Electromechanical coupling in piezoelectric nanobeams due to the flexoelectric effect

Abstract

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.

Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting

“…For the flexoelectric material, the general electric Gibbs free energy density function can be expressed as [25,36]…”

“…It should be noted that the external work done by the surface hyper-stress tractions was ignored in this case. For the flexoelectric material, the general electric Gibbs free energy density function g 1 can be expressed as [25,36]…”

“…According to the constitutive equations of the flexoelectric beams, the non-zero stress, higher-order stress, and electric displacement are [25,32,36]…”

“…where σ 113 is defined as the higher order stress or the moment stress [25,36], which is induced by the electric field due to the flexoelectric effect, c 1111 is an elastic modulus, e 311 is a piezoelectric coefficient, µ 3113 is a flexoelectric coefficient, and κ 33 is a dielectric coefficient.…”

“…Using Equations (5) and (6), the variational expression of general electric Gibbs free energy can be written as [36]…”