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Abstract This study investigates the use of the J-integral to compute the statistics of the energy release rate of a random elastic medium. The spatial variability of the elastic modulus is modeled as a homogeneous lognormal random field. Within the framework of Monte Carlo simulation, a modified contour integral is applied to evaluate the first and second statistical moments of the energy release rate. These results are compared with the energy release rate calculated from the potential energy function. The comparison shows that, if the random field of elastic modulus is homogenous in space, the path independence of the classical J-integral remains valid for calculating the mean energy release rate. However, this path independence does not extend to the higher order statistical moments. The simulation further reveals the effect of the correlation length of the spatially varying elastic modulus on the energy release rate of the specimen. 

Abstract The investigation of statistical scaling in localization-induced 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 weakest-link model. However, abundant experimental observations on various localization-induced 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 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 independent discrete random variables. 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 quasi-brittle structures, failure statistics of micro-electromechanical 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 reliability-based structural design are discussed. In particular, the recent development of the size-dependent safety factors is reviewed.