134
4 Visco-Elasticity
in the time interval t ∈ [0, t max = 10], whereby N = 100 time steps with t = 0.1
are computed.
Figure 4.31b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.99, a harmonic signal with amplitude
σ a = E ∞
[1 + τ
2
k ω 2 ]/[1 + c 2 τ
2
k ω 2 ] a ≈ 3.66 after an initial transient phase.
The viscous strain v (t), which—after a slight initial transient phase—is also a
(phase shifted) harmonic signal is demonstrated in Fig. 4.31c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the slight initial transient phase is highlighted in Fig. 4.31d.
Finally, Fig. 4.31e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to higher and lower strain rates
|˙ (t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with
linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙ (t)| → ∞ or
E ∞ = 0.5 for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the Standard-Linear-Solid Kelvin model to a prescribed Ramp strain
history is documented in Fig. 4.32a–f.
Figure 4.32a depicts 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 4.32b showcases the resulting stress history σ(t) that especially displays
relaxation to the equilibrium response σ(t) → E ∞ a = 2.5 during the holding phase
and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 3.5
(σ(t) ≈ −1) at the end of the loading (unloading) phase.
The viscous strain v (t), which approaches v (t) → 2.5 during the holding phase
and v (t) ≈ 1.5 ( v (t) ≈ 1) at the end of the loading (unloading) phase is demonstrated in Fig. 4.32c.
The resulting σ = σ() diagram is highlighted in Fig. 4.32d.
Finally, Fig. 4.32e, f depict the resulting σ = σ() diagrams for 100 times smaller
and 100 times larger t 1 , t 2 , t 3 corresponding to higher and lower strain rates |˙ (t)|,
respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙ (t)| → ∞ or
E ∞ = 0.5 for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the Standard-Linear-Solid Kelvin model to a prescribed Zig-Zag
stress history is documented in Fig. 4.33a–f.
Figure 4.33a 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.
4 Visco-Elasticity
in the time interval t ∈ [0, t max = 10], whereby N = 100 time steps with t = 0.1
are computed.
Figure 4.31b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.99, a harmonic signal with amplitude
σ a = E ∞
[1 + τ
2
k ω 2 ]/[1 + c 2 τ
2
k ω 2 ] a ≈ 3.66 after an initial transient phase.
The viscous strain v (t), which—after a slight initial transient phase—is also a
(phase shifted) harmonic signal is demonstrated in Fig. 4.31c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the slight initial transient phase is highlighted in Fig. 4.31d.
Finally, Fig. 4.31e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to higher and lower strain rates
|˙ (t)|, respectively. They clearly demonstrate an elastic solid-like behaviour with
linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙ (t)| → ∞ or
E ∞ = 0.5 for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the Standard-Linear-Solid Kelvin model to a prescribed Ramp strain
history is documented in Fig. 4.32a–f.
Figure 4.32a depicts 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 4.32b showcases the resulting stress history σ(t) that especially displays
relaxation to the equilibrium response σ(t) → E ∞ a = 2.5 during the holding phase
and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 3.5
(σ(t) ≈ −1) at the end of the loading (unloading) phase.
The viscous strain v (t), which approaches v (t) → 2.5 during the holding phase
and v (t) ≈ 1.5 ( v (t) ≈ 1) at the end of the loading (unloading) phase is demonstrated in Fig. 4.32c.
The resulting σ = σ() diagram is highlighted in Fig. 4.32d.
Finally, Fig. 4.32e, f depict the resulting σ = σ() diagrams for 100 times smaller
and 100 times larger t 1 , t 2 , t 3 corresponding to higher and lower strain rates |˙ (t)|,
respectively. They clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for |˙ (t)| → ∞ or
E ∞ = 0.5 for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the Standard-Linear-Solid Kelvin model to a prescribed Zig-Zag
stress history is documented in Fig. 4.33a–f.
Figure 4.33a 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.