Traditionally, ultrasonic waves in heterogeneous elastic materials are modeled using displacement fields through Cauchy's elastodynamics and Hooke's law. This approach creates equations with spatial gradients on the stiffness tensor, forming the basis for canonical models like Weaver (1990), Stanke and Kino (1984), and Hirsekorn (1982). This work presents an alternative stress-based formulation where elastic waves are modeled as propagating and scattering stress fields rather than displacements. The derived stress equations feature spatial gradients on density rather than stiffness. For single-phase polycrystals with uniform density, this yields a standard tensorial wave equation without density heterogeneity complications. Numerical solutions using finite difference schemes on Dream3D-generated polycrystalline materials demonstrate the method's effectiveness in predicting grain scattering effects on wave attenuation and dispersion. Results are compared against analytical predictions from Weaver's model using Von Kármán statistics. The stress-based approach offers significant advantages, particularly simplified Dirichlet-type boundary conditions compared to the complex Von Neumann conditions required in displacement-based simulations.
Kube et al. (Wed,) studied this question.