Second-order Willis metamaterials: Gradient elasto-momentum coupling in flexoelectric composites

Willis materials are composites whose the overall constitutive relations exhibit coupling between momentum and strain. Recently, Willis materials that are piezoelectric are studied that allow the macroscopic momentum to be additionally coupled to non-mechanical stimulus. Such metamaterials classified as first-order Willis materials generate cross-couplings due to their asymmetric microstructures in order to realize novel phenomena in wave propagation. In this work, we study Willis materials that are flexoelectric, offer electric field induced by strain gradient. We show that in the case of flexoelectric Willis materials, the momentum also gets coupled to the strain gradient term under an effective description. Hereby, an ensemble averaging-based dynamic homogenization theory is developed for flexoelectric composites to compute constitutive relations of the macroscopic fields. This second-order Willis metamaterial offers a novel coupling termed as gradient elasto-momentum coupling. The presence of non-uniform strain that can break inversion symmetry of a unit cell is thus significant to generate the imaginary portion of all cross-couplings in the absence of asymmetric microstructures.