254
5 Plasticity
Figure 5.21b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = E ˙
(t) 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. 5.21a, b
coincide), and ˙
σ(t) = [E − E
2
/[E + K ]] ˙
(t) = ˙
(t)/11 in the plastic phases where
|σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σ y = 1.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.21c. It is easy to verify that the stress varies between [0, 1] , ±15/11 and
∓7/11 in the elastic phases.
Figure 5.21d 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 ] = 5 ×
10/11, 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. 5.21e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/11 during the plastic phases and constant in time
during the elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80/11 = 390/11 (for
4.875 plastic phases of plastic arc-length 80/11 each).
Prescribed Strain History: Sine
The response of the specific Prandtl kinematic hardening model to a prescribed Sine
strain history is documented in Fig. 5.22a, b, c, d, e.
Figure 5.22a 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. Plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 5.22b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the
corresponding curve segments representing elastic loading/unloading in Fig. 5.22a,
b are affine), and ˙
σ(t) = ˙
(t)/11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.22c. Due to the finite sized time step t and corresponding finite sized strain
increment the expected σ = σ() behavior is only approximately captured in
the elastic-plastic transition. It is easy to verify that the stress varies between [0, 1],
±15/11 and ∓7/11 in the elastic phases.
Figure 5.22d 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 ] = |˙ (t)| ×
10/11, 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. 5.22e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 10/11 during the plastic phases
and constant in time during the elastic phases, thus κ max = [0.5 + 4 + 0.375] ×
80/11 = 390/11 (for 4.875 plastic phases of plastic arc-length 80/11 each).
5 Plasticity
Figure 5.21b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = E ˙
(t) 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. 5.21a, b
coincide), and ˙
σ(t) = [E − E
2
/[E + K ]] ˙
(t) = ˙
(t)/11 in the plastic phases where
|σ(t) − K κ(t)| = |σ(t) − 0.1 κ(t)| = σ y = 1.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.21c. It is easy to verify that the stress varies between [0, 1] , ±15/11 and
∓7/11 in the elastic phases.
Figure 5.21d 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 ] = 5 ×
10/11, 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. 5.21e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 5 × 10/11 during the plastic phases and constant in time
during the elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80/11 = 390/11 (for
4.875 plastic phases of plastic arc-length 80/11 each).
Prescribed Strain History: Sine
The response of the specific Prandtl kinematic hardening model to a prescribed Sine
strain history is documented in Fig. 5.22a, b, c, d, e.
Figure 5.22a 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. Plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 5.22b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the
corresponding curve segments representing elastic loading/unloading in Fig. 5.22a,
b are affine), and ˙
σ(t) = ˙
(t)/11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.22c. Due to the finite sized time step t and corresponding finite sized strain
increment the expected σ = σ() behavior is only approximately captured in
the elastic-plastic transition. It is easy to verify that the stress varies between [0, 1],
±15/11 and ∓7/11 in the elastic phases.
Figure 5.22d 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 ] = |˙ (t)| ×
10/11, 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. 5.22e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 10/11 during the plastic phases
and constant in time during the elastic phases, thus κ max = [0.5 + 4 + 0.375] ×
80/11 = 390/11 (for 4.875 plastic phases of plastic arc-length 80/11 each).