346
6 Visco-Plasticity
Figure 6.20d 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. 6.20e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 21.25 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna isotropic hardening model to a prescribed Sine
strain history is documented in Fig. 6.21a–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. 5.15a–e and 6.13a–e, respectively.)
Figure 6.21a 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 6.21b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.21c. Due to isotropic
hardening its resulting rounded parallelogram-type format has constant amplitude in
the direction and is isotropically expanding in the σ direction.
Figure 6.21d 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.
Finally, the strain arc-length κ(t) in Fig. 6.21e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 23 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna isotropic hardening model to a prescribed Ramp
strain history is documented in Fig. 6.22a–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. 5.16a–e and 6.14a–e, respectively.)
Figure 6.22a 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 6.22b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙
(t) = ±5 in the loading and the unloading phases.
6 Visco-Plasticity
Figure 6.20d 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. 6.20e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 21.25 (from visual inspection).
Prescribed Strain History: Sine
The response of the specific Perzyna isotropic hardening model to a prescribed Sine
strain history is documented in Fig. 6.21a–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. 5.15a–e and 6.13a–e, respectively.)
Figure 6.21a 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 6.21b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.21c. Due to isotropic
hardening its resulting rounded parallelogram-type format has constant amplitude in
the direction and is isotropically expanding in the σ direction.
Figure 6.21d 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.
Finally, the strain arc-length κ(t) in Fig. 6.21e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 23 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna isotropic hardening model to a prescribed Ramp
strain history is documented in Fig. 6.22a–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. 5.16a–e and 6.14a–e, respectively.)
Figure 6.22a 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 6.22b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙
(t) = ±5 in the loading and the unloading phases.