220
5 Plasticity
Prescribed Strain History: Sine
The response of the specific Prandtl model to a prescribed Sine strain history is
documented in Fig. 5.10a, b, c, d, e. (These shall be compared to the corresponding
response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.5a, b, c,
d, e.)
Figure 5.10a 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. Plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 5.10b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t)| < 1 (thus the corresponding
curve segments representing elastic loading/unloading in Fig. 5.10a, b are affine), and
˙
σ(t) = 0 in the plastic phases where |σ(t)| = 1.
The resulting σ = σ() diagram is highlighted in Fig. 5.10c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format 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 5.10d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)|, whereas p (t) stays
constant with | p (t)| = 4 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.10e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| during the plastic phases and constant in
time during the elastic phases, thus κ max = [0.5 + 4 + 0.375] × 8 = 39 (for 4.875
plastic phases of plastic arc-length 8 each).
Prescribed Strain History: Ramp
The response of the specific Prandtl model to a prescribed Ramp strain history is
documented in Fig. 5.11a, b, c, d, e. (These shall be compared to the corresponding
response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.6a, b, c,
d, e.)
Figure 5.11a 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 5.11b showcases the resulting stress history σ(t) that displays a trapezoidal
signal with ˙
σ(t) = ˙
(t) in the two elastic phases where |σ(t)| < 1 (thus the slopes
in the two elastic phases in Fig. 5.11a, b coincide), and ˙
σ(t) = 0 in the two plastic
phases where |σ(t)| = 1. In particular during the holding phase σ(t) = 1 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 4 = 1.
5 Plasticity
Prescribed Strain History: Sine
The response of the specific Prandtl model to a prescribed Sine strain history is
documented in Fig. 5.10a, b, c, d, e. (These shall be compared to the corresponding
response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.5a, b, c,
d, e.)
Figure 5.10a 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. Plastic time steps are emphasized by larger hollow circles, whereas
elastic time steps are indicated by smaller filled circles.
Figure 5.10b showcases the resulting stress history σ(t) that displays a periodic
signal with ˙
σ(t) = ˙
(t) in the elastic phases where |σ(t)| < 1 (thus the corresponding
curve segments representing elastic loading/unloading in Fig. 5.10a, b are affine), and
˙
σ(t) = 0 in the plastic phases where |σ(t)| = 1.
The resulting σ = σ() diagram is highlighted in Fig. 5.10c. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram-type format 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 5.10d demonstrates the plastic strain history p (t): during the plastic phases
p (t) evolves in parallel to the total strain with |˙ p (t)| = |˙ (t)|, whereas p (t) stays
constant with | p (t)| = 4 (or as initial value p (t) = 0) during the elastic phases.
Finally, the plastic arc-length κ(t) in Fig. 5.10e follows as a sequence of sine waves
segments from integrating ˙
κ(t) = |˙ p (t)| during the plastic phases and constant in
time during the elastic phases, thus κ max = [0.5 + 4 + 0.375] × 8 = 39 (for 4.875
plastic phases of plastic arc-length 8 each).
Prescribed Strain History: Ramp
The response of the specific Prandtl model to a prescribed Ramp strain history is
documented in Fig. 5.11a, b, c, d, e. (These shall be compared to the corresponding
response of the underlying, rigid-plastic, specific St. Venant model in Fig. 5.6a, b, c,
d, e.)
Figure 5.11a 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 5.11b showcases the resulting stress history σ(t) that displays a trapezoidal
signal with ˙
σ(t) = ˙
(t) in the two elastic phases where |σ(t)| < 1 (thus the slopes
in the two elastic phases in Fig. 5.11a, b coincide), and ˙
σ(t) = 0 in the two plastic
phases where |σ(t)| = 1. In particular during the holding phase σ(t) = 1 results as
the response to the elastic strain e (t) = (t) − p (t) = 5 − 4 = 1.