Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

Abstract

• Extended theory of piezoelectricity is used to investigate flexoelectricity effect on vibrational behavior of nanobeam. • The von-Kármán strain–displacement relationship, electric Gibbs free energy and Hamilton's principle are employed. • Multiple scales method is used to obtain a closed form solution of derived nonlinear governing equations of motion. • The effects of changes in parameters on free vibration and postbuckling behavior of nanobeam are discussed comprehensively. • Theoretical range of flexoelectricity constant is estimated through buckling analysis . This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.