142
4 Visco-Elasticity
Note that the energetic stress σ
coincides here with the elastic stress σ e 1 := σ
in
the first generic Kelvin element and that, due ∂ ψ 1 = −∂ v k ψ 1 , the elastic stresses σ e k
in the remaining (k = 2, . . . , K ) generic Kelvin elements follow as σ e k := σ
+ σ
v k
.
In analogy the definitions σ v 1 := σ
and σ v k := σ
+ σ
v k
(k = 2, . . . , K ) are introduced for the viscous stresses in the generic Kelvin elements in terms of the dissipative
stress σ
and dissipative viscous stresses σ
v k
.
Recall finally that the energetic and the dissipative stress are constitutively related
to the total stress σ (that enters the equilibrium condition) by σ = σ
+ σ
which is
likewise engraved in the relation between the energetic and the dissipative viscous
stresses σ
v k
= −σ
v k
resulting in σ e 1 + σ v 1 = σ e k + σ v k ≡ σ. Thus the sum of the
elastic stress contribution σ e k and the viscous stress contribution σ v k for each generic
Kelvin element (k = 2, . . . , K ) equals likewise the total stress σ (the equilibrium
stress).
Furthermore, for the generic Generalized-Kelvin model the convex and smooth
(non-quadratic) dissipation and dual dissipation potentials follow as the sums
π ( ˙
, { ˙
v j } \1 ) = ¯
π ( ˙
v 1 , { ˙
v j } \1 ) = ¯
π 1 ( ˙
v 1 ) +
K
k=2
¯
π k ( ˙
v k ),
(4.134a)
π
∗
(σ
, {σ
v j
} \1 ) = ¯
π
∗
(σ v 1 , {σ v j } \1 ) = ¯
π
∗
1 (σ v 1 ) +
K
k=2
¯
π
∗
k (σ v k ),
(4.134b)
with abbreviations {˙ v j } \1 := {˙ v 2 , . . . , ˙
v k , . . . , ˙
v K } for the set of viscous strain rates,
˙
v 1 := ˙
− ˙
v for the rate of the total viscous strain and {σ
v j
} \1 := {σ
v 2
, . . . , σ
v k
, . . . ,
σ
v K
} for the set of dissipative viscous stresses, {σ v j } \1 := {σ v 2 , . . . , σ v k , . . . , σ v K } for
the set of viscous stresses (with σ v 1 := σ
and σ v k := σ
+ σ
v k
).
Thereby, the convex and smooth (non-quadratic) dissipation and dual dissipation
potentials for the generic Generalized-Kelvin model introduced as π = π(˙ , {˙ v j } \1 )
and π
∗
= π
∗
(σ
, {σ
v j
} \1 ), respectively, are related via corresponding Legendre transformations (note that σ
˙
+
K
k=2 σ
v k
˙
v k = σ v 1 ˙
v 1 +
K
k=2 σ v k ˙
v k )
π ( ˙
, { ˙
v j } \1 ) = max
σ ,{σ
v j } \1
{σ
˙
+
K
k=2
σ
v k
˙
v k − π
∗
(σ
, {σ
v j
} \1 )},
(4.135a)
π
∗
(σ
, {σ
v j
} \1 ) = max
˙
,{ ˙
v j } \1
{σ
˙
+
K
k=2
σ
v k
˙
v k − π ( ˙
, { ˙
v j } \1 )}.
(4.135b)
The stationarity conditions corresponding to Eqs. 4.135a and 4.135b are the constitutive relations
˙
(•) = ∂ σ π
∗
(•) and ˙
v k (•) = ∂ σ
v k
π
∗
(•) (• = σ
, {σ
v j
} \1 ),
(4.136a)