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{\bf Homogenization of Fiber Reinforced Microstructures:\\
The Extremal Cases}

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Marco Barchiesi\\
Department of Mathematical Sciences\\
Carnegie Mellon University \\
Pittsburgh, PA  15213\\
marcob@andrew.cmu.edu 

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and

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Gianni Dal Maso\\
SISSA\\
Trieste, Italy\\
dalmaso@sissa.it 
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\noindent {\bf Abstract}:  We analyze the as umptotic behavior of the
antiplane deformations of a {\it fragile} material reinforced by a reticulated
{\it elastic} structure.  The microscopic geometry of this material is
described by means of two ``small'' parameters:  The size $\epsilon$ of the
periodic grid and the ratio $\delta$ between the thickness of each of the
fibers and their period of distribution.  We show that this behavior depends
dramatically on the relative size of the parameters.  Indeed, in the two
considered cases, {\it i.e.} $\epsilon \ll \delta$ and $\epsilon \gg \delta$,
we obtain two different limit models:  a perfectly elastic model and an
elastic model with macroscopic cracks, respectively.

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