## 2 Materials and Methods
### Numerical Modeling  
PFC is a numerical simulation software based on the particle discrete element method. The particle discrete element method simplifies the complex mechanical behavior of reality to the mechanical representation of a large number of small particles. The linear parallel bond model is a mathematical method for describing the mechanical behavior between discrete particles. The core of the linear parallel bond model is the bond between the numerous particles. The bond transmits not only forces but also moments. As shown in Figure 6, two bonded particles may be subjected to a force in different directions in the linear parallel bond model (Linear Parallel Bond Model). These parameters mainly determine the bond's strength: Pb_ten, Pb_coh, Emod, and Pb_fa. If this force exceeds the set bond strength, various types of ruptures occur in the bond between the particles. PFC can record this rupture and label it as different types of damage. A large number of small breaks eventually form macroscopic cracks. PFC3D requires many more particles than PFC2D to show the crack extension process accurately. The particle increase consumes much computer memory, leading to computational inefficiency. PFC2D was chosen as the simulation tool based on research objectives and computational cost savings. The numerical model was modeled according to the actual parameters. A rectangular specimen of 50 mm \(\times\) 100 mm was constructed in PFC2D, where the particle size range was 0.25-0.5 mm, and the total number of particles generated ranged between 9000 and 10,000. We classify particles smaller than 0.30 mm as water particles and particles from 0.30 mm to 0.50 mm as rock particles. To reduce simulation errors, the ratio of the model (RA) [28] was calculated so that the particle size did not affect the macromechanical properties. The ratio of the numerical model was calculated using Equation (4). \[RA=\frac{R_{max}}{R_{min}} \tag{4}\]  
where \(R_{max}\) is the maximum particle size, [mm];  
\(R_{min}\) is the minimum particle size, [mm]. The particle numbers meet the requirements of numerical calculations, the RA = 2.0 (the recommended value is above 1.25) was calculated by substituting the specific values. The prefabricated crack was generated after the particles were cemented, and the linear parallel bond model (Linear Parallel Bond Model) was selected for the cementation model. Some scholars [29] have shown that this bond model can very well simulate the properties of the rock. Figure 6: Schematic diagram of particle force and damage mechanism. The selection of numerical model parameters needs to be based on experimental parameters. To obtain the mechanical parameters needed for modeling, we took three sandstone specimens without freeze-thaw cycles for uniaxial compression tests. The experimental results are shown in Table 1.  
According to the experimental results, the trial-and-error method was used to select the model parameters. Subsequently, uniaxial compression simulation was carried out on the model. The stress-strain curve was made according to the simulation results and compared with the experimentally derived stress-strain curve (Figure 7). The model generated by PFC software does not have a compaction phase during uniaxial compression. In contrast, sandstone in its natural state has a complex composition, and its interior has certain pore spaces, joints, and other defects. Sandstone cannot be regarded as a purely ideal elastic body, and its stress-strain curve has a compacting phase. Therefore, the simulated curves do not precisely overlap with the measured curves [30]. Parameter calibration aims to confirm whether the uniaxial compressive strength and modulus of elasticity are consistent. From the fitting results of the two curves, the effect of parameter calibration meets the basic requirements of numerical simulation. The specific parameters of the numerical simulation are shown in Table 2. In Table 2, Pb_fa refers to the angle of internal friction of the rock.