On bandgap sensitivity to three-to-one internal resonances between acoustic and optical waves in metamaterials

We investigate the nonlinear equations governing wave propagation across a metamaterial consisting of a cellular periodic structure hosting resonators with linear and cubic springs. The resulting system of two coupled equations with cubic nonlinearity is Hamiltonian, with the origin being an elliptic equilibrium characterized by two distinct linear frequencies associated with the acoustic and optical wave modes. Understanding this complex wave propagation problem requires the explicit derivation of analytical formulas for the nonlinear frequencies as functions of the relevant physical parameters. In the small wave amplitude regime and away from internal resonances, by using Hamiltonian Perturbation Theory, we obtain the first-order nonlinear correction to the linear frequencies. Additionally, we analytically estimate the threshold for the applicability of the perturbative solution, identifying the maximum admissible amplitude at which the obtained formulas remain valid. This threshold decreases significantly in the presence of internal resonances, particularly when the ratio between the optical and acoustic frequencies is close to 3 due to a 3:1 resonance which affects the correction. Furthermore, we analytically evaluate the remainder after the first step of the perturbation scheme. The specific application deals with the wave propagation equations obtained for metamaterial honeycombs with periodically distributed nonlinear one-dof resonators. Our analysis provides quantitative insights into how internal resonances affect the bandgap. Specifically, we identify a 3:1 resonant curve in the parameter space represented by the resonator modal mass and stiffness. We also identify a region away from the above resonant curve where the nonlinear bandgap for softening resonators springs is significantly larger than the linear counterpart.