Willis Coupling Research Articles

Bianisotropic acoustic metasurfaces for wavefront transformation

Acoustic metasurfaces are thin, engineered structures that can control the local reflection and transmission phase of acoustic waves, and thus enable a high degree of flexibility for wave manipulation. Here, we describe our recent work on so-called perfect metasurfaces, in which elements are designed that can control the local transmission and reflection amplitude and phase response of the metasurface. Controlling both the amplitude and the resulting asymmetric phase response requires a fundamentally asymmetric surface impedance for the unit cells. The resulting ideal surface properties thus have strong links to the concept of bianisotropy in electromagnetics and Willis coupling in elastodynamics. We have developed a shunt resonator-based element design that contains the needed degrees of freedom to independently control the surface impedance on both sides of the metasurface. Using this perfect metasurface design approach, we demonstrate the design and experimentally measurement of high efficiency wide-angle transmissive beam steering at levels beyond what is possible with phase control alone.

Guided waves at bianisotropic fluid interfaces

Willis fluids are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This effective dynamic response coupling may arise due to microstructural asymmetry, long range order, or time-varying material properties and has been shown to be analogous to electromagnetic bianisotropic media [Phys. Rev. B 96, 104303 (2017)]. In this study, we report on the existence of guided waves at the interface between two fluids when at least one displays Willis coupling. Criteria for the existence of these waves are discussed in terms of the material properties, frequency, and wave number, and expressions for the dispersion relation and rate of spatial decay away from the interface are obtained analytically. We demonstrate that interface waves are supported when one of the fluids possesses Willis coupling, in contrast to an interface between two classical isotropic fluids, which cannot support interface waves. Special cases are highlighted via numerical examples. [Work supported by ONR, NSF, the Applied Research Laboratories at The University of Texas at Austin, and the NRC Research Associateship Program.]

Willis coupling in underwater elastic Helmholtz resonators

Helmholtz resonators (HRs) have been central in many acoustic metamaterial devices. For example, one can use an array of HRs to obtain negative effective bulk modulus or a panel of HRs to achieve total absorption of low-frequency sound. The aforementioned applications are based on the local monopolar resonance of each HR. However, the HR behaves differently in water. The elastic modulus of the resonator wall can become comparable to the modulus of water making the resonant frequency much lower than the rigid wall case. Moreover, considering the wall elasticity and mass leads to large structural asymmetry and induces cross-coupling between pressure and velocity fields. In this talk, we provide a lumped element model in order to predict the resonant frequency of the elastic HR. The model is demonstrated by sound scattering from an elastic HR in the context of acoustic bianisotropy, or Willis behavior, following a recent paper [Li Quan et al., Phys. Rev. Lett., 2018]. It is found that both the pressure and velocity fields can generate monopole and dipole responses from an elastic HR. The explicit example of an elastic HR in a one-dimensional waveguide will be discussed.Helmholtz resonators (HRs) have been central in many acoustic metamaterial devices. For example, one can use an array of HRs to obtain negative effective bulk modulus or a panel of HRs to achieve total absorption of low-frequency sound. The aforementioned applications are based on the local monopolar resonance of each HR. However, the HR behaves differently in water. The elastic modulus of the resonator wall can become comparable to the modulus of water making the resonant frequency much lower than the rigid wall case. Moreover, considering the wall elasticity and mass leads to large structural asymmetry and induces cross-coupling between pressure and velocity fields. In this talk, we provide a lumped element model in order to predict the resonant frequency of the elastic HR. The model is demonstrated by sound scattering from an elastic HR in the context of acoustic bianisotropy, or Willis behavior, following a recent paper [Li Quan et al., Phys. Rev. Lett., 2018]. It is found that both the pressure and v...

