Viscoelastic functions spectra

Note that the quantities in bold are vectors represented in the complex plane. In Fig. 12.5 a transfer function spectrum obtained in 0.25 s is shown for 100 frequencies around 10 MHz. It can be used for real time evaluation of the quartz electro-acoustical impedance when the viscoelastic properties change. 

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

Once these parameters are known, the Gross frequency relaxation spectrum can be calculated (see Eqs 7.85-7.87) and as a result all linear viscoelastic functions. 

The dynamic tests at small amplitude in parallel plates or cone-and-plate geometry are simple and reproducible. From the experimental values of storage and loss shear moduli, G and G", respectively, first the yield stress ought to be extracted and then the characteristic four material parameters in Eq. (2.13), rjo, r, mi, and m2, might be calculated. Next, knowing these parameters one may calculate the Gross frequency relaxation spectrum (see Eqs. (2.31) and (2.32)) and then other linear viscoelastic functions. 

A group of Maxwell elements in parallel represents a discrete spectrum of relaxation times, each time r,- being associated with a spectral strength G/. Since in a parallel arrangement the forces (or stresses) are additive, it can readily be shown that for the Maxwell model. Fig. 1-9, the viscoelastic functions G(/), G (aj), G"(w), and r] ((x)) are obtained simply by summing the expressions in equations 2 to 5 over all the parallel elements thus, if there are n elements,... 

A group of Voigt elements in series represents a discrete spectrum of retardation times, each time t/ being associated with a spectral compliance magnitude Since in a series arrangement the strains are additive, it turns out that for the Voigt model. Fig. 1-10, the viscoelastic functions /(/), J ((x>), and are obtained by... 

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. ... 

With increasing proportion of diluent, the monomeric friction coefficient fo is normally diminished, as evidenced by displacement of logarithmic plots of viscoelastic functions in the transition zone to higher frequencies or shorter times with relatively little change in shape. Examples are shown in Fig. 17-2 for the relaxation spectrum of poly( -bulyl methacrylate), and in Fig. 17-3 for the creep compliance of poly(vinyl acetate), both diluted to varying extents with diethyl phthalate. (In the latter figure, we focus attention now on the transition zone, where log J t) < -6.5 the other zones will be discussed later.) Introduction of diluent displaces the time scale by many orders of magnitude. Similar results were obtained in an ex-... 

To show the effect of diluent o.n the detailed shapes of viscoelastic functions, it is convenient to employ corresponding-state plots as in Section C of Chapter 12. For the relaxation spectrum, we plot log H - log Tc/Mq against log t - log a o/kT). Of course, for a single polymer and its solutions the only variables are fo and c (which in the pure polymer becomes p). In Fig. 17-9, poly(vinyl acetate) is compared in this manner with its 50% solution in tri-m-cresyl phosphate. The values of log fo at 40°C for these two systems are 1.75 and -5.25, respectively—the diluent reduces the local friction coefficient by a factor of 10. The curves after reduction coincide at the bottom of the transition zone because this is fixed by the corresponding-state conditions, and are rather similar in shape throughout. However, the diluent causes the spectrum to rise somewhat more sharply from the theoretical slope of -5 at short times but at still shorter times it crosses the spectrum of the pure polymer, and its entrance into the glassy zone involves a broader maximum than the latter. 

Synthetic binder 2 (Figure 10.2) exhibits a behaviour equivalent to synthetic binder 1 it shows the same 3 regions of the mechanical spectrum as a function of temperature. At low temperatures, the transition from the glassy to rubbery phase is observed. In this interval, a crossover between the linear viscoelastic functions is present, as is a displacement of the crossing-point towards higher frequencies as the temperature... 

It should be noted that (3.7.23) is essentially as general as an arbitrary discrete or continuous spectrum model, in the present context. This is because all quantities must be linear in the viscoelastic functions, so that the results for a more general model are simply sums of terms of the form that will now be derived. Explicit results can be obtained for the problem with friction, in terms of Whittaker functions. However, these will not be introduced in the present work. We refer to Golden (1979a, 1986a) for further details. In the frictionless case (from (3.7.11) we see that d,o = 0) ... 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

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 ... 

Thirion (239a) has suggested that the plateau and terminal regions are the result of diffuse interchain interactions in a viscoelastic medium. He obtains a modified Rouse spectrum by replacing the subchain frictional coefficient by a time dependent micro-memory function. The theory is partly phenomenological since the memory function is not specified. However, reasonable choices lead to forms for G (co) and G"(o>) which are similar to those observed experimentally. 

A series of modulus-time curves was also made for temperatures covering the entire viscoelastic spectrum. At higher temperatures, the instrument was first allowed to reach the desired temperature and was held there for half an hour. Samples were then quickly inserted. After 10 minutes moduli were measured as a function of time up to 1000 seconds. Total measuring time for each isotherm thus was kept to less than 30 minutes to minimize chemical decomposition and plasticizer evaporation. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

The main feature about molten high polymers (molecular weights higher than about 104) concerns the broadness of the relaxation spectrum that characterises the viscoelastic response of these systems. This broad two-dispersion spectrum may spread over a range of relaxation times going from about 10 9 up to several seconds [4]. It is well illustrated from the modulus of relaxation observed after applying a sudden stress to the polymer the resulting sudden deformation of the sample is then kept constant and the applied stress is released in order to avoid the flow of the polymer. For example, the release of the constraint oxy(t) is expressed as a function of the shear modulus of relaxation Gxy(t) ... 

From the concept of separability, the memory function of the linear viscoelasticity is required. This memory function can be related to a discrete relaxation time spectrum, available firom dynamic experiments, given by ... 

In the transient compliance function, J(t), the retardation spectrum L(x) is modulated by the function 1 — exp(—t/x) [see Eq. (9.15)]. Plotting this function against In t/x gives the sigmoidal curve shown in Figure 9.6. We should note that the time of observation ( ) in the first quadrant is greater than the retardation times, and as a result x varies between zero and t. Then the creep compliance function for viscoelastic liquids is approximately given by (1,2)... 

The crystalline phase affects the viscoelastic dynamic functions describing the glass-rubber relaxation. For example, the location of this absorption in the relaxation spectrum is displaced with respect to that of the amorphous polymer and greatly broadened. Consequently, the perturbing effects of crystal entities in dynamic experiments propagate throughout the amorphous fraction. The empirical Boyer-Beaman law (32)... 

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