## 3 Results and Discussion
### DVC Analysis
#### 3.2.4 Localization Factor  
It can be summarized from the analysis in Section 3.2.2 that the internal damage of the specimen is ultimately caused by the large maximum principal strain. Therefore, the distribution range of the maximum principal strain of all units is analyzed by the probability density function, as shown in Figure 16. The maximum principal strain has a narrow distribution range under 30 MPa, and the strain values concentrate in the range of 0-0.006, indicating that most areas inside the specimen are in the stage of small deformation. When the stress increases to 50 MPa, the distribution range of the maximum principal strain becomes wide and presents a normal distribution, indicating that the deformation of the specimen develops further. These arbitrary values of the specimen were selected to correspond to approximately 90% of the cumulative distributions as the strain threshold [30]. \(\varepsilon_{\text{thr}}\) = 0.006 and \(\varepsilon_{\text{thr}}\) = 0.021 were selected as the strain thresholds under 30 MPa and 50 MPa, respectively. Strains with \(\varepsilon_{\text{max}}>\varepsilon_{\text{thr}}\) will be referred to as \"high-magnitude strains\". It can be seen from Figure 16 that the number of high-amplitude strains under  
Figure 15: Gray value and damage index of different section position. different stress states is small; however, these strains are the main impact factor in forming strain localization. Figure 17 presents the space distributions of high-magnitude strains under different stress stages. When the stress is 30 MPa, there are many units with high-amplitude strain, and their distribution is relatively uniform, as shown in Figure 17a. With the increase in stress, the number of high-amplitude strain units decreases. The positions of some units are roughly the same as those at 30 MPa; otherwise, high-amplitude strain units also appear within the range of 150-300 voxels near the \(z\)-axis, as shown in Figure 17b, indicating that prominent localized damage occurred in the specimen at this time. This phenomenon corresponds to the failure process of the specimen. The high-amplitude strain is mainly caused by cracks inside the specimen. Therefore, the spatial distribution of the high-amplitude strain unit can reflect the failure process of the specimen under uniaxial compression. The coordinate position of high-amplitude strain units was analyzed, and a correlation coefficient [31,32] was introduced as Equation (3):  
\[C_{xyz}=\frac{|\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})(z_{i}- \overline{z})|}{\sqrt{\sum_{i}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i}^{n} (y_{i}-\overline{y})^{2}}\sqrt{\sum_{i}^{n}(z_{i}-\overline{z})^{2}}} \tag{3}\]  
where \(\overline{x}\), \(\overline{y}\), and \(\overline{z}\) are the average coordinate values in the X-, Y-, and Z-directions, respectively; \(n\) is the number of units. Figure 16: Probability density functions of the maximum principal strains \(\varepsilon\)max. Figure 17: Space distribution of high-magnitude strains units under different stress stages. The localization factor was defined as Equation (4):  
\[L_{f}=1-C_{xyz} \tag{4}\]  
The values of the correlation coefficient and the localization factor ranged within [0, 1]; however, they were inversely proportional. Hence, the larger value of \(L_{f}\), the smaller value of \(C_{xyz}\), and the more dispersed the space distribution of the high-amplitude strain units are, indicating that the damage of the cement mortar was relatively small; otherwise, the smaller the value of \(L_{f}\), the larger the value of \(C_{xyz}\), and the more concentrated the space distribution of the strain units, implying greater damage. Figure 18 shows how the localization factor changes at different stages of stress. The value of \(L_{f}\) was 1 under 0 MPa, indicating that no damage occurred in the mortar at this time. When the stress increased to 30 MPa, the value of \(L_{f}\) decreased to 0.89, suggesting that the mortar began to fracture.