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 epsilon. We show that the asymptotic behavior as epsilon -> 0(+) and delta -> 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.