Optimal discrete-time Prony series fitting method for viscoelastic materials

The most important characteristics of the behaviour of viscoelastic materials are the time and temperature dependence of their properties. Viscoelastic models based on Prony series are usually used due to easy implementation in finite element analysis (FEA) codes. The experimental data are fitted to a Prony series using a user-convenience number of terms represented by two coefficients. The time coefficients τ are previously fixed in the time scale in order to determine the second parameters of the model. Usually, an homogeneous distribution in logarithmic-time scale is used for τ, which produces accurate fittings when a large number of terms in the Prony series are used as well as when the material presents a uniform sigmoidal viscoelastic curve along several decades of time. When short-time curves must be fitted or the relaxation curve shape is not so uniform distributed along time, the homogeneous distribution of time coefficients could be a significant drawback since a large number of coefficients could be needed or even a reasonable fitting with a Prony series model is not possible.
In this study, an optimized τi distributed method for fitting master curves of viscoelastic materials based on Prony series model is proposed. The method is based on an optimization algorithm strategy to allocate the time coefficients along the time scale in order to obtain the best fit. The method is validated by using experimental data of temporomandibular joint (TMJ) disc, which is a biological material that presents a short-time and high relaxation rate viscoelastic curve. The results show that the method improves significantly the fitting of the viscoelastic curves when compared with uniform distributed time fittings.
Furthermore, the optimized coefficients are also used to obtain the complex moduli of the material using an analytical conversion, which is compared with the experimental complex moduli curves of the material.

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11043-018-9394-z