In the present paper, the material parameters of the isotropic relaxed micromorphic model derived for a specific metamaterial in a previous contribution are used to model its transmission properties. Specifically, the reflection and transmission coefficients at an interface between a homogeneous solid and the chosen metamaterial are analyzed by using both the relaxed micromorphic model and a direct FEM implementation of the detailed microstructure. The obtained results show excellent agreement between the transmission spectra derived via our enriched continuum model and those issued by the direct FEM simulation. Such excellent agreement validates the indirect measure of the material parameters and opens the way towards an efficient metastructural design.
DeymierPA.Acoustic metamaterials and phononic crystals. In: DeymierPA (ed.) Springer series in solid-state sciences. New York: Springer, 2013, vol. 173.
2.
LiuZZhangXMaoYet al. Locally resonant sonic materials. Science2000; 289(5485): 1734–1736.
3.
PhamKKouznetsovaVGGeersMGD. Transient computational homogenization for heterogeneous materials under dynamic excitation. J Mech Phys Solids2013; 61(11): 2125–2146.
4.
SridharAKouznetsovaVGGeersMGD. Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Comput Mech2016; 57(3): 423–435.
AuriaultJLBoutinC.Long wavelength inner-resonance cut-off frequencies in elastic composite materials. Int J Solids Struct2012; 49(23-24): 3269–3281.
7.
BoutinCHansS.Homogenisation of periodic discrete medium: Application to dynamics of framed structures. Comput Geotech2003; 30(4): 303–320.
8.
BoutinCHansSChesnaisC.Generalized beams and continua. Dynamics of reticulated structures. New York: Springer, 2010 pp. 131–141.
9.
BoutinCSoubestreJ.Generalized inner bending continua for linear fiber reinforced materials. Int J Solids Struct2011; 48(3–4): 517–534.
10.
dell’IsolaFAndreausUPlacidiL.At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20(8): 887–928.
11.
dell’IsolaFDella CorteAGiorgioI.Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids2017; 22(4): 852–872.
12.
AuffrayNdell’IsolaFEremeyevVAet al. Analytical continuum mechanics a la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math Mech Solids2015; 20(4): 375–417.
13.
MadeoANeffPGhibaIDet al. Band gaps in the relaxed linear micromorphic continuum. Z Angew Math Mech2014; 95(9): 880–887.
14.
MadeoANeffPGhibaIDet al. Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model. J Mech Phys Solids2016; 95: 441–479.
15.
MadeoABarbagalloGd’AgostinoMVet al. First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model. Proc R Soc Lond A2016; 472(2190): 20160169.
16.
MadeoAColletMMiniaciMet al. Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia. J Elasticity. Epub ahead of print 24March2017. DOI: 10.1007/s10659-017-9633-6.
17.
MadeoANeffPd’AgostinoMVet al. Complete band gaps including non-local effects occur only in the relaxed micromorphic model. C R Mec2016; 344(11–12): 784–796.
18.
MadeoANeffPAifantisECet al. On the role of micro-inertia in enriched continuum mechanics. Proc R Soc Lond A2017; 473(2198): 20160722.
19.
MadeoANeffPBarbagalloGet al. A review on wave propagation modeling in band-gap metamaterials via enriched continuum models. In: dell’IsolaFSofoneaMSteigmannDJ (eds.) Mathematical modelling in solid mechanics. Singapore: Springer, 2017, pp. 89–105.
20.
PlacidiLAndreausUGiorgioI.Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J Eng Math2017; 103(1): 1–21.
21.
FanYColletMIchchouMet al. Energy flow prediction in built-up structures through a hybrid finite element/wave and finite element approach. Mech Syst Signal Process2016; 66–67: 137–158.
22.
YiKColletMIchchouMet al. Flexural waves focusing through shunted piezoelectric patches. Smart Mater Struct2016; 25(7): 075007.
23.
MadeoANeffPGhibaIDet al. Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Continuum Mech Thermodyn2015; 27(4–5): 551–570.
24.
NeffPGhibaIDMadeoAet al. A unifying perspective: The relaxed linear micromorphic continuum. Continuum Mech Thermodyn2014; 26(5): 639–681.
25.
d’AgostinoMVBarbagalloGGhibaIDet al. A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model. Z Angew Math Mech. Epub ahead of print 16May2017. DOI: 10.1002/zamm.201600227.
26.
ColquittDJCrasterRVAntonakakisTet al. Rayleigh-Bloch waves along elastic diffraction gratings. Proc R Soc Lond A2014; 471(2173): 20140465.
27.
BarbagalloGMadeoAd’AgostinoMVet al. Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics. Int J Solids Struct2017; 120: 7–30.
28.
NeffPGhibaIDLazarMet al. The relaxed linear micromorphic continuum: Well-posedness of the static problem and relations to the gauge theory of dislocations. Q J Mech Appl Math2015; 68(1): 53–84.
29.
GhibaIDNeffPMadeoAet al. The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics. Math Mech Solids2015; 20(10): 1171–1197.
30.
NeffPPaulyDWitschKJ.Poincaré meets Korn via Maxwell: extending Korn’s first inequality to incompatible tensor fields. J Differ Equations2015; 258(4): 1267–1302.
31.
NeffPPaulyDWitschKJ.Maxwell meets Korn: A new coercive inequality for tensor fields in Rn×n with square-integrable exterior derivative. Math Meth Appl Sci2012; 35(1): 65–71.
32.
NeffPPaulyDWitschKJ.A canonical extension of Korn’s first inequality to H(Curl) motivated by gradient plasticity with plastic spin. C R Math2011; 349(23): 1251–1254.
33.
BauerSNeffPPaulyDet al. New Poincaré-type inequalities. C R Math2014; 352(2): 163–166.
34.
BauerSNeffPPaulyDet al. Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM Contr Optim Calculus Var2016; 22(1): 112–133.
35.
NeffPMadeoABarbagalloGet al. Real wave propagation in the isotropic-relaxed micromorphic model. Proc R Soc Lond A2017; 473(2197): 20160790.
36.
LucklumRKeMZubtsovM.Two-dimensional phononic crystal sensor based on a cavity mode. Sens Actuators B2012; 171–172: 271–277.
