## 2 Equipartition Shear Zone Model of Orthogonal Cutting Based on [PERSON]'s Theory
### Shear Zone Analysis  
The schematic diagram of the orthogonal cutting shear zone is shown in Figure 1. A pair of equilibrium forces is formed by the normal force \(F_{N}\) at the shear plane and the cutting force \(F_{s}\) at the shear plane along with the normal force \(F_{n}\) at the tool-chip interface and the friction force \(F_{f}\) at the tool-chip interface. The chip formation force \(F_{R}\) is decomposed into \(F_{c}\) and \(F_{t}\). The cutting-force components and the chip thickness \(t_{2}\) can be calculated using the following formula:  
\[F_{c}=F_{R}\cos(\lambda-\alpha) \tag{1}\]  
\[F_{t}=F_{R}\sin(\lambda-\alpha) \tag{2}\]  
\[F_{f}=F_{R}\sin\lambda \tag{3}\]  
\[F_{n}=F_{R}\cos\lambda \tag{4}\]  
\[F_{R}=\frac{F_{s}}{\cos\theta}=\frac{\sigma_{AB}t_{1}w}{\sin\varphi\cos\theta} \tag{5}\]  
\[t_{2}=t_{1}\cos(\varphi-\alpha)/\sin\varphi \tag{6}\]where \(\alpha\) is the rake angle of the tool; \(\varphi\) is the shear angle; \(\lambda\) is the friction angle; \(t_{1}\) is the undeformed chip thickness; \(w\) is the cutting width; \(\theta\) is the angle between \(F_{R}\) and \(AB\); \(\sigma_{AB}\) represents the average flow stress on the \(AB\) plane. In the Oxley model, there is a relationship between the velocity of any particle in the first deformation zone and the average shear strain rate in the shear zone expressed as follows:  
\[\dot{\gamma}_{arg}=\frac{V\cos\alpha}{\Delta S_{1}\cos(\varphi-\alpha)} \tag{7}\]  
where \(V\) is the orthogonal cutting speed, and \(\Delta S_{1}\) is the thickness between two parallel bands in the shear zone. After the shear angle \(\varphi\) is determined, the strain and strain rate at any position on the shear plane can be obtained using the [PERSON] stress yield criterion. \[\varepsilon_{AB}=\frac{1}{2\sqrt{3}}\cdot\frac{\cos\alpha}{\sin\varphi\cos( \varphi-\alpha)} \tag{8}\]  
\[\dot{\varepsilon}_{AB}=\frac{1}{\sqrt{3}}\cdot\frac{V\cos\alpha}{\Delta S_{1} \cos(\varphi-\alpha)} \tag{9}\]  
According to the velocity vector relationship in Figure 1, the flow velocity of the chip material \(V_{c}\) and the material flow velocity \(V_{s}\) of the shear surface can be obtained as follows:  
\[ \begin{cases}V_{c}=\frac{\sin\varphi}{\cos(\varphi-\alpha)}V\\ V_{s}=\frac{\cos\alpha}{\cos(\varphi-\alpha)}V\end{cases}  \tag{10}\]  
The friction angle is calculated as \(\lambda=\theta+\alpha-\varphi\). According to [PERSON]'s theory, the shear angle and the included angle satisfy the following relationship:  
\[\tan\theta=1+2(\frac{\pi}{4}-\varphi)-C_{0} \tag{11}\]  
The modified strain rate constant \(C_{0}\)[17] takes into account the effect of material strain and is expressed as follows:  
\[C_{0}=C_{0 xlog}n\frac{B\varepsilon_{AB}^{n}}{A+B\varepsilon_{AB}^{n}} \tag{12}\]  
where \(C_{Oxley}\) is the length-width ratio of the shear band; \(A\), \(B\), and \(n\) are Johnson-Cook constitutive parameters, respectively. According to the Boothroyd temperature model [18], the average temperature is expressed as follows:  
\[T_{AB}=T_{r}+\eta\frac{(1-\beta)F_{s}V_{s}}{\rho_{w}C_{w}V_{1}w\sin\varphi} \tag{13}\]  
Figure 1: Diagram of the orthogonal cutting shear zone. where \(T_{r}\) is the workpiece temperature, \(\eta\) is the average temperature coefficient, 0.9 [19]; \(\rho_{w}\) is the material density; \(C_{w}\) is the specific heat capacity; and \(\beta\) is the heat distribution coefficient [17]. \[\beta=\left\{ \begin{matrix}0.5-0.35\lg(R_{0}\tan\varphi)&0.04\leq R_{0}\tan \varphi\leq 10\\ 0.3-0.15\lg(R_{0}\tan\varphi)&R_{0}\tan\varphi{>}10\end{matrix} \right. \tag{14}\]  
\[R_{0}=\frac{\rho_{w}C_{w}Vt_{1}}{K_{w}} \tag{15}\]  
where \(K_{w}\) is the thermal conductivity. In high-speed cutting, the material undergoes a process of large strain, with a high strain rate and high temperature. S32760 duplex stainless steel shows an obvious viscous effect with a strain rate above \(10^{4}\) s\({}^{-1}\)[20].