Homogenization of a strongly heterogeneous periodic elastic medium

We consider the homogenization of a periodic elastic medium made of two connected components having constants elasticity tensors separated by a third material with thickness of the same order (epsilon)than the basic periodicity cell, and with stiffness tensor proportional to (epsilon)^r, r>0. In the case where r<=2, we identify the homogenized problem. If r>2, we have to assume moreover that there are no body forces in the third material.