Two scale homogenization of a row of locally resonant inclusions - the case of anti-plane shear waves
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 (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 (Fig. 1). The mass density ρ and the shear modulus G are spatially varying parameters being piecewise constants; in the matrix, and in the inclusions . In the harmonic regime, the elastic displacement U has a time dependence 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.
References (43)
- et al.
Long wavelength inner-resonance cut-off frequencies in elastic composite materials
Int. J. Solids Struct.
(2012) - et al.
Homogenization of the 3d Maxwell system near resonances and artificial magnetism
Comptes Rendus Mathématique
(2009) - et al.
Approximate models for wave propagation across thin periodic interfaces
Journal de mathématiques pures et appliquées
(2012) Elastic body with defects distributed near a surface
Homogenization Techniques for Composite Media
(1987)- et al.
Experimental observation of locally-resonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars
Phys. Rev. B
(2011) Acoustics of heterogeneous media: macroscopic behavior by homogenization
Curr. Top. Acoust. Res. I
(1994)- et al.
Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials
(1989) - et al.
Simulation of Muffler’s transmission losses by a homogenized finite element method
J. Comp. Acoust.
(2004) - et al.
Enhanced and reduced transmission of acoustic waves with bubble meta-screens
Appl. Phys. Lett.
(2011) - et al.
Soft acoustic metamaterials
Science
(2013)
Complex modes and artificial magnetism in three-dimensional periodic arrays of titanium dioxide microspheres at millimeter waves
J. Opt. Soc. Am. B
A non-periodic two scale asymptotic method to take account of rough topographies for 2-d elastic wave propagation
Geophys. J. Int.
Extraordinary absorption of sound in porous lamella-crystals
Sci. Rep.
Tuning the wavelength of spoof plasmons by adjusting the impedance contrast in an array of penetrable inclusions
Appl. Phys. Lett.
Homogenized interface model describing inhomogeneities located on a surface
J. Elast.
High-order asymptotics for the electromagnetic scattering by thin periodic layers
Math. Methods Appl. Sci.
Metamaterial with simultaneously negative bulk modulus and mass density
Phys. Rev. Lett.
Acoustic band gap formation in metamaterials
Int. J. Mod. Phys. B
Impedance operator description of a metasurface with electric and magnetic dipoles
Math. Prob. Eng.
Theory of mesoscopic magnetism in photonic crystals
Phys. Rev. Lett.
Comparison of the sound attenuation efficiency of locally resonant materials and elastic band-gap structures
Phys. Rev. B
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