## 3 Results and Discussion
### Case 1: Intact Specim
This can be described by the following exponential function:  
\[f(x)=0.00139*e^{x/0.17259}+0.01527 \tag{1}\]  
It is a one-phase exponential decay function with a time constant parameter. The theoretical curve is shown as the red dashed line. The fluctuations of the SD curve in Figure 4 indicate that the MC strain rates have a larger dispersion degree as the total strain reaches closer to the peak stress. Therefore, there is a significant correlation between the dispersion of the MC strain rate and the degree of specimen failure. The coefficient of variation (CV) for each strain rate field was calculated and plotted as a red curve in Figure 5. Then, the averages of 20 contiguous CVs were calculated and plotted as a blue curve in Figure 5. These two curves could be used to analyze the relationship between the dispersion of the strain rate data and the damage evolution of the intact specimen.In this figure, the blue curve has more obvious characteristics that can distinguish the damage stages of pre-peak loading. From 30% to 70% strain, the curve of the strain-contrast RA of the CVs ascends slowly with some slight waves; from 70% to 80% strain, the waves become larger; between 0.8 and \(\sim\)1.0 strain, the extremely volatile growth of the blue curve correspond with the rapid damage accumulation in the BPM. To further analyze the evolution of the MC strain rate distribution during the loading process, eight contour maps of the strain rate field at different moments were investigated, as shown in Figure 6. The Z-axes values illustrate the volume strain rate of each measurement circle. A positive strain rate indicates an increase in volume strain, and vice versa. Under uniaxial compression, negative strain rates are the main component of the strain rate field; however, a positive strain rate might occur in situations such as particle ejection at the edge of the specimen, shear dilation, and tensile failure of bonds. Figure 5: Coefficient of variation of strain rate field and its regional average (case 1). In Figure 6, at 20% peak strain, the strain rate field has slight waves uniformly distributed over the whole surface with negative values. At this moment, the intact specimen is in the stage of crack closure. At 40%, 60%, and 70% strain, a few high ratios of MC strain rates randomly emerge in the strain rate field, corresponding to the growth pattern of microcracks in Figure 2:0 From 40% to 70% strain, the specimen is in the stage of stable crack growth. From 80% to 100% strain, the extreme values of the MC strain rate are larger than ever before and concentrate in the left-bottom side of the sample. This indicates that the damages concentrate in that particular area, and the specimen moves into the stage of unstable crack growth. Figure 6 represents the failure process of this intact BPM. From scattered to centralized, the damage distribution changed by the load increment and finally collapsed because of damage localization at the left-bottom side of the BPM. The variation of these contour maps confirms the micro-crack increase tendency and other statistical data curves. Figure 2:1 Strain rate field at different strains (case 1). As mentioned above, the pre-peak failure process of the intact sample could be divided into three or four sections: the first section is before 30% peak strain; section 2 can be seen as two parts: one is between 30 and 60% peak strain and the other is between 60 and 80% peak strain; the last section is between 80 and 100% strain. In section 1, the micro-crack increment occasionally and separately emerged; all three statistics (average, SD, and CV) of the MC strain rates are close to zero and have minor up and down patterns; the RA values of the range are decreasing and close to zero; the strain rate field displays regular undulation with negative values. These performances indicate that section 1 corresponds to the crack closure stage and linear elastic deformation stage in which the damage accumulation is essentially close to zero.