Optimal spacing and penetration of cracks in a shrinking slab

A method based on energy minimization is used to determine the spacing and penetration of a regular array of cracks in a slab that is shrinking due to a changing temperature field. The results show a range of different crack propagation behavior dependent on a single dimensionless parameter, being the ratio of the slab thickness and a characteristic length for the material. At low parameter values the minimum energy state can be achieved by continually adding more cracks until a steady state is achieved. At higher values, a minimum crack spacing is reached at finite time, beyond which the cracks are constrained to propagate with the minimum spacing. In the latter case, the uniform propagation is potentially unstable to a spatial period doubling, leading to increasingly complex crack penetration patterns. The energy minimization combined with the period doubling instability provides a means of determining the minimum energy state of cracks for all time. The problem considered here can be seen as a paradigm for cracking phenomena that occur on a large range of scales, from planetary to microscopic.