The flexoelectric effect is a coupling of polarization and strain gradient, which exists in a wide variety of materials and may lead to strong size-dependent properties at the nanoscale. Based on an extension to the classical beam model, this paper investigates the electromechanical coupling response of piezoelectric nanobeams with different electrical boundary conditions including the effect of flexoelectricity. The electric Gibbs free energy and the variational principle are used to derive the governing equations with three types of electrical boundary conditions. Closed-form solutions are obtained for static bending of cantilever beams. The results show that the normalized effective stiffness increases with decreasing beam thickness in the open circuit electrical boundary conditions with or without surface electrodes. The induced electric potential due to the flexoelectric effect is obtained under the open circuit conditions, which may be important for sensing or energy harvesting applications. An intrinsic thickness depending on the material properties is identified for the maximum induced electric potential. The present results also show that flexoelectricity has a more significant effect on the electroelastic responses than piezoelectricity at the nanoscale. Our analysis in the present study can be useful for understanding of the electromechanical coupling in nanobeams with flexoelectricity.

With the attention focused on harvesting energy from the ambient environment for nanoscale electronic devices, electromechanical coupling effects in materials have been studied for many potential applications. Flexoelectricity can be observed in all dielectric materials, coupling the strain gradients and polarization, and may lead to strong size-dependent effects at the nanoscale. This paper investigates the flexoelectric energy harvesting under the harmonic mechanical excitation, based on a model similar to the classical Euler–Bernoulli beam theory. The electric Gibbs free energy and the generalized Hamilton’s variational principle for a flexoelectric body are used to derive the coupled governing equations for flexoelectric beams. The closed-form electromechanical expressions are obtained for the steady-state response to the harmonic mechanical excitation in the flexoelectric cantilever beams. The results show that the voltage output, power density, and mechanical vibration response exhibit significant scale effects at the nanoscale. Especially, the output power density for energy harvesting has an optimal value at an intrinsic length scale. This intrinsic length is proportional to the material flexoelectric coefficient. Moreover, it is found that the optimal load resistance for peak power density depends on the beam thickness at the small scale with a critical thickness. Our research indicates that flexoelectric energy harvesting could be a valid alternative to piezoelectric energy harvesting at micro- or nanoscales.