This paper applies a semi-analytical polynomial method (SAPM) to solve the mechanical governing equations of nonlinear static and dynamic deformations, free and forced vibrations, and buckling analyses of nano-sized spherical functionally graded structures. Due to the nano-sized structure, the constitutive equations of motion are obtained based on the nonlocal elasticity theory. The governing equations are solved to obtain the deformations, critical buckling loads, and natural frequencies of the analyzed structure. The SAPM is based on polynomial functions (with no boundary conditions), including unknown finite coefficients. The spherical coordinate system is assumed to obtain several geometrical structures: cylindrical, conical, circular, sectorial, and rectangular. The nonlinear strains and the first-order shear deformation theory (FSDT) are employed to model the structure. The research considers the von Kármán assumptions to formulate the nonlinear strain components in the spherical coordinate system. The obtained results are compared with those in other articles to examine the accuracy of the applied solution method. The formulation's generality and simplicity and the appropriate accuracy of the SAPM's results distinguish this solution method for analyzing all mechanical aspects.

本文应用半解析多项式方法 (SAPM) 求解非线性静态和动态变形、自由和受迫振动的力学控制方程，以及纳米球形功能梯度结构的屈曲分析。由于纳米尺寸的结构，运动的本构方程是基于非局部弹性理论获得的。求解控制方程以获得所分析结构的变形、临界屈曲载荷和固有频率。SAPM 基于多项式函数（无边界条件），包括未知的有限系数。假定球坐标系获得几种几何结构：圆柱形、圆锥形、圆形、扇形和矩形。采用非线性应变和一阶剪切变形理论 (FSDT) 对结构进行建模。该研究考虑了 von Kármán 假设来制定球坐标系中的非线性应变分量。将获得的结果与其他文章中的结果进行比较，以检验所应用的求解方法的准确性。该公式的通用性和简单性以及 SAPM 结果的适当准确性使这种用于分析所有机械方面的解决方案方法脱颖而出。