Size-dependent direct and converse flexoelectricity around a micro-hole

In this paper, the complete explicit solution for physical fields around a micro-hole is solved with the simultaneous consideration of the strain gradient elasticity, the direct flexoelectricity, and the converse flexoelectricity. First, the higher-order Navier-like governing equations are proposed for an isotropic flexoelectric solid by using an extended linear theory of flexoelectric materials considering the coupling between the strain gradient and the polarization, and conversely between the polarization gradient and the strain. Second, the displacement, the electric potential, and the polarization are successfully obtained by solving the corresponding boundary value problems for a micro-hole model. Finally, the influence of the strain gradients and the flexoelectric effect on the mechanical fields and the electric responses are studied around the micro-hole in flexoelectric solids, and the size effects of the direct flexoelectricity and the converse flexoelectricity are also investigated. The results indicate that an electric response can be obviously induced by a mechanical loading due to the direct flexoelectricity. Conversely, a mechanical strain can be produced by an electric field through the converse flexoelectricity. The size dependence of both the direct flexoelectricity and the converse flexoelectricity is also successfully predicted in this paper, in the form that the flexoelectricity significantly increases with the decrease in the sample size.