Dynamic Analysis Of Three Constitutive Models For Viscoelastic Material Structures Based On Shifted Legendre Polynomials

With the development of fractional differential operators,fractional viscoelastic constitutive models have received more and more attention.These models are widely used to describe the viscoelastic behavior of materials.These fractional differential operators can describe the memory characteristics of viscoelastic materials well.The study of viscoelastic material structure is of great significance for solving practical engineering problems.At present,most scholars use Laplace transform to solve the displacement problem of viscoelastic material structure.This paper uses different constitutive models of viscoelastic materials to establish the fractional governing equations of three viscoelastic material structures,and uses the shifted Legendre polynomial algorithm to directly analyze the dynamics in the time domain,thus avoiding Laplace The complexity of transformation and its inverse transformation.Firstly,based on the integral-order motion control equations,variable-fractional constitutive model and geometric relations of the viscoelastic material arch,the motion control equations of the nonlinear variable-fractional viscoelastic material arch are derived and obtained in the time domain using the shifted Legendre polynomial numerical solution.According to the definition of the Caputo-type variable fractional derivative and the definition of the shifted Legendre polynomial,the integer-order and variable-fractional differential operator matrices are derived.The nonlinear variable-fractional governing equation is converted into the form of matrix product,and then the discrete variable method is used discretization,and finally obtain the displacement solution of the viscoelastic material arch in the time domain by the least square method.Numerical example proves the accuracy and effectiveness of the algorithm.The numerical solution of the displacement of the viscoelastic material arch under different loads and the performance of the two viscoelastic material arches are compared.Secondly,according to the fractional Kelvin-Voigt model and geometric relationship,the fractional governing equation of the viscoelastic material column is established,and the differential operator matrix for solving the fractional governing equation in the time domain is derived.The displacement solutions of the columns of viscoelastic materials under different uniform loads and harmonic loads are analyzed,and the numerical solutions of the displacement,stress and strain of the columns of two viscoelastic materials are compared.The effectiveness and accuracy of the algorithm are proved by numerical example.Finally,a new fractional viscoelastic constitutive model describing the physical behavior of materials is established,and the parameters in the model are obtained through physical and mechanical analysis.Using the proposed fractional viscoelastic constitutive model,Hamiltonian principle and geometric relationship,the fractional governing equation of PMMA viscoelastic material beam is established.The shifted Legendre polynomial algorithm is used for time-domain numerical solution.At the same time,the dynamic analysis of the PMMA viscoelastic material beam is carried out,and the displacement solutions of the PMMA viscoelastic material beam under different loads and temperatures are compared.