Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres

We describe the asymptotic behaviour of a cylindrical elastic body, reinfor ced along identical epsilon -periodically distributed fibres of size r(epsi lon), with 0 < r(epsilon) < epsilon, filled in with some different elastic material, when this small parameters goes to 0. The case of small deformati ons and small strains is considered. We exhibit a critical size of the fibr es and a critical link between the radius of the fibres and the size of the Lame coefficients of the reinforcing elastic material. Epi-convergence arg uments are used in order to prove this asymptotic behaviour. The proof is e ssentially based on the construction of appropriate test functions.