Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach-Mayergoyz space model

Summary The existence of pores, cracks, and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress-strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach-Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress-strain and modulus-stress relationships under different stress protocols. Additionally, we analyze the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach-Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain-displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wavefront discontinuities and uneven energy distribution in the 2-D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.