88
4 Visco-Elasticity
Figure 4.6b showcases the resulting stress history σ(t) that displays a periodic
block signal with σ(t) = ±η ˙
(t) = ±5 since ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 4.6d. 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 ∞ when t → 0.
Finally, Fig. 4.6e, f depict the resulting σ = σ() diagrams for a 10 and 100 times
longer period T corresponding to a 10 and 100 times lower strain rate |˙ (t)|, respectively. They clearly demonstrate a viscous fluid-like behaviour with vanishing stress
for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Newton model to a prescribed Sine strain history is
displayed in Fig. 4.7a–f.
Figure 4.7a, c depict the prescribed Sine (viscous) 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 4.7b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.18, a cosine signal with σ(t) = η ˙
(t) =
η ω ω a cos(ω t) and amplitude η ω ω a = 2.5π ≈ 7.85.
The resulting, slightly distorted σ = σ() diagram is highlighted in Fig. 4.7d. Due
to the finite sized time step t and corresponding finite sized strain increment
the expected slope of the σ = σ() diagram at = 0 is only approximately captured;
however, it obviously tends to ∞ when t → 0. The numerical integration error due
to the finite sized time step t explains likewise the slight distortion of the σ = σ()
ellipsoidal path.
Finally, Fig. 4.7e, f depict the resulting σ = σ() diagrams for a 10 and 100 times
longer period T corresponding to lower strain rates |˙ (t)|, respectively. They clearly
demonstrate a viscous fluid-like behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Newton model to a prescribed Ramp strain history is
displayed in Fig. 4.8a–f.
Figure 4.8a, c depict the prescribed Ramp (viscous) 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 4.8b showcases the resulting stress history σ(t) that displays a block-type
signal with σ(t) = η ˙
(t) = ±5 when ˙
(t) = ±5 in the loading and the unloading
phases and σ(t) = η ˙
(t) = 0 when ˙
(t) = 0 during the holding phase, and |σ(t)| ∈
[0, 5].
4 Visco-Elasticity
Figure 4.6b showcases the resulting stress history σ(t) that displays a periodic
block signal with σ(t) = ±η ˙
(t) = ±5 since ˙
(t) = ±5.
The resulting σ = σ() diagram is highlighted in Fig. 4.6d. 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 ∞ when t → 0.
Finally, Fig. 4.6e, f depict the resulting σ = σ() diagrams for a 10 and 100 times
longer period T corresponding to a 10 and 100 times lower strain rate |˙ (t)|, respectively. They clearly demonstrate a viscous fluid-like behaviour with vanishing stress
for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Newton model to a prescribed Sine strain history is
displayed in Fig. 4.7a–f.
Figure 4.7a, c depict the prescribed Sine (viscous) 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 4.7b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.18, a cosine signal with σ(t) = η ˙
(t) =
η ω ω a cos(ω t) and amplitude η ω ω a = 2.5π ≈ 7.85.
The resulting, slightly distorted σ = σ() diagram is highlighted in Fig. 4.7d. Due
to the finite sized time step t and corresponding finite sized strain increment
the expected slope of the σ = σ() diagram at = 0 is only approximately captured;
however, it obviously tends to ∞ when t → 0. The numerical integration error due
to the finite sized time step t explains likewise the slight distortion of the σ = σ()
ellipsoidal path.
Finally, Fig. 4.7e, f depict the resulting σ = σ() diagrams for a 10 and 100 times
longer period T corresponding to lower strain rates |˙ (t)|, respectively. They clearly
demonstrate a viscous fluid-like behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Newton model to a prescribed Ramp strain history is
displayed in Fig. 4.8a–f.
Figure 4.8a, c depict the prescribed Ramp (viscous) 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 4.8b showcases the resulting stress history σ(t) that displays a block-type
signal with σ(t) = η ˙
(t) = ±5 when ˙
(t) = ±5 in the loading and the unloading
phases and σ(t) = η ˙
(t) = 0 when ˙
(t) = 0 during the holding phase, and |σ(t)| ∈
[0, 5].