4.2 Kelvin Model
109
respectively. They clearly demonstrate an elastic solid-like behaviour with linear
σ = σ() relation for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Kelvin model to a prescribed Sine strain history is
documented in Fig. 4.18a–f.
Figure 4.18a, c depicts the prescribed Sine (viscous) 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 4.18b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.55, a harmonic signal with σ(t) = E (t) +
η ˙
(t) = E a sin(ω t) + η ω ω a cos(ω t) and amplitude σ a = E
√
1 + τ 2 ω 2 a
≈ 9.31.
The resulting σ = σ() diagram is highlighted in Fig. 4.18d. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
slope of the σ = σ() diagram at = 0 is only approximately captured, however it
obviously tends to ∞ when t → 0.
Finally, Fig. 4.18e, f depict the resulting σ = σ() diagrams for a 10 and 100
times longer period T corresponding to lower strain rates |˙ (t)|, respectively. They
clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for
|˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Kelvin model to a prescribed Ramp strain history is
documented in Fig. 4.19a–f.
Figure 4.19a, c 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 4.19b showcases the resulting stress history σ(t) that displays a distorted
block-type signal with ˙
σ(t) = E ˙
(t) = ±5 when ˙
(t) = ±5 in the loading and the
unloading phases and σ(t) = E (t) = 5 when ˙
(t) = 0 during the holding phase,
and |σ(t)| ∈ [0, 10].
The resulting σ = σ() diagram is highlighted in Fig. 4.19d. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram format of the σ = σ() diagram with vertical slopes at = 0 and =
±5 is only approximately captured, however the slopes at = 0 and = 5 obviously
tend to ∞ with t → 0.
Finally, Fig. 4.19e, f depict the resulting σ = σ() diagrams for 10 and 100 times
larger t 1 , t 2 , t 3 corresponding to lower strain rates |˙ (t)|, respectively. They clearly
demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙ (t)| →
0.
109
respectively. They clearly demonstrate an elastic solid-like behaviour with linear
σ = σ() relation for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the specific Kelvin model to a prescribed Sine strain history is
documented in Fig. 4.18a–f.
Figure 4.18a, c depicts the prescribed Sine (viscous) 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 4.18b showcases the resulting stress history σ(t) that displays, in accordance with the analytical solution in Eq. 4.55, a harmonic signal with σ(t) = E (t) +
η ˙
(t) = E a sin(ω t) + η ω ω a cos(ω t) and amplitude σ a = E
√
1 + τ 2 ω 2 a
≈ 9.31.
The resulting σ = σ() diagram is highlighted in Fig. 4.18d. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
slope of the σ = σ() diagram at = 0 is only approximately captured, however it
obviously tends to ∞ when t → 0.
Finally, Fig. 4.18e, f depict the resulting σ = σ() diagrams for a 10 and 100
times longer period T corresponding to lower strain rates |˙ (t)|, respectively. They
clearly demonstrate an elastic solid-like behaviour with linear σ = σ() relation for
|˙ (t)| → 0.
Prescribed Strain History: Ramp
The response of the specific Kelvin model to a prescribed Ramp strain history is
documented in Fig. 4.19a–f.
Figure 4.19a, c 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 4.19b showcases the resulting stress history σ(t) that displays a distorted
block-type signal with ˙
σ(t) = E ˙
(t) = ±5 when ˙
(t) = ±5 in the loading and the
unloading phases and σ(t) = E (t) = 5 when ˙
(t) = 0 during the holding phase,
and |σ(t)| ∈ [0, 10].
The resulting σ = σ() diagram is highlighted in Fig. 4.19d. Due to the finite
sized time step t and corresponding finite sized strain increment the expected
parallelogram format of the σ = σ() diagram with vertical slopes at = 0 and =
±5 is only approximately captured, however the slopes at = 0 and = 5 obviously
tend to ∞ with t → 0.
Finally, Fig. 4.19e, f depict the resulting σ = σ() diagrams for 10 and 100 times
larger t 1 , t 2 , t 3 corresponding to lower strain rates |˙ (t)|, respectively. They clearly
demonstrate an elastic solid-like behaviour with linear σ = σ() relation for |˙ (t)| →
0.