## 2 Finite Element Model of Micro-Cutting
### Heat Conduction Model of the Workpiece-Tool  
In the micro-cutting process of Ti6 Al4V, a large amount of heat is generated during the cutting process, and this causes the tool temperature to increase, and the material to soften. Heat is caused by plastic deformation in the primary deformation zone. It is difficult to transfer heat in a short time, and this increases the temperature of the workpiece and tool and affects the tool life, processed surface, and processing efficiency. The partial differential equation of plane orthogonal cutting heat conduction in the primary deformation zone is expressed as Equation (8) [22]:  
\[\lambda\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{ \partial y^{2}}\right)-\rho C\left(\mu\frac{\partial T}{\partial x}+v\frac{ \partial T}{\partial y}\right)+\dot{Q}=0 \tag{8}\]  
\[\dot{Q}=\kappa\sigma_{ij}\dot{\varepsilon}_{ij} \tag{9}\]  
where \(\lambda\) denotes the thermal conductivity coefficient, \(\rho\) denotes the material density, \(C\) denotes the specific heat of the material, \(\dot{Q}\) denotes the heat generation rate per unit volume of mechanical energy converted to thermal energy, \(x\), and \(y\) are cartesian coordinates, \(\mu\) and \(v\) denote the velocity component of the moving heat source in the \(x\) and \(y\) direction, \(\sigma_{ij}\) denotes the material stress, \(\dot{\varepsilon}_{ij}\) denotes the material strain, and \(\kappa\) denotes the work-heat conversion coefficient.  
In the finite element simulation cutting process, \(\kappa\) is typically assumed such that 90% of the plastic deformation can be converted into heat and transferred to the workpiece and tool [61].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(D_{1}\) & \(D_{2}\) & \(D_{3}\) & \(D_{4}\) & \(D_{5}\) \\
**Initial Failure Strain** & **Exponent Factor** & **Triaxiality Factor** & **Strain Rate Factor** & **Temperature Factor** \\ \hline \(-0.09\) & \(0.25\) & \(-0.5\) & \(0.014\) & \(3.87\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Johnson-Cook damage parameters [65].  
The secondary deformation zone is formed since the rake face is affected by chip crushing and friction. The heat generated by the friction increase as the temperature of the chip contact surface and increases the cutting heat generated in the process. It directly affects the wear and service life of the tool. During the micro-cutting process, the friction between the rake face and chips causes severe tool wear. [PERSON] [66] proposed a viscous-slip friction model without considering temperature effects in 1963. The model obeys the coulomb's law of friction. The equation is as Equation (10):  
\[\tau=\left\{\begin{array}{l}\mu\sigma_{n}\;\mu\sigma_{n}<\tau_{max}\\ \tau_{max}\;\mu\sigma_{n}\geq\tau_{max}\end{array}\right. \tag{10}\]  
where \(\mu\) denotes the coefficient of friction, \(\sigma_{n}\) denotes the positive pressure at the tool-chip interface, and \(\tau_{max}\) denotes the ultimate shear stress of the material, and the coefficient of friction \(\mu\) is 0.3.  
The heat in the secondary deformation zone is mainly due to the friction between the rake face and chips. The heat generated by the friction is expressed as Equation (11):  
\[Q=\tau_{f}v_{f} \tag{11}\]  
where \(\tau_{f}\) denotes the shear stress between the chip and rake face, and \(v_{f}\) denotes the relative speed between the chip and rake face.