Varying cross-section and axial strain-gradient effects in flexoelectric cantilevers at submicron thickness levels

In elastic dielectrics, piezoelectricity is the polarization response 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 nanoscale. Flexoelectricity is controlled by a fourth-rank tensor and is therefore allowed in materials of any symmetry. In this work, we explore the effects of varying crosssection and axial strain gradient on bending vibrations on flexoelectric cantilevers. The focus is placed on the development of governing electroelastodynamic flexoelectric equations for a cantilever with varying cross-sectional widths for energy harvesting. The coupled governing equations are analyzed to obtain the frequency response and study the effects of various axial geometry profiles on the electromechanical coupling. The effect of axial strain gradient was also studied and found to be negligible for the geometries and various cross-sections studied here. Varying cross-section profile (with a reduced tip width) yields increased flexoelectric coupling.