Statistical theory for predicting the failure of brittle materials

Statistical models for predicting failure probability of brittle materials are investigated. A formula is derived from a physical consideration of 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 finite element structural analysis program SAP4 is modified; some new subroutines are developed so that the failure analyses can be made using statistical models. A truss element, a triangular element and a quadrilateral element are each modified and several subroutines are developed to calculate the risk of rupture and the failure probability for each element and for the whole structure. The triangular element and quadrilateral elements are capable of doing failure analyses for plane stress, plane strain and axisymmetric problems. Several sample analyses are also provided to verify the program.

The statistical model is also used in a failure analysis of the rounded 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% failure probability ) are calculated for the specimen at different temperatures. The results are compared with experimental data; good agreement is obtained.