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

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES

卷:120

Article

Barbagallo, Gabriele^1,2  Madeo, Angela²  d'Agostino, Marco Valerio²  Abreu, Rafael³  Ghiba, Ionel-Dumitrel^4,5,6  Neff, Patrizio⁷ 

[1] Univ Lyon, INSA Lyon, LaMCoS CNRS, 20 Ave Albert Einstein, F-69621 Villeurbanne, France

[2] Univ Lyon, INSA Lyon, LGCIE, 20 Ave Albert Einstein, F-69621 Villeurbanne, France

[3] Westfalische Wilhelms Univ Munster, Inst Geophys, Corrensstr 24, D-48149 Munster, Germany

[4] Univ Duisburg Essen, Fak Math, Chair Nonlinear Anal & Modelling, Thea Leymann Str 9, D-45127 Essen, Germany

[5] Alexandru Ioan Cuza Univ, Dept Math, Blvd Carol 1,11, Iasi 700506, Romania

[6] Romanian Acad, Iasi Branch, Octav Mayer Inst Math, D-700505 Iasi, Germany

[7] Univ Duisburg Essen, Fak Math, Chair Nonlinear Anal & Modelling, Math Carree, Thea Leymann Str 9, D-45127 Essen, Germany

关键词: Relaxed micromorphic model;    Anisotropy;    Arithmetic mean;    Geometric mean;    Harmonic mean;    Reuss-bound;    Voigt-bound;    Generalized continuum models;    Homogenization;    Multi-scale modeling;   

DOI  :  10.1016/j.ijsolstr.2017.01.030

来源: Elsevier

【 摘 要 】

In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micromorphic 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 L-c -> 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. (C) 2017 Elsevier Ltd. All rights reserved.

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