## 3 FE Modeling of the Diamond Micro-Milling of Cu  
Figure 2 shows the FE model of micro-milling, which is composed of a Cu workpiece and a diamond milling tool. The workpiece has dimensions of 10 um in height, 100 um \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Unit** & **Values** \\ \hline Feed speed & mm/s & 10 \\ Spindle speed & rpm & 6000, 12,000, 18,000, 20,000, 30,000 \\ DOC & μm & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of micro-milling. Figure 1: (a) Experimental layout of diamond micro-milling; (b) Optical image of the single-edge diamond ball milling tool. in width and 120 m in length, and the bottom is constrained against any movement in each direction. Since the stiffness of diamond is much higher than that of cemented carbide, it is believed that the deformation of the tool shank is the main factor causing the tool chatter during the milling process. Consistent with the experimental configuration, the milling tool is composed of a deformable tool shank and a rigid diamond tool insert. The tool insert has a length of 1.3 mm, a clearance angle of 7\({}^{\circ}\) and an edge radius of 0 nm. The tool shank has a length of 2 mm, and the contact length between the tool shank and tool insert is 0.7 mm. The rotary axis of the diamond tool insert and the carbide tool shank both coincide with the X axis of the global XYZ coordinate system in the FE model. There is an inclination angle of 15\({}^{\circ}\) between the YZ plane of the workpiece surface and the milling tool. The coordinates on the rotary axis of the milling tool do not change during the FE simulation of the micro-milling process, so the center point of the tool ball head is treated as the output point to characterize the tool chatter during the micro-milling process, as shown in Figure 2a. The profile of the milled groove bottom is extracted from the bottom, where the tool insert contacts with the Cu workpiece along the feed direction. In the as-established FE model of micro-milling, the upper half layer of the Cu workpiece with a thickness of 5 m is meshed by eight-node linear hexahedron elements (C3D8) with a global size of 5 nm. A coarser mesh with a mesh size of 6-10 nm is used at the bottom of the Cu workpiece to improve computational efficiency, as shown in Figure 1. The interaction between the tool shank and tool insert is modeled by a tie constraint. Table 2 lists the mechanical properties of Cu [18], cemented carbide [19] and diamond [20]. The Cu workpiece is considered as isotropic material in the FE model, and the Johnson-Cook (J-C) constitutive model is used to characterize its elastoplastic deformation behavior:  
\[\overline{\sigma}=\Big{[}A+B\Big{(}\overline{\varepsilon}^{pl}\Big{)}^{n} \Big{]}\Big{[}1+C\ln\dot{\xi}\Big{]}\left(1-\dot{\theta}^{m}\right) \tag{1}\]  
where \(\overline{\sigma}\) is the yield stress, \(\overline{\varepsilon}^{pl}\) is the equivalent plastic strain, \(\dot{\xi}\) is the dimensionless plastic strain. \(A\), \(B\), \(n\), \(m\) and \(\dot{\theta}\) stand for the yield stress, strain hardening modulus, hardening  
Figure 2: FE model of the micro-milling of Cu considering milling tool deformation. index, thermal softening index and nondimensional temperature, respectively. In order to determine the thermal softening index \(m\), the function \(f_{3}(T^{*})\) is expressed as Equation (2):  
\[f_{3}(T^{*})=\big{(}1-\dot{\theta}^{m}\big{)}=\frac{\overline{\sigma}^{(T^{*})}} {\overline{\sigma}^{(T)}} \tag{2}\]  
where \(\overline{\sigma}^{(T^{*})}\) and \(\overline{\sigma}^{(T)}\) are the yield stress at the current temperature \(T^{*}\) and the reference temperature \(T\), respectively. Finally, the value of \(m\) is obtained by fitting \(f_{3}(T^{*})\). Table 3 lists the corresponding parameters of the J-C constitutive model for Cu [18].