Geometric rules for the annihilation dynamics of step lines on fracture fronts

The roughness of a fracture surface records a crack's complex path through a material and can affect the resultant frictional or fluid transport properties of the broken medium. For brittle fractures, some of the most prominent surface features are long, step-like discontinuities called step lines. In heterogeneous materials, the mean crack surface roughness created by these step lines is well captured by a simple, one-dimensional ballistic annihilation model, which assumes the creation of these steps is a random processes with a single probability that depends on the heterogeneity of the material, and that their destruction occurs via pairwise interactions. Here, through an exhaustive study of experimentally generated crack surfaces in brittle hydrogels, we examine step interactions and show that interaction outcomes depend on the geometry of the incoming steps. The rules that govern step interactions can be categorized into three unique classes and are fully described, providing a complete framework for predicting fracture roughness.