5.3 Prandtl Hardening Model
259
Figure 5.24d demonstrates the plastic strain history p (t): during the plastic phase
p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 55 − 5/1 = 50, whereas p (t) stays
constant with | p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.24e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 50 during the plastic phases and constant in time during the
elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases
of plastic arc-length 80 each).
Prescribed Stress History: Sine
The response of the specific Prandtl kinematic hardening model to a prescribed Sine
stress history is documented in Fig. 5.25a, b, c, d, e.
Figure 5.25a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 5.25b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the
corresponding curve segments representing elastic loading/unloading in Fig. 5.25a,
b are affine), and ˙
(t) = ˙
σ(t) × 11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1.
Thus a = 45 denotes the corresponding strain amplitude.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.25c. 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 strain varies between [0, 1], and
±[45, 43] in the elastic phases.
Figure 5.25d demonstrates the plastic strain history p (t): during the plastic
phase p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E|, whereas p (t) stays constant with
| p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.25e follows as sinusoidal in time from
integrating ˙
κ(t) = |˙ p (t)| during the plastic phases and constant in time during the
elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases
of plastic arc-length 80 each).
Prescribed Stress History: Ramp
The response of the specific Prandtl kinematic hardening model to a prescribed Ramp
stress history is documented in Fig. 5.26a, b, c, d, e.
Figure 5.26a depicts the prescribed Ramp stress history σ(t) with maximum σ 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
259
Figure 5.24d demonstrates the plastic strain history p (t): during the plastic phase
p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 55 − 5/1 = 50, whereas p (t) stays
constant with | p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.24e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 50 during the plastic phases and constant in time during the
elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases
of plastic arc-length 80 each).
Prescribed Stress History: Sine
The response of the specific Prandtl kinematic hardening model to a prescribed Sine
stress history is documented in Fig. 5.25a, b, c, d, e.
Figure 5.25a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 5.25b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) in the elastic phases where |σ(t) − 0.1 κ(t)| < 1 (thus the
corresponding curve segments representing elastic loading/unloading in Fig. 5.25a,
b are affine), and ˙
(t) = ˙
σ(t) × 11 in the plastic phases where |σ(t) − 0.1 κ(t)| = 1.
Thus a = 45 denotes the corresponding strain amplitude.
The resulting cyclic parallelogram-type σ = σ() diagram is highlighted in
Fig. 5.25c. 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 strain varies between [0, 1], and
±[45, 43] in the elastic phases.
Figure 5.25d demonstrates the plastic strain history p (t): during the plastic
phase p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E|, whereas p (t) stays constant with
| p (t)| = 40 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.25e follows as sinusoidal in time from
integrating ˙
κ(t) = |˙ p (t)| during the plastic phases and constant in time during the
elastic phases, thus κ max = [0.5 + 4 + 0.375] × 80 = 390 (for 4.875 plastic phases
of plastic arc-length 80 each).
Prescribed Stress History: Ramp
The response of the specific Prandtl kinematic hardening model to a prescribed Ramp
stress history is documented in Fig. 5.26a, b, c, d, e.
Figure 5.26a depicts the prescribed Ramp stress history σ(t) with maximum σ 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