General restrictions on flexoelectric Willis material properties

As a generalization of Willis materials, it is recently discovered that flexoelectric composites offer a new form of effective constitutive relations that are nonlocal in time and space. These materials possess gradient elasto-momentum coupling in addition to electro-momentum coupling and Willis coupling. Since a standard flexoelectric medium admits a range of material properties, these admissible constraints should be analogously determined for Willis counterparts. In this work, a mathematical derivation is made to show general restrictions on the material properties of such flexoelectric Willis media due to reciprocity, passivity and causality. The resultant constraints imposed by reciprocity and passivity will be extended to obtain generalized bounds that are helpful for the design of functional flexoelectric devices to realize optimal effective properties.