Wave scattering by cracks in inhomogeneous continua using BIEM

Abstract

Elastic wave scattering by cracks in inhomogeneous geological continua with depth-dependent material parameters and under conditions of plane strain is studied in this work. A restricted case of inhomogeneity is considered, where Poisson's ratio is fixed at one-quarter, while both shear modulus and density profile vary with depth, but proportionally to each other. Furthermore, time-harmonic conditions are assumed to hold. For this type of material, the body wave speeds remain macroscopically constant and it becomes possible to recover exact Green functions for the crack-free inhomogeneous continuum by using an algebraic transformation method proposed in earlier work as a first step. In a subsequent step, the complete elastodynamic fundamental solution along with its spatial derivatives and an asymptotic expansion for small argument, are all derived in closed-form using the Radon transform. Next, a non-hypersingular, traction-based boundary integral equation method (BIEM) formulation for solving boundary-value problems with internal cracks is presented. Specifically, the BIEM is used for computing stress intensity factors and scattered wave displacement amplitudes for an inclined line crack in an otherwise inhomogeneous continuum that is being swept by either pressure waves or vertically polarized shear waves at an arbitrary angle of incidence with respect to the horizontal direction. The numerical results obtained herein reveal substantial differences between homogeneous and continuously inhomogeneous materials containing a crack in terms of their dynamic response, where the latter case is assumed to give a more realistic representation of geological deposits compared to the former one. Finally, these types of examples plus the benchmark case used for validation, serve to illustrate the present approach and to show its potential for solving complex problems not just in geophysics, but also in mechanics.