## 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
According to a previous study [43], when the width of the flank wear band is small, elastic contact is found between machined surface and wear band, while a plastic flow phenomenon can occur when the wear band width is large. These may be due to the high temperatures and pressures existing near the cutting edge, such as plastic flow contact. As the referred to previous research [44] and coordinate transformation shown in Figure 8, the distribution of normal stress \(\sigma_{n-f}(x^{\prime\prime})\) over the wear band on the flank is determined, as shown in Figure 8 and Equation (31). Figure 7: Heat source in the FWZ. \[\sigma_{n-f}(x^{\prime\prime})=\left\{ \begin{array}{ll}\sigma_{tip}&,\quad x^{ \prime\prime}\in(VB_{CR},VB_{1})\\ \sigma_{tip}\Big{(}\frac{x^{\prime\prime}}{VB_{CR}^{-}}\Big{)}^{2}&,\quad x^{ \prime\prime}\in(0,VB_{CR})\end{array} \right. \tag{31}\]  
In Equation (31), \(VB_{CR}\) is the critical point for both the plastic flow area and elastic contact area, while it also represents the width of the elastic contact area. The relationship between the width of the plastic flow area and the width of the wear band on the flank is fitted to previously published experimental data [45]. According to a prior research [46], when \(VB_{1}\)\(>\)\(VB_{CR}\), the normal stress \(\sigma_{tip}\) on the tip of microelement can be calculated, as shown in Equation (32). \[\sigma_{tip}\)= K\(\left[1+\frac{\pi}{2}-2\rho-2\phi+2\eta_{w}+\sin(2\eta_{w})\right] \tag{32}\]  
where \(\eta_{w}=0.5\)cos\({}^{-1}\)(m\({}_{w}\)). In Equation (32), \(K\) is the ratio of the shear stress on the cutting edge to the shear flow stress on the workpiece, which is equal to the friction coefficient \(\mu_{c}\) of the cutting edge near the flank. Due to the bonding nature of contact between the wear band on the flank and the machined surface, a previous study [47] has suggested that \(\mu_{c}\) is uniform. \(m_{w}\) is the slip line field angle of the wear band on the flank; its numerical value is equal to the friction coefficient \(\mu_{c}\) of the cutting edge near the flank. The value of the Ti6 Al4V workpiece obtained from the current study is \(m_{w}=\mu_{c}=0.85\). According to a previous research [46], if the proportion of undeformed chip thickness to cutting width is more than 5%, it is considered \(\rho=0^{\circ}\), and the obtained value is \(0.8^{\circ}\). According to [PERSON]'s principle (\(\phi+\overline{\lambda}_{r}-\gamma_{0}\)= \(45^{\circ}\)), the direction of principal stress and the direction of maximum shear stress are adjusted to \(45^{\circ}\), in order to determine the shear angle \(\phi\). As shown in Equation (33), the formula for shear angle \(\phi\) is expressed as follows in Equation (33). \[\phi\)= \(45^{\circ}+\gamma_{0}-\overline{\lambda}_{r} \tag{33}\]  
Basing on a prior research [48], the heat distribution ratio in the cutting process can be calculated by Equation (34). \[B\)= \(1\)-\(\left(1+\frac{\pi VB_{1}}{2 hP_{e}\ln(2w/VB_{1})}\right)^{-1}\) (34)  
Figure 8: Distribution of normal stress over the wear band on the flank. Owing to the undeformed chip thickness during the peripheral milling process and the flank length (\(VB_{1}\)) are time-varying, respectively, which results in the result that the heat distribution ratio of the heat source in the FWZ (\(B_{III\_flank}\) (0)) is time-varying. the Equation (44) is substituted into Equation (34) to calculate the heat distribution ratio.