GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.3 765 The problem that we desire to tackle here is finding a link between static and dynamic elastic moduli. The idea is to find a rock physics model that predicts the dynamic elastic moduli given the static elastic moduli. In the present work, we limit to: isotropic compression, unconsolidated sands, dry conditions, and estimation of bulk modulus. In this framework, both irreversible plastic deformation and hysteretic nonlinear elasticity occur. Consequently, our model must deal with both these phenomena. The Preisach-Mayergoyz model is a classic model of hysteretic nonlinear elasticity which consists in representing a medium macroscopically as a collection of mesoscopic hysteretic elastic units defined by a pair of pressures: P c and P o . A density distribution of the units in the ( P c , P o ) space is used to estimate static and dynamic bulk moduli, as well as the stress-strain relation. The space is also called PM-space. The PM-space was used, e.g. in (Mccall and Guyer, 1994), to represent a consolidated rock with cracks that close/open under imposed pressure conditions. (Guyer, Mccall, Boitnott, Hilbert Jr., and Plona, 1997) proposed a discretized version of the PM-space model, such that all equations can be re-casted in matrix notation. The authors performed a stochastic inversion to determine the PM-space that best fits the static bulk modulus, and used the model to infer the stress-strain relation and the dynamic bulk modulus of the medium. In granular media, due to rotation and sliding of grains and grain crushing, a significant component of the strain is plastic, and this is not accounted for in the PMmodel. (Zimmer, 2003) developed an adapted PM space analysis to unconsolidated natural sands and glass beads, by separating the elastic contribution to the plastic contribution in the total strain. The plastic strain is fitted on observed data to match the measured stress-strain curves. In this work, we propose Fig. 1 - a) mechanism of hysteresis of a hysteretic mesoscopic elastic unit (HMEU) in the classical Preisach-Mayergoyz model. The unit contracts from lc to lo during compression at P c and expands from lo to lc during dilatation at P o . The difference between P c and P o causes hysteresis. b) the medium is represented as a collection of HMEU with different ( P c , P o ) , the negative P o represent the plasticity (coloured in red). c) discretized version of (b) following Guyer et al. (1997).