# 3-D Numerical Study on Progressive Failure Characteristics of Marbles under Unloading Conditions
## 2 Numerical Methodology and Parameter Calibrations
### Parameter Calibration  
In PB-DEM, microscopic parameters commonly cannot directly correspond to the macroscopic parameters of geo-materials. Thus, microscopic parameters in PB-DEM require calibration through comparison of failure patterns and mechanical responses between numerical simulations the laboratory physical tests. According to the previous studies [39; 61; 62; 64], a series of uniaxial and triaxial compression experiments with different parameters was employed. The calibration of the marble model parameters was performed using a trial-and-error method that compares the numerical simulation results with the laboratory experiments results [65]. The schematic diagram of the compression test is presented in Figure 2(a),b.  
Figure 3: Schematics of intact rock specimen under (**a**) uniaxial compression and (**b**) triaxial compression. (**c**) Geometry of pre-fissured marble specimens. Figure 2: Schematic of particle-based discrete element method (PB-DEM) in (**a**) 2-D and in (**b**) 3-D cases. To avoid influence of model interface size on macroscopic parameters, cylindrical and rectangular models were used for calibration experiments. The ultimate failure patterns of intact marble specimens obtained from the present numerical predictions and the previous experimental observations are presented in Figure 4. It can be observed from Figure 4c that the main crack in the previous laboratory experiments results from the diagonal direction of the sample and a secondary crack in the opposite direction, which is consistent with the present numerical experimental predictions, as shown in Figure 4d,e. The flaking damage at the top and bottom of the sample in the numerical predictions, as shown in Figure 4a,b, is in agreement with the bottom rock spalling presented in Figure 4c. The uniaxial compressive axial stress-strain curves of marble specimens obtained from the present numerical simulations and the previous laboratory experiments are shown in Figure 5. It can be observed from Figure 5 that the marble specimen performs significantly brittle failure after peak strength. The stress-strain curve of a physical experiment has a concave shape due to rock compaction at the beginning of the curve, which is different from the numerical predictions. The numerical model is dense inside due to the expansion during particle generation. Therefore, the axial stress-strain curve directly enters the elastic phase. Under the condition of confining pressure of 30 MPa, the results of experiments and numerical predictions are consistent. Moreover, it can be observed from Figure 5 that peak strength of 85.9 (160.5) MPa and deformation modulus \((E)\) of 35.9 (53.2) GPa obtained from the present numerical predictions are similar to values of 83.5 (154.4) MPa and 36.0 (53.6) GPa \((E)\) in the laboratory tests. Since the unloading speed was a large value in the experiment, the contact damper coefficient was set to 0.5 in order to solve the dynamic effect of particles. The mode of damping is full normal and full shear, so the interaction between particles is viscous. The Poisson ratio of the intact sample obtained by parameter calibration was 0.21, but no clear corresponding parameter was found in previous laboratory experiments [65]. Thus, the micro-parameters after calibration of particle and contact are listed in Table 1, which can be used for a series of loading and unloading experiments on marble specimens in the present study.