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

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December 2013, 8(4): 913-941. doi: 10.3934/nhm.2013.8.913

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, France
2.	I.R.M.A.R., Université de Rennes 1, Rennes Cedex, France

Received February 2013 Published November 2013

Abstract
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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.

Keywords: two-phase composites, strong field, columnar structures., high-contrast conductivity, Homogenization.

Mathematics Subject Classification: Primary: 35B27, 35J25; Secondary: 74Q2.

Citation: Mohamed Camar-Eddine, Laurent Pater. Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures. Networks & Heterogeneous Media, 2013, 8 (4) : 913-941. doi: 10.3934/nhm.2013.8.913

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. Google Scholar
[2]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland Pub. Co., (1978). Google Scholar
[3]	D. J. Bergman, Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites,, Percolation Structures and Processes, 5 (1983), 297. Google Scholar
[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). doi: 10.1103/PhysRevB.71.035120. Google Scholar
[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. doi: 10.1103/PhysRevB.60.13016. Google Scholar
[6]	D. J. Bergman and Y. M. Strelniker, Strong-field magnetotransport of conducting composites with a columnar microstructure,, Phys. Rev. B, 59 (1999), 2180. doi: 10.1103/PhysRevB.59.2180. Google Scholar
[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. doi: 10.1103/PhysRevLett.80.3356. Google Scholar
[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. doi: 10.1103/PhysRevB.61.6288. Google Scholar
[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. doi: 10.1016/S0378-4371(97)00095-2. Google Scholar
[10]	M. Briane, Nonlocal effects in two-dimensional conductivity,, Arch. Rational Mech. Anal., 182 (2006), 255. doi: 10.1007/s00205-006-0427-4. Google Scholar
[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. doi: 10.1137/S0036141001398666. Google Scholar
[12]	M. Briane, Homogenization of non-uniformly bounded operators: Critical barrier for nonlocal effects,, Arch. Rational Mech. Anal., 164 (2002), 73. doi: 10.1007/s002050200196. Google Scholar
[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. doi: 10.1080/03605300600910423. Google Scholar
[14]	M. Briane and D. Manceau, Duality results in the homogenization of two-dimensional high-contrast conductivities,, Networks and Heterogeneous Media, 3 (2008), 509. doi: 10.3934/nhm.2008.3.509. Google Scholar
[15]	M. Briane, D. Manceau and G. W. Milton, Homogenization of the two-dimensional Hall effect,, J. Math. Anal. Appl., 339 (2008), 1468. doi: 10.1016/j.jmaa.2007.07.044. Google Scholar
[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. doi: 10.1007/s00205-008-0200-y. Google Scholar
[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. doi: 10.1016/j.jmaa.2011.12.059. Google Scholar
[18]	M. Briane and N. Tchou, Fibered microstructures for some nonlocal Dirichlet forms,, Ann. Sc. Norm. Super. Pisa, 30 (2001), 681. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.1137/080721455. Google Scholar
[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. doi: 10.1137/080721455. Google Scholar
[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. Google Scholar
[24]	E. H. Hall, On a new action of the magnet on electric currents,, Amer. J. Math., 2 (1879), 287. doi: 10.2307/2369245. Google Scholar
[25]	E. Ya. Khruslov, Homogenized models of composite media,, Composite Media and Homogenization Theory (Trieste, 5 (1991), 159. doi: 10.1007/978-1-4684-6787-1_10. Google Scholar
[26]	E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations,, Progress in Mathematical Physics, (2006). Google Scholar
[27]	L. Landau and E. Lifshitz, Électrodynamique des Milieux Continus,, Éditions Mir, (1969). Google Scholar
[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. doi: 10.1103/PhysRevB.38.11296. Google Scholar
[29]	G. W. Milton, The Theory of Composites,, Cambridge Monographs on Applied and Computational Mathematics, (2002). doi: 10.1017/CBO9780511613357. Google Scholar
[30]	F. Murat and L. Tartar, H-convergence,, Mimeographed notes, (1978). Google Scholar
[31]	F. Murat and L. Tartar, H-convergence,, Topics in the Mathematical Modelling of Composite Materials, (1998), 21. Google Scholar
[32]	M. A. Omar, Elementary Solid State Physics: Principles and Applications,, World Student Series Edition, (1975). Google Scholar
[33]	L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar
[34]	S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571. Google Scholar

