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A Two-Dimensional Labile Aether Through Homogenization

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Abstract

Homogenization in linear elliptic problems usually assumes coercivity of the accompanying Dirichlet form. In linear elasticity, coercivity is not ensured through mere (strong) ellipticity so that the usual estimates that render homogenization meaningful break down unless stronger assumptions, like very strong ellipticity, are put into place. Here, we demonstrate that a L2-type homogenization process can still be performed, very strong ellipticity notwithstanding, for a specific two-phase two dimensional problem whose significance derives from prior work establishing that one can lose strong ellipticity in such a setting, provided that homogenization turns out to be meaningful. A striking consequence is that, in an elasto-dynamic setting, some two-phase homogenized laminate may support plane wave propagation in the direction of lamination on a bounded domain with Dirichlet boundary conditions, a possibility which does not exist for the associated two-phase microstructure at a fixed scale. Also, that material blocks longitudinal waves in the direction of lamination, thereby acting as a two-dimensional aether in the sense of e.g. Cauchy.

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Acknowledgements

G.F. acknowledges the support of the the National Science Fundation Grant DMS-1615839. The authors also thank Giovanni Leoni for his help in establishing Remark 3.2, Patrick Gérard for fruitful insights into propagation in the absence of coercivity and Lev Truskinovsky for introducing us to the fascinating history of the elastic aether.

The authors are also grateful to the referees for various comments and suggestions which have improved the presentation of the paper.

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Authors and Affiliations

  1. Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, 35000, Rennes, France

    Marc Briane

  2. LAGA, Université Paris-Nord & Courant Institute of Mathematical Sciences, New York University, New York, USA

    Gilles A. Francfort

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  1. Marc Briane
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  2. Gilles A. Francfort
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Correspondence to Gilles A. Francfort.

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Communicated by C. De Lellis

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Briane, M., Francfort, G.A. A Two-Dimensional Labile Aether Through Homogenization. Commun. Math. Phys. 367, 599–628 (2019). https://doi.org/10.1007/s00220-019-03333-7

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  • DOI: https://doi.org/10.1007/s00220-019-03333-7

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