The effective conductivity equation for a highly heterogeneous periodic medium

摘要

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.