## 2 Test Preparation
### RFPA\({}^{3D}\) Feasibility Verification  
Rock is a heterogeneous material. The heterogeneity plays an important role in determining the crack initiation and fracture mode of rock mass. The influence of heterogeneity is pronounced in the progressive failure process. The Weibull distribution is used to consider the heterogeneity of rock in the RFPA method. RFPA is also suitable for the fracture analysis of brittle materials. Compared with ABAQUS and COMSOL, RFPA calculates crack initiation and propagation without considering the convergence problem. At present, a large number of papers have been published on numerical simulation analysis of rock materials using RFPA, and the principle can be found in the relevant papers [32; 33; 34; 35; 36; 37; 38]. RFPA\({}^{3D}\) was used to perform the splitting experiments of the frozen rock specimens. The aim of the simulation was to obtain the internal microcrack initiation and extension, and the energy accumulation and release in the process of frozen rock failure. The size of the model was \(\phi\) = 50 \(\times\) 25 mm, and the mesh size was 0.25 \(\times\) 0.5 \(\times\) 0.25 mm. Cardboard was added at both ends to reduce the end effect. The thickness of the cardboard was 5 mm and the loading rate was 0.0005 mm/step. The material parameters of the model (Table 1) were obtained from the uniaxial compression of intact frozen rock and Brazilian splitting experiments, and converted by RFPA empirical formula [39; 40]. Figure 8 shows the numerical simulation results of the Brazilian splitting of intact frozen rock. Compared with the experimental results, the frozen rock fracture mode was similar, and macroscopic cracks were formed in the loading direction. The intact frozen rock specimen showed a typical brittle failure form. AE events increased sharply before macroscopic cracks occurred. The frozen rock specimen fractured instantly and released a large amount of energy (Figure 9). The load capacity of the intact frozen rock specimen was 8.15 kN in the numerical simulation and 8.88 kN in the experiment, which were similar to each other. Figure 10 shows the numerical simulation results of the Brazilian splitting of frozen rock with different ice-filled crack angles. Red and blue circles in the AE distribution indicate the shear and tensile failure of the element. Compared with the experimental results (Figure 6), the splitting fracture modes of the frozen rock model were similar. The ice in the crack supported and transferred some force during the loading. However, the ice strength was lower than the rock strength, which induced the stress concentration at the ice-filled crack. When loaded to a certain extent, wing cracks occurred at the tip and penetrated through the whole model. Figure 10: _Cont._  
Figure 9: The relationship curve between load, AE counts and accumulated AE counts for intact frozen rock. Figure 10: Numerical simulation results of Brazilian splitting of frozen rock in different ice-filled crack angles: (**a**) minimum principal stress, (**b**) model slice; (**c**) AE. Figure 11 shows the relationship curve between the load, AE counts and accumulated AE counts for frozen rock with different ice-filled crack angles. The peak load of 0\({}^{\circ}\) ice-filled crack frozen rock was 7.62 kN, and splitting failure occurred at step 67. The peak load of 15\({}^{\circ}\) ice-filled crack frozen rock was 6.81 kN, and splitting failure occurred in step 87. The peak load of 30\({}^{\circ}\) ice-filled crack frozen rock was 4.11 kN, and splitting failure occurred at step 38. The peak load of the 45\({}^{\circ}\) fractured frozen rock specimen was 2.59 kN, and splitting failure occurred in step 29. The frozen rock specimens with 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) ice-filled crack showed typical brittle failure mode. At the initial stage of loading, the internal microcracks of the frozen rock specimen accumulated continuously. AE events were fewer with no macroscopic cracks on the surface.