Surface effects on the nonlinear forced vibration response of third-order shear deformable nanobeams

Abstract

Given the high surface to volume ratio, the nonlinear forced vibration behavior of third-order shear deformable nanobeams in the presence of the both effects of surface stress and that of surface inertia is investigated. Gurtin–Murdoch elasticity theory is utilized within the framework of third-order shear deformation beam theory to develop a novel non-classical beam model to incorporate surface effects into the forced vibration analysis of nanobeams. A cubic variation through the thickness of nanobeam is considered for the normal stress component of the bulk in order to satisfy the surface equilibrium equations. Hamilton’s principle is used to derive size-dependent nonlinear governing differential equations of motion. The equations are solved numerically using generalized differential quadrature method with an iterative algorithm on the basis of shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, based on the Galerkin’s technique, the set of nonlinear partial differential equations are reduced into a time-varying set of ordinary differential equations of Duffing type. At the end, the pseudo arc-length method is employed to solve the set of nonlinear equations of the time domain. It is observed that by increasing the beam thickness, surface effects on the nonlinear forced vibration behavior of nanobeam diminish which leads to increasing the deviation from the linear response.