A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity

• Published in: International journal of solids and structures Vol. 202; pp. 398 - 417
• Main Authors: Patnaik, Sansit, Sidhardh, Sai, Semperlotti, Fabio
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.10.2020; Elsevier BV
• Subjects: Boundary conditions; Boundary value problems; Continuum modeling; Euler-Bernoulli beams; Exact solutions; Finite element method; Fractional Calculus; Mathematical models; Nonlocal beams; Nonlocal elasticity; Variational calculus; Variational principles; Fractional Calculus; Finite element method; Variational calculus; Nonlocal beams
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2020.05.034

•Self-adjoint positive-definite fractional-order nonlocal elastic model established.•Paradoxical nonlocal interactions addressed by the fractional-order continuum model.•Ritz finite element model for the fractional-order BVP developed.•Singularities in the kernel of fractional derivative circumvented.•Validation and convergence tests of the fractional-Finite Element Model accomplished. We present the analytical formulation and the finite element solution of a fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing consistent definitions for the fractional-order kinematic relations, the governing equations and the associated boundary conditions are derived based on variational principles. Remarkably, the fractional-order nonlocal model gives rise to a self-adjoint and positive-definite system accepting a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this boundary value problem, a finite element model for the fractional-order governing equations is presented. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the nonlocal response of a Euler-Bernoulli beam subjected to various loading and boundary conditions. The fractional-order positive definite system will be used here to address some paradoxical results obtained for nonlocal beams through classical integral approaches to nonlocal elasticity. Although presented in the context of a 1D Euler-Bernoulli beam, the f-FEM formulation is very general and could be extended to the solution of any general fractional-order boundary value problem.