Acoustic scattering by a Willis-coupled fluid cylinder

We present an integral equation based approach to analyze an acoustic scattering from fluid cylinders exhibiting Willis coupling. The Willis-coupled cylinder is characterized by a bulk modulus and mass density as well as a coupling vector. The coupling vector relates the pressure and momentum density to the volume strain and particle velocity simultaneously. We consider integral representations obtained using the free-space Green's function for the exterior fluid medium. The integral equations are evaluated using the T-matrix method. The computation of scattered pressure wave due to an obliquely incident wave on bi-anisotropic fluid cylinder immersed in an external homogeneous fluid will be described. The method will be illustrated giving examples for fluid Willis cylinders.

Acoustic focusing via guided waves in bianisotropic fluid layers

Willis fluids display a complex, effective dynamic response stemming from microstructural asymmetry, long range order, and/or time-varying material properties. These materials are characterized by constitutive relations that couple the pressure and momentum density to both the particle velocity and the volume strain. This coupling has been shown to be analogous to electromagnetic bianisotropic media, which exhibit coupled electric and magnetic fields [Phys. Rev. B 96, 104303 (2017)]. In a recent study, an acoustic lens composed of an array of Willis fluid layers was demonstrated, where the pressure phase imparted by each array element was tuned by varying the direction of Willis polarization and was determined using finite element analysis. In this work, we report on analytical solutions for guided waves in layers of fluid displaying Willis coupling and derive dispersion relations for layers with rigid and pressure-release boundary conditions. Special cases highlighting the effect of Willis polarization direction are presented via numerical examples. The acoustic lens is revisited with array phases obtained from analytical solutions, eliminating the need for finite element methods. Lenses designed for focusing and steering incident pressure waves are demonstrated using fully analytical models. [Work supported by NSF, ONR, and ARL:UT.]

Maximum Willis Coupling in Acoustic Scatterers.

Willis coupling in acoustic materials defines the cross-coupling between strain and velocity, analogous to bianisotropic phenomena in electromagnetics. While these phenomena have been garnering significant attention in recent years, to date their effects have been considered mostly perturbative. Here, we derive general bounds on the Willis response of acoustic scatterers, show that these bounds can be reached in suitably designed scatterers, and outline a systematic venue for the realistic implementation of maximally bianisotropic acoustic inclusions. We then employ these inclusions to realize acoustic metasurfaces for bending and steering of sound with unitary efficiency.

Bi-layer plate-type acoustic metamaterials with Willis coupling

Dynamic effective negative parameters are principal to the representation of the physical properties of metamaterials. In this paper, a bi-layer plate-type unit was proposed with both a negative mass density and a negative bulk modulus; moreover, through analysis of these bi-layer structures, some important problems about acoustic metamaterials were studied. First, dynamic effective mass densities and the bulk modulus of the bi-layer plate-type acoustic structure were clarified through both the direct and the retrieval methods, and, in addition, the intrinsic relationship between the sound transmission (absorption) characteristics and the effective parameters was analyzed. Furthermore, the properties of dynamic effective parameters for an asymmetric bi-layer acoustic structure were further considered through an analysis of experimental data, and the modified effective parameters were then obtained through consideration of the Willis coupling in the asymmetric passive system. In addition, by taking both the clamped and the periodic boundary conditions into consideration in the bi-layer plate-type acoustic system, new perspectives were presented for study on the effective parameters and sound insulation properties in the range below the cut-off frequency. The special acoustic properties established by these effective parameters could enrich our knowledge and provide guidance for the design and installation of acoustic metamaterial structures in future sound engineering practice.

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

In addition to conventional acoustic properties (mass density and compressibility), recent research has shown that inhomogeneous acoustic media may require coupling tensors between effective volume strain and momentum fields. These coupling tensors imply conversion between monopolar and dipolar motion in the effective material response. Intrinsic coupling between strain and momentum is known as Willis coupling in elastodynamics, and it is analogous to bianisotropy in electromagnetism. This paper investigates the microscale and mesoscale effects that lead to Willis coupling, studies the proper homogenization of acoustic metamaterials with non-negligible Willis coupling, and demonstrates the impact of Willis coupling on defining physically meaningful effective material parameters.

