# Numerical Simulation of Rock-Breaking Mechanism by Spherical Tooth Impact in Granite Formation
## 2 Model and Methodology  
### Spherical Tooth Impact Model  
Contact stress and contact area are fundamental parameters used to describe the impact fragmentation process. The geometric description of contact is shown in Figure 1. Another important parameter during the impact process is the average contact stress \(\sigma_{avg}\), which is related to the contact stress \(L\) and the contact radius \(a\). Here, the contact radius \(a\) refers to the distance from the particle's center axis to the edge of the actual contact area between the particle and the rock. The relationship is expressed as follows:  
\[\sigma_{avg}=\frac{L}{\pi a^{2}} \tag{1}\]  
Based on elastic-plastic indentation analysis and Newton's law, the differential relationship between force and displacement is obtained:  
\[mvdv=-Ldr \tag{2}\]  
where \(m\) is the quality of the spherical tooth (apportion the quality of the bit); \(v\) is the impact velocity of the spherical tooth; \(r\) represents the depth of the impact crater. Based on the Brinell hardness experiments conducted by [PERSON] et al. (2010), a nonlinear elastoplastic impact analysis model was developed. The model provides a detailed description of the contact stress and derives its corresponding expression:  
\[H=\frac{L}{\pi a^{2}}=K\bigg{(}\frac{a}{2R_{0}}\bigg{)}^{n} \tag{3}\]  
where \(H\) represents Meyer hardness; \(P\) represents contact stress; \(K\) is the Meyer law constant; \(n\) is the strain hardening exponent of the material; \(R_{0}\) is the radius of the spherical tooth. Under the impact of the spherical tooth, the stress and strain generated inside the rock propagate in the form of a stress wave spherical wave. As shown in Figure 1, the spherical tooth impacts the rock at interface point \(O\), causing the stress wave to propagate from the surface of the spherical tooth into the rock. Due to the symmetry of the spherical wave, in the spherical coordinate system \((r,\theta,\varphi)\), the wave is independent of \(\theta\) and \(\varphi\), so \(\frac{\partial}{\partial\varphi}=\frac{\partial}{\partial\theta}=0\), the longitudinal displacement \(v\) and the lateral displacement \(w\) are both zero, \(v=w=0\), and the radial displacement \(u\) is not zero. According to the spherical coordinates, the stress-strain relationship is:  
\[\varepsilon_{r}=\frac{\partial\sigma_{r}}{\partial r}dr=\frac{\partial u}{ \partial r} \tag{4}\]  
Figure 1: Diagram of spherical coordinates of stress waves. \[\epsilon_{\theta}=\epsilon_{\varphi}=\frac{u}{r} \tag{5}\]  
where \(\epsilon_{r}\), \(\epsilon_{\theta}\) are the radial and circumferential strain of the rock in a spherical coordinate system:  
According to the generalized [PERSON]'s law, assuming that the rock is an elastic body before being damaged, the mathematical expressions of the radial and circumferential stresses of the stress wave are obtained:  
\[\sigma_{r}=\frac{E}{(1+\mu)(1-2\mu)}\bigg{[}(1-\mu)\frac{\partial u}{\partial r }+2\mu\frac{u}{r}\bigg{]} \tag{6}\]  
\[\sigma_{\theta}=\frac{E}{(1+\mu)(1-2\mu)}\bigg{[}\mu\frac{\partial u}{\partial r }+\frac{u}{r}\bigg{]} \tag{7}\]  
where \(E\) is the [PERSON]'s modulus of the rock; \(\mu\) is the [PERSON]'s ratio of the rock. The calculation of the rock stress field can be regarded as the solution to the motion equation of spherical cavity compression. Because the stress wave is spherically symmetric, the wave field variables are only wave radius r and time t, and the only variable is along the radial direction, which is set as \(u(r,t)\). The boundary conditions and initial conditions are:  
\[\left\{ \begin{array}{l}r\geq\frac{R_{0}}{2T},t\geq 0,u(r,t)=u^{\prime}(r,t)=0\\ r=\frac{R_{0}}{2},t=0,\sigma_{r}=-P\end{array} \right. \tag{8}\]  
The negative sign represents the pressure and the conservation of the momentum of the spherical coordinate infinitesimal \(d\theta,dr\).