## 2 Calculation Model of Particle Impact Strength Corrosion
### Influence Ranges of Particle Impact Strength Erosion  
The particle impacting behavior can be simplified as the concentrated force \(F=m_{p}a_{p}\) (where \(a_{p}\) is the acceleration of the impact particle), shown in Figure 3.  
According to Figure 3, the distance between any point \(M(x,y,z)\) in the rock mass and the impact point is \(r\); the distance from the impact acting surface is \(z\); and the projected length on the impact action surface is \(\rho\). Then, the stress at point \(M\) is:  
\[\begin{split}\sigma_{x}&=-\frac{F}{2\pi r^{2}} \bigg{\{}\frac{3x^{2}z}{r^{3}}-(1-2\mu_{T})\bigg{[}\frac{z}{r}-\frac{r}{r+2}+ \frac{x^{2}(2r+z)}{r(r+z)^{2}}\bigg{]}\bigg{\}}\\ \sigma_{y}&=-\frac{F}{2\pi r^{2}}\bigg{\{}\frac{3y^ {2}z}{r^{3}}-(1-2\mu_{T})\bigg{[}\frac{z}{r}-\frac{r}{r+2}+\frac{y^{2}(2r+z)}{ r(r+z)^{2}}\bigg{]}\bigg{\}}\\ \sigma_{z}&=-\frac{3 Fz^{3}}{2\pi r^{5}}\end{split} \tag{7}\]  
To establish a numerical calculation model for broken rock mass considering the impact effect, it is necessary to define the range of bond strength affected by the stress corrosion phenomenon. If we consider the process of a small particle impacting a large rock block as the squeezing of the rock block by the particle with the same force as the inertial force of accelerating motion, then [PERSON]'s theory can be applied. Assuming that the collision phenomenon during model loading conforms to [PERSON]'s contact theory, the pressure distribution in the impact zone is in a semi-spherical form, with maximum stress at the point of impact. Then, it is not necessary to consider the horizontal stresses, and we can calculate the radius of the semi-spherical by only considering the vertical stress. The maximum stress can be calculated as:  
\[q_{\max}=-\frac{3F}{2\pi a^{2}} \tag{8}\]  
where \(F\) denotes the force acting on the impact particle and \(a\) represents the radius of the contact area, which can be computed as:  
\[a^{3}=\frac{3}{2}\frac{1-\mu_{T}^{2}}{E_{T}}r_{p}F \tag{9}\]  
where \(r_{p}\) denotes the radius of the impact particle and \(\mu_{T}\) is the Poisson ratio of the rock mass.  
Hence, by combining Equations (8) and (9), the equation of \(q_{\max}\) can be obtained and expressed as follows:  
Figure 3: Schematic diagram for calculating the impact ranges of impact effects on rock blocks. (**a**) Distribution diagram of fractured rock mass in goat; (**b**) Schematic diagram of the impact of rock particles on the broken rock mass in goat; (**c**) Simplified diagram of the force on the rock block affected by impact erosion.  
\[q_{\max}=0.364F^{\frac{1}{3}}\bigg{(}\frac{1-\mu_{T}}{E_{T}}r_{p}\bigg{)}^{-\frac{2} {3}} \tag{10}\]  
By analogy with Hertz, the range of stress corrosion phenomena is also assumed to be hemispherical in the target rock mass. Similarly, the strength corrosion phenomenon no longer occurs in the target material rock mass when the vertical stress at point \(M\) is less than \(0.1\,q_{\max}\). By combining Equations (7) and (10), the strength erosion area can be obtained as follows:  
\[r<3.623\bigg{(}\frac{1-\mu_{T}}{E_{T}}r_{p}n_{p}\frac{\partial v_{p}}{\partial 2^{2}}\bigg{)}^{\frac{1}{3}} \tag{11}\]