Equivalence principle superposition principles

The WLF equation can be widely applied, and demonstrates the equivalence of time and temperature, the so-called time-temperature superposition principle, on the mechanical relaxations of an amorphous polymer. The equation holds up to about 100° above the glass transition temperature, but after that begins to break down. 

The Boltzmann superposition principle states that the response of a material to a given load is independent of the response of the material to ary load that is already on the material. Another consequence of this principle is that the deformation of a specimen is directly proportional to the applied stress when all deformations are compared at equivalent, times... 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

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.

Polymers are generally assumed to obey the Boltzmann superposition principle in the domain of small strains. When there are changes of loading conditions, the effects of these changes are additive when the corresponding responses are considered at equivalent times. For instance, if different stresses a0, CT, a2, are applied at different times 0, t], t2,, respectively, the... 

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

In the preceding sections, we have looked at the various types of relaxation processes that occur in polymers, focusing predominantly on properties like stress relaxation and creep compliance in amorphous polymers. We have also seen that there is an equivalence between time (or frequency) and temperature behavior. In fact this relationship can be expressed formally in terms of a superposition principle. In the next few paragraphs we will consider this in more detail. First, keep in mind that there are a number of relaxation processes in polymers whose temperature dependence we should explore. These include ... 

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. 

The major features of linear viscoelastic behavior that will be reviewed here are the superposition principle and time-temperature equivalence. Where they are valid, both make it possible to calculate the mechanical response of a material under a wide range of conditions from a limited store of experimental information. 

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. 

Suppose now that a periodic field is equivalent to a continuous series of static fields, each being applied during an infinitely short time du between and u- -du. It is found by applying the superposition principle that the distribution function at a given moment is... 

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. 

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. 

Although Eq. (2.10) fulfils the relativistic condition of space-time equivalence, it does not fulfil the quantum requirement of linearity so that the superposition principle, probability density formula and uncertainty principle could apply [5,6]. The third step was to look for an analogous equation linear in all that is,... 

The aforesaid extrapolations make use of a time-temperature superposition principle which is based on the fact that time and temperature have essentially equivalent effects on the modulus values of amorphous polymers. Figure 3.19 shows modulus data taken at several temperatures for poly(methyl methacrylate) [8]. Because of the equivalent effect of time and temperature, data at different... 

In summary, if G t), which is contained in Eqs. (4.30), (4.34)-(4.37), (4.49)-(4.51), (4.63) and (4.73), is known, all the linear viscoelastic quantities can be calculated. In other words, all the various viscoelastic properties of the polymer are related to each other through the relaxation modulus G t). This result is of course the consequence of the generalized Maxwell equation or equivalently Boltzmann s superposition principle. The experimental results of linear viscoelastic properties of various polymers support the phenomenological principle. Some viscoelastic properties play more important roles than the others in certain rheological processes related to... 

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. 

The Boltzman Superposition Principle is one starting point for inclusion of structural relaxation losses. An equally valid starting point is to include in equation 9 time derivatives (first-order and higher) of stress and strain. It can he shown that this approach is equivalent to the above integral representation (10). Finally, modified stress-strain relations, to describe viscoelastic response, have also been formulated using fractional derivatives (11). 

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,... 

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... 

What is applied here is known in the literature as the time-temperature superposition principle . The result of the synthesis is called a master-curve . For a thermally activated Debye-process, the basis of the principle is easily seen. According to Eq. (5.65), the dynamic compliance and the dynamic modulus here are functions of the product ljt, or equivalently, of log ljt. If we also use Eq. (5.93), we may then represent the compliance as a function of a sum of terms... 

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