Application of $J$-Integral to a Random Elastic Medium

TL;DR

This paper explores the application of the $J$-integral to evaluate the statistical properties of energy release rates in a random elastic medium with spatially variable elastic modulus, using Monte Carlo simulations.

Contribution

It demonstrates that the classical $J$-integral maintains path independence for mean energy release rate in homogeneous random fields, but not for higher moments.

Findings

Path independence holds for mean energy release rate.

Higher moments of energy release rate are path dependent.

Correlation length influences energy release rate statistics.

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 homogeneous 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.