A theoretical link between gradient and nonlocal elasticity models, including higher order boundary conditions

The paper presents a recently developed rational derivation of the strain gradient elasticity model from the nonlocal (or integral) model. This kind of derivations are generally recovered just by an expansion into a Taylor series of the nonlocal strain field up to a certain order, and then operating the integration (or averaging) over the spatial interaction domain. The latter procedure is fully consistent when the analysis is performed over an unbounded domain, but when a classical bounded domain is analyzed it lacks in reproducing the so-called higher-order boundary conditions. In the present contributions the complete derivation is achieved employing an extended version of the Principle of Virtual Power (PVP), written in a special form in order to comply with the nonsimple nature of the nonlocal material. Namely, extra terms are invoked in both internal and external virtual power.