238
5 Plasticity
10/11, whereas p (t) stays constant with | p (t)| = {0, 40/11, 360/11
2
, 3240/11
3
,
29160/11
4
, 262440/11
5
} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.15e follows as a sequence of sine
waves segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 10/11 during the plastic
phases and constant in time during the elastic phases, thus κ max = 2 × [40/11 +
360/11
2
+ 3240/11
3
+ 29160/11
4
] + 262440/11
5
= 3817640/11
5
≈ 24.
Prescribed Strain History: Ramp
The response of the specific Prandtl isotropic hardening model to a prescribed Ramp
strain history is documented in Fig. 5.16a, b, c, d, e.
Figure 5.16a 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. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.16b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t)| < 1 + 0.1 κ(t) (thus the slopes in the two elastic
phases in Fig. 5.16a, b coincide), and ˙
σ(t) = ˙
(t)/11 in the two plastic phases where
|σ(t)| = 1 + 0.1 κ(t). In particular during the holding phase σ(t) = 15/11 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11.
The resulting σ = σ() diagram is highlighted in Fig. 5.16c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (isotropically expanding in the σ direction) of the σ =
σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at = 0 and = 5 obviously tend to the elastic modulus E = 1 for
t → 0.
Figure 5.16d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E + H ] = 50/11
(or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 0.8 ×
50/11 = 40/11 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.16e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/11, 0, 50/11} over the time
interval t ∈ [0, t max = 10], thus κ max = 40/11 + 250/11
2
= 690/11
2
≈ 5.7.
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.17a, b, c, d, e.
Figure 5.17a 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.
5 Plasticity
10/11, whereas p (t) stays constant with | p (t)| = {0, 40/11, 360/11
2
, 3240/11
3
,
29160/11
4
, 262440/11
5
} during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.15e follows as a sequence of sine
waves segments from integrating ˙
κ(t) = |˙ p (t)| = |˙ (t)| × 10/11 during the plastic
phases and constant in time during the elastic phases, thus κ max = 2 × [40/11 +
360/11
2
+ 3240/11
3
+ 29160/11
4
] + 262440/11
5
= 3817640/11
5
≈ 24.
Prescribed Strain History: Ramp
The response of the specific Prandtl isotropic hardening model to a prescribed Ramp
strain history is documented in Fig. 5.16a, b, c, d, e.
Figure 5.16a 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. Plastic time steps are emphasized by larger hollow
circles, whereas elastic time steps are indicated by smaller filled circles.
Figure 5.16b showcases the resulting stress history σ(t) with ˙
σ(t) = ˙
(t) in the
two elastic phases where |σ(t)| < 1 + 0.1 κ(t) (thus the slopes in the two elastic
phases in Fig. 5.16a, b coincide), and ˙
σ(t) = ˙
(t)/11 in the two plastic phases where
|σ(t)| = 1 + 0.1 κ(t). In particular during the holding phase σ(t) = 15/11 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 0.8 × 50/11 = 15/11.
The resulting σ = σ() diagram is highlighted in Fig. 5.16c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format (isotropically expanding in the σ direction) of the σ =
σ() diagram is only approximately captured in the elastic-plastic transition, however the slopes at = 0 and = 5 obviously tend to the elastic modulus E = 1 for
t → 0.
Figure 5.16d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)| E/[E + H ] = 50/11
(or ˙
p (t) = 0 in the holding phase), whereas p (t) stays constant with p (t) = 0.8 ×
50/11 = 40/11 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.16e follows constant-linear-constantlinear in time from integrating ˙
κ(t) = |˙ p (t)| = {0, 50/11, 0, 50/11} over the time
interval t ∈ [0, t max = 10], thus κ max = 40/11 + 250/11
2
= 690/11
2
≈ 5.7.
Prescribed Stress History: Zig-Zag
The response of the specific Prandtl isotropic hardening model to a prescribed ZigZag stress history is documented in Fig. 5.17a, b, c, d, e.
Figure 5.17a 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.