Willis coupling in acoustic and elastic metamaterials
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An acoustic metamaterial is a mechanical or material system with a subwavelength structure designed to generate specific effective material properties in the homogenization limit. Most acoustic metamaterials are assumed to be properly and fully described using two material properties: mass density and stiffness. However, Willis showed that for general elastodynamic homogenization theory another material property emerges [Willis, Wave Motion 3 (1981)]. This additional material property is necessary to describe what have come to be known as Willis materials, and couples the stress and momentum equations in a phenomenon called Willis coupling. The purpose of this dissertation is to expand the understanding of Willis coupling. First, a derivation of the restrictions on the range of elastic Willis material properties imposed by assuming reciprocity, passivity, and causality is presented. Reciprocity is found to impose symmetry conditions on the Willis coupling coefficients and passivity is found to impose bounds on the range of the real part of the Willis coupling coefficients, for Willis constitutive relations in the standard form. Causality is used to suggest an alternative, more physically meaningful, formulation of a Willis material's constitutive equations. A physical interpretation of local Willis coupling is then presented in the context of one-dimensional systems. Local Willis coupling is shown to arise from systems with inherently asymmetric microstructures, which leads to a misalignment of the centroid of an effective material element and its center of mass. These examples also present a new theoretical method for determining the effective material properties of such systems, which is then compared with experimentally determined effective material properties and found to be in good agreement. A method for determining macroscopic effective material properties of a three-dimensional elastic matrix material with small, randomly distributed inclusions is then derived. While approximate, this homogenization method is very general, accounting for Willis coupling among other phenomena. This homogenization method demonstrates that subwavelength elements exhibiting Willis coupling may lead to macroscopic Willis coupling, and it provides a means to quantify this effect. Finally, a summary of the results of this research is presented, and topics related to Willis coupling that should receive further research are discussed.