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.

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

The study of the Weyl and Dirac topological materials (topological semimetals, insulators, superfluids and superconductors) opens the route for the investigation of the topological quantum vacua of relativistic fields. The symmetric phase of the standard model (SM), where both electroweak and chiral symmetry are not broken, represents the topological semimetal. The vacua of the SM (and its extensions) in the phases with broken electroweak symmetry represent the topological insulators of different types. We discuss in detail the topological invariants in both the symmetric and broken phases and establish their relation to the stability of vacuum.