216
5 Plasticity
σ
n
p = −E [
n
p −
n
] =: σ
p − E
n
p .
(5.77)
Here the trial plastic stress σ
p is computable exclusively from known quantities at
the beginning of the time step and follows as
σ
p := −E [
n−1
p
−
n
].
(5.78)
Incorporating the discretized evolution law for the plastic strain then renders
σ
n
p = σ
p − E λ
σ
n
p
|σ n
p |
.
(5.79)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress
|σ
n
p | + E λ
σ
n
p
|σ n
p |
= σ
p .
(5.80)
As an immediate consequence the equivalent stress and its trial value are related via
|σ
n
p | = |σ
p | − E λ.
(5.81)
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 |
.
(5.82)
Eventually, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
p | − σ y = φ
− E λ.
(5.83)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
p | − σ y .
(5.84)
Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is
computed in closed form from
λ =
φ
E
≥ 0.
(5.85)
Once λ is computed all other variables may be updated. In particular, the plastic
stress at the end of the time step reads
5 Plasticity
σ
n
p = −E [
n
p −
n
] =: σ
p − E
n
p .
(5.77)
Here the trial plastic stress σ
p is computable exclusively from known quantities at
the beginning of the time step and follows as
σ
p := −E [
n−1
p
−
n
].
(5.78)
Incorporating the discretized evolution law for the plastic strain then renders
σ
n
p = σ
p − E λ
σ
n
p
|σ n
p |
.
(5.79)
This relation is regrouped in order to separate the unknowns at the end of the time
step from the known trial stress
|σ
n
p | + E λ
σ
n
p
|σ n
p |
= σ
p .
(5.80)
As an immediate consequence the equivalent stress and its trial value are related via
|σ
n
p | = |σ
p | − E λ.
(5.81)
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 |
.
(5.82)
Eventually, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
p | − σ y = φ
− E λ.
(5.83)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
p | − σ y .
(5.84)
Thus the Lagrange multiplier λ ≥ 0 (enforcing the admissibility constraint) is
computed in closed form from
λ =
φ
E
≥ 0.
(5.85)
Once λ is computed all other variables may be updated. In particular, the plastic
stress at the end of the time step reads