## 5 Discussion
### The Relationship Between the Height of Extrusion Zone and Cutting Depth  
The key distinction between the proposed new dynamic cutting model (NDCM) and the traditional dynamic cutting model (DCM) lies in the size of the extrusion zone. To quantitatively analyze this distinction, the ratio of the extrusion zone depth \(h_{c}\) to the cutting depth \(h\) is defined as the geometric difference coefficient (\(GDC\)). \[GDC=\frac{h_{c}}{h} \tag{15}\]  
The closer the geometric difference coefficient is to 1, the smaller the difference between NDCM and DCM. Based on this study, the variation in GDC can be analyzed from the following three perspectives. Firstly, the theoretical change rule of the geometric difference coefficient is investigated. \(h_{c}\) in Equation (15) is replaced by Equation (12), yielding the following:  
\[GDC=\frac{h_{c}}{h}=\sqrt{\frac{4E_{s}}{E_{c}\sin\psi\tan a}}\frac{1}{\sqrt{h}} \tag{16}\]  
As long as the relevant parameters in Equation (16) are determined, the geometric difference coefficient can be obtained. Since Equation (12) is based on the principle of energy conservation, this method of deriving \(GDC\) is referred to as the energy analysis method. Once the characteristic parameters of the rock and the rake angle are determined, the term within the brackets in Equation (3) becomes constant. Additionally, there exists a power function relationship between \(GDC\) and cutting depth \(h\), as shown in Figure 17. As the cutting depth increases, \(GDC\) exhibits a nonlinear trend, initially decreasing and then gradually stabilizing. Moreover, the geometric difference coefficient can also be determined based on the proposed new dynamic cutting model (NDCM) and the cutting frequency \(f_{c}\). According to the geometric relationship shown in Figure 4, the height of the extrusion zone can be expressed as follows:  
\[h_{c}=\frac{v\tan\psi}{f_{c}} \tag{17}\]  
\(\Psi\) can be obtained according to the following equations [22; 27]. Figure 17: An example of the diagram of the relationship between \(GDC\) and \(h\). \[\left\{ \begin{array}{l}\tan\varphi^{\prime}=\frac{\pi}{2}\tan\varphi\\ \psi=\frac{\pi}{4}-\frac{\varphi^{\prime}-\varphi}{2}\end{array} \right. \tag{18}\]  
where \(\varphi\) is the internal friction angle of the rock, and \(\varphi^{\prime}\) is the friction angle between the cuttings in the extrusion zone and the rock. The value of \(f_{c}\) can be obtained from the tangential force sampling data through fast Fourier transform (FFT). By substituting Equation (17) into Equation (15), the geometric difference coefficient based on cutting force (referred to as \(GDC_{f}\)) is given as follows:  
\[GDC_{f}=\frac{v\tan\psi}{hf_{c}} \tag{19}\]  
Figure 14 shows that the linear slope \(k\) of the relationship between \(f_{c}\) and \(v\) is related to the cutting depth. Based on the fitting equation in Table 3, the distribution of \(k\) with \(h\) is presented in Figure 18. The fitting equation demonstrates that \(k\) is proportional to \(h^{-0.5}\), with a fitting degree of 0.97. Therefore, by combining the data from Table 3, the relationship between cutting frequency, cutting speed, and cutting depth can be expressed as follows:  
\[f_{c}=mv\frac{1}{\sqrt{h}} \tag{20}\]  
where \(m\) is the slope of the fitted linear relationship between \(k\) and \(h^{-0.5}\) in Figure 18.  
By substituting Equation (20) into Equation (19), we obtain the following:  
\[GDC_{f}=\frac{\tan\psi}{m\sqrt{h}} \tag{21}\]  
Clearly, the relationship between \(GDC\) and \(h\) in Equation (21) is similar to that in Equation (16), with both equations illustrating that rock characteristic parameters and cutting depth are the decisive factors influencing \(GDC\). Additionally, once the size of any boundary of the extrusion zone is determined, the height of the extrusion zone \(h_{c}\) and \(GDC\) can be obtained based on Figure 4. According to previous studies [35], it is feasible to measure or monitor the boundary size of the extrusion zone.