GNGTS 2018 - 37° Convegno Nazionale

768 GNGTS 2018 S essione 3.3 First, for the sake of comparison, we applied the classic elastic PM model to the data. Since the data display a plastic deformation during the two cycles we limit the elastic analysis to a single loading/unloading cycle. We chose the 8 th cycle in which the amount of plastic deformation is smaller. For this inversion, ρ is purely elastic, hence it is a 4×4 matrix with 10 unknowns. Fig. 2a shows the observed and predicted static bulk moduli for the 8 th cycle. In the legend L stands for load, U stands for unloading, and 8 indicates the cycle number. Note that there is a reasonable fit between observed and predicted data, in spite of the limited number of observed data available in the high-pressure regime (P>1 MPa). Some minor mismatches can be explained, e.g., the “PM U8” curve is lower than the respective observed curve, this can be understood by looking at the stress-strain curves in Fig. 2b. In fact, because of the purely elastic approximation, the predicted strain is forced to be zero at zero pressure, this causes the predicted K to be softer than the observed K during the unloading. Fig. 2c display the elastic ρ pred . Fig. 2d displays the predicted and observed dynamic bulk moduli. The PM model overpredicts the dynamic modulus, a similar overprediction is observed also by (Zimmer, 2003). Next, we applied the elasto-plastic PM space to the 7 th and 8 th cycles which display a significant plastic strain. Fig. 3a compares the fitted and observed static bulk moduli in the two cycles. The mismatch for the 8 th cycle is analogous to the elastic case of the first example, it is minor in the unloading path (U8) due to the inclusion of the plastic strain, whereas it is larger in the loading (L8), likely due to the approximation of purely elastic behaviour of the pre-compacted which constrains L8 to meet L7 at 15 MPa in the stress-strain path (Fig. 3b). Fig. 3 - Results of the genetic algorithm inversion on the 7 th and 8 th cycles of the Gulf of Mexico beach sand using the plastic PM space proposed in this work. a) observed and predicted static bulk moduli for loading and unloading in the two cycles, b) observed and predicted stress-strain curves for the two cycles. c) density in the PM space discretized as a n × n ( n +1) matrix, in which the plastic deformation is limited to the bottom row of the matrix, d) observed and predicted dynamic bulk moduli. Note that the model correctly accounts for the plastic deformation which remains at zero pressure after the two unloading paths. The model is able to incorporate the plastic deformation in the predicted stress-strain curve.