Foreshocks, aftershocks, and remote triggering in quasi-static fault models

We consider a crack pattern penetrating into a solid material, forming a columnar structure as observed in basalts, tuffs, and also in starches. Using the finite element method, we calculate the strain field ahead of the crack front, in the presence of a uniform lateral expansion simulating the strain induced by cooling (in rocks) or drying (in starches). We observe the tendency of the cracks to adapt dynamically in order to reach a stable polygonal pattern. The polygons have different numbers of sides and different areas, with slightly curved borders. We find that the tendency of the cracks to shift laterally upon penetration depends on the geometry of the neighbor cracks, but it is independent of the global scale of the pattern. We suggest that the mean width of the columns is determined by two different effects: the typical size of the (irregular) crack pattern at the free surface, from which the polygonal pattern develops upon penetration, and the further possibility of terminate some cracks during advance, which leads to a thickening of the columns as a function of depth.