Multiple scattering of local nonlinear resonators on a thin plate

Developing nonlinear resonant metamaterial plates requires an effective wave simulation method to analyze the wave interactions between multiple nonlinear scatterers. The multiple scattering method has the advantages of high computational efficiency and closed-form displacement solutions in modeling the finitely periodic scatterer array. However, the multiple scattering method of nonlinear scatterers is absent in the available studies of plate structures. In this paper, a nonlinear multiple scattering approach is formulated and analyzed for thin plates with nonlinear resonant scatterers. The resonant scatterer consists of an inner plate with a local resonator modeled by the nonlinear mass-spring system. The motion equation of resonant scatterers is solved using the perturbation technology and wave expansion method. The T-matrix of the resonant scatterer of each order is then derived at the arbitrary harmonic frequency from continuous interfacial conditions of resonant scatterers. Finally, the multiple scattering is formulated for a finitely periodic array of resonant scatterers on a thin plate. Numerical simulations capture the scattering characteristics and stop-band formations of fundamental and second-harmonic waves. Furthermore, the finitely periodic array of nonlinear and linear resonant scatterers achieves the nonreciprocal transmission of flexural waves. These numerical results demonstrate the potential applications of the proposed method in designing metamaterial-based plates with complex transmission properties.