Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

Gabriele Barbagallo (1, 2) , Angela Madeo (2) , Marco Valerio d'Agostino (2) , Rafael Abreu (3) , Ionel-Dumitrel Ghiba (4) , Patrizio Neff (5)

1 LaMCoS - Laboratoire de Mécanique des Contacts et des Structures [Villeurbanne]
2 LGCIE - Laboratoire de Génie Civil et d'Ingénierie Environnementale
3 WWU - Westfälische Wilhelms-Universität Münster = University of Münster
4 UAIC - Alexandru Ioan Cuza University of Iași = Universitatea Alexandru Ioan Cuza din Iași
5 Universität Duisburg-Essen = University of Duisburg-Essen [Essen]

In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micro-morphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial the meso-and micro-coefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

Mots clés: harmonic mean; geometric mean; homogenization; multi-scale modeling; generalized continuum models; parameter identification; relaxed micromorphic model; non-redundant model; long wavelength limit; Voigt-bound; anisotropy; Cauchy continuum; Reuss-bound; macroscopic consistency; arithmetic mean

Domaines: Physique mathématique [math-ph]; Mécanique des solides [physics.class-ph]

Gabriele Barbagallo, Angela Madeo, Marco Valerio d'Agostino, Rafael Abreu, Ionel-Dumitrel Ghiba, et al.. Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics. International Journal of Solids and Structures, 2017, 120, pp.7-30. ⟨10.1016/j.ijsolstr.2017.01.030⟩. ⟨hal-01719592⟩