## 3 Fractal Model Based on Optical Method  
The contour of the pore section of concrete is complicated and has obvious fractal features. Previous studies have shown that a fractal model based on an optical method is constructed mainly around the solution of the fractal dimension of the pore section contour, e.g., by the perimeter-area method [36]. The fractal model of the concrete pore section is established according to the principle of the perimeter-area method, and the calculation formula is given as follows:  
\[\lg P=0.5D_{p}\lg A+C \tag{1}\]  
Figure 2: (**a**) Test instrument. (**b**) Test specimen.  
where \(P\) is the perimeter, \(A\) is the area, \(D_{p}\) is the fractal dimension of pore section contour, \(C\) is a constant.  
[PERSON] and [PERSON] used the above-described model to study the fractal dimension of concrete [37; 38]. Their results demonstrated that the fractal dimension obtained by this model could effectively describe the roughness of concrete, but the correlation between the fractal dimension and the macro-performance of concrete was very low. Therefore, [PERSON] improved the model and proposed a new fractal dimension for calculating the pore size distribution [26]; the calculation formula is given as follows:  
\[\lg{n}=D_{d}\lg{d}+C \tag{2}\]  
where \(d\) is the diameter of a pore, \(n\) is the number of pores with diameter larger than or equal to d, \(D_{d}\) is the fractal dimension of the pore size distribution, and \(C\) is a constant.  
The fractal dimension can effectively describe the pore size distribution of concrete by analyzing the experimental data, and it has a good linear relationship with the compressive strength of concrete. According to the test method and the characteristics of the test data, [PERSON] et al. introduced the box dimension into the previous model. The box dimension is determined by the covering of the same shape set, and it is relatively simple to calculate [39]. The mathematical expression of the box dimension is given as follows: F is an arbitrary nonempty subset of R\({}^{\text{n}}\) and N\({}_{\delta}\)(F) is the minimum number of sets that have a maximum diameter of \(\delta\) and that can cover set \(F\); then, the box dimension of \(F\) can be calculated by formula (3):  
\[Dim_{B}F=\lim_{\delta\to 0}\frac{\lg{N_{\delta}(F)}}{\lg(1/\delta)} \tag{3}\]  
The pore structure analyzer for hardened concrete measures the number of circular pores at a given circular degree under the assumption that the tested pore in concrete is a regular circular air-hole, and n circular box are selected for measurement according to the definition of the box dimensions. These boxes are used to cover pores whose diameter is greater than or equal to d\({}_{\text{i}}\). According to the principle of equal area, the pore with a diameter greater than d\({}_{\text{i}}\) is transformed into that with the diameter of d\({}_{\text{i}}\), and the number of converted pores, \(N_{\text{ci}}\), with diameter \(d_{\text{i}}\) is obtained. As a result, the data set (d\({}_{\text{1}}\), N\({}_{\text{c1}}\)), (d\({}_{\text{2}}\), N\({}_{\text{c2}}\)), (d\({}_{\text{3}}\), N\({}_{\text{c3}}\)),..., (d\({}_{\text{i}}\), N\({}_{\text{ci}}\)) is obtained. Finally, data on the pore diameter and the number of converted pores plotted on the double-logarithmic axis are subjected to linear regression analysis, and the slope of the regression line is the box dimension. The corresponding mathematical expression is given in Equation (4):  
\[\lg{N_{c}}=-D_{d}\cdot\lg{d}+C \tag{4}\]  
where \(N_{c}\) is the number of converted pores, \(D_{d}\) is the fractal dimension of the pore size distribution, \(d\) is the pore diameter, and \(C\) is a constant.