Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures

Mohamed Camar-Eddine,Laurent Pater

doi:10.3934/nhm.2013.8.913

Mohamed Camar-Eddine, Laurent Pater

Open Access

https://doi.org/10.3934/nhm.2013.8.913

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Abstract

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.

Highlights

At the end of the 19th century, it was discovered [24] that a constant magnetic field h modifies the symmetric conductivity matrix σ of a conductor into a non symmetric matrix σ(h)
In the Maclaurin series of the perturbed resistivity (σ(h))−1 the zeroth-order term coincides with the resistivity σ−1 in the absence of a magnetic field [27]
We consider the idealized situation when the induced non symmetric part is proportional to the applied magnetic field: σ(h) = αI3 + βE (h), where α and β are two constant real numbers

Summary

Introduction

At the end of the 19th century, it was discovered [24] that a constant magnetic field h modifies the symmetric conductivity matrix σ of a conductor into a non symmetric matrix σ(h). The case of high-contrast conductivities is very different since non classical phenomena, such as nonlocal terms, may appear in the limit problem as shown, for instance, in [19, 25, 1, 18, 11, 26] This does not happen in dimension two if the sequence σn is uniformly bounded from below. The key ingredient of this approach is a fundamental compactness result (see Lemma 3.1) based on a control of high conductivities in thin structures through weighted Poincaré-Wirtinger type inequalities This compactness lemma, combined with the two-dimensional results of [17] and the cylindrical structure of the composite allows us to obtain an explicit formula of σ∗(h), once again, in terms of the transversal homogenized conductivity σ∗(h) and of some bounded function θ which, in some sense, takes account of the distribution of the highly conducting phase Ωn in Ω (see Theorem 3.1).

General notations and definitions

The periodic case

The non periodic case

Circular fibres with variable radius

Thin squared grids