# Crack modelling

Fracture mechanics theory in college was never my forte, whether that was because it was taught by a particularly dull professor who thought he had a loud voice or because it was just a dry subject. Rule 1: If a crack forms in a solid material, it’s because the material is stressed, and the crack will propagate in the direction that reduces that stress. Rule 2: Cracks will move towards each other. This sounds simple enough; it’s when you try to show simple things in math that they get both complicated and boring.

Now, what would happen when a material is stressed such that two collinear cracks form (with a small offset) and move towards each other? You’d expect the cracks to meet, form one even bigger crack and break the material in two if need be. That’s exactly what doesn’t happen: as the two cracks meet head-on or nearly head-on, they briefly repel each other – as if they were electric charges – move some distance away and then resume their attractive relationship.  Fracture mechanics theory doesn’t account for this behaviour, although it accounts for a lot otherwise, and it’s been an outstanding problem in mechanical engineering.

Physicists from France and Russia have found a potential way out in the form of scale-dependent interactions. They created computer simulations of cracks and then deployed finite element analysis – a technique that divvies up a material into really small constituents, individually measures the forces acting on each one of them and then integrates them all to generate the big picture. They found that the repulsive behaviour was a product of mechanical forces that depended on two numbers only: the length of the cracks before they entered the repulsive mode (called L) and the distance, or offset, between the cracks (called x and y for the respective axes).

The repulsive mode comes into play if the distance between the cracks as they approach each other becomes smaller than 1% of L. At this point, how much they repel each other by, represented by an angle θ, depends on x and y. Using their predictions, the physicists were able to model cracks with θ up to 18º – a good sign in case anyone was wondering whether the scale-dependent interactions they’d found actually worked, although independent experiments will have to verify it. Nonetheless, that fracture behaviour depends on the scale across which interactions occur and not on the overall size of the cracks is fascinating because one can now explain features both very large and very small using the same theory.

… our analysis could help in understanding why spreading centres observed in geological situations and involving hundreds of kilometres long ridges interacting on a scale of a few hundred meters, commonly exhibit a repulsive deviation of their trajectories before overlapping [9,30,31]. The ability to model at small scales the attraction-repulsion transition during propagation of en passant cracks is especially relevant for industrial applications that involve a control of cracking behaviour such as in mechanical sensors [28] or stretchable electronics [27].

The study was published in the journal Physical Review Letters on June 20, 2018.

I could find this paper on arXiv, which doesn’t happen often, so thanks to Sci-Hub for getting me covered!