Two scale homogenization of a row of locally resonant inclusions - the case of anti-plane shear waves

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Abstract

We present a homogenization model for a single row of locally resonant inclusions. The resonances, of the Mie type, result from a high contrast in the shear modulus between the inclusions and the elastic matrix. The presented homogenization model is based on a matched asymptotic expansion technique; it slightly differs from the classical homogenization which applies for thick arrays with many rows of inclusions (and thick means large compared to the wavelength in the matrix). Instead of the effective bulk parameters found in the classical homogenization, we end up with interface parameters entering in jump conditions for the displacement and for the normal stress; among these parameters, one is frequency dependent and encapsulates the resonant behavior of the inclusions. Our homogenized model is validated by comparison with results of full wave calculations. It is shown to be efficient in the low frequency domain and accurately describes the effects of the losses in the soft inclusions.

Introduction

Metamaterial structures are constructed by repeating, most often periodically, a unit cell. They may have a lattice periodicity of the same order of magnitude than the working wavelength, resulting in a structure of the crystal type; such structures present resonant behaviors due to Bragg scattering mechanism. Starting in the 1990’s with the pioneering work of Auriault (1994), resonant structures with subwavelength unit cells have been proposed in the context of elasticity (Auriault, Boutin, 2012, Liu, Zhang, Mao, Zhu, Yang, Chan, Sheng, 2000), in the context of electromagnetism (Bouchitté, Bourel, Felbacq, 2009, Felbacq, Bouchitté, 2005, O’Brien, Pendry, 2002), and in a unified mathematical context (Zhikov, 2005, Zhikov, 2000). In this case, the resonances are attributable to an inclusion placed in the unit cell and presenting a high contrast in its material properties with respect to the surrounding matrix. These resonances often referred to as Mie resonances occur at frequencies producing a wavelength in the inclusion comparable to the inclusion size (and this size is much smaller than the wavelength in the matrix). The ability of these so-called locally resonant structures to forbid the wave propagation has been exhibited and the forbidden band gaps have been interpreted in terms of an effective negative parameter being the mass density in elasticity and the permeability in electromagnetism. Since then, locally resonant materials have been intensively studied for applications including the design of efficient wave shields (Goffaux, Sánchez-Dehesa, Yeyati, Lambin, Khelif, Vasseur, Djafari-Rouhani, 2002, Ho, Cheng, Yang, Zhang, Sheng, 2003), absorbers of small thicknesses (Brunet, Leng, Mondain-Monval, 2013, Goffaux, Sánchez-Dehesa, Lambin, 2004, Zhao, Wen, Yu, Wen, 2010) and subwavelength waveguides (Achaoui, Khelif, Benchabane, Robert, Laude, 2011, Jin, Fernez, Pennec, Bonello, Moiseyenko, Hémon, Pan, Djafari-Rouhani, 2016).

Because of their subwavelength scales, such materials are well described by homogenization approaches. When the thickness of the structure made of the resonant material is large compared to any wavelength outside the inclusions, the bulk response is dominant. In this context, extensions of the classical homogenization to high contrast versions have been proposed (Zhikov, 2005, Zhikov, 2000), in particular in the context of elasticity (Auriault, 1994, Auriault, Boutin, 2012) and of electromagnetism (Bouchitté, Bourel, Felbacq, 2009, Felbacq, Bouchitté, 2005). These high contrast homogenizations provide effective bulk parameters among which the effective mass density or permeability is frequency dependent and may change in sign as a result. It is worth noting that, in the two contexts of waves, the constitutive equations being respectively the Navier equations and the Maxwell equations differ but the conclusions are of the same nature and conform to the expectations (we refer to “expectations” the expected negative parameters previously obtained using retrieval methods). Note also alternative methods based on computational homogenization (Pham et al., 2013). A priori, the story should end here.

