## 2 Establishment of Milling Temperature Model
### Average Temperature: The Contact Area of the Rear Cutter Surface  
According to the literature, the average temperature of the rear surface contact area can be calculated [14]. The calorific value per unit time and per unit area of the friction heat source on the rear face is \(q_{3}\):  
\[q_{3}=\frac{F_{fa}\cdot V}{I_{micro-rak}\cdot\overline{a_{e}}}, \tag{12}\]  
where \(F_{fa}\) is the rear face friction force (N), \(V\) is the cutting speed (m/min), and \(l_{f\text{ micro-rear}}\) is the contact length between the cutter and the back face in micro-element cutting of a ball-end milling cutter.  
(1) The average temperature of the rear face calculated from the workpiece's perspective.  
The friction heat source is a moving surface heat source for the workpiece, and the workpiece is regarded as semi-infinite. The moving surface heat source theory is applied to obtain the average temperature of the rear face as follows:  
\[\overline{\theta_{w}}=\frac{0.752R_{3}\cdot q_{3}}{k_{4}}\sqrt{\frac{\alpha_ {3}l_{micro-rak}}{V}}+\theta_{0}, \tag{13}\]  
where \(R_{3}\) is the ratio of the heat generated in the contact area (between rear face and workpiece) that flows into the workpiece, \(k_{4}\) is the thermal conductivity coefficient [W/(m\(\cdot\)k)] when the temperature of the workpiece material is the average value of \(\overline{\theta_{w}}\) and \(\theta_{0}\), and \(\alpha_{3}\) is the thermal conductivity coefficient when the temperature of the workpiece material is the average value of \(\overline{\theta_{0}}\) and \(\overline{\theta_{w}}\).  
(2) The average temperature of the rear face calculated from the tool's perspective.  
The friction heat source is a fixed continuous heat source for the rear face. The heat source is located using a semi-infinite principle, regarding the tool with the 1/4 infinite principle (two-dimensional cutting) or 1/8 infinite principle (three-dimensional cutting). According to the heat source formula of the static continuous action surface, the average temperature \(\overline{\theta_{w}}\) of the tool rear face is as follows:  
\[\overline{\theta_{w}}=\frac{(1-R_{3})q_{3}l_{micro-rak}}{k_{5}}\overline{A}+ \theta^{\prime}_{0}, \tag{14}\]  
where \(k_{5}\) is the thermal conductivity when the temperature of the tool material is the average of \(\overline{\theta_{w}}\) and \(\theta^{\prime}_{0}\).  
Given Equations (13) and (14) are equal, \(R_{3}\) can be defined as follows:  
\[R_{3}=\frac{\frac{F_{fa}\cdot V\cdot\overline{A}}{a_{e}k_{5}}+\theta^{\prime} _{0}-\theta_{0}}{\frac{0.752F_{fa}}{k_{4}k_{4}}\sqrt{\frac{\alpha_{3}\cdot v}{ l_{micro-rak}}+\frac{F_{fa}\cdot V\cdot\overline{A}}{a_{e}k_{5}}}}. \tag{15}\]From Equations (10) and (14), it can be seen the average cutting temperature of the tool is as follows:  
\[\overline{\theta}=\frac{(1-R_{2})\cdot q_{2}\cdot l_{micro-nake}}{k_{3}}\cdot \overline{A}+\frac{(1-R_{3})\cdot q_{3}\cdot l_{micro-nake}}{k_{5}}\cdot \overline{A}+\theta^{\prime}_{0}+\theta_{0}. \tag{16}\]  
Substituting Equations (8) and (12) into Equation (16) results in the following:  
\[\overline{\theta}=\frac{(1-R_{2})F_{f}\cdot v_{c}}{k_{3}a_{e}}\cdot\overline{ A}+\frac{(1-R_{3})\cdot F_{fa}\cdot V}{k_{5}a_{e}}\cdot\overline{A}+\theta^{ \prime}_{0}+\theta_{0}. \tag{17}\]