A unified integro-differential nonlocal model

In this paper a unified integro-differential nonlocal elasticity model is presented and its use in the bending analysis of Euler-Bernoulli beams is illustrated. A general (for an elastic continuum) finite element formulation for the two-phase integro-differential form of Eringen nonlocal model is provided. The equations are specialized for the case of the Euler-Bernoulli beam theory. Several numerical examples, including the paradoxical cantilever beam problem that eluded other researchers, are provided to show how the present nonlocal model affects the transverse displacement of beams. The examples show that Eringen nonlocal constitutive relation has a softening effect on the beam, except for the case of the simply supported beam. A brief discussion on the applicability of the integro-differential model to other problems is also presented. Finally, the transition from the stiffened nonlocal simply supported beam to the softened nonlocal clamped beam is also investigated. (C) 2015 Elsevier Ltd. All rights reserved.