Authors: Z.Z. He, H.Y. Zhao, Y.C. Zhou, B.X. Xu, W.Q. Chen, L. Zhang

Flexoelectricity, induced by strain gradients, has emerged as a promising phenomenon with broad potential applications. Fully exploring its engineering potential requires efficient and accurate modeling methods. This paper proposes a novel beam-type mixed finite element method (M-FEM) for predicting and modeling the multi-field coupling mechanical behaviors of flexoelectric beams coupled with the external RLC circuits. A generalized Timoshenko beam theory is presented incorporating flexoelectricity and external RLC circuits using the lumped element model characterized by dual variables of voltage and charge. Building on this foundation, a beam-type M-FEM formulation is established for flexoelectric beams coupled with external RLC circuits by introducing two auxiliary variables to reduce continuity requirements. The proposed beam-type M-FEM is rigorously validated against the two-dimensional (2D) M-FEM, demonstrating superior computational efficiency while maintaining high accuracy in capturing the multi-field coupling mechanical behaviors of flexoelectric beam-like structures. Numerical results show that the external RLC circuits play a significant role in tuning the multi-field coupling responses and mechanical behaviors, including static bending, linear static buckling, and forced vibration of flexoelectric beams. This study provides a robust and efficient tool for advancing flexoelectric research and enabling practical engineering applications of flexoelectric structures.