Dual audible-range band gaps in three-dimensional locally resonant phononic crystals

We describe a simple and transparent physical model for local acoustic resonances and their interactions in three-dimensional phononic crystals. The widely quoted point-mass-in-a-box representation of acoustic resonators is systematically generalized to an extended rigid body connected to massive springs, exhibiting multiple coupled local resonances. Millimeter-sized acoustic resonators over the audible frequency range typically consist of dense cores coupled to stiff shells through elastically soft material. When the local resonator is small compared to the acoustic wavelength, an elastostatic equilibrium approximation yields closed-form rational functions for its frequency-dependent, effective mass and moment of inertia. Our representation allows intuitive and quantitative analyses of the coupled acoustic modes of lattices of interacting resonators. The existence and the frequency range of local resonance band gaps are predicted by the concurrence of negative effective mass and moment of inertia. A large local resonance gap may occur in spectral proximity to a distinct gap arising from Bragg scattering. The band structure and density of states are determined by solving computationally inexpensive 6x6 matrix eigenvalue equations. These agree with the exact band structures obtained by finite-element method within 3.71%, 2.32%, and 2.38% errors for the simple cubic, body-centered cubic, and face-centered cubic arrangements of the resonators, respectively. Our model enables precise design of locally resonant phononic crystals with large dual band gaps spanning a significant fraction of the audible spectrum. By increasing the mass contrast between the core and the shell in spherical resonators, we demonstrate, using a specific phononic crystal, a local resonance band gap with 126.7% gap-to-midgap ratio. Our model is further extended to a lattice of dumbbellshaped resonators, resulting in a dense collection of flat bands over a narrow, predetermined frequency range.