Mesh‐Distortion Immune Trefftz Element for Flexoelectric Kirchhoff Plates Within Consistent Couple Stress Theory

ABSTRACT Flexoelectricity, an electromechanical coupling phenomenon induced by strain gradients, finds plate structures to be particularly suitable due to their inherent susceptibility to developing such gradients. However, practical exploitation requires efficient numerical tools for modeling geometrically complex thin‐walled structures. This work presents a novel 4‐node quadrilateral Trefftz element for analyzing flexoelectric thin plates within consistent couple stress theory (CCST). To accomplish this, the non‐classical flexoelectric Kirchhoff plate model is first established based on CCST, and Trefftz functions satisfying the governing equations are systematically derived. These Trefftz functions subsequently serve as basis functions for constructing the displacement field, and generalized conforming conditions at collocation points are used to link them to nodal degrees of freedom. The performance of the proposed element is rigorously evaluated through benchmark problems involving static bending and free vibration analyses. Numerical results are compared against analytical solutions or three‐dimensional finite element simulations. It is shown that the element achieves high precision in capturing the size‐dependent flexoelectric responses, which are prominent at the micro/nano‐scale, and exhibits superior robustness against mesh distortion. This establishes the method as an efficient numerical tool for the design of flexoelectric plates, such as those used in micro‐sensors and energy harvesters.