6.3 Perzyna Hardening Model
393
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.29a–e and 6.13a–e, respectively.)
Figure 6.35a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude 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. Visco-plastic time steps are emphasized by larger hollow circles,
whereas elastic time steps are indicated by smaller filled circles.
Figure 6.35b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.35c. Due to mixed
(isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the direction and is isotropically expanding and
kinematically shifting back and forth in the σ direction.
Figure 6.35d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant during the elastic phases with decreasing amplitude after each
half-period (and eventually vp (t) → 1.6).
Finally, the strain arc-length κ(t) in Fig. 6.35e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 24 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Ramp strain history is documented in Fig. 6.36a–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. 5.30a–e and 6.14a–e, respectively.)
Figure 6.36a depicts the prescribed Ramp strain history (t) with maximum 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 6.36b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙
(t) = ±5 in the loading and the unloading phases.
During the holding phase with ˙
(t) = 0 the stress relaxes to σ(t) → σ y + 0.1 κ(t) +
0.1 vp (t) ≈ 1.66 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.36c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation to σ = 1.66 during the holding phase is clearly visible at = 5.
Figure 6.36d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.3 and vp (t) → 1.6 in the loading and unloading phase, respectively
(from visual inspection).
393
and visco-plastic, specific Prandtl mixed (isotropic and kinematic) hardening and
Perzyna models in Figs. 5.29a–e and 6.13a–e, respectively.)
Figure 6.35a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude 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. Visco-plastic time steps are emphasized by larger hollow circles,
whereas elastic time steps are indicated by smaller filled circles.
Figure 6.35b showcases the resulting stress history σ(t) that displays a nonperiodic, increasing signal.
The resulting σ = σ() diagram is highlighted in Fig. 6.35c. Due to mixed
(isotropic and kinematic) hardening its resulting rounded parallelogram-type format has constant amplitude in the direction and is isotropically expanding and
kinematically shifting back and forth in the σ direction.
Figure 6.35d demonstrates the corresponding visco-plastic strain history vp (t):
during the visco-plastic phases vp (t) evolves in parallel to the strain signal, whereas
vp (t) stays constant during the elastic phases with decreasing amplitude after each
half-period (and eventually vp (t) → 1.6).
Finally, the strain arc-length κ(t) in Fig. 6.35e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max = 24 (from visual inspection).
Prescribed Strain History: Ramp
The response of the specific Perzyna mixed (isotropic and kinematic) hardening
model to a prescribed Ramp strain history is documented in Fig. 6.36a–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. 5.30a–e and 6.14a–e, respectively.)
Figure 6.36a depicts the prescribed Ramp strain history (t) with maximum 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 6.36b showcases the resulting stress history σ(t) that displays an in/decreasing signal whenever ˙
(t) = ±5 in the loading and the unloading phases.
During the holding phase with ˙
(t) = 0 the stress relaxes to σ(t) → σ y + 0.1 κ(t) +
0.1 vp (t) ≈ 1.66 (from visual inspection).
The resulting σ = σ() diagram is highlighted in Fig. 6.36c. The elastic slope
(E = 1) in the loading and unloading phase are easy to verify. Likewise the stress
relaxation to σ = 1.66 during the holding phase is clearly visible at = 5.
Figure 6.36d demonstrates the corresponding visco-plastic strain history vp (t)
with vp (t) → 3.3 and vp (t) → 1.6 in the loading and unloading phase, respectively
(from visual inspection).