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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 $\varepsilon$ of the grid and the ratio $\delta$ between the thickness of the fibers and the period $\varepsilon$. We show that the asymptotic behavior as $\varepsilon\to0^+$ and $\delta\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\varepsilon\ll\delta$ and $\varepsilon\gg\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.