A collocation mixed finite element method for the analysis of flexoelectric solids. (15th May 2021) - Tian, Xinpeng; Sladek, Jan; Sladek, Vladimir; Deng, Qian; Li, Qun
Abstract: A collocation mixed finite element method (MFEM) for direct and converse flexoelectricity in piezoelectric materials is developed for 2D problems. The size-effect phenomenon in micro/nano structures is considered by the strain- and electric intensity vector-gradient effects. C 0 continuous finite element method is inadequate to treat flexoelectricity problems involving the size-effect. To this end, the MFEM with Lagrangian multipliers to treat these solids has been reported recently. With existing MFEM, the computational efficiency is low due to the additional nodal degrees of freedom (DOFs) for the Lagrangian multipliers. In this study, a new collocation MFEM is proposed, in which the number of the DOFs, when compared to the traditional Lagrangian approach, can be reduced. At the same time, the kinematic constraints between the displacement and strain are guaranteed. These kinematic constraints are satisfied by the collocation method at some internal points in the finite elements. The present collocation MFEM can be used to solve flexoelectricity problems with higher efficiency. Its accuracy is verified by comparing the numerical results with available analytical solutions for the bending of a cantilever beam and the compression of a truncated pyramid, respectively. The results indicate that flexoelctricity is strongly related to the geometry of the physical problem. It is shown that flexoelectricity increases significantly with the decrease of the sample size. The same occurs when, for the beam problem, the ratio of the length to depth dimensions increases; similarly, for the truncated pyramid problem, when the ratio of the width of the bottom and top surfaces increases.