## 4 Multi-Source Monitoring Function Based on DIC-AE Fusion
### Analysis on Premonitory Information Based on DIC-AE Fusion  
During the loading and failure processes of red sandstone specimens, the fitting function values of the multi-source information change nonlinearly (Figure 16). By comparing the stress-strain curves and the multi-source information function curves during the failure process, it was found that the points at which the concavity and monotonicity of the multi-source information function curves change may be the dividing points between different failure stages.  
In order to further explore the change of the multi-source information function and characterize the process and law of red sandstone failure, the first-order and second-order derivatives of the function were worked out. The curves of the first-order and second-order derivative functions are presented in Figure 16. When the curves of the derivative functions  
\begin{table}
\begin{tabular}{c c} \hline \hline
**i** & \(\mathbf{R^{2}}\) \\ \hline
1 & 0.98698 \\
2 & 0.99538 \\
3 & 0.94148 \\
4 & 0.97489 \\
5 & 0.9108 \\
6 & 0.93581 \\
7 & 0.96927 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the fitting function.  
Figure 16: Derivation curves of the multi-source information: (**a**) first-order derivative image and (**b**) second-order derivative image.  
intersect the \(X\)-axis, the value of the derivative function is 0. This intersection represents a stagnation point or an inflection point.  
By comparing the first-order and second-order derivative functions, it can be seen that functions \(f_{1}(\mathbf{x})-f_{7}(\mathbf{x})\) all have extreme points and inflection points. Subsequently, the failure stages are divided according to the extreme points and inflection points in the derivation functions, and the time points were modified. The modified stage division results are listed in Table 3. The premonitory characteristics of the specimens can be divided into four stages. To be specific, Stage I witnesses the closure of primary cracks at the initial stage of loading; Stage II is the precursor to the stable development into the elastic stage; Stage III is the precursor of the transition from the elastic stage to the nonlinear development stage; and Stage IV is the precursor of the overall failure. On this basis, red sandstone failure can be forewarned by levels. In Figure 17, the divided stages are combined with the stress-strain curves of the specimens. As depicted in Figure 17, Stages II and III last relatively longer, during which the specimens experience the transition from the elastic stage to the crack expansion stage and accumulate energy for the upcoming failure. Stages I and IV occupy a smaller proportion of the time. Stage IV represents the process in which the cracks gradually develop and expand to reach the limit that the specimens can bear, leading to their rapid failure. This stage corresponds to the crack expansion and post-peak failure stages during red sandstone failure. Hence, it is of great significance to focus on the monitoring and forewarning of Stages III and IV.  
According to the results of the derivative functions, it was determined that the pre-monitory limit time when the specimens enter the failure period (Stage IV) is 398 s. In comparison, the experimental results show that the macroscopic crack expansion appears at 502 s and that the overall failure occurs at 543 s when the bearing limit is reached (Table 4). The initial crack expansion time and failure limit time measured by the derivative functions advance by 20.72% and 26.71%, respectively. The time node obtained by multi-source information fusion monitoring, which is remarkably earlier than the time node of the actual macroscopic failure, can be utilized as the premonitory information for forewarning the specimen failure.