Elastic Metamaterials and Dynamic Homogenization: A Review

TL;DR

This review discusses recent progress in elastic metamaterials, focusing on their unusual properties, homogenization theories like Willis relations, and applications such as transformation elastodynamics inspired by optics.

Contribution

It synthesizes recent developments in understanding and applying elastic metamaterials, emphasizing the role of homogenization and transformation techniques.

Findings

Willis relations describe inhomogeneous elastodynamics.

Metamaterials exhibit negative density and tensorial properties.

Transformation elastodynamics connects metamaterials with optical principles.

Abstract

In this paper we review the recent advances which have taken place in the understanding and applications of acoustic/elastic metamaterials. Metamaterials are artificially created composite materials which exhibit unusual properties which are not found in nature. We begin with presenting arguments from discrete systems which support the case for the existence of unusual material properties such as tensorial and/or negative density. The arguments are then extended to elastic continuums through coherent averaging principles. The resulting coupled and nonlocal homogenized relations, called the Willis relations, are presented as the natural description of inhomogeneous elastodynamics. They are specialized to Bloch waves propagating in periodic composites and we show that the Willis properties display the unusual behavior which is often required in metamaterial applications such as the Veselago lens. We finally present the recent advances in the area of transformation elastodynamics, charting its inspirations from transformation optics, clarifying its particular challenges, and identifying its connection with the constitutive relations of the Willis and the Cosserat types.