Homogenization of fiber reinforced brittle materials: The extremal cases

Abstract

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 t he 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 3≫ δ, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively. © 2009 Society for Industrial and Applied Mathematics.