Nonlocal elasticity models are tackled with a general formulation in terms of source and target fields belonging to dual Hilbert spaces. The analysis is declaredly focused on small movements, so that a geometrically linearised approximation is assumed to be feasible. A linear, symmetric and positive definite relation between dual fields, with the physical interpretation of stress and elastic states, is assumed for the local elastic law which is thus governed by a strictly convex, quadratic energy functional. Genesis and developments of most referenced theoretical models of nonlocal elasticity are then illustrated and commented upon. The purpose is to enlighten main assumptions, to detect comparative merits and limitations of the nonlocal models and to focus on still open problems. Integral convolutions with symmetric averaging kernels, according to both strain-driven and stress-driven perspectives, homogeneous and non-homogeneous elasticity models, together with stress gradient, strain gradient, peridynamic models and nonlocal interactions between beams and elastic foundations, are included in the analysis.

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关于非局部弹性问题的表述

非局部弹性模型通过属于双 Hilbert 空间的源场和目标场的一般公式处理。分析明确集中在小运动上，因此假设几何线性化近似是可行的。双场之间的线性、对称和正定关系，以及应力和弹性状态的物理解释，被假定为局部弹性定律，因此由严格凸的二次能量函数控制。然后说明和评论了大多数参考的非局部弹性理论模型的起源和发展。目的是启发主要假设，检测非局部模型的比较优点和局限性，并关注仍然存在的问题。具有对称平均内核的积分卷积，