Viscoelastic response curves

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Isothermal measurements of the dynamic mechanical behavior as a function of frequency were carried out on the five materials listed in Table I. Numerous isotherms were obtained in order to describe the behavior in the rubbery plateau and in the terminal zone of the viscoelastic response curves. An example of such data is shown in Figure 6 where the storage shear modulus for copolymer 2148 (1/2) is plotted against frequency at 10 different temperatures. 

Even though time crosslink density correspondence can be used very efficiently to generate viscoelastic response curves of some non-reacting polymer networks, our findings make it clear that this technique is not generally applicable to all systems even in the case of limited extents of chemical reaction. 

In most, if not all, textbooks on the viscoelasticity of polymers, once the location of Tg is known, the glass transition and its phenomenology become subjects of peripheral interest. At best, one is reminded of the glass transition in discussions of the temperature dependence of the shift factor ar used to construct the master viscoelastic-response curve by time-temperature superposition of experimental data, especially so if Tg is chosen as the reference temperature To in the WLF equation (2.16) for To justify time-temperature superposition of data, one makes the assumption that all viscoelastic mechanisms are governed by one and... 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

PP bead foams of a range of densities were compressed using impact and creep loading in an Instron test machine. The stress-strain curves were analysed to determine the effective cell gas pressure as a function of time under load. Creep was controlled by the polymer linear viscoelastic response if the applied stress was low but, at stresses above the foam yield stress, the creep was more rapid until compressed cell gas took the majority of the load. Air was lost from the cells by diffusion through the cell faces, this creep mechanism being more rapid than in extruded foams, because of the small bead size and the open channels at the bead bonndaries. The foam permeability to air conld be related to the PP permeability and the foam density. 15 refs. 

The viscoelastic response of a series of CMIMx copolymers has been determined as a function of temperature [79]. The loss modulus, El, at 1 Hz, of a series of copolymers with molar content of maleimide units varying from 5 to 25% is plotted in Fig. 127 for comparison the curve for pure PMMA is also drawn. 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

The master curves and shift factors of transient and dynamic linear viscoelastic responses are calculated for linear, semi-crystalline, and cross-linked polymers. The transition from a WLF dependence to an Arrhenius temperature dependence of the shift factor in the vicinity of Tg is predicted and is related to the temperature dependence of physical aging rate. 

Figure 20.6 Creep curves for viscous, elastic, and viscoelastic responses.

Influence of the Strain Rate on Viscoelastic Response. In high-shear-strain experiments, we observe significant differences only with type 2 and type 3 blends (Figures 9b and 9c). The complex viscosity still depends on frequency, but it decreases as phase separation begins. Moreover, a crossover of the tan 8 curves appears at this moment. The differences reported in this land of experiment can be found at two levels ... 

Assuming that the WLF equation does indeed describe the time-temperature shifts, the complete viscoelastic response of any polymer under any experimental conditions may be obtained from knowledge of any two of the following three functions the master curve at any temperature, the modulus-temperature curve at any time, and the shift factors relative to some reference temperature. For example, suppose we are given the constants Cj, and C2 for a polymer whose master curve is known. (The values given for C, and C2 are those that result from fitting equation (4-6) to the aT vs. T data.5) For simplicity, we can assume that the master curve is at the same reference temperature as that in the WLF equation, perhaps Tg. Suppose it is desired to calculate the 10-second modulus-versus-temperature curve for this polymer. 

Intermediate Response. Figure 6 is a double logarithmic plot of o/e vs. time in seconds at three different strain rates for the samples as a function of H O content. To extend the time scale and to correlate results at various , we have used the reduced-variables procedure shown to be applicable in describing the viscoelastic response of rubbery materials (8) as well as of several glassy polymers (6). (To compensate for the effect of different e we plot a/e vs. e/e the latter is simply the time, t.) Superposition over the entire time scale for 0% H2O (upper curve) is excellent except for times close to the fracture times of the materials tested e higher strain rates. For example, a deviat ipn occurs at 10 sec for the material at e = 3.3 x 10 sec... 

In spite of these complications, the viscoelastic response of an amorphous polymer to small stresses turns out to be a relatively simple subject because of two helpful features (1) the behavior is linear in the stress, which permits the application of the powerful superposition principle and (2) the behavior often follows a time-temperature equivalence principle, which permits the rapid viscoelastic response at high temperatures and the slow response at low temperatures to be condensed in a single master curve. 

By use of the time-temperature equivalence principle, the viscoelastic response of a given polymeric material over a wide temperature range can be accommodated in a single master curve. By use the superposition principle, this master curve can be used to estimate the time-dependent response to time-dependent stresses in simple tensile or shear specimens or to nonhomogeneous time-dependent stresses arising in stressed objects and structures. 

Figure 3. Stress-strain curves in nonlinear elastic and nonlinear viscoelastic responses...

Time-temperature superposition is applicable to a wide variety of viscoelastic response tests, as are creep and stress relaxation. We illustrate the principle by considering stress relaxation test data. As a result of time-temperature correspondence, relaxation curves obtained at different temperatures can be superimposed on data at a reference temperature by horizontal shifts along the time scale. This generates a simple relaxation curve outside a time range easily accessible in laboratory experiments. This is illustrated in Figure 14.13 for polyisobutylene. Here, the reference temperature has been chosen arbitrarily to be 25°C. Data obtained at temperature above 25°C are shifted to the right, while those obtained below 25°C are shifted to the left. 

T > To are shifted to longer times, and measurements for T < Tq aie shifted to shorter times. A well-defined reduced curve means the viscoelastic response is thermorheologically simple (Schwarzl and Staverman, 1952). It represents log Jp(t) at To over an extended time range. The time scale shift factors aj that were used in the reduction of the creep compliance curves to obtain the reduced curve constitute the temperature dependence, ar is fitted to an analytical form, which is often chosen to be the Williams-Landel-Ferry (WLF) equation (Ferry, 1980),... 

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