Retardation Spectrum from Creep Compliance

The analog of the Ferry-Williams calculation, as shown by Stern, is i 

The second approximation method of Schwarzl and Staverman gives 

Alternative second approximations, and third and higher approximations in alternative forms, are provided by the method of Tschoegl. -  

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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. 

The Lo(t) that was obtained is shown in Fig. 12.22 with the retardation spectrum obtained from the shear creep compliance function, J(t). The levels of the short time behavior are matched. Three feamres should be noted. The functionality at short times is the same within experimental uncertainty. The 1/3 slope of log L(t) at short time indicates that the response is dominated by motions that contribute to Andrade creep. [66,85-87,141,142]... 

Fig. 2.31. The logarithm of the retardation spectrum L of poly(methyl methacrylate) as a function of the logarithm of the reduced retardation time r/ar. The solid curve was calculated from the reduced Jr(t) curve obtained from creep data taken at lower temperatures (14.4-34.7 °C) and longer times (10° s < r < 10 s) and shifted to 13.1° C. The dashed line was calculated from the dynamic compliances obtained by Williams and Ferry at higher temperatures and frequencies To was chosen to be 10.8 °C. From [217] by permission.

The retardation spectrum corresponding to the Rouse theory is also discrete, equivalent to a generalized Voight model, but not with equal magnitudes of compliance contributions the pth contribution Jp to the compliance falls ofif rapidly with increasing p, being approximately proportional to the retardation time itself as well as to l/nkT. They have been calculated by Betry the retardation times are spaced between the relaxation times, as shown in the lower panel of Fig. 9-7. From the discrete retardation times and Jp values, the creep compliance can be calculated in principle by equation 18 of Qrapto 3. 

FIG. 15-3. Recoverable shear creep compliance and retardation spectrum of l 3 5-tri-a-naphthyl benzene, reduced to 64.2°C from measurements at that temperature and above (open circles) and at four other temperatures as indicated. (Points for spectrum calculated in several different ways.) (Plazek and Magill. )... 

The relaxation time spectrum can be calculated exactly from the measured stress relaxation modulus using Fourier or Laplace transform methods, and similar eonsiderations apply to the retardation time spectrum and the creep compliance. It is more convenient to consider these transformations at a later stage, when the final representation of linear viscoelasticity, that of the complex modulus and complex compliance, has been discussed. 

In a creep experiment, the stress, rather than the strain, is increased suddenly from zero to a constant value (Tq at time t = 0. The resulting data are interpreted in terms of the creep compliance / = y t)lretardation spectrum . All these terms are defined later in this section. Plazek and Echeverria [5] have argued that compliance data are more useful than stress relaxation data in revealing various relaxation mechanisms. 

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