Microscopic features and their hysteretic behavior can be used to predict the macroscopic response of materials in dynamic experiments. Preisach modeling of hysteresis provides a refined procedure to obtain the stress–strain relation under arbitrary conditions, depending on the pressure history of the material. For hysteretic materials, the modulus is discontinuous at each stress–strain reversal which leads to difficulties in obtaining an analytic solution to the wave equation. Numerical implementation of the integral Preisach formulation is complicated as well. Under certain conditions an analytic expression of the modulus can be deduced from the Preisach model and an elementary description of elastic wave propagation in the presence of hysteresis can be obtained. This approach results in a second-order partial differential equation with discontinuous coefficients. Classical nonlinear representations used in acoustics can be found as limiting cases. The differential equation is solved in the frequency domain by application of Green’s function theory and perturbation methods. Limitations of this quasi-analytic approach are discussed in detail. Model examples are provided illustrating the influence of hysteresis on wave propagation and are compared to simulations derived from classical nonlinear theory. Special attention is given to the role of hysteresis in nonlinear attenuation. In addition guidance is provided for inverting a set of experimental data that fall within the validity region of this theory. This work will lead to a new type of NDT characterization of materials using their nonlinear response.

Much is known empirically about the qualitative and quantitative nonlinear response of rocks. Still, the mechanism of nonlinear response in porous media is only speculation. In this work we illustrate evidence from several types of experiments indicating that fluid plays an important role in the nonlinear response. For instance, measurements at low degrees of water saturation indicate that molecular layers of adsorbed moisture as well as condensation fluids significantly influence the linear and nonlinear response at dynamic strain levels due to the activation of internal molecular forces. In rock, we measure a significant increase in the nonlinear response, especially in the saturation range of 1–20%. This is consistent with observed changes in the linear response, but the extent of the variation is larger in the nonlinear measurements, especially in rocks containing small pore systems. These modifications can be attributed to an increased fluid‐solid interaction upon wetting causing the material to expand and to soften. Concurrently, the microscopic and mesoscopic hysteretic entities, which give rise to the macroscopically observed nonlinear response, are activated at saturation‐dependent opening and closing pressures. As a consequence of the moisture‐induced forces, the fraction of active hysteretic units increases with saturation. The fact that the nonlinear response increases with water saturation, especially in the low saturation range, implies that the presence of moisture plays a major role in the nonlinear mechanism or, more precisely, in the activation of that mechanism.

Recent studies show that a broad category of materials share “nonclassical” nonlinear elastic behavior much different from “classical” (Landau-type) nonlinearity. Manifestations of “nonclassical” nonlinearity include stress–strain hysteresis and discrete memory in quasistatic experiments, and specific dependencies of the harmonic amplitudes with respect to the drive amplitude in dynamic wave experiments, which are remarkably different from those predicted by the classical theory. These materials have in common soft “bond” elements, where the elastic nonlinearity originates, contained in hard matter (e.g., a rock sample). The bond system normally comprises a small fraction of the total material volume, and can be localized (e.g., a crack in a solid) or distributed, as in a rock. In this paper a model is presented in which the soft elements are treated as hysteretic or reversible elastic units connected in a one-dimensional lattice to elastic elements (grains), which make up the hard matrix. Calculations are performed in the framework of the local interaction simulation approach (LISA). Experimental observations are well predicted by the model, which is now ready both for basic investigations about the physical origins of nonlinear elasticity and for applications to material damage diagnostics.