Analysis of Landau–Lifshitz and neo-Hookean models for static and dynamic acoustoelastic testing

Abstract

A comparison of three different isotropic non-linear elastic models uncovers subtle but important differences in the acoustoelastic responses of a material slab that is subjected to dynamic deformations during a pump-probe experiment. The probe wave deformations are small and are superimposed on larger underlying deformations using three different models: Landau–Lifshitz (using its fourth-order extension), compressible neo-Hookean model (properly accounting for volumetric deformations), and an alternative neo-Hookean model (fully decoupled energies due to distortional isochoric and volumetric deformations). The analyses yield elasticity tensors and respective expressions for the propagation speeds of P-wave and S-wave probes for each model. Despite having many similarities, the different models give different predictions of which probe wave types will have speeds that are perturbed by different pump wave types. The analyses also show a conceptual inconsistency in the Landau–Lifshitz model, that a simple shear deformation induces a stress and a shear wave probe speed that depend on the second-order elastic constant λ, which controls resistance to volumetric changes and thus should not be present in the expressions for shear stress and shear wave probe speeds. Thus, even though the Landau–Lifshitz model is widely used, it may not always be the best option to model experimental data.