Exact solutions for functionally graded flexoelectric micro-cylinders

The flexoelectric effect implies a wide application potential in micro- and nanoscale electromechanical systems, where cylinders are widely used due to their wide range of applications. At the same time, functionally graded materials combine the advantages of different materials to achieve optimized material properties. Motivated by these considerations, we use the generalized power series method for the first time to derive exact solutions to functionally graded flexoelectric cylinder problems, including pressure and shear scenarios. This research systematically investigates the effects of material gradation, characteristic length parameters, and flexoelectric coefficients on the intricate electromechanical coupling behavior of functionally graded flexoelectric micro-cylinders. In addition, a comparative analysis between the exact and mixed finite element solutions demonstrates remarkable agreement. In particular, this investigation pioneers the extension of the Lamé problem, a cornerstone of classical elasticity, into the advanced realm of higher-order electroelasticity in inhomogeneous materials. This advance holds great promise for the design and optimization of micro- and nanoscale electromechanical systems based on the principles of flexoelectricity.