Faults Self-organized by Repeated Earthquakes in a Quasi-static Antiplane Crack Model

We study a 2D quasi-static discrete crack anti-plane model of a tectonic plate with long range elastic forces and quenched disorder. The plate is driven at its border and the load is transfered to all elements through elastic forces. This model can be considered as belonging to the class of self-organized models which may exhibit spontaneous criticality, with four additional ingredients compared to sandpile models, namely quenched disorder, boundary driving, long range forces and fast time crack rules. In this ”crack” model, as in the ”dislocation” version previously studied, we find that the occurrence of repeated earthquakes organizes the activity on well-defined fault-like structures. In contrast with the ”dislocation” model, after a transient, the time evolution becomes periodic with run-aways ending each cycle. This stems from the ”crack” stress transfer rule preventing criticality to organize in favor of cyclic behavior. For sufficiently large disorder and weak stress drop, these large events are preceded by a complex space-time history of foreshock activity, characterized by a Gutenberg-Richter power law distribution with universal exponent B = 1±0.05. This is similar to a power law distribution of small nucleating droplets before the nucleation of the macroscopic phase in a first-order phase transition. For large disorder and large stress drop, and for certain specific initial disorder configurations, the stress field becomes frustrated in fast time : out-of-plane deformations (thrust and normal faulting) and/or a genuine dynamics must be introduced to resolve this frustration. Résumé : Nous étudions un modèle quasi-statique discret anti-plan de fissures dans une plaque tectonique en présence de forces élastiques à longue portée et de désordre gelé. La plaque est soumise sur ces bords à un taux de cisaillement constant faible qui est retransmis sur tous les éléments de la plaque par les interactions élastiques. Ce modèle peut être vu comme un exemple de modèles auto-organisés qui peuvent donner lieu à une criticalité spontanée, possédant les ingrédients additionnels suivants : désordre gelé, forçage de frontière, transfert à longues distances et règles d’évolution en ”temps rapide” de type ”fissure”. Dans ce modèle de type