Wave propagation properties of one-dimensional acoustic metamaterials with nonlinear diatomic microstructure

Acoustic metamaterials are artificial microstructured media, typically characterized by a periodic locally resonant cell. The cellular microstructure can be functionally customized to govern the propagation of elastic waves. A one-dimensional diatomic lattice with cubic inter-atomic coupling—described by a Lagrangian model—is assumed as minimal mechanical system simulating the essential undamped dynamics of nonlinear acoustic metamaterials. The linear dispersion properties are analytically determined by solving the linearized eigenproblem governing the free wave propagation in the small-amplitude oscillation range. The dispersion spectrum is composed by a low-frequency acoustic branch and a high-frequency optical branch. The two frequency branches are systematically separated by a stop band, whose amplitude is analytically derived. Superharmonic 3:1 internal resonances can occur within a wavenumber-dependent locus defined in the mechanical parameter space. A general asymptotic approach, based on the multiple scale method, is employed to determine the nonlinear dispersion properties. Accordingly, the nonlinear frequencies and waveforms are obtained for the two fundamental cases of non-resonant and superharmonically 3:1 resonant or nearly resonant lattices. Moreover, the invariant manifolds associated with the nonlinear waveforms are parametrically determined in the space of the two principal coordinates. Finally, some examples of non-resonant and resonant lattices are selected to discuss their nonlinear dispersion properties from a qualitative and quantitative viewpoint.