Nonlinear plane-wave expansion method for analyzing dispersion properties of piezoelectric metamaterial lattices with encapsulated resonators

square lattices encapsulating identical nonlinear resonators in each cell are investigated through an asymptotic treatment of the wave propagation equations. The nonlinear effects of the resonators, composed of suspended piezoelectric membranes with a central mass, are investigated through the introduction of a generalized nonlinear version derived from the plane-wave expansion (PWE) method. This method leads to nonlinear wave propagation equations and the analytical derivation of nonlinear dispersion functions using the method of multiple scales. Numerical simulations verify the validity of the analytical solutions. The proposed nonlinear PWE method is shown to overcomes the limitations of the popular approach based on the enforcement of the Floquet-Bloch theorem in the context of the cell projection method. While the latter provides the dispersion curves of the fundamental propagation mode, the nonlinear PWE delivers the nonlinear dispersion curves of all modes, offering a broader perspective into the design process for semi-adaptively programmable metamaterials aimed at controlling wave propagation.