## 2 Transient Temperature Field Model of Wear Band on the Flank of End Mills
### Temperature Rise Model of the Wear Band Affected by Heat Source in the FWZ  
During the peripheral milling process, the heat source in the FWZ contacts with the tool, so the transient temperature field of wear land on the flank of end mills will inevitably be affected by the heat source in the FWZ. According to [PERSON] et al. [32], the temperature rise at any point of the tool flank make is calculated by the heat source model in the FWZ, but the tool rake angle effect on the temperature rise is not considered. As mentioned in Section 2.2, this paper considers the influence of rake angle on cutting temperature; the heat source in the FWZ is composed of both native and mirror heat sources, and their heat intensity are relatively similar (Figure 7). The temperature rises \(\Delta\)T\({}_{flank\text{-}III}\) (\(x^{\prime\prime}\), 0, 0) at any point \(P_{III}\) (\(x^{\prime\prime}\), 0, 0) along \(X^{\prime\prime}\) direction can be calculated by Equations (25)-(27). \[\Delta T_{III}(x^{\prime\prime},0,0)=\frac{q_{III}(x^{\prime\prime})B_{III-flank }(\theta)}{2\pi\lambda_{t}}\int_{0}^{VB_{1}}dx_{i-3}^{\prime\prime}\int_{- \frac{\
u^{\prime}}{2}}^{\frac{\
u^{\prime}}{2}}\frac{1}{R_{i-3}}dy_{i-3}^{ \prime\prime} \tag{25}\]  
\[\Delta T_{III^{\prime}}(x^{\prime\prime},0,0)=\frac{q_{III}(x^{\prime\prime})B _{III-flank}(\theta)}{2\pi\lambda_{t}}\int_{VB_{1}}^{VB_{1}[1+\cos(2\gamma_{n}) ]}\frac{1}{\cos(2\gamma_{n})}dx_{i-3}^{\prime\prime}\int_{-\frac{\
u^{\prime} }{2}}^{\frac{\
u^{\prime}}{2}}\frac{n_{f}}{R_{i-3}}dy_{i-3}^{\prime\prime} \tag{26}\]  
\[\Delta T_{flank\text{-}III}(x^{\prime\prime},0,0)=\Delta T_{III}(x^{\prime\prime },0,0)+\Delta T_{III^{\prime}}(x^{\prime\prime},0,0) \tag{27}\]  
Figure 6: Chip outflow diagram. From Equation (25) to (27), \(\Delta T_{III}\) (\(x^{\prime\prime}\), 0, 0) and \(\Delta T_{III^{\prime}}\) (\(x^{\prime\prime}\), 0, 0) are the temperature rise caused by native heat source and mirror heat source in the FWZ, respectively. The distance from any point \(P_{III}\) (\(x^{\prime\prime}\), 0, 0) along \(X^{\prime\prime}\) direction to the points of native and mirror heat sources in the FWZ can be measured by Equation (28). \[\left\{ \begin{array}{c}R_{i-3}=\sqrt{\left(x^{\prime\prime}-x_{i-3}^{\prime \prime}\right)^{2}+\left(y_{i-3}^{\prime\prime}\right)^{2}}\\ R_{i-3^{\prime}}=\sqrt{\left(x^{\prime\prime}-x_{i-3^{\prime\prime}}^{\prime \prime}\right)^{2}+\left(y_{i-3}^{\prime\prime}\right)^{2}+\frac{\left(x_{i +3^{\prime\prime}}^{\prime\prime}-VB_{1}\right)^{2}}{\cos(2\gamma_{n})}}\\ w^{\prime}=\frac{w}{\cos A_{s}}\end{array} \right. \tag{28}\]  
As previously described in Section 2.2, \(n_{f}=0.5\). Due to the non-uniformity of normal stress distribution in the wear band during the peripheral milling process, a non-uniform heat intensity is found in the FWZ. Further, according to the physical definition of heat intensity, the heat intensity of the FWZ (\(q_{III}\)(\(x^{\prime\prime}\))) is calculated using Equation (29). \[q_{III}(x^{\prime\prime})=\frac{f_{\rm C}(x^{\prime\prime})V}{(VB_{1}w)/\cos A _{s}} \tag{29}\]  
The friction between flank and machined surface is mostly sliding friction type. In order to simplify it to Coulomb friction type, the friction on the interface between tool and workpiece is obtained using Equation (30). \[f_{\rm C}(x^{\prime\prime})=\overline{\mu}_{f}\sigma_{n-f}(x^{\prime\prime}) \tag{30}\]  
In Equation (30), \(\overline{\mu}_{f}\) is the average friction coefficient on the interface between tool and workpiece, which is related to the average temperature of the interface between tool and workpiece. It is well accepted [41] that both of them satisfy the following empirical formula: \(\overline{\mu}_{f}=1-3.44\times 10^{-4}\overline{T}_{f}\), \(\overline{T}_{f}\in[25^{\circ}{\rm C},95^{\circ}{\rm C}]\). Moreover, it has been previously reported [42] that \(\overline{T}_{f}\approx 500^{\circ}{\rm C}\), and thus \(\overline{\mu}_{f}\approx 0.828.\sigma_{n-f}(x^{\prime\prime})\) is the normal stress distribution in the middle of wear band on the flank.