A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects

Abstract

A micro scale Timoshenko beam is developed for the piezoelectricity with flexoelectricity and surface effects. By virtue of the variational principle and Hamilton’s principle, the governing equations and related boundary conditions are obtained. As the shear strain is involved for the Timoshenko beam, the flexoelectric effect induced by the shear strain gradient is considered to make the new model more general. Based on this new Timoshenko beam model, the static bending and free vibration problems of a simply supported piezoelectric nanobeam with uniform load applied are solved respectively. The numerical results reveal that both the flexoelectricity and surface effects have remarkable influence on the deformations of the beam. Only flexoelectricity considered may cause a big difference between the theoretical prediction and the experiment results. The shear strain gradient also has considerable effect on the electromechanical response of static bending problem. Moreover, the change of surface properties not only directly affects the static bending but also significantly changes the natural frequency of the beam. This study provides a guideline for the analysis of piezoelectric nanobeam, and also is helpful to understand how the material characteristics affect the electromechanical response of nanobeam.