Page 105 - Plastics Engineering

88 Mechanical Behaviour of Plastics

Stress-Strain Relations
These are the same as the Maxwell Model and are given by equations (2.27)
and (2.28).

Equilibrium Equation
For equilibrium of forces it can be seen that the applied load is supported
jointly by the spring and the dashpot, so

a = u1 + a2 (2.36)

Geometry of Deformation Equation
In this case the total strain is equal to the strain in each of the elements, i.e.

&=E1 =E2 (2.37)
From equations (2.27), (2.28) and (2.36)

a = 6 * El + qb2

or using equation (2.37)
a=f*&+q*& (2.38)

This is the governing equation for the Kelvin (or Voigt) Model and it is
interesting to consider its predictions for the common time dependent defor-
mations.

(3 c=p
If a constant stress, a,, is applied then equation (2.38) becomes

a, = 6 *E + qi
and this differential equation may be solved for the total strain, E, to give

where the ratio q/( is referred to as the retardation time, TR.
This indicates an exponential increase in strain from zero up to the value,
ao/t, that the spring would have reached if the dashpot had not been present.
This is shown in Fig. 2.37. As for the Maxwell Model, the creep modulus may
be determined as
(2.39)