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This paper develops the equilibrium equations describing the flexoelectriceffect in soft dielectrics under large deformations. Previous works havedeveloped related theories using a flexoelectric coupling tensor of mixedmaterial-spatial character. Here, we formulate the model in terms of aflexoelectric tensor completely defined in the material frame, with the samesymmetries of the small-strain flexocoupling tensor and leading naturally toobjective flexoelectric polarization fields. The energy potential andequilibrium equations are first expressed in terms of deformation andpolarization, and then rewritten in terms of deformation and electricpotential, yielding an unconstrained system of fourth order partialdifferential equations (PDEs). We further develop a theory of geometricallynonlinear extensible flexoelectric rods under open and closed circuitconditions, with which we examine analytically cantilever bending and bucklingunder mechanical and electrical actuation. Besides being a simple and explicitmodel pertinent to slender structures, this rod theory also allows us to testour general theory and its numerical implementation using B-splines. Thisnumerical implementation is robust as it handles the electromechanicalinstabilities in soft flexoelectric materials.

Modeling flexoelectricity in soft dielectrics at finite deformation