## 2 Finite Element Model of Micro-Cutting
### Material Model  
Considering the micro-cutting simulation modeling of Ti6 Al4V, selecting a reliable material for the constitutive model to evaluate the dynamic mechanical behavior of the material under different cutting conditions is crucial [58]. An accurate material flow stress model is necessary to represent the constitutive behavior of the working material under high strain rate deformation conditions. Therefore, formulating a material model that can truly reflect the stress-strain characteristics and the corresponding relationship between the strain rate strengthening and temperature of the material being processed constitutes the basis and key to ensure the accuracy of the micro-cutting simulation. In the micro-cutting simulation model, the J-C constitutive parameters of Ti6 Al4V are shown in Table 1, and their physical and mechanical properties are shown in Table 2. The physical and mechanical properties of the tool are listed in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Density** **(kg/m\({}^{3}\))**} & **Elastic** **Modulus** **(GPa)** & **Poisson’s** **Ratio** & **Thermal Conductivity (W/m \({}^{\circ}\)C)** & **Specific Heat** **(J/kg \({}^{\circ}\)C)** & **Thermal** **Competition** **(\({}^{\circ}\)C)** \\ \hline \multirow{4}{*}{4430} & 109 (50 \({}^{\circ}\)C) & & 6.8 (20 \({}^{\circ}\)C) & 611 (20 \({}^{\circ}\)C) & \multirow{4}{*}{9.0 \(\times\) 10\({}^{-6}\)} & \multirow{4}{*}{1605} \\  & 91 (250 \({}^{\circ}\)C) & & 7.4 (100 \({}^{\circ}\)C) & 624 (100 \({}^{\circ}\)C) & & \\ \cline{1-1}  & 75 (750 \({}^{\circ}\)C) & & 9.8 (300 \({}^{\circ}\)C) & 674 (300 \({}^{\circ}\)C) & & \\ \cline{1-1}  & & & 11.8 (500 \({}^{\circ}\)C) & 703 (500 \({}^{\circ}\)C) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Physical and mechanical properties of Ti6 Al4V [22, 59, 60]. Figure 1: Geometric model and boundary conditions of the orthogonal micro-cutting simulation. In the cutting simulation process of Ti6 Al4V, the cutting process undergoes three stages [62]. The first stage is initially linear elastic (a-b), the second stage is plastic yielding with strain hardening (b-c), and the third stage is the material damage evolution stage (c-d). Among the three stages, the J-C constitutive model, which represents the plastic deformation stage of the material, belongs to the second stage. As shown in Figure 2, the solid line \(l_{abcd}\) denotes the stress-strain relationship considering material damage and the dashed line \(l_{abcf}\) denotes the stress-strain relationship without considering material damage. The strength of the material begins to decrease at the initial critical point \(c\) indicating material damage. When it reaches point \(d\), it reaches the fracture quasi-side wherein the material stiffness is completely lost and the material is fractured [63]. The material begins to fail after the plastic strain of the material reaches the initial damage point \(c\). In the Ti6 Al4V cutting simulation process, the equivalent plastic strain based on the integral point of the element is used to define the damage parameter \(J\). When the damage parameter \(J>1\), the workpiece material is fractured, the local material stiffness is completely lost, the element mesh is deleted, and chips begin to form. The J-C shear failure criterion and shear strain criterion are commonly expressed as Equation (2) [64]:  
\[J=\sum\frac{\Delta\overline{z}}{\overline{z}_{JC}} \tag{2}\]  
where \(J\) denotes the damage initiation parameter, \(\Delta\overline{z}\) denotes the equivalent plastic strain increment, and \(\overline{z}_{JC}\) denotes the critical equivalent plastic strain.