider consisting of a parallel arrangement of (i) a
frictional slider, (ii) a viscous dashpot and (iii) a hardening spring (see the sketch of
the specific Perzyna hardening model in Fig. 6.18).
The basic kinematic assumption of the Perzyna hardening model is the additive
decomposition of the total strain into the elastic strain e (representing the elongation
of the elastic spring) and the visco-plastic strain vp (representing the elongation of
the hardening viscous frictional slider), i.e.
= e + vp .
(6.118)