Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures

In this paper we determine, in dimension three, the effective conductivities of non periodic high-contrast two-phase cylindrical composites, placed in a constant magnetic field, without any assumption on the geometry of their cross sections. Our method, in the spirit of the H-convergence of Murat-Tartar, is based on a compactness result and the cylindrical nature of the microstructure. The homogenized laws we obtain extend those of the periodic fibre-reinforcing case of [M. Briane and L. Pater. Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three. J. Math. Anal. Appl., 393 (2) (2012), 563 -589] to the case of periodic and non periodic composites with more general transversal geometries.Comment: 28 page

Homogenization of two-dimensional elasticity problems with very stiff coefficients

International audienceIn this paper we study the asymptotic behaviour of a sequence of two-dimensional linear elasticity problems with equicoercive elasticity tensors. Assuming the sequence of tensors is bounded in L^1, we obtain a compactness result extending to the elasticity the div-curl approach of [12] for the conduction. In the periodic case this compactness result is refined replacing the L^1-boundedness by a less restrictive condition involving the oscillations period. We also build a sequence of isotropic elasticity problems with L^1-unbounded Lamé's coefficients, which converges to a second gradient limit problem. This loss of compactness shows a gap in the limit behaviour between the very stiff problems of elasticity and those of conduction. Indeed, in the conduction case a compactness result was proved in [13] without assuming any bound from above for the conductivities

An optimal condition of compactness for elasticity problems involving one directional reinforcement

25 pagesInternational audienceThis paper deals with the homogenization of a homogeneous elastic medium reinforced by very stiff strips in dimension two. We give a general condition linked to the distribution and the stiffness of the strips, under which the nature of the elasticity problem is preserved in the homogenization process. This condition is sharper than the one used in [M. Briane & M. Camar-Eddine, Homogenization of two-dimensional elasticity problems with very sti coefficients, J. Math. Pures et Appl., 88, 483-505 (2007)] and is shown to be optimal in the case where the strips are periodically arranged. Indeed, a fourth-order derivative term appears in the limit equation as soon as the condition is no more satisfied. In the periodic case the influence of oscillations in the medium surrounding the strips is also considered. The homogenization method is based both on a two-scale convergence for the strips and the use of suitable oscillating test functions. This allows us to obtain a distributional convergence of two of the three entries of the stress tensor contrary to the Gamma-convergence approach of [M. Briane & M. Camar-Eddine, Homogenization of two-dimensional elasticity problems with very sti coefficients, J. Math. Pures et Appl., 88, 483-505 (2007)

A metamaterial having a frequency dependent elasticity tensor and a zero effective mass density

Within the context of linear elasticity we show that a two-terminal network of springs and masses, can respond exactly the same as a normal spring, but with a frequency dependent spring constant. As a consequence a network of such springs can have a frequency dependent effective elasticity tensor but zero effective mass density. The internal masses influence the elasticity tensor, but do not contribute to the effective mass density at any frequency.Comment: 5 pages 1 figur

Fermeture des fonctionnelles de diffusion et de l'élasticité linéaire pour la topologie de la Mosco-convergence

Les membres du jury de la thèse: M. ALIBERT Jean-Jacques , ANAM, Université de Toulon et du Var, Examinateur. M. BOUCHITTE Guy, ANAM, Université de Toulon et du Var , Examinateur. M. BRIANE Marc, I.N.S.A de Rennes, Rapporteur. M. CHAMBOLLE Antonin, CEREMADE-CNRS, Paris-Dauphine, Examinateur. M. DAL MASO Gianni, SISSA, Trieste, Rapporteur. M. SEPPECHER Pierre, ANAM, Université de Toulon et du Var, Directeur de thèse. M. SUQUET Pierre, LMA-CNRS, Marseille, Président. M. BUTTAZZO Giuseppe, Université de Pise, Italy, membre invité.The purpose of this thesis is to characterize all possible Mosco-limits of sequences of diffusion functionals or isotropic elasticity ones. It is a well-known fact that, when the diffusion coefficients in the scalar case, or the elasticity coefficients in the vectorial one, are not uniformly bounded, non local terms and killing terms can appear in the limit functional, despite the strong local nature of any element of those sequences. In the vectorial case, the limit functional can even involve some second derivative of the displacement. From a mechanical point of view, the effective properties of a composite material can differ fundamentally from those of its components. Umberto Mosco has shown that any limit of a sequence of diffusion functionals has to be a Dirichlet form. The contribution of the first part of this work provides a positive answer to the inverse problem. We show that any Dirichlet form is the Mosco-limit of some sequence of diffusion functionals. In a crucial step, we exhibit an explicit composite diffusive material, the effective properties of which contain an elementary non-local interaction. Then, using a step by step approach, we reach at each step a more general non-local interaction until obtaining all the Dirichlet forms. The second part of this work deals with the vectorial case. We show that the Mosco-closure of the set of isotropic elasticity functionals coincides with the set of all non-negative lower semi-continuous quadratic functionals which are objective. The proof of this result, which is far from being a simple generalisation of the scalar case, is based, at the start, on a result which is comparable to the scalar case. Then a fundamentally different approach is necessary.L'objectif de cette thèse est l'identification de toutes les limites possibles, vis-à-vis de la Mosco-convergence, des suites de fonctionnelles de diffusion ou de l'élasticité linéaire isotrope. Bien que chaque élément de ces suites soit une fonctionnelle fortement locale, il est bien connu que, sans hypothèse de majoration uniforme sur les coefficients de diffusion, dans le cas scalaire, ou d'élasticité dans le cas vectoriel, la limite peut contenir un terme non-local et un terme étrange. Dans le cas vectoriel, il peut même arriver que la fonctionnelle limite dépende du second gradient du déplacement. D'un point de vue mécanique, les propriétés effectives d'un matériau composite peuvent radicalement différer de celles de ces différents constituants. Umberto Mosco a montré que toute limite d'une suite de fonctionnelles de diffusion est une forme de Dirichlet. La contribution des travaux présentés dans la première partie de cette thèse apporte une réponse positive au problème inverse. Nous montrons que toute forme de Dirichlet est limite d'une suite de fonctionnelles de diffusion. Une étape cruciale consiste en la construction explicite d'un matériau composite dont les propriétés effectives contiennent une interaction non-locale élémentaire. Puis, on obtient progressivement des interactions plus complexes, pour finalement atteindre toutes les formes de Dirichlet. La deuxième partie de nos travaux traite du cas vectoriel. On y démontre que la fermeture des fonctionnelles de l'élasticité linéaire isotrope est l'ensemble de toutes les formes quadratiques positives, objectives et semi-continues inférieurement. La preuve de ce résultat qui est loin d'être une simple généralisation du cas scalaire s'appuie, au départ, sur un résultat comparable au cas scalaire. Elle nécessite ensuite une approche complétement différente

Fermeture des fonctionnelles de diffusion et de l'élasticité linéaire pour la topologie de la Mosco-convergence

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CLOSURE OF THE SET OF DIFFUSION FUNCTIONALS WITH RESPECT TO THE MOSCO-CONVERGENCE

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Non-local interactions resulting from the homogenization of a linear diffusive medium

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