4.5 Generalized-Maxwell Model
183
Prescribed Strain History: Ramp
The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp
strain history is documented in Fig. 4.56a–f.
Figure 4.56a 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 4.56b showcases the resulting stress history σ(t) that especially displays
relaxation to the equilibrium response σ(t) → E ∞ a = 5 during the holding phase
and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 8
(σ(t) ≈ −3) at the end of the loading (unloading) phase.
The viscous strain v (t), which approaches v (t) → 5 during the holding phase
and v (t) ≈ 2 ( v (t) ≈ 3) at the end of the loading (unloading) phase is demonstrated
in Fig. 4.56c.
The resulting σ = σ() diagram is highlighted in Fig. 4.56d.
Finally, Fig. 4.56e, 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 = 2 for |˙ (t)| → ∞ or E ∞ = 1
for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag
stress history is documented in Fig. 4.57a–f.
Figure 4.57a 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.
Figure 4.57b showcases the resulting strain history (t) that displays a periodic,
distorted zigzag or rather sawtooth-like signal after an initial transient phase.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.57c.
The viscous strain v (t), which—after an initial transient phase— is also a (phase
shifted) periodic signal, is demonstrated in Fig. 4.57d.
Finally, Fig. 4.57e, f depict the resulting σ = σ() diagrams for a 10 and 100
times shorter period T corresponding to a 10 and 100 times higher stress rate | ˙
σ(t)|,
respectively. They clearly demonstrate an elastic solid-type behaviour with linear
σ = σ() relation and stiffness approaching E 0 = 2 for | ˙
σ(t)| → ∞.
183
Prescribed Strain History: Ramp
The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp
strain history is documented in Fig. 4.56a–f.
Figure 4.56a 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 4.56b showcases the resulting stress history σ(t) that especially displays
relaxation to the equilibrium response σ(t) → E ∞ a = 5 during the holding phase
and nonlinear stress response during the un/loading phases approaching σ(t) ≈ 8
(σ(t) ≈ −3) at the end of the loading (unloading) phase.
The viscous strain v (t), which approaches v (t) → 5 during the holding phase
and v (t) ≈ 2 ( v (t) ≈ 3) at the end of the loading (unloading) phase is demonstrated
in Fig. 4.56c.
The resulting σ = σ() diagram is highlighted in Fig. 4.56d.
Finally, Fig. 4.56e, 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 = 2 for |˙ (t)| → ∞ or E ∞ = 1
for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the Standard-Linear-Solid Maxwell model to a prescribed Zig-Zag
stress history is documented in Fig. 4.57a–f.
Figure 4.57a 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.
Figure 4.57b showcases the resulting strain history (t) that displays a periodic,
distorted zigzag or rather sawtooth-like signal after an initial transient phase.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.57c.
The viscous strain v (t), which—after an initial transient phase— is also a (phase
shifted) periodic signal, is demonstrated in Fig. 4.57d.
Finally, Fig. 4.57e, f depict the resulting σ = σ() diagrams for a 10 and 100
times shorter period T corresponding to a 10 and 100 times higher stress rate | ˙
σ(t)|,
respectively. They clearly demonstrate an elastic solid-type behaviour with linear
σ = σ() relation and stiffness approaching E 0 = 2 for | ˙
σ(t)| → ∞.