## 4 Numerical Simulation Analysis of Knife Handle Working Face
### Energy Evolution Characteristics of Overlying Rock in Knife Handle Working Face  
Based on the comprehensive histogram of the borehole, a FLAC\({}^{\text{3D}}\) numerical calculation model was established to study the energy distribution characteristics when the working face width changes and to analyze the energy accumulation and release rules during the mining process. The total input energy of coal and rock mass units in the energy field generated by the stope stress environment is expressed as:  
\[e=\int_{0}^{\epsilon_{1}}\sigma_{1}\text{d}\epsilon_{1}+\int_{0}^{\epsilon_{2 }}\sigma_{2}\text{d}\epsilon_{2}+\int_{0}^{\epsilon_{3}}\sigma_{3}\text{d} \epsilon_{3} \tag{14}\]  
Figure 8: Vertical stress distribution characteristics of the working face when mining 400 m. (**a**) Vertical stress distribution cloud diagram of surrounding rock. (**b**) Vertical stress distribution characteristics of coal mass in front of the working face. Figure 7: Vertical stress distribution characteristics of the working face when mining 200 m. (**a**) Vertical stress distribution cloud diagram of surrounding rock. (**b**) Vertical stress distribution characteristics of coal mass in front of the working face. In the formula: \(e\) is the total work performed by the principal stress in the principal strain direction(J); \(\sigma_{1}\) is the maximum principal stress (MPa); \(\sigma_{2}\) is the intermediate principal stress (MPa); \(\sigma_{3}\) is the minimum principal stress (MPa); \(\varepsilon_{1}\) is the maximum principal stress elastic strain in the direction of stress; \(\varepsilon_{2}\) is the elastic strain in the direction of the intermediate principal stress; \(\varepsilon_{3}\) is the elastic strain in the direction of the minimum principal stress. [PERSON]'s law in elastic mechanics is introduced to deduce the \"stress-energy\" relationship of the coal and rock mass, and the elastic strain energy expression is obtained [34] as:  
\[e=\int_{0}^{\frac{1}{2}[\sigma_{1}-\mu(\sigma_{2}+\sigma_{3})]}\frac{\sigma_{ 1}}{E}d\sigma_{1}+\int_{0}^{\frac{1}{2}[\sigma_{2}-\mu(\sigma_{1}+\sigma_{3})] }\frac{\sigma_{2}}{E}d\sigma_{2}+\int_{0}^{\frac{1}{2}[\sigma_{3}-\mu(\sigma _{2}+\sigma_{1})]}\frac{\sigma_{3}}{E}d\sigma_{3} \tag{15}\]  
\[U_{e}=\frac{1}{2E}\Big{[}\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}-2\mu( \sigma_{1}\sigma_{2}+\sigma_{1}\sigma_{3}+\sigma_{2}\sigma_{3})\Big{]} \tag{16}\]  
In the formula, \(E\) is the elastic modulus (GPa); \(\mu\) is Poisson's ratio; \(Ue\) is the releasable elastic strain energy of the coal and rock mass (J/m\({}^{3}\)). Using the FLAC\({}^{3\text{D}}\) numerical simulation software for secondary development, the energy distribution cloud diagram when the working face width changes is output, and the energy accumulation area during the mining process of the working face is determined. When the working surface advances every 20 m, the fish language is used to export the energy density cloud diagram, and the strain energy density of the I010206 working surface is sliced and combined, as shown in Figure 9. During the advancing process of the working surface, when the width of the working surface is 85 m, energy accumulation areas appear on both sides of the working face and the head-on side. As the working face continues to advance, the energy accumulation area further increases; the overall distribution pattern is relatively similar. When the working face width is 137.8 m, the strain energy density of the working face at different mining distances has peak values of 1.21 \(\times\) 10\({}^{5}\) J/m\({}^{3}\), 1.26 \(\times\) 10\({}^{5}\) J/m\({}^{3}\), 1.30 \(\times\) 10\({}^{5}\) J/m\({}^{3}\), 1.38 \(\times\) 10\({}^{5}\) J/m\({}^{3}\), and 1.78 \(\times\) 10\({}^{5}\) J/m\({}^{3}\). As the width of the working surface increases, the peak strain energy density shows an increasing trend. As the working face continues to advance, the basic roof breaks periodically, the roof of the gob sinks further, and the strain energy density of the protected coal pillar area at the corner further increases.