Fracture statistics of brittle materials - parametric model validity

Abstract

For brittle materials, a parametric approach is commonly used in the estimation of the (population) strength distribution (very often simply assumed to be Weibull); validity relies implicitly on the assumed distribution being correct. A number of models leading to failure distributions have been claimed to be justifiable by the physical processes involved (i.e. as having a theoretical basis). These models have been claimed to have been developed by a consideration of aspects such as flaw characteristics, microstructure, fracture mechanics and statistics; observed data are not taken into account. Objectives: to examine critically the theoretical bases proposed for these models (and hence, of the derived distribution) and the means of selection in general. Methods: The validity of all aspects of all known models was examined in detail through a dissection of the arguments, careful identification of the underlying assumptions (both stated and unstated), and a thorough evaluation of their compatibility with materials science, fracture mechanics and statistics. Secondly, the ability to check if an assumed distribution is correct was examined. Results: No argument examined can be considered sound: continuum models violate linear elastic fracture mechanics (LEFM); the Weibull distribution does not follow extreme value theory and does not legitimately derive from LEFM; lattice-based models rely on several restrictive and unlikely requirements. The only certain information about strength distributions is that they are bounded above and below. The power of goodness-of-fit tests is limited for (typically used) small sample sizes, and thus the reverse support for the validity of an assumed distribution from experimental data is weak. Conclusions: No distribution can be assumed a priori. Consequently, in each case, the experimental data must be employed in choosing an appropriate distribution, which must therefore pass an explicit statistical goodness-of-fit test before being accepted. However, sample sizes larger than those commonly used are required.