Onset of scale-invariant pattern in growth processes: the cracking problem

We discuss a possible mechanism for the onset of scale-invariant pattern when a cracking structure propagates in a continuous medium. We show that sufficiently close to the tip of an evolving arm the stress field is insensitive to the shear on the boundaries far away. We predict the number of major arms given the local relation between the growth rate and the stress field. We find oscillatory modes periodic in the logarithm of the distance from the tip of the pattern. We argue that these solutions lead to initiation of log-periodic corrugations which are unstable and hence develop into fully grown sidebranches with the same spacing pattern. This pattern is scale-invariant and hence this analysis provides a mechanism for the onset of self-similarity in these structures, a phenomenon observed in many simulated and real systems. The relation to the pattern formed by a diffusion controlled growth is discussed.