A remark about the periodic homogenization of certain composite fibered media

François Murat; Ali Sili; François Murat; Ali Sili

doi:10.3934/nhm.2020006

Networks and Heterogeneous Media

2020, Volume 15, Issue 1: 125-142. doi: 10.3934/nhm.2020006

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A remark about the periodic homogenization of certain composite fibered media

François Murat ^{1
,},
Ali Sili ^{2
,
,}

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

Received: 01 May 2019 Revised: 01 September 2019
Primary: 35B25, 35B27, 35B40; Secondary: 76M50, 74K10

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.
- Composite media,
- fibers,
- strange term,
- nonlocal problem,
- homogenization,
- dimension reduction
Citation: François Murat, Ali Sili. A remark about the periodic homogenization of certain composite fibered media[J]. Networks and Heterogeneous Media, 2020, 15(1): 125-142. doi: 10.3934/nhm.2020006

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Abstract

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.

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