Abstract: The paper reviews some newconcepts and models that are proposed RECENTLY to study the mechanicalproperties of the so called nonlinear mesoscopic elastic materials (NMEmaterials) represented by sandstone. First, a physical model, thePreisach-Mayergoyz (P-M) model, is introduced. P-M model is then usedto simulate hysteresis based upon the so-called hysteretic units. Thesimulation procedure and the results for the nonlinear hysteretic stressans stainbehaviour of rocks using the model are depicted in detail. Themacroscopic hysteretic behaviour and discrete memory in thestress-strain relation of rocks can be well simulated by thisphenomenological model. Then, the paper performed the analysis onmechanical mechanism of energy dissipation. At the end , a mathematicmethod is based on the general elastic wave propagation law, and used toanalyze the influence of special properties from NME materials onelastic wave propagation. Therefore, the reason the elastic wavepropagating unusually in NME materials was revealed.