We apply computer simulation techniques to obtain the clastic moduli of a matrix containing circular holes. As the area fraction of holes increases, the Young's modulus of the composite decreases from E0 until it eventually vanishes at the percolation threshold. We study three distinct geometries: (a) periodically centered circular holes on a honeycomb lattice, (b) periodically centered circular holes on a triangular lattice, and (c) randomly centered circular holes. All three cases have the same dilute limit that can be calculated exactly. By examining the narrow necks between adjacent circles, we have calculated the critical behavior for the regular cases and obtain critical exponents of 12 or 32, depending on the local breakdown mode at the necks. For (c) we compare our results with an effective-medium theory, which predicts that the Poisson's ratio tends to 13 as the percolation threshold is approached, independent of its value in the pure system. Our results are also compared with recent experimental results. Based on this work, we propose that the relative Young's modulusEE0of a two-dimensional sheet containing circular holes, overlapping or not, is the same for ail materials, independent of the Poisson's ratio v0, for any prescribed geometry.