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.

Erigen 的（各向同性）弹性材料的微分非局部梁的理论与原始积分公式无关。梁问题在\(C^{(0)}-\)连续位移框架允许偏转梁轴的坡度不连续，并在每个横截面形成弯曲铰链，其中施加横向集中外力，载荷或反作用力。文献中鲜为人知的概念在本文中用于更一般的上下文，其中梁被设想为宏观梁，其微观结构能够呈现由载荷和约束条件决定的尺寸相关的初始曲率。事实上，初始曲率似乎是一种将尺寸效应注入微米和纳米光束的有效分析工具。所提出的理论应用于一组基准梁问题，表明始终可以预测软化行为，而不会出现自相矛盾的情况。还介绍了与其他理论的比较。