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Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model

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Abstract

In this paper, we explore the use of micromorphic-type interface conditions for the modeling of a finite-sized metamaterial. We show how finite-domain boundary value problems can be approached in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized tractions, as well as additional conditions on the micro-distortion tensor and on the double-traction. The case of a metamaterial slab of finite width is presented, its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to a direct finite-element simulation encoding all details of the selected microstructure. The reflection and transmission coefficients obtained via the two methods are presented as a function of the frequency and of the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies going from the long-wave limit to frequencies beyond the first band-gap and for angles of incidence ranging from normal to near-parallel incidence. The present paper sets the basis for a new viewpoint on finite-size metamaterial modeling enabling the exploration of meta-structures at large scales.

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Notes

  1. It is worth noticing that the parameters of the relaxed micromorphic model are “constant” in the sense that they do not depend on frequency, as it is common to see when considering homogenized models for metamaterials. This implies that, once an estimate of the parameters is made for a given metamaterial, it will be possible to study the metamaterial’s response for any type of applied external load. In some sense, one obtains a true “metamaterial’s characterization” via the determination of few constant parameters that are known once for all.

  2. For example, \((A\cdot v)_i=A_{ij}v_j\), \((A\cdot B)_{ik}=A_{ij}B_{jk}\), \(A:B=A_{ij}B_{ji}\), \((C\cdot B)_{ijk}=C_{ijp}B_{pk}\), \((C:B)_{i}=C_{ijp}B_{pj}\), \(\left\langle v,w\right\rangle = v\cdot w = v_i w_i\), \(\left\langle A, B \right\rangle = A_{ij}B_{ij}\), etc.

  3. The operators \(\nabla \), \({{\,\mathrm{curl}\,}}\) and \({{\,\mathrm{Div}\,}}\) are the classical gradient, curl and divergence operators. In symbols, for a field u of any order, \((\nabla u)_i=u_{,i}\), for a vector field v, \(({{\,\mathrm{curl}\,}}v)_i = \epsilon _{ijk}v_{k,j}\) and for a field w of order \(k>1\), \(({{\,\mathrm{Div}\,}}w)_{i_1 i_2\ldots i_{k-1}} = w_{i_1 i_2\ldots i_k,i_k}\)

  4. The use of the letter t is reserved for the time variable and will not be used to denote the components of tensors. Analogously, the subscript t will denote derivation with respect to the scalar variable t. Finally, the subscript tt will indicate the second derivative with respect to time.

  5. We could consider more complex expressions for the curvature term, which would also include anisotropy and in fact, such expressions will be explored in further works. Here, we want to show that curvature terms provide small corrections to the overall behavior of the metamaterial and so, we limit ourselves to a simplified isotropic expression.

  6. For example, the fourth-order tensor \(\mathbb {C}_e\) is written as \(\widetilde{\mathbb {C}}_e\) in Voigt notation.

  7. As we will show in the following, \(k_2\), which is the second component of the wave-number, is always supposed to be known and is given by Snell’s law when imposing interface conditions on a given interface.

  8. Here again, as in the case of a Cauchy medium, \(k_2\) will be fixed and given by Snell’s law when imposing interface conditions.

  9. In a more general perspective, mixed interface conditions in which traction and double-traction are proportional to displacement and micro-distortion, respectively, could also be envisaged (e.g., [18]). Nevertheless, the direct interest of this kind of interface conditions is not evident for the case under study and will not be considered in the present work.

  10. We denote by \(\widetilde{v}\) the first two components of the micromorphic field v defined in Eq. (20).

  11. We remark that, in the relaxed micromorphic model, only the tangent part of the double-traction in (29) or of the micro-distortion tensor in (30) must be assigned, see [30, 31] for more details.

  12. The index \(j\in \{L,S\}\) for the elastic field \(u^j\) indicates the fact that \(u^j\) is the solution field generated by an L- or S-incident wave, respectively. Nevertheless, \(u^j\) typically contains in itself coupled L and S waves as a result of scattering.

  13. We limit ourselves to the plane-strain case, so that we set \(u^j_3=0\) and \(u^j_{i,3}=0\), \(j\in \{L,s\}\), \(i \in \{1,2,3\}\).

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Acknowledgements

Angela Madeo and Domenico Tallarico acknowledge funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006). Angela Madeo acknowledges support from IDEXLYON in the framework of the “Programme Investissement d’Avenir” ANR-16-IDEX-0005. All the authors acknowledge funding from the “Région Auvergne-Rhône-Alpes” for the “SCUSI” project for international mobility France/Germany.

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Authors and Affiliations

  1. GEOMAS, INSA-Lyon, Université de Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne Cedex, France

    Alexios Aivaliotis, Marco-Valerio d’Agostino, Ali Daouadji & Angela Madeo

  2. Lab. Acoustics/Noise Control, Empa, Swiss Federal Laboratories for Materials Science and Technology, Überlandstrasse 129, 8600, Dübendorf, Switzerland

    Domenico Tallarico

  3. Head of Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Mathematik-Carrée, Thea-Leymann-Straße 9, 45127, Essen, Germany

    Patrizio Neff

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  1. Alexios Aivaliotis
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Aivaliotis, A., Tallarico, D., d’Agostino, MV. et al. Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model. Arch Appl Mech 90, 1073–1096 (2020). https://doi.org/10.1007/s00419-019-01651-9

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