| In Part 1, we formulate a dynamic, statistical lattice model of microscopic fractures based on the physics of 1-D fractures and Glauber's dynamic Ising model. We include spatial heterogeneities, interactions between fractures and interaction of fractures with external forces. We describe the stress-corrosion growth and healing of microfractures, and with the use of Percolation Theory and Dynamic Renormalization Group Theory, we scale to larger scales and describe the fusion of smaller fractures into larger ones and the eventual failure of the material as a phase transition. We also investigate the fractal dimension of the fractures generated by our model and compare it to experimental results. Finally, we investigate the magnitude and temporal distribution of foreshocks as defined by our model. Our results are compared to previous theoretical and experimental studies.; In Part 2, we adapt our model to describe macroscopic fracture propagation in 1-D along a fault plane. Asperities are modeled by a heterogeneous distribution of breaking strengths. We investigate the various regions of the behavior of the fracture, including the propagating, slowing and the non-propagating regimes, and the dynamics of the distribution of the released elastic energy. We compare these results to analogous continuum elastic models and show the advantages of our discrete lattice model. Finally, we investigate the penetration of a front into a region of high barrier strengths through a two region model. |
