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Space-time modulation of material parameters offers new possibilities for manipulating elastic wave propagation by exploiting time-reversal symmetry breaking. Here we propose and validate a general framework based on the multiple scattering theory to model space-time modulated elastic metamaterials, namely elastic waveguides equipped with modulated resonators. The formulation allows to consider an arbitrary distribution of resonators with a generic space-time modulation profile and compute the wavefield within and outside the resonators' region. Additionally, under appropriate assumptions, the same framework can be exploited to predict the waveguide dispersion relation. We demonstrate the capabilities of our formulation by revisiting the dynamics of two representative space-time modulated systems, e.g. the non-reciprocal propagation of (i) flexural waves along a metabeam and (ii) surface acoustic waves along a metasurface. Given its flexibility, the proposed method can pave the way towards the design of novel devices able to realize unidirectional transport of elastic energy for vibration isolation, signal processing and energy harvesting purposes.

In this work, we investigate the dynamic behavior and the topological properties of quasiperiodic elastic metasurfaces, namely arrays of mechanical oscillators arranged over the free surface of an elastic half-space according to a quasiperiodic spatial distribution. An ad-hoc multiple scattering formulation is developed to describe the dynamic interaction between Rayleigh waves and a generic array of surface resonators. The approach allows to calculate the spectrum of natural frequencies of the quasiperiodic metasurface which reveals a fractal distribution of the frequency gaps reminiscent of the Hofstadter butterfly. These gaps have nontrivial topological properties and can host Rayleigh-like edge modes. We demonstrate that such topologically protected edge modes can be driven from one boundary to the opposite of the array by a smooth variation of the phason, a parameter which modulates the geometry of the array. Topological elastic waveguides designed on these principles provide new opportunities in surface acoustic wave engineering for vibration control, energy harvesting, and lossless signal transport, among others.

We provide an analytical formulation to model the propagation of elastic waves in a homogeneous half-space supporting an array of thin plates. The technique provides the displacement field obtained from the interaction between an incident wave generated by a harmonic source and the scattered fields induced by the flexural motion of the plates. The scattered field generated by each plate is calculated using an ad-hoc set of Green's functions. The interaction between the incident field and the scattered fields is modeled through a multiple scattering formulation. Owing to the introduction of the multiple scattering formalism, the proposed technique can handle a generic set of plates arbitrarily arranged on the half-space surface. The method is validated via comparison with finite element simulations considering Rayleigh waves interacting with a single and a collection of thin plates. Our framework can be used to investigate the interaction of vertically polarized surface waves and flexural resonators in different engineering contexts, from the design of novel surface acoustic wave devices to the interpretation of urban vibrations problems.

In this work, we experimentally and numerically investigate the propagation and attenuation of vertically polarized surface waves in an unconsolidated granular medium equipped with small-scale metabarriers of different depths, i.e., arrays composed of one, two, and three embedded layers of sub-wavelength resonators. Our findings reveal how such a multi-layer arrangement strongly affects the attenuation of the surface wave motion within and after the barrier. When the surface waves collide with the barriers, the wavefront is back-scattered and steered downward underneath the oscillators. Due to the stiffness gradient of the granular medium, part of the wavefield is then rerouted to the surface level after overcoming the resonant array. Overall, the in-depth insertion of additional layers of resonators leads to a greater and broader band wave attenuation when compared to the single layer case.

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.

We propose an analytical framework to model the effect of single and multiple mechanical surface oscillators on the dynamics of vertically polarized elastic waves propagating in a semi-infinite medium. The formulation extends the canonical Lamb's problem, originally developed to obtain the wavefield induced by a harmonic line source in an elastic half-space, to the scenario where a finite cluster of vertical oscillators is attached to the medium surface. In short, our approach utilizes the solution of the classical Lamb's problem as Green's function to formulate the multiple scattered fields generated by the resonators. For an arbitrary number of resonators, arranged atop the elastic half-space in an arbitrary configuration, the displacement fields are obtained in closed-form and validated with numerics developed in a two-dimensional finite element environment.

We investigate how Rayleigh waves interact with modulated resonators located on the free surface of a semi-infinite elastic medium. We begin by studying the dynamics of a single resonator with time-modulated stiffness. In particular, we evaluate the accuracy of an analytical approximation of the resonator response and identify the parameter ranges in which its behavior remains stable. Then, we develop an analytical model to describe the interaction between surface waves and an array of resonators with spatio-temporally modulated stiffness. By combining our analytical models with full-scale numerical simulations, we demonstrate that spatio-temporal stiffness modulation of this elastic metasurface leads to the emergence of non-reciprocal features in the Rayleigh wave spectrum. Specifically, we show how the frequency content of a propagating signal can be filtered and converted when traveling through the modulated medium, and illustrate how surface-to-bulk wave conversion plays a role in these phenomena. Throughout this article, we indicate bounds of modulation parameters for which our theory is reliable, thus providing guidelines for future experimental studies on the topic.

In this article the physics of horizontally polarized shear waves travelling across a locally resonant metasurface in an unconsolidated granular medium is experimentally and numerically explored. The metasurface is comprised of an arrangement of sub-wavelength horizontal mechanical resonators embedded in silica microbeads. The metasurface supports a frequency-tailorable attenuation zone induced by the translational mode of the resonators. The experimental and numerical findings reveal that the metasurface not only backscatters part of the energy, but also redirects the wavefront underneath the resonators leading to a considerable amplitude attenuation at the surface level, when all the resonators have similar resonant frequency. A more complex picture emerges when using resonators arranged in a so-called graded design, e.g., with a resonant frequency increasing/decreasing throughout the metasurface. Unlike Love waves propagating in a bi-layer medium, shear waves localized at the surface of our metasurface are not converted into bulk waves. Although a detachment from the surface occurs, the depth-dependent velocity profile of the granular medium prevents the mode-conversion, steering again the horizontally polarized shear waves towards the surface. The outcomes of our experimental and numerical studies allow for understanding the dynamics of wave propagation in resonant metamaterials embedded in vertically inhomogeneous soils and, therefore, are essential for improving the design of engineered devices for ground vibration and seismic wave containment., Comment: Manuscript submitted to Phys. Rev. Appl