Surface energy effects on the free vibration characteristics of postbuckled third-order shear deformable nanobeams

The prime aim of the current study is to predict the free vibration behavior of third-order shear deformable nanobeams in the vicinity of postbuckling configuration and in the presence of surface effects which includes surface elasticity, residual surface stress and surface inertia. To accomplish this end, Gurtin–Murdoch elasticity theory within the framework of third-order shear deformation beam theory is employed. In order to satisfy the balance conditions between the bulk and surfaces of nanobeam, a cubic distribution is considered for the normal stress through the thickness. By using Hamilton’s principle, the non-classical governing differential equations of motion including von Karman geometric nonlinearity are derived. After using generalized differential quadrature (GDQ) method to discretize the governing equations on the basis of Chebyshev–Gauss–Lobatto grid points, the pseudo-arc length continuation technique is utilized to solve the eigenvalue problem. The natural frequencies of nanobeam corresponding to the both prebuckling and postbuckling domains are obtained for various buckling mode shapes based on the numerical solution strategy. It is demonstrated that in the prebuckling domain of the first vibration mode shape, increasing of beam thickness leads to lower natural frequency for all types of boundary conditions, but this behavior becomes reverse in the postbuckling domain.