Nonlocal free vibration of graded nanobeams resting on a nonlinear elastic foundation using DQM and LaDQM

This paper presents a study of the free vibration response of a nonlocal nonlinear functionally graded (FG) Euler–Bernoulli nanobeam resting on a nonlinear elastic foundation. A power-law distribution is used to describe the material distribution along the thickness of the beam. Eringen's nonlocal elasticity model with a material length scale is adopted to account for material behavior at the nano-scale along with a modified version of the von Kármán geometric nonlinearity that in turn accounts for moderate rotations. The derived equation of motion is solved using the well-known Differential Quadrature Method (DQM) in addition to the more numerically stable Locally adaptive Differential Quadrature Method (LaDQM). The obtained nonlocal nonlinear frequencies of the nanobeam are first validated based on published analytical results that use linear mode shapes. The use of LaDQM is helpful in assessing the effect of the nonlinearities on the modes shapes which in turn was used to explain the discrepancy between the numerical and analytical results. This study aims to investigate the effects of the nonlocal parameter, and power-law index as well as linear and the nonlinear stiffnesses of the elastic foundation on the nonlinear fundamental frequency of the nanobeam for the selected boundary conditions.