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 r ε , which are built with another material, which are ε -periodically distributed in the plane { x 3 = 0 } , but which make a variable angle with respect to this plane. We describe the asymptotic behaviour of this material when ε goes to 0. We exhibit the limit law of the material using the Γ -convergence method. Depending indeed on the size of r ε 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.