Dislocations with corners in an elastic body with applications to fault detection

TL;DR

This paper develops a mathematical framework for modeling elastic dislocations with corners, providing well-posedness results and methods for identifying dislocation surfaces and jump vectors, with applications to fault detection in geophysics.

Contribution

It introduces a variational approach to prove well-posedness and characterizes corner singularities, enabling the inverse problem of fault surface reconstruction.

Findings

Proved well-posedness of the dislocation problem with general Lamé parameters.

Characterized jump vectors at corner points of dislocation surfaces.

Established unique reconstruction methods for dislocation curves and jump vectors.

Abstract

This paper focuses on an elastic dislocation problem that is motivated by applications in the geophysical and seismological communities. In our model, the displacement satisfies the Lam\'e system in a bounded domain with a mixed homogeneous boundary condition. We also allow the occurrence of discontinuities in both the displacement and traction fields on the fault curve/surface. By the variational approach, we first prove the well-posedness of the direct dislocation problem in a rather general setting with the Lam\'e parameters being real-valued $L^\infty$ functions and satisfy the strong convexity condition. Next, by considering that the Lam\'e parameters are constant and the fault curve/surface possesses certain corner singularities, we establish a local characterization of the jump vectors at the corner points over the dislocation curve/surface. In our study, the dislocation is geometrically rather general and may be open or closed. We establish the unique results for the inverse problem of determining the dislocation curve/surface and the jump vectors for both cases.