A Sign Control Method for Fitting and Interconverting Material Functions for Linearly Viscoelastic Solids

The multidata method was originally proposed to fit a Prony exponentialseries function to experimental viscoelastic modulus and compliance data;this was accomplished by the application of a linear least squares solver.This paper considers a similar approach, but extended in two key ways.First, it has been applied to the solution of convolution integralequations; specifically those used for material function interconversion.Second, it has been modified to force the signs of the Prony seriescoefficients to be positive; this is an essential criterion for the properphysical interpretation of a Prony series material function, which istypically not satisfied by multidata method solutions. Sign control isimplemented by an iterative Levenberg–Marquardt solution algorithm withan appropriate constraint, and can be used for both fitting experimentaldata and interconversion. To use the method, a Prony series must becapable of adequately representing the applicable functions. The method isdemonstrated by first fitting and converting experimental modulus data.Formulation for a composite lamina is also shown, in which a system ofintegral equations is reduced to a single integral equation; this can thenbe solved using the new method. Finally, application of the new method tofrequency domain transformations is demonstrated, along with comparisonsto other techniques.