Nonlinear electroelasticity via inverse motion: A gauge-invariant formulation with explicit variational structure

We develop a variational formulation of nonlinear electroelasticity based on inverse motion and gauge invariance. Motivated by Ericksen’s reconsideration of elastic dielectrics from the viewpoint of energy minimization, the proposed framework describes both mechanical and electromagnetic fields directly in the current configuration, providing a unified setting for electroelastic interac- tions. We discuss the relation between the inverse-motion formulation and the classical theories of Toupin and Ericksen, showing that standard electroelastic models are recovered in the small- strain regime. Particular attention is devoted to flexoelectric effects, interpreted as manifestations of latent microstructure and incorporated through higher-gradient contributions. The variational implications of the theory are illustrated through explicit coercivity conditions ensuring stability in a representative quadratic model and through an existence theorem for minimizers of a broad class of nonlinear electroelastic functionals. The resulting framework provides a unified nonlinear setting for electroelasticity and flexoelectricity while making explicit the connection between electroelastic couplings, stability, and energy minimization.