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Building on Willis' homogenization framework, recent work has revealed that heterogeneous conductors exhibit macroscopic thermal bianisotropy, in which the macroscopic heat flux and entropy are nonlocally coupled to both temperature and temperature gradient. Existing numerical examples, however, are limited to the subwavelength regime. Here, we provide the first explicit demonstration of this spatial nonlocality by computing the effective kernels of a periodic laminate using three independent homogenization methods. The three approaches yield consistent nonlocal cross-coupling terms, clarifying the roles of spatial asymmetry and averaging choice. We also calculate the corresponding thermal impedance and show that it is direction-dependent, highlighting a physical signature of thermal bianisotropy relevant to thermal metamaterials.

The existence and uniqueness of weak solutions is shown for a system related to the Willis model of elastodynamics. Both the whole space case and the case of a bounded smooth domain are studied. To this end the equations are reformulated as a linear symmetric hyperbolic system of first order and the existing theory for such systems is applied. If the initial and boundary data is regular enough, classical solutions are obtained. The possibility to transform the problem to a linear symmetric hyperbolic system hinges on a new symmetry condition on the Willis coupling tensor S, not yet considered in the literature. This condition demands that S is a totally symmetric third-order tensor.

Electromagnetic bi-anisotropy finds an analogy in acoustic metamaterial science as Willis coupling. Its impact and emergence in the field of elastodynamic metamaterials is not as well understood however, given the coupling between compressional and shear waves. Here we discuss the emergence of Willis coupling in heterogeneous elastic slabs embedded in an acoustic fluid. The microstructure of the slab comprises circular cylindrical voids and asymmetry is present via two neighbouring line arrays, each with a repeating void of differing radius. The slab matrix is soft, with Poisson ratio close to $1/2$ so that the voids act as Giant Monopole Resonators, and induce a strong dynamic response at low frequency. The incorporation of Willis constitutive coupling ensures that a unique set of effective material properties can be assigned to the slab up to a maximum frequency defined by the periodic spacing of the voids and the elastic properties of the substrate. Including loss in the elastic medium via its shear modulus induces strong directional-dependent absorption at low frequency, whilst of course maintaining reciprocity.

This article introduces a methodology for inducing wavenumber bandgaps via alternating Willis coupling signs. A non-reciprocal wave equation of Willis-type is first considered, and its wave dispersion analyses are carried out via the transfer matrix method. By creating unit cells from two identical Willis-type elastic layers, yet with reversed Willis-coupling signs, a reciprocal band structure peculiarly emerges, although each layer exhibits non-reciprocity if considered individually. Wavenumber bandgaps open due to such unit cell configuration, and their width and limits are analytically quantified. Similarities between materials with reversed-sign Willis coupling and bi-layered phononic crystals are noted, followed by concluding remarks.

Broadband underwater sound focusing in the low-frequency range is essential for various applications such as battery-free environmental monitoring and sensing. However, achieving low-frequency underwater focusing typically necessitates bulky, heavy structures that hinder practical deployment. Here, we introduce a three-dimensional underwater lens comprising cavity-based locally resonant asymmetric structures, enabling the efficient manipulation of low-frequency waterborne sound through a densely packed lattice configuration. We experimentally validated its broadband focusing performance over a range of 20-35 kHz. In addition, we observed that our lens exhibits asymmetric backscattering-a distinctive effect arising from its bianisotropic nature-which we term the Willis lens. Unlike conventional underwater lenses that rely on fully filled structures, our design employs cavity-based scatterers, achieving a lighter yet robust focusing performance. With its lightweight, efficient, and reliable design, the Willis lens provides a promising platform for underwater sensor networks and future advancements in on-demand waterborne sound focusing.

Willis elasticity is an effective medium theory for linearly elastic composites that incorporates an unusual coupling between stress and velocity, as well as between momentum and strain. Interest in the theory peaked following the discovery that its formulation is invariant under curvilinear changes of coordinates and that, consequently, it can be used to inverse-design ``invisibility'' cloaks for elastodynamics. That said, the microstructure-property relationships in Willis elasticity are poorly understood and, in particular, the mechanics that underlie the coupling are largely unknown. Thus, no such cloaks were constructed. Here, we put forward the idea that Willis elasticity is a particular microcontinuum field theory where the (generalized) micro-displacements have been eliminated in favor of the macroscopic displacement field as if by Schur completion. The field theory is special in that it features an inertial coupling between the micro- and macro-displacements that, upon completion, re-emerges as the coupling term in Willis elasticity. Concretely, we analyze an asymptotic regime where mechanical lattices exhibit a kinematic enrichment with a strong (leading-order) inertial c

Willis materials are complex media characterized by four macroscopic material parameters, the conventional mass density, and bulk modulus and two additional Willis coupling terms, which have been shown to enable unsurpassed control over the propagation of mechanical waves. However, virtually all previous studies on Willis materials involved passive structures which have been shown to have limitations in terms of achievable Willis coupling terms. In this article, we show experimentally that linear active Willis metamaterials breaking these constraints enable highly non-reciprocal sound transport in very subwavelength structures, a feature unachievable through other methods. Furthermore, we present an experimental procedure to extract the effective material parameters expressed in terms of acoustic polarizabilities for media in which the Willis coupling terms are allowed to vary independently. The approach presented here will enable a new generation of Willis materials for enhanced sound control and improved acoustic imaging and signal processing.

