Optimal discrete-time Prony series fitting method for viscoelastic materials

  • Eva BarrientosEmail author
  • Fernández Pelayo
  • Álvaro Noriega
  • María Jesús Lamela
  • Alfonso Fernández-Canteli
  • Eiji Tanaka


Viscoelastic models based on Prony series are usually used due to easy implementation in finite element analysis codes. The experimental data are fitted to a Prony series using a user-chosen number of terms represented by two coefficients. The time coefficients \(\tau _{i}\) are previously fixed in the time scale to determine the second parameter of the model. Usually, a homogeneous distribution in the logarithmic-time scale is used for \(\tau _{i}\). When short-time curves must be fitted or the relaxation curve shape is not uniformly distributed in time, the homogeneous distribution of time coefficients could be a significant drawback, since a large number of coefficients might be needed or even a reasonable fitting is not possible.

In this study, an optimized \(\tau _{i}\) distribution 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 to obtain the best fit. The method is validated by using experimental data of temporomandibular joint disc, which presents a short-time and high relaxation rate viscoelastic curve. The method improves significantly the fitting of the viscoelastic curves when compared with uniformly distributed time fittings.

Furthermore, the optimized coefficients are also used to obtain the complex moduli of the material using an analytical conversion, which are then compared with the experimental complex moduli curves of the material.


Viscoelastic Prony series Optimization Relaxation Soft materials Viscoelastic behaviour 



This research was supported in part by Grants-in-Aid 26293436 (E.T.) for Science Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The funder had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The authors would like to acknowledge the support of the CajAstur Fellowship-University of Oviedo 2011 program.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Construction and Manufacturing EngineeringUniversity of OviedoOviedoSpain
  2. 2.Department of Orthodontics and Dentofacial Orthopedics, Institute of Biomedical SciencesTokushima University Graduate SchoolTokushimaJapan

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