250
5 Plasticity
σ
n
p = −E [
n
p −
n
] =: σ
p − E
n
p ,
(5.161)
σ
n
hk = −K
n
hk
=: σ
hk − K
n
hk .
Here the trial plastic stress σ
p and the trial kinematic-hardening stress σ
hk are computable exclusively from known quantities at the beginning of the time step and
follow as
σ
p := −E [
n−1
p
−
n
],
(5.162)
σ
hk := −K
n−1
hk
.
Combining the plastic stress and the kinematic-hardening stress at the end of the
time step and incorporating the discretized evolution laws for the plastic strain and
the kinematic-hardening strain then renders
σ
n
p + σ
n
hk = σ
p + σ
hk − [E + K ] λ
σ
n
p + σ
n
hk
|σ n
p + σ
n
hk |
.
(5.163)
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 + σ
n
hk | + [E + K ] λ
σ
n
p + σ
n
hk
|σ n
p + σ
n
hk |
= σ
p + σ
hk .
(5.164)
As an immediate consequence the equivalent stress and its trial value are related via
|σ
n
p + σ
n
hk | = |σ
p + σ
hk | − [E + K ] λ.
(5.165)
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
hk
|σ n
p + σ
n
hk |
≡
σ
p + σ
hk
|σ
p + σ
hk |
.
(5.166)
Eventually, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
p + σ
n
hk | − σ y = φ
− [E + K ] λ.
(5.167)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
p + σ
hk | − σ y .
(5.168)
Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is
computed in closed form from