6.3 Perzyna Hardening Model
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6.3.6 Specific Perzyna Kinematic Hardening Model:
Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna kinematic hardening model to a prescribed
Zig-Zag strain history is documented in Fig. 6.27a–e. (These shall be compared
to the corresponding response of the underlying, elasto-plastic and visco-plastic,
specific Prandtl kinematic hardening and Perzyna models in Figs. 5.21a–e and 6.12a–
e, respectively.)
Figure 6.27a 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 6.27b showcases the resulting stress history σ(t) that displays a periodic, saw-tooth-type signal with σ(t) = σ y + η ˙
(t) + K vp (t) = 1.375 + 0.1 vp (t)
whenever σ(t) > 1 + 0.1 vp (t) and with ˙
(t) = 5, thus σ max ≈ 1.715 (from visual
inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.27c. Due to kinematic
hardening its resulting rounded parallelogram-type format has constant amplitude in
both the and σ direction.
Figure 6.27d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant with | vp (t)| ≈ 3.4 (from visual inspection) during the elastic
phases.
Finally, the strain arc-length κ(t) in Fig. 6.27e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 65 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna kinematic hardening model to a prescribed Sine
strain history is documented in Fig. 6.28a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl
kinematic hardening and Perzyna models in Figs. 5.22a–e and 6.13a–e, respectively.)
Figure 6.28a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T
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 6.28b showcases the resulting stress history σ(t) that displays a periodic
signal.
367
6.3.6 Specific Perzyna Kinematic Hardening Model:
Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna kinematic hardening model to a prescribed
Zig-Zag strain history is documented in Fig. 6.27a–e. (These shall be compared
to the corresponding response of the underlying, elasto-plastic and visco-plastic,
specific Prandtl kinematic hardening and Perzyna models in Figs. 5.21a–e and 6.12a–
e, respectively.)
Figure 6.27a 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 6.27b showcases the resulting stress history σ(t) that displays a periodic, saw-tooth-type signal with σ(t) = σ y + η ˙
(t) + K vp (t) = 1.375 + 0.1 vp (t)
whenever σ(t) > 1 + 0.1 vp (t) and with ˙
(t) = 5, thus σ max ≈ 1.715 (from visual
inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.27c. Due to kinematic
hardening its resulting rounded parallelogram-type format has constant amplitude in
both the and σ direction.
Figure 6.27d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant with | vp (t)| ≈ 3.4 (from visual inspection) during the elastic
phases.
Finally, the strain arc-length κ(t) in Fig. 6.27e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 65 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna kinematic hardening model to a prescribed Sine
strain history is documented in Fig. 6.28a–e. (These shall be compared to the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl
kinematic hardening and Perzyna models in Figs. 5.22a–e and 6.13a–e, respectively.)
Figure 6.28a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T
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 6.28b showcases the resulting stress history σ(t) that displays a periodic
signal.