Combined piezoelectric and flexoelectric effects in resonant dynamics of nanocantilevers

We establish and analyze an analytical framework by accounting for both the piezoelectric and flexoelectric effects in bimorph cantilevers. The focus is placed on the development of governing electroelastodynamic piezoelectric-flexoelectric equations for the problems of resonant energy harvesting, sensing, and actuation. The coupled governing equations are analyzed to obtain closed-form frequency response expressions via modal analysis. The combined piezoelectric-flexoelectric coupling coefficient expression is identified and its size dependence is explored. Specifically, a typical atomistic value of the flexoelectric constant for barium titanate is employed in the model simulations along with its piezoelectric constant from the existing literature. It is shown that the effective electromechanical coupling of a piezoelectric material, such as barium titanate, is significantly enhanced for thickness levels below 100 nm. The electromechanical coupling coefficient of a barium titanate bimorph cantilever increases from the bulk piezoelectric value of 0.065 to the combined piezoelectric-flexoelectric value exceeding 0.3 toward nanometer thickness level. Electromechanical frequency response functions for resonant power generation and dynamic actuation also capture the size-dependent enhancement of the electromechanical coupling. The analytical framework given here can be used for parameter identification and design of nanoscale cantilevers that can be used as energy harvesters, sensors, and actuators.