Vector $J$-Integral Analysis of Crack Interaction With Pre-existing Singularities

In this paper, the mechanics of a semi-infinite crack interacting with near crack-tip singularities (e.g., dislocations) in two-dimensional solids is investigated using the concept of potential energy release rate. The spontaneous relationship between the crack potential energy release rate and the well-known vector conservative integral $Ji$$(i=1,2)$ is derived. It is demonstrated that $J1$ and $J2$ integrals are equally important in solving crack problems. This allows a more rational criterion to be proposed, based on the criterion of maximum energy release rate, to assess the so-called shielding/amplification effect on the crack tip due to the presence of the singularities. It is shown that the new criterion can not only assess the shielding/amplification effect under pure mode I or mode II remote loading, but also efficiently assess crack-singularity interaction under mixed mode remote loading. Simultaneously, it is found by re-examining the $Ji$ integrals that there exists a simple but universal relation among the three values of the vector $Ji$ integral corresponding separately to the contributions induced from the semi-infinite crack tip, the singularity, and the remote loading. Next, a multi-singularity-crack interaction model is addressed, and the closed-form solution is obtained. Finally, as an example, the problem of a single dislocation interacting with a main crack is solved to demonstrate the validity of the proposed model and the new criterion.