Effective medium theory for elastic metamaterials in two dimensions

An effective medium theory is developed which goes beyond the quasistatic limit to accurately predict the unusual properties of certain elastic metamaterials in two dimensions. The theory's validity is numerically verified through band-structure calculations for three different elastic metamaterials. The theory shows that the effective bulk modulus κe, shear modulus μe, and mass density ρe can be made negative near resonances by choosing appropriate resonant scatterers, leading to eight possible types of wave propagation. The theory not only provides a convenient tool to search for new metamaterials with desired properties, it also gives a unified physical picture of these properties. Two examples are presented: one possesses large band gaps at low frequencies; the other exhibits two regions of negative refraction, i.e., one for both longitudinal and transverse waves and the other for longitudinal waves only.