390
6 Visco-Plasticity
dimension the algorithmic tangent trivially coincides with its continuous counterpart.
It shall be noted, however, that this is at variance with the corresponding result in
two and three dimensions.
The algorithmic step-by-step update for the specific Perzyna hardening model
capturing mixed hardening is summarized in Table 6.12.
6.3.9 Specific Perzyna Mixed Hardening Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Zig-Zag strain history is documented in Fig. 6.34a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.28a–e and 6.12a–e, respectively.)
Figure 6.34a 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 6.34b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σ y + η ˙
(t) + H κ(t) + K vp (t) = 1.375 +
0.1 κ(t) + 0.1 vp (t) whenever σ(t) > 1 + 0.1 κ(t) + 0.1 vp (t) and with ˙
(t) = 5,
thus σ max ≈ 3.55 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.34c. Due to mixed
(isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the direction and is isotropically expanding and
kinematically shifting back and forth in the σ direction.
Figure 6.34d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant during the elastic phases with decreasing amplitude after each
half-period (and eventually vp (t) → 1.5).
Finally, the strain arc-length κ(t) in Fig. 6.34e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 20.25 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Sine strain history is documented in Fig. 6.35a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic
6 Visco-Plasticity
dimension the algorithmic tangent trivially coincides with its continuous counterpart.
It shall be noted, however, that this is at variance with the corresponding result in
two and three dimensions.
The algorithmic step-by-step update for the specific Perzyna hardening model
capturing mixed hardening is summarized in Table 6.12.
6.3.9 Specific Perzyna Mixed Hardening Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Zig-Zag strain history is documented in Fig. 6.34a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.28a–e and 6.12a–e, respectively.)
Figure 6.34a 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 6.34b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal with σ(t) = σ y + η ˙
(t) + H κ(t) + K vp (t) = 1.375 +
0.1 κ(t) + 0.1 vp (t) whenever σ(t) > 1 + 0.1 κ(t) + 0.1 vp (t) and with ˙
(t) = 5,
thus σ max ≈ 3.55 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.34c. Due to mixed
(isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the direction and is isotropically expanding and
kinematically shifting back and forth in the σ direction.
Figure 6.34d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant during the elastic phases with decreasing amplitude after each
half-period (and eventually vp (t) → 1.5).
Finally, the strain arc-length κ(t) in Fig. 6.34e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 20.25 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Sine strain history is documented in Fig. 6.35a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic