6.2 Perzyna Model
327
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.15c.
Figure 6.15d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal with vp (t) ∈ [0, 42. ¯
6].
Finally, the strain arc-length κ(t) in Fig. 6.15e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 213. ¯
3.
Figure 6.15d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.7d and e since they are both exposed to the
identical stress history.
Prescribed Stress History: Sine
The response of the specific Perzyna model to a prescribed Sine stress history is documented in Fig. 6.16a–e. (These shall be compared to the corresponding response of
the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham
models in Figs. 4.45a–f and 6.8a–e, respectively.)
Figure 6.16a 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. Visco-plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 6.16b showcases the resulting strain history (t) that displays a periodic
signal with (t) ∈ [0, 47] (from visual inspection).
The resulting (lentil-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.16c.
Figure 6.16d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal with vp (t) ∈ [0, 59.9].
Finally, the strain arc-length κ(t) in Fig. 6.16e follows from integrating ˙
κ(t) =
|˙ (t)| over two and a half periods and approaches κ max = 299.6.
Figure 6.16d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.8d and e since they are both exposed to the
identical stress history.
Prescribed Stress History: Ramp
The response of the specific Perzyna model to a prescribed Ramp stress history is
documented in Fig. 6.17a–e.
Figure 6.17a 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.
327
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.15c.
Figure 6.15d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal with vp (t) ∈ [0, 42. ¯
6].
Finally, the strain arc-length κ(t) in Fig. 6.15e follows from integrating ˙
κ(t) =
|˙ vp (t)| over two and a half periods and approaches κ max ≈ 213. ¯
3.
Figure 6.15d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.7d and e since they are both exposed to the
identical stress history.
Prescribed Stress History: Sine
The response of the specific Perzyna model to a prescribed Sine stress history is documented in Fig. 6.16a–e. (These shall be compared to the corresponding response of
the underlying, visco-elastic and rigid-visco-plastic, specific Maxwell and Bingham
models in Figs. 4.45a–f and 6.8a–e, respectively.)
Figure 6.16a 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. Visco-plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 6.16b showcases the resulting strain history (t) that displays a periodic
signal with (t) ∈ [0, 47] (from visual inspection).
The resulting (lentil-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 6.16c.
Figure 6.16d demonstrates the corresponding visco-plastic strain history vp (t),
which is also a periodic signal with vp (t) ∈ [0, 59.9].
Finally, the strain arc-length κ(t) in Fig. 6.16e follows from integrating ˙
κ(t) =
|˙ (t)| over two and a half periods and approaches κ max = 299.6.
Figure 6.16d and e follow exactly the corresponding signals of the rigid-viscoplastic, specific Bingham model in Fig. 6.8d and e since they are both exposed to the
identical stress history.
Prescribed Stress History: Ramp
The response of the specific Perzyna model to a prescribed Ramp stress history is
documented in Fig. 6.17a–e.
Figure 6.17a 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.