6.2 Perzyna Model
321
6.2.3 Specific Perzyna Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna model to a prescribed Zig-Zag strain history is documented in Fig. 6.12a–e. (These shall be compared to the corresponding
response of the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell, Bingham and Prandtl models in Figs. 4.41a–f, 6.4a–e and 5.9a–e,
respectively.)
Figure 6.12a 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 6.12b showcases the resulting stress history σ(t) that displays a periodic
signal with |σ(t)| → σ y + η |˙ (t)| = 1.375 (since ˙
(t) = ±5) at the Zig-Zag turning
points.
The resulting σ = σ() diagram is highlighted in Fig. 6.12c. The initial elastic
slope (E = 1) and the elastic slope upon strain reversal at |σ(t)| = 1.375 are easy to
verify.
Figure 6.12d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 5 − 1.375/1 = 3.625.
Finally, the strain arc-length κ(t) in Fig. 6.12e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 35 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna model to a prescribed Sine strain history is documented in Fig. 6.13a–e. (These shall be compared to the corresponding response of
the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell,
Bingham and Prandtl models in Figs. 4.42a–f, 6.5a–e and 5.10a–e, respectively.)
Figure 6.13a 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 6.13b showcases the resulting stress history σ(t) that displays—after an
initial phase—a (slightly upwards shifted) periodic signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.13c. The initial elastic
slope (E = 1) and the elastic slope upon strain reversal at |(t)| = 5 are easy to
verify.
321
6.2.3 Specific Perzyna Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna model to a prescribed Zig-Zag strain history is documented in Fig. 6.12a–e. (These shall be compared to the corresponding
response of the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell, Bingham and Prandtl models in Figs. 4.41a–f, 6.4a–e and 5.9a–e,
respectively.)
Figure 6.12a 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 6.12b showcases the resulting stress history σ(t) that displays a periodic
signal with |σ(t)| → σ y + η |˙ (t)| = 1.375 (since ˙
(t) = ±5) at the Zig-Zag turning
points.
The resulting σ = σ() diagram is highlighted in Fig. 6.12c. The initial elastic
slope (E = 1) and the elastic slope upon strain reversal at |σ(t)| = 1.375 are easy to
verify.
Figure 6.12d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 5 − 1.375/1 = 3.625.
Finally, the strain arc-length κ(t) in Fig. 6.12e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 35 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna model to a prescribed Sine strain history is documented in Fig. 6.13a–e. (These shall be compared to the corresponding response of
the underlying, visco-elastic, rigid-visco-plastic and elasto-plastic, specific Maxwell,
Bingham and Prandtl models in Figs. 4.42a–f, 6.5a–e and 5.10a–e, respectively.)
Figure 6.13a 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 6.13b showcases the resulting stress history σ(t) that displays—after an
initial phase—a (slightly upwards shifted) periodic signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.13c. The initial elastic
slope (E = 1) and the elastic slope upon strain reversal at |(t)| = 5 are easy to
verify.