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

Citation
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
Citations number
14
Categorie Soggetti
Apllied Physucs/Condensed Matter/Materiales Science
Journal title
JOURNAL OF PHYSICS D-APPLIED PHYSICS
ISSN journal
00223727 → ACNP
Volume
33
Issue
10
Year of publication
2000
Pages
1219 - 1229
Database
ISI
SICI code
0022-3727(20000521)33:10<1219:ADNRMF>2.0.ZU;2-D
Abstract
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.