Materials with asymmetric microstructure can constitutively couple macroscopic fields from different physics. Examples include piezoelectric materials that couple mechanical and electric fields and Willis materials that anomalously couple dynamic and elastic fields, for example velocity and stress. Recently, it was shown that anomalous coupling between the elastodynamic and electric field emerges when piezoelectricity is incorporated into Willis materials. Here, we investigate one-dimensional asymmetric piezoelectric Willis elements using heuristic homogenization, long-wavelength asymptotic analysis and numerical experiments. We show that in order to describe the heterogeneous scatterer using a homogenized description that respects reciprocity and energy conservation, anomalous electromomentum moduli must be included. Our findings elucidate the origins of this electromomentum coupling and provide insight for the future design of this new class of coupled-field metamaterials.

具有不对称微观结构的材料可以本构耦合来自不同物理场的宏观场。示例包括耦合机械和电场的压电材料和异常耦合动态和弹性场（例如速度和应力）的 Willis 材料。最近，研究表明，当将压电性结合到 Willis 材料中时，会出现弹性动力学和电场之间的异常耦合。在这里，我们使用启发式均匀化、长波长渐近分析和数值实验研究一维非对称压电 Willis 元件。我们表明，为了使用尊重互易性和能量守恒的均匀描述来描述异质散射体，必须包括异常电动动量模量。