## 2 Drill Core Fidelity Performance Model
### Analysis of Forces on Cutting Teeth  
As the drill bit continues to drill downwards, the damage of the cutting force of the cutter is gradually transmitted to the core part. With increasing stress inside the rock, when these stresses surpass their tolerance limits, the rock fractures, forming a core. The core created during the drill bit's cutting process is therefore influenced by the cutting forces exerted by the drill bit and the rock damage mechanism.  
For the sake of calculation convenience, the fracture surface of the rock is simplified as a plane. It can be deduced from force analysis that the stress distribution on this fracture is maximum at the bottom edge and zero at the free surface. Based on this boundary condition, let us assume that the stress distribution on the fracture follows Equation (1) [25].  
\[p=p_{0}(\frac{d}{\sin\alpha}-r)^{n} \tag{1}\]  
Figure 2: The direction of the back-rake angle and side-rake angle.  
In the provided equation, \"\(\tau\)\" represents the distance from any point on the fracture face to the tip of the cutting tooth; \"\(d\)\" is the depth of cut; \"\(\alpha\)\" is the angle between the fracture face of the rock and the cutting plane of the PDC piece; and \"\(n\)\" is the stress distribution coefficient, which is influenced by the back-rake angle \(\theta\) and the geometry and size of the drill bit. By integrating the above equation, the force acting on the cut core can be simplified as shown in Figure 3 without considering the motion effect. The resultant external load on the fracture surface should be equal to the external force of the PDC acting on the cut core, that is, the average cutting force, as shown in Equations (2) and (3).  
\[p_{0}\int_{0}^{d/\sin\alpha}\left(\frac{d}{\sin\alpha}-r\right)^{n}dr=P_{N} \tag{2}\]  
\[p_{0}=\frac{(n+1)P_{N}}{\left(d/\sin\alpha\right)^{n+1}} \tag{3}\]  
Bringing the stress distribution of the structure into Equation (1) above, this results in a maximum stress for the tip of the rupture face as shown in Equation (4).  
\[p_{\max}=\frac{(n+1)P_{N}\sin\alpha}{d} \tag{4}\]  
At this point the direction of the maximum stress \(p\_\)max load is parallel to the cutting surface and the angle \(\gamma\) is the shear fracture angle forming the rupture surface, then the positive and tangential stresses \(\sigma\) and \(\tau\) on the shear breaking surface of the rock, where:  
\[\sigma=p_{\max}\sin\alpha=p_{\max}\sin\left(\frac{\pi}{2}-\theta-\gamma\right) =p_{\max}\cos(\theta+\gamma) \tag{5}\]  
\[\tau=p_{\max}\cos\alpha=p_{\max}\cos\left(\frac{\pi}{2}-\theta-\gamma\right) =p_{\max}\sin(\theta+\gamma) \tag{6}\]  
The damage analysis of the rock is based on the M-C shear damage theory, which takes into account the effects of positive and shear stresses [26]:  
\[\tau=\sigma\tan\varphi+c \tag{7}\]  
where \(\varphi\) is the angle of internal friction, \(c\) is the cohesive force, and the simplification gives:  
\[P_{N}=d\cdot c\cdot\frac{\cos\varphi}{(n+1)\cdot\cos(\theta+\gamma)\cdot\sin( \theta+\gamma-\varphi)} \tag{8}\]  
Figure 3: Analysis of forces on cutting teeth.  
The relationship between the cutting force and the structural parameters of the drill bit is shown in Equation (8), and the core's shear strength and the drill bit's structural parameters are shown in Equation (9).  
\[\tau=\sin(\theta+\gamma)\cdot c\cdot\frac{\cos\varphi}{\sin(\theta+\gamma- \varphi)} \tag{9}\]  
Based on the analysis presented above, it is demonstrated that the shear stress distribution in the PDC drill core depends not only on the axial and tangential loads, as well as the internal friction angle and cohesion of the rock, but also on factors such as the back-rake and side-rake angles of the drill bit's structural parameters. Since the internal friction angle and cohesion of the rock remain constant during the cutting process, this paper proceeds with simulation experiments to analyze the impact of the back and side rake angles of the drill bit's structural parameters on the fidelity performance of the drill bit.