## 2 Prediction of Cutter Edge Temperature in Milling
### Calculation of Cutter Edge Temperature in Milling  
In milling, the cutter is heated up mainly due to the heat that flows into the cutter through chip-tool contact zone. It has been verified that the heat convection effect is relatively insignificant if no coolant is used during machining, as demonstrated by [PERSON] et al. [32]. [PERSON] et al. [33] pointed out that the heat radiation generated by cutting heat was generally weak; it had little impact on machining process. Therefore, heat convection and heat radiation effects are neglected in this study. Thus, it is reasonable to assume the boundary planes \(x=0\), \(y=0\), and \(z=0\) depicted in Figure 5 are insulated. To calculate the cutter temperature, a local coordinate system is established (see Figure 5). For a location \(A\) inside the cutter, if the thermal properties of the tool are homogeneous and are regarded as constant, its temperature \(T\) satisfies  
\[\frac{\partial T}{\partial t}=K\cdot\left(\frac{\partial^{2}T}{\partial x^{2 }}+\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial^{2}T}{\partial z^{2}}\right) \tag{10}\]  
where \(T=T(x,y,z,t)\) is the temperature at position \((x,y,z)\) and time \(t\). \(K\) is the cutter thermal diffusivity. Equation (10) represents a 3-dimensional transient temperature problem, and its definite condition is  
\[\left\{\begin{array}{c}-k\cdot\frac{\partial T}{\partial x}=q_{c}(t)\\ x=0\\ 0\leq y\leq l_{cn}\\ 0\leq z\leq a_{p}\end{array}\right. \tag{11}\]  
where \(k\) is the cutter thermal conductivity, and \(q_{c}(t)\) is the heat flux that flows into the cutter through chip-tool contact zone on the cutter rake. In this research, \(q_{c}(t)\) is assumed to be evenly distributed on the contact zone.  
To solve Equations (10) and (11), [PERSON]'s [34] theory is adopted. Then, the temperature \(T\) is obtained by a triple integral  
\[T=\frac{K}{k}\cdot\int_{0}^{t}\int_{0}^{l_{cn}}\int_{0}^{a_{p}}\theta_{G}(x,y, z,y_{p},z_{p},D)\cdot q_{c}(\tau)\cdot dy_{p}\cdot dz_{p}\cdot d\tau \tag{12}\]  
where \(\theta_{G}\) is green function and it represents the temperature rise at location \(A\) of coordinate (\(x\),\(y\),\(z\)) at time \(t\) due to an instantaneous point heat source of unit strength at location \(B(0,y_{p},z_{p})\) at time \(t=\tau\). Green function is expressed as  
Figure 4: The relationship between the un-deformed chip thickness \(h\) and contact length \(l_{cn}\).  
\[\begin{array}{ll}&\theta_{G}(x,y,z,y_{p},2_{p},D)\\ &=\frac{2}{(\sqrt{\pi D})^{3}}\exp\left[\frac{-x^{2}}{D^{2}}\right]\cdot\left[ \exp\left(\frac{-(y+y_{p})^{2}}{D^{2}}\right)+\exp\left(\frac{-(y-y_{p})^{2}}{D ^{2}}\right)\right]\\ &\left[\exp\left(\frac{-(z+z_{p})^{2}}{D^{2}}\right)+\exp\left(\frac{-(z-z_{p}) ^{2}}{D^{2}}\right)\right]\end{array} \tag{13}\]  
where \(D=2\sqrt{K\cdot(t-\tau)}\).  
Because Equation (12) is complicated and cannot be evaluated analytically, the numerical methods are often employed, and these methods are usually time-consuming. To solve the temperature effectively, a simplification is conducted here. Since \(q_{c}(t)\) is evenly distributed, which is independent of location, Equation (12) is simplified into an integral as  
\[T=\frac{K}{k}\int_{0}^{t}A(x,y,z,D)\cdot q_{c}(\tau)\cdot d\tau \tag{14}\]  
where  
\[\begin{array}{ll}A(x,y,z,D)&=\int_{0}^{t_{cn}}\int_{0}^{q_{p}}\theta_{G} \left(x,y,z,0,y_{p},z_{p},D\right)\cdot dy_{p}\cdot dz_{p}\\ &=\frac{1}{2\sqrt{\pi D}}\exp\left(\frac{-x^{2}}{D^{2}}\right)\cdot\left[erf \left(\frac{t_{cn}+y}{D}\right)+erf\left(\frac{t_{cn}-y}{D}\right)\right]\\ &\left[erf\left(\frac{q_{p}+z}{D}\right)+erf\left(\frac{q_{p}-z}{D}\right) \right]\end{array} \tag{15}\]  
and  
\[erf(u)=\frac{2}{\sqrt{u}}\cdot\int_{0}^{u}e^{-\tau^{2}}\cdot d\eta \tag{16}\]  
By combining the engagement patterns between cutting edge and material, Equation (14) can be used to calculate cutter edge temperature in trochoidal milling.