94
4 Visco-Elasticity
Finally, Fig. 4.10e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Newton model to a prescribed Ramp stress history is
documented in Fig. 4.11a–f.
Figure 4.11a 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 4.11b, d showcase the resulting (viscous) strain history (t) that displays a
monotonic signal with ˙
(t) = σ(t)/η ∝ ±5t when σ(t) ∝ ±5t in the loading and the
unloading phases, thus leading to piecewise quadratic (t), and ˙
(t) = σ(t)/η = 5
when σ(t) = 5 during the holding phase, thus leading to piecewise linear (t). Taken
together, (t 3 ) = 2 × 2.5 + 5 × 8 = 45.
The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted
in Fig. 4.11c.
Finally, Fig. 4.11e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
4.1.4 Generic Newton Model: Formulation
A generic formulation of the Newton can be obtained from generalizing the specific
Newton model in Fig. 4.1 by assuming the viscous dashpot as nonlinear.
For the generic Newton model the free energy density ψ vanishes identically
ψ() ≡ 0.
(4.33)
Thus the energetic stress σ
vanishes identically as well
σ
() ≡ 0.
(4.34)
Recall that the energetic and the dissipative stresses are constitutively related to
the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
, thus (with
σ
≡ 0) the total stress σ ≡ σ
will exclusively be used in the sequel.
Furthermore, for the generic Newton model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ )
and π
∗
= π
∗
(σ), respectively, which are related via corresponding Legendre trans-
4 Visco-Elasticity
Finally, Fig. 4.10e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
Prescribed Stress History: Ramp
The response of the specific Newton model to a prescribed Ramp stress history is
documented in Fig. 4.11a–f.
Figure 4.11a 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 4.11b, d showcase the resulting (viscous) strain history (t) that displays a
monotonic signal with ˙
(t) = σ(t)/η ∝ ±5t when σ(t) ∝ ±5t in the loading and the
unloading phases, thus leading to piecewise quadratic (t), and ˙
(t) = σ(t)/η = 5
when σ(t) = 5 during the holding phase, thus leading to piecewise linear (t). Taken
together, (t 3 ) = 2 × 2.5 + 5 × 8 = 45.
The resulting σ = σ() diagram of nonlinear parallelogram format is highlighted
in Fig. 4.11c.
Finally, Fig. 4.11e, 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 a rigid behaviour with vanishing strain for | ˙
σ(t)| → ∞.
4.1.4 Generic Newton Model: Formulation
A generic formulation of the Newton can be obtained from generalizing the specific
Newton model in Fig. 4.1 by assuming the viscous dashpot as nonlinear.
For the generic Newton model the free energy density ψ vanishes identically
ψ() ≡ 0.
(4.33)
Thus the energetic stress σ
vanishes identically as well
σ
() ≡ 0.
(4.34)
Recall that the energetic and the dissipative stresses are constitutively related to
the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
, thus (with
σ
≡ 0) the total stress σ ≡ σ
will exclusively be used in the sequel.
Furthermore, for the generic Newton model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ )
and π
∗
= π
∗
(σ), respectively, which are related via corresponding Legendre trans-