Sub-surface crack in an inhomogeneous half-plane: Wave scattering phenomena by BEM

Elastic wave propagation in a cracked inhomogeneous half-plane is studied herein with the aid of the Boundary Element Method (BEM). More specifically, inhomogeneity arises in the half-plane because of spatial dependence of its material parameters on the depth coordinate. Furthermore, conditions of plane strain are assumed to hold, while the external load is an incident time-harmonic pressure (P) wave. More specifically, inhomogeneity is restricted to the case where both shear modulus and density profiles are quadratic functions of depth, but vary proportionally to each other, while Poisson's ratio is fixed at one-quarter. This way, body wave speeds remain macroscopically constant and it becomes possible to recover appropriate fundamental solution using a functional transformation method in conjunction with the Radon transform. Subsequently, a non-hypersingular, traction-based BEM is developed for solution of this particular problem. The usual quadratic boundary elements are used for discretization of all surfaces, supplemented by special edge-type boundary elements to model crack-tips. The present methodology is validated against standard examples appearing in the literature, and is subsequently used to study boundary-value problems (BVP) involving a sub-surface crack in an inhomogeneous half-plane swept by time-harmonic P-waves. The results of this detailed parametric study demonstrate that surface wave-fields are sensitive to the degree of material inhomogeneity, to the characteristics of the incident wave and especially to the relative position of crack versus free-surface.