The Time-Temperature Equivalence Principle

The shift factor is philosophically based on the concept that the viscoelastic response is a result of the ability of the polymer chains to respond to stress or deformation. Largely this response is temperature or conversely rate dependent. However, one of the factors often overlooked is that the ability of a polymer to respond is also a function of the volume available for the polymer chain to deform, which is ultimately related to density. This results in an additional vertical shift of the raw experimental data. This shift is often overlooked for two reasons. The first reason is that many practitioners are unaware that it exists and the more valid reason is that the time-temperature transformation (TTT) of the raw data is at best often only an order of magnitude predictor of the response over long times. 

For Kelvin solid-type materials, the viscosities can be represented by the Arrhenius equation, and can be calculated from an equation of the form 

FIGURE 3.14 Storage compliance of poly( -octyl methacrylate) determined under isothermal conditions. (Adapted from Ferry, J. D., Viscoelastic Properties of Polymers, New York, NY John Wiley Sons, 1980.) 

Another shift factor that has been used is based on the work of Williams, Landel, and Ferry (WLF), and referred to as the WLF shift factor 

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We shall presently examine the physical significance of the shift factors, since they quantitatively embody the time-temperature equivalence principle. For the present, however, we shall regard these as purely empirical parameters. The following Ust enumerates some pertinent properties of a ... 

Dynamic mechanical experiments yield both the elastic modulus of the material and its mechanical damping, or energy dissipation, characteristics. These properties can be determined as a function of frequency (time) and temperature. Application of the time-temperature equivalence principle [1-3] yields master curves like those in Fig. 23.2. The five regions described in the curve are typical of polymer viscoelastic behavior. 

It is well established that between Tg and about Tg + 50 K, the relaxation kinetics obeys the WLF law (Williams et al., 1955). If Pr is a property depending on the macromolecular mobility (relaxation modulus, complex modulus, viscosity, diffusion rate, etc.), the time-temperature equivalence principle may be formulated as... 

It was initially stated that Cf are Cf were universal constants (Cf 17 Cf 50 K), but Cf can vary between 2 and 50 and Cf between 14 and 250 K (Mark, 1996). Epoxy values have been found in the low part of these intervals Cf 10, Cf 40 15 K (Gerard et al., 1991), whereas unsaturated polyester values can be relatively high Cf/Cf = 15-55 = 73-267 K (Shibayama and Suzuki, 1965). There is, to our knowledge, no synthetic study on the ideality and crosslinking effects on Cfand Cf. The time-temperature equivalence principles will be examined in detail in Chapter 11, which is devoted to elasticity and viscoelasticity. 

As discussed in Chapter 10, network polymers - as linear polymers - obey the time-temperature equivalence principle in the domain where they are stable, both chemically (no postcure, no thermal degradation), and physically (no orientation relaxation, water desorption, physical aging, etc.). 

The time-temperature equivalence principle can also be applied to other viscoelastic functions in a similar way. Again, this leads to shift factors that are identical with those obtained from stress relaxation ... 

When the rate of elongation is increased, the tensile strength and the modulus also increase the elongation to break generally decreases (except in rubbers). Normally an increase of the speed of testing is similar to a decrease of the temperature of testing. To lightly cross-linked rubbers even the time-temperature equivalence principle can be applied. The rate dependence will not surprise in view of the viscoelastic nature and the influence of the Poisson ratio on the ultimate properties. 

For the tensile strength of a rubber to follow the time-temperature equivalence principle of linear viscoelasticity it is necessary that the extension at break also follow it. This is most easily verified by use of Equation (23), i.e., with the simplifying assumption of strain-time factorization. In an experiment conducted at fixed rate of strain, i = constant, the stress at any temperature and strain may be shown to be (200) ... 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

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. 

STUDY ON THE TIME-TEMPERATURE EQUIVALENT PRINCIPLE FOR ROCKS... 

Abstract Based on the theory of irreversible process thermodynamics, non-linear stress-strain-temperature equations are derived, together with an expression for time-temperature equivalence. In addition, an equation of shift factor for time-temperature equivalence is also obtained. The parameters in the equations are experimentally determined and the main curves for creep compliance and cohesion of TOP granite are obtained by a series of creep tests. As a result, it is proved that both deformation and strength of the TOP granite follow the time-temperature equivalent principle. 

