4.1 Newton Model
91
The resulting σ = σ() diagram is highlighted in Fig. 4.8d. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = 5 obviously tend to ∞ when t → 0.
Finally, Fig. 4.8e, 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 a viscous fluid-like behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the specific Newton model to a prescribed Zig-Zag stress history is
displayed in Fig. 4.9a–f.
Figure 4.9a 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 4.9b, d showcase the resulting (viscous) strain history (t) that displays a
periodic signal with ˙
(t) = σ(t)/η ∝ ±5t, thus leading to piecewise quadratic (t)
with |(t)| ∈ [0, 5].
The resulting, slightly tilted, σ = σ() diagram is highlighted in Fig. 4.9c. The
numerical integration error due to the finite sized time step t explains the slight
tilting of the σ = σ() path.
Finally, Fig. 4.9e, 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 Newton model to a prescribed Sine stress history is
displayed in Fig. 4.10a–f.
Figure 4.10a 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 4.10b, d showcase the resulting (viscous) strain history (t) that displays, in
accordance with the analytical solution in Eq. 4.20, a (negative and upwards shifted)
cosine signal with ˙
(t) = σ(t)/η = σ a /η sin(ω t) oscillating about the mean value
a = σ a /η/ω = 10/π ≈ 3.18 with amplitude a ≈ 3.18.
The resulting, slightly tilted σ = σ() diagram is highlighted in Fig. 4.10c. The
numerical integration error due to the finite sized time step t explains the slight
tilting of the σ = σ() ellipsoidal path.
91
The resulting σ = σ() diagram is highlighted in Fig. 4.8d. Due to the finite sized
time step t and corresponding finite sized strain increment the expected rectangular format of the σ = σ() diagram is only approximately captured, however the
slopes at = 0 and = 5 obviously tend to ∞ when t → 0.
Finally, Fig. 4.8e, 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 a viscous fluid-like behaviour with vanishing stress for |˙ (t)| → 0.
Prescribed Stress History: Zig-Zag
The response of the specific Newton model to a prescribed Zig-Zag stress history is
displayed in Fig. 4.9a–f.
Figure 4.9a 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 4.9b, d showcase the resulting (viscous) strain history (t) that displays a
periodic signal with ˙
(t) = σ(t)/η ∝ ±5t, thus leading to piecewise quadratic (t)
with |(t)| ∈ [0, 5].
The resulting, slightly tilted, σ = σ() diagram is highlighted in Fig. 4.9c. The
numerical integration error due to the finite sized time step t explains the slight
tilting of the σ = σ() path.
Finally, Fig. 4.9e, 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 Newton model to a prescribed Sine stress history is
displayed in Fig. 4.10a–f.
Figure 4.10a 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 4.10b, d showcase the resulting (viscous) strain history (t) that displays, in
accordance with the analytical solution in Eq. 4.20, a (negative and upwards shifted)
cosine signal with ˙
(t) = σ(t)/η = σ a /η sin(ω t) oscillating about the mean value
a = σ a /η/ω = 10/π ≈ 3.18 with amplitude a ≈ 3.18.
The resulting, slightly tilted σ = σ() diagram is highlighted in Fig. 4.10c. The
numerical integration error due to the finite sized time step t explains the slight
tilting of the σ = σ() ellipsoidal path.