6.1 Bingham Model
305
Figure 6.7a 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. Visco-plastic time steps are emphasized by
larger hollow circles, whereas rigid time steps are indicated by smaller filled circles.
Figure 6.7b showcases the resulting strain history (t) that displays a periodic
signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and
with either (t) = 0 or (t) = 42. ¯
6 for the rigid phases with |σ(t)| ∈ [0, 1].
The resulting (lens-shaped) σ = σ() diagram is highlighted in Fig. 6.7c.
Figure 6.7d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.7e follows from integrating ˙
κ(t) =
|˙ (t)| during the visco-plastic phases and constant in time during the rigid phases,
thus κ max = 5 × 42. ¯
6 = 213. ¯
3.
Prescribed Stress History: Sine
The response of the specific Bingham model to a prescribed Sine stress history is
documented in Fig. 6.8a–e.
Figure 6.8a 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
rigid time steps are indicated by smaller filled circles.
Figure 6.8b showcases the resulting strain history (t) that displays a periodic
signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and
with either (t) = 0 or (t) ≈ 59.9 for the rigid phases with |σ(t)| ∈ [0, 1].
The resulting (lentil-shaped) σ = σ() diagram is highlighted in Fig. 6.8c.
Figure 6.8d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.8e follows from integrating ˙
κ(t) =
|˙ (t)| during the visco-plastic phases and constant in time during the rigid phases,
thus κ max ≈ 5 × 59.9 ≈ 299.6.
Prescribed Stress History: Ramp
The response of the specific Bingham model to a prescribed Ramp stress history is
documented in Fig. 6.9a–e.
Figure 6.9a 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 6.9b showcases the resulting strain history (t) that displays an initial rigid
phase with (t) = 0 for σ(t) ∈ [0, 1], subsequent visco-plastic loading, creep and
visco-plastic unloading phases for σ(t) ∈ [1, 5], σ(t) = 5 and σ(t) ∈ [5, 1] with
305
Figure 6.7a 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. Visco-plastic time steps are emphasized by
larger hollow circles, whereas rigid time steps are indicated by smaller filled circles.
Figure 6.7b showcases the resulting strain history (t) that displays a periodic
signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and
with either (t) = 0 or (t) = 42. ¯
6 for the rigid phases with |σ(t)| ∈ [0, 1].
The resulting (lens-shaped) σ = σ() diagram is highlighted in Fig. 6.7c.
Figure 6.7d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.7e follows from integrating ˙
κ(t) =
|˙ (t)| during the visco-plastic phases and constant in time during the rigid phases,
thus κ max = 5 × 42. ¯
6 = 213. ¯
3.
Prescribed Stress History: Sine
The response of the specific Bingham model to a prescribed Sine stress history is
documented in Fig. 6.8a–e.
Figure 6.8a 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
rigid time steps are indicated by smaller filled circles.
Figure 6.8b showcases the resulting strain history (t) that displays a periodic
signal with non-constant (t) for the visco-plastic phases with |σ(t)| ∈ [1, 5] and
with either (t) = 0 or (t) ≈ 59.9 for the rigid phases with |σ(t)| ∈ [0, 1].
The resulting (lentil-shaped) σ = σ() diagram is highlighted in Fig. 6.8c.
Figure 6.8d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.8e follows from integrating ˙
κ(t) =
|˙ (t)| during the visco-plastic phases and constant in time during the rigid phases,
thus κ max ≈ 5 × 59.9 ≈ 299.6.
Prescribed Stress History: Ramp
The response of the specific Bingham model to a prescribed Ramp stress history is
documented in Fig. 6.9a–e.
Figure 6.9a 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 6.9b showcases the resulting strain history (t) that displays an initial rigid
phase with (t) = 0 for σ(t) ∈ [0, 1], subsequent visco-plastic loading, creep and
visco-plastic unloading phases for σ(t) ∈ [1, 5], σ(t) = 5 and σ(t) ∈ [5, 1] with