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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

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), 407-436.  Google Scholar

[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.  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, Annals of the Israel Physical Society, (eds. G. Deutscher, R. Zallen and J. Adler), Israel Physical Society, Jerusalem, 5 (1983), 297-321.  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), 035120. 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), 13016-13027. 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), 2180-2198. 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), 3356-3359. 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), 6288-6297. 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), 278-283. 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), 255-267. 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), 33-60. 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), 73-101. 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), 935-969. 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), 509-522. 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), 1468-1484. 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), 715-736. 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), 563-589. 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, Cl. Sci., IV. Ser., 30 (2001), 681-711.  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), 24-27.  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), 300-353.  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), 456-462. 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), 973-1024. 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), 623-632.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46, Birkhäuser, Boston, 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), 11296-11303. doi: 10.1103/PhysRevB.38.11296.  Google Scholar

[29]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[30]

F. Murat and L. Tartar, H-convergence, Mimeographed notes, Séminaire d'Analyse Fonctionnelle et Numérique, Universitéd'Alger, Boston 1978, (English translation in [31]).  Google Scholar

[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.  Google Scholar

[32]

M. A. Omar, Elementary Solid State Physics: Principles and Applications, World Student Series Edition, Addison-Wesley, Reading, MA, 1975. Google Scholar

[33]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. 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), 571-597; Errata, Ibid. (3).  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), 407-436.  Google Scholar

[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.  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, Annals of the Israel Physical Society, (eds. G. Deutscher, R. Zallen and J. Adler), Israel Physical Society, Jerusalem, 5 (1983), 297-321.  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), 035120. 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), 13016-13027. 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), 2180-2198. 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), 3356-3359. 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), 6288-6297. 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), 278-283. 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), 255-267. 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), 33-60. 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), 73-101. 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), 935-969. 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), 509-522. 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), 1468-1484. 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), 715-736. 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), 563-589. 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, Cl. Sci., IV. Ser., 30 (2001), 681-711.  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), 24-27.  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), 300-353.  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), 456-462. 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), 973-1024. 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), 623-632.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations, Progress in Mathematical Physics, 46, Birkhäuser, Boston, 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), 11296-11303. doi: 10.1103/PhysRevB.38.11296.  Google Scholar

[29]

G. W. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.  Google Scholar

[30]

F. Murat and L. Tartar, H-convergence, Mimeographed notes, Séminaire d'Analyse Fonctionnelle et Numérique, Universitéd'Alger, Boston 1978, (English translation in [31]).  Google Scholar

[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.  Google Scholar

[32]

M. A. Omar, Elementary Solid State Physics: Principles and Applications, World Student Series Edition, Addison-Wesley, Reading, MA, 1975. Google Scholar

[33]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292. 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), 571-597; Errata, Ibid. (3).  Google Scholar

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