## 3 Results and Discussion
### Analysis of Damage Evolution  
Figure 4 shows the distribution of broken bonds visualized by particles colored according to their percentage of broken bonds. The count of broken bonds divided by the count of total bonds is employed as a data attribute of each particle to visualize damaged zones of the sample, where particles become weakly tied to their neighbors. In order to better examine the differences between the crack propagations, the frontal surface is segmented by horizontal and vertical planes that intersect in the center of the surface. The resulting four regions are numbered as follows: \"I\", \"II\", \"III\", and \"IV\". According to Figure 4, four time intervals, [\(t_{1}\); \(t_{2}\)], [\(t_{2}\); \(t_{3}\)], [\(t_{3}\); \(t_{4}\)], and [\(t_{4}\); \(t_{5}\)], are considered, where \(t_{1}\) = 1.0 \(\times\) 10\({}^{-5}\) s, \(t_{2}\) = 1.9 \(\times\) 10\({}^{-5}\) s, \(t_{3}\) = 2.4 \(\times\) 10\({}^{-5}\) s, \(t_{4}\) = 2.5 \(\times\) 10\({}^{-5}\) s, and \(t_{5}\) = 2.6 \(\times\) 10\({}^{-5}\) s. In Figure 4a,b, orange ellipsoids and polygons mark some cracks that do not propagate in time period [\(t_{2}\); \(t_{3}\)]. Red arrows show the main shear-dominated microcracks, whose lengths are more than four times larger than the distance between the centers of particles. Crack propagation can be investigated taking into account the time variation rate of the broken bond count. In the time interval [\(t_{1};t_{2}\)], it can be computed by using the formula \(\frac{\Delta N_{bb}}{\Delta t}=\frac{N_{bb}\left(t_{1}\right)-N_{bb}\left(t_{ 2}\right)}{t_{j}-t_{1}}\), where the difference of the broken bond count \(\Delta N_{bb}\) is divided from the time interval length \(\Delta t\). The proposed indicator is relevant to the average propagation speed of a unit length crack in micromechanics. In the first time interval [\(t_{1};t_{2}\)] of 9 us, the strain rises from 0.010 to 0.0197 (Figure 4a), while the count of broken bonds increases from 54 to 2740. The time variation rate of the broken bond count is equal to 305 1/ us, which indicates the time interval of slow microcrack propagation. In the second time interval [\(t_{2};t_{3}\)] of 5 us, the strain grows from 0.0197 (Figure 4a) to 0.025 (Figure 4b), while \(N_{bb}\) increases from 2740 to 7147. The time variation rate of the broken bond count increases till 881 1/ us, which is 2.9 times faster than that in the above introduced time interval of slow microcrack propagation. Thus, it can be called the time interval of medium microcrack propagation. Since individual bonds have a discrete nature, the microcracks formed in the neighborhood are also discrete and might be not connected. In the third time interval [\(t_{3};t_{4}\)] of 1 us, the strain grows from 0.025 (Figure 4b) to 0.0261 (Figure 4c), while \(N_{bb}\) increases from 7147 to 9146. The time variation rate of the broken bond count reaches 2028 1/ us, which is 6.7 times faster than that in the time interval of slow microcrack propagation and 2.3 times faster than that in the time interval of medium microcrack propagation. Figure 4c shows the formation of macroracks that are mainly concentrated in the left upper region \"I\". Therefore, it can be considered as the time interval of microcrack transfer to macroracks. In the fourth time interval [\(t_{4};t_{5}\)] of 1 us, the strain rises from 0.0261 (Figure 4c) to 0.0271 (Figure 4d), which leads to the fastest growth of the broken bond count, from 9146 to 19,453. The time variation rate of the broken bond count equals 10,307 1/ us, which is 5.1 times faster than that in the previous time interval of microcrack transfer to macroracks. Macrorack propagation is caused by the high increase of the broken bond count in the whole macrorack propagation volume; therefore, the last interval can be called the time interval of fast macro crack propagation. We made a detailed qualitative analysis of the accumulation of shearing cracks.