## 4 Results and Discussion
### Evolution of the Deformation Localization Band  
The length and width of the DLB were measured to quantitatively analyze its evolution. The correlation between the relative displacement rate on both sides of the DLB and the evolutions of its length and width was analyzed. Figure 6 shows the stress, length, and width of the DLB vs. time in the deformation localization stage. According to the analysis in Section 4.1, the deformation localization started when the loading time was \(5.96\times 10^{3}\) s. In stage I, the evolution of the DLB was mainly characterized by the growth of its length. The length of the DLB at the end of stage I was about 45.46 mm, while the width was basically unchanged at about 2.53 mm. In stage II, both the width and length of the DLB did not increase. In stage III, the width of the DLB remained constant, but the length further increased to 100.12 mm. In stage IV, DLB1 was completely formed; the length of the DLB grew slowly to 108.93 mm, but the width increased rapidly to 5.22 mm. According to the evolution of the deformation field, the increases in the width and length of the DLB before stage V were mainly due to the formation of DLB1. In stage V, the appearance and evolution of DLB2 rapidly increased the length and width of the DLB to 169.43 mm and 7.16 mm, respectively. The substantial drop in stress indicates that the rapid development of the DLB reduced the load-bearing capacity of the specimen. After stage V, the length and width of the DLB continued to grow. At the peak stress, the length and width of the DLB reached 200.91 mm and 9.63 mm, respectively. Because of the damage accumulation, macro-cracks appeared at the position of the DLB, and the rock specimen rapidly failed after a short period of stress adjustment. Figure 5: Strain fields corresponding to the marked points in Figure 4: (**a**) corresponding to point 1, (**b**) corresponding to point 2, (**c**) corresponding to point 3, (**d**) corresponding to point 3, (**e**) corresponding to point 5, (**f**) corresponding to point 6, (1–4) are the marker of deformation concentration area. The rate of relative displacement on both sides of the DLB were calculated as follows [53; 54]: First, the strain field and failure mode presented in Section 4.1 were used to determine the location of the DLB. Second, on both sides of the DLB, two spatial-temporal subsets with their connecting line perpendicular to the DLB were selected with a spatial subset size of 15 \(\times\) 15 pixels and a temporal subset size of 5, as shown in Figure 7a. The displacements in the \(x\) and \(y\) directions of the center points \(P_{1}\) and \(P_{2}\) of the two spatial-temporal subsets were calculated, where \(P_{1}\) and \(P_{2}\) are a pair of measurement points. As shown in Figure 7a, five pairs of measurement points were arranged along DLB1 and DLB2, respectively. Third, the relative displacements \(\Delta u\) and \(\Delta v\) of each pair of measurement points were calculated. As shown in Figure 7b, both \(\Delta u\) and \(\Delta v\) were decomposed along the tangent and normal directions of the DLB, where the vector sums along the tangent and normal directions of the DLB were the relative sliding displacement and relative opening displacement, respectively. Finally, the average value of the five pairs of measurement points was taken as the relative displacement of each DLB, and the sliding rate and opening rate of each DLB were obtained by differentiating the relative displacements with respect to time. According to a preliminary analysis, DLB1 was dominated by sliding, while DLB2 was dominated by opening, as shown in Figure 7c. Therefore, only the sliding rate of DLB1 and the opening rate of DLB2 are presented in this paper. Figure 8 plots the curves of the normalized length and width of the DLB vs. time and the relative displacement rate vs. time in the deformation localization stage for comparison. To be clear, the length value is the sum of the lengths of all the DLB, and the width value is the maximum width of the DLB.