6.3 Perzyna Hardening Model
343
|σ
n
vp | + E λ
σ
n
vp
|σ n
vp |
= σ
vp .
(6.152)
As an immediate consequence the equivalent stress and its trial value are related
via
|σ
n
vp | = |σ
vp | − E λ.
(6.153)
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
vp |
≡
σ
vp
|σ
vp |
.
(6.154)
Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore
σ
n
hi = σ
hi − H λ.
(6.155)
Consequently, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
vp | − σ y + σ
n
hi = φ
− [E + H ] λ.
(6.156)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
vp | − σ y + σ
hi .
(6.157)
Next for visco-plastic loading with λ > 0 the definition for the incremental
visco-plastic multiplier is regrouped to render
φ
n
− λ
η
t n = 0.
(6.158)
Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form
from
λ =
φ
E + H + η//t n ≥ 0.
(6.159)
Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t
n
→ 0. Once λ is computed
all other variables may be updated. In particular, the visco-plastic stress at the end
of the time step reads
σ
n
vp = σ
vp − E λ
σ
vp
|σ
vp |
.
(6.160)
The sensitivity of σ
n
vp = σ
n with respect to
n is denoted the algorithmic tangent
E a (thus dσ = E a d) and is computed from the product rule while noting that λ
343
|σ
n
vp | + E λ
σ
n
vp
|σ n
vp |
= σ
vp .
(6.152)
As an immediate consequence the equivalent stress and its trial value are related
via
|σ
n
vp | = |σ
vp | − E λ.
(6.153)
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
vp |
≡
σ
vp
|σ
vp |
.
(6.154)
Incorporating the discretized evolution law for the isotropic-hardening strain renders furthermore
σ
n
hi = σ
hi − H λ.
(6.155)
Consequently, the yield function at the end of the time step is expressed as
φ
n
:= |σ
n
vp | − σ y + σ
n
hi = φ
− [E + H ] λ.
(6.156)
Here the trial value of the yield function φ
has been defined as
φ
:= |σ
vp | − σ y + σ
hi .
(6.157)
Next for visco-plastic loading with λ > 0 the definition for the incremental
visco-plastic multiplier is regrouped to render
φ
n
− λ
η
t n = 0.
(6.158)
Thus the incremental visco-plastic multiplier λ ≥ 0 is computed in closed form
from
λ =
φ
E + H + η//t n ≥ 0.
(6.159)
Observe that λ degenerates to the plastic case for η → 0, likewise λ degenerates to zero in the limit of very fast processes with t
n
→ 0. Once λ is computed
all other variables may be updated. In particular, the visco-plastic stress at the end
of the time step reads
σ
n
vp = σ
vp − E λ
σ
vp
|σ
vp |
.
(6.160)
The sensitivity of σ
n
vp = σ
n with respect to
n is denoted the algorithmic tangent
E a (thus dσ = E a d) and is computed from the product rule while noting that λ