Acoustic metamaterials are structures with exotic acoustic properties, having promising applications in acoustic beam steering, focusing, impedance matching, absorption and isolation. Recent work has shown that the efficiency of many acoustic metamaterials can be enhanced by controlling an additional parameter known as Willis coupling, which is analogous to bianisotropy in electromagnetic metamaterials. The magnitude of Willis coupling in an acoustic meta-atom has been shown theoretically to have an upper limit, however the feasibility of reaching this limit has not been experimentally investigated. Here we introduce a meta-atom with Willis coupling which closely approaches this theoretical limit, that is much simpler and less prone to thermo-viscous losses than previously reported structures. We perform two-dimensional experiments to measure the strong Willis coupling, supported by numerical calculations. Our meta-atom geometry is readily modeled analytically, enabling the strength of Willis coupling and its peak frequency to be easily controlled. Together with its ease of fabrication, this will facilitate the design of future high efficiency acoustic devices.

Brillouin-zone (BZ) definition in a class of non-reciprocal Willis monatomic lattices (WMLs) is analytically quantified. It is shown that BZ boundaries only shift in response to non-reciprocity in one-dimensional WMLs, implying a constant BZ width, with asymmetric dispersion diagrams exhibiting unequal wavenumber ranges for forward and backward going waves. An extension to square WMLs is briefly discussed, analogously demonstrating the emergence of shifted and irregularly shaped BZs, which maintain constant areas regardless of non-reciprocity strength.

Bianisotropy is common in electromagnetics whenever a cross-coupling between electric and magnetic responses exists. However, the analogous concept for elastic waves in solids, termed as Willis coupling, is more challenging to observe. It requires coupling between stress and velocity or momentum and strain fields, which is difficult to induce in non-negligible levels, even when using metamaterial structures. Here, we report the experimental realization of a Willis metamaterial for flexural waves. Based on a cantilever bending resonance, we demonstrate asymmetric reflection amplitudes and phases due to Willis coupling. We also show that, by introducing loss in the metamaterial, the asymmetric amplitudes can be controlled and can be used to approach an exceptional point of the non-Hermitian system, at which unidirectional zero reflection occurs. The present work extends conventional propagation theory in plates and beams to include Willis coupling, and provides new avenues to tailor flexural waves using artificial structures.

A material that exhibits Willis coupling has constitutive equations that couple the pressure-strain and momentum-velocity relationships. This coupling arises from subwavelength asymmetry and non-locality in heterogeneous media. This paper considers the problem of the scattering of a plane wave by a cylinder exhibiting Willis coupling using both analytical and numerical approaches. First, a perturbation method is used to describe the influence of Willis coupling on the scattered field to a first-order approximation. A higher-order analysis of the scattering based on generalized impedances is then derived. Finally, a finite element method-based numerical scheme for calculating the the scattered field is presented. These three analyses are compared and show strong agreement for low to moderate levels of Willis coupling.

We propose a concept called acoustic amplifying diode in combining both signal isolation and amplification in a single device. The signal is exponentially amplified in one direction with no reflection and is completely absorbed in another. In this case, the reflection is eliminated from the device in both directions due to impedance matching, preventing backscattering to the signal source. Here, we experimentally demonstrate the amplifying diode using an active metamaterial with non-reciprocal Willis coupling. We also discuss the situation with the presence of both reciprocal and non-reciprocal Willis couplings for more flexibility in implementation. The concept of acoustic amplifying diode will enable applications in sound isolation, sensing and communication, in which non-reciprocity can play an important role.

