324
6 Visco-Plasticity
Figure 6.13d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 4 (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.13e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 37.5 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna model to a prescribed Ramp strain history
is documented in Fig. 6.14a–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. 4.43a–f, 6.6a–e and 5.11a–e,
respectively.)
Figure 6.14a depicts the prescribed Ramp strain history (t) with maximum 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
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 6.14b showcases the resulting stress history σ(t) that displays an increasing
signal with |σ(t)| → σ y + η |˙ (t)| = 1.375 whenever ˙
(t) = ±5 in the loading and
the unloading phases. During the holding phase with ˙
(t) = 0 the stress relaxes to
σ(t) → σ y = 1.
The resulting σ = σ() diagram is highlighted in Fig. 6.14c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation from σ = 1.375 to σ = 1 during the holding phase is clearly visible at
= 5.
Figure 6.14d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 4 and | vp (t)| → 1.3 in the loading and unloading phase, respectively
(from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.14e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 6.7 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna model to a prescribed Zig-Zag stress history is
documented in Fig. 6.15a–e. (These shall be compared to the corresponding response
of the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham models in Figs. 4.44a–f and 6.7a–e, respectively.)
Figure 6.15a 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 6.15b showcases the resulting strain history (t) that displays a periodic
signal with (t) ∈ [0, 45] (from visual inspection).
6 Visco-Plasticity
Figure 6.13d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 4 (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.13e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 37.5 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna model to a prescribed Ramp strain history
is documented in Fig. 6.14a–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. 4.43a–f, 6.6a–e and 5.11a–e,
respectively.)
Figure 6.14a depicts the prescribed Ramp strain history (t) with maximum 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
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 6.14b showcases the resulting stress history σ(t) that displays an increasing
signal with |σ(t)| → σ y + η |˙ (t)| = 1.375 whenever ˙
(t) = ±5 in the loading and
the unloading phases. During the holding phase with ˙
(t) = 0 the stress relaxes to
σ(t) → σ y = 1.
The resulting σ = σ() diagram is highlighted in Fig. 6.14c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation from σ = 1.375 to σ = 1 during the holding phase is clearly visible at
= 5.
Figure 6.14d demonstrates the corresponding visco-plastic strain history vp (t)
with | vp (t)| → 4 and | vp (t)| → 1.3 in the loading and unloading phase, respectively
(from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.14e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 6.7 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna model to a prescribed Zig-Zag stress history is
documented in Fig. 6.15a–e. (These shall be compared to the corresponding response
of the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham models in Figs. 4.44a–f and 6.7a–e, respectively.)
Figure 6.15a 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 6.15b showcases the resulting strain history (t) that displays a periodic
signal with (t) ∈ [0, 45] (from visual inspection).