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Encyclopedia of Physical Science and Technology EN012c-598 July 26, 2001 15:59

710 Polymers, Mechanical Behavior

reciprocal time, increasing this variable essentially short- using low-strain deformations that are carried out over a
ens the observation time scale and hence decreases the range of both temperature and frequency. The basics of
denominator in Eq. (19). Now whereas the denominator the technique are best explained by first considering the
is controlled specifically by the observation time, which response of a pure elastic body to an oscillatory dynamic
is inversely related to deformation rate, the numerator is strain such as is expressed by
controlled by many factors, many of which extend from
ε(t) = ε 0 sin(ωt), (20)
internal variables such as molecular weight, cross-linking,
molecular architecture, chain stiffness as related through where ω = 2π frequency. A linear elastic body would be
chain chemistry, and crystallinity. As mentioned earlier, it that of a Hookean spring whose relationship of strain to
is important to recognize the balance between three types stress in tension is given by
of energy: intermolecular, intramolecular, and thermal en-
ergy. Indeed, the numerator in Eq. (19) is highly dependent σ 0 (t) = E ε(t). (21)
on the balance of these energies. In summary, the Deborah
This relationship clearly shows that the stress on the sys-
number concept is important for rationalizing the mechan-
tem is a function only of the displacement and carries
ical behavior of a polymeric material when tested under
no other time-dependent characteristics. Hence, with
different loading rates and environmental conditions.
Eq. (20) for the time-dependent strain, the time-dependent
Utilizing our earthworm concept of polymeric species,
Hookean stress becomes
let us strengthen our understanding of the effect of defor-
mation rate. Suppose we hold temperature constant, for σ 0 (t) = E ε 0 sin(ωt). (22)
example, at ambient temperature, and carry out a series
Hence, a general plot of these two functions together
of stress–strain experiments, but the changing variable is
would appear as shown in Fig. 15a. Note that the amplitude
now deformation rate, be it in shear or extension. There is
little doubt that again the response expected would mimic of the stress function varies in proportion to its modulus
what is displayed in Fig. 10. At low rates of deformation, but that both the stress and strain oscillate in phase; that
due to the backbone motion of the worms, they could move is, a zero phase angle exists.
within the time frame of a slow deformation experiment, To represent a linear viscous body, Newton’s law of
thereby leading to considerable strain and low-modulus viscosity suffices, which can be stated in tension as
behavior. On the other hand, at very high rates of defor-
σ 0 (t) = η(d ε/dt), (23)
mation, there is little doubt as to what the result would be!
Crude as the analogy seems, the response of the worms where η represents a Newtonian viscosity; that is, it is in-
would now provide a higher modulus due to the inabil- dependent of the deformation rate. Utilizing this equation
ity to disentangle, thereby leading to chain scission and in conjunction with Eq. (20) for the time-dependent strain
more brittle-like behavior similar to what we discussed shows that the time-dependent stress for a purely viscous
earlier with reference to the “silly putty” polymeric ma- material is
terial. Indeed, one can show that chain scission occurs
σ 0 (t) = ηωε 0 cos(ωt), (24)
in real macromolecular systems under suitable rates and
temperatures by utilizing electron spin resonance studies whichisstrikinglydifferentinitstime-dependentbehavior
to reveal the presence of free radicals generated by the from the Hookean stress elastic response. In particular, the
breaking of covalent bonds. amplitude now varies linearly with the deformation rate
With this somewhat simplistic but useful earthworm (frequency), a factor that had no importance in the elas-
analogy we have a firm grasp of the general interplay be- tic or Hookean behavior. Particularly important is the fact
tween temperature and loading rate. Our desire now is to that, for a purely viscous response, the stress and strain
◦
quantify the relative degree of storage of mechanical en- are out of phase by 90 (see Fig. 15b). (The stress will
ergy or (elastic behavior) caused by an entangled network- “lead” that of the strain, since the material will not un-
like behavior in contrast to the dissipation of mechani- dergo deformation without sensing stress.) It then should
cal energy (viscous dissipation) through chain slippage be obvious that, if a viscoelastic material is placed under
and general segmental frictional forces that are related to an oscillatory deformation at small strains where its re-
important mechanical properties, such as hysteresis, dis- sponse should be linear, and if one can monitor the stress
cussed earlier. A common way to quantify the relative and strain response via suitable electronic techniques, this
degree of the viscous and the elastic components in these will allow a direct measurement of the phase angle be-
viscoelastic materials is by means of dynamic mechanical tween these two functions. This measurement will clearly
analysis (DMA) techniques. This approach allows a di- provide a direct measure of the viscous response. As one
rect measurement of these two components generally by might expect, a viscoelastic polymer undergoing cyclic