Size-dependent modeling of the free vibration characteristics of postbuckled third-order shear deformable rectangular nanoplates based on the surface stress elasticity theory

Abstract

The present study deals with the nonlinear postbuckling and free vibration of third-order shear deformable rectangular nanoplate with various edge supports in the pre- and post-buckling regimes incorporating the surface effects. The Gurtin–Murdoch surface stress elasticity theory in conjunction with the third-order shear deformation plate theory is used for the size-dependent mathematical modeling of the nanoplates. The von Kármán-type kinematic nonlinearity is used to consider the nonlinear behavior of the nanoplate subjected to the in-plane loadings. The normal stress is assumed to be changed cubically through the thickness direction of nanoplate to satisfy the equilibrium conditions between on the surfaces and bulk layers. The size-dependent coupled in-plane and out-of-plane governing differential equations of motion and corresponding boundary conditions are derived by means of an energy method based on Hamilton's principle. The generalized differential quadrature (GDQ) method and pseudo-arc length continuation are employed to obtain the postbuckling load-deflection curves of nanoplates with various edge supports. A time-dependent small disturbance around the buckled configuration is considered to analyze the free vibration of postbuckled nanoplates. The effects of thickness and surface parameters on the postbuckling path and free vibration characteristics of nanoplates in the pre- and post-buckling regimes are studied. Also, a comparison is made between the results based upon the surface stress elasticity and classical continuum theories so as to show the significance of surface effects.