## 2 Prediction of Cutter Edge Temperature in Milling
### Calculation of Heat Flux Into the Cutter  
As depicted in Figure 1, the cutting power consumed in the second shear zone is divided into two parts. The part that flows into the cutter increases the temperature of the cutting edge, and the corresponding heat flux \(q_{c}\) is calculated as  
\[q_{c}=R_{1}\cdot Q_{f}/A_{s} \tag{1}\]  
where \(R_{1}\) is the energy partition coefficient representing the fraction of heat that flows into the cutter, and its calibration is introduced in Section 3.2. \(Q_{f}\) is friction power consumed in the rake face, and  
\[Q_{f}=F_{u}\cdot V_{c} \tag{2}\]  
where \(F_{u}\) and \(V_{c}\) are the friction force and chip velocity. For equilibrium, \(F_{u}\) satisfies  
\[F_{u}=F_{fc}\cdot\sin(\alpha_{r})+F_{fc}\cdot\cos(\alpha_{r}) \tag{3}\]  
Figure 1: Cutting power consumption in metal cutting. where \(F_{tc}\) is the main cutting force along cutter speed \(V\), \(F_{fc}\) is perpendicular to \(F_{tc}\) (see Figure 1), and \(\alpha_{r}\) is rake angle. In accordance with the principles of kinematics, the chip velocity is obtained by  
\[V_{c}=\frac{V\cdot\sin(\phi_{c})}{\cos(\phi_{c}-\alpha_{r})} \tag{4}\]  
where \(\phi_{c}\) is shear angle. In Equation (1), \(A_{s}\) is the area of chip-tool contact zone, and it is computed by  
\[A_{s}=l_{cn}\cdot a_{p} \tag{5}\]  
where \(l_{cn}\) and \(a_{p}\) are the contact length and axial depth of cut. Notice that by Equation (1), the heat flux \(q_{c}\) is assumed to be distributed on the contact zone evenly. In milling, the relationships of the cutting force, shear angle, and chip-tool contact length with respect to the un-deformed chip thickness \(h\) are quite complicated. To represent the relationships, a commercial machining simulation software Deform 2D v11 is employed. The cutter is modelled as a rigid and heat transferable body, and the workpiece as an elastoplastic body. In one cut of an edge, the un-deformed chip thickness continually varies between 0 and feed per tooth \(f_{z}\), so several ones of thickness \(h_{i}\) in a cut are sampled  
\[h_{i}=i\cdot f_{z}/m,\ i\ =\ 1,\ 2,\ \ldots,\ m. \tag{6}\]  
Each \(h_{i}\) is fed into the machining simulation software, together with other parameters, the cutting speed, the rake angle, the friction coefficient between the cutter rake face and chip, etc., to evaluate the shear angle, the contact length, and the cutting forces. The input parameters and evaluation results are listed in Table 1.  
To represent the cutting forces as a function of un-deformed chip thickness, a well-established mechanistic cutting force model presented by [PERSON] et al. [31] is introduced. The cutting forces are modelled as  
\[\left\{ \begin{array}{l}F_{tc}(h)=a_{p}\cdot(K_{tc}\cdot h+K_{te})\\ F_{fc}(h)=a_{p}\cdot(K_{fc}\cdot h+K_{fe})\end{array} \right. \tag{7}\]  
where \(K_{tc}\), \(K_{te}\), \(K_{fc}\) and \(K_{fe}\) are the coefficients and they are determined by least square fitting to fit the model to the \(m\) sampled thickness and the corresponding cutting forces. An example is plotted in Figure 2. \begin{table}
\begin{tabular}{c c c} \hline \hline  & & **Input Parameters** \\ \hline
**Symbol** & **Unit** & **Remark** \\ \hline \(h_{i}\) & mm & Sampled Un-deformed chip thickness, \(i\) = 1, 2, \(\ldots\), \(m\) \\ \(V\) & m/s & Cutting speed \\ \(\alpha_{r}\) & \({}^{\circ}\) & Rake angle \\ \(\mu\) & & Friction coefficient between rake face and chip \\ \hline \hline  & & **Evaluation Results** \\ \hline
**Symbol** & **Unit** & **Remark** \\ \hline \(F_{tc}{}^{i}\) & N & The cutting force in horizontal direction of cutting speed, under a specified \\ \(F_{fc}{}^{i}\) & N & The cutting force in vertical direction of cutting speed, under a specified \\ \(l_{cn}{}^{i}\) & mm & The chip tool contact under a specified un-deformed thickness \(h_{i}\) \\ \(\phi_{c}{}^{i}\) & \({}^{\circ}\) & The shear angle of the given tool-workpiece pair under the given cutter \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The input parameters and evaluation results.