6.3 Perzyna Hardening Model
373
Figure 6.30a depicts the prescribed Zig-Zag stress history σ(t) with amplitude
σ 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 6.30b showcases the resulting strain history (t) that displays a periodic
signal after an initial transient phase.
The resulting (distorted lens-shaped) σ = σ() diagram that also displays the
initial transient phase is highlighted in Fig. 6.30c.
Figure 6.30d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal after the initial transient phase.
Finally, the strain arc-length κ(t) in Fig. 6.30e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 162 (from visual inspection).
Prescribed Stress History: Sine
The response of the specific Perzyna kinematic hardening model to a prescribed Sine
stress history is documented in Fig. 6.31a–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. 5.25a–e and 6.16a–e, respectively.)
Figure 6.31a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 6.31b showcases the resulting strain history (t) that displays a periodic
signal after the initial transient phase.
The resulting (distorted lentil-shaped) σ = σ() diagram is highlighted in
Fig. 6.31c.
Figure 6.31d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal after the initial transient phase.
Finally, the strain arc-length κ(t) in Fig. 6.31e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 218 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna kinematic hardening model to a prescribed
Ramp stress history is documented in Fig. 6.32a–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. 5.26a–e and 6.17a–e,
respectively.)
Figure 6.32a depicts the prescribed Ramp stress history σ(t) with maximum σ a =
5, loading phase during t ∈ [t 0 = 0, t 1 = 1), holding phase during t ∈ [t 1 = 1, t 2 =
9], and unloading phase during t ∈ (t 2 = 9, t 3 = 10], whereby N = 100 time steps
373
Figure 6.30a depicts the prescribed Zig-Zag stress history σ(t) with amplitude
σ 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 6.30b showcases the resulting strain history (t) that displays a periodic
signal after an initial transient phase.
The resulting (distorted lens-shaped) σ = σ() diagram that also displays the
initial transient phase is highlighted in Fig. 6.30c.
Figure 6.30d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal after the initial transient phase.
Finally, the strain arc-length κ(t) in Fig. 6.30e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 162 (from visual inspection).
Prescribed Stress History: Sine
The response of the specific Perzyna kinematic hardening model to a prescribed Sine
stress history is documented in Fig. 6.31a–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. 5.25a–e and 6.16a–e, respectively.)
Figure 6.31a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 6.31b showcases the resulting strain history (t) that displays a periodic
signal after the initial transient phase.
The resulting (distorted lentil-shaped) σ = σ() diagram is highlighted in
Fig. 6.31c.
Figure 6.31d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal after the initial transient phase.
Finally, the strain arc-length κ(t) in Fig. 6.31e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 218 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna kinematic hardening model to a prescribed
Ramp stress history is documented in Fig. 6.32a–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. 5.26a–e and 6.17a–e,
respectively.)
Figure 6.32a depicts the prescribed Ramp stress history σ(t) with maximum σ a =
5, loading phase during t ∈ [t 0 = 0, t 1 = 1), holding phase during t ∈ [t 1 = 1, t 2 =
9], and unloading phase during t ∈ (t 2 = 9, t 3 = 10], whereby N = 100 time steps