Numerical analysis of fractional-order Euler–Bernoulli beam model under composite model

The primary objective of this study is to develop a new constitutive model by combining a fractional-order Kelvin–Voigt model with an Abel dashpot element in parallel. Subsequently, this new model will be incorporated into the Euler–Bernoulli beam's governing equation, utilizing shifted Legendre polynomials as basis functions, a classical orthogonal polynomial system, to solve the fractional-order partial differential equations. By comparing the numerical solutions with the analytical solutions, we aim to evaluate the applicability of shifted Legendre polynomials in solving such problems and the accuracy of the obtained numerical solutions. Furthermore, we will investigate the performance of viscoelastic HDPE beams under different loading conditions and conduct a comparative analysis of the displacements of HDPE beams under the new constitutive model and the traditional fractional-order Kelvin–Voigt model. Through this research, we hope to gain a deeper understanding of the characteristics of fractional-order phenomena and provide more accurate and efficient numerical simulation and analysis methods for the field of structural mechanics, promoting the development of related engineering applications.