Flexoelectric effects on the natural frequencies for free vibration of piezoelectric nanoplates

Flexoelectricity is an electromechanical coupling phenomenon between polarization and strain gradient. Based on the Kirchhoff thin plate theory, the electromechanical coupling responses of nanoplates with the piezoelectric and flexoelectric effects are studied in this paper. Free vibration of a piezoelectric nanoplate with consideration of flexoelectricity is analyzed with emphasis on the influence of the dynamic flexoelectric effect on the natural frequencies. By means of Hamilton's variational principle, the governing equation of rectangular plates together with associated boundary conditions is derived. The natural frequencies are evaluated for a nanoplate simply supported at two opposite edges, and exact frequency equations are obtained for the other two opposite edges being simply supported, clamped-clamped, clamped-free, simply supported-free, or clamped-simply supported. The influence of dynamic flexoelectricity on the natural frequencies is elucidated. The results show that the dynamic flexoelectric effect is also size-dependent; the smaller the plate thickness is, the more obvious the dynamic flexoelectric effect is. The results also show that the dynamic flexoelectric effect is more pronounced when the order of vibration modes is higher and nanoplate's side ratio is larger. The positive and negative choice of static and dynamic flexoelectric coefficients have completely different effects on the natural frequencies. The influence of the dynamic flexoelectric effect on the natural frequencies is closely related to the side constraint and geometry of the plate. The piezoelectric effect does not alter the natural frequencies for free vibration of a homogeneous nanoplate.