Homogenization of Fiber Reinforced Brittle Materials: The Extremal Cases

We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period ε of the grid and the ratio δ between the thickness of the fibers and the period ε. We show that the asymptotic behavior as ε → 0 and δ → 0 depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., ε ≪ δ and ε ≫ δ, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.