, Volume 61, Issue 2, pp 231-244
Date: 17 Jan 2012

The effective conductivity equation for a highly heterogeneous periodic medium

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Abstract

We consider the homogenization of a conductivity equation for a medium made up of a set \({F_\varepsilon}\) ( \({\varepsilon}\) being the size of the period of the medium) of highly conductive vertical fibers surrounded by another material (the matrix) assumed to be a poor conductor. The conductivity coefficients in the fibers behave as \({\frac{1}{\varepsilon^2}}\) while whose of the matrix behave as \({\varepsilon^2}\) . We show that the homogenized problem consists of an equality of the kind u(x) = m(x) f (x) where u denotes the macroscopic temperature, f the source term and m(x) a coefficient given by solving some cell equation.

Communicated by Editor in Chief.