Generalized linear sampling method for the inverse elastic scattering of fractures in finite bodies

In this work, we consider a mathematical foundation for elastic-wave reconstruction of multiply-connected fractures in a finite solid body by way of the Generalized Linear Sampling Method. To allow for the presence of partially-closed fractures as is often the case in non-destructive testing applications, we describe the contact behavior of 'hidden' fractures by the Schoenberg's linear slip model. We further assume that the elastic body (probed for fractures) is supported by either by Robin or Dirichlet boundary conditions, while the rest of its boundary is taken to be of Neumann type and used for both fracture illumination and gathering of sensory data. To cater for the application of GLSM to laboratory ultrasonic sensing of fractures in slab-like rock specimens, we assume the finite elastic body to be a subset of ℝ² however the results of this study are mostly dimension-generic and allow for a straightforward extension to inverse elastic scattering in finite subsets of ℝ³.