6.3 Perzyna Hardening Model
339
σ vp
σ v
−[σ y − σ hi ]
+[σ y − σ hi ]
σ vp
σ p
+[σ y − σ hi ]
−[σ y − σ hi ]
Fig. 6.19 Specific Perzyna isotropic hardening model: The visco-plastic stress σ vp = σ v + σ p is
the sum of the viscous overstress σ v and the plastic stress σ p . The viscous damper is only activated
once the load carrying capacity of the isotropic-hardening frictional slider is exceeded. Accordingly
the viscous overstress is identically zero σ v ≡ 0 for |σ vp | − [σ y − σ hi ] ≤ 0 (left), while the plastic
stress remains constant (at a particular σ hi that may be considered to expand along a third dimension
perpendicular to the plane displayed in the above) with |σ p | = σ y − σ hi for |σ vp | − [σ y − σ hi ] > 0
(right)
π
∗
(σ vp , σ hi ) = max
˙
vp ,˙ hi
σ vp ˙
vp + σ hi ˙
hi
(6.140)
−
1
2
η |˙ vp |
2
− [σ y + H hi ] |˙ vp | + H hi ˙
hi
then reads
4
4 The expressions for the evolution of the visco-plastic and the isotropic-hardening strains in
Eqs. 6.138, 6.139 result in
σ vp ˙
vp (σ vp , σ hi ) + σ hi ˙
hi (σ vp , σ hi ) =
|σ vp | + σ hi
⎧
⎪ ⎨
⎪ ⎩
0
|σ vp | < σ y − σ hi
for
|σ vp | − [σ y − σ hi ]
η
|σ vp | ≥ σ y − σ hi
⎫
⎪ ⎬
⎪ ⎭
and
1
2
η |˙ vp (σ vp , σ hi )|
2 =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0
|σ vp | < σ y − σ hi
for
1
2
|σ vp | − [σ y − σ hi ]
2
η
|σ vp | ≥ σ y − σ hi
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
and
[σ y + H hi (σ hi )] |˙ vp (σ vp , σ hi )| − H hi (σ hi ) ˙
hi (σ vp , σ hi ) =