The evolutionary history of elastic materials subjected to the shear stress and the high confining pressures encountered in seismological applications is simulated by a set of avalanche models. Some models exhibit bimodal fracture-size distributions reminiscent of a first-order phase transition, while others have scaling behavior. We argue that this scaling behavior arises as a consequence of the high degree of self-generated roughness. In all models the scaling region broadens as surface roughness is increased. An epidemic growth model, closely related to the avalanche models, is introduced to understand the statistical properties obtained in the simulations. By means of these alternative formulations we interpret the scaling in the size distributions of the avalanche models as a finite-size effect; the system self-organizes, tuning itself to a size-dependent steady state which exhibits an apparent criticality. A value for the scaling exponent, determined from the epidemic model by way of correlated percolation theory, is in good agreement with the numerical result. These problems have relevance to sandpile and earthquake models and to aspects of self-organized criticality.

• Received 31 October 1991

DOI:https://doi.org/10.1103/PhysRevA.45.2211