Experimental evidence of Willis coupling in a one-dimensional effective material element

The primary objective of acoustic metamaterial research is to design subwavelength systems that behave as effective materials with novel acoustical properties. One such property couples the stress–strain and the momentum–velocity relations. This response is analogous to bianisotropy in electromagnetism, is absent from common materials, and is often referred to as Willis coupling after J.R., Willis, who first described it in the context of the dynamic response of heterogeneous elastic media. This work presents two principal results: first, experimental and theoretical demonstrations, illustrating that Willis properties are required to obtain physically meaningful effective material properties resulting solely from local behaviour of an asymmetric one-dimensional isolated element and, second, an experimental procedure to extract the effective material properties from a one-dimensional isolated element. The measured material properties are in very good agreement with theoretical predictions and thus provide improved understanding of the physical mechanisms leading to Willis coupling in acoustic metamaterials.

A discussion of macroscopic properties for acoustic metamaterials: Models and measurements

Macroscopic material properties are useful to describe long wavelength dynamics of inhomogeneous media. Analytically, these properties are often determined by weighted field averages, which define the effective fields of a representative unit cell. The relations between these effective fields provide macroscopic properties. In addition to traditional properties (wavenumber, impedance, density, and compressibility), recent research has shown that inhomogeneous media require coupling parameters between effective volume-strain and momentum fields, known as Willis coupling or bianisotropy. However in the absence of embedded sources, metamaterial properties are non-unique allowing for macroscopic descriptions which only include traditional properties or traditional properties and coupling parameters. Many acoustic metamaterial measurements extract macroscopic properties using reflection and transmission coefficients of finite samples. Unfortunately, this widely used technique returns properties which relate boundary fields, not effective fields as usually assumed. Even though boundary fields may well approximate effective fields for very long wavelengths, the extracted properties are at best Bloch properties of a periodic medium and in general only apply to the exact measurement setup. The aim of this talk is to provide discussion on the issues of non-uniqueness and measurements of macroscopic properties in light of the importance of physically meaningful properties.

Scattering from a fluid cylinder with strain-momentum coupled constitutive relations

Many important applications of acoustics are associated with the principle of scattering. For example, biomedical ultrasound and sonar both make use of acoustic field scattering for localization, imaging, and identification of objects. While the theory of acoustic scattering from fluid and elastic materials is well established and has been validated with numerical and physical experiments, no work has been published to describe scattering from a more general class of acoustic materials known as Willis materials. Willis materials are characterized by a bulk modulus and mass density as well as a vector that couples the pressure-strain relationship with the momentum density-particle velocity relationship. The coupling vector is the result of microstructural asymmetry. We present a theoretical description of acoustic scattering of a plane wave incident upon a cylinder exhibiting weak Willis coupling using a perturbation approach. The scattered field depends upon the orientation of the Willis coupling vector and is therefore anisotropic despite the symmetry of the geometry. The analytical model is validated through comparison with a finite element-based numerical experiment.

Analysis of one-dimensional wave phenomena in Willis materials

The primary benefit of treating a complicated acoustic system as an acoustic metamaterial (AMM) is that once the effective material properties are determined, well-established mathematical analyses may be used to describe wave propagation within the system. However, many standard analyses are not well understood for the class of materials known as Willis materials. Willis materials are characterized by constitutive relations that couple both the pressure and momentum density to both the particle velocity and the volume strain. This work presents the mathematical analysis of the propagation of a velocity pulse of finite duration within a one-dimensional Willis material in the time-domain. In particular, the propagation of the pulse is described in the context of (i) an infinite Willis material, (ii) two half-spaces where one or both display Willis coupling, and (iii) a thin coupled partition in an uncoupled background. Willis coupling is shown to affect the relationship between incident and scattered waves via a convolution rather than a simple multiplication, as is the case with uncoupled media.

