280
5 Plasticity
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 5.32d demonstrates the plastic strain history p (t): during the two plastic
phases p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E|, 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. 5.32e 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.
Prescribed Stress History: Ramp
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Ramp stress history is documented in Fig. 5.33a, b, c, d, e.
Figure 5.33a 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
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 5.33b showcases the resulting strain history (t) that displays an initial
elastic phase with ˙
(t) = ˙
σ(t), a subsequent plastic phase with ˙
(t) = ˙
σ(t) × 6 (thus
in particular during the holding phase (t) = 1/1 + 4 × 6 = 25), and a final elastic
phase with ˙
(t) = ˙
σ(t), respectively.
The resulting σ = σ() diagram is highlighted in Fig. 5.33c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is linear elastic
with slope E = 1. It is easy to verify that the strain varies between [25, 20] in the
final elastic phase.
Figure 5.33d demonstrates the plastic strain history p (t): during the plastic phase
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. 5.33e follows constant-linear-constant in
time from integrating ˙
κ(t) = |˙ p (t)| = {0, 25, 0} over the time interval t ∈ [0, t max =
10], thus κ max = 20.
5.3.10 Generic Prandtl Hardening Model: Formulation
A generic formulation of the Prandtl hardening model can be obtained from generalizing of the specific Prandtl hardening model in Fig. 5.12 by assuming the elastic
spring or/and the hardening spring or/and the frictional slider as nonlinear.
For the generic Prandtl hardening model the free energy density ψ is expressed
as a non-quadratic but convex function of − p (the elastic strain e ) and ε h (the
hardening strain)
5 Plasticity
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 5.32d demonstrates the plastic strain history p (t): during the two plastic
phases p (t) evolves with |˙ p (t)| = |˙ (t) − ˙
σ(t)/E|, 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. 5.32e 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.
Prescribed Stress History: Ramp
The response of the specific Prandtl mixed (isotropic and kinematic) hardening model
to a prescribed Ramp stress history is documented in Fig. 5.33a, b, c, d, e.
Figure 5.33a 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
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 5.33b showcases the resulting strain history (t) that displays an initial
elastic phase with ˙
(t) = ˙
σ(t), a subsequent plastic phase with ˙
(t) = ˙
σ(t) × 6 (thus
in particular during the holding phase (t) = 1/1 + 4 × 6 = 25), and a final elastic
phase with ˙
(t) = ˙
σ(t), respectively.
The resulting σ = σ() diagram is highlighted in Fig. 5.33c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is linear elastic
with slope E = 1. It is easy to verify that the strain varies between [25, 20] in the
final elastic phase.
Figure 5.33d demonstrates the plastic strain history p (t): during the plastic phase
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. 5.33e follows constant-linear-constant in
time from integrating ˙
κ(t) = |˙ p (t)| = {0, 25, 0} over the time interval t ∈ [0, t max =
10], thus κ max = 20.
5.3.10 Generic Prandtl Hardening Model: Formulation
A generic formulation of the Prandtl hardening model can be obtained from generalizing of the specific Prandtl hardening model in Fig. 5.12 by assuming the elastic
spring or/and the hardening spring or/and the frictional slider as nonlinear.
For the generic Prandtl hardening model the free energy density ψ is expressed
as a non-quadratic but convex function of − p (the elastic strain e ) and ε h (the
hardening strain)