Finite difference simulation of interacting resonances and pulse waves in a nonlinear material

Abstract

The field of nonlinear acousto-elastic behaviour in materials such as rocks is an area of active research, applicable to phenomena such as earthquakes or material fatigue. This nonlinearity arises from the rock microstructure, notably through cracks, and appears in the form of a nonlinear relation between the stress and strain fields within the rock. We study how this nonlinearity manifests when the sample is in either a resonant or a transient state. To do this, we numerically model a sample including a crack and broadcast a low frequency pump wave and a high frequency probe wave through the sample. We use a fourth order finite difference scheme to model the evolution of wave velocity, stress, and strain, then use a form of averaging to represent the cracked, heterogeneous model with an effective homogeneous model. Calculating the nonlinear interactions between the two waves allows us to compare the resonant and transient behaviour. We demonstrate differences in the effective wave velocity, and in the travel time delays between effective velocities with and without a pump source.