354
6 Visco-Plasticity
computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 6.24b 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. 6.24c. 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 6.24d 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. 6.24e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 40 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna isotropic hardening model to a prescribed Ramp
stress history is documented in Fig. 6.25a–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. 5.19a–e and 6.17a–e, respectively.)
Figure 6.25a 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
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 6.25b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 45 and a purely
elastic behaviour in the unloading phase with (t) → 40. The nonlinear creep during
the holding phase saturates due to the isotropic hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.25c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is purely elastic with
σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 40. Likewise
the creep towards = 45 during the holding phase is clearly visible at σ = 5.
Figure 6.25d demonstrates the corresponding smoothly and monotonically increasing visco-plastic strain history vp (t) with vp (t) → 40.
Finally, the strain arc-length κ(t) in Fig. 6.25e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max = 40.
6.3.4 Specific Perzyna Kinematic Hardening Model:
Formulation
The specific Perzyna kinematic hardening model, similar to that displayed in Fig.
6.18 (however with the hardening modulus H and the hardening strain ε h coinciding
6 Visco-Plasticity
computed. Visco-plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 6.24b 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. 6.24c. 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 6.24d 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. 6.24e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 40 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna isotropic hardening model to a prescribed Ramp
stress history is documented in Fig. 6.25a–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. 5.19a–e and 6.17a–e, respectively.)
Figure 6.25a 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
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 6.25b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 45 and a purely
elastic behaviour in the unloading phase with (t) → 40. The nonlinear creep during
the holding phase saturates due to the isotropic hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.25c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is purely elastic with
σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 40. Likewise
the creep towards = 45 during the holding phase is clearly visible at σ = 5.
Figure 6.25d demonstrates the corresponding smoothly and monotonically increasing visco-plastic strain history vp (t) with vp (t) → 40.
Finally, the strain arc-length κ(t) in Fig. 6.25e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max = 40.
6.3.4 Specific Perzyna Kinematic Hardening Model:
Formulation
The specific Perzyna kinematic hardening model, similar to that displayed in Fig.
6.18 (however with the hardening modulus H and the hardening strain ε h coinciding