# Mechanical Properties of Marble Under Triaxial and Cyclic Loading Based on Discrete Elements
## 4 Numerical Simulation
### Strength and Deformation Characteristics
This leads to enhanced stiffness, resulting in the minimal influence of the unloading process on the internal fractures. With increasing cycles, the elastic modulus gradually decreases, with a rapid decline occurring near the peak strength. Once the rock specimen enters the residual strength stage, the elastic modulus stabilizes. Overall, with the increase in the circumferential pressures, the elastic modulus increases; however, the rate of the increase becomes less pronounced at higher circumferential pressures. Specifically, taking the sixth cycle as an example, when the circumferential pressures increases from 10 MPa to 20 MPa, the loading elastic modulus increases from 37.78 GPa to 39.55 GPa, an increase of 4.69%, and the unloading elastic modulus increases from 38.30 GPa to 39.62 GPa, an increase of 3.45%. When the circumferential pressures increases from 30 MPa to 40 MPa, the loading elastic modulus increases from 39.95 GPa to 40.30 GPa, with an increase of 0.88%, and the unloading elastic modulus increases from 40.04 GPa to 40.38 GPa, with an increase of 0.85%. From the overall trend of the post-peak elastic modulus, it can be seen that the decrease in the modulus of elasticity under loading and unloading is significantly reduced at higher circumferential pressures, indicating that an increase in the circumferential pressures effectively mitigates the damage to the rock specimens. Additionally, the Poisson's ratio increases with the number of cycles under varying circumferential pressures (Figure 11c). To further investigate the influence of the circumferential pressures on the elastic modulus and Poisson's ratio, the average values of these parameters were calculated, as shown in Figure 12. It can be observed that the average elastic modulus increases with the rise in the circumferential pressures, while the average Poisson's ratio increases linearly with increasing circumferential pressure. Figure 9: _Cont._  
Figure 10: Relationship between the mechanical parameters and the circumferential pressures under cyclic loading conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Circumferential Pressures/MPa** & **Peak Strength/MPa** & **Residual Strength/MPa** & **Peak Axial Strain/\%** \\ \hline
10 & 161.57 & 30.55 & 0.449 \\
20 & 176.35 & 36.25 & 0.499 \\
30 & 187.62 & 42.81 & 0.545 \\
40 & 201.26 & 46.22 & 0.595 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mechanical parameters under cyclic loading conditions. Figure 9: Stress-strain curve of the cyclic loading numerical test: (**a**) 10 MPa; (**b**) 20 MPa; (**c**) 30 MPa; and (**d**) 40 MPa. Figure 11: Relationship between the cyclic loading deformation parameters and the cycle times under varying circumferential pressures: (**a**) loading elastic modulus; (**b**) unloading elastic modulus; and (**c**) Poisson’s ratio. Figure 12: Variation of the elastic modulus and Poisson’s ratio with the circumferential pressures. Figure 13 presents a comparison of the peak strength, Poisson's ratio, and elastic modulus between conventional triaxial and cyclic loading at various circumferential pressures. It can be seen that the difference in peak strength between conventional triaxial and cyclic loading under the same circumferential pressure conditions is not significant, being higher under conventional triaxial. During the cyclic loading process, the rock specimens are continuously compacted and the peak strength is improved to some extent, However, a higher cyclic stress level will cause damage to the rock specimen and decrease the peak strength, so the peak strength under the two loading methods is relatively close (Figure 13a). As shown in Figure 13b, the Poisson's ratio under cyclic loading is higher at each circumferential pressure compared to conventional triaxial loading. This is primarily due to the repeated loading and unloading, which facilitates the formation and propagation of new cracks within the rock specimen, leading to greater lateral deformation than in conventional triaxial testing.