5.3 Prandtl Hardening Model
241
Figure 5.17b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 (i.e. a purely elastic phase) after the initial
elastic and the subsequent plastic phase with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 and ˙
(t) =
˙
σ(t)/[E − E
2
/[E + H ]] = ˙
σ(t) × 11, respectively. Thus (1) = σ y /1 + [σ a − σ y ]
× 11 = 1/1 + 4 × 11 = 45 at the end of the plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.17c. Once the plastic phase
is completed the cyclic σ = σ() behavior in the remaining elastic phase is linear
elastic with slope E = 1. It is easy to verify that the strain varies between [45, 35]
in the remaining elastic phase.
Figure 5.17d 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. 5.17e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 50 during the plastic phase and constant in time during the
elastic phases, thus κ max = 40.
Prescribed Stress History: Sine
The response of the specific Prandtl isotropic hardening model to a prescribed Sine
stress history is documented in Fig. 5.18a, b, c, d, e.
Figure 5.18a 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 5.18b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) (i.e. a purely elastic phase) after the initial elastic and the
subsequent plastic phase with ˙
(t) = ˙
σ(t) and ˙
(t) = ˙
σ(t) × 11, respectively. Thus
(1) = 1/1 + 4 × 11 = 45 at the end of the plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.18c. 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.
Once the plastic phase is completed the cyclic σ = σ() behavior in the remaining
elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain
varies between [45, 35] in the remaining elastic phase.
Figure 5.18d 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. 5.18e follows as sinusoidal in time from
integrating ˙
κ(t) = |˙ p (t)| during the plastic phase and constant in time during the
elastic phases, thus κ max = 40.
241
Figure 5.17b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 (i.e. a purely elastic phase) after the initial
elastic and the subsequent plastic phase with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 and ˙
(t) =
˙
σ(t)/[E − E
2
/[E + H ]] = ˙
σ(t) × 11, respectively. Thus (1) = σ y /1 + [σ a − σ y ]
× 11 = 1/1 + 4 × 11 = 45 at the end of the plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.17c. Once the plastic phase
is completed the cyclic σ = σ() behavior in the remaining elastic phase is linear
elastic with slope E = 1. It is easy to verify that the strain varies between [45, 35]
in the remaining elastic phase.
Figure 5.17d 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. 5.17e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 50 during the plastic phase and constant in time during the
elastic phases, thus κ max = 40.
Prescribed Stress History: Sine
The response of the specific Prandtl isotropic hardening model to a prescribed Sine
stress history is documented in Fig. 5.18a, b, c, d, e.
Figure 5.18a 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 5.18b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) (i.e. a purely elastic phase) after the initial elastic and the
subsequent plastic phase with ˙
(t) = ˙
σ(t) and ˙
(t) = ˙
σ(t) × 11, respectively. Thus
(1) = 1/1 + 4 × 11 = 45 at the end of the plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.18c. 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.
Once the plastic phase is completed the cyclic σ = σ() behavior in the remaining
elastic phase is linear elastic with slope E = 1. It is easy to verify that the strain
varies between [45, 35] in the remaining elastic phase.
Figure 5.18d 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. 5.18e follows as sinusoidal in time from
integrating ˙
κ(t) = |˙ p (t)| during the plastic phase and constant in time during the
elastic phases, thus κ max = 40.