Homogenization and optimal design of an elastic material reinforced by means of quasi-periodic fibres of higher rigidity

We consider an elastic material reinforced by means of identical fibres of size rεrε, which are built with another material, which are εε-periodically distributed in the plane {x3=0}{x3=0}, but which make a variable angle with respect to this plane. We describe the asymptotic behaviour of this material when εε goes to 0. We exhibit the limit law of the material using the ΓΓ-convergence method. Depending indeed on the size of rεrε and of the Lamé coefficients of the elastic material building the fibres, we exhibit different limit functionals. We finally study an optimal design problem with respect to the angle.