On the Path-Dependence of the $J$-Integral Near a Stationary Crack in an Elastic-Plastic Material

The path-dependence of the $J$-integral is investigated numerically via the finite-element method, for a range of loadings, Poisson’s ratios, and hardening exponents within the context of $J2$-flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This construct allows for a dense finite-element mesh within the plastic zone and accurate far-field boundary conditions. Details of the crack tip field that have been computed previously by others, including the existence of an elastic sector in mode I loading, are confirmed. The somewhat unexpected result is that $J$ for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for mode II. The applications of $T$- or $S$-stress, whether applied proportionally with the $K$-field or prior to $K$⁠, have only a modest effect on the path-dependence.