Effective Description of Anisotropic Wave Dispersion in Mechanical Band-Gap Metamaterials via the Relaxed Micromorphic Model

  • Marco Valerio d’AgostinoEmail author
  • Gabriele Barbagallo
  • Ionel-Dumitrel Ghiba
  • Bernhard Eidel
  • Patrizio Neff
  • Angela Madeo


In this paper the relaxed micromorphic material model for anisotropic elasticity is used to describe the dynamical behavior of a band-gap metamaterial with tetragonal symmetry. Unlike other continuum models (Cauchy, Cosserat, second gradient, classical Mindlin–Eringen micromorphic etc.), the relaxed micromorphic model is endowed to capture the main microscopic and macroscopic characteristics of the targeted metamaterial, namely, stiffness, anisotropy, dispersion and band-gaps.

The simple structure of our material model, which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent and thus truly constitutive parameters, valid for both static and wave-propagation analyses in the plane. The static macro- and micro-parameters are identified by numerical homogenization in static tests on the unit-cell level in Neff et al. (J. Elast.,, 2019, in this volume).

The remaining inertia parameters for dynamical analyses are calibrated on the dispersion curves of the same metamaterial as obtained by a classical Bloch–Floquet analysis for two wave directions.

We demonstrate via polar plots that the obtained material parameters describe very well the response of the structural material for all wave directions in the plane, thus covering the complete panorama of anisotropy of the targeted metamaterial.


Anisotropy Dispersion Planar harmonic waves Relaxed micromorphic model Enriched continua Dynamic problems Micro-elasticity Size effects Wave propagation Band-gaps Parameter identification Effective properties Unit-cell Micro-macro transition 

Mathematics Subject Classification

74A10 74A30 74A35 74A60 74B05 74M25 74Q15 74J05 



Angela Madeo acknowledges funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006) and the 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. The work of I.D. Ghiba was supported by a grant of the "Alexandru Ioan Cuza" University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2017-10. B. Eidel acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) within the Heisenberg program (grant no. EI 453/2-1).

The authors thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.GEOMAS, INSA-LyonUniversité de LyonVilleurbanne cedexFrance
  2. 2.Department of MathematicsAlexandru Ioan Cuza University of IaşiIaşiRomania
  3. 3.Octav Mayer Institute of Mathematics of the Romanian AcademyIaşiRomania
  4. 4.Institut für Mechanik, Heisenberg-groupUniversität SiegenSiegenGermany
  5. 5.Head of Chair for Nonlinear Analysis and Modelling, Fakultät für MathematikUniversität Duisburg-Essen, Mathematik-CarréeEssenGermany

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