Linear and Non Linear Homogenization of a Composite Medium

In this book we consider in the first time the homogenization of a heat transfer linear problem between two periodic connected media exchanging a heat fux throughout their common interface. The interfacial exchange coeffcient is assumed to tend to zero or to infinity when the size of the basic cell tends to zero. Three homogenized problems are determined according to some critical value. In the second time we study the homogenization of a nonlinear problem posed in a fibre-reinforced composite with matrix-fibres interfacial condition. Using Γ-convergence methods three homogenized problems are determined. The main result is that the effective constitutive relations reveal non-local terms associated with the microscopic interactions between the matrix and the fibers.