146
4 Visco-Elasticity
π(˙ v ) =
1
2
η |˙ v |
2
.
(4.144)
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 ˙
v .
(4.145)
Recall that the energetic and the dissipative viscous stresses 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
(4.146)
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 |˙ v |
2
}
(4.147)
then reads
π
∗
(σ v ) =
1
2
1
η
|σ v |
2
.
(4.148)
The evolution law for the viscous strain then follows as partial derivative of the
dual dissipation potential with respect to its conjugated variable
˙
v (σ v ) = ∂ σ v π
∗
(σ v ) =
1
η
σ v .
(4.149)
Table 4.10 Summary of the specific Maxwell model
(1) Strain
= e + v
(2) Energy ψ =
1
2 E [ − v ] 2
(3) Stress
σ = E [ − v ] ≡ σ ≡ −σ
v
(4) Potential π =
1
2 η |˙ v | 2
(5) Stress
σ v = η ˙
v ≡ σ
v
or
(4) Potential π ∗ =
1
2
1
η
|σ v |
2
(5) Evolution ˙
v =
1
η
σ v