Viscoelastic Deformation PDF

## Viscoelastic Deformation An amorphous polymer may behave like a glass at low temperatures, a rubbery solid at intermediate temperatures, and a viscous liquid as the temperature is further raised. For relatively small deformations, the mechanical behavior at low temperatures may be elastic; that...

## Viscoelastic Deformation An amorphous polymer may behave like a glass at low temperatures, a rubbery solid at intermediate temperatures, and a viscous liquid as the temperature is further raised. For relatively small deformations, the mechanical behavior at low temperatures may be elastic; that is, in conformity to Hooke's law, σ = Ee. At the highest temperatures, viscous or liquid like behavior prevails. For intermediate temperatures the polymer is a rubbery solid that exhibits the combined mechanical characteristics of these two extremes; this condition is termed viscoelasticity. **Figure 14.4 Schematic tensile stress-strain curve for a semicrystalline polymer. Specimen contours at several stages of deformation are included.** - Stress - Strain **Elastic deformation is instantaneous**, which means that total deformation (or strain) occurs the instant the stress is applied or released. The strain is independent of time. Upon release of the external stress, the deformation is totally recovered. The specimen assumes its original dimensions. This behavior is represented in Figure 14.5b as strain versus time for the instantaneous load-time curve, shown in Figure 14.5a. **By way of contrast, for totally viscous behavior**, deformation or strain is not instantaneous; that is, in response to an applied stress, deformation is delayed or dependent on time. This deformation is not reversible and is not completely recovered after the stress is released. This phenomenon is demonstrated in Figure 14.5d. **For the intermediate viscoelastic behavior**, the imposition of a stress in the manner of Figure 14.5a results in an instantaneous elastic strain, which is followed by a viscous, time-dependent strain, a form of anelasticity. This behavior is illustrated in Figure 14.5c. A familiar example of these viscoelastic extremes is found in a silicone polymer that is sold as a novelty and known by some as "silly putty". When rolled into a ball and dropped onto a horizontal surface, it bounces elastically. The rate of deformation during the bounce is very rapid. On the other hand, if pulled in tension with a gradually increasing applied stress, the material elongates or flows like a highly viscous liquid. The rate of strain determines whether the deformation is elastic or viscous. ## Viscoelastic Relaxation Modulus The viscoelastic behavior of polymeric materials is dependent on both time and temperature. Several experimental techniques may be used to measure and quantify this behavior. Stress relaxation measurements represent one possibility. With these tests, a specimen is initially strained rapidly in tension to a predetermined and relatively low strain level. The stress necessary to maintain this strain is measured as a function of time while temperature is held constant. Stress is found to decrease with time due to molecular relaxation processes that take place within the polymer. We may define a relaxation modulus E,(t), a time-dependent elastic modulus for viscoelastic polymers as: $E_r(t) = \frac{\sigma(t)}{\epsilon_0}$ where σ(t) is the measured time-dependent stress and εo is the strain level, which is maintained constant. Furthermore, the magnitude of the relaxation modulus is a function of temperature; and to more fully characterize the viscoelastic behavior of a polymer, isothermal stress relaxation measurements must be conducted over a range of temperatures. Figure 14.6 is a schematic log E,(t)-versus-log time plot for a polymer that exhibits viscoelastic behavior. Curves generated at a variety of temperatures are included. Key features of this plot are that (1) the magnitude of E,(t) decreases with time (corresponding to the decay of stress, Equation 14.1), and (2) the curves are displaced to lower E,(t) levels with increasing temperature. To represent the influence of temperature, data points are taken at a specific time from the log E,(t)-versus-log time plot, for example, t1 in Figure 14.6, and then cross-plotted as log E, (t₁) versus temperature. Figure 14.7 is such a plot for an amorphous (atactic) polystyrene. In this case, t₁ was arbitrarily taken 10 s after the load application. Several distinct regions may be noted on the curve shown in this figure. At the lowest temperatures, in the glassy region, the material is rigid and brittle, and the value of E,(10) is that of the elastic modulus, which initially is virtually independent of temperature. Over this temperature range, the strain-time characteristics are as represented in Figure 14.5b. On a molecular level, the long molecular chains are essentially frozen in position at these temperatures. As the temperature is increased, E, (10) drops abruptly by about a factor of 10³ within a 20°C (35°F) temperature span; this is sometimes called the leathery, or glass transition region, and the glass transition temperature (T, Section 14.13) lies near the upper temperature extremity; for polystyrene (Figure 14.7), T₁ = 100°C (212°F). Within this temperature region, a polymer specimen will be leathery; that is, deformation will be time dependent and not totally recoverable on release of an applied load, characteristics that are depicted in Figure 14.5c. Within the rubbery plateau temperature region (Figure 14.7), the material deforms in a rubbery manner. Both elastic and viscous components are present, and deformation is easy to produce because the relaxation modulus is relatively low. The final two high-temperature regions are rubbery flow and viscous flow. Upon heating through these temperatures, the material experiences a gradual transition to a soft rubbery state, and finally to a viscous liquid. In the rubbery flow region, the polymer is a very viscous liquid that exhibits both elastic and viscous flow components. Within the viscous flow region, the modulus decreases dramatically with increasing temperature. The strain-time behavior is as represented in Figure 14.5d. From a molecular standpoint, chain motion intensifies so greatly that for viscous flow, the chain segments experience vibration and rotational motion largely independent of one another. At these temperatures, any deformation is entirely viscous and essentially no elastic behavior occurs. Normally, the deformation behavior of a viscous polymer is specified in terms of viscosity, a measure of a material's resistance to flow by shear forces. Viscosity is discussed for the inorganic glasses in Section 12.12. The rate of stress application also influences the viscoelastic characteristics. Increasing the loading rate has the same influence as lowering temperature. The log E,(10)-versus-temperature behavior for polystyrene materials having several molecular configurations is plotted in Figure 14.8. The curve for the amorphous material (curve C) is the same as in Figure 14.7. For a lightly crosslinked atactic polystyrene (curve B), the rubbery region forms a plateau that extends to the temperature at which the polymer decomposes; this material will not experience melting. For increased crosslinking, the magnitude of the plateau E,(10) value will also increase. Rubber or elastomeric materials display this type of behavior and are ordinarily used at temperatures within this plateau range. Also shown in Figure 14.8 is the temperature dependence for an almost totally crystalline isotactic polystyrene (curve A). The decrease in E,(10) at T is much less pronounced than the other polystyrene materials since only a small volume fraction of this material is amorphous and experiences the glass transition. Furthermore, the relaxation modulus is maintained at a relatively high value with increasing temperature until its melting temperature T,, is approached. From Figure 14.8, the melting temperature of this isotactic polystyrene is about 240°C (460°F). ## Viscoelastic Creep Many polymeric materials are susceptible to time-dependent deformation when the stress level is maintained constant. Such deformation is termed viscoelastic creep. This type of deformation may be significant even at room temperature and under modest stresses that lie below the yield strength of the material. For example, automobile tires may develop flat spots on their contact surfaces when the automobile is parked for prolonged time periods. Creep tests on polymers are conducted in the same manner as for metals (Chapter 11); that is, a stress (normally tensile) is applied instantaneously and is maintained at a constant level while strain is measured as a function of time. Furthermore, the tests are performed under isothermal conditions. Creep results are represented as a time-dependent creep modulus Ex(t), defined by: $E_c(t) = \frac{\sigma_0}{e(t)}$ wherein σo is the constant applied stress and e(t) is the time-dependent strain. The creep modulus is also temperature sensitive and diminishes with increasing temperature.