Thermorheologically simple material

For linear thermorheologically simple materials a single temperature-dependent shift factor, aT T), can be used to predict the transient thermal response [20]. The mechanical response is history dependent and involves the use of reduced times, ( ) and (t), which can be found from the shift factor as... 

Materials to which time-temperature superposition is applicable, are sometimes termed thermorheologically simple materials. The amount of shift, log aT, which is required to bring measurements at a given temperature, T, into superposition with measurements at a reference temperature, Tr, is described, usually within a range of temperatures Tg < T < (Tg + 100), by the WLF equation (6) ... 

Many amorphous homopolymers and random copolymers show thermorheologically simple behavior within the usual experimental accuracy. Plazek (23,24), however, found that the steady-state viscosity and steady-state compliance of polystyrene cannot be described by the same WLF equation. The effect of temperature on entanglement couplings can also result in thermorheologically complex behavior. This has been shown on certain polymethacrylate polymers and their solutions (22, 23, 26, 31). The time-temperature superposition of thermorheologically simple materials is clearly not applicable to polymers with multiple transitions. The classical study in this area is that by Ferry and co-workers (5, 8) on polymethacrylates with relatively long side chains. In these the complex compliance is the sum of two contributions with different sets of relaxation mechanisms the compliance of the chain backbone and that of the side chains, respectively. 

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

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. 

The combining of time and temperature dependence in the manner described above is referred to as the time-temperature superposition principle. Materials which have this property are sometimes referred to as thermorheologically simple materials. 

Problem 1.7.1 Show that the complex moduli of thermorheologically simple materials have frequency and temperature dependence combined into the variable u = coa(7). 

Homogeneity can be re-established, in terms of the pseudo-time variable given by (1.7.6), for a thermorheologically simple material, in the constitutive equation at least. In the light of (1.7.8), we rewrite (1.8.24a) as... 

Where R is the gas constant, 8.314E-3 kJ/(K mol), AHj, and AH are the relaxation or retardation activation energy and is the reference temperature in °K. Therefore, in this model it is assumed that the elastic modulus, which is time-independent, changes with temperature manifest a vertical shift. Furthermore, vertical shifting also affects the viscoelastic components, which are simultaneously subjected to horizontal shifting (in time). Consequently, this form does comply with the definition given by Delay and Plazek [157] for thermorheologically simple materials. 

Finally, like any polymer, UHMWPE is a viscoelastic material, even at room temperature, i.e. it exhibits creep and stress relaxation under static and dynamic loading conditions. Within certain limits, it was shown that UHMWPE is a thermorheologically simple material [12]. Remarkably, not many published papers are devoted to study the viscoelastic behavior of UHMWPE [11]. However this is an important subject since the components are mounted with tight tolerances and UHMWPE excessive creep limits the long-term survival of TJA [155]. Again, the importance of minimizing the UHMWPE viscoleastic behavior is very clear. 

Increasing temperature invariably reduces the magnitude of the relaxation time, by as much as a factor of 10 . Despite this enormous effect on the time of relaxation, for a thermorheologically simple material, temperature has no effect on the rate at which the stress decays (i.e., the shape of the relaxation function). At least for linear polymers, the terminal relaxation function and height of the rubbery plateau are minimally affected by temperature. Segmental... 

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. 

R is the gas constant and Ea the flow activation energy. The latter is a material-specific factor that is not dependent on the molar mass of the polymer and for thermorheological simple polymers also is not dependent on the shear stress. The activation energy Ea for polymer melts varies between 25 and 80 kj/mol. It can be determined from the slope of the line in the Arrhenius plot. 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

The most common means to extend the frequency scale is to invoke time-temperature superpositioning (Ferry, 1980). If all motions of a polymer contributing to a particular viscoelastic response are affected the same by temperature, then changes in temperature only alter the overall time scale such a material is thermorheologically simple. Thermorheological simplicity means conformance to the time-temperature superposition principle, whereby lower and higher strain rate data can be obtained from measurements at higher and lower temperatures, respectively. 

If the modulus data of a material satisfy Equation (3.27), this material is referred to as thermorheologically simple and obeys the (viscoelastic) time-temperature superposition. 

It should be noted that the use of creep and shear compliances as material property input allows Poisson s ratio to be time-dependent. Hence, the present formulation is applicable to any thermorheologically simple isotropic viscoelastic material over any length of time. 

Staverman and Schwarzl [19] call these materials thermorheologically simple, and Lee and his collaborators [20] have worked out the theoretical consequences of this assumption, so that complex problems concerning the deformation of viscoelastic solids in variable temperature situations can be solved. 

VII. Thermorheologically Simple Viscoelastic Materials. For these materials, a temperature change has the effect of scaling the relaxation and creep functions with respect to time. Equations (1.11.1, 2) are replaced by... 

As pointed out in Sect. 1.7, the viscoelastic functions of many materials depend strongly on temperature. The simplest realistic way of incorporating this dependence is to assume that the material is thermorheologically simple (TRS) in the sense defined in Sect. 1.7. This implies a non-linear dependence on the temperature field which renders the solution of most problem categories very difficult, in particular those where the temperature field is not given a priori but must be determined as part of the solution. A way out of this is to adopt a fully linear theory, as developed for example by Christensen (1982), Chap. 3. The assumption behind such a theory is that the effects of temperature variation on the viscoelastic functions is sufficiently small that its product with the field variables can be neglected. In many cases, this is very restrictive on the allowed range of temperature variation. A fully linear theory will not be considered here. We remark however that such a theory is susceptible to treatment by the Correspondence Principle-based methods, discussed in Chap. 2. 

There have been a few reports [54] that the dependency of bj on temperature is in fact weaker than that indicated by the Rouse model (Eq. 4.70) and depends on the polymer species, and there is a theory that predicts this [61]. However, the dependency is quite weak in any case. To summarize, data obtained at a temperature, T, can be reduced to superpose on those at a reference temperature, Tq, if the reduced modulus is plotted as a function of the reduced time, tj. When data obtained at several temperatures are plotted in this way, the result is called a master curve. And a material for which data can be reduced to a master curve in this way is said to be thermorheologically simple. This terminology was introduced by Schwarzl and Staverman [62]. 

Often the assumption is made that a material is thermorheologically simple, meaning all mechanisms contributing to the response have the same temperature dependence. In this situation, temperature only changes the relrixation time, not the shape of the relaxation function. Accordingly, the principle of time-temperature superpositioning can be applied. The measured quantities are shifted... 

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