## 2 Characterization of Surface Structure and Properties of Titanium Alloy Turning Operations
### Transient Cutting Layer and Mechanical Behavior  
To elucidate the cutting layer and its mechanical response during the turning of titanium alloy, we characterize the instantaneous force and cutting layer at the point where the lowest workpiece surface point and the tool's transition arc segment make contact with the workpiece, as depicted in Figure 3.  
In Figure 3, point \(m\) is the highest point on the cutting edge, point \(p\) is an arbitrary point on the transition arc of the cutting edge, \(\gamma_{0}\) is the front angle of the tool, \(\alpha_{0}\) is the back angle of the tool, \(\beta_{0}\) is the friction angle on the front face, \(\phi_{p}\) is the shear angle of point \(p\),\(f\) is the feed per revolution, \(\kappa_{rp}\) is the instantaneous main deflection angle of point \(p\), \(h_{D}\) is the thickness of the cutting layer, \(ds\) is the instantaneous micrometer width of the cutting layer of point \(p\), \(A_{D}\) is the actual cutting layer area, \(\Delta A_{D}\) is the cutting residual area, \(R_{m}\) is the radius of gyration at point \(m\), \(R_{p}\) is the radius of gyration at point \(p\), \(v_{c}\) is the cutting speed, \(F_{c}\) is the main cutting force, \(v_{m}\) is the friction speed, \(F_{\gamma}\) is the friction force on the front blade surface, \(F_{\gamma N}\) is the normal force of the friction force on the front blade surface, \(F_{a}\) is the friction force on the rear blade surface, \(F_{aN}\) is the normal force of the friction force on the rear blade surface, \(v_{sh}\) is the shear speed, \(F_{sh}\) is the shear force, \(F_{shN}\) is the shear force in the normal force, \(F\) is the combined force of front and rear tool face friction and its normal force, the combined force of main cutting force and feed force.  
The instantaneous cutting layer thickness solution for any point \(p\) is shown in Equation (6).  
\[h_{D}=f\cdot\sin\kappa_{rp}\cdot\cos\lambda_{s} \tag{6}\]  
The cutting layer area \(A_{D}\) solution is shown in Equation (7).  
\[A_{D}=\int_{e}^{m}h_{D}(x,y,z)\cdot ds\cdot\cos\lambda_{s} \tag{7}\]  
From Figure 3, Equations (6) and (7), \(\tau\) denotes the shear stress on the shear surface, and the solution of each force at point \(p\) is shown in Equations (8)-(11).  
\[F_{sh}=\frac{\tau\cdot A_{D}}{\sin\phi_{p}} \tag{8}\]  
\[F_{sh}=F\cdot\cos(\phi_{p}+\beta_{0}-\gamma_{0}) \tag{9}\]  
\[F=\frac{F_{sh}}{\cos(\phi_{p}+\beta_{0}-\gamma_{0})}=\frac{\tau\cdot A_{D}}{ \sin\phi_{p}\cdot\cos(\phi_{p}+\beta_{0}-\gamma_{0})} \tag{10}\]  
\[F_{c}=F\cdot\cos(\beta_{0}-\gamma_{0})=\frac{\tau\cdot A_{D}\cdot\cos(\beta_{0 }-\gamma_{0})}{\sin\phi_{p}\cdot\cos(\phi_{p}+\beta_{0}-\gamma_{0})} \tag{11}\]  
Figure 3: Instantaneous cutting layer and instantaneous force on turning surface: (**a**) Instantaneous force at the lowest point of the workpiece surface; (**b**) Cutting layer parameters; (**c**) Instantaneous force at any point of the transition arc.  
According to the cutting layer and its mechanical behavior in the turning process, the instantaneous force on any point of the cutting edge and the instantaneous dynamic change characteristics of the cutting layer can be revealed.