Determination of crack spacing and penetration due to shrinkage of a solidifying layer

A method based on energy minimisation is used to determine the spacing and penetration of a regular array of cracks in a layer of material whose thickness is increasing as it solidifies from a liquid. After solidification, the slab shrinks and subsequently cracks due to internal stresses. A simple Stefan solidification model is used to determine the thickness of the slab as time progresses as well as the temperature profile in the slab. The key feature of the results is that a minimum crack spacing occurs early in the solidification process and this minimum defines a basic spacing for the crack array. The minimum spacing occurs for a range of constraints (boundary conditions) and thermal profiles in the material, indicating the robustness of the phenomenon. Cracks propagating with the unique minimum spacing are subject to a period doubling instability that acts to coarsen the crack pattern, which brings the crack spacing close to the minimum energy state for later time. Good numerical comparison between the crack spacings predicted by energy minimisation and those observed in basalt columns is demonstrated.