On power-law tail distribution of strength statistics of brittle and quasibrittle structures

Understanding the tail behavior of the probability distribution of structural strength is of paramount importance for reliability-based design of engineering structures. For brittle and quasibrittle structures, the tail distribution not only determines the design strength at a low failure probability, but also governs the functional form of the strength distribution of large-size structures. There exists clear evidence that the Weibull distribution is applicable to brittle structures. Based on the theory of extreme value statistics, the applicability of Weibull distribution suggests that the tail distribution must follow a power law. The justification of the power-law tail distribution was first proposed by Freudenthal based on an assumed type of flaw statistics and linear elastic fracture mechanics of non-interacting flaws. A series of recent studies suggested that the power-law tail can be explained by the transition rate theory governing the statistics of material failure at the nano-scale. In this study, we investigate the tail distribution by considering the randomness in both material strength and applied stress field. The present analysis adopts a nonlocal strength-based failure criterion, in which the spatial variability of the material strength is represented by an autocorrelated random field. The failure statistics of the structure is calculated as a first passage probability. We analyze this problem in 1D, 2D and 3D settings, and the results indicate that in all cases the tail distribution of the nominal structural strength follows a power law. It is shown that the power-law tail behavior of strength distribution stems from the tail distribution of material strength. The flaw statistics introduces additional randomness to the nominal structural strength, but does not dictate the power-law form of its tail distribution.