156
4 Visco-Elasticity
behaviour with linear σ = σ() relation for |˙ (t)| → ∞ and a viscous fluid-like
behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Maxwell model to a prescribed Sine strain history is
documented in Fig. 4.42a–f.
Figure 4.42a 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 4.42b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.158, a harmonic signal with amplitude
σ a = E τ ω/
√
1 + τ 2 ω 2 a ≈ 4.21 after an initial transient phase.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which—after an
initial transient phase—is also a harmonic signal (phase shifted by π/2 with respect
to she stress signal) with amplitude σ a /η/ω ≈ 2.68, is demonstrated in Fig. 4.42c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.42d.
Finally, Fig. 4.42e, 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 for |˙ (t)| → ∞ and a viscous fluid-like behaviour with vanishing
stress for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Maxwell model to a prescribed Ramp strain history is
documented in Fig. 4.43a–f.
Figure 4.43a 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 4.43b showcases the resulting stress history σ(t) that especially displays
complete relaxation during the holding phase and nonlinear stress response during
the un/loading phases approaching |σ(t)| → 3.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, 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. 4.43c.
The resulting σ = σ() diagram is highlighted in Fig. 4.43d.
Finally, Fig. 4.43e, 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 for |˙ (t)| → ∞ and a viscous fluid-like behaviour with vanishing
stress for |˙ (t)| → 0.
4 Visco-Elasticity
behaviour with linear σ = σ() relation for |˙ (t)| → ∞ and a viscous fluid-like
behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Maxwell model to a prescribed Sine strain history is
documented in Fig. 4.42a–f.
Figure 4.42a 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 4.42b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.158, a harmonic signal with amplitude
σ a = E τ ω/
√
1 + τ 2 ω 2 a ≈ 4.21 after an initial transient phase.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which—after an
initial transient phase—is also a harmonic signal (phase shifted by π/2 with respect
to she stress signal) with amplitude σ a /η/ω ≈ 2.68, is demonstrated in Fig. 4.42c.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.42d.
Finally, Fig. 4.42e, 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 for |˙ (t)| → ∞ and a viscous fluid-like behaviour with vanishing
stress for |˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Maxwell model to a prescribed Ramp strain history is
documented in Fig. 4.43a–f.
Figure 4.43a 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 4.43b showcases the resulting stress history σ(t) that especially displays
complete relaxation during the holding phase and nonlinear stress response during
the un/loading phases approaching |σ(t)| → 3.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, 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. 4.43c.
The resulting σ = σ() diagram is highlighted in Fig. 4.43d.
Finally, Fig. 4.43e, 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 for |˙ (t)| → ∞ and a viscous fluid-like behaviour with vanishing
stress for |˙ (t)| → 0.