Theory for dielectrics considering the direct and converse flexoelectric effects and its finite element implementation

Flexoelectricity describes the linear energy coupling between the strain gradient and the electric polarization in a solid crystalline material. Based on strain-gradient theory, a new boundary-value model is developed for dielectrics when both the direct and converse flexoelectric effects are considered; here, the electric polarization is considered to be related to both the symmetrical and rotational strain gradients. Then, its finite element implementation is realized by establishing an equivalent-energy weak form of the problem. For this higher-order elastic problem, C¹-continuous interpolations of the displacement and electric variables are required in the conventional displacement-based approach. In the present work, we construct an equivalent-energy weak form of the problem by introducing additional nodal degrees of freedom and enforcing the kinematic constraints between displacement and strain in the bulk and surface integration using Lagrange multipliers. The C¹-continuous problem then reduces to a C⁰-continuous form. Using standard C⁰-continuous shape functions, some mixed-type finite elements are developed herein for dielectrics, in which the unknown variables, i.e., displacement, strain, rotational component and polarization intensity, are interpolated as independent nodal degrees of freedom. These elements are tested and applied to solve certain electromechanical problems, and their good convergence and accuracy are demonstrated via analysis of the mechanical and electrical properties of a dielectric.