Electromechanical coupling in piezoelectric nanobeams due to the flexoelectric effect

The flexoelectric effect is a coupling of polarization and strain gradient, which exists in a wide variety of materials and may lead to strong size-dependent properties at the nanoscale. Based on an extension to the classical beam model, this paper investigates the electromechanical coupling response of piezoelectric nanobeams with different electrical boundary conditions including the effect of flexoelectricity. The electric Gibbs free energy and the variational principle are used to derive the governing equations with three types of electrical boundary conditions. Closed-form solutions are obtained for static bending of cantilever beams. The results show that the normalized effective stiffness increases with decreasing beam thickness in the open circuit electrical boundary conditions with or without surface electrodes. The induced electric potential due to the flexoelectric effect is obtained under the open circuit conditions, which may be important for sensing or energy harvesting applications. An intrinsic thickness depending on the material properties is identified for the maximum induced electric potential. The present results also show that flexoelectricity has a more significant effect on the electroelastic responses than piezoelectricity at the nanoscale. Our analysis in the present study can be useful for understanding of the electromechanical coupling in nanobeams with flexoelectricity.