## 3 Experimental Results and Analysis
### Fractal Features of the AE Signal Points  
The fractal dimension can be used to quantitatively evaluate the development process of complex phenomena. A larger fractal dimension indicates greater chaos at this stage; conversely, a reduced fractal dimension suggests improved order. Thus, the fractal dimension can, to some extent, describe the relationship between the local laws of phenomena and their overall development process. The development of acoustic emission signal points was analyzed in the previous section. However, a mere qualitative analysis of the development chart for acoustic emission signal points yields only a general trend of specimen damage and crack development. To further quantitatively analyze the development of acoustic emission signal points in rock, the coordinates of these points are plotted in both top and side views. Additionally, the correlation dimension is introduced to explore the fractal development of cracks in the X-Y and X-Z planes (Figure 8). The projection of acoustic emission signal points onto the top (XOY plane) and side (XOZ plane) views involves measuring the distance, L, from the coordinate to the origin on the side view, indicating the signal point's proximity to the lower end of the rock specimen. Similarly, the distance, R, from the coordinate to the origin on the top view indicates the distance from the center of the test specimen. Here, the X- and Y-axes are situated on the bottom surface of the cylinder, with the origin of the XOY plane located at the center of the cylinder's bottom surface and the \(Z\)-axis running parallel to the cylinder's axis and extending upward from the base. The G-P correlation dimension method is utilized, employing a function called MATLAB 2020, to calculate the fractal dimensions of the R and L values of the acoustic emission signal points from the origin. This calculation aims to analyze the fractal characteristics of the signal point development within the rock under various loading rates and temperatures. The core principle of the G-P correlation dimension method involves ordering the distances of each acoustic emission signal point from the origin according to their temporal sequence, as follows:  
\[X=[x_{1}x_{2}x\cdots x_{n}] \tag{4}\]  
Figure 8: Projection and three-dimensiona coordinate position of acoustic emission signal points on the X-Z and X-Y planes. The time series' first vector of distance from the origin for m elements, denoted as \(X_{1}\), can construct an m-dimensional phase space based on the sequential time delay interval, where (\(N=n-m+1\)), and \(m\)-dimensional vectors are constructed as follows:  
\[X_{i}=[X_{i},X_{i+1}\cdots X_{i+m-1}] \tag{5}\]  
The G-P function is defined by the space as follows:  
\[C_{m}(r)=\frac{1}{N^{2}}{\sum\limits_{i}^{N}{\sum\limits_{i}^{N}{H\big{[}r- \big{\|}X_{i}-X_{j}\big{\|}}\big{]}}} \tag{6}\]  
where \(C\) represents the correlation function, \(H\) denotes the Heaviside function, and \(r\) is the scale factor, expressed as follows:  
\[H=\left\{ \begin{array}{l}0,x<0\\ 1,x\geq 0\end{array} \right. \tag{7}\]  
\[r(k)=k\frac{i}{N^{2}}{\sum\limits_{i=1}^{N}{\sum\limits_{j=1}^{N}{\big{|}X_{i }-X_{j}\big{|}}}} \tag{8}\]  
where \(k\) is the observation coefficient, and based on the value of \(k\), the value of \(r\) is determined in the interval to obtain the association dimension \(D\), where the expression of \(D\) is as follows:  
\[D=\lim\limits_{r\to 0}\frac{\ln{C(r)}}{\ln{r}} \tag{9}\]  
The linear regression of the calculation outcome [ln\(r\), lnC] more closely resembles a straight line, which implies a tendency toward fractal characteristics in the data. Points are taken at 15 s intervals, and the minimum embedding dimension \(m=1\) is selected for the calculation. Each correlation dimension \(D\) in the calculation results is fitted using 20 [ln\(r\), lnC] points, ensuring a fit interval across all the data results within the range of 95-99%.