Derivative based algorithms for continuous relaxation spectrum recovery

Abstract

Historically, it has always been understood that the relaxation spectra of linear viscoelastic materials are continuous. Nevertheless, because of their ease of implementation computationally, delta function recovery methods have been and continue to be important, even though they do not generate continuous approximations. Derivative based recovery techniques were popular in the pre-computer days because they engendered simple formulas for continuous relaxation spectra approximation and estimation. They also represent a practical basis for continuous relaxation spectra estimation from oscillatory shear data. Here, using local Fourier deconvolution, we give precise formulae which generalize certain classical derivative based approximations to the relaxation spectra of linear viscoelastic materials using oscillatory shear data. We also present new formulae in this class. Finally we present a stable iterative algorithm, of the type proposed by Gureyev, which circumvents the calculation of very high order derivatives. The importance of the proposed derivative based approximations are that they are local and therefore are appropriate for the experimental situation where the oscillatory shear data is only available for a finite range of frequencies. Results are presented for both exact and experimental data.