Analytic model of phononic crystals with local resonances

A simple analytic model is presented to describe the low-frequency effective mass densities of three-component phononic crystals with local resonances. We show that the effective mass densities can turn negative close to the local resonances. Expressions for the effective mass densities are derived for both three-dimensional systems with coated spheres embedded in a host matrix, and two-dimensional systems with coated cylinders embedded in a host matrix.