Reciprocity, passivity and causality in Willis materials

Abstract

Materials that require coupling between the stressstrain and momentum-velocity constitutive relations were first proposed by Willis (Willis 1981 Wave Motion 3, 1-11. (doi:10.1016/0165-2125(81)90008-1)) and are now known as elastic materials of the Willis type, or simply Willis materials. As coupling between these two constitutive equations is a generalization of standard elastodynamic theory, restrictions on the physically admissible material properties for Willis materials should be similarly generalized. This paper derives restrictions imposed on the material properties of Willis materials when they are assumed to be reciprocal, passive and causal. Considerations of causality and low-order dispersion suggest an alternative formulation of the standard Willis equations. The alternative formulation provides improved insight into the subwavelength physical behaviour leading to Willis material properties and is amenable to time-domain analyses. Finally, the results initially obtained for a generally elastic material are specialized to the acoustic limit.

Acoustic meta-atom with experimentally verified maximum Willis coupling

“…Willis coupling is a term in the acoustic and elastic constitutive relations that couples potential and kinetic energy [11][12][13], analogous to the bianisotropy parameter in electromagnetism [14,15]. The Willis coupling and bianisotropy parameters can only be non-zero in structures which lack mirror symmetry about one of their major axes, and their inclusion into the constitutive relations has been shown to resolve violations of causality and passivity in metamaterial homogenization [16,17]. Recent work has demonstrated that the incorporation of Willis coupling or bianisotropy into metamaterial structures of sub-wavelength thickness, known as metasurfaces, can improve their efficiency when refracting at large angles [18][19][20].…”

Retrieval method for the bianisotropic polarizability tensor of Willis acoustic scatterers

“…By proper design of the inclusions one can achieve a variety of exotic effective material properties such as negative bulk modulus and density, 1-3 zero index, 4,5 anisotropic and pentamode properties. [6][7][8] Numerous applications have been demonstrated, e.g. acoustic cloaking, 9,10 negative refraction, 11,12 and nonreciprocal transmission.…”

“…13,14 The medium of particular interest in this paper is known as a Willis material, which requires strain-velocity cross-coupling in elasticity or pressure-velocity coupling in acoustics. [15][16][17][18][19] It is well-known that monopole and dipole moments are dominant sources of the scattered field for subwavelength scatterers. For Willis type acoustic scatterers, the scattering requires cross-coupling between pressure and velocity fields.…”

“…Phases of the transmitted and reflected waves for the two incident directions calculated using Eq (19)…”

Willis Metamaterial on a Structured Beam

“…Another attempt is the proposal to realize Willis media 28 . These media introduce new constitutive terms not only in Hooke's law but also in Newton's second law [29][30][31][32][33] . These terms, which introduce coupling between stress and velocity and between momentum and strain, are typically small perturbations and difficult to realize.…”