98
4 Visco-Elasticity
4.2.1 Specific Kelvin Model: Formulation
The specific Kelvin model, displayed in Fig. 4.12, consists of a parallel arrangement
of (1) a linear elastic spring with stiffness E and (2) a linear viscous dashpot with
viscosity η.
Direct Representation
For the specific Kelvin model the free energy density ψ is expressed as a quadratic
(and thus convex) function of (the total strain)
ψ() =
1
2
E
2
.
(4.39)
Then the energetic stress σ
, which is conjugated to the total strain , follows as
σ
() = ∂ ψ() = E .
(4.40)
Furthermore, for the specific Kelvin model the convex and smooth (quadratic)
dissipation potential π is chosen as
π(˙ ) =
1
2
η |˙ |
2
.
(4.41)
Observe that (i) π does depend on ˙
, thus the dissipative stress σ
= 0 for ˙
= 0,
and that (ii) π is positively homogenous of degree two in ˙
and obviously smooth at
the origin ˙
= 0. Consequently, the dissipative stress σ
computes as partial derivative
of the dissipation potential with respect to its conjugated variable
σ
(˙ ) = ∂ ˙
π(˙ ) = η ˙
.
(4.42)
Recall that the total stress σ applied to the rheological model (that enters the
equilibrium condition), the energetic stress σ
=: σ e (the stress in the elastic spring),
and the dissipative stress σ
=: σ v (the stress in the viscous dashpot) are constitutively
related by
σ = σ
+ σ
=: σ e + σ v with σ e := σ
and σ v := σ
.
(4.43)
The corresponding dual dissipation potential π
∗ , as determined from a Legendre
transformation
π
∗
(σ v ) = max
˙
{σ v ˙
−
1
2
η |˙ |
2
}
(4.44)
then reads