Acoustic bianisotropy, also known as the Willis parameter, expands the field of acoustics by providing nonconventional couplings between momentum and strain in constitutive relations. Sharing the common ground with electromagnetics, the realization of acoustic bianisotropy enables the exotic manipulation of acoustic waves in cooperation with a properly designed inverse bulk modulus and mass density. While the control of entire constitutive parameters substantiates intriguing theoretical and practical applications, a Willis metamaterial that enables independently and precisely designed polarizabilities has yet to be developed to overcome the present restrictions of the maximum Willis bound and the nonreciprocity inherent to the passivity of metamaterials. Here, by extending the recently developed concept of virtualized metamaterials, we propose acoustic Willis metamaterials that break the passivity and reciprocity limit while also achieving decoupled control of all constitutive parameters with designed frequency responses. By instituting basis convolution kernels based on parity symmetry for each polarization response, we experimentally demonstrate bianisotropy beyond the limit of p

Acoustophoresis deals with the manipulation of sub-wavelength scatterers in an incident acoustic field. The geometric details of manipulated particles are often neglected by replacing them with equivalent symmetric geometries such as spheres, spheroids, cylinders or disks. It has been demonstrated that geometric asymmetry, represented by Willis coupling terms, can strongly affect the scattering of a small object, hence neglecting these terms may miss important force contributions. In this work, we present a generalized formalism of acoustic radiation force and radiation torque based on the polarizability tensor, where Willis coupling terms are included to account for geometric asymmetry. Following Gorkov's approach, the effects of geometric asymmetry are explicitly formulated as additional terms in the radiation force and torque expressions. By breaking the symmetry of a sphere along one axis using intrusion and protrusion, we characterize the changes in the force and torque in terms of partial components, associated with the direct and Willis Coupling coefficients of the polarizability tensor. We investigate in detail the cases of standing and travelling plane waves, showing how t

The WILLI detector, built in IFIN-HH Bucharest, in collaboration with KIT Karlsruhe, is a rotatable modular detector for measuring charge ratio for cosmic muons with energy $<$ 1 GeV. It is under construction a mini-array for measuring the muon charge ratio in Extensive Air Showers. The EAS simulations have been performed with CORSIKA code. The values of the muon flux, calculated with semi-analytical formula, and simulated with CORSIKA code, based on DPMJET and QGSJET models for the hadronic interactions, are compared with the experimental data determined with WILLI detector. No significant differences between the two models and experimental data are observed. The measurements of the muon charge ratio for different angles-of-incidence, (performed with WILLI detector) shows an asymmetry due to the influence of magnetic field on muons trajectory; the values are in agreement with the simulations based on DPMJET hadronic interaction model. The simulations of muon charge ratio in EAS performed with CORSIKA code based on three hadronic interaction models (QGSJET2, EPOS and SYBILL) show relative small difference between models for H and for the Fe showers; the effect is more pronounced

Acoustic materials displaying coupling between pressure and momentum are known as Willis materials. The simplest Willis materials are comprised of sub-wavelength scatterers that couple monopoles to dipoles and {\it vice versa}, with the interaction defined by a polarizability tensor. We propose a method for retrieving the polarizability tensor for sub-wavelength Willis acoustic scatterers using a finite set of scattering amplitudes. We relate the polarizability tensor to standard T-matrix and S-matrix scattering formalisms. This leads to an explicit method for retrieving the components of the polarizability tensor in terms of a small set of scattered pressure data in the near- or far-field. Numerical examples demonstrate the retrieval method for one and two dimensional configurations.

In an effective medium description of acoustic metamaterials, the Willis coupling plays the same role as the bianisotropy in electromagnetism. Willis media can be described by a constitutive matrix composed of the classical effective bulk modulus and density and additional cross-coupling terms defining the acoustic bianisotropy. Based on an unifying theoretical model, we unite the properties of acoustic Willis coupling with $\mathcal{PT}$ symmetric systems under the same umbrella and show in either case that an exceptional point hosts a remarkably pronounced scattering asymmetry that is accompanied by one-way zero reflection for sound waves. The analytical treatment is backed up by experimental input in asymmetrically side-loaded wavesguides showing how gauge transformations and loss biasing can embrace both Willis materials and non-Hermitian physics to tailor unidirectional reflectionless acoustics, which is appealing for purposeful sound insulation and steering.

Acoustic meta-atoms serve as the building blocks of metamaterials, with linear properties designed to achieve functions such as beam steering, cloaking and focusing. They have also been used to shape the characteristics of incident acoustic fields, which led to the manipulation of acoustic radiation force and torque for development of acoustic tweezers with improved spatial resolution. However, acoustic radiation force and torque also depend on the shape of the object, which strongly affects its scattering properties. We show that by designing linear properties of an object using metamaterial concepts, the nonlinear acoustic effects of radiation force and torque can be controlled. Trapped objects are typically small compared to the wavelength, and are described as particles, inducing monopole and dipole scattering. We extend such models to a polarizability tensor including Willis coupling terms, as a measure of asymmetry, capturing the significance of geometrical features. We apply our model to a three-dimensional, sub-wavelength meta-atom with maximal Willis coupling, demonstrating that the force and the torque can be reversed relative to an equivalent symmetrical particle. By con