A probabilistic approach to modelling ultrasonic shear wave propagation in locally anisotropic heterogeneous media

This article considers the propagation of a high-frequency time harmonic, elastic wave in a spatially heterogeneous, randomly layered material. The material is locally anisotropic, and the material properties change from one layer to the next by a random rotation of the associated slowness surface in the plane of wave propagation. The layer thicknesses and this rotation follow a stochastic (Markovian) process. This situation is found in ultrasonic wave propagation in polycrystalline materials; for example, in the ultrasonic non-destructive testing of welds and additively manufactured metallic components. This work focuses on monochromatic shear waves propagating in a two-dimensional plane. Using the differences in length scales between the ultrasound wavelength, the mean layer size, and the wave propagation distance, a small parameter is identified in the stochastic differential equation that emerges. Its infinitesimal generator leads to a Fokker-Planck equation via limit theorems involving this small parameter. A weak form of the Fokker-Planck equation is derived and then solved via a finite element package. The numerical solution to the Fokker-Planck equation is used to compute statistical moments of the power transmission coefficient. Finally, a parametric study on the effect of the degree of anisotropy (asphericity of the slowness surface) of the material on the transmitted energy is performed.