CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.3
Viscous flow analogy.
FIGURE 9.4
Representations of Kelvin and Maxwell models for a viscoelastic solid and fluid, respectively.
where δ˙
is the time-rate of extension. This analogy is illustrated in
Figure 9-3.
Based upon the two fundamental elements described above, it is easy to
construct viscoelastic models by suitable combinations of this pair of elements.
Two especially simple combinations immediately come to mind. The
first, that of the spring and dashpot in parallel, Figure 9.4a, portrays the
Kelvin solid for which Eq 9.3-1 becomes
{ }
σ = + η ∂ γ
12 G t 12
(9.3-6)
where the partial derivative with respect to time is denoted by ∂ t ≡ ∂/∂t. The
second, the spring and the dashpot connected in series, Figure 9.4b, represents
the Maxwell fluid having the constitutive equation
⎧ 1⎫
⎨∂t
+ ⎬σ
= { G∂t}
γ
⎩ τ ⎭
12 12
(9.3-7)