5.1 St. Venant Model
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Figure 5.4a depicts the prescribed Zig-Zag strain history (t) with amplitude 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.
Figure 5.4b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±σ y = ±1 whenever ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 5.4c. 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 5.4d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint p (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 5.4e follows linear in time from integrating ˙
κ(t) = |˙ (t)| = 5 over two and a half periods, thus κ max = 50.
Prescribed Strain History: Sine
The response of the specific St. Venant model to a prescribed Sine strain history is
documented in Fig. 5.5a, b, c, d, e.
Figure 5.5a 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 5.5b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±1 whenever ˙
(t) = a ω cos(ω t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 5.5c. 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 5.5d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint p (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 5.5e 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 St. Venant model to a prescribed Ramp strain history
is documented in Fig. 5.6a, b, c, d, e.
Figure 5.6a 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 5.6b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±1 whenever ˙
(t) = ±5 in the loading and the unloading phases.
In particular during the holding phase σ(t) = σ p = +1 results as a reaction to the
kinematic constraint p (t) ≡ (t).
205
Figure 5.4a depicts the prescribed Zig-Zag strain history (t) with amplitude 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.
Figure 5.4b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±σ y = ±1 whenever ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 5.4c. 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 5.4d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint p (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 5.4e follows linear in time from integrating ˙
κ(t) = |˙ (t)| = 5 over two and a half periods, thus κ max = 50.
Prescribed Strain History: Sine
The response of the specific St. Venant model to a prescribed Sine strain history is
documented in Fig. 5.5a, b, c, d, e.
Figure 5.5a 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 5.5b showcases the resulting stress history σ(t) that displays a block signal
with σ(t) = ±1 whenever ˙
(t) = a ω cos(ω t) = 0.
The resulting σ = σ() diagram is highlighted in Fig. 5.5c. 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 5.5d clearly demonstrates that the augmented Lagrange multiplier format
effectively enforces the constraint p (t) ≡ (t).
Finally, the strain arc-length κ(t) in Fig. 5.5e 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 St. Venant model to a prescribed Ramp strain history
is documented in Fig. 5.6a, b, c, d, e.
Figure 5.6a 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 5.6b showcases the resulting stress history σ(t) that displays a block signal with σ(t) = ±1 whenever ˙
(t) = ±5 in the loading and the unloading phases.
In particular during the holding phase σ(t) = σ p = +1 results as a reaction to the
kinematic constraint p (t) ≡ (t).