232
5 Plasticity
σ
n
p = −E [
n
p −
n
] =: σ
p − E
n
p ,
(5.125)
σ
n
hi = −H
n
hi
=: σ
hi − H
n
hi .
Here the trial plastic stress σ
p 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
p := −E [
n−1
p
−
n
],
(5.126)
σ
hi := −H
n−1
hi
.
Incorporating the discretized evolution law for the plastic strain then renders
σ
n
p = σ
p − E λ
σ
n
p
|σ n
p |
.
(5.127)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stresses
|σ
n
p | + E λ
σ
n
p
|σ n
p |
= σ
p .
(5.128)
As an immediate consequence the equivalent stress and its trial value are related via
|σ
n
p | = |σ
p | − E λ.
(5.129)
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
p
|σ n
p |
≡
σ
p
|σ
p |
.
(5.130)
Incorporating the discretized evolution law for the isotropic-hardening strain renders
furthermore
σ
n
hi = σ
hi − H λ.
(5.131)
Consequently, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
p | − σ y + σ
n
hi = φ
− [E + H ] λ.
(5.132)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
p | − σ y + σ
hi .
(5.133)