Elasticity in consolidated granular materials exhibits non-classical nonlinearity and slow dynamics. These effects are typically analyzed through separate experimental methods and theoretical models. Our research aims to unify these descriptions by introducing a model based on non-equilibrium strain, incorporating a wide range of relaxation times to account for both fast nonlinear effects and slow conditioning and relaxation of elastic properties. Utilizing observations from dynamic acoustoelastic testing of sandstone, we propose a time-delay model that accurately reflects the observed experimental behaviors, including log-time relaxation and hysteresis. Our findings indicate that the slow and fast dynamics in sandstone are intrinsically coupled, and the model provides a comprehensive framework for understanding these complex interactions. This model, which is validated by fitting experimental data including conditioning loops and relaxation curves, offers a tool for predicting the elastic behavior under various loading conditions.

Materials with grain contacts or partially closed cracks exhibit anomalous elastic behavior: hysteresis in quasi-static experiments and slow dynamics in fast dynamic ones. Albeit the behavior in the two cases (which correspond to very different strain ranges) appears different, it should stem from the same physics and, thus, could be modeled by a universal equation of state. We propose a modification of the standard acoustoelastic theory, introducing the concept of conditioning induced non-equilibrium strain, which allows us to predict the evolution of elastic wave velocity in both quasi-static and dynamic ranges, including the velocity anisotropy induced by external uniaxial loading.

Hysteretic nonlinear elasticity is often observed in consolidated granular media, including concrete, mortar, sandstones, or rocks. Nonlinearity is frequently quantified using Nonlinear Resonant Ultrasonic Spectroscopy (NRUS), which provides tools to define nonlinear parameters for both fast and slow dynamic effects, often observed when analyzing the propagation velocity dependence on strain in such materials. The dependence of these parameters on temperature was studied with the aim of using NRUS to quantify the induced thermal damage; thus, experiments were performed spanning a wide temperature range. However, since most of these materials are used in construction (concrete and sandstone, mostly), it is of interest to understand how sensitive the measured nonlinear parameters are to small environmental temperature fluctuations. In this paper, the dependence on temperature of elastic parameters is investigated, both linear (wave velocity and damping) and nonlinear (the slope and hysteresis of the curves describing the strain dependence of wave velocity and residual conditioning effect on wave velocity), separating the slow from the fast dynamic properties of nonlinearity. The observations reported here denote a different behavior for concrete and Berea sandstone.