Effective medium theory for elastic metamaterials in two dimensions

TLDR

The study develops an effective medium theory to predict unusual properties of two‑dimensional elastic metamaterials beyond the quasistatic limit. The theory extends beyond the quasistatic limit, enabling accurate modeling of elastic metamaterials in two dimensions. Numerical band‑structure calculations confirm that the theory predicts negative effective bulk modulus, shear modulus, and mass density near resonances, yielding eight wave‑propagation types, large low‑frequency band gaps, and dual negative‑refraction regimes in two example metamaterials.

Abstract

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 ${\ensuremath{\kappa}}_{e}$, shear modulus ${\ensuremath{\mu}}_{e}$, and mass density ${\ensuremath{\rho}}_{e}$ 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.