6.3 Perzyna Hardening Model
399
The resulting σ = σ() diagram is highlighted in Fig. 6.38c. Once the initial elastic
and visco-plastic phase is completed the σ = σ() behavior displays less and less
hysteresis in the remaining cycles and approaches a purely elastic response.
Figure 6.38d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a nearly periodic signal with decreasing amplitude after the initial
elastic and visco-plastic phase.
Finally, the strain arc-length κ(t) in Fig. 6.38e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 40 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Ramp stress history is documented in Fig. 6.39a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.33a–e and 6.17a–e, respectively.)
Figure 6.39a 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. Visco-plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 6.39b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 25 and a purely
elastic behaviour in the unloading phase with (t) → 20. The nonlinear creep during
the holding phase saturates due to the mixed (isotropic and kinematic) hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.39c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is purely elastic with
σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 20. Likewise
the creep towards = 25 during the holding phase is clearly visible at σ = 5.
Figure 6.39d demonstrates the corresponding smoothly and monotonically
increasing visco-plastic strain history vp (t) with vp (t) → 20.
Finally, the strain arc-length κ(t) in Fig. 6.39e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max = 20.
6.3.10 Generic Perzyna Hardening Model: Formulation
A generic formulation of the Perzyna hardening model can be obtained from generalizing the specific Perzyna hardening model in Fig. 6.18 by assuming the elastic
spring or/and the hardening spring or/and the viscous dashpot or/and the frictional
slider as nonlinear.
For the generic Perzyna hardening model the free energy density ψ is expressed
as a non-quadratic but convex function of − vp (the elastic strain e ) and ε h (the
399
The resulting σ = σ() diagram is highlighted in Fig. 6.38c. Once the initial elastic
and visco-plastic phase is completed the σ = σ() behavior displays less and less
hysteresis in the remaining cycles and approaches a purely elastic response.
Figure 6.38d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a nearly periodic signal with decreasing amplitude after the initial
elastic and visco-plastic phase.
Finally, the strain arc-length κ(t) in Fig. 6.38e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 40 (from visual inspection).
Prescribed Stress History: Ramp
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Ramp stress history is documented in Fig. 6.39a–e. (These
shall be compared to the corresponding response of the underlying, elasto-plastic
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.33a–e and 6.17a–e, respectively.)
Figure 6.39a 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. Visco-plastic time steps are emphasized by larger
hollow circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 6.39b showcases the resulting strain history (t) that displays a smoothly
increasing signal in the loading and holding phases with (t) → 25 and a purely
elastic behaviour in the unloading phase with (t) → 20. The nonlinear creep during
the holding phase saturates due to the mixed (isotropic and kinematic) hardening.
The resulting σ = σ() diagram is highlighted in Fig. 6.39c. Once the holding
phase is completed the σ = σ() behavior in the unloading phase is purely elastic with
σ(t) ∈ [5, 0] and slope E = 1, whereby the strain approaches (t) → 20. Likewise
the creep towards = 25 during the holding phase is clearly visible at σ = 5.
Figure 6.39d demonstrates the corresponding smoothly and monotonically
increasing visco-plastic strain history vp (t) with vp (t) → 20.
Finally, the strain arc-length κ(t) in Fig. 6.39e follows from integrating ˙
κ(t) =
|˙ (t)| over the time interval t ∈ [0, t max = 10] and approaches κ max = 20.
6.3.10 Generic Perzyna Hardening Model: Formulation
A generic formulation of the Perzyna hardening model can be obtained from generalizing the specific Perzyna hardening model in Fig. 6.18 by assuming the elastic
spring or/and the hardening spring or/and the viscous dashpot or/and the frictional
slider as nonlinear.
For the generic Perzyna hardening model the free energy density ψ is expressed
as a non-quadratic but convex function of − vp (the elastic strain e ) and ε h (the