Acoustic modes of locally resonant phononic crystals: Comparison with frequency-dependent mass models

Locally resonant acoustic materials were proposed to control sound using structures with feature size orders of magnitude smaller than the acoustic wavelength in air. We analytically derive the effective, frequency-dependent mass densities of resonant oscillators consisting of a heavy mass within a light shell, embedded in foam, in two dimensions, using a rigid core-shell approximation. The effective mass is expressible in closed form by elementary functions, enabling a mapping of the low-frequency physics to a simple one-dimensional model involving a point mass harmonically coupled within a box. The acoustic band structure of a two-dimensional square lattice of these oscillators is evaluated in the effective, frequency-dependent mass density model and compared with the exact solution. For the out-of-plane oscillation within this two-dimensional phononic crystal, the nonlinear eigenvalue equation resulting from the frequency-dependent density is solved with the Cutting Surface Method to yield the acoustic band structure. This agrees with the exact band structure obtained by finite element method calculations within $2%$ numerical error. For the in-plane oscillations involving internal translations or rotations within the core-shell resonators, the one-dimensional mass-in-a-box model recaptures the exact acoustic modes at only certain high-symmetry points of the two-dimensional Brillouin zone. The description of more complex resonances within acoustic metamaterials requires a fundamental generalization of current effective mass models.