180
4 Visco-Elasticity
4.5.3 Standard-Linear-Solid Maxwell Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag
strain history is documented in Fig. 4.54a–f.
Figure 4.54a 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 4.54b showcases the resulting stress history σ(t) that displays a periodic,
distorted zig-zag or rather sawtooth-type signal after an initial transient phase.
The viscous strain v (t), which—after an initial transient phase—is also a periodic
signal, is demonstrated in Fig. 4.54c.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.54d.
Finally, Fig. 4.54e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to a 100 times higher and a 100 times
lower strain rate |˙ (t)|, respectively. They clearly demonstrate an elastic solid-like
behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 2 for
|˙ (t)| → ∞ or E ∞ = 1 for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine
strain history is documented in Fig. 4.55a–f.
Figure 4.55a 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 4.55b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.204, a harmonic signal with amplitude
σ a = E ∞
[1 + e 2 τ 2
m ω 2 ]/[1 + τ 2
m ω 2 ] a ≈ 8.85 after an initial transient phase.
The viscous strain v (t), which—after an initial transient phase—is also a (phase
shifted) harmonic signal is demonstrated in Fig. 4.55c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.55d.
Finally, Fig. 4.55e, 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 = 2 for |˙ (t)| → ∞ or
E ∞ = 1 for |˙ (t)| → 0.
4 Visco-Elasticity
4.5.3 Standard-Linear-Solid Maxwell Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag
strain history is documented in Fig. 4.54a–f.
Figure 4.54a 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 4.54b showcases the resulting stress history σ(t) that displays a periodic,
distorted zig-zag or rather sawtooth-type signal after an initial transient phase.
The viscous strain v (t), which—after an initial transient phase—is also a periodic
signal, is demonstrated in Fig. 4.54c.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.54d.
Finally, Fig. 4.54e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to a 100 times higher and a 100 times
lower strain rate |˙ (t)|, respectively. They clearly demonstrate an elastic solid-like
behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 2 for
|˙ (t)| → ∞ or E ∞ = 1 for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine
strain history is documented in Fig. 4.55a–f.
Figure 4.55a 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 4.55b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.204, a harmonic signal with amplitude
σ a = E ∞
[1 + e 2 τ 2
m ω 2 ]/[1 + τ 2
m ω 2 ] a ≈ 8.85 after an initial transient phase.
The viscous strain v (t), which—after an initial transient phase—is also a (phase
shifted) harmonic signal is demonstrated in Fig. 4.55c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.55d.
Finally, Fig. 4.55e, 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 = 2 for |˙ (t)| → ∞ or
E ∞ = 1 for |˙ (t)| → 0.