4.1 Newton Model
79
Thus the energetic stress σ
, which is conjugated to the total strain , vanishes
identically as well
σ
() ≡ 0.
(4.3)
Furthermore, for the specific Newton model the convex and smooth (quadratic)
dissipation potential π is chosen as
π(˙ ) =
1
2
η |˙ |
2
.
(4.4)
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 ˙
.
(4.5)
Recall that the total stress σ applied to the rheological model (that enters the
equilibrium condition) and the energetic and dissipative stresses are constitutively
related by σ = σ
+ σ
, thus (since here σ
≡ 0) the value
σ ≡ σ
(4.6)
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) = max
˙
σ ˙
−
1
2
η |˙ |
2
(4.7)
then reads
π
∗
(σ) =
1
2
1
η
|σ|
2
.
(4.8)
The evolution law for the total strain then follows as partial derivative of the dual
dissipation potential with respect to its conjugated variable
˙
(σ) = ∂ σ π
∗
(σ) =
1
η
σ.
(4.9)
Obviously the expressions in Eqs. 4.5 and 4.9 are inverse relations. The smooth
dissipation and dual dissipation potentials π = π(˙ ) and π
∗
= π
∗
(σ) together with
the resulting smooth constitutive relations σ = σ(˙ ) and ˙
= ˙
(σ) are displayed in
Fig. 4.2.
The specific Newton model is summarized in Table 4.1.