## 3 Results and Discussion
### Case 2: Specim with an Open Flaw  
To study the influence of the pre-existing flaw on damage process, the corresponding data of specimen with 45\({}^{\circ}\) flaw have been calculated and statistically analyzed. Figure 7 has shown the strain rate extremums of the MC and micro-crack increment in the pre-peak stage of the flawed specimen, with the same legends as Figure 3. The micro-crack increment (red bars) appears intermittently before 50% peak strain; then, the red bars become crowded till 80% strain; at last, the red bars become much longer and more intense. Between 80% strain and 100% strain, there are four peaks on the maximum curve, accompanied by the rapid growth of micro-cracks. The average, standard deviation (SD), and range RA of the MC strain rate at the pre-peak part of the flawed specimen are shown in Figure 8 (the legends are consistent with Figure 3). The SD of the strain rate field is close to zero before 60% strain and has three peaks between 60 and \(\sim\)100% strain. The average curve shows the same tendency as the SD curve. In Figure 8, the rise height of the range RA segments becomes higher as the loading strain increases, and the skip distance between the last two red segments is obviously larger than before. Compared to the intact BPM, the red segments of the flawed BPM become well regulated and have more uplifts after 50% strain. By fitting the range RA data with an exponential function, we obtain the following exponential function:  
\[f(x)=0.0006*e^{x/0.17259}+0.00774 \tag{2}\]  
The regional average data fits into the exponential function, and the theoretical curve is shown as the black dash curve. Equations (1) and (2) both are a one-phase exponential decay function with a time constant parameter. These two equations have the same time constant 0.17259, but the parameters of the offset and amplitude both reduce to 50% in Equation (2). Compared with the theoretical curve of the single-flaw specimen in Figure 8, the curve of the intact model in Figure 4 is steeper. It indicates that the regional averages of  
Figure 8: The standard deviation, average, and range regional averages of MC volume strain rate (case 2). Figure 7: Extremum values and average deviation of MC strain rate and micro-crack increment (case 2). the strain rate range of the intact model are higher than that of the flawed model when the strain proportions were coincided in the two cases. Figure 9 shows the coefficient of variation in case 2 (the red curve) and the average of 20 contiguous CVs (the blue curve). The blue curve can be loosely divided into two parts: the line part before 60% US strain and the wave part with three waves. In Figure 9, between 60% and 90% strain, the blue curve has two waves with clear peaks and troughs; meanwhile, the CV curve has fluctuations that appear at intervals. From approximately 87% strain to 100% strain, the last wave on the blue curve has fluctuated upwards higher than ever before. The blue curve shape in case 2 is significantly different from the corresponding curve in case 1. The existence of the flaw has made the variation pattern of the CV and its regional average more regular and more distinguished. Eight contour maps of MC strain rates at critical moments are shown in Figure 10, providing help to determine the damage localization of the flawed sample. In contour maps of 20% and 40% strain, the high ratio strain rates surround the pre-existing flaw, and there is basically no change in other areas. This shows that the strain localization appears near the flaw during the initial stage of loading. It is different from the first stage of case 1 that the damage concentration begins from the initial loading period, omitting the scattered damage stage. In the contour map of 60% peak strain, the areas with high ratio strain rates move to the flaw tips. When the strain reaches 70% and 80% peak strain, the extreme values of the strain rate field move with crack propagation.