show all references

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. Google Scholar
[2]	A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland Pub. Co., (1978). Google Scholar
[3]	D. J. Bergman, Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites,, Percolation Structures and Processes, 5 (1983), 297. Google Scholar
[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). doi: 10.1103/PhysRevB.71.035120. Google Scholar
[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. doi: 10.1103/PhysRevB.60.13016. Google Scholar
[6]	D. J. Bergman and Y. M. Strelniker, Strong-field magnetotransport of conducting composites with a columnar microstructure,, Phys. Rev. B, 59 (1999), 2180. doi: 10.1103/PhysRevB.59.2180. Google Scholar
[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. doi: 10.1103/PhysRevLett.80.3356. Google Scholar
[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. doi: 10.1103/PhysRevB.61.6288. Google Scholar
[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. doi: 10.1016/S0378-4371(97)00095-2. Google Scholar
[10]	M. Briane, Nonlocal effects in two-dimensional conductivity,, Arch. Rational Mech. Anal., 182 (2006), 255. doi: 10.1007/s00205-006-0427-4. Google Scholar
[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. doi: 10.1137/S0036141001398666. Google Scholar
[12]	M. Briane, Homogenization of non-uniformly bounded operators: Critical barrier for nonlocal effects,, Arch. Rational Mech. Anal., 164 (2002), 73. doi: 10.1007/s002050200196. Google Scholar
[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. doi: 10.1080/03605300600910423. Google Scholar
[14]	M. Briane and D. Manceau, Duality results in the homogenization of two-dimensional high-contrast conductivities,, Networks and Heterogeneous Media, 3 (2008), 509. doi: 10.3934/nhm.2008.3.509. Google Scholar
[15]	M. Briane, D. Manceau and G. W. Milton, Homogenization of the two-dimensional Hall effect,, J. Math. Anal. Appl., 339 (2008), 1468. doi: 10.1016/j.jmaa.2007.07.044. Google Scholar
[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. doi: 10.1007/s00205-008-0200-y. Google Scholar
[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. doi: 10.1016/j.jmaa.2011.12.059. Google Scholar
[18]	M. Briane and N. Tchou, Fibered microstructures for some nonlocal Dirichlet forms,, Ann. Sc. Norm. Super. Pisa, 30 (2001), 681. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. doi: 10.1137/080721455. Google Scholar
[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. doi: 10.1137/080721455. Google Scholar
[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. Google Scholar
[24]	E. H. Hall, On a new action of the magnet on electric currents,, Amer. J. Math., 2 (1879), 287. doi: 10.2307/2369245. Google Scholar
[25]	E. Ya. Khruslov, Homogenized models of composite media,, Composite Media and Homogenization Theory (Trieste, 5 (1991), 159. doi: 10.1007/978-1-4684-6787-1_10. Google Scholar
[26]	E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations,, Progress in Mathematical Physics, (2006). Google Scholar
[27]	L. Landau and E. Lifshitz, Électrodynamique des Milieux Continus,, Éditions Mir, (1969). Google Scholar
[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. doi: 10.1103/PhysRevB.38.11296. Google Scholar
[29]	G. W. Milton, The Theory of Composites,, Cambridge Monographs on Applied and Computational Mathematics, (2002). doi: 10.1017/CBO9780511613357. Google Scholar
[30]	F. Murat and L. Tartar, H-convergence,, Mimeographed notes, (1978). Google Scholar
[31]	F. Murat and L. Tartar, H-convergence,, Topics in the Mathematical Modelling of Composite Materials, (1998), 21. Google Scholar
[32]	M. A. Omar, Elementary Solid State Physics: Principles and Applications,, World Student Series Edition, (1975). Google Scholar
[33]	L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar
[34]	S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571. Google Scholar

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