  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.
