# Stability Index of Surrounding Rock during Deep Rock Excavation Considering Energy Release Speed
## 2 Local Energy Release Effectiveness
### Energy Release Effectiveness  
Under the conditions above, combined with the local energy release rate index proposed by [PERSON] et al. [8], this study proposed a new index, that is local energy release effectiveness. The index considers both the energy released by the rock mass per unit volume and the energy released speed when the strain energy accumulated in the surrounding rock exceeds the energy storage limit of the rock mass. It is a quantitative index that can reflect the rockburst location and strength. Based on the above considerations, the following steps can be used to obtain the local energy release effectiveness index through numerical simulation: Use a constitutive model that reflects the brittleness of the rock mass in the simulation. The strain energy density of every element in the model during the excavation process is recorded by custom variables defined with the Fish language. Then, the maximum strain energy density (\(E_{\rm{Imax}}\)), \begin{table}
\begin{tabular}{c c} \hline \hline
**Name of Combustible** & **Reaction Heat (kJ/kg)** \\ \hline Coal & 8960 \\ Nitroglycerin & 6217 \\ Nitro cotton & 4291 \\ Trinitrofluence (TNT) & 4187 \\ Black powder & 2784 \\ Ammonium nitrate & 4228 \\ Fulminating mercury & 1733 \\ Lead azide & 1536 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the reaction heat of coal and some explosive substances. the minimum strain energy density (\(E_{\text{imin}}\)), the maximum time (\(t_{i\text{max}}\)), and the minimum time (\(t_{i\text{min}}\)) are identified. The local energy release rate (\(LERR\)) can be calculated with Equation (3):  
\[LERR_{i}=E_{i\text{max}}-E_{i\text{min}} \tag{3}\]  
The local energy release speed (\(LERS\)) can be calculated with Equation (4):  
\[LERS_{i}=\frac{E_{i\text{max}}-E_{i\text{min}}}{t_{i\text{min}}-t_{i\text{max}}} \tag{4}\]  
The local energy release effectiveness (\(LERE\)) can be calculated with Equation (5):  
\[LERE_{i}=(E_{i\text{max}}-E_{i\text{min}})\times f(LERS_{i}) \tag{5}\]  
In Equation (5), there is no theoretical support for the relationship between the energy release speed and energy release effectiveness. Therefore, for the expression of \(f(LERS)\), this study drew lessons from the relationship between the average fragment size of rock after blasting and the explosive intensity. The explosive intensity is closely related to its ability to crush and destroy the objects in contact with it during the explosion, which is the embodiment of its instantaneous power, and the fragment size of rock can reflect the crack growth density of rock to a certain extent. The specific description is interpreted as follows. Sandia National Laboratory of the United States carried out experiments to study the relationship between the average fragment size of rock dynamic crushing and the loading rate, the relationship is shown in Equation (6):  
\[d=\left(\frac{\sqrt{20}K_{lc}}{\rho c\hat{c}}\right)^{2/3} \tag{6}\]  
where \(d\) is the fragment size, \(K_{lc}\) is the fracture toughness, \(\rho\) is the density, \(c\) is the elastic wave velocity, and \(\hat{c}\) is the strain rate of loading. According to the research results of [PERSON] et al. [25], the rock strength under different loading rates can be approximately expressed as Equation (7):  
\[\hat{\sigma}=k\hat{\epsilon}^{1/3} \tag{7}\]  
where \(\hat{\sigma}\) is the strength of the rock and \(k\) is the coefficient related to the rock properties.