GNGTS 2018 - 37° Convegno Nazionale

GNGTS 2018 S essione 3.3 767 (3) The dynamic modulus is given by the diagonal elements of elastic part of rho, that is (4) Example. We applied the plastic PMmodel to observed data measured on a cylindric sample of a Gulf-of-Mexico beach-sand (Zimmer, 2003). The sample was put through 9 loading/ unloading cycles with increasing maximum pressure, and the measured data consisted in stress- strain measurements and ultrasonic velocity data, from which both static and dynamic elastic moduli were estimated. For simplicity, we focus on two loading cycles (the 7 th and the 8 th ): the 7 th loading cycle spans from approximately 0 to 15 MPa, whereas the 8 th loading cycle spans from 0 to 20 MPa. For the discretization, we used Δ P = 5 MPa as the pressure step, resulting in a 4×5 ρ matrix, containing 14 unknowns (10 for the elastic part, and 4 for the plastic part). Analogously with (Guyer et al., 1997), we performed an inversion procedure in which we fitted the observed K static curves using a stochastic iterative method to determine the density matrix ρ pred . Next, we used ρ pred to infer the dynamic bulk modulus and the stress-strain curve. We used the genetic algorithms for the stochastic optimization. To increase the exploration of the model space we performed 10 inversions each with 300 generations. From this set of inversions, we extracted the solution with minimum misfit. Fig. 2 - Results of the genetic algorithm inversion on the 8 th cycle of the Gulf of Mexico beach sand using the elastic PM space. a) observed and predicted static bulk moduli for loading and unloading, b) observed and predicted stress- strain curves. Note that the model predicts that the final strain is the same as the initial strain since it approximates the medium as purely elastic. c) density in the PM space discretized as a n × n matrix according to the (Guyer et al. , 1997) representation, d) observed and predicted dynamic bulk moduli. Note that the PM space overpredict the observed dynamic modulus.