DETERMINATION OF RELAXATION-TIME SPECTRA BY ANALYTICAL INVERSION USING A LINEAR VISCOELASTIC MODEL WITH FRACTIONAL DERIVATIVES

Recently, Friedrich proposed an empirical model for linear viscoelastic fluids corresponding to a constitutive equation with fractional derivatives [Phil. Mag. Lett, 66, 287 (1992)]. For this model, the relaxation modulus, the dynamic moduli, the relaxation time spectrum, and other material functions have been explicitly calculated as a function of the few parameters that characterize a viscoelastic fluid within this model. By fitting this model to experimental data, the model parameters can be determined and other material functions, in particular the relaxation time spectrum, can be calculated immediately. This paper reports to what extent this method, which may be called analytical inversion, is appropriate for the determination of relaxation time spectra. For that pupose, the spectra of a number of very different polymeric materials are determined with this method. The spectra calculated in this way are compared with the spectra obtained by nonlinear regularization. It turns out that the empirical model describes the linear viscoelastic properties of a variety of different materials with high accuracy. Keeping in mind that the determination of the relaxation time spectrum requires the solution of an ill-posed problem, the agreement between the relaxation time spectra obtained by analytical inversion and by regularization is satisfactory for these materials.