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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

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.
Citation: Mohamed Camar-Eddine, Laurent Pater. Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures. Networks and 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.

[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).

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.

[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).

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