Exact Solutions for Flexoelectric Response in Nanostructures

This paper is concerned with the derivation of the exact solutions for the static responses of the simply supported flexoelectric nanobeams subjected to the applied mechanical load and applied voltage difference across the thickness of the beams. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements and the electric potential in the beams. Due to the converse flexoelectric effect, the active beams significantly counteract the applied mechanical load. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect in the beams resulting in the coupling of bending and stretching deformations in the beams. For the particular values of the length of the beam and the applied voltage, the deflection of the nanobeam due to the converse flexoelectric effect significantly increases with the decrease in the thickness of the beam. But the deflection of the beam remains invariant with the change in length of the beam for the particular values of the thickness of the beam and the applied voltage. Also, for the particular values of the thickness of the beam and the applied mechanical load, the induced transverse electric polarization on the surface of the beam is independent of the variation of the length of the beam and the value of the polarization increases with the decrease in the thickness of the beam. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric nanobeams may be effectively exploited for advanced applications as smart sensors and actuators at nanoscale.