Euler\u2013Bernoulli elastic beam models of Eringen\u2019s differential nonlocal type revisited within a mathbf{C }^{0}-continuous displacement framework

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A theory of the Erigen’s differential nonlocal beams of (isotropic) elastic material is prospected independent of the original integral formulation. The beam problem is addressed within a C^{(0)}-continuous displacement framework admitting slope discontinuities of the deflected beam axis with the formation of bending hinges at every cross section where a transverse concentrated external force is applied, either a load or a reaction. Concepts sparsely known from the literature are in this paper used within a more general context, in which the beam is envisioned as a macro-beam whose microstructure is able to take on a size dependent initial curvature dictated by the loading and constraint conditions. Indeed, initial curvature seems to be an effective analytical tool to inject size effects into micro- and nano-beams. The proposed theory is applied to a set of benchmark beam problems showing that a softening behaviour is always predicted without the appearance of paradoxical situations. Comparisons with other theories are also presented.