160
4 Visco-Elasticity
Prescribed Stress History: Zig-Zag
The response of the specific Maxwell model to a prescribed Zig-Zag stress history
is documented in Fig. 4.44a–f.
Figure 4.44a 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 4.44b showcases the resulting strain history (t) that displays a periodic,
distorted zigzag or rather sawtooth-like signal.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.44c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
periodic signal (phase shifted by π/2 with respect to she stress signal) with v (t) ∈
[0, 5], is demonstrated in Fig. 4.44d.
Finally, Fig. 4.44e, 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 for | ˙
σ(t)| → ∞.
Prescribed Stress History: Sine
The response of the specific Maxwell model to a prescribed Sine stress history is
documented in Fig. 4.45a–f.
Figure 4.45a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 4.45b showcases the resulting strain history (t) that displays, in accordance with the analytical solution in Eq. 4.160, a harmonic signal oscillating about
σ a /τ /ω ≈ 3.18 with amplitude a = σ a
√
1 + τ 2 ω 2 /τ /ω/E ≈ 5.93 thus rendering
max(min)(t) ≈ 9.11(−2.74).
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.45c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
harmonic signal (phase shifted by π/2 with respect to she stress signal) that oscillates
about σ a /η/ω ≈ 3.18 with amplitude σ a /η/ω ≈ 3.18 thus rendering max v (t) ≈
6.37, is demonstrated in Fig. 4.45d.
Finally, Fig. 4.45e, f depict the resulting σ = σ() diagrams for a 10 and 100
times shorter period T corresponding to higher stress rates | ˙
σ(t)|, respectively. They
clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation for
| ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Maxwell model to a prescribed Ramp stress history is
documented in Fig. 4.46a–f.
4 Visco-Elasticity
Prescribed Stress History: Zig-Zag
The response of the specific Maxwell model to a prescribed Zig-Zag stress history
is documented in Fig. 4.44a–f.
Figure 4.44a 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 4.44b showcases the resulting strain history (t) that displays a periodic,
distorted zigzag or rather sawtooth-like signal.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.44c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
periodic signal (phase shifted by π/2 with respect to she stress signal) with v (t) ∈
[0, 5], is demonstrated in Fig. 4.44d.
Finally, Fig. 4.44e, 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 for | ˙
σ(t)| → ∞.
Prescribed Stress History: Sine
The response of the specific Maxwell model to a prescribed Sine stress history is
documented in Fig. 4.45a–f.
Figure 4.45a depicts the prescribed Sine stress history σ(t) = σ a sin(ω t) with
amplitude σ 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 4.45b showcases the resulting strain history (t) that displays, in accordance with the analytical solution in Eq. 4.160, a harmonic signal oscillating about
σ a /τ /ω ≈ 3.18 with amplitude a = σ a
√
1 + τ 2 ω 2 /τ /ω/E ≈ 5.93 thus rendering
max(min)(t) ≈ 9.11(−2.74).
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted is highlighted in Fig. 4.45c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
harmonic signal (phase shifted by π/2 with respect to she stress signal) that oscillates
about σ a /η/ω ≈ 3.18 with amplitude σ a /η/ω ≈ 3.18 thus rendering max v (t) ≈
6.37, is demonstrated in Fig. 4.45d.
Finally, Fig. 4.45e, f depict the resulting σ = σ() diagrams for a 10 and 100
times shorter period T corresponding to higher stress rates | ˙
σ(t)|, respectively. They
clearly demonstrate an elastic solid-type behaviour with linear σ = σ() relation for
| ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Maxwell model to a prescribed Ramp stress history is
documented in Fig. 4.46a–f.