Non-Riemannian and fractal geometries of fracturing in geomaterials

The mechanism of earthquakes is presented by use of the elastic dislocation theory. With consideration of the continuous dislocation field, the general problem of medium deformation requires analysis based on non-Riemannian geometry with the concept of the continuum with a discontinuity ("no-more continuum"). Here we derive the equilibrium equation (Navier equation) for the continuous dislocation field by introducing the relation between the concepts of the continuous dislocation theory and non-Riemannian geometry. This equation is a generalization of the Laplace equation, which can describe fractal processes like diffusion limited aggregation (DLA) and dielectric breakdown (DB). Moreover, the kinematic compatibility equations derived from Navier equation are the Laplace equations and the solution of Navier equation can be put in terms of functions which satisfy the biharmonic equation, suggesting a close formal connection with fractal processes. Therefore, the relationship between the non-Riemannian geometry and the fractal geometry of fracturing (damage) in geomaterials as earthquakes can be understood by using the Navier equation. Moreover, the continuous dislocation theory can be applied to the problem of the earthquake formation with active folding related with faulting (active flexural-slip folding related to the continuous dislocation field).