This paper presents exact elasticity solutions for nano-plane structures subjected to any distribution of inplane body forces. In deriving the plane stress solutions, three different models are used. They are a lattice elasticity model called the Hencky bar-grid model (eHBM), the continualised nonlocal plane model (CNM) and Eringen’s nonlocal plane model (ENM). eHBM is a physical structural model comprising a system of rigid bar grids with bars connected by axial and torsional springs. CNM is a nonlocal model derived by continualising the governing discrete equations of the eHBM. ENM is a stress gradient nonlocal model. The use of three models allows independent confirmation of the solutions as well as providing a better understanding of the phenomenological similarities between them. Based on the exact solutions for a nano-plane structure under a partial uniformly inplane body force, it is found that by setting the bar grid length of eHBM to be equal to the characterisitc length \(\ell \) of CNM and ENM, the maximum inplane displacements predicted by eHBM and CNM are in exact agreement when CNM small length scale coefficient \(c_{{0}}=1/\sqrt{12} \). However, the ENM maximum inplane displacements are in agreement with eHBM solutions only when ENM’s small length scale coefficient \(e_{{0}}\) lies between \(1/\sqrt{50} \) and \(1/\sqrt{10} .\) These results confirm some phenomenological similarities among eHBM, CNM and ENM; with CNM being closely related to the eHBM physical structure model.

本文提出了受任何平面内力分布影响的纳米平面结构的精确弹性解决方案。在推导平面应力解时，使用了三种不同的模型。它们是称为 Hencky 条形网格模型 (eHBM)、连续非局部平面模型 (CNM) 和 Eringen 非局部平面模型 (ENM) 的晶格弹性模型。eHBM 是一个物理结构模型，包括一个刚性杆网格系统，杆由轴向和扭转弹簧连接。CNM 是通过继续控制 eHBM 的控制离散方程而导出的非局部模型。ENM 是应力梯度非局部模型。使用三个模型可以独立确认解决方案，并更好地理解它们之间的现象学相似性。\(\ell \) CNM 和ENM，当CNM 小长度比例系数\(c_{{0}}=1/\sqrt{12} \)时，eHBM 和CNM 预测的最大面内位移完全一致。然而，仅当 ENM 的小长度尺度系数\(e_{{0}}\)位于\(1/\sqrt{50} \)和\(1/\sqrt )之间时，ENM 的最大面内位移才与 eHBM 解一致{10} .\)这些结果证实了 eHBM、CNM 和 ENM 之间的一些现象学相似性；CNM 与 eHBM 物理结构模型密切相关。