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

Abstract

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 new homogenization 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.

Active Willis metamaterials for ultracompact nonreciprocal linear acoustic devices

“…A key advantage of our active Willis material is its ability to control the Willis polarizabilities α md and α dm independently, which is not possible in passive media 6,11 . We compute the acoustic polarizabilities by inverting Eqs.…”

“…where the acoustic pressure p is related to the volumetric strain u k,k via the bulk modulus B and to the particle velocity vector v k via the Willis coupling vector S k . The momentum density vector µ i is related to the particle velocity vector v j via the second order mass density tensor The coupling between adjacent unit cells inside a metamaterial means that the dynamics of the unit cells inside the middle of the metamaterial are slightly different from the dynamics of edge unit cells when the unit cell size is larger than roughly a tenth of the operating wavelength 6 . Therefore, the description of the metamaterial acoustic behavior in terms of macroscopic material parameters becomes insufficient.…”

Maximum Willis Coupling in Acoustic Scatterers

“…In direct analogy, Willis coupling has been explored in elastodynamics [19]- [22] and acoustics [23]- [25] since the 80s, and it has been recently become of interest in the context of acoustic metamaterials. Willis coupling describes the connection between acoustic pressure and particle velocity in some acoustic media, and so far it has been treated as a higher-order perturbative phenomenon [22]- [24]. In order to exploit to its full extent bianisotropy in elastodynamics and acoustics, it would be ideal to realize metamaterial inclusions with large, ideally maximum, Willis coupling.…”

Coupled Decorated Membrane Resonators with Large Willis Coupling

“…Muhlestein et al further analyzed the reciprocity, passivity and causality in Willis materials [6], and predicted that the Willis cross coupling coefficient must have the line shapes that follow the Kramers-Kronig relation dictated by causality. Sieck et al further developed a source driven multiple scattering scheme [7] that emphasizes the physical origin, in particular the asymmetry of the building blocks of the media, and pointed out the necessity of including the Willis coupling in retrieving the macroscopic dynamic parameters of inhomogeneous media that are consistent with reciprocity, passivity and causality. Quan et al derived general bounds on the response of acoustic scatterers including the Willis coupling coefficient based on energy conservation [8].…”