## 2 Construction of New Dynamic Rock-Cutting Model and Analysis of Cutting Periodicity  
### Construction of a New Dynamic Cutting Model  
It can be seen from the photos of the PDC coring bits shown in Figure 1 that the drilling process results from the superposition of multiple single-tooth cutting and rock-breaking behaviors. To investigate the mechanism of rock cutting, previous studies proposed a quasi-static cutting model (QCM), as shown in Figure 2[22]. This model represents the cutting state under the limiting condition of shear surface formation, referred to as the 'limit cutting state' in this study. The region formed between the cutter and the rock by the accumulation of cuttings is known as the extrusion zone or crushed zone. Since the formation of the failure surface follows the shear criterion [18; 19; 39], it is referred to as the shear surface in this study. To determine the geometric characteristics of the extrusion zone, the height of the extrusion zone, denoted as \(\mathbf{h_{c}}\), is introduced to represent the length of its vertical projection. Once \(\mathbf{h_{c}}\) is determined, the other dimensions of the extrusion zone can be obtained through geometric relationships. As shown in Figure 2, \(\mathbf{h_{c}}\) in the QCM is considered equal to the cutting depth \(\mathbf{h}\). To analyze the dynamic characteristics of the cutting process, we expand the quasi-static cutting model (QCM) in Figure 2 from the perspectives of time and space, resulting in a dynamic cutting model (DCM) shown in Figure 3. The DCM illustrates the positional relationship between the cutter and the rock during two consecutive limit cutting states. Similarly to the QCM, the extrusion zone height \(h_{c}\) is considered equal to the cutting depth \(h\) in the DCM. However, studies have shown that the height of the extrusion zone is not always equal to the cutting depth [35]. To address this, the DCM is further modified into a new dynamic cutting model (NDCM), as shown in Figure 4. Compared to the DCM, the most significant feature of the NDCM is that \(h_{c}\) is treated as a variable, not equal to \(h\). Figure 1: Photo of the PDC bit. Figure 3: The dynamic rock-cutting model (DCM) with \(h_{c}=h\). Figure 2: The quasi-static rock-cutting model (QCM) with \(h_{c}=h\). The Relationship Between the Variation Characteristics of Tangential Cutting Force \(F_{c}{}^{t}\) and the Dynamic Rock-Cutting Process  
Before analyzing the tangential cutting force in the dynamic cutting model, the following assumptions are made, based on previous literature [26]:  
1. To simplify the problem, the cutter is considered a rigid body, without considering its wear and thermal effects. 2. Except for the rock debris forming the crushing zone, the remaining debris can be cleared promptly and will not affect the cutting due to accumulation in front of the cutter. 3. To avoid the influence of uneven distribution of rock components on the cutting force, the rock is considered an isotropic, homogeneous material. 4. In the analysis, the problem is simplified to a two-dimensional orthogonal cutting model. 5. The effects of temperature and humidity on the cutting process are not considered. In NDCM, \(t_{\mathbf{0}}\) is the starting time of cutting, when the rake face of the tool begins to contact the fresh rock below the existing shear surface. As the cutter moves at speed \(v\), the rock ahead of the rake face is broken by the tooth. Some of the generated cuttings escape directly, while others accumulate near the rake face to form an extrusion zone. During this period, the tangential cutting force \(F_{c}{}^{t}\) increases with the growth of the effective cutting depth. By \(t_{\mathbf{1}}\), the stress state of the potential shear surface satisfies the shear failure condition. At this point, cutting enters a limit state, and the tangential cutting force \(F_{c}{}^{t}\) reaches a local maximum. The extrusion zone reaches its maximum size, and the shear plane is rapidly formed.