On formulation of nonlocal elasticity problems

Abstract Nonlocal elasticity models are tackled with a general formulation in terms of source and target fields belonging to dual Hilbert spaces. The analysis is declaredly focused on small movements, so that a geometrically linearised approximation is assumed to be feasible. A linear, symmetric and positive definite relation between dual fields, with the physical interpretation of stress and elastic states, is assumed for the local elastic law which is thus governed by a strictly convex, quadratic energy functional. Genesis and developments of most refer- enced theoretical models of nonlocal elasticity are then illustrated and commented upon. The purpose is to enlighten main assumptions, to detect comparative merits and limitations of the nonlocal models and to focus on still open problems. Integral convolutions with symmetric averaging kernels, according to both strain-driven and stress-driven perspectives, homoge- neous and non-homogeneous elasticity models, together with stress gradient, strain gradient, peridy- namic models and nonlocal interactions between beams and elastic foundations, are included in the analysis.