A dynamic nonlinear regression method for the determination of the discrete relaxation spectrum

The relaxation spectrum is an important tool for studying the behaviour of viscoelastic materials. The most popular procedure is to use data from a small-amplitude oscillatory shear experiment to determine the parameters in a multi-mode Maxwell model. However, the discrete relaxation times appear nonlinearly in the mathematical model for the relaxation modulus. The indirect calculation of the relaxation times is an ill-posed problem and its numerical solution is fraught with difficulties. The ill-posedness of the Linear regression approach, in which the relaxation times are specified apriori and the minimization is performed with respect to the elastic moduli, is well documented. A nonlinear regression technique is described in this paper in which the minimization is performed with respect to both the discrete relaxation times and the elastic moduli. In this technique the number of discrete modes is increased dynamically and the procedure is terminated when the calculated values of the model parameters are dominated by a measure of their expected values. The sequence of nonlinear least-squares problems, solved using the Marquardt-Levenberg procedure, is shown to be robust and efficient. Numerical calculations on model and experimental data are presented and discussed.