A consistent derivation of soil stiffness from elastic wave speeds

Elastic wave speeds are fundamental in geomechanics and have historically been described by an analytic formula that assumes linearly elastic solid medium. Empirical relations stemming from this assumption were used to determine nonlinearly elastic stiffness relations that depend on pressure, density, and other state variables. Evidently, this approach introduces a mathematical and physical disconnect between the derivation of the analytical wave speed (and thus stiffness) and the empirically generated stiffness constants. In our study, we derive wave speeds for energy-conserving (hyperelastic) and non-energy-conserving (hypoelastic) constitutive models that have a general dependence on pressure and density. Under isotropic compression states, the analytical solutions for both models converge to previously documented empirical relations. Conversely, in the presence of shear, hyperelasticity predicts changes in the longitudinal and transverse wave speed ratio. This prediction arises from terms that ensure energy conservation in the hyperelastic model, without needing fabric to predict such an evolution, as was sometimes assumed in previous investigations. Such insights from hyperelasticity could explain the previously unaccounted-for evolution of longitudinal wave speeds in oedometric compression. Finally, the procedure used herein is general and could be extended to account for other relevant state variables of soils, such as grain-size, grain-shape, or saturation.