132
4 Visco-Elasticity
Table 4.8 Algorithmic update for the Standard-Linear-Solid Kelvin model
Input
n n−1
v
Trial Strain
˜
e = n − n−1
v /[1 − c]
Update Strain n
v =
˜ t n
τ 0 + ˜ t n [1 − c] ˜
e +
n−1
v
Update Stress σ n =
η k
τ 0 + ˜ t n [1 − c] ˜
e + E ∞
n
Tangent
E n
a =
η k
τ 0 + ˜ t n [1 − c] + E ∞
Output
σ n n
v E n
a
4.3.3 Standard-Linear-Solid Kelvin Model: Response
Analysis
Prescribed Strain History: Zig-Zag
The response of the Standard-Linear-Solid Kelvin model to a prescribed Zig-Zag
strain history is documented in Fig. 4.30a–f.
Figure 4.30a 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 4.30b showcases the resulting stress history σ(t) that displays a periodic,
distorted zig-zag or rather sawtooth-type signal after a slight initial transient phase.
The viscous strain v (t), which—after a slight initial transient phase—is also a
periodic signal, is demonstrated in Fig. 4.30c.
The resulting (lens-shaped) σ = σ() diagram that is (elastically) tilted and that
also displays the slight initial transient phase is highlighted in Fig. 4.30d.
Finally, Fig. 4.30e, 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
behaviour with linear σ = σ() relation and stiffness approaching either E 0 = 1 for
|˙ (t)| → ∞ or E ∞ = 0.5 for |˙ (t)| → 0.
Prescribed Strain History: Sine
The response of the Standard-Linear-Solid Kelvin model to a prescribed Sine strain
history is documented in Fig. 4.31a–f.
Figure 4.31a depicts the prescribed Sine strain history (t) = a sin(ω t) with
amplitude a = 5, period T = 4 and corresponding angular frequency ω = 2π/T