## 2 Transient Temperature Field Model of Wear Band on the Flank of End Mills  
During the peripheral milling process, each tooth of the helical end mills is discretized into many slices along the axis (\(Z\) direction). Helical angle changes the end mill into an independent oblique cutting edge, and thus the peripheral milling process is transformed into an oblique cutting process (Figure 1).  
Figure 1: The conversion diagram of oblique cutting for end mills.  
Because milling involves intermittent tool-workpiece contact, when the edge is in contact with the workpiece, heat source that results in temperature rise is primarily generated in three different deformation zones at the time of metal cutting. The PSZ generates a high temperature due to its plastic deformation on the shear surface, thus softening the workpiece material and leading to greater workpiece deformation. This is a process of mutual coupling between heat and deformation. The heat generated in the RCZ is attributed to chip deformation, bonding and sliding friction between the tool and chip. The third deformation zone contains the heat generated by sliding friction and extrusion between the machined workpiece surface and wear band on the flank face. When the edge is out of contact with the workpiece, a dissipating heat source that results in temperature drop primarily is generated in wear land on the flank of end mills. Among these three zones, the PSZ and RCZ are mainly affected by cutting conditions, whereas the FWZ is largely affected by the wear of the flank face, dissipating heat source mainly affected by the cooling medium wear land on the flank of end mills. In this study, the flank of end mills was worn out, and thus the influences of the PSZ, RCZ, FWZ, and dissipating heat source on the transient temperature field of flank wear band on the end mills were comprehensively considered.  
The following five assumptions were made for the transient temperature field model of the wear band on the flank of end mills.  
1. The heat flow generation and temperature distribution are stable.
2. Deformation energy refers to the deformation energy within the shear zone, deformation energy at the rake-chip interface due to friction and extrusion, and deformation energy at the flank-workpiece interface due to friction and extrusion. All deformation energy is involved in mechanical processes and is converted into cutting heat. This negligible part is stored in the deformed metal in the form of potential. The heat loss along the contact surface as well as the tool, chip and workpiece surface are neglected.
3. The heat sources in the RCZ are not affected by crater wear on the rake face.
4. The cutting-edge radius is zero.
5. The temperature of end mills is not related to the milling depth.
6. The yield strength of Ti6 Al4V is affected by its temperature, strain rate and stress state. In order to simplify the calculation, only the effect of temperature on the yield strength of Ti6 Al4V is considered.  
As shown in Figure 2, three different coordinate systems were established, by considering the influence of four heat sources on the temperature field of the wear band on the flank of end mills.  
Figure 2: Four heat sources and heat distribution systems.  
To calculate the increased temperatures caused by heat sources in the PSZ, RCZ, FWZ and dissipating heat source, the coordinate systems of heat sources in the PSZ and RCZ were converted into the coordinate system of heat source in the FWZ (or dissipating heat source), \(\beta_{0}=90^{\circ}-\gamma_{n}\), as shown in Equation (1).  
\[\left\{\begin{array}{l}\begin{array}{l}X=X^{\prime\prime}-VB_{1}\\ Y=-Y^{\prime\prime}\\ Z=-Z^{\prime\prime}\\ X^{\prime}=l_{c}-\frac{VB_{1}-X^{\prime\prime}}{\cos\beta_{0}}\\ Y^{\prime}=-Y^{\prime\prime}\\ Z^{\prime}=\frac{VB_{1}-l_{c}\cos\beta_{0}-X^{\prime\prime}}{\sin\beta_{0}} \\ \end{array}\right. \tag{1}\]