Finite element modelling of nonlocal beams

Based on a high-order Euler–Bernoulli nonlocal beam theory, a nonlocal finite element method (NFEM) is consistently developed to evaluate the displacement and the bending moment of nanobeams. As a benchmark a simply supported nanobeam under a uniform external load is considered and the numerical solution obtained by means of the proposed NFEM is compared with the exact nonlocal solution obtained by solving a sixth-order differential equation. The comparison shows that the NFEM provides an exact solution of the nonlocal problem, for any value of the internal length parameter, with a coarse mesh. The proposed NFEM does not show pathological behaviours such as mesh dependence, numerical instability or boundary effects. Moreover a cantilever nanobeam subjected to an intermediate applied force is addressed. Contrary to what is reported in the literature, the proposed methodology shows that the nonlocal effects are apparent to both left and right of the application point of the external force.