# American Institute of Mathematical Sciences

March  2020, 15(1): 125-142. doi: 10.3934/nhm.2020006

## A remark about the periodic homogenization of certain composite fibered media

 1 Laboratoire Jacques-Louis Lions, Boite courrier 187, Sorbonne Université, 4 place Jussieu, 75252 Paris cedex 05, France 2 Institut de Mathématiques de Marseille (I2M), UMR 7373, Aix-Marseille Univ, CNRS, Centrale Marseille, CMI, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France

* Corresponding author: Ali Sili

Received  May 2019 Revised  September 2019 Published  December 2019

We explain in this paper the similarity arising in the homogenization process of some composite fibered media with the problem of the reduction of dimension $3d-1d$. More precisely, we highlight the fact that when the homogenization process leads to a local reduction of dimension, studying the homogenization problem in the reference configuration domain of the composite amounts to the study of the corresponding reduction of dimension in the reference cell. We give two examples in the framework of the thermal conduction problem: the first one concerns the reduction of dimension in a thin parallelepiped of size $\varepsilon$ containing another thinner parallelepiped of size $r_ \varepsilon \ll \varepsilon$ playing a role of a "hole". As in the homogenization, the one-dimensional limit problem involves a "strange term". In addition both limit problems have the same structure. In the second example, the geometry is similar but now we assume a high contrast between the conductivity (of order $1$) in the small parallelepiped of size $r_ \varepsilon : = r \varepsilon$, for some fixed $r$ ($0 < r < \frac{1}{2}$) and the conductivity (of order $\varepsilon^2$) in the big parallelepiped of size $\varepsilon$. We prove that the limit problem is a nonlocal problem and that it has the same structure as the corresponding periodic homogenized problem.

Citation: François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media. Networks and Heterogeneous Media, 2020, 15 (1) : 125-142. doi: 10.3934/nhm.2020006
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Analysis, 21 (1990), 823-836.  doi: 10.1137/0521046. [3] M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 26 (1998), 407-436. [4] A. Boughammoura, Homogenization and correctors for composite media with coated and highly anisotropic fibers, Elect. J. Differential Equations, (2012), 27 pp. [5] A. Braides, M. Briane and J. Casado-Diaz, Homogenization of non-uniformly bounded periodic diffusion energies in dimension two, Nonlinearity, 22 (2009), 1459-1480.  doi: 10.1088/0951-7715/22/6/010. [6] A. Braides, V.-C. Piat and A. Piatnitski, A variational approach to double-porosity problems, Asympt. Analysis, 39 (2004), 281-308. [7] D. Caillerie and B. Dinari, A perturbation problem with two small parameters in the framework of the heat conduction of a fiber reinforced body, Partial Differential Equations, Warsaw, 19 (1987), 59-78. [8] A. Brillard & M. El Jarroudi, Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibers, IMA J. Appl. Math., 66 (2001), 567-590.  doi: 10.1093/imamat/66.6.567. [9] J. Casado-Diaz, Two-scale convergence for nonlinear Dirichlet problems, Proceed. Royal. Soc. Edinburgh Sect. A, 130 (2000), 249-276.  doi: 10.1017/S0308210500000147. [10] K. D. Cherednichenko and V. P. Smyshlyaev & V. V. Zhikov, Nonlocal limits for composite media with highly anisotropic periodic fibers, Proceed. Royal. Soc. Edinburgh Sect. A, 136 (2006), 87-144.  doi: 10.1017/S0308210500004455. [11] D. Cioranescu & F. Murat, Un terme étrange venu d'ailleurs, In Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vols Ⅱ end Ⅲ (ed. H. Brézis and J.-L. Lions). Research Notes in Mathematics, 60 (1982), 98–138 and 78–154, English translation: A strange term coming from nowhere, Topics in the mathematical modelling of composite materials, ed. by A. Cherkaev & R.V. Kohn, Progress in nonlinear Differential Equations and their Applications, 31 (1997), Birkhäuser, Boston, 44–93. [12] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043. [13] F. Murat and A. Sili, Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes, C.R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1199-1204. [14] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, Jour. Funct. Anal., 18 (1975), 27-59.  doi: 10.1016/0022-1236(75)90028-2. [15] A. Sili, Homogénéisation dans des cylindres minces, C.R. Acad. Sci. Paris Sér. I Math., 332 (2001), 777-782.  doi: 10.1016/S0764-4442(01)01902-4. [16] A. Sili, Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci., 14 (2004), 329–353. doi: 10.1142/S0218202504003258. [17] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084. [2] T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Analysis, 21 (1990), 823-836.  doi: 10.1137/0521046. [3] M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 26 (1998), 407-436. [4] A. Boughammoura, Homogenization and correctors for composite media with coated and highly anisotropic fibers, Elect. J. Differential Equations, (2012), 27 pp. [5] A. Braides, M. Briane and J. Casado-Diaz, Homogenization of non-uniformly bounded periodic diffusion energies in dimension two, Nonlinearity, 22 (2009), 1459-1480.  doi: 10.1088/0951-7715/22/6/010. [6] A. Braides, V.-C. Piat and A. Piatnitski, A variational approach to double-porosity problems, Asympt. Analysis, 39 (2004), 281-308. [7] D. Caillerie and B. Dinari, A perturbation problem with two small parameters in the framework of the heat conduction of a fiber reinforced body, Partial Differential Equations, Warsaw, 19 (1987), 59-78. [8] A. Brillard & M. El Jarroudi, Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibers, IMA J. Appl. Math., 66 (2001), 567-590.  doi: 10.1093/imamat/66.6.567. [9] J. Casado-Diaz, Two-scale convergence for nonlinear Dirichlet problems, Proceed. Royal. Soc. Edinburgh Sect. A, 130 (2000), 249-276.  doi: 10.1017/S0308210500000147. [10] K. D. Cherednichenko and V. P. Smyshlyaev & V. V. Zhikov, Nonlocal limits for composite media with highly anisotropic periodic fibers, Proceed. Royal. Soc. Edinburgh Sect. A, 136 (2006), 87-144.  doi: 10.1017/S0308210500004455. [11] D. Cioranescu & F. Murat, Un terme étrange venu d'ailleurs, In Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vols Ⅱ end Ⅲ (ed. H. Brézis and J.-L. Lions). Research Notes in Mathematics, 60 (1982), 98–138 and 78–154, English translation: A strange term coming from nowhere, Topics in the mathematical modelling of composite materials, ed. by A. Cherkaev & R.V. Kohn, Progress in nonlinear Differential Equations and their Applications, 31 (1997), Birkhäuser, Boston, 44–93. [12] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043. [13] F. Murat and A. Sili, Problèmes monotones dans des cylindres de faible diamètre formés de matériaux hétérogènes, C.R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1199-1204. [14] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, Jour. Funct. Anal., 18 (1975), 27-59.  doi: 10.1016/0022-1236(75)90028-2. [15] A. Sili, Homogénéisation dans des cylindres minces, C.R. Acad. Sci. Paris Sér. I Math., 332 (2001), 777-782.  doi: 10.1016/S0764-4442(01)01902-4. [16] A. Sili, Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci., 14 (2004), 329–353. doi: 10.1142/S0218202504003258. [17] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009.
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