4.3 Generalized-Kelvin Model
119
σ
σ
1 ≡ e
2 ≡ v
E 0
η k
E k
Fig. 4.24 Standard-Linear-Solid Kelvin model
ψ( v ) =
1
2
E 0 [ − v ]
2
+
1
2
E k
2
v .
(4.83)
Then the energetic stress σ
, which is conjugated to the total strain , and the
energetic viscous stress σ
v , which is conjugated to the viscous strain v , follow as
σ
( v ) = ∂ ψ( v ) = E 0 [ − v ]
,
(4.84a)
σ
v ( v ) = ∂ v ψ( v ) = −E 0 [ − v ] + E k v .
(4.84b)
Note that the total stress σ applied to the rheological model (that enters the equilibrium condition) coincides identically with the energetic (or rather instantaneous)
stress, σ
:= σ 0 ≡ σ, and, due to the serial arrangement of the (elastic instantaneous)
Hooke element and the Kelvin element, also with the difference of the elastic stress in
the Kelvin element, σ e := E k v , and the energetic viscous stress, σ e − σ
v = σ 0 ≡ σ.
Furthermore, for the Standard-Linear-Solid Kelvin model the convex and smooth
(quadratic) dissipation potential π is chosen as
π(˙ v ) =
1
2
η k |˙ v |
2
.
(4.85)
Observe that (i) π does not depend on ˙
thus the dissipative stress σ
= σ − σ
≡
0 vanishes identically (however the dissipative viscous stress σ
v = 0 for ˙
v = 0),
and that (ii) π is positively homogenous of degree two in ˙
v and obviously smooth at
the origin ˙
v = 0. Consequently, the dissipative viscous stress σ
v computes as partial
derivative of the dissipation potential with respect to its conjugated variable
σ
v (˙ v ) = ∂ ˙
v π(˙ v ) = η k ˙
v .
(4.86)