Bernstein Polynomial Algorithm For Solving Variable Fractional Differential Equations Of Three Types Of Viscoelastic Material Structures

In recent years,the research of fractional calculus has been continuously developed in many fields such as science and engineering and it has attracted wide attention of scholars.Compared with the integer order,the fractional order has better time memory and can use fewer parameters to more accurately simulate the constitutive model of viscoelastic materials.However,in the latest research,it has been found that the fractional order constitutive model cannot simulate the creep behavior of viscoelastic materials with high accuracy under large deformation conditions.Scholars have proposed variable fractional order model to describe the dynamics of viscoelastic materials in engineering.The current research on the variable fractional model of viscoelastic materials only focuses on the establishment of the model.In the numerical algorithms of variable fractional order model,most methods transform the problem to the frequency domain to get the feasible solution and then transform it back to the time domain for analysis.Based on these,this paper introduces time parameters to establish variable fractional order governing equations for three kinds of viscoelastic materials and applies the shifted Bernstein polynomial algorithm to directly perform numerical analysis in the time domain.Firstly,based on the variable fractional model of the viscoelastic material,the dynamic equation of the beam fixed at both ends and the relationship between the displacement and strain of the beam,the variable fractional displacement governing equation of viscoelastic material fixed beam at both ends is established.According to the definition of variable fractional differential operator and Bernstein polynomial approximation theory,the integer order and variable fractional order operator matrices of the polynomial matrix are derived.The variable fractional displacement governing equation of viscoelastic material fixed beam at both ends is transformed into the form of matrix product.Through discrete variables,the displacement governing equation is transformed into algebraic equations again.Numerical solutions of displacement of fixed beams at both ends are directly obtained in the time domain.The comparison between the numerical solution and the exact solution of the numerical example verifies the effectiveness of the proposed algorithm in solving the governing equation of variable fractional viscoelastic material beam.The algorithm is applied to the governing equation of high-density polyethylene(HDPE)beam to solve the numerical solution of displacement and the numerical solutions are compared with the existing results in the literature to further prove the feasibility of the algorithm proposed in this paper.By calculating the displacement solutions of polyurea beam and polyethylene terephthalate polymer(PET)beam under different loads,the mechanical properties of polyurea and PET materials are analyzed.Secondly,according to Hamilton's principle and variable fractional viscoelastic material model,the displacement governing equation of viscoelastic rotating beam is established.Using the shifted Bernstein polynomial algorithm,the numerical solution of the displacement of the variable fractional rotating beam is directly simulated in the time domain.The displacement changes of rotating beams of polyurea and PET are analyzed under uniform load,linear load and harmonic load.Finally,based on the variable fractional viscoelastic material model,the dynamic equation of the string and the strain-displacement relationship,the displacement governing equation of the nonlinear viscoelastic string is established.Shifted Bernstein polynomials are used as the basis functions to approximate the displacement function of the string.Combined with the collocation method and the least square method,the numerical solutions of the displacement of the string are obtained in the time domain.Based on the convergence analysis,the displacement,velocity and acceleration of viscoelastic string are calculated under different parameters.