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21.01.2020 | Original

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

verfasst von: Alexios Aivaliotis, Domenico Tallarico, Marco-Valerio d’Agostino, Ali Daouadji, Patrizio Neff, Angela Madeo

Erschienen in: Archive of Applied Mechanics | Ausgabe 5/2020

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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.

Vorheriger Artikel Worm-like motion enabled by changing the position of mass center in the anisotropic environment

Nächster Artikel A mixed stabilized finite element formulation for finite deformation of a poroelastic solid saturated with a compressible fluid

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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.

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.

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 \(k>1\), \(({{\,\mathrm{Div}\,}}w)_{i_1 i_2\ldots i_{k-1}} = w_{i_1 i_2\ldots i_k,i_k}\)

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.

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.

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

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.

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.

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.

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

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.

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.

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

Aivaliotis, A., Daouadji, A., Barbagallo, G., Tallarico, D., Neff, P., Madeo, A.: Low-and high-frequency Stoneley waves, reflection and transmission at a Cauchy/relaxed micromorphic interface. arXiv preprint arXiv:1810.12578 (2018)

Aivaliotis, A., Daouadji, A., Barbagallo, G., Tallarico, D., Neff, P., Madeo, A.: Microstructure-related Stoneley waves and their effect on the scattering properties of a 2D Cauchy/relaxed-micromorphic interface. Wave Motion 90, 99–120 (2019)MathSciNetCrossRef

Aivaliotis, A., Tallarico, D., Daouadji, A., Neff, P., Madeo, A.: Relaxed micromorphic broadband scattering for finite-size meta-structures—a detailed development. arXiv preprint (2019)

Auld, B.A.: Acoustic Fields and Waves in Solids, vol. I. Wiley, New York (1973)

Barbagallo, G., Madeo, A., d’Agostino, M.V., Abreu, R., Ghiba, I.-D., Neff, P.: Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics. Int. J. Solids Struct. 120, 7–30 (2017). https://doi.org/10.1016/j.ijsolstr.2017.01.030 CrossRefMATH

Barbagallo, G., Tallarico, D., d’Agostino, M.V., Aivaliotis, A., Neff, P., Madeo, A.: Relaxed micromorphic model of transient wave propagation in anisotropic band-gap metastructures. Int. J. Solids Struct. 162, 148–163 (2019)CrossRef

Basu, U., Chopra, A.K.: Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput. Methods Appl. Mech. Eng. 192(11–12), 1337–1375 (2003)CrossRef

Bloch, F.: Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift für Physik 52(7–8), 555–600 (1929)CrossRef

Chen, H., Chan, C.T.: Acoustic cloaking and transformation acoustics. J. Phys. D Appl. Phys. 43(11), 113001 (2010)CrossRef

10.

Craster, R.V., Guenneau, S.: Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking, vol. 166. Springer, Berlin (2012)

11.

d’Agostino, M.V., Barbagallo, G., Ghiba, I.-D., Eidel, B., Neff, P., Madeo, A.: Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. J. Elast. 25, 22 (2019). https://doi.org/10.1007/s10659-019-09753-9. (Accepted)CrossRef

12.

Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0555-5 CrossRefMATH

13.

Floquet, G.: Sur les equations differentielles lineaires. Ann. ENS [2] 12(1883), 47–88 (1883)MATH

14.

Geers, M.G.D., Kouznetsova, V.G., Brekelmans, W.A.M.: Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 234(7), 2175–2182 (2010)CrossRef

15.

Kadic, M., Bückmann, T., Schittny, R., Wegener, M.: Metamaterials beyond electromagnetism. Rep. Progr. Phys. 76(12), 126501 (2013)CrossRef

16.

Krushynska, A.O., Kouznetsova, V.G., Geers, M.G.D.: Towards optimal design of locally resonant acoustic metamaterials. J. Mech. Phys. Solids 71, 179–196 (2014)CrossRef

17.

Leckner, J.: Theory of Reflection: Reflection and Transmission of Electromagnetic, Particle and Acoustic Waves. Wiley, New York (1973)

18.

Lurie, S., Solyaev, Y., Volkov, A., Volkov-Bogorodskiy, D.: Bending problems in the theory of elastic materials with voids and surface effects. Math. Mech. Solids 23(5), 787–804 (2018)MathSciNetCrossRef

19.

Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Band gaps in the relaxed linear micromorphic continuum. Zeitschrift für Angewandte Mathematik und Mechanik 95(9), 880–887 (2014)MathSciNetCrossRef

20.

Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., Rosi, G.: Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27(4–5), 551–570 (2015). https://doi.org/10.1007/s00161-013-0329-2 MathSciNetCrossRefMATH

21.

Madeo, A., Barbagallo, G., Agostino, M.V., Placidi, L., Neff, P.: First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model. Proc. R. Soc. A Math. Phys. Eng. Sci. 472(2190), 20160169 (2016). https://doi.org/10.1098/rspa.2016.0169 MathSciNetCrossRefMATH

22.

Madeo, A., Neff, P., d’Agostino, M.V., Barbagallo, G.: Complete band gaps including non-local effects occur only in the relaxed micromorphic model. Comptes Rendus Mécanique 344(11–12), 784–796 (2016). https://doi.org/10.1016/j.crme.2016.07.002 CrossRef

23.

