Due to unphysical coupling induced by the material inhomogeneity, FG (functionally graded) nanobeam problems were formulated in a very complex way so that they cannot be analytically solved. In this paper, an uncoupled theory is proposed for FG nanobeams considering their small size effects. First, with the aid of the neutral axis, the axial displacement is expressed in terms of generalized displacements for FG nanobeams. Based on the nonlocal strain gradient theory, the generalized stresses and strains are accordingly defined and uncoupled constitutive relations are derived. Based on the principle of virtual work, an uncoupled theory is eventually established, including governing equations and boundary conditions. Within the present framework, analytical solutions to FG nanobeams are obtained for the first time for general boundary conditions. These solutions not only re-evaluate the previous results but shed light on the small size effects of FG nanobeams.

由于材料不均匀性引起的非物理耦合，因此以非常复杂的方式解决了FG（功能梯度）纳米束问题，因此无法进行分析解决。本文针对FG纳米束的小尺寸效应提出了一种解耦理论。首先，借助中性轴，以FG纳米束的广义位移表示轴向位移。基于非局部应变梯度理论，据此定义了广义应力和应变，并导出了非耦合本构关系。根据虚拟工作原理，最终建立了一个包括理论方程和边界条件在内的非耦合理论。在目前的框架内，首次针对一般边界条件获得了FG纳米束的分析解决方案。