Calculus of Variations and Geometric Measure Theory

M. Barchiesi - G. Dal Maso

Homogenization of fiber reinforced brittle materials: the extremal cases

created by barchiesi on 12 Aug 2008
modified on 12 Jan 2012


Published Paper

Inserted: 12 aug 2008
Last Updated: 12 jan 2012

Journal: SIAM J. Math. Anal.
Volume: 41
Pages: 1874-1889
Year: 2009


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 $\varepsilon\to 0^+$ and $\delta\to 0^+$ 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.

Keywords: Homogenization, Gamma-convergence, fracture mechanics, free discontinuity, multiscale analysis