Madeo, A., Neff, P., Ghiba, I.-D., Rosi, G.: Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J. Mech. Phys. Solids 95, 441–479 (2016). https://doi.org/10.1016/j.jmps.2016.05.003 MathSciNetCrossRef

24.

Madeo, A., Barbagallo, G., Collet, M., d’Agostino, M.V., Miniaci, M., Neff, P.: Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: new perspectives towards metastructural design. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517728423 CrossRefMATH

25.

Madeo, A., Collet, M., Miniaci, M., Billon, K., Ouisse, M., Neff, P.: Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia. J. Elast. 130(1), 59–83 (2017). https://doi.org/10.1007/s10659-017-9633-6 MathSciNetCrossRefMATH

26.

Madeo, A., Neff, P., Aifantis, E.C., Barbagallo, G., d’Agostino, M.V.: On the role of micro-inertia in enriched continuum mechanics. Proc. R. Soc. A Math. Phys. Eng. Sci. 473(2198), 20160722 (2017). https://doi.org/10.1098/rspa.2016.0722 MathSciNetCrossRefMATH

27.

Madeo, A., Neff, P., Barbagallo, G., d’Agostino, M.V., Ghiba, I.-D.: A review on wave propagation modeling in band-gap metamaterials via enriched continuum models. In: dell’Isola, F., Sofonea, M., Steigmann, D.J. (eds.) Mathematical Modelling in Solid Mechanics, Advanced Structured Materials, pp. 89–105. Springer, Berlin (2017). https://doi.org/10.1007/978-981-10-3764-1_6 CrossRefMATH

28.

Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRef

29.

Misseroni, D., Colquitt, D.J., Movchan, A.B., Movchan, N.V., Jones, I.S.: Cymatics for the cloaking of flexural vibrations in a structured plate. Sci. Rep. 6, 23929 (2016)CrossRef

30.

Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26(5), 639–681 (2014). https://doi.org/10.1007/s00161-013-0322-9 MathSciNetCrossRefMATH

31.

Neff, P., Ghiba, I.-D., Lazar, M., Madeo, A.: The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q. J. Mech. Appl. Math. 68(1), 53–84 (2015). https://doi.org/10.1093/qjmam/hbu027 MathSciNetCrossRefMATH

32.

Neff, P., Madeo, A., Barbagallo, G., d’Agostino, M.V., Abreu, R., Ghiba, I.-D.: Real wave propagation in the isotropic-relaxed micromorphic model. Proc. R. Soc. A Math. Phys. Eng. Sci. 473(2197), 20160790 (2017). https://doi.org/10.1098/rspa.2016.0790 MathSciNetCrossRefMATH

33.

Neff, P., Eidel, B., d’Agostino, M.V., Madeo, A.: Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization. J. Elast., 1–30 (2019). https://doi.org/10.1007/s10659-019-09752-w

34.

Norris, A.N.: Acoustic cloacking. Acoust. Today 11(1), 38–46 (2015)

35.

Owczarek, S., Ghiba, I.-D., d’Agostino, M.-V., Neff, P.: Nonstandard micro-inertia terms in the relaxed micromorphic model: well-posedness for dynamics. Math. Mech. Solids 24(10), 3200–3215 (2019)MathSciNetCrossRef

36.

Platts, S.B., Movchan, N.V., McPhedran, R.C., Movchan, A.B.: Two-dimensional phononic crystals and scattering of elastic waves by an array of voids. Proc. R. Soc. A 458(2026), 2327–2347 (2002)MathSciNetCrossRef

37.

Rokoš, O., Ameen, M.M., Peerlings, R.H.J., Geers, M.G.D.: Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields. J. Mech. Phys. Solids 123, 119–137 (2019)MathSciNetCrossRef

38.

Sridhar, A., Kouznetsova, V.G., Geers, M.G.D.: A general multiscale framework for the emergent effective elastodynamics of metamaterials. J. Mech. Phys. Solids 111, 414–433 (2018)MathSciNetCrossRef

39.

Srivastava, A., Willis, J.R.: Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach. Proc. R. Soc. A 473(2200), 20160765 (2017)MathSciNetCrossRef

40.

Willis, J.R.: Dynamics of composites. In: Continuum Micromechanics, pp. 265–290. Springer (1997)

41.

Willis, J.R.: Exact effective relations for dynamics of a laminated body. Mech. Mater. 41(4), 385–393 (2009)CrossRef

42.

Willis, J.R.: Effective constitutive relations for waves in composites and metamaterials. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 467, pp. 1865–1879. The Royal Society (2011)

43.

Willis, J.R.: The construction of effective relations for waves in a composite. Comptes Rendus Mécanique 340(4–5), 181–192 (2012)CrossRef

44.

Willis, J.R.: Negative refraction in a laminate. J. Mech. Phys. Solids 97, 10–18 (2016)MathSciNetCrossRef

Titel: Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model
verfasst von: Alexios Aivaliotis
Domenico Tallarico
Marco-Valerio d’Agostino
Ali Daouadji
Patrizio Neff
Angela Madeo
Publikationsdatum: 21.01.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Archive of Applied Mechanics / Ausgabe 5/2020
Print ISSN: 0939-1533
Elektronische ISSN: 1432-0681
DOI: https://doi.org/10.1007/s00419-019-01651-9

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