Wave propagation and multi-stopband behavior of metamaterial lattices with nonlinear locally resonant membranes

Wave propagation and stopband behavior in square lattices hosting nonlinear resonators made of suspended piezoelectric membranes with a central mass are investigated by treating asymptotically the wave propagation equations obtained via a projection method combined with the Floquet-Bloch theorem. The metamaterial membrane resonators made of a piezoelectric polymer are designed to exploit the piezoelectric effect which regulates the membrane stretching thus shifting the frequencies and, in turn, the bandgaps. Moreover, the resonators multi -modal behavior is tailored for its potential of generating multiple bandgaps for enhanced wave propagation control. The nonlinear dispersion functions of the metamaterial are obtained in closed form, and a numerical parametric analysis offers important hints on the design optimization. Numerical validations are also presented to verify the soundness of the analytical solutions and highlight the potential application of the semi -adaptively programmable metamaterial for wave propagation control.