## 5 Rockburst Warning Based on a Mechanical Model
### Mechanism of Two-Body Interaction  
After the surrounding rock is excavated and unloaded, the surrounding rock shows an increase in the tangential stress. Because of the heterogeneous characteristics of the rock mass, the loading stress will be concentrated in the relatively weak position, resulting in microfracture precursors. When the local microfracture accumulates to a certain extent, the elastic energy accumulated in the local damaged area and around the relatively intact rock mass is released together, resulting in a rockburst. This process can be explained by a two-body interaction system [65]. The intact surrounding rock mass in the elastic deformation stage is called Body I, and the local damaged area that interacts with Body I is called Body II. When a load \(P\) is applied to the two-body system, in the loading process, Body I is always in an elastic state, while Body II undergoes a softening process. The two-body system can be represented by a spring with stiffness \(k\) and deformation \(a-u\) (as shown in Figure 6a):  
\[P=k(a-u) \tag{10}\]  
The load of Body II can be expressed as:  
\[F(u)=\lambda ue^{-u/u_{0}} \tag{11}\]  
where \(\lambda\) is the initial stiffness; \(u_{0}\) is the deformation value corresponding to the peak load. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Date**} & **Total Rockburst** & **Maximum** & **Model Prediction** & **Rockburst Level** & **Risk Level Predicted** \\  & **Range** & **Rockburst Level** & **Range** & **Predicted by the Model** & **by the Model** \\ \hline
2020/3/9 & K40+688.0–694.0 & Strong & K40+680–700 & Strong & Extremely high \\
2020/3/10 & K40+694.0–697.0 & Strong & K40+687–707 & Strong & Extremely high \\
2020/3/11 & K40+697.0–700.0 & Strong & K40+690–710 & Strong & Extremely high \\
2020/3/12 & K40+700.0–704.0 & Strong & K40+693–713 & Strong & High \\
2020/3/13 & K40+704.0–710.0 & Strong & K40+696–716 & Moderate & High \\
2020/3/14 & K40+710.0–717.0 & Moderate & K40+703–723 & Moderate & Extremely high \\
2020/3/15 & K40+717.0–723.2 & Strong & K40+712–732 & Strong & Extremely high \\
2020/3/16 & K40+723.0–727.0 & Strong & K40+717–737 & Strong & Extremely high \\
2020/3/17 & K40+726.8–730.0 & Strong & K40+721–741 & Strong & Extremely high \\
2020/3/18 & K40+730.0–734.0 & Strong & K40+724–744 & Moderate & Extremely high \\
2020/3/19 & K40+734.0–738.0 & Strong & K40+727–747 & Strong & Extremely high \\
2020/3/20 & K40+738.0–741.0 & Strong & K40+731–751 & Strong & Extremely high \\
2020/3/21 & K40+741.0–744.4 & Strong & K40+734–754 & Strong & Extremely high \\
2020/3/22 & K40+746.4–751.4 & Strong & K40+737–757 & Strong & Extremely high \\
2020/3/23 & K40+751.4–755.0 & Strong & K40+742–762 & Strong & Extremely high \\
2020/3/24 & K40+755.0–759.4 & Strong & K40+746–766 & Strong & Extremely high \\
2020/3/25 & — & No rockburst & K40+755–771 & Mild & Low \\
2020/3/26 & — & No rockburst & K40+755–775 & No rockburst & No rockburst \\
2020/3/27 & — & No rockburst & K40+759–779 & No rockburst & No rockburst \\
2020/3/28 & K40+771.9 & Mild & K40+764–784 & Mild & High \\
2020/3/29 & — & No rockburst & K40+768–788 & No rockburst & No rockburst \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the real performance of the model applied to the early warning of rockbursts (a part). The instability conditions of the system are as follows:  
\[\frac{dF(u)}{du}+k<0 \tag{12}\]  
The stability of the two-body interaction system can be divided into three stages, as shown in Figure 6b. In the 0-A stage, both Body I and Body II are in the stage of elastic energy storage. Because of the low stress level, the microcracks propagate steadily, resulting in some low-intensity microseismic events. In the A-B stage, the load gradually increases to the maximum, the Body II is in the inelastic deformation stage, and Body I is still in the elastic stage or has a slight inelastic deformation. At this stage, \(\frac{dF(u)}{du}\) gradually decreases to 0, and \(\frac{dF(u)}{du}+k>0\), Therefore, the system is stable at this stage.