## 3 Numerical Simulation
### Model Calibration and Validation  
To simulate the macroscopic phenomena of the experimental test in numerical modeling, repeated calculation and adjustment were required for model calibration. The numerical model was constructed and calibrated against the experimental results. In this paper, parameter adjusting methods were referred to the PFC official handbook. Studies have shown that the size of elementary particles has a certain effect on the mechanical properties of the numerical model [43; 44]. When the elementary particle size is too large, the macroscopic mechanical properties of the model are unstable. The macroscopic mechanical properties of the model are not affected when the size of the elementary particle is sufficiently small. When the elementary particle size is too small, the number of particles in a numerical model will greatly increase to a point where the computational complexity will be unacceptable. We set the minimum diameter of elementary particles to 1.0 mm and the ratio of maximum to minimum size to 1.6. The ratio of model diameter to median particle size (\(L/d\)) is 38.5, and the effect on the simulation is very small, according to [PERSON] [45]. The unconfined compression test (UCT) and splitting test (ST) were used to calibrate and validate the microparameters of the model material. For comparison with the experiment, the numerical model was set to be the same size as the experimental specimen (see Figure 8). Each contact was initially bonded, gapped or slit (see Figure 9) in a flat-jointed material, where \(g_{0}\) is the initial gap between the notional surfaces. The material microstructure parameters are the fraction of bonded (\(\phi_{B}\)) and gapped (\(\phi_{C}\)) contacts. We assumed that there is no initial crack in an artificial gypsum specimen; thus, all the contacts were set to Type B. The interfaces were discretized into three elements in the circumferential direction (\(N_{a}=3\)), and there was one element in the radial direction (\(N_{r}=1\)). The radius multiplier was set to be fixed (\(C_{\lambda}=0\)) so that all flat-joint contact radius multipliers were set equal to the specific value (\(\lambda_{v}=1.0\)). The flat-joint contact model was set  
Figure 8: Numerical models: (**a**) unconfined compression test; (**b**) splitting test. Figure 7: A typical flat-joint contact. to be installed at all ball-ball contacts with a gap less than or equal to 0.0002 m (\(g_{i}=2.0\times 10^{-4}\) m). The porosity-considered bulk density was set to 2650 kg/m\({}^{3}\). The above parameters were fixed during the calibration process. The calibration process involves the trial-and-error adjustment of the following micromechanical parameters: effective modulus \(E^{*}\) (flat joint) and \(E^{*}_{lmm}\) (linear), normal-to-shear stiffness ratio \(\kappa^{*}\) (flat joint) and \(\kappa^{*}_{lmm}\) (linear), friction coefficient \(\mu\) (flat joint) and \(\mu_{lmm}\) (linear), tensile strength \(\sigma_{c}\), cohesion \(c\), and friction angle \(\phi\). All the tested values of these micromechanical parameters are listed in Table 2, and the calibration work was conducted using orthogonal experimental design. Note that in this study, \(E^{*}\)= \(E^{*}_{lmm}\), \(\kappa^{*}\) = \(\kappa^{*}_{lmm}\), and \(\mu\) = \(\mu_{lmm}\). The normal and shear stiffnesses were set based on a specified deformability by defining the effective modulus (\(E^{*}\)) and normal to shear stiffness ratio (\(\kappa^{*}\)):  
\[k_{n}=\frac{E^{*}}{L},\;k_{s}=\frac{k_{n}}{\kappa^{*}}\qquad\text{with}\quad L =\left\{ \begin{array}{ll}R^{(1)}+R^{(2)},&\text{ball}-\text{ball}\\ R^{(1)},&\text{ball}-\text{facet}\end{array} \right. \tag{1}\]  
The macroproperties used for model calibration include unconfined compressive strength, tensile strength, elastic modulus, and Poisson's ratio. The model sensitivity to key micromechanical parameters was analyzed with a multifactor Analysis of Variance (ANOVA) at 5% significance level (\(\alpha=0.05\)).