Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization

Willis fluids, or more generally Willis materials, are homogenized composites that exhibit coupling between momentum and strain. This coupling is intrinsic to inhomogeneous media and can play a significant role in the overall response in acoustic metamaterials. In this paper, we draw connections between bianisotropy in electromagnetism and Willis coupling in elastodynamics to provide a qualitative understanding. Building upon these analogies, we introduce a homogenization technique for acoustic metamaterials based on a source-driven, multiple scattering approach that highlights the physical origins of Willis coupling. Moreover, through numerical examples, we compare several macroscopic material descriptions of acoustic metamaterials with non-negligible Willis coupling. The descriptions neglecting Willis coupling may not satisfy restrictions stemming from reciprocity, passivity, and causality, which suggests that including Willis coupling in macroscopic descriptions is necessary to realize physically meaningful macroscopic parameters.