## 2 Establishment of Milling Temperature Model
### Average Temperature: The Shear Surface  
Most of the heat generated on the shear surface passes to the chip and part to the workpiece. The heat generated \(q_{1}\) per unit time per unit area on the shear surface is as follows:  
\[q_{1}=\frac{F_{s}\cdot v_{s}\cdot\sin\varphi}{a_{p}\cdot a_{e}}, \tag{2}\]  
where \(F_{s}\) is the shear force, \(v_{s}\) is the shear velocity along the shear plane, and \(\varphi\) is the shear angle.  
The average temperature of the three surface heat sources is solved (sheer plane, rake face, rear). First, the average temperature of the shear plane is solved. It is assumed that the surface of the semi-infinite object has a moving surface heat source with a width of \(2l_{m}\) and an infinite length. If the heat per unit area per unit time is \(q\), the average temperature \(\overline{\theta}\) in the surface heat source can be expressed as follows:  
\[\overline{\theta}=0.752\frac{q}{k_{0}}\sqrt{\frac{2 kl_{m}}{c_{0}\rho_{0}v}}+\theta, \tag{3}\]  
where \(c_{0}\) is the specific heat of the object (J/(kg\(\cdot\)K)), \(\rho_{0}\) is the object density (kg/m\({}^{3}\)), \(v\) is the moving speed of the surface heat source (m/s), \(k_{0}\) is the thermal conductivity of the object [W/(m\(\cdot\)K)], \(l_{m}\) is the half (m) of the surface heat source width, and \(\theta\) is the initial temperature of the object (\({}^{\circ}\)C).  
(1) The average temperature of the chip-side shear surface.  
For the chip, the heat source produced by shearing is a continuous fixed heat source. According to the equation of heat balance, the average temperature of the shear zone is as follows:  
\[\overline{\theta_{s}}=\frac{R_{1}\cdot q_{1}}{c_{1}\cdot\rho_{1}\cdot v\cdot \sin\varphi}+\theta_{0}, \tag{4}\]  
where \(R_{1}\) is the ratio of heat to chip in the first deformation region, \(c_{1}\) is the specific heat capacity of temperature \(\left(\overline{\theta_{s}}+\theta_{0}\right)/2\), \(\rho_{1}\) is the density of the workpiece material, and \(\theta_{0}\) is the initial temperature of the workpiece.  
(2) The average temperature of the workpiece-side shear surface.  
The average temperature of the shear surface on the workpiece can be calculated according to the temperature field of the moving surface heat source. The average temperature of the workpiece-side shear plane is obtained as follows:  
\[\overline{\theta_{s}}=0.752\frac{(1-R_{1})q_{1}}{k_{1}}\sqrt{\frac{\alpha_{1} \cdot q_{p}}{v_{s}\cdot\sin\varphi}}+\theta_{0}, \tag{5}\]  
where \(k_{1}\) is the thermal conductivity of the workpiece material at the temperature \((\overline{\theta}_{s}+\theta_{o})/2\), and \(\alpha_{1}\) is the thermal conductivity coefficient of the workpiece material at temperature \((\overline{\theta}_{s}+\theta_{o})/2\).  
Equation (4) is equal to Equation (5). According to the chip shear deformation, the relative slip \(\varepsilon_{slip}\) is as follows:  
\[\varepsilon_{slip}=\frac{\cos\gamma_{nke}}{\sin\varphi\cos(\varphi-\gamma_{ nke})}, \tag{6}\]  
where \(\gamma_{nke}\) is the rake angle of the micro-element cutting of the ball-end milling cutter.  
Given \(c_{1}p_{1}=\frac{k_{1}}{a_{1}}\) and \(v_{s}=\frac{\
u\cos\gamma_{nke}}{\cos(\varphi-\gamma_{nke})}\), the ratio \(R_{1}\) of the heat transfer from the shear surface to chip is as follows:  
\[R_{1}=\frac{1}{1+1.33\sqrt{\frac{a_{1}\cdot\varepsilon_{slip}}{v\cdot a_{c}}}}, \tag{7}\]  
where \(a_{c}\) is the actual cutting depth.