## 3 Friction Force Prediction at TCI Based on TPF
### Prediction of Friction Component at TPI  
3.1.1. Prediction of Normal Force in Unit SiC (\(F_{N-S}\))  
Reinforced particles are assumed to be circular spheres with radius \(R\) and fixed-point angle. Therefore, the stress model of elemental reinforcement analysis can be obtained from [PERSON] and [PERSON] et al. [1,13], as shown in Figure 3. P-Q-M is the rake face where the tool contacts a single SiC-rein forced particle. Since P-Q-M is a circular arc, this also provides evidence for sliding friction between the two bodies.  
The normal force of a unit particle can be obtained by the following equation:  
\[F_{N-S}=\frac{29\pi R\sigma_{y(tool)}\delta_{PO}}{10} \tag{12}\]  
\(\delta_{PO}\) is the maximum critical value of relative encroachment of reinforced particles when full plastic deformation occurs on the rake face of the tool.  
\[\delta_{PO}=R\left(\frac{9\pi}{4}\right)^{2}\left(\frac{\sigma_{y(tool)}}{E^{ *}}\right)^{2} \tag{13}\]  
\(E^{*}\) is the compound mode can be obtained from the following Equation (14):  
\[\frac{1}{E^{*}}=\frac{\left(1-v_{1}\right)^{2}}{E_{1}}+\frac{\left(1-v_{2} \right)^{2}}{E_{2}} \tag{14}\]  
\[r_{(grave)}=\sqrt{R^{2}-\left(R^{2}-\delta_{PO}\right)} \tag{15}\]  
3.1.2. Prediction of Two-Body Sliding Friction Component at TPI (\(F_{S}\))  
Based on the two-body sliding friction behavior of the tool-particle interface, the two-body sliding friction component can be given by the following equation.  
\[F_{S}=F_{N-C}\cdot\zeta_{S}\cdot\lambda \tag{16}\]  
Assuming that the hard particle volume fraction is evenly distributed over the contact length, the normal stress should be the integral of the \(L_{T-C}\), \(f(x,q_{x})\) is the function of the contact length on the particle volume fraction \(q_{x}(0\leq x\leq L_{S})\). The two-body friction component can be obtained from the following equation:  
\[F_{S}=\iint F_{N-S}\cdot\zeta_{S}\cdot\lambda\cdot f(x,q_{x})\mathrm{d}x \mathrm{d}A \tag{17}\]  
Combining Equation (6):  
\[F_{S}=\int_{0}^{L_{T-C}}\int_{t_{1}}^{t_{2}}P(t)F_{N-S}\cdot\zeta_{S}\cdot \lambda\cdot f(x,q_{x})\mathrm{d}t\mathrm{d}x \tag{18}\]  
Figure 3: Particle element analysis.  
3.1.3. Prediction of Three-Body Rolling Friction Component at TPI (\(F_{R}\))  
The particles are firmly constrained in the chip when sharpening, causing sliding friction between the two bodies. Three-body rolling friction is caused by deboned particles rolling between the rake face and the Al matrix [26].  
\[F_{R}=F_{N-S}\cdot\zeta_{S}\cdot\eta\cdot\mu_{3} \tag{19}\]  
Assuming that the hard particle volume fraction is evenly distributed over the \(L_{T-C}\), the normal stress in the EVC should be the integral of the tool-chip contact length (\(L_{S}\leq x\leq L_{T-C}\)). The two-body friction component can be obtained from the following equation:  
\[F_{R}=\iint F_{N-S}\cdot\zeta_{S}\cdot\eta f(x,q_{x})\cdot\mu_{3}\mathrm{d}x \mathrm{d}A \tag{20}\]  
\[F_{R}=\int_{0}^{L_{T-C}}\int_{t_{1}}^{t_{2}}P(t)F_{N-S}\cdot\zeta_{S}\cdot \eta\cdot f(x,q_{x})\mu_{3}\mathrm{d}t\mathrm{d}x \tag{21}\]  
\(\mu_{3}\) is the friction coefficient of three-body friction [27], which can be obtained from the following equations:  
\[\mu_{3}=\frac{k_{(tool)}}{\pi\cdot H_{t}}\cdot\left(\frac{2R}{r_{(groove)}} \right)^{2}\cdot\left\{1-\left[1-\left(\frac{r_{(groove)}}{R}\right)^{2} \right]^{\frac{1}{2}}\right\} \tag{22}\]  
\[k_{(tool)}=\frac{H_{t}}{6}=\frac{\sigma_{y(tool)}}{2} \tag{23}\]