Size-dependent bending and vibration behaviour of piezoelectric nanobeams due to flexoelectricity

Flexoelectricity, representing a spontaneous electric polarization induced by a non-uniform strain field (or strain gradient), is believed to become manifest and be responsible for the size-dependent properties of dielectric materials at the nanoscale. In this paper, the influence of the flexoelectric effect on the static bending and free vibration of a simply supported piezoelectric nanobeam is investigated based on the extended linear piezoelectricity theory and the Timoshenko beam model. The governing equations of the piezoelectric nanobeam with non-homogeneous boundary conditions are obtained from Hamilton's principle. Explicit expressions of the beam deflection and resonant frequency are derived to show the size-dependency of the flexoelectric effect. It is found that the flexoelectricity has a significant effect on the deflection of the bending beam and may reverse the deflection direction under certain loading conditions. Simulation results also indicate that the influence of the flexoelectricity on the vibration behaviour of the piezoelectric nanobeam is more prominent for beams with smaller thickness. Thus, it is suggested that possible frequency tuning of piezoelectric nanobeams by adjusting the applied electrical load should incorporate the flexoelectric effect. The current study can be claimed as helpful for qualitatively characterizing the trend of the flexoelectric effect on the mechanical responses of piezoelectric nanobeams.