Attenuation of Rayleigh waves by a nonlinear metamaterial with serial-connected resonators

There has been a rising interest in utilizing metamaterials to manipulate the propagation of surface waves, including Rayleigh waves. These novel materials have a diverse range of applications, from micro-scale sensors and actuators to macro-scale seismic protection systems. In the field of seismic engineering, they are commonly referred to as seismic metamaterials. While various linear seismic metamaterials have been developed, incorporating nonlinearity into the design of seismic metamaterials could reveal novel phenomena and greatly expand their potential applications. In the present study, we propose a nonlinear metamaterial to block the propagation of low-frequency Rayleigh waves. The proposed metamaterial consists of a linear elastic substrate and nonlinear resonant units periodically attached to the surface of the substrate. Each resonant unit has two Duffing oscillators connected in series. We use the first-order harmonic balance method to derive analytical solutions for the dispersion of Rayleigh waves, considering both linear and various nonlinear cases. Our findings demonstrate that by coupling the motion of an elastic substrate with the dynamics of attached masses, a linear metamaterial with serial-connected resonators can achieve two band gaps. Furthermore, introducing softening nonlinearity can facilitate the attainment of a low-frequency band gap, while introducing hardening nonlinearity may result in the closure of the original linear band gaps. Our study broadens the range of applications for elastic metamaterials and potentially contributes to the development of more effective seismic wave blockers.