The investigation of statistical scaling in localizationinduced failures dates back to da Vinci's speculation on the length effect on rope strength in 1500 s. The early mathematical description of statistical scaling emerged with the birth of the extreme value statistics. The most commonly known mathematical model for statistical scaling is the Weibull size effect, which is a direct consequence of the infinite weakestlink model. However, abundant experimental observations on various localizationinduced failures have shown that the Weibull size effect is inadequate. Over the last two decades, two mathematical models were developed to describe the statistical size effect in localizationinduced failures. One is the finite weakestlink model, in which the random structural resistance is expressed as the minimum of a set of independent discrete random variables. The other is the level excursion model, a continuum description of the finite weakestlink 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 microelectromechanical systems (MEMS) devices, breakdown statistics of high– k gate dielectrics, and probability distribution of buckling pressure of spherical shells containing random geometric imperfections. In addition, the implications of statistical scaling for the stochastic finite element simulations and the reliabilitybased structural design are discussed. In particular, the recent development of the sizedependent safety factors is reviewed.
This content will become publicly available on January 1, 2025
 Award ID(s):
 2029641
 NSFPAR ID:
 10525831
 Publisher / Repository:
 Elsevier
 Date Published:
 Journal Name:
 Journal of the Mechanics and Physics of Solids
 Volume:
 182
 Issue:
 C
 ISSN:
 00225096
 Page Range / eLocation ID:
 105479
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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