Viscoelastic functions exact

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

These relations enable one to relate the shear viscoelastic functions to their tensile counterparts. At high compliance levels, rubbers are highly incompressible, and the proportional relation between the tensile and shear moduli and compliances holds. However, at lower compliances approaching Jg, the Poison ratio fi (which in an elongational deformation is -(Mw/dM, where w is the specimen s width and / is its length) is less than Eqs. (28) and (29) are then no longer exact. For a glass ju T. When G(t) = K(t), E t) = 2.25 Gif). 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

EXACT INTERRELATIONS AMONG VISCOELASTIC FUNCTIONS CH. 3 B. THE RELAXATION AND RETARDATION SPECTRA... 

If, within a particular zone of viscoelastic behavior, an empirical equation can be used to fit a viscoelastic function, an exact expression for the corresponding spectrum can sometimes be derived, as shown by Smith. ... 

Generally, the approximation methods have an analytical foundation based oh the properties of the integrands of the corresponding exact equations. Such an integrand is usually the product of the viscoelastic function initially known and an... 

Exact matching of the shapes of adjacent curves has already been cited as one criterion for the applicability of reduced variables. Two others which should be applied to any experimental example when possible are (a) the same values of ar must superpose all the viscoelastic functions (b) the temperature dependence of flr niust have a reasonable form consistent with experience. 

The shift factor ap can be used to combine time-dependent or frequency-dependent data measured at different pressures, exactly as ap is used for different temperatures in Section A above, and with a shift factor ar,p data at different temperatures and pressures can be combined. It is necessary to take into account the pressure dependence of the limiting values of the specific viscoelastic function at high and low frequencies, of course, in an analogous manner to the use of a temperature-dependent Jg and the factor Tp/Topo in equations 19 and 20. The pressure dependence of dynamic shear measurements has been analyzed in this way by Zosel and Tokiura. A very comprehensive study of stress relaxation in simple elongation, with the results converted to the shear relaxation modulus, of several polymers was made by Fillers and Tschoegl. An example of measurements on Hypalon 40 (a chlorosulfonated polyethylene lightly filled with 4% carbon black) at pressures from 1 to 4600 bars and a constant temperature of 25°C... 

The corresponding-state plot of Fig. 17-10 implies, insofar as the curves coincide in the transition region, that all relaxation times are proportional to ib and that the magnitudes of contributions to H are all proportional to c as well as to T. This is, of course, exactly what the bead-spring chain theories predict, as in equations 18 and 20 of Chapter 9, for example (recalling that n is proportional to c). On this basis, a scheme of reduced variables can be devised to combine measurements at different concentrations as well as temperatures, without actually calculating fo and without requiring that the detailed shapes of the viscoelastic functions conform to the simple theories. 

The exact formal relationships between the various viscoelastic functions are conveniently expressed using Fourier or Laplace transform methods (cf. Section 5.4.2). However, it is often adequate to use simple approximations due to Alfrey in which the exponential term for a single Kelvin or Maxwell unit is replaced by a step function, as shown schematically in Figure 5.18. 

Since time-temperature shifting, as illustrated in Figs. 2-4, enjoys widespread use in linear viscoelasticity, it is important to have a consistent set of criteria for its validity. It is advisable that one have nearly exact matching of shapes of adjacent curves with over more than half-range overlap, that the shift factor have a reasonable form (e.g. WLF, Arrhenius) and possess the same value for all of the viscoelastic functions. A sharp test of how well one has time-temperature... 

Both selection and design of a fibre for a given end-use must take into account a wide range of properties. However, satisfactory mechanical performance is always essential. Various aspects of the mechanical properties of polymers are discussed in detail in other chapters of this work and elsewhere." " For fibres with cylindrical symmetry, the fundamental approach requires determination of five independent viscoelastic functions," which is prohibitively time-consuming. Besides, in practice, the fibres assembled in an end-product are often deformed in a complex way. The magnitudes and directions of the applied forces vary with time in a manner which precludes an exact mathematical description. At the same time, the fibres may be exposed to a changing environment. Consequently, it is often necessary to use various end-use oriented tests for a practical evaluation of a fibre. 

The Dirac delta function clearly provides one form of spectra which has an analytical transform to the viscoelastic experimental regimes discussed so far. An often overlooked function was developed by Tobolsky6 and Smith.7 They noted that particular forms of the relaxation or retardation spectra have exact analytical transforms. These functions give well defined spectra and provide good fits to experimental data. The relaxation spectrum is defined by the function ... 

Preliminary nanoindentation results on other teeth (premolars, incisors and canines) indicate variations in mechanical properties as large as those discussed for molars [unpubl. data]. In each case the exact distribution of mechanical properties within the enamel appears to correlate with the extent of mechanical loading experienced by the tooth during mastication. However, there appears to be an increase in the viscoelasticity (loss modulus) for the enamel of anterior teeth when compared to posterior teeth, again this may be related to their function. 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

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