A numerical model for the nonlinear interaction of elastic waves with cracks

Abstract

It is reasonably well accepted that cracks play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here, we simulate wave propagation to understand these mechanisms. Following existing techniques, we formulate the stress in terms of its linear and nonlinear contributions. The linear stress is the generalized Hooke’s law involving only the fourth-rank elastic stiffness tensor. The nonlinear stress comes from the product of the fourth- and sixth-rank tensors, and the spatial derivatives of the displacement vector. In a nonlinear isotropic medium, we show that the speeds of P- and S-waves generated by a time-harmonic source-function are not constant over time. In an anisotropic medium, P-wave speed is commonly estimated using effective medium theory. In the linear slip theory, we represent a crack by a displacement discontinuity embedded in an isotropic background. In a cracked medium, the estimated wave speeds show nonlinear behaviors similar to the ones estimated using the direct nonlinear approach. When the particle motion is parallel to the normal of the crack, the variation in P-wave speed is large, indicating that the crack is clapping (opening and closing).