Finite element modeling and analysis of flexoelectric plates using gradient electromechanical theory

This work presents the development of a two-way coupled flexoelectric plate theory starting from a 3D gradient electromechanical theory. The gradient electromechanical theory considers three mechanical length scale parameters and two electric length scale parameters to account for both mechanical and electrical size effects. Variational formulation is used to derive the plate governing equations and boundary conditions considering Kirchhoff’s assumptions. A computationally efficient \(C^2\) continuous non-conforming finite element is developed to solve the resulting plate equations. To assess the accuracy of the non-conforming finite element framework, the results are compared with Navier-type analytical solution for a simply supported flexoelectric plate. The finite element framework is also validated with experimental results in the existing literature for a passive micro-plate. The results show excellent agreement with both analytical and experimental results. Furthermore, computational efficiency of the non-conforming element is compared with the standard conforming element, which contains greater degrees of freedom and continuity across all elemental edges. It was observed that the non-conforming element is almost twice as fast as the conforming element without a significant loss of accuracy. The 2D finite element formulation is subsequently used to analyze the size-dependent response of flexoelectric composite plates operating in both sensor and actuator modes. Various parametric studies are performed to analyze the effect of boundary conditions, length scale parameters, size of the plate, flexoelectric layer thickness ratio, etc., on the response of flexoelectric plate-type sensors and actuators. It is found that the effective electromechanical coupling increases in a flexoelectric plate at microscale (due to the size effects), and it is higher than standard piezoelectric materials for plate thickness \(h \le 8\,{{\upmu }}\)m.