A pure stress formulation for modeling elastic waves using central finite differences
A pure stress-based finite difference formulation is introduced for modeling elastic wave propagation in linear elastic solids with spatial heterogeneity. The approach derives from the strong form of the elastodynamic equation of motion, in which stress is the only dependent variable. A standard second-order central difference scheme is applied to discretize the equation of motion, allowing the space-time-dependent evolution of stress components to be modeled. Numerical dispersion analysis is performed for homogeneous, elastically isotropic materials. Simulations are then carried out for a spatially heterogeneous case consisting of a bimaterial with stiffness heterogeneity. This bimaterial case allows comparison with known closed-form solutions for reflection and transmission coefficients and with an analogous displacement-based finite difference model. Simulations are executed on modern graphics processing unit architectures, enabling stress-based modeling of large-scale three-dimensional problems exceeding one billion degrees of freedom. The approach shows promise for ultrasonic simulations in materials with stiffness heterogeneity and uniform mass density, conditions common in polycrystalline metals used in engineering applications. The formulation offers a potential alternative means of modeling wave propagation and scattering in heterogeneous materials, with possible applications in nondestructive evaluation, materials characterization, biomedical ultrasound, and geosciences.