In this study a formulation based on the meshless method is developed to study the dynamic behavior of 2D-functionally graded nanobeams. The First order shear deformation theory is employed to model the behavior of the functionally graded beam and the small size effect is considered by employing the nonlocal theory of elasticity. The material properties are functionally graded both in the thickness and length of the nanobeam. The governing equations of the 2D-FG nanobeam are derived based on the Hamilton's principle. A meshless formulation based on the weak form of the governing equations and the point interpolation method with radial basis function is derived for discretization and solution of the problem and the mathematical details are presented. The convergence of predictions is investigated, and the predictions of presented solution are validated by comparing with the available results in the literature for special cases, and good agreements are achieved. Various numerical results are presented and the emphasis is placed on investigating the effect of several parameters such as small scale effects, the material distribution profile, mode number, length to thickness ratio and boundary conditions on the normalized natural frequencies of 1D- and 2D-FG nanobeams.

在这项研究中，开发了一种基于无网格方法的配方，以研究二维功能梯度纳米束的动力学行为。采用一阶剪切变形理论对功能梯度梁的行为进行建模，并采用非局部弹性理论来考虑小尺寸效应。在纳米束的厚度和长度上，材料特性在功能上均分级。2D-FG纳米束的控制方程是根据汉密尔顿原理导出的。推导了基于控制方程的弱形式的无网格公式和具有径向基函数的点插值方法进行离散化和问题求解，并给出了数学上的细节。研究了预测的收敛性，通过与文献中特殊情况下可获得的结果进行比较，验证了所提出解决方案的预测，并取得了良好的共识。给出了各种数值结果，并着重于研究几个参数的影响，例如小尺度效应，材料分布曲线，模式编号，长度与厚度之比以及边界条件对一维和二维FG归一化自然频率的影响纳米束。