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5 Plasticity
Prescribed Strain History: Ramp
The response of the specific Prandtl kinematic hardening model to a prescribed Ramp
strain history is documented in Fig. 5.23a, b, c, d, e.
Figure 5.23a 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. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.23b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the slopes in the two elastic
phases in Fig. 5.23a, b coincide), and ˙
σ(t) = ˙
(t)/11 in the two plastic phases where
|σ(t) − 0.1 κ(t)| = 1. In particular during the holding phase σ(t) = 15/11 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11.
The resulting parallelogram-type σ = σ() diagram is highlighted in Fig. 5.23c.
It is easy to verify that the stress varies between 15/11 and −7/11 in the second
elastic phase.
Figure 5.23d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E + K ] = 50/11
(or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 40/11
(or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.23e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/11, 0, 50/11} over the time
interval t ∈ [0, t max = 10], thus κ max = 40/11 + 30/11 = 70/11.
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl kinematic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.24a, b, c, d, e.
Figure 5.24a 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. Plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.24b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 in the elastic phases where |σ(t) − K κ(t)| =
|σ(t) − 0.1 κ(t)| < σ y = 1 (E = 1, thus the slopes in the elastic phases in Fig. 5.24a,
b coincide), and ˙
(t) = ˙
σ(t)/[E − E
2
/[E + K ]] = ˙
σ(t) × 11 in the plastic phases
where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σ y = 1. Thus a = σ y /1 + [σ a − σ y ]
× 11 = 1/1 + 4 × 11 = 45 denotes the corresponding strain amplitude.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.24c. It is easy to verify that the strain varies between [0, 1], and ±[45, 43]
in the elastic phases.
5 Plasticity
Prescribed Strain History: Ramp
The response of the specific Prandtl kinematic hardening model to a prescribed Ramp
strain history is documented in Fig. 5.23a, b, c, d, e.
Figure 5.23a 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. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.23b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the slopes in the two elastic
phases in Fig. 5.23a, b coincide), and ˙
σ(t) = ˙
(t)/11 in the two plastic phases where
|σ(t) − 0.1 κ(t)| = 1. In particular during the holding phase σ(t) = 15/11 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11.
The resulting parallelogram-type σ = σ() diagram is highlighted in Fig. 5.23c.
It is easy to verify that the stress varies between 15/11 and −7/11 in the second
elastic phase.
Figure 5.23d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E + K ] = 50/11
(or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 40/11
(or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.23e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/11, 0, 50/11} over the time
interval t ∈ [0, t max = 10], thus κ max = 40/11 + 30/11 = 70/11.
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl kinematic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.24a, b, c, d, e.
Figure 5.24a 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. Plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.24b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 in the elastic phases where |σ(t) − K κ(t)| =
|σ(t) − 0.1 κ(t)| < σ y = 1 (E = 1, thus the slopes in the elastic phases in Fig. 5.24a,
b coincide), and ˙
(t) = ˙
σ(t)/[E − E
2
/[E + K ]] = ˙
σ(t) × 11 in the plastic phases
where |σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σ y = 1. Thus a = σ y /1 + [σ a − σ y ]
× 11 = 1/1 + 4 × 11 = 45 denotes the corresponding strain amplitude.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.24c. It is easy to verify that the strain varies between [0, 1], and ±[45, 43]
in the elastic phases.