Relationship between direct and converse flexoelectric coefficients

Flexoelectric effect, as a universal electromechanical coupling, has drawn lots of interests in dielectric materials. However, due to the restrictions of present measurement techniques, only part of coefficients has been experimentally examined. In this study, we derived the coordinate-dependent Gibbs free energy density function in the inhomogeneous spatial field to investigate the relationship between the direct and converse flexoelectric coefficients. In crystalline mediums and systems, the direct and converse flexoelectric coefficients are proved to equivalent according to the Maxwell relation. These results will broaden the application of the Maxwell relation into non-linear spatial field, and provide the guideline for experimental measurement and prediction of flexoelectric coefficients.