Shifted Legendre polynomials algorithm used for the dynamic analysis of PMMA viscoelastic beam with an improved fractional model

Chaos, Solitons & Fractals, 2020, 141, pp.110342 (9)

Jiawei Cao 1, Yiming Chen 1,2, Yuanhui Wang 1, Gang Cheng 3, Thierry Barrière 2


1 College of Sciences, Yanshan University, China

2 Université de Bourgogne Franche-Comté, FEMTO-ST Institute, CNRS/ENSMM/UTBM, Department of Applied Mechanics, Besançon 25000, France

3 INSA Centre Val de Loire, Université de Tours, Université d'Orléans, LaMé, CS 23410, Blois 41034, France


In this paper, a fractional viscoelastic model is proposed to describe the physical behaviour of polymeric material. The material parameters in the model are characterized by the experimental data obtained in the dynamical mechanical analysis. The proposed model is integrated into the fractional governing equation of polymethyl methacrylate (PMMA) above its glass transition temperature. The numerical algorithm based on the shifted Legendre polynomials is retained to solve the fractional governing equations in the time-domain. The accuracy and effectiveness of the algorithm are verified according to the mathematical examples. The advantage of this method is that Laplace transform and the inverse Laplace transform commonly used in fractional calculus are avoided. The dynamical response of the viscoelastic PMMA beam is determined with several loading conditions (uniformly distributed load and harmonic load). The effects of the loading condition and the temperature on the dynamic response of the beam are investigated in the results. The proposed approach shows great potentials for the high-precision calculation in solving the fractional equations in the science and engineering.


Fractional calculus
Fractional partial differential equation
Dynamic analysis
Polymethyl methacrylate
Viscoelastic model
Shifted Legendre polynomial
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