Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

A clear relationship between the population of brittle-fracture controlling flaws generated in a manufactured material and the distribution of strengths in a group of selected components is established. Assumptions regarding the strength-flaw size relationship, the volume of the components, and the number in the group, are clarified and the contracting effects of component volume and truncating effects of group number on component strength empirical distribution functions highlighted. A simple analytical example is used to demonstrate the forward prediction of population distribution and the more important reverse procedure of empirical strength distribution underlying flaw population. Three experimental examples are given of the application of the relationships to state-of-the-art micro- and nano-scale strength distributions to experimentally determine flaw populations: two on etched microelectromechanical systems (MEMS) structures and one on native and oxidized silicon nanowires. In all examples, the minimum threshold strength and conjugate maximum flaw size are very well estimated and the complete flaw population, including the minimum flaw size, are very poorly estimated, although etching, bimodal, and oxidation effects were clearly discernible. The results suggest that the best use of strength distribution information for MEMS manufacturers and designers might be in estimation of the strength threshold.