350
6 Visco-Plasticity
During the holding phase with ˙
(t) = 0 the stress relaxes to σ(t) → σ y + 0.1 κ(t) ≈
1.375 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.22c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation to σ = 1.375 during the holding phase is clearly visible at = 5.
Figure 6.22d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.75 and vp (t) → 1.925 in the loading and unloading phase, respectively (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.22e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 5.5 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 6.23a–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. 5.17a–e and 6.15a–e, respectively.)
Figure 6.23a 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 6.23b showcases the resulting strain history (t) that displays a nearly
periodic signal after the initial elastic and visco-plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 6.23c. Once the initial elastic
and visco-plastic phase is completed the σ = σ() behavior displays less and less
hysteresis in the remaining cycles and approaches a purely elastic response.
Figure 6.23d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a nearly periodic signal with decreasing amplitude after the initial
elastic and visco-plastic phase.
Finally, the strain arc-length κ(t) in Fig. 6.23e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 35 (from visual inspection).
Prescribed Stress History: Sine
The response of the specific Perzyna isotropic hardening model to a prescribed Sine
stress history is documented in Fig. 6.24a–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. 5.18a–e and 6.16a–e, respectively.)
Figure 6.24a 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
6 Visco-Plasticity
During the holding phase with ˙
(t) = 0 the stress relaxes to σ(t) → σ y + 0.1 κ(t) ≈
1.375 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.22c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation to σ = 1.375 during the holding phase is clearly visible at = 5.
Figure 6.22d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.75 and vp (t) → 1.925 in the loading and unloading phase, respectively (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.22e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 5.5 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 6.23a–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. 5.17a–e and 6.15a–e, respectively.)
Figure 6.23a 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 6.23b showcases the resulting strain history (t) that displays a nearly
periodic signal after the initial elastic and visco-plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 6.23c. Once the initial elastic
and visco-plastic phase is completed the σ = σ() behavior displays less and less
hysteresis in the remaining cycles and approaches a purely elastic response.
Figure 6.23d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a nearly periodic signal with decreasing amplitude after the initial
elastic and visco-plastic phase.
Finally, the strain arc-length κ(t) in Fig. 6.23e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 35 (from visual inspection).
Prescribed Stress History: Sine
The response of the specific Perzyna isotropic hardening model to a prescribed Sine
stress history is documented in Fig. 6.24a–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. 5.18a–e and 6.16a–e, respectively.)
Figure 6.24a 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