Statistical size effect in quasi-brittle structures: II. Nonlocal theory

The failure probability of structures must be calculated from the stress field that exists just before failure, rather than the initial elastic field. Accordingly, fracture-mechanics stress solutions are utilized to obtain the failure probabilities. This leads to an amalgamated theory that combines the size effect due to fracture energy release with the effect of random variability of strength having Weibull distribution. For the singular stress field of linear elastic fracture mechanics, the failure-probability integral diverges. Convergent solution, however, can be obtained with the nonlocal-continuum concept. This leads to nonlocal statistical theory of size effect. According to this theory, the asymptotic size-effect law for very small structure sizes agrees with the classical-power law based on Weibull theory. For very large structures, the asymptotic size-effect law coincides with that of linear elastic fracture mechanics of bodies with similar cracks, and the failure probability is dominated by the stress field in the fracture-process zone while the stresses in the rest of the structure are almost irrelevant. The size-effect predictions agree reasonably well with the existing test data. The failure probability can be approximately calculated by applying the failure-probability integral to spatially averaged stresses obtained, according to the nonlocal-continuum concept, from the singular stress field of linear elastic fracture mechanics. More realistic is the use of the stress field obtained by nonlinear finite element analysis according to the nonlocal-damage concept.