## 3 Effects of F-T Cycles on the Mechanical Properties and Microscopic Parameters
### Effect of F-T Cycles on Particle Morphological Characteristics
#### 3.2.2 Effect of F-T Cycles on Particle Size Fractal Dimension  
Due to the irregularity and complexity of the shape dimension of things in nature, the fractal dimension (D) is used to describe the complexity of a fractal set and represents the continuous non-integer value. Generally speaking, D is mostly based on the self-similarity of the graph, so the graph can be divided into several similar parts according to the scale. D of the graph can be expressed by Equation (1).  
\[\mathrm{D}=-\lim_{\varepsilon\to 0}\frac{\ln\mathrm{N}(\varepsilon)}{\ln\varepsilon} \tag{1}\]  
where: \(\varepsilon\) is the scale, \(\mathrm{N}(\varepsilon)\) is the metric value under the scale, and D is the fractal dimension of the study object.  
The particle size fractal dimension (\(\mathrm{D}_{\mathrm{ps}}\)) is used to describe the degree of unevenness of the particle size (r), which is characterized by the distribution characteristic of the cumulative number \(\mathrm{N}(\leq\mathrm{r})\) of particles smaller than a certain particle size and expressed by the morphological characteristics of the \(\mathrm{r}-\mathrm{N}(\leq\mathrm{r})\). curve. It can be known from the mass distribution feature [38] that the two have a good power function correspondence.  
\[\mathrm{N}(\leq\mathrm{r})\propto\mathrm{r}^{-\mathrm{D}} \tag{2}\]  
\[\mathrm{N}(\geq\mathrm{r})=\mathrm{M}-\mathrm{N}(\leq\mathrm{r}) \tag{3}\]  
where \(\mathrm{N}(\leq\mathrm{r})\) is the number of particles larger than a certain particle size and \(\mathrm{M}\) is the total number of particles, which is a constant.  
For the \"particle image\" where \(\mathrm{M}\) is certain, then \(\mathrm{N}(\leq\mathrm{r})\) has a fixed correspondence with \(\mathrm{N}(\geq\mathrm{r})\), so the following relationship can be considered.  
\[\mathrm{N}(\mathrm{r})\propto\mathrm{r}^{-\mathrm{D}} \tag{4}\]  
Equation (4) is identical in form to Equation (1), so D is defined as \(\mathrm{D}_{\mathrm{ps}}\) whose meaning is equivalent to the uneven coefficient \(\mathrm{C}_{\mathrm{u}}\) of the coarse-grained soil. The larger the \(\mathrm{D}_{\mathrm{ps}}\) is, the worse the degree of homogenization of particles is. The scatter plot of the two is plotted under the double logarithmic coordinates where r is the abscissa and \(\mathrm{N}(\leq\mathrm{r})\) is the ordinate. If the relationship between the particle size content and the particle size is linear, as well as the slope of the line is b, then \(\mathrm{D}_{\mathrm{ps}}=3-\mathrm{b}\). The changes in the \(\mathrm{D}_{\mathrm{ps}}\) of S0 and S2 under F-T cycles are drawn in Figure 5.  
Figure 5: The curve of \(\mathrm{D}_{\mathrm{ps}}\) and F-T cycles.  
Figure 5 illustrates that the D\({}_{\text{ps}}\) of S0 increased as F-T cycles continue. The D\({}_{\text{ps}}\) of S2 was between 1.88 and 2.15. The D\({}_{\text{ps}}\) of S2 was obviously smaller than that of S0, and this demonstrated that the properties of particles were relatively stable under F-T after cement was mixed. The particles of S0 only recombined simply due to water migration after F-T, so the particles after recombination showed a non-uniform state. However, because of the existence of the skeleton structure in S2, more small particles attached to the surface of large particles through water migration to form larger particle structures, making the particles uniform, which was consistent with the result discussed above.