## 3 Results and Discussion
### Superiority Verification of the Optimized DDARF Program
#### 3.2.3 Stress Comparison  
Eight measuring points are set at the top, left and right sides of the border of the three case models to measure the displacements and stresses of the models in the compression simulation experiments, after the simulation, according to the measured values of the measuring points.Stress-strain curves of the three cases models can be obtained, as shown in Figure 14 (taking compression as positive and tension as negative). Figure 12: Failure process of Case 3 in uniaxial compression simulation experiment. (**a**) The ends of the preset joints cracked and wing cracks appeared; (**b**) Wing cracks expanded; (**c**) The width of the wing crack increased and fine cracks and some secondary joints appeared; (**d**) A group of new oblique parallel cracks appeared; (**e**) The fine cracks became connected with the wing cracks and preset joints. Figure 13: Expansion law of joints simulated in RFPA-2D. As can be seen from the curves in Figure 14, the three cases all experienced three stages in the experimental process of the model:  
1. Compaction stage The preset joints in the models displayed closed compaction, and the whole models are axially compressed and laterally expanded, so the axial strain was positive strain (axial compression) and the lateral strain was negative strain (lateral expansion) on the stress-strain curve. 2. Elastic deformation stage In this stage, the curves showed linear growth, the axial and lateral strains continued to increase, and the slope increased compared with the compaction stage. Near the end of this stage, the preset joint began to crack. 3. Steady crack growth stage The main cracks continued to expand, and secondary cracks appeared in other parts of the models. The slope of the curve changed a lot in this stage, mainly caused by the increase in the number of cracks in the model. After the stress reached the peak, the curves began to decline, indicating that the models were damaged. At the end of the experiment, the lateral strains of the models under the three cases were greater than the axial strains, indicating the dilatation phenomenon of the models. Although the stress-strain curves of the three cases vary in the same way, the stress at the initiation of preset joints (crack initiation stress) and the maximum stress (peak stress) of the three cases were both different; crack initiation stress and peak stress are very important for the study of mechanical properties of rock mass with joints [31; 32; 33; 34]. The crack initiation stress and peak stress in the three cases are shown in Table 3:  
From Table 3, the crack initiation stress and peak stress of Case 1 are both the largest among the three cases, while the crack initiation stress of Case 2 and Case 3 is similar, as is the peak stress, which shows that the calculation accuracy of Case 2 and Case 3 is approximately the same, suggesting that the optimized DDARF's calculation accuracy in the simulation experiment is in agreement with the model with the greater number of fine uniform grids (Case 3). According to the cracking algorithm principle of the DDARF, virtual joints provide channels for the expansion of the joints and cracks--that is to say, the more virtual joints  
Figure 14: Stress-strain curves of the three case models in uniaxial compression simulation experiments. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Crack Initiation Stress (MPa)** & **Peak Stress (MPa)** & **Crack Initiation Stress/Peak Stress (\%)** \\ \hline
1 & 21.19 & 41.56 & 50.99\% \\
2 & 17.36 & 34.43 & 50.42\% \\
3 & 16.84 & 32.87 & 51.23\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Crack initiation stress and peak stress of the three cases. there are, the more joints expand where they are supposed to expand. In Case 1, due to the small number of grids and large area of each grid, the cracking and expansion of the joints are hindered, and the crack initiation stress and peak stress of joints are relatively large.