168
4 Visco-Elasticity
Furthermore, for the Standard-Linear-Solid Maxwell model the convex and
smooth (quadratic) dissipation potential π is chosen as
π(˙ v ) =
1
2
η m |˙ v |
2
.
(4.189)
Observe that (i) π does not depend on ˙
, thus the dissipative stress σ
= σ − σ
≡
0 vanishes identically, and that (ii) π is positively homogenous of degree two in ˙
v
and obviously smooth at the origin ˙
v = 0. Consequently, the dissipative viscous
stress σ
v computes as partial derivative of the dissipation potential with respect to
its conjugated variable
σ
v (˙ v ) = ∂ ˙
v π(˙ v ) = η m ˙
v .
(4.190)
Recall that the energetic and the dissipative viscous stresses (in the elastic spring
and the viscous dashpot of the Maxwell element) are constitutively related by σ
v +
σ
v = 0, thus the notion of viscous stress defined as the value
σ v := σ
v = −σ
v
(4.191)
will exclusively be used in the sequel for convenience of exposition.
The corresponding dual dissipation potential π
∗ , as determined from a Legendre
transformation
π
∗
(σ v ) = max
˙
v
{σ v ˙
v −
1
2
η m |˙ v |
2
}
(4.192)
then reads
Table 4.13 Summary of the Standard-Linear-Solid Maxwell model
(1) Strain
= e + v
(2) Energy ψ =
1
2 E m [ − v ] 2 +
1
2 E ∞ 2
(3) Stress
σ = E m [ − v ] + E ∞ = σ v + σ ∞ ≡
σ
(4) Stress
σ v = E m [ − v ]
= σ − σ ∞ ≡ −σ
v
(5) Potential π =
1
2 η m |˙ v | 2
(6) Stress
σ v = η m ˙
v
≡
σ
v
or
(5) Potential π ∗ =
1
2
1
η m
|σ v |
2
(6) Evolution ˙
v =
1
η m
σ v