Statistical Scaling in Localization-Induced Failures

Investigation of statistical scaling in localization-induced failures dates back to da Vinci's speculation on the length effect on the rope strength in 1500s. The early mathematical description of statistical scaling stems from the birth of the extreme value statistics. The most commonly known mathematical model for statistical scaling is the Weibull size effect, a direct consequence of the infinite weakest-link model. However, abundant experimental observations on different localization-induced failures showed that the Weibull size effect is inadequate. Over the last two decades, two mathematical models were developed to describe the statistical size effect on localization-induced failures. One is the finite weakest-link model, in which the random structural resistance is expressed as the minimum of a set of discrete independent random variables, and the other is the level excursion model, a continuum description of the finite weakest-link model, in which the structural failure probability is calculated as the probability of the upcrossing of a random field over a barrier. This paper reviews the mathematical formulation of these two models, and their applications to various engineering problems including the strength distributions of quasibrittle structures, failure statistics of micro-electro-mechanical systems (MEMS) devices, breakdown statistics of highk gate dielectrics, and probability distribution of buckling pressure of spherical shells containing random geometric imperfections. The implications of statistical scaling for the stochastic finite element simulations and the reliability-based structural design are discussed. In particular, the recent development of the size-dependent safety factors is reviewed.