## 2 Equipartition Shear Zone Model of Orthogonal Cutting Based on [PERSON]'s Theory
### Modeling of Force of Chip Formation  
Since the shear angle of the shear zone \(\varphi\), the deformation coefficient of the shear zone \(C_{Oxley}\), and the ratio of the thickness of the second deformation zone to the chip thickness \(\delta\) vary with cutting conditions, material properties, and tool geometry, these three variables are iterated over in ranges of \(\varphi\in[5^{\circ},45^{\circ}]\), \(C_{Oxley}\in[2,10]\), and \(\delta\in[0.005,0.2]\), respectively. The calculation will be terminated when the output satisfies three equilibrium conditions: (i) the stress balance at the tool-chip interface; (ii) the stress balance at the tool tip; and (iii) the minimum principle of the cutting force (\(F_{c}\)) [27].  
Assuming that the tool-chip interface stress is uniformly distributed, the expressions of the tool-chip interface stress \(\tau_{\text{int}}\) and stress \(\sigma_{N}\) at point B are given as follows:  
\[\begin{cases}\tau_{\text{int}}=\frac{F_{f}}{h_{tc}w}\\ \sigma_{N}=\frac{F_{c}}{h_{tc}w}\end{cases} \tag{39}\]  
The normal stress \(\sigma_{N}^{\prime}\) at point B near the tip in Figure 1 can be obtained by combining the average shear flow stress of the shear plane given as follows:  
\[\sigma_{N}^{\prime}=\sigma_{AB}(1+\frac{\pi}{2}-2\alpha-2C_{0}) \tag{40}\]  
In the cutting-force prediction model, when solving the temperature of the shear zone, the flow stress of the shear zone is determined at a given initial temperature. The temperature of the shear surface is then updated according to the flow stress and replaced with the initial temperature to start a new calculation. This process is repeated until the difference between the initial temperature and the calculated temperature is less than 1 \({}^{\circ}\)C. When determining the tool-chip interface temperature, the sum of the increases in room temperature and the temperature of the first deformation zone is used as the initial temperature, and then the maximum increase in temperature and the average increase in the temperature of the chip are used as the temperature increment to determine the new chip temperature and replace it with the initial temperature to start a new calculation. The process is repeated until the difference between the initial temperature and the calculated temperature is less than 1 \({}^{\circ}\)C. The tangential stress of the tool-chip interface (\(\tau_{\text{int}}\)) and the flow stress of the tool-chip interface (\(\sigma_{chip}\)) are compared, and the difference between the absolute values of the two is taken as the basis for judgment. The shear angle (\(\varphi\)) takes the value of whichever has the smallest absolute value and is then used. The calculation of the various parameters in the shear zone is repeated. The normal stress of the tool tip (\(\sigma_{N}\)), the normal stress of the B point near the tool tip (\(\sigma_{N}^{\prime}\)), and the difference between the absolute values of the two are computed and used as the basis for judgment. The shear zone deformation coefficient (\(C_{Oxley}\)) takes the value of whichever has the smallest absolute value, and this shear zone deformation coefficient is then used. The calculation of the various parameters in the shear zone is then repeated and finally compared with the cutting force (\(F_{c}\)). The minimum value is selected to determine \(\delta\).  
Thus, considering different cutting conditions, material properties, and tool geometries using the chip formation force model, the calculation flowchart of the orthogonal cutting process is shown in Figure 3.