190
4 Visco-Elasticity
cous stress −σ
v ≡ σ. Moreover the energetic and the dissipative viscous stresses are
constitutively related by σ
v + σ
v = 0, thus the notion of viscous stress defined as
σ v := σ
v = −σ
v will exclusively be used in the sequel. Analogous statements hold
for the stresses related to each generic Maxwell element (m = 1, . . . , M).
Furthermore, for the generic Generalized-Maxwell model the convex and smooth
(non-quadratic) dissipation and dual dissipation potentials follow as the sums
π ({ ˙
v j }) =
M
m=1
π m ( ˙
v m ),
(4.237a)
π
∗
({σ v j }) =
M
m=1
π
∗
m (σ v m ),
(4.237b)
with abbreviations {˙ v j } := {˙ v 1 , . . . , ˙
v m , . . . , ˙
v M } for the set of viscous strain rates
and {σ v j } := {σ v 1 , . . . , σ v m , . . . , σ v M } for the set of viscous stresses. Thereby, the
convex and smooth (non-quadratic) dissipation and dual dissipation potentials for
each generic Maxwell element (m = 1, . . . , M) introduced as π m = π m (˙ v m ) and
π
∗
= π
∗
(σ v m ), respectively, are related via corresponding Legendre transformations
π m ( ˙
v m ) = max
σ vm
{σ v m ˙
v m − π
∗
m (σ v m )},
(4.238a)
π
∗
m (σ v m ) = max
˙
vm
{σ v m ˙
v m − π m ( ˙
v m )}.
(4.238b)
The stationarity conditions corresponding to Eqs. 4.238a and 4.238b are the constitutive relations
˙
v m (σ v m ) = ∂ σ vm π
∗
(σ v m ),
(4.239a)
σ v m ( ˙
v m ) = ∂ ˙
vm π ( ˙
v m ).
(4.239b)
Obviously, the relations in Eqs. 4.239a and 4.239b determine entirely the dissipative behavior of the generic Generalized-Maxwell model, thus the formulation is
completed at this stage.
Finally, as a further interesting aspect, the dissipation for the generic Generalized-Maxwell model follows as the sum
d({σ v j }, {˙ v j }) =
M
m=1
d m (σ v m , ˙
v m ) ≥ 0.
(4.240)
Thereby, the dissipation d m = σ v m ˙
v m (m = 1, . . . , M) for each generic Maxwell
element is alternatively expressed from Eqs. 4.238a and 4.238b in terms of the dissipation potential π m and the dual dissipation potential π
∗
m as