Forced vibration of piezoelectric and flexoelectric Euler–Bernoulli beams by dynamic Green’s functions

This work presents the dynamic response of piezoelectric and flexoelectric Euler–Bernoulli beams subjected to mechanical loads by means of Green’s functions. Exact solutions in closed form for the electromechanical coupling behaviors are derived. The present solutions will reduce to those of classical piezoelectric beam models by neglecting the flexoelectric effect. Numerical results for a BaTiO ₃ cantilever show that the tip deflection of the beam will be separated when the beam thickness reduces to one micrometer; therefore, pure mechanical experiments can be used to determine the flexoelectric effect by a bending test under different electrical boundary conditions. In addition, the dynamic output voltage under open-circuit condition and the dynamic output charge under short-circuit condition are dominated by the flexoelectric effect. The dynamic output voltage and charge become negligible when the beam thickness reduces to one micrometer. By reducing the length–thickness ratio or introducing the effective piezoelectric coefficient, the electric response can be more easily detected at microscale. These results demonstrate that the flexoelectric effect can be strong enough to dominate the electromechanical coupling behavior, coordinating with the ‘ever green’ trend to miniaturization, and the flexoelectric effect can be applicable for energy harvesters, sensing, and others.