In this paper, the bending behaviour of small-scale Bernoulli–Euler beams is investigated by Eringen’s two-phase local/nonlocal theory of elasticity. Bending moments are expressed in terms of elastic curvatures by a convex combination of local and nonlocal contributions, that is a combination with non-negative scalar coefficients summing to unity. The nonlocal contribution is the convolution integral of the elastic curvature field with a suitable averaging kernel characterized by a scale parameter. The relevant structural problem, well-posed for non-vanishing local phases, is preliminarily formulated and exact elastic solutions of some simple beam problems are recalled. Limit behaviours of the obtained elastic solutions, analytically evaluated, studied and diagrammed, do not fulfil equilibrium requirements and kinematic boundary conditions. Accordingly, unlike alleged claims in literature, such asymptotic fields cannot be assumed as solutions of the purely nonlocal theory of beam elasticity. This conclusion agrees with the known result which the elastic equilibrium problem of beams of engineering interest formulated by Eringen’s purely nonlocal theory admits no solution.

在本文中，利用Eringen的两相局部/非局部弹性理论研究了小型Bernoulli-Euler梁的弯曲行为。弯矩由弹性曲率表示，是局部和非局部贡献的凸组合，即与非负标量系数加总为1的组合。非局部贡献是弹性曲率场的卷积积分具有以比例参数为特征的合适平均内核。初步制定了适合于不消失的局部相的相关结构问题，并召回了一些简单梁问题的精确弹性解。经分析评估，研究和图解说明的所得弹性解的极限行为不满足平衡要求和运动学边界条件。因此，与文献中声称的权利要求不同，不能将这种渐近场假定为梁弹性的纯粹非局部理论的解决方案。该结论与已知结果一致，即由Eringen的纯非局部理论提出的工程关注梁的弹性平衡问题不予解决。