4.2 Kelvin Model
117
Table 4.6 Summary of the generic Kelvin model
(1) Strain
≡ e ≡ v
(2) Energy ψ = ψ()
(3) Stress
σ e = ∂ ψ =: σ − σ v ≡ σ
(4) Potential π = π(˙ )
(5) Stress
σ v = ∂ ˙
π =: σ − σ e ≡ σ
or
(4) Potential π ∗ = π ∗ (σ v )
(5) Evolution ˙
= ∂ σv π ∗
Finally, as a further interesting aspect, the dissipation d = σ v ˙
is alternatively
expressed from Eqs. 4.78a and 4.78b in terms of the dissipation potential π and the
dual dissipation potential π
∗ as
d = π(˙ ) + π
∗
(σ v ) ≥ 0.
(4.80)
The generic Kelvin model is summarized in Table 4.6.
4.3 Generalized-Kelvin Model
The Generalized-Kelvin model of a visco-elastic solid/fluid (in short the
Generalized-Kelvin model) consists of a serial arrangement of K Kelvin elements
each (k = 1, . . . , K ) consisting of a parallel arrangement of (i) an elastic spring and
(ii) a viscous dashpot (see the sketc.h of the specific Generalized-Kelvin model in
Fig. 4.23).
The basic kinematic assumption of the kth Kelvin element is the equality of the
total strain contribution k , the elastic strain contribution e k (representing the elongation of the kth elastic spring), and the viscous strain contribution v k (representing
the elongation of the kth viscous dashpot), i.e.
k ≡ e k ≡ v k ,
(4.81)
whereby the overall total strain adds up from all corresponding contributions
117
Table 4.6 Summary of the generic Kelvin model
(1) Strain
≡ e ≡ v
(2) Energy ψ = ψ()
(3) Stress
σ e = ∂ ψ =: σ − σ v ≡ σ
(4) Potential π = π(˙ )
(5) Stress
σ v = ∂ ˙
π =: σ − σ e ≡ σ
or
(4) Potential π ∗ = π ∗ (σ v )
(5) Evolution ˙
= ∂ σv π ∗
Finally, as a further interesting aspect, the dissipation d = σ v ˙
is alternatively
expressed from Eqs. 4.78a and 4.78b in terms of the dissipation potential π and the
dual dissipation potential π
∗ as
d = π(˙ ) + π
∗
(σ v ) ≥ 0.
(4.80)
The generic Kelvin model is summarized in Table 4.6.
4.3 Generalized-Kelvin Model
The Generalized-Kelvin model of a visco-elastic solid/fluid (in short the
Generalized-Kelvin model) consists of a serial arrangement of K Kelvin elements
each (k = 1, . . . , K ) consisting of a parallel arrangement of (i) an elastic spring and
(ii) a viscous dashpot (see the sketc.h of the specific Generalized-Kelvin model in
Fig. 4.23).
The basic kinematic assumption of the kth Kelvin element is the equality of the
total strain contribution k , the elastic strain contribution e k (representing the elongation of the kth elastic spring), and the viscous strain contribution v k (representing
the elongation of the kth viscous dashpot), i.e.
k ≡ e k ≡ v k ,
(4.81)
whereby the overall total strain adds up from all corresponding contributions