The tensile strength of ceramic and metal matrix composites is subject to an important role of the fiber/matrix interface. The mechanical properties of this interface dictate the stress concentration that develops in fibers that surround a failed fiber. An analysis of this phenomenon is used to illustrate interface conditions that sufficiently diminish the stress concentration that a global load sharing criterion may be used to prescribe the contribution of the fibers to the composite strength. This, in turn, leads to a criterion for the transition to failure by local load sharing.