GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.3 769 The 7 th cycle (PM L7 and PM U7), which is completely free to behave plastically displays a satisfactory fit. Fig. 3b shows the observed and predicted stress-strain curves, note the correct reconstruction of the strain path for cycle 7 and the fair reconstruction for cycle 8. Fig. 3c shows the density in the ( P c , P o ) space which constitute the model that represent our granular material. In the matrix, the bottom row represents the plastic deformation. Let us compare it with Fig. 2c, which is its elastic counterpart. We expect that, column-wise, the sum of the entries of the two matrices is similar. In fact, the consequence of adding a plastic bottom row in ρ is that the density spreads column-wise partially emptying the elastic entries of ρ to fill the respective element in the plastic bottom row. Fig. 3d compares the predicted and observed dynamic moduli. Note that, even if the match is not perfect, it constitutes a relevant improvement with respect to the elastic prediction (Fig. 2d), which is limited by fact that it does not handle the plastic deformation. As a side note, in this particular case, logarithmically spaced pressure steps in the density distribution would be beneficial, since observed data are logarithmically sampled. This aspect is it is not described here, but related work is currently in progress. In general, it is often difficult to obtain a good prediction of the dynamic modulus since it relies on the sole diagonal part of ρ , which is not well constrained with a limited amount of data. Conclusions. The method that we propose extends the classic Preisach-Mayergoyz (PM) model of hysterestic nonlinear elasticity by including plastic strain in the first compaction, whereas approximating the pre-compacted material as purely elastic. This is obtained by simply adding a bottom row in the PM-space density matrix. We applied this elasto-plastic PM space to model static and dynamic elastic moduli of an unconsolidated sand sample. We performed a stochastic inversion, in which we fitted the static bulk modulus to predict the dynamic bulk modulus. Compared with the classical purely elastic PM-space inversion, our elasto-plastic PM-space model better constrains the dynamic bulk modulus. In future applications, this model can be extended to consolidated rocks containing multiple cracks. Acknowledgement. We thank Prof. R. A. Guyer for helpful comments and suggestions. References Aleardi, M., & Ciabarri, F. (2017). Assessment of different approaches to rock-physics modeling : A case study from offshore Nile Delta. Geophysics , 82 (1). https://doi.org/10.1190/geo2016-0194.1 Guyer, R. A., Mccall, K. R., Boitnott, G. N., Hilbert Jr., L. B., & Plona, T. J. (1997). Quantitative implementation of Preisach-Mayergoyz space to find static and dynamic elastic moduli in rock. Journal of Geophysical Research , 102 (B3), 5281–5293. Mccall, K. R., & Guyer, R. A. (1994). Equation of state and wave propagation in hysteretic nonlinear elastic materials. Journal of Geophysical Research , 99 (B12), 887–897. Zimmer, M. A. (2003). Seismic velocities in unconsolidated sands: Measurements of pressure, sorting, and compaction effects . Stanford University.