Derivation of wave speed for dry and saturated nonlinearly elastic models

Elastic wave speeds are crucial in geomechanics for measuring elastic stiffness. Traditionally, these speeds are calculated using an analytic formula based on a linearly elastic solid medium. However, empirical evidence suggests that stiffness is state-dependent, creating a mismatch between the theoretical assumptions and the observed stiffness constants. Recent advancements have addressed this gap by deriving wave speeds for hyperelastic (energy conserving) and hypoelastic (non-energy-conserving) constitutive models that are dependent on pressure and density. These new derivations align with conventional empirical findings for isotropic states. However, the hyperelastic model predicts variations in the ratio of longitudinal to transverse wave speeds, as observed in experiments and discrete element simulations. This prediction emerges from energy-conservation terms in the model, eliminating the need for fabric assumptions previously considered in research. Our study expands on this by exploring various density-dependent relationships and extending wave speed calculations to saturated granular media. This research enhances our understanding of granular media stiffness in dry and saturated scenarios, offering new insights for interpreting wave speed experiments.