## 3 Results and Discussion  
### Case 1: Intact Specim  
During the process of compression, the volume strain rate of each measurement circle has been automatically recorded every 200 steps. Statistical analysis of the volume strain rates were made to get the average, the standard deviation (SD), the range, and the average deviation (AD) at each recorded time. From the statistics, it has been found that the occurrence of floater particles can cause extremely high values of the MC strain rate and significantly impact the statistics at corresponding moments. In subsequent analysis, the measurement circle with floater particles was removed from the data array. The extremum MC strain rates before the peak stress are plotted in Figure 3. The black curve and blue curve indicate the maximum and minimum value of the strain rate, respectively; the red lines represent the crack increment; and the purple curve represents the average deviation of the strain rate. The bottom axis reflects the normalized strain (from 0% to 100%); and the 100% normalized strain matches the total strain at the peak strength of the intact numerical model. In the magnified part of Figure 3, the corresponding gaps between the strain rate extremum curves, i.e., the strain rate ranges, are obviously larger when the micro-cracks increase. This indicates that the variation patterns of the strain rate largely reflect the micro-crack generation and damage evolution of the BPM. Thus, the statistical analysis of the MC strain rates might be used to identify the failure process of specimens and to explore methods for failure predicting. In Figure 3, before the 30% peak strain, the micro-crack increment emerges occasionally; between 30 and -60% strain, its value and frequency both change a little; in the last section, its curve has sharp rises. The AD curve is similar to the micro-crack increment. All curves  
Figure 2: Measurement circle arrangement (**left**) and failure modes of numerical simulations (**right**). (**a**) Intact BPM; (**b**) 45° flaw BPM. in Figure 3 indicate that the damage accumulation of the intact specimen accelerates after 60% strain and becomes much larger after 80% strain. Figure 4 shows the average and standard deviation of the MC strain rate at the pre-peak part of the intact specimen simulation. The SD curve (the green one) has larger fluctuations, compared with the average curve. Based on the tendency of the green curve, the SD variation can be split into four phases: phase 1, at 0-40% strain, the values of the SD are close to zero; at 40-60% strain, the green curve has some peaks; at 60-80% strain, fluctuations of the SD curve become denser and larger than before; at 80-100% strain, the larger fluctuations emerge intensively. For a further investigation of the damage stage of the intact BPM, the regional averages for the MC strain rate ranges (RA) have been calculated at each 10% strain, which appear  
Figure 4: The standard deviation, average, and range regional averages of MC volume strain rate (case 1). Figure 3: Extremum values and average deviation of MC strain rate and micro-crack increment (case 1). as red segments in Figure 4. Those segments can be characterized as four groups: 0\(\sim\)40% strain is group 1, in which part of the regional average has slightly changed; 40-60% strain is group 2, in which part of the red segments gently increase; 60-80% strain is group 3, in which part of the RA has an obvious integral uplift; and 80-100% strain is group 4, in which part of the RA has a significant growth. By comprehensively analyzing the trends of the RA and SD curves, it can be concluded that the damage evolution of the intact BPM can be seen as three sections: the first section is between 0 and -40% strain; the second section is between 40 and -80%; the section is between 80 and -100% strain. Through data fitting, it has been found that the relationship of the RA and loading strain fit into the exponential function.