5.3 Prandtl Hardening Model
277
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Zig-Zag stress history is documented in Fig. 5.31a, b, c, d, e.
Figure 5.31a 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 5.31b 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 an initial series
of elastic and plastic phases with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 and ˙
(t) = ˙
σ(t)/[E −
E
2
/[E + H + K ]] = ˙
σ(t) × 6, respectively. Thus (1) = σ y /1 + [σ a − σ y ] × 6 =
1/1 + 4 × 6 = 25 at the end of the first plastic phase and (3) = (1) − [σ a + 1]/1 −
[σ a − 1] × 6 = 25 − 6 − 4 × 6 = −5 at the end of the second plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.31c. Once the second
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 ∓5 in the remaining elastic phase.
Figure 5.31d demonstrates the plastic strain history p (t): during the two plastic
phases p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 30 − 5/1 = 25, whereas p (t)
stays constant with p (t) = 20 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.31e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 25 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 mixed (isotropic and kinematic) hardening model
to a prescribed Sine stress history is documented in Fig. 5.32a, b, c, d, e.
Figure 5.32a 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 5.32b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) (i.e. a purely elastic phase) after an initial series of elastic and
plastic phases with ˙
(t) = ˙
σ(t) and ˙
(t) = ˙
σ(t) × 6, respectively. Thus (1) = 1/1 +
4 × 6 = 25 at the end of the first plastic phase and (3) = 25 − 6 − 4 × 6 = −5 at
the end of the second plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.32c. 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 second plastic phase is completed the cyclic σ = σ() behavior in the
277
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Zig-Zag stress history is documented in Fig. 5.31a, b, c, d, e.
Figure 5.31a 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 5.31b 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 an initial series
of elastic and plastic phases with ˙
(t) = ˙
σ(t)/E = ˙
σ(t)/1 and ˙
(t) = ˙
σ(t)/[E −
E
2
/[E + H + K ]] = ˙
σ(t) × 6, respectively. Thus (1) = σ y /1 + [σ a − σ y ] × 6 =
1/1 + 4 × 6 = 25 at the end of the first plastic phase and (3) = (1) − [σ a + 1]/1 −
[σ a − 1] × 6 = 25 − 6 − 4 × 6 = −5 at the end of the second plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.31c. Once the second
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 ∓5 in the remaining elastic phase.
Figure 5.31d demonstrates the plastic strain history p (t): during the two plastic
phases p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E| = 30 − 5/1 = 25, whereas p (t)
stays constant with p (t) = 20 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.31e follows linear in time from integrating ˙
κ(t) = |˙ p (t)| = 25 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 mixed (isotropic and kinematic) hardening model
to a prescribed Sine stress history is documented in Fig. 5.32a, b, c, d, e.
Figure 5.32a 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 5.32b showcases the resulting strain history (t) that displays a periodic
signal with ˙
(t) = ˙
σ(t) (i.e. a purely elastic phase) after an initial series of elastic and
plastic phases with ˙
(t) = ˙
σ(t) and ˙
(t) = ˙
σ(t) × 6, respectively. Thus (1) = 1/1 +
4 × 6 = 25 at the end of the first plastic phase and (3) = 25 − 6 − 4 × 6 = −5 at
the end of the second plastic phase.
The resulting σ = σ() diagram is highlighted in Fig. 5.32c. 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 second plastic phase is completed the cyclic σ = σ() behavior in the