Rayleigh wave propagation in nonlinear metasurfaces

TL;DR

The paper addresses amplitude-dependent Rayleigh-wave dispersion on elastic substrates coated with nonlinear surface resonators. It develops a leading-order effective-medium model, treating the resonator interaction as a cubic (Duffing-like) nonlinearity and incorporating energy loss, to derive closed-form dispersion relations that predict how hardening and softening nonlinearities shift or remove band gaps, and how damping modulates these effects. Key contributions include analytical expressions for the amplitude-dependent dispersion, characterization of hardening vs softening behavior, and demonstration of damping-driven upper bounds and potential spatial gaps, all validated by 2D FE simulations that also reveal third-harmonic generation. The results enable tunable surface-wave filtering and offer insights for seismic metasurfaces, with implications for design of adaptive wave control devices in elastic media.

Abstract

We investigate the propagation of Rayleigh waves in a half-space coupled to a nonlinear metasurface. The metasurface consists of an array of nonlinear oscillators attached to the free surface of a homogeneous substrate. We describe, analytically and numerically, the effects of nonlinear interaction force and energy loss on the dispersion of Rayleigh waves. We develop closed-form expressions to predict the dispersive characteristics of nonlinear Rayleigh waves by adopting a leading-order effective medium description. In particular, we demonstrate how hardening nonlinearity reduces and eventually eliminates the linear filtering bandwidth of the metasurface. Softening nonlinearity, in contrast, induces lower and broader spectral gaps for weak to moderate strengths of nonlinearity, and narrows and eventually closes the gaps at high strengths of nonlinearity. We also observe the emergence of a spatial gap (in wavenumber) in the in-phase branch of the dispersion curves for softening nonlinearity. Finally, we investigate the interplay between nonlinearity and energy loss and discuss their combined effects on the dispersive properties of the metasurface. Our analytical results, supported by finite element simulations, demonstrate the mechanisms for achieving tunable dispersion characteristics in nonlinear metasurfaces.

Figures (11)

• Figure 1: Schematic of the setup.
• Figure 2: Flow chart depicting the procedure to calculate the amplitude-dependent dispersive properties of the nonlinear metasurface.
• Figure 3: Panel (a): dispersion relation for Rayleigh waves with linear interaction force, $c_L/c_T=1.5$ and $m\omega_R/\rho A c_T=0.15$. The dash-dotted curve corresponds to $\omega = c_R q$, the dispersion curve for Rayleigh waves in the absence of surface resonators. The dashed curve corresponds to shear wave dispersion, $\omega = c_T q$, above which waves are not bound to the surface. The horizontal dotted lines indicate the lower bound ($\omega=\omega_R$) and the upper bound ($\omega=\omega^\star$) of the band gap. The vertical dotted line indicates $q=q^\star$, the onset of the out-of-phase branch of the dispersion curve. Panel (b): steady-state dynamics of an undamped surface resonator, $|Y/B_W|$, as a function of the frequency of the incoming Rayleigh wave, $\omega$, with linear ($\beta=0$) and nonlinear ($\beta=\pm 1$) interaction forces. The incoming wave amplitude is $B_W=0.005$ for both the hardening and softening types of nonlinearity. The unstable portions of the response curves are depicted using dashed lines. The vertical dash-dotted line indicates $\omega=\omega^\star$.
• Figure 4: Panel (a): influence of hardening, cubic nonlinearity ($\beta=1$, $k_2=0$) on the dispersion of Rayleigh waves for different values of the incoming wave amplitude, $B_W$. The onset of the out-of-phase branch remains almost unchanged for small values of $B_W$. The inset highlights the effect of nonlinearity on the onset of the out-of-phase branch at higher values of $B_W$. Panel (b): influence of softening, cubic nonlinearity ($\beta=-1$, $k_2=0$) on the dispersion of Rayleigh waves for different values of the incoming wave amplitude, $B_W$. The onset of the out-of-phase branch remains almost unchanged for small values of $B_W$. The inset highlights the effect of nonlinearity on onset of the out-of-phase branch at higher values of $B_W$. In both the panels, the linear dispersion curve from Fig. \ref{['f:disp_linear']} is included for comparison and the unstable portions of the response curves are depicted using dashed lines.
• Figure 5: Influence of energy loss on the linear dispersion of Rayleigh waves for increasing values of damping ratio. Panels (a) and (b) show, respectively, the real and imaginary parts of the dispersion curves. The imaginary part is shown with a negative sign. Notice that the dispersion curve of the undamped system ($\zeta=0$) is real-valued.