Page 127 - Plastics Engineering

110 Mechanical Behaviour of Plastics

Therefore after the 11th cycle the total creep time is 11 x 100 = 1.1 x io3
hours. If the total strain at this time is plotted on Fig. 2.51 then a straight line
can be drawn through this point and the point E,(T), and this line may be
extrapolated to any desired number of cycles. For the case in question the line
must be extrapolated to (1001 x 100) hours at which point the total strain may
be obtained as 1.09%. Thus the accumulated residual strain after lo00 cycles
would be 1.09 - 0.747 = 0.343% as calculated on the computer.
Of course it should always be remembered that the solutions obtained in this
way are only approximate since the assumptions regarding linearity of relation-
ships in the derivation of equation (2.64) are inapplicable as the stress levels
increase. Also in most cases recovery occurs more quickly than is predicted
by assuming it is a reversal of creep. Nevertheless this approach does give a
useful approximation to the strains resulting from complex stress systems and
as stated earlier the results are sufficiently accurate for most practical purposes.

2.13 Dynamic Loading of Plastics
So far the deformation behaviour of plastics has been considered for situations
where (i) the stress (or strain) is constant (ii) the stress (or strain) is changing
relatively slowly or (iii) the stress (or strain) is changed intermittently but the
frequency of the changes is small. In practice it is possible to have situations
where the stress and strain are changing quite rapidly in a regular manner. This
can lead to fatigue fracture which will be discussed in detail in later. However,
it is also interesting to consider the stresdstrain relationships when polymers
are subjected to dynamic loading.
The simplest dynamic system to analyse is one in which the stress and
strain are changing in a sinusoidal fashion. Fortunately this is probably the
most common type of loading which occurs in practice and it is also the basic
deformation mode used in dynamic mechanical testing of plastics.
When a sinusoidally varying stress is applied to a material it can be repre-
sented by a rotating vector as shown in Fig. 2.53. Thus the stress at any moment
in time is given by
(T = 00 sinwt (2.66)
where w is the angular velocity of the vector (= 2x f = 2x/T, where f is the
cyclic frequency in hertz (H,) and T is the period of the sinusoidal oscillations).
If the material being subjected to the sinusoidal stress is elastic then there
will be a sinusoidal variation of strain which is in phase with the stress, i.e.
E = EO sinwt
However, for a viscoelastic material the strain will lag behind the stress. The
strain is thus given by
E = EO sin(wt - 6) (2.67)
where 6 is the phase lag.