## 2 Establishment of Milling Temperature Model  
### Analysis of Milling Heat Source of Ball-End Milling Cutter  
A simplified diagram of the heat source model for ball-end milling cutter is illustrated in Figure 1. According to the structure of the ball-end milling cutter and milling principle, when calculating the cutter temperature, the ball-end milling cutter can be decomposed into numerous uniform micro units along the axial direction, and the cutting of each micro unit can be simplified as oblique cutting. The milling heat for this kind of cutter is mainly generated in three areas. The first heat source area is the shear surface, where the heat is generated by the deformation work done to overcome the plastic deformation of the metal. The second heat source area is the rake face, where the heat is generated by the friction between the chip and the rake face. The third heat source area is the rear, where the heat is generated by the friction between the rear and the workpiece. When calculating the temperature field of the workpiece, the surface heat sources of the three deformation zones are equivalent to the line heat sources at the cutting edge. The ball end milling cutter can be discretized into a collection of uniform micro-elements along the axis, and the cutting process of each micro element can be simplified as oblique cutting. The discrete micro-element line heat sources are \(dz\) in height, \(dL\) in length, and \(\beta_{1}\) in helix angle of the milling cutter, in the following:  
\[dL=dz/\cos\beta_{1}. \tag{1}\]  
During the milling process of ball-end milling cutter with micro-texture, the contact state between the cutter and the workpiece is constantly changing, and the heat flux density is non-uniform distribution. During the cutting process, the length of spiral heat source changes continuously, which  
Figure 1: Simplification of a heat source model for ball-end milling cutters.  
is a process from zero to the longest and from the longest to zero. Therefore, the heat transfer process of the micro-textured ball-end milling cutter can be simplified as an intermittent heating process of the periodicity on the workpiece processing surface by the spiral heat source with variable length, non-uniform heat flux density, and constant change [12]. As the helix angle of the ball-end milling cutter with micro-texture used in this paper is zero, it is not necessary to consider the influence of helix angle on the variation of linear heat source. In order to simplify the analysis, the ball-end milling cutter is divided into numerous uniform micro-elements along the cutting edge and the cutting process of each micro-element is simplified to a right-angle cutting form.  
During the milling process of a ball-end milling cutter, because the cutting edge is a circular arc, the cutting depth at different positions of the cutting edge exhibits a state of constant change from zero to the maximum value. From the classical cutting force formula, it can be seen that the change of the cutting depth causes slight changes in the main cutting force, which can be ignored. Therefore, each cutting element can be integrated and the influence of the cutting edge form on the cutting temperature can be obtained.