Acoustic wave phenomena in Willis metamaterials

The design of acoustic metamaterials requires accurate modeling of the dynamics present at all relevant time and length scales. Recent work has shown that the effective dynamic properties of inhomogeneous elastic materials can result in constitutive relations that couple strain to momentum and velocity to stress, which is often referred to as Willis coupling [Willis, Wave Motion, 3(1), 1-11, (1981)]. The current work will examine macroscale acoustic propagation for waves on different time scales in a material by solving the coupled first-order equations of motion with the constitutive relations that account for Willis coupling. Specifically, second-order perturbation theory will be used to examine the classic problem of a high-frequency, low-amplitude “signal” wave superposed on a low-frequency, high-amplitude “pump” wave. Of particular interest is the slowly changing momentum bias generated by the pump wave and implications on dynamic control of signal wave propagation. Analysis and discussion will be restricted to one-dimensional wave motion.

Modulated phononic crystals: Non-reciprocal wave propagation and Willis materials

Research on breaking time-reversal symmetry in wave phenomena is a growing area of interest in the field of phononic crystals and metamaterials aiming to realize one-way propagation devices which have many potential technological applications. Here we investigate wave propagation in phononic crystals, periodic laminates in particular, where both elastic moduli and mass density are modulated in space and time in a wave-like fashion. The modulation introduces a bias which breaks time-reversal symmetry and reciprocity. A full characterization of how the dispersion curve transforms due to wave-like modulations is given in analytical and geometrical terms for both low (subsonic) and high (supersonic) modulation speeds. Theoretical findings are supported by numerical simulations. More specific to low frequencies, the macroscopic constitutive law of 1, 2 and 3D modulated laminates is proven to be of the Willis type with a non-negligible Willis coupling in the strictly scale-separated homogenization limit. The existence of a macroscopic stress-velocity and momentum-strain Willis coupling is in fact directly related to the breaking of reciprocity. Finally, closed form expressions of the macroscopic constitutive parameters are obtained and some elementary yet insightful energy bounds are derived and discussed.

Experimental measurement of the Willis coupling coefficient in a one-dimensional system

The primary objective of acoustic metamaterial research is to design subwavelength systems that behave as effective materials with novel acoustical properties. Willis coupling is one such property. Initially described by J. R. Willis [Wave Motion 3, pp. 1-11 (1981)], subwavelength structural asymmetry and/or nonlocal effects couple the stress-strain relation to the momentum-velocity relation. This coupling is absent in typical materials. While various theoretical models have predicted the existence of Willis coupling, experimental observation of Willis coupling has not, to our knowledge, been reported in the literature. We present here the first experimental evidence of Willis coupling in a one-dimensional unit cell consisting of a 0.12-mm-thick polyimide membrane under tension, a 5.9-mm-thick layer of air, and a perforated sheet of 0.45-mm-thick paper (perforation surface fraction 0.48, circular holes 3.1 mm in diameter). The properties of the unit cell were extracted using reflection and transmission da...

Iterative approaches to extend dilute homogenization predictions of Willis materials to higher volume fractions

Dynamic homogenization of generally heterogeneous elastic materials leads to three different effective material properties: stiffness, density, and parameters that couple momentum to strain and stress to velocity. This formulation was first proposed by Willis [Wave Mot. 3, 1–11, (1981)] and is thus commonly referred to as Willis coupling. While there are some homogenization methods which account for Willis coupling, those models rely on either the assumption of periodic media or on dilute concentrations of coupled inhomogeneities in an otherwise homogeneous matrix. This work presents the use of dilute models with iterative micromechanical methods to predict general material properties in materials with no long-range order and elevated volume fractions of inhomogeneity. Two such methods, self-consistent and differential effective medium models, will be presented for the homogenization of composites of arbitrary anisotropy and Willis coupling resulting from the presence of Willis coupled inclusions in an el...