6.3 Perzyna Hardening Model
345
∂ φ
−
E + H +
η
t n
∂ ((λ) =
(6.163)
σ
vp
|σ
vp |
E −
E + H +
η
t n
∂ ((λ)
.
= 0.
As a conclusion the algorithmic tangent E a is thus finally expressed as
E
n
a = E − H 0 ((λ)
E
2
E + H + η//t n .
(6.164)
Note that, consequently, the algorithmic tangent degenerates to the plastic case
for η → 0 and λ > 0, likewise it degenerates to E a = E for t
n
→ 0. In one
dimension the algorithmic tangent trivially coincides with its continuous counterpart.
It shall be noted, however, that this is at variance with the corresponding result in
two and three dimensions.
The algorithmic step-by-step update for the specific Perzyna hardening model
capturing isotropic hardening is summarized in Table 6.8.
6.3.3 Specific Perzyna Isotropic Hardening Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag strain history is documented in Fig. 6.20a–e. (These shall be compared to the
corresponding response of the underlying, elasto-plastic and visco-plastic, specific
Prandtl isotropic hardening and Perzyna models in Figs. 5.14a–e and 6.12a–e, respectively.)
Figure 6.20a depicts the prescribed Zig-Zag strain history (t) with amplitude
a = 5 and period T = 4 in the time interval t ∈ [0, t max = 10], whereby N = 100
time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized
by larger hollow circles, whereas elastic time steps are indicated by smaller filled
circles.
Figure 6.20b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σ y + η ˙
(t) + H κ(t) = 1.375 + 0.1 κ(t)
whenever σ(t) > 1 + 0.1 κ(t) and with ˙
(t) = 5, thus σ max ≈ 3.5 (from visual
inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.20c. Due to isotropic
hardening its resulting rounded parallelogram-type format has constant amplitude in
the direction and is isotropically expanding in the σ direction.
345
∂ φ
−
E + H +
η
t n
∂ ((λ) =
(6.163)
σ
vp
|σ
vp |
E −
E + H +
η
t n
∂ ((λ)
.
= 0.
As a conclusion the algorithmic tangent E a is thus finally expressed as
E
n
a = E − H 0 ((λ)
E
2
E + H + η//t n .
(6.164)
Note that, consequently, the algorithmic tangent degenerates to the plastic case
for η → 0 and λ > 0, likewise it degenerates to E a = E for t
n
→ 0. In one
dimension the algorithmic tangent trivially coincides with its continuous counterpart.
It shall be noted, however, that this is at variance with the corresponding result in
two and three dimensions.
The algorithmic step-by-step update for the specific Perzyna hardening model
capturing isotropic hardening is summarized in Table 6.8.
6.3.3 Specific Perzyna Isotropic Hardening Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag strain history is documented in Fig. 6.20a–e. (These shall be compared to the
corresponding response of the underlying, elasto-plastic and visco-plastic, specific
Prandtl isotropic hardening and Perzyna models in Figs. 5.14a–e and 6.12a–e, respectively.)
Figure 6.20a depicts the prescribed Zig-Zag strain history (t) with amplitude
a = 5 and period T = 4 in the time interval t ∈ [0, t max = 10], whereby N = 100
time steps with t = 0.1 are computed. Visco-plastic time steps are emphasized
by larger hollow circles, whereas elastic time steps are indicated by smaller filled
circles.
Figure 6.20b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σ y + η ˙
(t) + H κ(t) = 1.375 + 0.1 κ(t)
whenever σ(t) > 1 + 0.1 κ(t) and with ˙
(t) = 5, thus σ max ≈ 3.5 (from visual
inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.20c. Due to isotropic
hardening its resulting rounded parallelogram-type format has constant amplitude in
the direction and is isotropically expanding in the σ direction.