# Research on the Development Mechanism of Three Kinds of Cracks in Rock Cutting
## 3 Results and Discussion
### Cutting-Force Analysis  
The multi-cutting PCF at various depths is derived from a numerical simulation of rock cutting. The single PCF is easily affected by the damaged crack, and the damaged area of the previous cutting is more unstable. It can be understood that the strain energy of the next rock cutting corresponding to the crack is released in advance, so the PCF of the same set of experiments is chosen for the analysis. The following three PCF models are chosen for investigation referring to the previous methods of studying peak cutting force [33; 34; 35; 36].  
[PERSON] [37; 38] proposed a theoretical model based on the maximum tensile strength criterion for the prediction of the cutting force of conical teeth, and it is considered that the formation of rock fragmentation is caused by tensile failure, which is shown in Equation (1).  
\[PCF_{EV}=\frac{16\pi W\sigma_{t}^{2}d^{2}}{cos^{2}\alpha*\sigma_{c}} \tag{1}\]  
where \(PCF_{EV}\) is the peak cutting force of [PERSON] model. \(d\) is the depth of cut. \(W\) is the tool width. \(\alpha\) is the semi-tip angle of cutting tool. \(\sigma_{c}\) and \(\sigma_{t}\) are uni-axial compressive strength (UCS) and the Brazilian tensile strength (BTS) values of the rock.  
[PERSON] [39; 40] modified [PERSON] model and took the friction between the rock and the tool into consideration, which was shown in Equation (2).  
\[PCF_{GO}=\frac{4\pi W\sigma_{t}d^{2}sin^{2}(\alpha+f)}{\cos(\alpha+f)} \tag{2}\]  
Figure 11: The Relationship Between the Number of Failure Particles and the Number of Normal-size Particles in Parallel Cutting.  
Figure 12: DOC = 3.5 mm Relationship Between Cutting Force and Number of Cracks.  
where \(PCF_{GO}\) is the peak cutting force of [PERSON]'s model. \(f\) is the friction angle between the rock and tool. The other notations are the same as those used in Equation (1).  
[PERSON] [32] believed that the rock failure was caused by shear cracks produced by drilling teeth, and a rock cutting model was proposed based on the theory of [PERSON]. [PERSON] pointed out that his theory applied to coarse cutting chips produced by the propagation of brittle fracture cracks, which was shown in Equation (3).  
\[PCF_{NISH}=\frac{2W\sigma_{\mathrm{s}}d}{12.3-0.18\beta}\times\frac{\cos \gamma}{1-\sin(\gamma-\beta+f)} \tag{3}\]  
where \(PCF_{NISH}\) is the peak cutting force of [PERSON]'s model. \(\sigma_{\mathrm{s}}\) is the shear strength of specimen rock. \(\gamma\) is the angle of internal friction. \(\beta\) is the fake angle of the cutting tool. The other notations are the same as those used above.  
As shown in Table 6, the parameters required in Equations (1)-(3) are calculated and converted from the previous section.  
Figure 13 displays the experimental peak forces in a scattered form together with three distinct cutting force models. The numerical simulation is a 2D model, thus the unit of measurement is N/mm (kN/m), which is better correlated with [PERSON]'s model than the experimental data. In [PERSON]'s theory, the first macro-crack is a shear crack, and the shear failure happens after the macro-crack, resulting in a sharp reduction in force. In order to verify whether the force drop is caused by the shear failure of the macro shear crack, the conventional size cuttings at different depths of cut are analyzed. The research is described below.