materials
Article Hybridized Love Waves in a Guiding Layer Supporting an Array of Plates with Decorative Endings
Kim Pham 1,*, Agnès Maurel 2 , Simon Félix 3 and Sébastien Guenneau 4 1 IMSIA, ENSTA ParisTech, 828 Bd des Maréchaux, 91732 Palaiseau, France 2 Institut Langevin, CNRS, ESPCI ParisTech, 1 rue Jussieu, 75005 Paris, France; [email protected] 3 LAUM, CNRS UMR 6613, Le Mans Université, avenue Olivier Messiaen, 72085 Le Mans, France; [email protected] 4 Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, 13013 Marseille, France; [email protected] * Correspondence: [email protected]
Received: 11 November 2019; Accepted: 27 December 2019; Published: 1 April 2020
Abstract: This study follows from Maurel et al., Phys. Rev. B 98, 134311 (2018), where we reported on direct numerical observations of out-of-plane shear surface waves propagating along an array of plates atop a guiding layer, as a model for a forest of trees. We derived closed form dispersion relations using the homogenization procedure and investigated the effect of heterogeneities at the top of the plates (the foliage of trees). Here, we extend the study to the derivation of a homogenized model accounting for heterogeneities at both endings of the plates. The derivation is presented in the time domain, which allows for an energetic analysis of the effective problem. The effect of these heterogeneous endings on the properties of the surface waves is inspected for hard heterogeneities. It is shown that top heterogeneities affect the resonances of the plates, hence modifying the cut-off frequencies of a wave mathematically similar to the so-called Spoof Plasmon Polariton (SPP) wave, while the bottom heterogeneities affect the behavior of the layer, hence modifying the dispersion relation of the Love waves. The complete system simply mixes these two ingredients, resulting in hybrid surface waves accurately described by our model.
Keywords: metamaterial; homogenization; elastic metasurface; time domain analysis; elastic energy
1. Introduction The problem of waves propagating in an elastic half-space supporting an array of beams or plates is well known in seismology, where the site–city interaction aims at understanding the interaction of seismic waves with a set of buildings. Starting with the seminal work of Housner [1] (see also [2]), the site–city interaction has been intensively studied numerically [3–5] and analytically [6–11]. In this context, seismic shields, or metabarriers, have been considered using resonators buried in soil [12–15] or arrays of trees with a gradient in their heights [16–18]. More generally, this configuration is the elastic analog of a corrugated interface able to support surface waves, studied in acoustics [19] and in electromagnetism [20,21], where they are known as Spoof Plasmon Polaritons (SPPs). SPPs play a fundament role in the extraordinary transmission of long wavelength electromagnetic waves through metallic gratings [22,23] and have been studied intensively in the past twenty years for their potential applications in subwavelength optics, data storage, light generation, microscopy, and bio-photonics; see, e.g., [24]. Such similarities between surface waves in electromagnetism and elastodynamics fuel research in seismic metamaterials [25], as they lead to simplified models that see behind the tree that hides the forest [26].
Materials 2020, 13, 1632; doi:10.3390/ma13071632 www.mdpi.com/journal/materials Materials 2020, 13, 1632 2 of 27
To describe classical SPPs, the homogenization of a stratified medium is an easy and efficient tool [27,28]; the analysis is valid in the low frequency regime, namely owing to the existence of a small parameter measuring the ratio of the array spacing to the typical wavelength, and it provides, at the dominant order, the dispersion relation of SPPs. Thanks to the mathematical analogy between the problem in electromagnetism and in elasticity, this approach was applied in [18] accounting for the presence of a guiding soil layer underlain by an elastic half-space. Simple dispersion relations have been obtained from the effective model for the resulting spoof Love waves, so-called because of the characteristics they share with classical Love waves (surface waves supported by the layer on its own) and SPPs. Next, to account for the presence of heterogeneities (a foliage) at the top of the plates (trees), a hybrid model was used where the homogenization was performed locally (near the top of the plates) at the second order. The present study generalizes and complements this study following two ways: (i) from a physical point of view, we include the effect of heterogeneities at the bottom of the plates (Figure1), and (ii) from a technical point of view, we derive the full model at second order. This produces a significant increase in the accuracy of the theoretical prediction: in the reported examples, the model at order two is accurate up to a 1–2% error margin, while the model previously used in [18], at order one, would be accurate up to 10–30%. The second order model (see Equations (2) and (3)) provides a one-dimensional problem along the z-direction with a succession of homogeneous layers: the substrate occupying a half-space, the guiding layer, and an effective anisotropic layer replacing the region of the plates (see Figure2). The effect of the heterogeneities at the bottom is encapsulated in transmission conditions, which tell us that the displacement and the normal stress are not continuous; this holds for plates without ending heterogeneities, a fact that was disregarded in [18]. The effect of the heterogeneities at the top is encapsulated in a boundary condition that differs from the usual stress free condition, as in [18]. We recover that for most of the frequencies, the plates do not interact efficiently with the layer; in the present case, it results that the surface wave resembles that of the layer only, hence a wave of the Love type. However, the resonances of the plates produce cut-off frequencies around which the dispersion relations are deeply affected. For simple plates, this can already produce drastic modifications in the dispersion relations (hybridization of the Love branches, avoided crossings at the cut-off frequencies of the plates). When heterogeneities at the endings of the plates are accounted for, additional changes happen. The heterogeneities at the bottom of the plates modify the behavior of the layer on its own, resulting in modified Love waves. The heterogeneities at the top of the plates modify the resonances of the plates, hence the cut-off frequencies. These two simple ingredients allow us to interpret qualitatively the various dispersion relations obtained in the configuration of the plates decorated at both ends. Next, the dispersion relations are accurately recovered by our homogenized model.
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Figure 1. Periodic array of plates decorated at their endings with spacing ` = 1, height hP, and thickness
ϕP`; the substrate occupying a half-space is surmounted by a guiding layer of thickness hL able to support Love waves. The insets show a zoom on the two endings with heterogeneity surfaces = ϕ h St t t and = ϕ h . Sb b b Materials 2020, 13, 1632 3 of 27
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