J-integral in random elastic medium

The J-integral is a well-known path integral used to calculate the energy release rate in an elastic, homogeneous medium with a localized crack. In this study, it is extended to a heterogeneous medium, where Young’s modulus varies spatially according to a lognormal random field. It is shown that the mean of the J-integral remains path-independent and accurately represents the mean energy release rate. However, in a random medium, the mean energy release rate differs slightly from the deterministic case based on the average modulus. The coefficient of variation (CoV) of the J-integral, on the other hand, is generally path-dependent. This means that the variability in the energy release rate is influenced by the specific path chosen for the calculation of the J-integral. The CoV of the J-integral matches the CoV of the energy release rate only when the domain enclosed by the contour is smaller than the correlation length of the elastic modulus field. When the contour encloses a larger domain, the J-integral significantly underestimates the variability in energy release rate. This indicates that, for larger domains, a simple path-based approach is inadequate to fully capture the statistical variations introduced by material heterogeneity. To accurately capture the energy release rate statistics for arbitrary contours, an additional term is derived for the J-integral. This new term is a domain integral that accounts for material heterogeneity. Furthermore, the analysis reveals that the correlation length of the elastic modulus field has minimal impact on the mean energy release rate but strongly influences its variability. Specifically, for a given specimen, the CoV of the energy release rate decreases significantly as the correlation length decreases.