112
4 Visco-Elasticity
Prescribed Stress History: Zig-Zag
The response of the specific Kelvin model to a prescribed Zig-Zag stress history is
documented in Fig. 4.20a–f.
Figure 4.20a 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 4.20b, d showcases the resulting (viscous) strain history (t) that displays
a periodic nonlinear signal after an initial transient phase.
The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.20c.
Finally, Fig. 4.20e, 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 a rigid behaviour with vanishing strain for
| ˙
σ(t)| → ∞.
Prescribed Stress History: Sine
The response of the specific Kelvin model to a prescribed Sine stress history is
documented in Fig. 4.21a–f
Figure 4.21a 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 4.21b, d showcases the resulting (viscous) strain history (t) that displays,
in accordance with the analytical solution in Eq. 4.57, a harmonic signal with amplitude a = σ a /E/
√
1 + τ 2 ω 2 ≈ 2.68 after an initial transient phase.
The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.21c.
Finally, Fig. 4.21e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Kelvin model to a prescribed Ramp stress history is
documented in Fig. 4.22a–f.
Figure 4.22a depicts the prescribed Ramp stress history σ(t) with maximum σ 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 4.22b, d showcases the resulting (viscous) strain history (t) that displays
a monotonic signal until the end of the holding phase with (t 2 ) → 5 when the creep
4 Visco-Elasticity
Prescribed Stress History: Zig-Zag
The response of the specific Kelvin model to a prescribed Zig-Zag stress history is
documented in Fig. 4.20a–f.
Figure 4.20a 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 4.20b, d showcases the resulting (viscous) strain history (t) that displays
a periodic nonlinear signal after an initial transient phase.
The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.20c.
Finally, Fig. 4.20e, 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 a rigid behaviour with vanishing strain for
| ˙
σ(t)| → ∞.
Prescribed Stress History: Sine
The response of the specific Kelvin model to a prescribed Sine stress history is
documented in Fig. 4.21a–f
Figure 4.21a 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 4.21b, d showcases the resulting (viscous) strain history (t) that displays,
in accordance with the analytical solution in Eq. 4.57, a harmonic signal with amplitude a = σ a /E/
√
1 + τ 2 ω 2 ≈ 2.68 after an initial transient phase.
The resulting, slightly tilted σ = σ() diagram that also displays the initial transient phase is highlighted in Fig. 4.21c.
Finally, Fig. 4.21e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Kelvin model to a prescribed Ramp stress history is
documented in Fig. 4.22a–f.
Figure 4.22a depicts the prescribed Ramp stress history σ(t) with maximum σ 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 4.22b, d showcases the resulting (viscous) strain history (t) that displays
a monotonic signal until the end of the holding phase with (t 2 ) → 5 when the creep