Nonlinear size-dependent vibration behavior of graphene nanoplate considering surfaces effects using a multiple-scale technique

摘要

In this article, a multiple-scale perturbation method is employed to analyze nonlinear free vibration of nanoplate incorporating surface effects. Eringen's nonlocal theory as well as surface elasticity theory of Gurtin and Murdoch is used to consider small scale effect by presenting the nonlocal parameter and the surface effects at the top and bottom of the bulk part of the nanoplate as a membrane with different mechanical properties, respectively. A multiple-scale perturbation method is suggested to consider the expansion representing the response to be a function of multiple independent variables, or scales, instead of a single variable. It also should be mentioned that graphene sheets are considered suitable candidates in nanoplates to increase the mechanical properties of the nanostructures. Employing the Hamilton's principle, the three coupled nonlinear equations of motion are obtained based on Hooke's law and the von Karman nonlinear strain-displacement relationship. To show the accuracy of the presented method results are compared with literature and a good agreement is observed. Effects of two kinds of boundary conditions SSSS1 and SSSS2, as well as nonlocal parameter, surface residual stress and surface elastic modulus parameter on nonlinear frequency of nanoplate are analyzed. The results illustrate that by increasing the nonlocal parameter the nonlinear frequency shows a decreasing behavior while by increasing the surface residual stress and surface elastic modulus parameter, nanoplate's stiffness increases therefore the nonlinear frequencies increase. Also, it can be mentioned that the surface effects have very small effects on nonlinear frequencies and can be ignored. Therefore, they don't play an important role on the nonlinear fundamental frequencies of nanoplate.