# A Macro-Micro Damage Model for Rock under Compression Loading
## 4 Results  
### Triaxial Tests and Parameters Determination  
The macro-micro damage model for rock materials has been established, and its feasibility and rationality still need to be verified by example analysis. A series of conventional triaxial compression tests were carried out to verify the proposed damage constitutive model in this study. The rock used in this study is red sandstone collected from a mining area in Hunan Province, China. The mineral components of the sandstone samples were detected by X-ray detection. The red sandstone samples are mainly composed of quartz, anorthite, gismondine and clay cements. The average dry density of the sandstone samples was 2.31 g/cm\({}^{3}\) at room temperature. The selected rock samples belong to the same rock block, which ensures the reliability of the test results. To ensure homogeneity of the sandstone samples, the sandstone samples were taken from a great thickness of sandstone strata, and have no obvious defects. The sandstone samples have good uniformity of appearance, no obvious stratigraphic structure, and no weak layers visible to the naked eye. At the same time, the P-wave of the sandstone samples was measured to single out the specimens with velocities which were too small or large. The rock samples have been processed as cylinders, with the height of 100 mm and the diameter of 50 mm according to size requirements of the International Society for Rock Mechanics (ISRM). The conventional triaxial compression tests were carried out on the RMT-301 material test system from Wuhan University in China. The machine's maximum axial compressive force is 1500 kN, and the maximum confining pressure is 70 MPa. The triaxial tests were conducted under the displacement control mode to capture the post peak behavior. During the test, the system directly records the load and the displacement. The conventional triaxial tests were conducted, which was divided into two steps. First, \(\sigma_{1}\) and \(\sigma_{3}\) were applied to a predetermined value (the confining pressure) by the hydrostatic pressure conditions. The predetermined value remained unchanged for 15 s. Second, \(\sigma_{3}\) remained unchanged. Then, the loading was switched to an axial displacement control, and the axial loading rate was set as 0.002 mm/s until rock fails. As shown in Figure 7, the failure modes of the red sandstone samples are mainly splitting failure (uniaxial compression) and compression shear failure (triaxial compression). Some micro-parameters in the current damage model were hardly measured directly by experiment. Therefore, the specific values of micro-parameters were obtained by comparing the experimental and theoretical data, which provides an important help for applying the proposed analytical model to real rock. This process was repeated until the theoretical results achieved a good agreement with the experimental results. The specific method for selection of model parameters is as follows. At low-stress levels, the initial crack does not slide when shear does not overcome friction (i.e., \(\mu|\sigma_{\rm n}|\)). The rock is in the elastic stage. When axial stress \(\sigma_{1}\) reaches the crack initiation stress \(\sigma_{1c}\), wing cracks start to form from the initial main crack tips. Substituting cracking angle \(\theta\)c = 0.392\(\pi\) = 70.5\({}^{\circ}\) and the effective shear stress \(\tau_{\it eff}\) acting on the main crack surface into Equation (9), the most dangerous crack is that lying at the main crack  
Figure 7: Failure modes of the red sandstone samples: (**a**) uniaxial compression; (**b**) triaxial compression. inclination angle \(\beta\) which maximizes \(K_{I}^{\text{infl}}\). Then, it can be found that \(\tan(2\beta)=-1/\mu\).