## 3 Numerical Experimentation and Analysis
### The Influence of Pore Structure on the Tensile Strength of Concrete  
In previous studies, scholars often used linear relationship [6; 40], exponential relationship [29; 50], and power function relationship [34; 52] to describe the relationship between total porosity and concrete strength. The relationship between the total porosity and the tensile strength obtained in this paper is shown in Figure 19, which is fitted with linear, exponential, and power functions, respectively. From the fitting results in Table 5, it can be seen that the exponential function fits the best. [PERSON] et al. [53] obtained the two-dimensional pore structure photos of concrete specimens through deep-learning-based image segmentation, and measured the relationship between the total porosity and tensile strength of concrete with the same aggregate gradation. The relationship exhibits an obvious exponential relationship, as shown in Figure 20, which is consistent with the conclusions of this paper.  
Figure 19: Relationship between the total porosity and the tensile strength.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Function Type** & **Fit Function** & **Correlation Coefficient (R\({}^{2}\))** \\ \hline Linear function & y = 3.197 \(-\) 0.322x & 0.866 \\ Exponential function & y = 1.721 + 1.858e\({}^{-\text{x}/2.121}\) & 0.934 \\ Power function & y = 328.12 \(-\) 325.22x\({}^{0.002}\) & 0.911 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Fitting results of the relationship between total porosity and tensile strength.  
Figure 18: Relationship between PSSA and elastic modulus.  
The maximum fluctuation amplitude of the tensile strength of the concrete specimens with the same aggregate structure and total porosity but different sub-porosities is 0.447 MPa, which occurs in specimens with total porosity of 1%, accounting for about 14.8% of its tensile strength. The maximum fluctuation amplitude of the tensile strength of the specimens with the same aggregate structure, total porosity, and sub-porosities but different pore distribution is 0.256 MPa, which occurs between specimen 1-1-L and specimen 1-2-L, accounting for about 8.66% of its tensile strength. Figure 21 shows the tensile strength of three types of concrete specimens. For specimens with the same total porosity, the larger the content of macropores, the smaller the number of pores. Large pores tend to reduce the strength of the specimen, and the decrease in the number of pores tends to increase the strength of the specimen. These two factors work together on the strength of the concrete, and neither one dominates, as shown in Figure 21.  
Figure 22 shows the relationship between the average pore radius and the tensile strength, while Figure 23 shows the relationship between the PSSA and the tensile strength. It can be seen from Figure 22 that for the specimens with the same total porosity, the average pore radius has little effect on the tensile strength of concrete, which is consistent with the research results of [PERSON] [52].  
Figure 20: Relationship between the total porosity and the tensile strength of specimens in Ref. [53].  
Figure 21: Tensile strength of three types of specimens.  
Polynomial function, exponential function, and power function are used to fit the relationship between tensile strength and PSSA. The fitting results are shown in Table 6, from which it can be seen that the quadratic polynomial function fits the best. It can be seen from Figure 23 that, for the specimens with the same total porosity, PSSA has no regular effect on the tensile strength.