## 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 PSZ  
During the peripheral milling process, the heat source in the PSZ contacts with the tool indirectly through the workpiece, so the transient temperature field of the wear land on the flank of end mills will inevitably be affected by the heat source in the PSZ. According to [PERSON] et al. [28], it is considered that the temperature rise at an arbitrary point on the workpiece can be calculated by the heat source model in the PSZ. This paper applies the heat source model in the PSZ to evaluate the temperature rise at any point on wear land on the flank of end mills, this model is based on the native heat source in the PSZ and its mirror heat source, and both of them exhibit the same heat source intensity (Figure 3). According to a previous study [29], it has been suggested that the temperature rise \(\Delta T_{P}\) of any point \(P\) in space is caused by a heat source point \(dl\) on an obliquely moving banded heat source, as demonstrated in Equation (2). \[\Delta T_{P}=\frac{q}{2\pi\lambda}\cdot e^{-\frac{\kappa_{y}V_{h}}{2\pi}} \cdot K_{0}\left(\frac{RV_{h}}{2\alpha}\right) \tag{2}\]  
In Equation (2), \(R_{V}\) is the length of the projection from heat source to an arbitrary point towards the motion of the banded heat source; and \(V_{h}\) is the velocity of the belt heat source. Figure 3 shows the relationship between the maximum length of the banded heat source in the first deformation zone and the undeformed chip thickness, as shown in Equation (3). \[l_{\text{max}}=\frac{h(\theta)}{\sin\phi_{n}} \tag{3}\]  
Further, the length of the projection from heat source to any point towards the motion of the banded heat source is calculated by Equation (4). \[R_{V}=-\left(l\cos\phi_{n}+\sqrt{x^{2}+z^{2}}\sin\gamma_{n}\right) \tag{4}\]  
Figure 3: Heat source in the PSZ. According to Equations (3) and (4), the maximum length of the banded heat source in the PSZ and the projection length of the distance from the heat source point to any point in the direction of the movement of the banded heat source are time-varying because of the time-varying characteristics of the undeformed chip thickness, which ultimately leads to the time-varying characteristics of the temperature rise of the transient temperature field in the flank wear zone on end mills affected by the heat source in the PSZ. Because the heat source in the PSZ is a banded heat source, the temperature rise of the transient temperature field in wear land on the flank of end mills affected by the heat source in the PSZ is regarded as the effect of the integral superposition of a finite number of heat source points. Further, according to the coordinate system established in Figure 3, the temperature rise at any point along the \(X\) direction can be calculated by Equations (5)-(7). \[\Delta T_{I}(x,0,0) = \frac{q_{I}}{2\pi\lambda_{t}}\int_{0}^{\frac{h(\theta)}{\sin\phi_ {n}}\cos\eta_{c}}e^{\frac{-(l\cos\phi_{n}+\sqrt{2+z^{2}}\sin\eta_{n})V_{sh}}{2 \alpha_{w}}}K_{0}\frac{V_{sh}R_{I}}{2\alpha_{w}}\cdot dl \tag{5}\] \[\Delta T_{I^{\prime}}(x,0,0) = \frac{q_{I}}{2\pi\lambda_{t}}\int_{0}^{\frac{h(\theta)}{\sin\phi_ {n}}\cos\eta_{c}}e^{\frac{-(l\cos\phi_{n}+\sqrt{2+z^{2}}\sin\eta_{n})V_{sh}}{2 \alpha_{w}}}K_{0}\frac{V_{sh}R_{I}^{\prime}}{2\alpha_{w}}\cdot dl^{\prime}\] (6) \[\Delta T_{flank-l}(x,0,0) = \Delta T_{I}(x,0,0)+\Delta T_{I^{\prime}}(x,0,0) \tag{7}\]  
From Equation (5) to (7), \(\Delta T_{I}\left(x,0,z\right)\) and \(\Delta T_{I^{\prime}}\left(x,0,z\right)\) are the temperature rises caused by native heat source and mirror heat source in the PSZ, respectively. The distance from any point \(P_{I}\left(x,0,z\right)\) along the \(X\) direction to the points of the native heat source \(dl\) and mirror heat source \(dl^{\prime}\) in the PSZ can be measured by Equation (8). \[\left\{ \begin{array}{rl}&R_{I}=\sqrt{\left(x-l\cos\phi_{n}\right)^{2}+\left( z+l\sin\phi_{n}\right)^{2}}\\ &R_{i}^{\prime}=\sqrt{\left(x-l\cos\phi_{n}\right)^{2}+\left(z-l\sin\phi_{n} \right)^{2}}\end{array} \right.