Page 112 - Plastics Engineering

Mechanical Behaviour of Plastics 95

where al, a,, bl and bo are all material constants. In the more modem theories
of viscoelasticity this type of equation or the more general form given in
equation (2.53) is favoured.

The models described earlier are special cases of this equation.

2.12 Intermittent Loading
The creep behaviour of plastics considered to date has assumed that the level of
the applied stress is constant. However, in service the material may be subjected
to a complex pattern of loading and unloading cycles. This can cause design
problems in that clearly it would not be feasible to obtain experimental data to
cover all possible loading situations and yet to design on the basis of constant
loading at the maximum stress would not make efficient use of material or be
economical. In these cases it is useful to have methods of predicting the extent
of the recovered strain which occurs during the rest periods of conversely the
accumulated strain after N cycles of load changes.
There are several approaches that can be used to tackle this problem and
two of these will be considered now.

2.12.1 Superposition Principle
The simplest theoretical model proposed to predict the strain response to a
complex stress history is the Boltzmann Superposition Principle. Basically this
principle proposes that for a linear viscoelastic material, the strain response to
a complex loading history is simply the algebraic sum of the strains due to each
step in load. Implied in this principle is the idea that the behaviour of a plastic
is a function of its entire loading history. There are two situations to consider.

(a) Step Changes of Stress
When a linear viscoelastic material is subjected to a constant stress, u1, at time,
tl, then the creep strain, &(t), at any subsequent time, t, may be expressed as

(2.54)

where E(t - tl) is the time-dependent modulus for the elapsed time (t - tl).
Then suppose that instead of this stress q, another stress, a2 is applied at
some arbitraxy time, t2, then at any subsequent time, t, the stress will have
been applied for a time (t - t2) so that the strain will be given by

1
&(t) = .fJ1
- t2)