6.3 Perzyna Hardening Model
387
σ
n
vp = −E [
n
vp −
n
] =: σ
vp − E
n
vp ,
(6.242)
σ
n
hk = −K
n
hk
=: σ
hk − K
n
hk ,
σ
n
hi = −H
n
hi
=: σ
hi − H
n
hi .
Here the trial visco-plastic stress σ
vp , the trial kinematic-hardening stress σ
hk
and the trial isotropic-hardening stress σ
hi are computable exclusively from known
quantities at the beginning of the time step and follow as
σ
vp := −E [
n−1
vp −
n
],
(6.243)
σ
hk := −K
n−1
hk
,
σ
hi := −H
n−1
hi
.
Combining the visco-plastic stress and the kinematic-hardening stress at the end
of the time step and incorporating the discretized evolution laws for the visco-plastic
strain and the kinematic-hardening strain then renders
σ
n
vp + σ
n
hk = σ
vp + σ
hk − [E + K ] λ
σ
n
vp + σ
n
hk
|σ n
vp + σ
n
hk |
.
(6.244)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress
|σ
n
vp + σ
n
hk | + [E + K ] λ
σ
n
vp + σ
n
hk
|σ n
vp + σ
n
hk |
= σ
vp + σ
hk .
(6.245)
As an immediate consequence the equivalent stress and its trial value are related
via
|σ
n
vp + σ
n
hk | = |σ
vp + σ
hk | − [E + K ] λ.
(6.246)
A direct further consequence that alleviates the computation of the flow direction
at the end of the time step in terms of trial values is then obviously
σ
n
vp + σ
n
hk
|σ n
vp + σ
n
hk |
≡
σ
vp + σ
hk
|σ
vp + σ
hk |
.
(6.247)
Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore
σ
n
hi = σ
hi − H λ.
(6.248)
Consequently, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
vp + σ
n
hk | − σ y + σ
n
hi = φ
− [E + H + K ] λ.
(6.249)