Size effects in piezoelectric cantilevers at submicron thickness levels due to flexoelectricity

In elastic dielectrics, piezoelectricity is the response of polarization to applied mechanical strain, and vice versa. Piezoelectric coupling is controlled by a third-rank tensor and is allowed only in materials that are non-centrosymmetric. Flexoelectricity, however, is the generation of electric polarization by the application of a non-uniform mechanical strain field, i.e. a strain gradient, and is expected to be pronounced at submicron thickness levels, especially at the nano-scale. Flexoelectricity is controlled by a fourth-rank tensor and is therefore allowed in materials of any symmetry. As a gradient effect, flexoelectricity is size dependent, while piezoelectric coupling has no size dependence. Any ordinary piezoelectric cantilever model developed for devices above micron-level thickness has to be modified for nano-scale piezoelectric devices since the effect of flexoelectric coupling will change the electroelastic dynamics at such small scales. In this work, we establish and explore a complete analytical framework by accounting for both the piezoelectric and flexoelectric effects. The focus is placed on the development of governing electroelastodynamic piezoelectric-flexoelectric equations for the problems of energy harvesting, sensing, and actuation. The coupled governing equations are analyzed to obtain the frequency response. The coupling coefficient for the bimorph configuration is identified and its size dependence is explored.