Statistical models for predicting failure probability of brittle materials are investigated. A formula is derived from a physical consideration for the fracture of microcracks in materials based on the general forms of a fracture criterion and a statistical distribution function incorporating the weakest link principle. The relationships of this model and other statistical models in the literature are discussed; they were found to be equivalent for isotropic materials in which microcracks are randomly distributed in all directions. The statistical model is also used in a failure analysis of the round-notch four-point bending specimen made of an AISI 1008 steel. The grain boundary carbide particles are considered to be microcracks in the plastic zone near the notch tip. The distribution function in the statistical theory is derived from the density and size distribution of carbide particles in the steel. The statistical theory for a triaxial stress state is used to predict the failure probability for any given load on the specimen. The failure loads (loads corresponding to 50 percent of failure probability) are calculated for the specimen at different temperatures. The results are compared with experimental data; good agreement is obtained.

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