A Dynamical Effective Medium Theory for Elastic Metamaterials

We develop a dynamical effective medium theory to accurately predict the unusual properties of elastic metamaterials in two dimensions near the resonant frequencies. The theory shows that the effective bulk modulus, shear modulus, and mass density can be made negative by choosing proper resonant scatterers, leading to 8 possible types of wave propagation. The theory not only provides a convenient tool to search for various metamaterials with desired properties, but also gives a unified physical picture of these properties. Here we demonstrate two examples. One possesses large band gaps at low frequencies. The other exhibits two regions of negative refraction: one for both longitudinal and transverse waves and the other for longitudinal waves only.