Willis materials are composites whose the overall constitutive relations exhibit coupling between momentum and strain. Recently, piezoelectric Willis materials have been studied, allowing the macroscopic momentum to be additionally coupled to the 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 and offer an electric field induced by a 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 gradient elasto-momentum coupling. The presence of non-uniform strain that can break the inversion symmetry of a unit cell is thus significant in generating the imaginary portion of all cross-couplings in the absence of asymmetric microstructures.

中文翻译：

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二阶 Willis 超材料：挠曲电复合材料中的梯度弹性动量耦合


Willis 材料是复合材料，其整体本构关系表现出动量和应变之间的耦合。最近，人们研究了压电 Willis 材料，允许将宏观动量额外耦合到非机械刺激。这种被归类为一阶 Willis 材料的超材料由于其不对称的微观结构而产生交叉耦合，以实现波传播中的新现象。在这项工作中，我们研究了 Willis 材料，这些材料是柔性的，并提供由应变梯度感应的电场。我们表明，在柔性电 Willis 材料的情况下，动量也与有效描述下的应变梯度项耦合。为此，为挠曲电复合材料开发了一种基于集成平均的动态均质化理论，以计算宏观场的本构关系。这种二阶 Willis 超材料提供了一种称为梯度弹性动量耦合的新型耦合。因此，在不存在不对称微结构的情况下，存在可以打破晶胞反转对称性的非均匀应变对于产生所有交叉耦合的虚部非常重要。