## 6 Conclusions  
(1) Based on the previous single-tooth cutting model, the idea that the height of the extrusion zone is not always equal to the cutting depth is introduced. By analyzing the continuous cutting rock-breaking behavior, a new dynamic cutting model is established, in which the extrusion zone is not equal to the cutting depth. Additionally, the definition of the single cutting process is presented, and it is emphasized that the rock-breaking process is composed of the repeated occurrence of the single cutting process. A new concept of cutting frequency \(f_{\epsilon}\) is also proposed to represent the speed of the single cutting process. (2) According to the new dynamic cutting model, the correlation between the fluctuations of tangential cutting force and the evolution of the single cutting process is analyzed. The results indicate that the two are synchronous, with the main frequency of the single-tooth tangential force signal, extracted using the fast Fourier transform (FFT) method,corresponding to the cutting frequency \(f_{c}\). Subsequently, a numerical simulation test of single-tooth rock cutting was conducted. Frame-by-frame analysis of the equivalent plastic zone cloud map revealed that the occurrence of the rock shear failure surface is periodic. Furthermore, the difference between the cutting frequency, based on the equivalent plastic zone cloud diagram, and the main frequency of the tangential cutting force is analyzed, with a root mean square error of 1.48 Hz. These results validate the periodicity of the single cutting process and confirm the effectiveness of the method used for extracting the cutting frequency. (3) Based on the numerical simulation of single-tooth rock breaking for Tuffeau, the influence of key operating parameters, such as rake angle, cutting speed, and cutting depth, on the cutting frequency is explored. The results show that the rake angle has little effect on the cutting frequency within the range of 10\({}^{\circ}\) to 20\({}^{\circ}\). The cutting frequency is positively proportional to the cutting speed. As the cutting depth increases, the cutting frequency decreases nonlinearly. Subsequently, the energy relationship between the formation of the extrusion zone and the formation of the shear failure surface in the single cutting process is analyzed, revealing the influence of rake angle, cutting speed, and cutting depth on the cutting frequency. (4) The concept of the geometric difference coefficient (\(\mathbf{GDC}\)) is introduced to characterize the ratio of the extrusion zone height to the cutting depth. Based on the principle of energy conservation, the influence of cutting depth on \(\mathbf{GDC}\) is theoretically analyzed. Additionally, utilizing the geometric relationship in the new dynamic cutting model, the relationship between \(\mathbf{GDC}\), cutting parameters, and cutting frequency is established. The reliability of the \(\mathbf{GDC}\) acquisition method, based on cutting frequency, and the influence of cutting depth on \(\mathbf{GDC}\) are then verified through the numerical simulation results of Tuffeau cutting. The findings show that as cutting depth increases, the \(\mathbf{GDC}\) initially decreases and then stabilizes. When the cutting depth is small, the \(\mathbf{GDC}\) is approximately 1.00, indicating that the height of the extrusion zone is nearly equal to the cutting depth. However, as cutting depth increases, the \(\mathbf{GDC}\) gradually decreases and eventually stabilizes within a smaller range. These results confirm the assumption that the depth of the extrusion zone is not always equal to the cutting depth, and offer new insights for determining the height of the extrusion zone and optimizing the application of cutting force, providing valuable reference for the optimization of the cutting model. Conceptualization, Y.H. and N.L. ; methodology, Y.H., N.L. and G.L. ; software, Y.H. ; validation, G.L. and Y.H. ; formal analysis, Y.H. and G.L. ; investigation, Y.H. and G.L. ; writing--original draft preparation, Y.H.