In this paper, the formulation of post-buckling of end-supported nanorods under self-weight was developed by the variational method. The surface stress effect was considered following the surface elasticity theory of Gurtin–Murdoch. The variational formulation involving the strain energy in the bulk material, the strain energy of the surface layer, and the potential energy due to self-weight was expressed in terms of the intrinsic coordinates. The variational formulation was accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method was used to derive a system of nonlinear equations resulting from the stationary of the total potential energy, and then, Newton–Raphson iterative procedure was applied to solve this system of equations. The post-buckled configurations of nanorods under self-weight due to various boundary conditions were presented and demonstrated that the variational formulation expressed in terms of intrinsic coordinate is highly recommended for post-buckling analysis of end-supported nanorods. In addition, the surface stress effect significantly influenced the post-buckling response of nanorods and exhibited higher stiffness in comparison with nanorods without surface stress. The model formulation presented in this study is of special interest in the design and application of advanced technological devices.

本文采用变分法研究了自重作用下末端支撑纳米棒的后屈曲公式。表面应力效应是根据Gurtin–Murdoch的表面弹性理论来考虑的。涉及散装材料中的应变能，表面层的应变能以及由于自重引起的势能的变化公式以固有坐标表示。通过引入拉格朗日乘数技术施加边界条件来完成变分公式化。有限元法被用来从总势能的平稳性推导一个非线性方程组，然后，牛顿-拉夫森迭代过程被用来求解该方程组。提出了由于各种边界条件而在自重作用下纳米棒的后屈曲构型，并证明了强烈建议将本征坐标表示的变化公式用于末端支撑纳米棒的后屈曲分析。另外，与没有表面应力的纳米棒相比，表面应力效应显着影响了纳米棒的屈曲后响应，并表现出更高的刚度。本研究中提出的模型公式对于先进技术设备的设计和应用特别感兴趣。与没有表面应力的纳米棒相比，表面应力效应显着影响了纳米棒的屈曲后响应，并表现出更高的刚度。本研究中提出的模型公式对于先进技术设备的设计和应用特别感兴趣。与没有表面应力的纳米棒相比，表面应力效应显着影响了纳米棒的屈曲后响应，并表现出更高的刚度。本研究中提出的模型公式对于先进技术设备的设计和应用特别感兴趣。