Variationally consistent Maxwell stress in flexoelectric structures under finite deformation and immersed in free space

Maxwell stress refers to the mechanical stress exerted on a dielectric material due to the presence of electric fields. It plays a significant role in the interaction between a dielectric material and the surrounding free space under finite deformation. Previous research on finite deformation of flexoelectricity mainly adopted a modified form of Maxwell stress, potentially not able to correctly capture some physical phenomena, such as the compression of a dielectric droplet in an electric field. In this work, we propose a consistent and complete variational principle for flexoelectricity, in which the Maxwell stress emerges naturally from the derivation, without introducing additional assumptions. An Isogeometric analysis-based numerical framework is developed accordingly and verified by both linear and nonlinear benchmark cases compared with experimental results. The present framework successfully captures and quantifies the behaviors of conductive liquids and soft dielectric solids subjected to an external electric field. Finally, a novel scenario is investigated in which a flexoelectric beam immersed in free space is analyzed, showing the interesting distribution of Maxwell stress-induced tractions at opposing boundaries. The test demonstrates that a higher dielectric constant can effectively enhance the material's stiffness in response to the external electric loading.