# Improved Cutting Force Modelling in Micro-Milling Aluminum Alloy LF 21 Considering Tool Wear
## 3 An Improved Cutting Force Model in Micro-Milling LF 21 Considering Tool Wear
### Micro-Milling Force Model
The plough area \(A_{p}\) is calculated as follows:  
\[A_{p}=\frac{1}{2}r_{e}^{2}(\alpha_{S}+\alpha_{0})+\frac{1}{2}r_{e}\cdot l_{AB} -\frac{1}{2}r_{e}\cdot l_{BO}\cdot\sin(\alpha_{S}+\alpha_{0}+\alpha_{P}) \tag{14}\]  
where, \(\alpha_{0}\) is the back edge angle; \(r_{e}\) is the radius of the cutting edge arc; \(\delta\) is the elastic recovery of material. \[\alpha_{S}=\cos^{-1}\left(\frac{r_{e}-t_{\min}}{r_{e}}\right) \tag{15}\]  
\[\alpha_{P}=\tan^{-1}\left(\frac{l_{AB}}{r_{e}}\right) \tag{16}\]  
\[l_{AB}=\frac{\delta-r_{e}(1-\cos\alpha_{0})}{\sin\alpha_{0}} \tag{17}\]  
\[l_{BO}=\sqrt{{r_{e}}^{2}+{l_{AB}}^{2}} \tag{18}\]  
Material elastic recovery \(\delta\) is calculated as Equation (19):  
\[\delta=\delta_{\max}=\frac{3\cdot\sigma_{\rm s}}{4\cdot E}\cdot r_{e}\cdot \left[2\exp\left(\frac{H}{\sigma_{\rm s}}-\frac{1}{2}\right)-1\right] \tag{19}\]  
where, \(\sigma_{\rm s}\) is tensile strength; \(E\) is elastic modulus; \(H\) is hardness. The limit cutting and cutting angles \(\theta_{s}\) and \(\theta_{e}\) are shown in Formulas (20) and (21), respectively:  
\[\theta_{s} =0 \tag{20}\] \[\theta_{e} =\pi \tag{21}\]  
The authors also built a micro-milling force model dominated by plough effect, as shown in Equation (22):  
\[\left\{ \begin{array}{l}F_{x}(t)=-(K_{tpp}\cdot V_{c}+K_{tw}\cdot\Phi_{c})-(K _{fpp}\cdot V_{s}+K_{fw}\cdot\Phi_{s})\\ F_{y}(t)=(K_{tpp}\cdot V_{s}+K_{tw}\cdot\Phi_{s})-(K_{fpp}\cdot V_{c}+K_{fw} \cdot\Phi_{c})\end{array} \right.\quad t_{c}\leq t_{\text{min}} \tag{22}\]  
where, \(K_{fpp}\), \(K_{tpp}\) and \(K_{app}\) are coefficients of radial, tangential and axial plow force, respectively, and other parameters are similar to Equation (10). The maximum elastic recovery \(\delta_{\text{max}}\) of the material is calculated as follows:  
\[\delta_{\text{max}}=\frac{3\cdot\sigma_{s}}{4\cdot E}\cdot r_{e}\cdot\left[2 \exp\left(\frac{H}{\sigma_{s}}-\frac{1}{2}\right)-1\right] \tag{23}\]  
Calculation of plough area \(A_{p}\) is slightly complicated and is divided into two cases as follows:  
1. If \(t_{c}>\delta_{\text{max}}\), the plough area can be calculated as Formula (24): \[A_{p}=\frac{1}{2}r_{e}^{2}(\alpha_{C}+\alpha_{0})+\frac{1}{2}r_{e}\cdot l_{AB} -\frac{1}{2}r_{e}\cdot l_{BO}\cdot\sin(\alpha_{C}+\alpha_{0}+\alpha_{P})\] (24) where, \[\alpha_{C}=\cos^{-1}\left(\frac{r_{e}-t_{c}}{r_{e}}\right)\] (25) \[\alpha_{P}=\tan^{-1}\left(\frac{l_{AB}}{r_{e}}\right)\] (26) \[l_{AB}=\frac{\delta_{\text{max}}-r_{e}(1-\cos\alpha_{0})}{\sin\alpha_{0}}\] (27) \[l_{BO}=\sqrt{r_{e}{}^{2}+l_{AB}{}^{2}}\] (28)
2. If \(t_{c}\leq\delta_{\text{max}}\), the plough area is calculated as follows: \[A_{p}=\frac{1}{2}r_{e}^{2}(\alpha_{D}+\alpha_{0})+\frac{1}{2}r_{e}\cdot l_{ AE}-\frac{1}{2}r_{e}\cdot l_{EO}\cdot\sin(\alpha_{D}+\alpha_{0}+\alpha_{Pe})\] (29) where, \[\alpha_{D}=\cos^{-1}\left(\frac{r_{e}-t_{c}}{r_{e}}\right)\] (30) \[\alpha_{Pe}=\tan^{-1}\left(\frac{l_{AE}}{r_{e}}\right)\] (31) \[l_{AE}=\frac{t_{c}-r_{e}(1-\cos\alpha_{0})}{\sin\alpha_{0}}\] (32) \[A_{p}=\frac{1}{2}r_{e}^{2}(\alpha_{C}+\alpha_{0})+\frac{1}{2}r_{e}\cdot l_{AB} -\frac{1}{2}r_{e}\cdot l_{BO}\cdot\sin(\alpha_{C}+\alpha_{0}+\alpha_{P})\]0 (33)