A 3D collocation MFEM for the analysis of flexoelectric solids

Abstract

In this paper, a three-dimensional (3D) collocation mixed finite element method (CMFEM) is proposed for the analysis of flexoelectric solids. In this 3D CMFEM, the independent approximations by quadratic interpolation polynomials are applied for the pair of elastic displacements and strains as well as for the pair of electric potential and electric field, with the compatibility of approximations for each pair being satisfied at the selected Gauss points inside each finite element. Thus, ${\text{C}}^0$ continuous approximations of strains and electric fields can be obtained based on standard finite elements without introducing any additional nodal degrees of freedom (DOFs) except the primary field variables. The accuracy of the present method is validated through comparing the numerical results with the analytical solutions for a cantilever beam and a truncated pyramid. Using the developed CMFEM, we also model the flexoelectric effect in the 3D Mode III crack and find that the flexoelectric field along the crack front line is nonuniform, which is different from previous 2D study and shows the importance of the proposed 3D CMFEM. Besides, numerical results indicate that with the increase of the thicknesses, the out-of-plane displacement and flexoelectric response would decrease.