The creep compliance of rocks reflects their deformability whereas the cohesion reflects their strength. In Section 1, the expressions are derived for the time-temperature equivalence for rocks and the testing results obtained in Sections 2 and 3 have shown clearly that not only the deformability of the TGP granite rock but also its strength follow the time-temperature equivalent principle. The establishment of shift factor eq. (25) and the determination of its parameters make it possible to predict correctly the long-term mechanical response of rock at lower temperatures according to its short-term mechanical behaviour at higher temperatures. 

GPa and = 1.00 GPa. The axial Young s modulus of the unit has already been determined by measurements of the shift of the meridional X-ray reflection resulting from an applied stress [32]. Using the time—temperature equivalence principle, it is estimated that the ultrasonic (10 MHz) " at room temperature (23" C) is equivalent to the quasi-static X-ray " of 137 GPa at — dO C. Using this " value and the above the four C values we obtained = 145 GPa. With the five known stiffnesses of the unit and the orientation parameters... 

From a time-temperature equivalence principle (see below), any material history may be represented by an isothermal equivalent, corresponding to a point U (tu, Tu) in the (t, T) graph. If the point U is below the TSC curve, the material will not undergo failure in the particular conditions. In contrast, if the point U is above the TSC curve, the material will undergo failure because its index % will change sign. 

Lightly cross-linked elastomers follow a simple pattern of ultimate behaviour. Smith (1958) has shown that the ultimate properties of this class of polymers follow a time-temperature equivalence principle just as the viscoelastic response to small non-destructive stresses does. 

Crystallisation accompanying stretching invalidates the simple time-temperature equivalence principle. 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

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. 

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. 

It is of a great significance to understand how the mechanical behaviours and properties of rock masses change with temperature, such as for nuclear waste repositories and deep mining at certain temperatures. The key to this problem is how to make predictions to long-term response of rocks based on mechanical models and test results within a short time of experiments. It is put forward in this paper that the problem can be resolved by means of time-temperature equivalent principle for rocks. 

Polymers show a similar response to temperature and strain rate (time), as might be expected from the time-temperature superposition principle (compare Figures 13.31 and 13.32). Specifically, the effect of decreasing temperature is equivalent to that of increasing the strain rate. As has become evident from our previous discussions, low temperature restricts molecular movement of polymers, and consequently they become rigid and brittle. Materials deform to relieve imposed stress. High strain rates preclude such deformation and therefore result in brittle failure. 

In the following sections we discuss the two superposition principles that are important in the theory of viscoelasticity. The first is the Boltzmann superposition principle, which is concerned with linear viscoelasticity, and the second is time-temperature superposition, which deals with the time-temperature equivalence. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

By application of the time-temperature superposition principle, a decade of frequency can be shown to correspond to a 6 or TC shift in 7. Noting that the normal acoustical range goes from 20 to 20,000 Hz, or three decades, it can be seen that the equivalent temperature range is 18-20°C. We then conclude that a properly chosen homopolymer can Just damp all acoustical frequencies at a single use temperature. 

It is noted that the cold-stretching induced microstmcture change depends on the temperature and stretching rate due to the famous time-temperature equivalent principle. During the stretching process, the molecule chains experience two competing processes. One is the extension, that is, the... 

It is commonly observed that the temperature and frequency dependence of polymer relaxations are related. This is expressed qualitatively as the time-temperature superposition principle, or the frequency-temperature equivalence,... 

The correspondence principles accomplish an important goal prediction of long-term behavior from short-term tests. However, some polymer scientists and engineers do not believe that the prediction methods work apparently, because they think that the time-temperature equivalence is the same thing as the so-called WLF equation of 1955 (63) for the shift factor ax- Ferry who co-created WLF warned (61) that the use of that equation is limited to a temperature range... 

Figure 10.14 (36-38) illustrates the time-temperature superposition principle using polyisobutylene data. The reference temperature of the master curve is 25°C. The reference temperature is the temperature to which all the data are converted by shifting the curves to overlap the original 25°C curve. Other equivalent curves can be made at other temperatures. The shift factor shown in the inset corresponds to the WLF shift factor. Thus the quantitative shift of the data in the range Tg to Tg x 50°C is governed by the WLF equation, and... 

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