## 3 Numerical Experimentation and Analysis
### The Influence of Pore Structure on the Stiffness of Concrete  
The 25 uniaxial tensile stress-strain curves obtained in this paper are shown in Figure 15. According to the suggestion of the famous scholar [PERSON] [48], the secant modulus at 50% of the peak stress in the ascending section is taken as the tensile elastic modulus of the concrete specimen. The macroscopic mechanical properties of the concrete specimens are shown in Table 4. The laws between the pore structure and the mechanical properties of concrete analyzed and discussed in the paper are limited to the range ofparameter values in this paper, such that the range of the total porosity is 0-4%, which is hereby explained, and no special explanation will be given later.  
In a previous experimental study [50] and in numerical simulation experimental studies [29,51], scholars used mainly linear and exponential relationships to describe the relationship between elastic modulus and total porosity, but these experiments have shortcomings in simulating the complex pore structure and applying the CVM accurately. The relationship between total porosity and the elastic modulus obtained in this paper is shown in Figure 16a. It can be seen that both the linear relationship and the exponential relationship can well reflect the relationship between the total porosity and the elastic modulus, but the exponential relationship is better than the linear relationship. Therefore, the change of aggregate structure has little effect on the functional relationship between total porosity and elastic modulus.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample** & **Modulus (MPa)** & **Strength (MPa)** & **Peak Strain (10\({}^{-3}\))** \\ \hline
0–1 & 33,841 & 3.532 & 0.168 \\
1-1-S & 29,632 & 3.149 & 0.2104 \\
1-2-S & 29,879 & 3.001 & 0.1925  
The maximum fluctuation amplitude of the elastic modulus of the concrete specimens with the same aggregate structure and total porosity but different sub-porosities is 1746 MPa, which occurs in specimens with a total porosity of 1%, accounting for about 5.7% of its elastic modulus. The maximum fluctuation amplitude of the elastic modulus of the specimens with the same aggregate structure, total porosity, and sub-porosities but different pore space distribution is 968 MPa, which occurs between specimen 3-1-M and specimen 3-2-M, accounting for about 4% of its elastic modulus. Figure 16b shows the elastic moduli of the three types of specimens established in this paper, from which it can be seen that, for the specimens with the same total porosity, the greater the content of macropores, the greater the elastic modulus. The reason for this may be that, for specimens with the same total porosity, the larger the content of macropores, the less the number of pores. While the number of pores decreases, the elastic modulus of concrete increases.  
Figures 17 and 18, respectively, show the relationship between the average pore radius and the elastic modulus and the relationship between the PSSA and the elastic modulus. It can be seen from Figure 17 that, for specimens with the same aggregate structure and total porosity, the elastic modulus is basically not affected by the size of the average pore radius. This finding is consistent with the conclusions of Ref. [29]. It can be seen from Figure 18 that the linear relationship and exponential relationship can well reflect the functional relationship that the elastic modulus decreases with the increase of PSSA, and the exponential relationship is better. For specimens with the same aggregate structure and total porosity, the elastic modulus has a tendency to decrease with the increasing PSSA.  
Figure 16: Relationship between total porosity and elastic modulus. (**a**) Tensile elastic modulus of all specimens; (**b**) Tensile elastic modulus of three types of specimens.  
Figure 17: Relationship between average pore radius and elastic modulus.