Smfds. Mustapha et Tn. Phillips, A dynamic nonlinear regression method for the determination of the discrete relaxation spectrum, J PHYS D, 33(10), 2000, pp. 1219-1229
The relaxation spectrum is an important tool for studying the behaviour of
viscoelastic materials. The most popular procedure is to use data from a sm
all-amplitude oscillatory shear experiment to determine the parameters in a
multi-mode Maxwell model. However, the discrete relaxation times appear no
nlinearly in the mathematical model for the relaxation modulus. The indirec
t calculation of the relaxation times is an ill-posed problem and its numer
ical solution is fraught with difficulties. The ill-posedness of the Linear
regression approach, in which the relaxation times are specified apriori a
nd the minimization is performed with respect to the elastic moduli, is wel
l documented. A nonlinear regression technique is described in this paper i
n which the minimization is performed with respect to both the discrete rel
axation times and the elastic moduli. In this technique the number of discr
ete modes is increased dynamically and the procedure is terminated when the
calculated values of the model parameters are dominated by a measure of th
eir expected values. The sequence of nonlinear least-squares problems, solv
ed using the Marquardt-Levenberg procedure, is shown to be robust and effic
ient. Numerical calculations on model and experimental data are presented a
nd discussed.