Beginning of Earthquakes Modeled with the Griffith's Fracture Criterion

We present a source model for the beginning of earthquakes based on the Griffith's fracture criterion. The initial state is a critical state of pre-existing circular fault, which is on the verge of instability. After the onset of instability, the fault grows with a progressively increasing rupture speed, satisfying the condition of fracture energy balance at the crack tip. We investigate the difference in rupture growth patterns in two classes of models that are considered to represent end-member cases. In the first model (Spontaneous model), we assume that the surface energy varies smoothly as a function of position in the crust. In this model, faults with small initial dimensions grow in regions with small surface energy, and those with large initial dimensions, in large surface energy. The rupture velocity increases progressively until it reaches its limiting value. The synthetic velocity seismogram at far field shows a weak initial phase during the transitional stage to limiting velocity. The time taken to reach the limiting velocity is proportional to the initial length of preexisting fault. Therefore, the duration of the weak initial phase scales with the initial length of fault. In the second model (Trigger model), we envisage that there are many pre-existing faults in the crust with various lengths. These faults are stable because they encounter some obstacle at their ends (e.g., fault segmentation, strong asperity, etc.). This situation is modeled with a local increase of surface energy near the ends of fault. An earthquake is triggered when the obstacle is suddenly removed (i.e., sudden weakening) or the stress is suddenly increased locally to overcome the obstacle. Once an earthquake is triggered, the fault growth is governed by the ambient surface energy. In this model, the rupture speed attains its limiting velocity almost instantly. The synthetic velocity seismogram at far field shows an abrupt, linear increase in amplitude without the weak initial phase that appears in the Spontaneous model. Both models can be unified using a trigger factor defined as a fractional perturbation of the surface energy at the ends of fault relative to the ambient surface energy. The Spontaneous model is characterized by a small trigger factor, and the Trigger model, by a large trigger factor. Thus, the seismic initiation phase with and without the slow initial phase can both occur depending on the trigger factor. The variability in the observed seismic initiation phase may represent a variation of surface energy (strength) distribution surrounding the pre-existing cracks. A theoretical consideration of rupture arrest by barriers using the Griffith' s fracture criterion does not support the scaling relation between the nucleation moment and the eventual size of earthquake.