Finite element method for stress-driven nonlocal beams

The bending behaviour of systems of straight elastic beams at different scales is investigated by the well-posed stress-driven nonlocal continuum mechanics. An effective computational methodology, based on nonlocal two-noded finite elements, is developed in order to take accurately into account long-range interactions present in the whole structural domain. The idea consists in partitioning the beam in subdomains and in observing that the nonlocal stress-driven convolution integral, equipped with the Helmholtz averaging kernel, can be equivalently formulated by expressing nonlocal bending interaction fields in terms of zero-th and second-order derivatives of elastic curvature fields which have to fulfil appropriate non-classical constitutive boundary and interface conditions. Relevant mesh-dependent shape functions governing the FEM technique are analytically detected. Each element is characterized by shape functions whose number is equal to four times the number of elements of the considered mesh. A simple analytical strategy to obtain nonlocal stiffness matrices and equivalent nodal forces of a finite element is exposed. The global nonlocal stiffness matrix is got by assembling the nonlocal element stiffness matrices accounting for long-range interactions among the elements. The proposed numerical approach is examined by exactly solving exemplar nonlocal case-problems of current interest in nano-engineering. The presented nonlocal strategy extends previous contributions on the matter and offers designers a consistent computational tool.