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

Mohamed Camar-Eddine; Laurent Pater

doi:10.3934/nhm.2013.8.913

Article Contents

2013, Volume 8, Issue 4: 913-941. Doi: 10.3934/nhm.2013.8.913

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

Mohamed Camar-Eddine^1, and
Laurent Pater^2,

1.
Centre de Mathématiques, I.N.S.A. de Rennes & I.R.M.A.R., Rennes Cedex
2.
I.R.M.A.R., Université de Rennes 1, Rennes Cedex

Received: January 31, 2013

Published: November 2013

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Abstract

In this paper we determine, in dimension three, the effective conductivities of non periodic and 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 [17] to the case of periodic and non periodic composites with more general transversal geometries.

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Mathematics Subject Classification: Primary: 35B27, 35J25; Secondary: 74Q20.

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References

[1]	M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Non local effects, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 407-436.
[2]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Pub. Co., Elsevier North-Holland, Amsterdam, New York, 1978.
[3]	D. J. Bergman, Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites, Percolation Structures and Processes, Annals of the Israel Physical Society, (eds. G. Deutscher, R. Zallen and J. Adler), Israel Physical Society, Jerusalem, 5 (1983), 297-321.
[4]	D. J. Bergman, X. Li and Y. M. Strelniker, Macroscopic conductivity tensor of a three-dimensional composite with a one- or two-dimensional microstructure, Phys. Rev. B, 71 (2005), 035120.doi: 10.1103/PhysRevB.71.035120.
[5]	D. J. Bergman and Y. M. Strelniker, Magnetotransport in conducting composite films with a disordered columnar microstructure and an in-plane magnetic field, Phys. Rev. B, 60 (1999), 13016-13027.doi: 10.1103/PhysRevB.60.13016.
[6]	D. J. Bergman and Y. M. Strelniker, Strong-field magnetotransport of conducting composites with a columnar microstructure, Phys. Rev. B, 59 (1999), 2180-2198.doi: 10.1103/PhysRevB.59.2180.
[7]	D. J. Bergman and Y. M. Strelniker, Duality transformation in a three dimensional conducting medium with two dimensional heterogeneity and an in-plane magnetic field, Phys. Rev. Lett., 80 (1998), 3356-3359.doi: 10.1103/PhysRevLett.80.3356.
[8]	D. J. Bergman, Y. M. Strelniker and A. K. Sarychev, Exact relations between magnetoresistivity tensor components of conducting composites with a columnar microstructure, Phys. Rev. B, 61 (2000), 6288-6297.doi: 10.1103/PhysRevB.61.6288.
[9]	D. J. Bergman, Y. M. Strelniker and A. K. Sarychev, Recent advances in strong field magneto-transport in a composite medium, Physica A, 241 (1997), 278-283.doi: 10.1016/S0378-4371(97)00095-2.
[10]	M. Briane, Nonlocal effects in two-dimensional conductivity, Arch. Rational Mech. Anal., 182 (2006), 255-267.doi: 10.1007/s00205-006-0427-4.
[11]	M. Briane, Homogenization of high-conductivity periodic problems: Application to a general distribution of one-directional fibers, SIAM Journal on Mathematical Analysis, 35 (2003), 33-60.doi: 10.1137/S0036141001398666.
[12]	M. Briane, Homogenization of non-uniformly bounded operators: Critical barrier for nonlocal effects, Arch. Rational Mech. Anal., 164 (2002), 73-101.doi: 10.1007/s002050200196.
[13]	M. Briane and J. Casado-Díaz, Two-dimensional div-curl results. Application to the lack of nonlocal effects in homogenization, Com. Part. Diff. Equ., 32 (2007), 935-969.doi: 10.1080/03605300600910423.
[14]	M. Briane and D. Manceau, Duality results in the homogenization of two-dimensional high-contrast conductivities, Networks and Heterogeneous Media, 3 (2008), 509-522.doi: 10.3934/nhm.2008.3.509.
[15]	M. Briane, D. Manceau and G. W. Milton, Homogenization of the two-dimensional Hall effect, J. Math. Anal. Appl., 339 (2008), 1468-1484.doi: 10.1016/j.jmaa.2007.07.044.
[16]	M. Briane and G. W. Milton, Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient, Arch. Ratio. Mech. Anal., 193 (2009), 715-736.doi: 10.1007/s00205-008-0200-y.
[17]	M. Briane and L. Pater, Homogenization of high-contrast two-phase conductivities perturbed by a magnetic field. Comparison between dimension two and dimension three, Journal of Mathematical Analysis and Applications, 393 (2012), 563-589.doi: 10.1016/j.jmaa.2011.12.059.
[18]	M. Briane and N. Tchou, Fibered microstructures for some nonlocal Dirichlet forms, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 30 (2001), 681-711.
[19]	V. N. Fenchenko, E. Ya. Khruslov, Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness, Dokl. AN Ukr. SSR, 4 (1981), 24-27.
[20]	Y. Grabovsky, G. W. Milton and D. S. Sage, Exact relations for effective tensors of polycrystals: Necessary conditions and sufficient conditions, Comm. Pure Appl. Math., 53 (2000), 300-353.
[21]	Y. Grabovsky, An application of the general theory of exact relations to fiber-reinforced conducting composites with Hall effect, Mechanics of Materials, 41 (2009), 456-462.doi: 10.1137/080721455.
[22]	Y. Grabovsky, Exact relations for effective conductivity of fiber-reinforced conducting composites with the Hall effect via a general theory, SIAM J. Math. Analysis, 41 (2009), 973-1024.doi: 10.1137/080721455.
[23]	Y. Grabovsky and G. W. Milton, Exact relations for composites: Towards a complete solution, Doc. Math. J. DMV Extra Volume ICM, III (1998), 623-632.
[24]	E. H. Hall, On a new action of the magnet on electric currents, Amer. J. Math., 2 (1879), 287-292.doi: 10.2307/2369245.
[25]	E. Ya. Khruslov, Homogenized models of composite media, Composite Media and Homogenization Theory (Trieste, 1990), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 5 (1991), 159-182.doi: 10.1007/978-1-4684-6787-1_10.
[26]	E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46, Birkhäuser, Boston, 2006.
[27]	L. Landau and E. Lifshitz, Électrodynamique des Milieux Continus, Éditions Mir, 1969.
[28]	G. W. Milton, Classical Hall effect in two-dimensional composites: A characterization of the set of realizable effective conductivity tensors, Phys. Rev. B, 38 (1988), 11296-11303.doi: 10.1103/PhysRevB.38.11296.
[29]	G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2002.doi: 10.1017/CBO9780511613357.
[30]	F. Murat and L. Tartar, H-convergence, Mimeographed notes, Séminaire d'Analyse Fonctionnelle et Numérique, Universitéd'Alger, Boston 1978, (English translation in [31]).
[31]	F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, eds. A. V. Cherkaev and R. V. Kohn, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston 1998, 21-43.
[32]	M. A. Omar, Elementary Solid State Physics: Principles and Applications, World Student Series Edition, Addison-Wesley, Reading, MA, 1975.
[33]	L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292.doi: 10.1007/BF00252910.
[34]	S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571-597; Errata, Ibid. (3).