Patterns and scaling in surface fragmentation processes

We consider a finite-element model for the fragmentation of a coating covering a bulk material. The coating breaks under a quasistatical, slowly increasing strain (induced, e.g., by temperature changes, by desiccation, or by mechanical deformations). We model the coating through an array of springs and account for its statistical inhomogeneities by assigning each spring a breakdown threshold taken from a given probability distribution (PD). The adhesion to the bulk is modeled through other springs, which connect the coating to the substratum. We consider the dependence on the strain of the mean fragment size and also the ensuing pattern of cracks. We find that the mean fragment size obeys a power-law dependence on the strain; the exponent of the power law is related to the strength of disorder (i.e.; the behavior of the assumed PD for breakdown thresholds in the vicinity of zero). Moreover, the mode of fragmentation also depends on the disorder's strengths: for small disorder (narrow PDs) the system fragments through crack propagation, for strong disorder (wide PD, starting from zero) the cracks are formed by the coalescence of initially independent point defects.