# 3-D Numerical Study on Progressive Failure Characteristics of Marbles under Unloading Conditions
## 3 Numerical Results and Discussions  
### Numerical Model Setup  
The complete marble model was generated using the microscopic parameters above, with a model size of \(100\times 50\times 50\) mm\({}^{3}\), containing 164,890 particles with a radius between 0.5 and 0.8 mm, the particles were connected by 820,396 parallel bond models. The geometry of the pre-cracked rock sample in the simulation was simulated by deleting the particle elements on the designated surface. The loading and geometrical conditions of the pre-existing fissures in the marble specimens are plotted in Figure 3b,c. Moreover, it can be observed from Figure 3c that \(a\) is crack length, \(\alpha\) is crack inclination angle, \(\beta\) is ligament inclination angle, and \(b\) is ligament length. The details of geometries of pre-fissured marble specimens are shown in Figure 6 and Table 2. It can be found from Figure 6 and Table 2 that crack inclination angles were equal to \(45^{\circ},30^{\circ},60^{\circ}\), and \(75^{\circ}\) in the pre-fissured marble specimens of Type-A, Type-B, Type-C, and Type-D, respectively. In the pre-fissured marble specimens of Type-A \(\sim\) Type-D, crack lengths were fixed as 24 mm, and ligament lengths were fixed as 33 mm, which resulted in various ligament inclination angles in these marble specimens, as shown in Figure 6 and Table 2. As shown in Figure 6 and Table 2, for Type-A, Type-E, Type-F, the initial fissure inclined angles and initial fissure lengths are fixed as \(45^{\circ}\) and 24 mm. The ligament lengths between two pre-existing fissures are 33 mm, 48 mm, and 24 mm in Type-A, Type-E, and Type-F marble specimens, respectively. Similarly, it can be also observed from Figure 6 and Table 2 that the initial fissure inclined angle and ligament length between the two pre-existing fissures were fixed as \(45^{\circ}\) and 33 mm in Type-A, Type-G, and Type-H marble specimens. Meanwhile, the initial lengths of pre-existing fissures in Type-A, Type-G and Type-H marble specimens equated to 24 mm, 16 mm, and 20 mm, respectively. Figure 5: Comparison of axial stress–strain curves obtained from the numerical simulations and the previous experimental data [65]. The stress redistribution caused by rock unloading usually manifests as the stress increase in one direction and a release in the orthogonal direction. Therefore, the stress phase of rock unloading is divided as follows: (i) Initial stress phase: At this stage, vertical stress \(\sigma_{z}\) and lateral stress \(\sigma_{x}\), \(\sigma_{y}\) were simultaneously increased to take the stress to initial stress \(\sigma_{0}\), and then the velocity of the particle is reset. (ii) Stress release phase: At this stage, the lateral stress \(\sigma_{x}\), \(\sigma_{y}\) were unloaded and the vertical stress \(\sigma_{z}\) increases until the sample reaches the peak stress state. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Geometries of Pre-Existing Fissure**} \\ \hline
**No.** & **\(\alpha\) (\({}^{\circ}\))** & **\(\beta\) (\({}^{\circ}\))** & **\(a\) (mm)** & **\(b\) (mm)** & **Note** \\ \hline Type A & 45 & 61 & 24 & 33 & \multicolumn{1}{c}{Fissure inclination angle} \\ Type B & 30 & 38 & 24 & 33 & \multicolumn{1}{c}{} \\ Type C & 60 & 75 & 24 & 33 & \multicolumn{1}{c}{} \\ Type D & 75 & 83 & 24 & 33 & \multicolumn{1}{c}{} \\ Type E & 45 & 70 & 24 & 48 & \multicolumn{1}{c}{} \\ Type F & 45 & 48 & 24 & 24 & \multicolumn{1}{c}{} \\ Type G & 45 & 34 & 16 & 33 & \multicolumn{1}{c}{} \\ Type H & 45 & 49 & 20 & 33 & \multicolumn{1}{c}{} \\ \hline \hline \multicolumn{6}{c}{Loading and unloading parameters} \\ \hline Initial confining & 0 & 0.6\(\sigma_{uCS}\) & 0.8\(\sigma_{uCS}\) & 1.0\(\sigma_{uCS}\) & 1.2\(\sigma_{uCS}\) \\ pressures (MPa) & & & & & \\ Unloading rates (mm/s) & 0.01 & 0.1 & 0.2 & 0.3 & 0.4 \\ Loading rates (mm/s) & & & 0.2 & \multicolumn{1}{c}{} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric and loading parameters of pre-fissured marble specimens. Figure 6: Geometric conditions of pre-fissured marbles.