96
4 Visco-Elasticity
formations
3
π ( ˙
= max
σ
{σ ˙
− π
∗
(σ)},
(4.35a)
π
∗
(σ) = max
˙
{σ ˙
− π ( ˙
(4.35b)
The stationarity conditions corresponding to Eqs. 4.35a and 4.35b are the constitutive relations
˙
= ∂ σ π
∗
(σ),
(4.36a)
σ( ˙
= ∂ ˙
π ( ˙
).
(4.36b)
Obviously, the relations in Eqs. 4.36a and 4.36b determine entirely the dissipative
behavior of the generic Newton model, thus the formulation is completed at this
stage.
Finally, as a further interesting aspect, the dissipation d = σ ˙
is alternatively
expressed from Eqs. 4.35a and 4.35b in terms of the dissipation potential π and the
dual dissipation potential π
∗ as
d = π(˙ + π
∗
(σ) ≥ 0.
(4.37)
The generic Newton model is summarized in Table 4.3.
Table 4.3 Summary of the generic Newton model
(1) Strain
≡ v
(2) Potential π = π(˙
(3) Stress
σ = ∂ ˙
π ≡ σ
or
(2) Potential π ∗ = π ∗ (σ)
(3) Evolution ˙
= ∂ σ π ∗
3 Remark on Legendre Transformation of Dissipation Potentials:
Consider the σ versus ˙
diagram (to the right), whereby the
dependence between σ and ˙
is either given by σ = σ(˙
or by ˙
= ˙
Then π(˙ follows as the integral π(˙ :=
˙
0 σ(˙ ) d˙ , i.e. the area under the σ = σ(˙ curve, thus
rendering ∂ ˙
π = σ(˙ whereas π ∗ (σ) follows as the integral π ∗ (σ) =
σ
0 ˙
) dσ , i.e. the area under the ˙
= ˙
curve, thus rendering ∂ σ π ∗ = ˙
Obviously, π(˙ and
π ∗ (σ) sum up to σ ˙
˙
σ
˙(σ)
σ(˙)
π
∗ (σ)
π(˙)
.