CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.5
Three-parameter standard linear solid and fluid models.
where τ = η G (The scalar τ is not to be confused with the shear stress tensor
components, τij.). Models composed of more than two elements are readily
constructed. When a Kelvin unit is combined in series with the linear spring
element, Figure 9.5a, the resulting model is said to represent the standard
linear solid. If the same Kelvin unit is joined in series with a dashpot,
Figure 9.5b, the model represents a three-parameter fluid. In general, the
model of a fluid has a “free dashpot” as one of its elements. Other threeparameter
models are easily imagined, for example, a Maxwell unit in parallel
with a spring, or a Maxwell unit in parallel with a dashpot.
Four-parameter and higher order models may also be constructed. There
are two basic patterns for systematically designing higher order models.
One, leading to the generalized Kelvin model, has n-Kelvin units in series,
Figure 9.6A. The second consists of n-Maxwell units in parallel and is called
the generalized Maxwell model, Figure 9.6B.
For these models the constitutive equations (Eq 9.3-1) in operator form are
σ 12 σ 12
σ 12
Kelvin: γ 12 = + + L +
(9.3-8a)
G + η∂ G + η∂
G + η ∂
1 1
t 2 2 t N N t
G1γ12 G2γ12 GNγ
12
Maxwell: σ 12 = + + L
+
(9.3-8b)
1 1 1
∂t
+ ∂t
+
∂t
+
τ τ
τ
1
2
In these generalized model equations, one or more of the constants G i and
η i may be assigned the values 0 or ∞ in order to represent behavior for a
particular material. Thus, with η 2 and all of the constants which follow it in
Eq 9.3-8a set equal to zero, the constitutive equation for the standard solid,
Figure 9.5a, will be given.
N