Numerical interconversion of linear viscoelastic material functions

Abstract

Calculation of the relaxation modulus in a manner which addresses the ill-posed nature of the problem specifically in the terminal and plateau regions is essential in order to subsequently determine a molecular weight distribution. A novel method to effect this result is demonstrated. The relaxation modulus is modeled as a discrete N element Maxwell (N≫1) line spectrum. The method incorporates additional independent rheological data into a constrained linear regression with regularization. Specifically, the zero shear viscosity and the steady-state recoverable compliance are used to impose integral moment equality constraints on the calculated relaxation modulus. Moment constraints necessarily generate a self-consistent conversion. All moduli are further constrained to be positive. The numerical method is robust and capable of extracting meaningful relaxation spectra from severely error infected and/or incomplete data sets. Imposing moment constraints dramatically reduces the error and dispersion of the calculated relaxation and retardation spectra in the terminal and plateau regions. Analytic conversion of the relaxation modulus to the compliance function is demonstrated through knowledge of the root sequence for a discrete Maxwell model.