6.1 Bingham Model
301
Figure 6.5a 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.
Figure 6.5b showcases the resulting stress history σ(t) that displays an alternating signal with σ(t) = ±[σ y + η |˙ (t)|] = ±[1 + 0.59 | cos(ω t)|] with σ max/min =
±1.59 whenever ˙
(t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 6.5c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected vertical
slopes of the σ = σ() diagram at = 0 and = ±5 are only approximately captured,
however they obviously tend to ∞ with t → 0.
Figure 6.5d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.5e follows as a sequence of (positive and
negative) sine quarter-waves from integrating ˙
κ(t) = |˙ (t)| = a ω | cos(ω t)| over
two and a half periods, thus κ max = 50.
Prescribed Strain History: Ramp
The response of the specific Bingham model to a prescribed Ramp strain history is
documented in Fig. 6.6a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.8a–f and 5.6a–e, respectively.)
Figure 6.6a 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.
Figure 6.6b showcases the resulting stress history σ(t) that displays a stepwise
signal with σ(t) = ±1.375 whenever ˙
(t) = ±5 in the loading and the unloading
phases. During the holding phase the relaxed σ(t) = σ vp = +1 results as a reaction
to the kinematic constraint vp (t) ≡ (t) with ˙
(t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 6.6c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = 5 obviously tend to ∞ with t → 0.
Figure 6.6d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.6e follows linear-constant-linear in time
from integrating ˙
κ(t) = |˙ (t)| = {5, 0, 5} over the time interval t ∈ [0, t max = 10],
thus κ max = 10.
Prescribed Stress History: Zig-Zag
The response of the specific Bingham model to a prescribed Zig-Zag stress history
is documented in Fig. 6.7a–e.
301
Figure 6.5a 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.
Figure 6.5b showcases the resulting stress history σ(t) that displays an alternating signal with σ(t) = ±[σ y + η |˙ (t)|] = ±[1 + 0.59 | cos(ω t)|] with σ max/min =
±1.59 whenever ˙
(t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 6.5c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected vertical
slopes of the σ = σ() diagram at = 0 and = ±5 are only approximately captured,
however they obviously tend to ∞ with t → 0.
Figure 6.5d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.5e follows as a sequence of (positive and
negative) sine quarter-waves from integrating ˙
κ(t) = |˙ (t)| = a ω | cos(ω t)| over
two and a half periods, thus κ max = 50.
Prescribed Strain History: Ramp
The response of the specific Bingham model to a prescribed Ramp strain history is
documented in Fig. 6.6a–e. (These shall be compared to the corresponding response
of the underlying, viscous and rigid-plastic, specific Newton and St. Venant models
in Figs. 4.8a–f and 5.6a–e, respectively.)
Figure 6.6a 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.
Figure 6.6b showcases the resulting stress history σ(t) that displays a stepwise
signal with σ(t) = ±1.375 whenever ˙
(t) = ±5 in the loading and the unloading
phases. During the holding phase the relaxed σ(t) = σ vp = +1 results as a reaction
to the kinematic constraint vp (t) ≡ (t) with ˙
(t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 6.6c. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = 5 obviously tend to ∞ with t → 0.
Figure 6.6d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint vp (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 6.6e follows linear-constant-linear in time
from integrating ˙
κ(t) = |˙ (t)| = {5, 0, 5} over the time interval t ∈ [0, t max = 10],
thus κ max = 10.
Prescribed Stress History: Zig-Zag
The response of the specific Bingham model to a prescribed Zig-Zag stress history
is documented in Fig. 6.7a–e.