Viscoelasticity time-temperature equivalence

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 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. 

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. 

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. 

In this section we are going to examine such viscoelastic properties in some detail and we will start by examining in turn three important mechanical methods of measurement creep, stress relaxation, and dynamic mechanical analysis. This will lead us to interesting things like time-temperature equivalence and a discussion of the molecular basis of what we have referred to as relaxation behavior. 

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. 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

The above phenomenological description of the viscoelastic behaviour of polymer melts and concentrated solutions leads to the following important conclusions if one focuses on the behavioiu- in the terminal region of relaxation, what is usually done for temperature (time-temperature equivalence) may also be done for the concentration effects and the effects of chain length one may define a "time-chain length equivalence" and "time-concentration equivalence"[4]. For monodisperse species, the various shifts along the vertical (modulus) axis and horizontal (time or frequency axis) are contained in two reducing parameters the... 

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) ... 

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. 

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. 

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. 

As already mentioned, the value of d> is usually far higher than that of W, and the energy dissipated can then be considered as the major contribution to the adhesion strength G. In the case of assemblies involving elastomers, it has been clearly shown in various studies [3,4,58,60-62] that the viscoelastic losses during peel experiments, and consequently, the function , follow a time-temperature equivalent law such as that of Williams et al. [63]. 

Master curve n. The curve one gets by applying the principle of time-temperature equivalence to viscoelastic data on, say, relaxation modulus or creep. 

Experimental Studies of Linear Viscoelastic Behaviour as a Function of Frequency and Temperature Time-Temperature Equivalence... 

Time-temperature equivalence in its simplest form implies that the viscoelastic behaviour at one temperature can be related to that at another temperature by a change in the time-scale only. Consider the idealized double logarithmic plots of creep compliance versus time shown in Figure 6.7(a). The compliances at temperatures T and T2 can be superimposed exactly by a horizontal displacement log at, where at is called the shift factor. Similarly (Figure 6.7(b)), in dynamic... 

For rubber-like materials, the viscoelastic losses vary with the strain rate and the temperature. The principle of time-temperature equivalence propose in 1955 by Williams et al (75) allows us to superimpose the experimental curves obtained at different temperatures through the known translation factor ar of the WLF transformation. As a consequence, at fixed geometry, the adherence forces provoking crack extensions at different speeds V can be studied as a function of the reduced parameter ajV. 

The idea of time-temperature equivalence introduced in Section 14.3.1 is of considerable practical importance because one would often like to predict the longterm response of materials on the basis of experiments carried out on a laboratory time scale. This is to some extent possible in polymers, for which it has been widely verified that viscoelastic functions determined at different T over a fixed range of oj or t, slightly adjusted to take into account the effect of T and density p on the elastic response [through Eq. (30)], superpose if the or t scale is multiplied by a shift factor, flr(T, Tr), where Tr is some convenient reference temperature. A typical master curve obtained in this way is shown in Figure 14.8 for stress relaxation data from polyisobutene, taking Tr = —66.5°C. The effect is to expand the t or scale of the measurement carried out at Tr, revealing the whole of the a transition in this case. 

The Time-Temperature Equivalence of the Glass Transition Viscoelastic Behaviour in Amorphous Polymers and the Williams, Landel and Ferry (WLF) Equation... 

In addition to the Boltzmann superposition principle, the second consequence of linear viscoelasticity is the time-temperature equivalence, which will be described in greater detail later on. This equivalence implies that functions such as a=/(s), but also moduli, behave at constant temperature and various exten-sional rates similarly to analogues that are measured at constant extensional rates and various temperatures. Time- and temperature-dependent variables such as the tensile and shear moduli (E, G) and the tensile and shear compliance (D, J) can be transformed from E =f(t) into E =f(T) and vice versa, in the limit of small deformations and homogeneous, isotropic, and amorphous samples. These principles are indeed not valid when the sample is anisotropic or is largely strained. 

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