370
6 Visco-Plasticity
The resulting σ = σ() diagram is highlighted in Fig. 6.28c. Due to kinematic
hardening its resulting rounded parallelogram-type format has constant amplitude in
both the and σ direction.
Figure 6.28d 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 with | vp (t)| ≈ 3.5 (from visual inspection) during the elastic
phases.
Finally, the strain arc-length κ(t) in Fig. 6.28e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 68 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna kinematic hardening model to a prescribed
Ramp strain history is documented in Fig. 6.29a–e. (These shall be compared to
the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.23a–e and 6.14a–e,
respectively.)
Figure 6.29a 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 6.29b showcases the resulting stress history σ(t) that displays an in/decreasing signal 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 + 0.1 κ(t) ≈
1.375 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.29c. 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 6.29d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.75 and vp (t) → 1.45 in the loading and unloading phase, respectively (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.29e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 6.05 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna kinematic hardening model to a prescribed
Zig-Zag stress history is documented in Fig. 6.30a–e. (These shall be compared
to the corresponding response of the underlying, elasto-plastic and visco-plastic,
specific Prandtl kinematic hardening and Perzyna models in Figs. 5.24a–e and 6.15a–
e, respectively.)
6 Visco-Plasticity
The resulting σ = σ() diagram is highlighted in Fig. 6.28c. Due to kinematic
hardening its resulting rounded parallelogram-type format has constant amplitude in
both the and σ direction.
Figure 6.28d 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 with | vp (t)| ≈ 3.5 (from visual inspection) during the elastic
phases.
Finally, the strain arc-length κ(t) in Fig. 6.28e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 68 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna kinematic hardening model to a prescribed
Ramp strain history is documented in Fig. 6.29a–e. (These shall be compared to
the corresponding response of the underlying, elasto-plastic and visco-plastic, specific Prandtl kinematic hardening and Perzyna models in Figs. 5.23a–e and 6.14a–e,
respectively.)
Figure 6.29a 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 6.29b showcases the resulting stress history σ(t) that displays an in/decreasing signal 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 + 0.1 κ(t) ≈
1.375 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.29c. 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 6.29d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.75 and vp (t) → 1.45 in the loading and unloading phase, respectively (from visual inspection).
Finally, the strain arc-length κ(t) in Fig. 6.29e follows from integrating ˙
κ(t) =
|˙ vp (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max ≈ 6.05 (from
visual inspection).
Prescribed Stress History: Zig-Zag
The response of the specific Perzyna kinematic hardening model to a prescribed
Zig-Zag stress history is documented in Fig. 6.30a–e. (These shall be compared
to the corresponding response of the underlying, elasto-plastic and visco-plastic,
specific Prandtl kinematic hardening and Perzyna models in Figs. 5.24a–e and 6.15a–
e, respectively.)