## 3 Numerical Experimentation and Analysis
### Failure Process Analysis
#### 3.1.1 The Failure Process of Concrete Specimen without Pores  
1. The relatively stable stage of micro-cracks (\(\sigma/\sigma_{\rm max}<\) 0.3-0.5);  
The damage degree of coh_mor and coh_CIE characterizes that of cement mortar and ITZ to some extent. At this stage, the internal stress of the concrete is small, and the visual cracking process shows that the damage of concrete first appears in the periphery of the specimen. Later in this stage, most of the damage variables of coh_mor are less than 0.45, and all of them are distributed in the periphery of the specimen. Figure 6a shows the distribution of CIEs with damage variable D = 0.12 when \(\sigma/\sigma_{\rm max}\approx 0.36\), where the black line segments represent CIEs, the green circle represents aggregate, and the yellow area represents the mortar. The damage variables of coh_ITZ are mainly distributed in the range of 0.46-0.59, and they are also mainly distributed in the periphery of the specimen. Figure 6b shows the distribution of CIEs with damage variable D = 0.52 when \(\sigma/\sigma_{\rm max}\approx 0.36\). This indicates that the damage degree of ITZ is greater than that of cement mortar at this stage, possible because the material strength of ITZ is only about 1/3 that of cement mortar, and the stress concentration at ITZ is stronger. At this stage, the damage of ITZ and cement mortar is mainly distributed in the periphery of the specimen and has not extended to the center of the specimen.  
2. The stable development stage of cracks (0.3-0.5 \(<\sigma/\sigma_{\rm max}<\) 0.75-0.9);  
Figure 7a,b respectively, show the distribution of CIEs with damage variable D = 0.34 and D = 0.7 when \(\sigma/\sigma_{\rm max}\approx 0.74\), which show that, at this stage, with the increase of stress, the damage of the cement mortar and ITZ develops and extends to the center of the specimen. The damage degree of ITZ is still larger than that of cement mortar, as shown in Figure 7b.  
Figure 5: Comparison of stress-strain curves in uniaxial tension [2,44].  
3. The unstable development stage of cracks (0.75-0.9 \(<\sigma/\sigma_{\rm max}\)).  
Figure 8a,b, respectively, show the distribution of CIEs with damage variable D = 0.7 and D = 0.96 when \(\sigma/\sigma_{\rm max}\approx 0.98\) (\(\varepsilon<\) peak strain \(\varepsilon_{\rm p}\)). Comparing Figures 7b and 8a, it is found that under the action of high-level stress, the micro-cracks in the cement mortar develop rapidly. Before the peak stress, \(\sigma_{\rm p}\), the damage variables of coh_ITZ mainly distributes in the range of 0.75-0.985, and the damage degree of ITZ is still larger than that of cement mortar, as shown in Figure 8b.  
Figure 8: The distribution of CIEs with D = 0.7 and D = 0.96 when \(\sigma/\sigma_{\rm max}\approx 0.98\), \(\varepsilon<\varepsilon_{\rm p}\). (**a**) D = 0.7; (**b**) D = 0.96.  
After reaching the peak stress, \(\sigma_{\rm p}\), the strain, \(\varepsilon\), continues to increase, and the micro-cracks in ITZ and cement mortar continue to develop, and the damage degree of cement mortar reaches the same level as that of ITZ, as shown in Figure 9, which shows the distribution of the CIEs with D = 0.98 when \(\sigma/\sigma_{\rm max}\approx 0.77\) (\(\varepsilon>\varepsilon_{\rm p}\)). With a further increase of strain, the micro-cracks in the cement mortar and ITZ are interconnected and penetrated to form macro-cracks, as shown in Figure 10, which shows a macroscopic penetrating crack when the specimen finally fails.