A THEoRY is presented to predict the ultimate tensile strength of brittle matrix composites as a function of underlying material parameters, and specifically to investigate the origin of the tough to brittle transition often observed in these materials as the fiber-matrix interfacial sliding resistance tau is increased. The theory relaxes the usual assumption of global load sharing of the load transfer from broken to unbroken fibers in the composite [CURTIN, W. A., J. Am. Ceram. Soc. 74, 2837 (1991)] by taking the load to be equally distributed among only N(f) fibers around a broken fiber (local load sharing). The composite is then modeled as a collection of independent fiber bundles with Nf fibers per bundle, and composite failure occurs when the weakest bundle fails. Composite strength is thus controlled by the strength distribution of size-N(f) bundles, which is calculated here by analytical and simulation techniques. As N(f) --textgreater infinity the global load sharing results for composite strengths are regained, but significant composite strength degradation is predicted for bundle sizes N(f) less-than-or-equal-to 100. An ansatz relating N(f) to material parameters is then proposed and calculations of the strengths of C-Nicalon composites agree well with experiment. Model calculations on a Nicalon-LAS glass composite show that local load sharing effects lead to a tough to brittle transition between 100 and 200 MPa, much lower than predicted by the global load sharing theory although still larger than found experimentally.