Numerical analysis for variable thickness plate with variable order fractional viscoelastic model

An accurate constitutive model for viscoelastic plates with variable thickness is crucial for understanding their deformation behaviour and optimizing the design of material-based devices. In this study, a variable order fractional model with a precise order function is proposed to effectively characterize the viscoelastic behaviour of variable thickness plates. The shifted Legendre polynomials algorithm is employed to solve the variable order fractional partial differential equation in the time domain, with a minimum absolute error of $1.521\times 10^{-8}$. The computational time is reduced by 30 % and the convergence rate is increased by over 50 % compared to the shifted Bernstein polynomials algorithm. Numerical analysis shows that viscoelastic plates with quadratic thickness variation exhibit the smallest displacement changes and the polyethylene terephthalate plates outperform the polyurethane plates in bending properties. These findings highlight the reliability and effectiveness of the numerical algorithm based on the shifted Legendre polynomials as a powerful tool for solving fractional equations, with significant potential in mechanical engineering.