186
4 Visco-Elasticity
Prescribed Stress History: Sine
The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine stress
history is documented in Fig. 4.58a–f.
Figure 4.58a 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 4.58b showcases the resulting strain history (t) that displays, in accordance with the analytical solution in Eq. 4.206, a harmonic signal with amplitude
a = σ a
[1 + τ 2
m ω 2 ]/[1 + e 2 τ 2
m ω 2 ]/E ∞ ≈ 2.82 after an initial transient phase.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.58c.
The viscous strain v (t), which—after an initial transient phase—is also a (phase
shifted) harmonic signal, is demonstrated in Fig. 4.58d.
Finally, Fig. 4.58e, 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 and
stiffness approaching E 0 = 2 for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp
stress history is documented in Fig. 4.59a–f.
Figure 4.59a 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 4.59b showcases the resulting strain history (t) that especially displays
nonlinear creep to the equilibrium response (t) → σ a /E ∞ = 5 during the holding
phase and nonlinear strain response during the un/loading phases.
The resulting σ = σ() diagram is highlighted in Fig. 4.59c.
The viscous strain v (t), which approaches v (t) → 5 during the holding phase,
is demonstrated in Fig. 4.59d.
Finally, Fig. 4.59e, f depict the resulting σ = σ() diagrams for 10 and 100 times
smaller t 1 , t 2 , t 3 corresponding to higher stress rates | ˙
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)| → ∞.
Equivalence to Standard-Linear-Solid Kelvin Model
The global response behaviour of the Standard-Linear-Solid Maxwell model with
parameters η m = 1.0, E m = 1.0 (thus τ m = 1.0), and E ∞ = 1.0 in terms of the total
strain and the total stress, as discussed in the above, is equivalent to the global
4 Visco-Elasticity
Prescribed Stress History: Sine
The response of the Standard-Linear-Solid Maxwell model to a prescribed Sine stress
history is documented in Fig. 4.58a–f.
Figure 4.58a 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 4.58b showcases the resulting strain history (t) that displays, in accordance with the analytical solution in Eq. 4.206, a harmonic signal with amplitude
a = σ a
[1 + τ 2
m ω 2 ]/[1 + e 2 τ 2
m ω 2 ]/E ∞ ≈ 2.82 after an initial transient phase.
The resulting (ellipsoidal) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.58c.
The viscous strain v (t), which—after an initial transient phase—is also a (phase
shifted) harmonic signal, is demonstrated in Fig. 4.58d.
Finally, Fig. 4.58e, 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 and
stiffness approaching E 0 = 2 for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the Standard-Linear-Solid Maxwell model to a prescribed Ramp
stress history is documented in Fig. 4.59a–f.
Figure 4.59a 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 4.59b showcases the resulting strain history (t) that especially displays
nonlinear creep to the equilibrium response (t) → σ a /E ∞ = 5 during the holding
phase and nonlinear strain response during the un/loading phases.
The resulting σ = σ() diagram is highlighted in Fig. 4.59c.
The viscous strain v (t), which approaches v (t) → 5 during the holding phase,
is demonstrated in Fig. 4.59d.
Finally, Fig. 4.59e, f depict the resulting σ = σ() diagrams for 10 and 100 times
smaller t 1 , t 2 , t 3 corresponding to higher stress rates | ˙
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)| → ∞.
Equivalence to Standard-Linear-Solid Kelvin Model
The global response behaviour of the Standard-Linear-Solid Maxwell model with
parameters η m = 1.0, E m = 1.0 (thus τ m = 1.0), and E ∞ = 1.0 in terms of the total
strain and the total stress, as discussed in the above, is equivalent to the global