Stress time-temperature correspondence

The time-temperature correspondence principle holds not only for the viscosity but also for the normal stresses. In the latter case, however, the... 

The time-temperature correspondence principle states that there are two methods to use to determine the polymer s behavior at longer (or shorter) times than those covered by a stress-relaxation experiment run at 7j. First, one may improve the experiment to measure directly the response at longer (shorter) times. For the longer times, however, this procedure rapidly becomes prohibitively time-consuming because the change is so slow (note that Figure 4-5 is plotted on a log scale). (For the shorter times, the limitations are equipment related, e.g., transducer response time, problems with instrument and sample inertia, etc.) An alternative, according to the time-temperature... 

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. 

We conclude that the addition of the PLC to PP strengthens the material in a predictable and quantifiable way. Surveying the above creep as well as stress relaxation, we find that the time-temperature correspondence principle is applicable here in spite of the multiphase character of the PLC. Hence the predictive capabilities, noted above for various specific instances, are quite extensive. It remains to be seen whether other PLC-containing blends behave similarly, although there is no a priori reason why they should behave differently. 

The correspondence principles allow us to achieve our goal prediction of long-term mechanical properties—and thus performance and reliability—from short-term tests. It is possible to predict behavior for, say, 16 decades of time from experiments each of which was made over four decades only examples will be given below. It is easy to see that, when using the time-temperature correspondence, essential is the capability to predict the temperature shift factor ariT). Similarly, when using the time-stress correspondence one needs the stress shift factor ao-ia). Starting from the Doolittle equation (24.8), it was possible to obtain a general equation [26]... 

Here v,ef pertains to the stress level of interest and is thus similar to T f. If we assume a constant stress level, we obtain an equation which allows us to apply the time-temperature correspondence ... 

Long-Term Predictions from a Minimum of Data. One problem with the use of the correspondence principles to make long-term prediction seemed unresolved the amount of data needed. Thus, in Reference 64 experimental creep and stress relaxation results for the PET/0.6PHB PLC were obtained at 10 temperatures. Similarly, in Reference 67 creep compliance was determined at 9 stress levels. Can we get away with doing experiments at two or three temperatures—or at two or three stress levels—and get valid predictions The answer is yes, and procedures for that purpose have been developed. When one works with the time-stress correspondence, then the minimum data procedure (68) is naturally based on equation 29. Similarly, when we use the time-temperature correspondence, the minimum data procedure (69) is based on equation 28. 

For a fiber immersed in water, the ratio of the slopes of the stress—strain curve in these three regions is about 100 1 10. Whereas the apparent modulus of the fiber in the preyield region is both time- and water-dependent, the equiUbrium modulus (1.4 GPa) is independent of water content and corresponds to the modulus of the crystalline phase (32). The time-, temperature-, and water-dependence can be attributed to the viscoelastic properties of the matrix phase. 

Product performance data Products subjected to a given load develop a corresponding predictable deformation. If it continues to increase without any increase in load or stress, the material is said to be experiencing creep or cold flow. Creep in any product is defined as increasing strain over time in the presence of a constant stress (Figs. 2-25 and 26). The rate of creep for any given plastic, steel, wood, etc. material depends on the basic applied stress, time, and temperature. 

The longest relaxation time. t,. corresponds to p = 1. The important characteristics of the polymer are its steady-state viscosity > at zero rate of shear, molecular weight A/, and its density p at temperature 7" R is the gas constant, and N is the number of statistical segments in the polymer chain. For vinyl polymers N contains about 10 to 20 monomer units. This equation holds only for the longer relaxation times (i.e., in the terminal zone). In this region the stress-relaxation curve is now given by a sum of exponential terms just as in equation (10), but the number of terms in the sum and the relationship between the T S of each term is specified completely. Thus... 

An attempt will now be made to determine the activation energies of the two processes, since the identification of the mechanism is based primarily on this factor a study of all the stress relaxation curves has revealed that within experimental error time-temperature superposition is valid above a modidus of 10 dynes/cm , and that the shift factors are of the WLF form. Therefore, we can determine the constants of the WLF equation which were listed in Table 3 and also the corresponding activation energies. The values of the activation energies for the process which is governed by these shift factors, calculated for T —Tg + 30, ranges from 150 to 230 Kcal. This mechanism will, for the moment, be called the "first mechanism. 

The aim of this work is to provide both experimental information and a corresponding formalization in order to elucidate structural propellant grain safety during ignition. The experimental data were obtained from uniaxial tensile tests and simple shear tests performed with an imposed hydrostatic pressure varying from atmospheric pressure to 15 MPa. It is well established that the materials studied exhibit time-temperature and pressure-sensitive properties. The ultimate properties reported here are formalized in a proposed stress-failure criterion capable of including the pressure effect. 

The behavior observed in the stress-strain curve corresponds to viscoelastic behavior that is typical of polymeric materials. The viscoelastic behavior is highly dependent on the temperature at which the test is performed and its relationship to the Tg of the sample. It is also dependent on the rate of deformation, as mentioned previously. In general, very rapid deformation does not allow time for molecular rearrangement to occur and results in behavior characteristic of a more brittle material. The effects of temperature and rate of testing on plastic materials are illustrated in Figs. 3.54 and 3.55, respectively. 

There are many stress-analysis problems involving viscoelastic materials that are of a statically determinate class, i.e., the stresses in the body depend only on the applied forces and moments and not specifically on the elastic properties of the body. Such problems can be solved by invoking the correspondence principle. Then, the time and temperature dependences of the strains and flexures in the body can be obtained through the time temperature-shift properties of the viscoelastic polymer. 

It was demonstrated in several papers [45-49] that increasing the stress leads to a shift of the time relaxation spectrum in a manner similar to increasing the temperature. Thus we have correspondence between time, temperature, frequency and also stress. This means, among other things, that the stress dependence of kernel K in equation (12.4) may be described by a function analogous to the temperature shift factor aj. Then the kernel K expressed as a sum of exponential functions may... 

Figure 16. The shear stress versus the shear rate for Ak = 1.25 aad

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