A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects

摘要

A non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed in this study. Based on Gurtin-Murdoch elasticity theory, the influence of surface stress is incorporated into the Euler-Bernoulli beam theory. A compact finite difference method (CFDM) of sixth order is employed for discretizing the non-classical governing differential equation to obtain the natural frequencies of nanobeams subject to different boundary conditions. To check the validity of the present numerical solution, based on an exact solution, an explicit formula for the fundamental frequency of simply-supported nanobeams is obtained. Good agreement between the results of exact and numerical solutions is achieved, confirming the validity and accuracy of the present numerical scheme. The comparison between the results generated by CFDM with those obtained by the conventional finite difference method (FDM) further reveals the advantages of the compact method over its classical counterpart. The influences of beam thickness, surface density, surface residual stress, surface elastic constants, and boundary conditions on the natural frequencies of nanobeams are also investigated. It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness.