Time-temperature superposition creep

If creep curves are available at only one temperature then the situation is a little more difficult. It is known that properties such as modulus will decrease with temperature, but by how much Fortunately it is possible to use a time-temperature superposition approach as follows ... 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

Time is the major (actor in determining the mechanical properties of a polymer. This is seen directly in creep and stress-relaxation experiments. These tests cover long periods of time, so that they are sensitive to the types of molecular motions that require long times. Tfrey give little direct information on the types of molecular motion that take place at short times. However, by using the time-temperature superposition principle and the WLF equations, access to these short times can be achieved even though they may not easily be attainable by direct experimentation. 

Time of flight (TOF), 75 660-661 Time-of-flight (ToF) mass analyzers, 24 109 Time of flight diffraction (TOFD), 79 486 Time-of-flight instrumentation, in particle counting, 78 150—151 Time-of-flight-SIMS technique, 24 109 Time-resolved fluorimetry, 74 148-149 Time-resolved spectra, analysis of, 74 613 Time standards, 75 749—750 Time-temperature parameters (TTP), 73 471, 478, 479 creep properties and, 73 480 Time-temperature superposition, 27 746-747... 

Time-temperature superposition is frequently applied to the creep of thermoplastics. As mentioned above, a simple power law equation has proved to be useful in the modelling of the creep of thermoplastics. However, for many polymers the early stages of creep are associated with a physical relaxation process in which the compliance (D t)) changes progressively from a lower limit (Du) to an upper limit (DR). The rate of change in compliance is related to a characteristic relaxation time (x) by the equation ... 

Time-temperature superposition is performed in the same empirical manner as for creep. 

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

Figure 3.4 Creep plot for T0 obtained using time-temperature superposition. (After J. Fried, Plastics Engineering, July 1982, with permission.)...

Since the relaxation mechanisms characteristic of the constituent blocks will be associated with separate distributions of relaxation times, the simple time-temperature (or frequency-temperature) superposition applicable to most amorphous homopolymers and random copolymers cannot apply to block copolymers, even if each block separately shows thermorheologically simple behavior. Block copolymers, in contrast to the polymethacrylates studied by Ferry and co-workers, are not singlephase systems. They form, however, felicitous models for studying materials with multiple transitions because their molecular architecture can be shaped with considerable freedom. We report here on a study of time—temperature superposition in a commercially available triblock copolymer rubber determined in tensile relaxation and creep. 

Because of the uncertainties involved in the decomposition, this procedure would not appear to be a practical way to determine the AHa value needed for Equation 8. It does, however, demonstrate three important points (1) it is the compliances of the mechanisms that are additive (2) T0 and AHa can be obtained from plots such as those shown in Figures 7 and 8 of shift data determined in either relaxation or creep experiments without decomposition of compliance master curves (3) Equation 8 describes time-temperature superposition in Kraton 102 adequately within the experimental accuracy. 

FIG. 13.48 Small-strain tensile creep of rigid PVC. Left short-time tests (t < 1000 s) at a te of 2 h after quenches from 90 °C to various temperatures (f/fe < 0.13). The master curve at 20 °C was obtained by time-temperature superposition (compare Section 13.4.8) the dashed curves indicate the master curves at other temperatures. Right, long-term tests (t = 2 x 106 s, fe = 1/2 h, t/te = 1100). The dashed lines are the master curves at 20 and 40 °C for a te of 1/2 h they were derived from the left-hand diagram. From Struik (1977,1978). Courtesy of the author and of Elsevier Science Publishers. 

Above Tg the stress relaxation and the creep behaviour of amorphous polymers obey the "time-temperature superposition (or equivalence) principle". 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

A. Briefly explain the time-temperature superposition principle and how it can be used to predict creep properties. 

Time-temperature superposition was first suggested by H. Leaderman who discovered that creep data can be shifted on the horizontal time scale in order to extrapolate beyond the experimentally measured time frame (9-10). The procedure was shown to be valid for any of the viscoelastic functions measured within the linear viscoelastic range of the polymer. The time-temperature superposition procedure was first explicitly applied to experimental data by... 

The successful application of time-temperature superposition [32] for polystyrene foam allows the prediction of long-term behavior from short-term measurements. This is valuable in construction applications, where creep is an important consideration. 

Whey protein gels (protein 87-143 Time-temperature superposition of creep Katsuta and Kinsella ... 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

In some epoxy systems ( 1, ), it has been shown that, as expected, creep and stress relaxation depend on the stoichiometry and degree of cure. The time-temperature superposition principle ( 3) has been applied successfully to creep and relaxation behavior in some epoxies (4-6)as well as to other mechanical properties (5-7). More recently, Kitoh and Suzuki ( ) showed that the Williams-Landel-Ferry (WLF) equation (3 ) was applicable to networks (with equivalence of functional groups) based on nineteen-carbon aliphatic segments between crosslinks but not to tighter networks such as those based on bisphenol-A-type prepolymers cured with m-phenylene diamine. Relaxation in the latter resin followed an Arrhenius-type equation. 

Creep failure by accelerated aging with elevated temperature. The physical assumption in using this method is that the time-temperature superposition principle holds for the mechanical properties of the polymer over the time and temperature ranges of interest. 

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. 

Thus viscoelasticity is characterized by dependencies on temperature and time, the complexities of which may be considerably simplified by the time-temperature superposition principle. Similarly the response to successively loadings can be simply represented using the applied Boltzmann superposition principle. Experimentally viscoelasticity is characterized by creep compliance quantified by creep compliance (for example), stress relaxation (quantified by stress relaxation modulus), and by dynamic mechanical response. 

Temperature or humidity fluctuations can be accelerated only to the point of maintaining uniform penetration that is likely in the end use environment. If creep or vibration is expected in service, time-temperature superposition may often be applied to accelerate laboratory testing. This technique mathematically predicts the material s response in service, based on laboratory characterization of the material over a... 

Despite the attractiveness of time-temperature superposition and the potential saving in time, the method has not in fact been widely used to obtain creep data for design. One good reason for doubt about the precision of the method is the existence of physical ageing (see Section 4.4.1). Nevertheless, general points well worth retaining in the mind are (i) creep deformation processes are speeded up at higher temperatures (ii) the effective time at a temperature Tg is t/a-p, where r is the time for the same mechanical effect at another temperature 7. 

Use the shear creep data in Figure 4.4, together with the method of time-temperature superposition, to estimate the shear creep compliance for linear polyethylene at 20°C and a creep time 10 s. Ust the assumptions that you make in this long extrapx>lation of the creep data. 

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