Page 113 - Plastics Engineering

96 Mechanical Behaviour of Plastics

Now consider the Situation in which the stress, 01, was applied at time, tl,
and an additional stress, 02, applied at time, r2, then Boltzmanns’ Superposition
Principle states that the total strain at time, t, is the algebraic sum of the two
independent responses.
1 1
E(t) = E(t - tl) a0 + E(r - 12) * 61

This equation can then be generalised, for any series of N step changes of
stress, to the form
i=N
e(t) = Eoi (2.55)
i=l
where ai is the step change of stress which occurs at time, ti.
To illustrate the use of this expression, consider the following example.
Example 2.13 A plastic which can have its creep behaviour described
by a Maxwell model is to be subjected to the stress history shown in
Fig. 2.43(a). If the spring and dashpot constants for this model are 20 GN/m2
and 1000 GNs/m2 respectively then predict the strains in the material after 150
seconds, 250 seconds, 350 seconds and 450 seconds.
Solution From Section 2.1 1 for the Maxwell model, the strain up to 100s is
given by
a
m
E(t) = - + -
ttl
Also the time dependent modulus E(t) is given by
E(t) = - - ttl (2.56)
O
~
- tl+ tt
Then using equation (2.54) the strains may be calculated as follows:
(i) at t = 150 seconds; ai = 10 MN/m2 at tl = 0, a2 = -10 MN/m2 at t2 =
100 s

= 0.002 - 0.001 = 0.1%
(ii) at 250 seconds; 61, a2 as above, a3 = 5 MN/m2 at t3 = 200 s

~(250) = 10 1
ttl
tl + * (250 - 200)

= 0.003 - 0.002 + 0.0005 = 0.15%