Vibrations of piezoelectric nanobeams considering flexoelectricity influence: a numerical approach based on strain-driven nonlocal differential/integral models

Abstract

In this work, a numerical study is presented for the free vibration behavior of piezoelectric Bernoulli–Euler nanoscale beams considering flexoelectric and nonlocal effects. Strain-driven nonlocal formulations of Eringen’s theory in both differential and integral forms are used to capture nonlocal influences. By Hamilton’s principle, the equation of motion and boundary conditions are obtained which are numerically solved. In the solution method, the finite difference and generalized differential quadrature (GDQ) methods are applied for discretizing the differential and integral equations, respectively. Also, matrix differential and integral operators are proposed. After investigating the convergence and validation of the approach, the influences of nonlocality, flexoelectric coefficient and slenderness ratio on the linear free vibrations of nanobeams subject to different end conditions are studied in the numerical results. It is revealed that the paradox related to the results of differential formulation for nanocantilevers is resolved by the proposed integral formulation. © 2022, The Author(s), under exclusive licence to The Brazilian Society of Mechanical Sciences and Engineering.