Predicting the direction of a crack submitted to mixed mode loading is key in a variety of domains ranging from earth science to structure safety. In the context of Linear Elastic Fracture Mechanics (LEFM), many bifurcation criteria have been proposed overtime: the Principle of Local Symmetry (PLS) [1], the Maximum Tangential Stress (MTS)[2], the Strain Energy Density (SED) criterion[3], ... The MTS and PLS are the most widely used and give extremely similar results [4].

Whether the theoretical framework of LEFM+PLS is sufficient to study propagation paths of interacting cracks is still under debate. The simplest case of crack interaction is sometimes referred in the literature as en-passant cracks pairs (EP-cracks): two initially straight, parallel and offset cracks which are submitted to far-field opening stress. It was observed experimentally many time that EP-cracks propagate straight ahead until their inner tips overlap, where they begin to attract one another. This behavior can be reproduced correctly assuming the PLS [5]. However, it has been observed that cracks may repel one another in some instances [6]. In this case, it seems that the PLS systematically underestimates the angle of repulsion [7].

We present a simple iterative method, based on FEM computation of the stress state at a given time and determination of the SIF after an infinitesimally small propagation step, to compute quickly and accurately the crack propagation direction in the context of LEFM and under the assumption of PLS. Applying it to the case of EP-cracks, we were able to determine under which conditions the trajectories were initially repulsive or attractive. We find surprising results, among which the fact that perfectly aligned cracks do not interact and that repulsion is a non-monotonous function of the inner tips' separation distances.

This method is robust and fast enough to be iterated in order to compute full crack paths. We numerically reproduced the trajectories observed in the experiments presented in an earlier paper [6]. This study allowed us to provide further insights into quantifying the impact of boundary and initial conditions on the shape of crack paths, and into the limitations of applying the LEFM+PLS framework to the case of interacting cracks.

References:

[1] B. Cotterell and J.R. Rice. Slightly curved or kinked cracks. International Journal of Fracture, 16(2):155–169, 1980.

[2] F Erdogan and G C Sih. On the crack extension in plane loading and transverse shear. Journal Basic Engr., 85:519–527, 1963.

[3] G.C. Sih. Some basic problems in fracture mechanics and new concepts. Engineering Fracture Mechanics, 5(2):365–377, 1973.

[4] J.B. Leblond. Mécanique de la rupture fragile et ductile. Hermes Science publication, Editions Lavoisier, 2003.

[5] Melissa L. Fender, Frédéric Lechenault, and Karen E. Daniels. Universal shapes formed by two interacting cracks. Physical Review Letters, 105(12):2–5, 2010.

[6] Marie-Julie Dalbe, Juha Koivisto, Loïc Vanel, Amandine Miksic, Osvanny Ramos, Mikko Alava, and Stéphane Santucci. Repulsion and Attraction between a Pair of Cracks in a Plastic Sheet. Physical review letters, 114(20):205501, 2015.