Modeling and simulation of functionally graded flexoelectric micro-cylinders based on the mixed finite element method

The direct flexoelectricity in solid dielectrics, as an electromechanical mechanism coupling polarization and strain gradient, exhibits strong size dependence and structures associated (geometry or microstructure). In this work, a novel asymmetric micro-cylinder composed of functionally graded (FG) materials is designed to evaluate computationally the role of flexoelectricity in the electromechanical response. However, analytical solutions involving strain gradient elasticity and flexoelectricity make the universal coupling phenomenon in multiple fields difficult for such asymmetric graded structure. Based on the mixed finite element method (FEM), we introduce Lagrange multipliers to enforce the kinematic relationship between the displacement field and its gradient, so as to help understand the flexoelectric performance of the proposed FG micro-cylinder. Code verification as well as validation of the mixed FEM for the present FG structure is obtained by comparing numerical results with analytical solutions for a traditional composite micro-cylinder. The simulated results show that the designed FG configuration highlights the critical role of material composition and material distribution on the effective electromechanical response. At the same time, a reasonable gradient function can greatly improve the mechanical and electrical properties of materials. Our results suggest that computer simulations can help to understand and quantify the physical properties of flexoelectric devices.