Linear and nonlinear modulus surfaces in stress space, from stress-strain measurements on Berea sandstone

The elastic response of many rocks to quasistatic stress changes is highly nonlinear and hysteretic, displaying discrete memory. Rocks also display unusual nonlinear response to dynamic stress changes. A model to describe the elastic behavior of rocks and other consolidated materials is called the Preisach-Mayergoyz (PM) space model. In contrast to the traditional analytic approach to stress-strain, the PM space picture establishes a relationship between the quasistatic data and a number density of hysteretic mesoscopic elastic elements in the rock. The number density allows us to make quantitative predictions of dynamic elastic properties. Using the PM space model, we analyze a complex suite of quasistatic stress-strain data taken on Berea sandstone. We predict a dynamic bulk modulus and a dynamic shear modulus surface as a function of mean stress and shear stress. Our predictions for the dynamic moduli compare favorably to moduli derived from time of flight measurements. We derive a set of nonlinear elastic constants and a set of constants that describe the hysteretic behavior of the sandstone.