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

Author

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  • Cao, Jiawei
  • Chen, Yiming
  • Wang, Yuanhui
  • Cheng, Gang
  • Barrière, Thierry

Abstract

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.

Suggested Citation

  • Cao, Jiawei & Chen, Yiming & Wang, Yuanhui & Cheng, Gang & Barrière, Thierry, 2020. "Shifted Legendre polynomials algorithm used for the dynamic analysis of PMMA viscoelastic beam with an improved fractional model," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    • Handle: RePEc:eee:chsofr:v:141:y:2020:i:c:s0960077920307372
    DOI: 10.1016/j.chaos.2020.110342
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    References listed on IDEAS

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    1. Li, Zheng & Wang, Hong & Xiao, Rui & Yang, Su, 2017. "A variable-order fractional differential equation model of shape memory polymers," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 473-485.
    2. Meng, Zhijun & Yi, Mingxu & Huang, Jun & Song, Lei, 2018. "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 454-464.
    3. Meng, Ruifan & Yin, Deshun & Yang, Haixia & Xiang, Guangjian, 2020. "Parameter study of variable order fractional model for the strain hardening behavior of glassy polymers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    4. Lei, Dong & Liang, Yingjie & Xiao, Rui, 2018. "A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 465-475.
    5. Xie, Jiaquan & Yao, Zhibin & Gui, Hailian & Zhao, Fuqiang & Li, Dongyang, 2018. "A two-dimensional Chebyshev wavelets approach for solving the Fokker-Planck equations of time and space fractional derivatives type with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 197-208.
    6. Wang, Lei & Chen, Yi-Ming, 2020. "Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
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    Cited by:

    1. Dang, Rongqi & Chen, Yiming, 2021. "Fractional modelling and numerical simulations of variable-section viscoelastic arches," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    2. Sun, Lin & Chen, Yiming & Dang, Rongqi & Cheng, Gang & Xie, Jiaquan, 2022. "Shifted Legendre polynomials algorithm used for the numerical analysis of viscoelastic plate with a fractional order model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 190-203.
    3. Sun, Lin & Chen, Yiming, 2021. "Numerical analysis of variable fractional viscoelastic column based on two-dimensional Legendre wavelets algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).

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