Materials for engineering, 3rd Edition - (Malestrom)

66
Materials for engineering
(a) (b) (c)
2.22 Spring-and-dashpot models of viscoelastic solids (a) Maxwell,
(b) Voigt and (c) four-element model.
same and the total strain or strain rate is determined from the sum of the two
components:
de
d = σ
+ 1d σ
t η E dt
Voigt model
When the spring and dashpot are in parallel (Fig. 2.22(b)), the strains in the
two elements are equal and the total stress on them is given by the sum:
de
σ T ( t) = Ee + η
dt
If the applied stress is constant, as in a creep test, this differential equation
may be directly integrated to give the strain as a function of time:
et ( ) =
σ –t/ τ
(1 – e )
E
where the time constant, τ = η/E.
Four-element model
A more realistic description of polymer behaviour is obtained with a fourelement
model consisting of Maxwell and Voigt elements in series (Fig.
2.22(c)). By combining the above equations, the total strain experienced by
this model is given by:
et ( ) =
σ
+
σ –t/ τ
(1 – e ) +
σ
t
E E
η
1 2