Integral and differential approaches to Eringen’s nonlocal elasticity models accounting for boundary effects with applications to beams in bending

The Eringen’s fully nonlocal elasticity model is known to lead to ill-posed boundary-value problems and to suffer some boundary effects arising from particle interactions impeded by the body’s boundary surface. An enhanced model is derived from the original fully nonlocal one by the addition of a regularizing non-homogeneous local phase which accounts for boundary effects and which leads to a Fredholm integral equation of the second kind, hence to wellposed boundary-value problems, without paradoxes, nor other drawbacks. The enhanced integral model applied to a beam in bending proves to be equivalent to a sixth order differential equation with variable coefficients, with extra nonlocality boundary conditions here also derived. Both the integral approach and the differential one lead to a same unique solution of the small-scale beam problem. An efficient numerical algorithm is presented in which the sixth order differential equation with variable coefficients is reduced to one of the second order, which is addressed by a finite difference method. The proposed theory is applied to a set of engineering beam problems, for each of which the inherent size effects are reported and graphically illustrated. The influence of the length scale parameter upon the beam’s response is highlighted by means of a function () representing the normalized maximum deflection of the beam as a function of the length scale parameter. It is shown that the enhanced model always predicts softening size effects no matter the boundary and loading conditions, and that the related response function () generally exhibits a waved pattern with positive slopes first, then negative, as the length scale parameter increases, with a limit asymptotic behavior like an atomic lattice model. A comparison with other theories is also presented together with possible future developments.