A variational method for post-buckling analysis of end-supported nanorods under self-weight with surface stress effect

Abstract

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.