Linear viscoelasticity functions

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

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. 

In spite of the often large contribution of secondary filler aggregation effects, measurements of the time-temperature dependence of the linear viscoelastic functions of carbon filled rubbers can be treated by conventional methods applying to unfilled amorphous polymers. Thus time or frequency vs. temperature reductions based on the Williams-Landel-Ferry (WLF) equation (162) are generally successful, although usually some additional scatter in the data is observed with filled rubbers. The constants C and C2 in the WLF equation... 

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] ... 

For linear viscoelastic functions near phase separation at low strains, Larson and Fredrickson... 

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. 

Rapid change of temperature to Tg causes phase separation by the mechanism known as spinodal decomposition (SD), which is characterized by the presence of two interpenetrating continuous phases that coarsen with time. Thus, the morphology within the metastable region is time dependent and in consequence so is the rheological response. Theoretically, instantaneous SD formation of interpenetrated structures causes the linear viscoelastic functions to go to infinity ]216, 217] ... 

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. 

Here it is emphasized that the definition of the elastic compliance J = 1/G is not valid for the viscoelastic compliance J(t) used in eqnation 10. Rather it is the complex compliance = l/G (.co). In addition, all the linear viscoelastic functions can be related one to the other. Full discussion of these relationships can be found in Ferry (9) and Tschoegl. (10). 

The equilibrium modulus and the memory function m(t-t ) can be obtained from measurements in the linear viscoelastic region. Oscillatory shear data are most appropriate to determine the linear viscoelastic functions We will not go further into this matter here, since the present article is concerned with the comparison of the shape of the nonlinear tensor functionals of different materials. 

FIG. 3-9. Relations among the linear viscoelastic functions with numbers of equations (in this chapter unless otherwise specified). 

The relations among the various types of linear viscoelastic functions are summarized in Fig. 3-9 with equation numbers identified. The equation numbers for calculation of viscoelastic constants are identified in Table 3-1. 

Approximate Interrelations among The Linear Viscoelastic Functions... 

Providing tests are performed at low strain amplitude, small enough for the complex modulus to exhibit no strain dependency, then dynamic testing yields in principle linear viscoelastic functions. This implies that, with an unknown material, a preliminary strain sweep test is performed in order to experimentally detect the maximum strain amplitude for a linear response to be observed [i.e. G lo, f(Y)]-As illustrated in Fig. 6 with data from Dick and Pawlowsky [20], such a requirement is practically never met within the available experimental window with filled rubber materials, whose linear region tends to move back to a lower and lower strain range as the filler content increases. 

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... 

Here we remind the reader that is a low shear rate, linear viscoelastic function of time only. Figure 4.2.4 shows that eqs. 4.2.6 and 4.2.7 work well for a sample of linear low density polyethylene... 

Figure 11.4.4 shows that experimentally t](y) and ti ico) and /7 (cu) areallvery similarfunctions. Other predictions for some of the linear viscoelastic functions are recoverable steady state shear compliance... 

In order to accomplish these objectives, we need methods of interrelating the various linear viscoelastic functions. The general techniques of obtaining one fimction from another have been discussed by Ferry [17]. In the present case, Baumgaertel and Winter have proposed a particularly simple method [18]. If we introduce Eq. (14.6.10) into Eqs. (14.7.3) and (14.7.4) and carry out the integrations, we get (see also Example 12.2). 

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