However, motivated by the design of compact metamaterial devices and following the intuitive argument that each local resonator vibrates almost like an independent unit, structures involving a single row of resonators or few rows have been thought, see e.g. Ho et al. (2003) and Zhao et al. (2010). In this case, interrogating the bulk response of the device becomes questionable. Indeed, when the number of cells is too small, the metamaterial device is dominated by boundary layer effects and the response of its bulk is not pertinent anymore; the failure of the effective medium theories for thick structures has been illustrated recently in Lapine et al. (2016) and in Marigo and Maurel (2016a; 2017) (non resonant structures were considered in these references). Homogenization approaches able to handle such thin structures have been developed originally in the context of solid mechanics (Bakhvalov, Panasenko, 1989, Marigo, Pideri, 2011, Sanchez-Palencia, 1987). In the context of wave propagation, they have been adapted in geophysics (Capdeville and Marigo, 2013), in acoustics (Bonnet-Bendhia, Drissi, Gmati, 2004, Marigo, Maurel, 2016, Marigo, Maurel, 2017) and in electromagnetism (Delourme, 2015, Delourme, Haddar, Joly, 2012, Marigo, Maurel, 2016, Maurel, Marigo, Ourir, 2016), see also Felbacq (2015). Here, we extend these works to the case of an array composed of a single row of locally resonant inclusions (Fig. 1). We restrict ourselves to the case of two dimensional shear wave propagation, which reduces our conclusions to a scalar case but make them applicable to the case of polarized electromagnetic waves as considered in Felbacq and Bouchitté (2005); note that the extension to the three dimensional case is quite easy in elasticity but more tricky in electromagnetism.

The homogenized model is presented in Section 2; it basically relies on the same ingredients than the classical homogenization, a two scale method and an asymptotic expansion of the solution. The solution is expressed in a rescaled, or renormalized, structure accounting for a small parameter ηkh1,(k is the wavenumber in the matrix and h the spacing between the inclusions in the row) and accounting for the high contrast in the shear modulus; this latter has to scale as 1/η2 to produce wavelengths in the inclusions of the order of h. In addition, it accounts for the small thickness of the device and this is done by using two different expansions near and far from the array, which are connected using so-called matching conditions. The homogenization is conduced up to O(η2) and it makes effective interface parameters to appear, which enter in jump conditions on the displacement and on the normal stress across an equivalent interface whose interior region is disregarded (Fig. 2). In the present case, we obtain six interface parameters among which one is frequency dependent. The homogenized problem is validated in Section 3 by comparison with full wave calculations in two cases of interest: first we inspect the scattering by such thin metafilms for a plane wave at oblique incidence, afterwards the ability of the array to support guided waves is regarded. In both cases, the effect of losses within the inclusions is considered (the effect of the losses in the matrix is trivial and it is disregarded). The homogenized problem can be solved explicitly, yielding the scattering coefficients and of the dispersion relation of the guided waves, respectively in the two problems.

Section snippets

The real and the homogenized problems

We consider two dimensional shear waves propagating in an elastic matrix containing a single row of periodically located elastic inclusions with spacing h and thickness e=O(h) (Fig. 1). The mass density ρ and the shear modulus G are spatially varying parameters being piecewise constants; in the matrix, ρ(X)=ρm,G(X)=Gm and in the inclusions ρ(X)=ρiG(X)=Gi. In the harmonic regime, the elastic displacement U has a time dependence eiωt with ω the frequency (the time dependence will be omitted in

Validation of the homogenized interface problem

In this section, we address the validity of our homogenized problem (3) with respect to the real one. This is done considering two classical problems for wave propagating in such an array of locally resonant inclusions: (i) the scattering of incident plane waves and (ii) the propagation of guided waves supported by the array. In both cases, the effect of losses in the inclusions is regarded and this is relevant in view of practical applications. In the former case, the absorption resulting from

Concluding remarks and perspectives

We have proposed a homogenization of an array of locally resonant inclusions with high contrast in its shear modulus compared to the matrix; this produces wavelengths inside the soft inclusions of the same order of magnitude as their typical size. Such homogenization leads to an equivalent problem in which jump conditions apply across the region originally occupied by the array. The homogenized problem has been validated by comparison with direct numerical calculations, and we addressed the

Acknowledgment

The authors thank the referees for their useful comments. The authors acknowledge the financial support of the french Mission Interdisciplainare du Centre National de la Recherche Scientifique (MI/CNRS) under grant INFYNITI/PomS.

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