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On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth
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 |
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.
|
[2] |
A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland Pub. Co., (1978).
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
M. Briane, Nonlocal effects in two-dimensional conductivity,, Arch. Rational Mech. Anal., 182 (2006), 255.
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.
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.
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.
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.
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.
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.
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.
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, 30 (2001), 681.
|
[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.
|
[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.
|
[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. |
[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. |
[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.
|
[24] |
E. H. Hall, On a new action of the magnet on electric currents,, Amer. J. Math., 2 (1879), 287.
doi: 10.2307/2369245. |
[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. |
[26] |
E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations,, Progress in Mathematical Physics, (2006).
|
[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. |
[29] |
G. W. Milton, The Theory of Composites,, Cambridge Monographs on Applied and Computational Mathematics, (2002).
doi: 10.1017/CBO9780511613357. |
[30] |
F. Murat and L. Tartar, H-convergence,, Mimeographed notes, (1978).
|
[31] |
F. Murat and L. Tartar, H-convergence,, Topics in the Mathematical Modelling of Composite Materials, (1998), 21.
|
[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. |
[34] |
S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571.
|
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.
|
[2] |
A. Bensoussan, J.-L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures,, North-Holland Pub. Co., (1978).
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[10] |
M. Briane, Nonlocal effects in two-dimensional conductivity,, Arch. Rational Mech. Anal., 182 (2006), 255.
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.
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.
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.
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.
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.
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.
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.
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, 30 (2001), 681.
|
[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.
|
[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.
|
[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. |
[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. |
[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.
|
[24] |
E. H. Hall, On a new action of the magnet on electric currents,, Amer. J. Math., 2 (1879), 287.
doi: 10.2307/2369245. |
[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. |
[26] |
E. Ya. Khruslov and V. A. Marchenko, Homogenization of Partial Differential Equations,, Progress in Mathematical Physics, (2006).
|
[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. |
[29] |
G. W. Milton, The Theory of Composites,, Cambridge Monographs on Applied and Computational Mathematics, (2002).
doi: 10.1017/CBO9780511613357. |
[30] |
F. Murat and L. Tartar, H-convergence,, Mimeographed notes, (1978).
|
[31] |
F. Murat and L. Tartar, H-convergence,, Topics in the Mathematical Modelling of Composite Materials, (1998), 21.
|
[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. |
[34] |
S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche,, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571.
|
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