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 $\epsilon$ of the grid and the ratio $\delta$ between the thickness of the fibers and the period ε. We show that the asymptotic behavior as ε → 0+ and $\delta \to 0+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\epsilon << \delta$ and $\epsilon >> \delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.