6.2.3 Time–Temperature Superposition Principle
In the stress-relaxation experiment, we can often define the time-dependent
modulus
EðtÞsðtÞ
e 0(6.23)
This modulus is determined not only by time, but also by the temperature at which
the experiment is conducted. As illustrated in Fig.6.11, the effect of increasing
temperature on the modulus of the polymer is equivalent to that of extending the
time scale at a given temperature. Examples of such widely exist in our daily life.
During the landing of an airplane, the mechanical responses experienced by the tire in
a very short instant is equivalent to that of the tires at a lower temperature over a
longer time. For the tires to maintain a rubbery-like response and to absorb the impact,
a very low glass transition temperature is required. If a small piece of stone skips
across the water surface with a high-enough speed from a small-enough angle (about
20 degree in optimum), it will feel like hitting a solid and bounces back several times;
if with a low-enough speed and a high-enough angle, it will feel like dropping into a
liquid and incurs weak resistance. With an improper posture of the body, diving from
the high board is likely to cause a severe hurt by hitting the water surface. The above
time-temperature equivalence of effects is often called thetime-temperature super-
position principle(Tobolsky and Andrews 1945 ). Therefore, there exists amaster
curve, displaying the mechanical responses of polymers, i.e. the sequential occur-
rence of the glass, rubber and fluid states upon temperature rise or time extension.
There is no clear criterion to separate the liquid and the solid states, because it
matters with the time scale of our observations. When the imposing time of the stress
is shorter than the relaxation time of the liquid, the liquid will mainly show an elastic
response, exhibiting the feature of a solid. On the contrary, when the imposing time
of the stress is longer than the relaxation time of the solid, the solid will experience a
Fig. 6.11Illustration of time-temperature superposition principle for the stress relaxation of
polymers. Theright-hand-sidemaster curve at a constant temperature is obtained by the parallel
shift of theleft-hand-sidecurves at various temperatures. The shift factor used to construct the
master curve follows the WLF equation
6.2 Relaxation of Polymer Deformation 103