164
4 Visco-Elasticity
Figure 4.46a 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 4.46b showcases the resulting strain history (t) that especially displays linear creep during the holding phase and nonlinear strain response during the un/loading
phases.
The resulting σ = σ() diagram is highlighted in Fig. 4.46c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
linear signal during the holding phase, is demonstrated in Fig. 4.46d.
Finally, Fig. 4.46e, 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 an elastic solid-type behaviour with linear σ = σ() relation for
| ˙
σ(t)| → ∞.
4.4.4 Generic Maxwell Model: Formulation
A generic formulation of the Maxwell model can be obtained from generalizing the
specific Maxwell model in Fig. 4.36 by assuming the elastic spring or/and the viscous
dashpot as nonlinear.
For the generic Maxwell model the free energy density ψ is expressed as a nonquadratic, yet convex, function of − v (the elastic strain e )
ψ(, v ) = ψ( − v ).
(4.181)
Note that ψ(, v ) and ψ( − v ) are different functions that return, however, the
same function value for the same values of and v . Then the energetic stress σ
and
the energetic viscous stress σ
v follow as
σ
(, v ) = ∂ ψ(, v ) = ∂ ψ( − v ),
(4.182a)
σ
v (, v ) = ∂ v ψ(, v ) = ∂ v ψ( − v ).
(4.182b)
Recall that the total stress σ (that enters the equilibrium condition) coincides
identically with the energetic stress σ
≡ σ and the negative of the energetic viscous stress −σ
v ≡ σ. Moreover the energetic and the dissipative viscous stresses are
constitutively related by σ
v + σ
v = 0, thus the notion of viscous stress defined as
σ v := σ
v = −σ
v will exclusively be used in the sequel.
Furthermore, for the generic Maxwell model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ v )
and π
∗
= π
∗
(σ v ), respectively, are related via corresponding Legendre transformations
π ( ˙
v ) = max
σ v
{σ v ˙
v − π
∗
(σ v )},
(4.183a)
4 Visco-Elasticity
Figure 4.46a 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 4.46b showcases the resulting strain history (t) that especially displays linear creep during the holding phase and nonlinear strain response during the un/loading
phases.
The resulting σ = σ() diagram is highlighted in Fig. 4.46c.
The viscous strain v (t) = (t) − σ(t)/E with ˙
v (t) = σ(t)/η, which is also a
linear signal during the holding phase, is demonstrated in Fig. 4.46d.
Finally, Fig. 4.46e, 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 an elastic solid-type behaviour with linear σ = σ() relation for
| ˙
σ(t)| → ∞.
4.4.4 Generic Maxwell Model: Formulation
A generic formulation of the Maxwell model can be obtained from generalizing the
specific Maxwell model in Fig. 4.36 by assuming the elastic spring or/and the viscous
dashpot as nonlinear.
For the generic Maxwell model the free energy density ψ is expressed as a nonquadratic, yet convex, function of − v (the elastic strain e )
ψ(, v ) = ψ( − v ).
(4.181)
Note that ψ(, v ) and ψ( − v ) are different functions that return, however, the
same function value for the same values of and v . Then the energetic stress σ
and
the energetic viscous stress σ
v follow as
σ
(, v ) = ∂ ψ(, v ) = ∂ ψ( − v ),
(4.182a)
σ
v (, v ) = ∂ v ψ(, v ) = ∂ v ψ( − v ).
(4.182b)
Recall that the total stress σ (that enters the equilibrium condition) coincides
identically with the energetic stress σ
≡ σ and the negative of the energetic viscous stress −σ
v ≡ σ. Moreover the energetic and the dissipative viscous stresses are
constitutively related by σ
v + σ
v = 0, thus the notion of viscous stress defined as
σ v := σ
v = −σ
v will exclusively be used in the sequel.
Furthermore, for the generic Maxwell model it is possible to introduce the convex
and smooth (non-quadratic) dissipation and dual dissipation potentials as π = π(˙ v )
and π
∗
= π
∗
(σ v ), respectively, are related via corresponding Legendre transformations
π ( ˙
v ) = max
σ v
{σ v ˙
v − π
∗
(σ v )},
(4.183a)