Geometrically Nonlinear Response of a Fractional-Order Nonlocal Model of Elasticity

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.