Geometrically nonlinear response of a fractional-order nonlocal model of elasticity

• Published in: International journal of non-linear mechanics Vol. 125; p. 103529
• Main Authors: Sidhardh, Sai, Patnaik, Sansit, Semperlotti, Fabio
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.10.2020; Elsevier BV
• Subjects: Boundary conditions; Boundary value problems; Differential equations; Euler-Bernoulli beams; Finite element method; Mathematical analysis; Mathematical models; Nonlinear equations; Nonlinear response; Variational principles
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2020.103529

This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler–Bernoulli beam. The finite nonlocal strains in the Euler–Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a mathematically well-posed formulation which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal Euler–Bernoulli beam are obtained using variational principles. Further, a nonlinear finite element model for the fractional-order system is developed in order to achieve the numerical solution of the integro-differential nonlinear governing equations. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the geometrically nonlinear response of a nonlocal beam subject to various loading and boundary conditions. Although presented in the context of a 1D beam, this nonlinear f-FEM formulation can be extended to higher dimensional fractional-order boundary value problems. •Finite (nonlinear) fractional-order strain–displacement relations developed.•Fractional-order beam theory for a geometrically nonlinear and nonlocal response established from variational principles.•Nonlinear fractional-finite element model for fractional-order governing equations developed.•Geometrically nonlinear and nonlocal elastic response of Euler–Bernoulli beams analyzed.