Curvature, a mechanical link between the geometrical complexities of a fault: application to bends, kinks and rough faults

Many recent studies have tried to determine the influence of geometry of faults in earthquake mechanics. However, it still remains largely unknown, and it is explored mainly with numerical models. In this paper, we will try to understand how exactly does the geometry come into play in the mechanics of an earthquake from analytical perspective. We suggest a new interpretation of the effect of geometry on the stress on a fault, where the curvatures of the fault that multiply the slip play a major role. Starting from the representation theorem, which links the displacement in a medium to the slip distribution on its boundary, and assuming a 3-D, homogeneous, infinite medium, a regularized boundary-element equation can be obtained. Using this equation, it is possible to separate the influence of geometry, as expressed by the curvatures and torsions of the field lines of slip on the fault surface, which multiply the slip, from the effect of the gradient of slip. This allows us to shed new light on the mechanical effects of geometrical complexities on the fault surface, with the key parameters being the curvatures and torsions of the slip field lines. We have used this new approach to show that, in 2-D static in-plane problems, the shear traction along the fault is mainly controlled by the gradient of slip along the fault, while the normal traction is mainly controlled by the slip that multiplies the curvature along the fault. Finally, we applied this new approach to re-interpret the effect of roughness (why there is a need for a minimum lengthscale in linear elasticity, how to study mechanically the difference of roughness measurements with the direction of slip, scaling of slip distribution versus geometry), bends and kinks (what is the difference between the two, are they sometimes equivalent), as well as to explain further the false paradox between smooth-and-abrupt-bends. This unified framework allows us to improve greatly our understanding of the effect of fault geometry on the mechanics of earthquakes.