4.4 Maxwell Model
155
Table 4.11 Algorithmic update for the specific Maxwell model
Input
n n−1
v
Trial Strain
e = n − n−1
v
Update Strain n
v =
t n
τ + t n
e +
n−1
v
Update Stress σ n
v =
η
τ + t n
e = σ
n
Tangent
E n
a =
η
τ + t n
Output
σ n n
v E n
a
processes (as compared to the relaxation time) the algorithmic tangent degenerates
to E a → 0, i.e. the response is infinitely soft.
The algorithmic step-by-step update for the specific Maxwell model is summarized in Table 4.11.
4.4.3 Specific Maxwell Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Maxwell model to a prescribed Zig-Zag strain history
is documented in Fig. 4.41a–f.
Figure 4.41a depicts the prescribed Zig-Zag strain history (t) with amplitude
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 4.41b showcases the resulting stress history σ(t) that displays a periodic,
distorted zig-zag or rather sawtooth-type signal after an initial transient phase.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which—after an
initial transient phase—is also a periodic signal (phase shifted with respect to she
stress signal), is demonstrated in Fig. 4.41c.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.41d.
Finally, Fig. 4.41e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to a 100 times higher and a 100 times
lower strain rate |˙ (t)|, respectively. They clearly demonstrate an elastic solid-like
155
Table 4.11 Algorithmic update for the specific Maxwell model
Input
n n−1
v
Trial Strain
e = n − n−1
v
Update Strain n
v =
t n
τ + t n
e +
n−1
v
Update Stress σ n
v =
η
τ + t n
e = σ
n
Tangent
E n
a =
η
τ + t n
Output
σ n n
v E n
a
processes (as compared to the relaxation time) the algorithmic tangent degenerates
to E a → 0, i.e. the response is infinitely soft.
The algorithmic step-by-step update for the specific Maxwell model is summarized in Table 4.11.
4.4.3 Specific Maxwell Model: Response Analysis
Prescribed Strain History: Zig-Zag
The response of the specific Maxwell model to a prescribed Zig-Zag strain history
is documented in Fig. 4.41a–f.
Figure 4.41a depicts the prescribed Zig-Zag strain history (t) with amplitude
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 4.41b showcases the resulting stress history σ(t) that displays a periodic,
distorted zig-zag or rather sawtooth-type signal after an initial transient phase.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which—after an
initial transient phase—is also a periodic signal (phase shifted with respect to she
stress signal), is demonstrated in Fig. 4.41c.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the initial transient phase is highlighted in Fig. 4.41d.
Finally, Fig. 4.41e, f depict the resulting σ = σ() diagrams for a 100 times shorter
and a 100 times longer period T corresponding to a 100 times higher and a 100 times
lower strain rate |˙ (t)|, respectively. They clearly demonstrate an elastic solid-like