## 3 Polishing Pressure Calculation Model
During the complete plastic contact deformation stage, the polishing pressure and deformation between the abrasive particle and the workpiece can be expressed as follows:  
\[\left\{ \begin{array}{l}F_{\text{mgs}}=A_{\text{mgs}}H_{\text{bw}}=2\pi R_{ \text{m}}H_{\text{bw}}\delta_{\text{mgs}}\\ \delta_{\text{h}}=110\delta_{0}\\ \delta_{\text{mg}}=\delta_{\text{mgs}}\end{array} \right. \tag{4}\]  
where \(\delta_{\text{h}}\) is the critical deformation for complete plastic deformation, \(\delta_{\text{mgs}}\) is the deformation between the abrasive particle and workpiece caused by the plastic deformation, and \(F_{\text{mgs}}\) is the force between the abrasive particle and workpiece in the complete plastic deformation stage. Figure 3: Elastic contact between single abrasive particle and workpiece. When \(\delta_{0}\leq\delta_{\text{mg}}<\delta_{\text{h}}\), the polishing pressure and deformation between the abrasive particle and the workpiece can be expressed as follows:  
\[\left\{ \begin{array}{l}F_{\text{ts}}=F_{\text{tsp}}A_{\text{ts}}=\pi R_{\text{ m}}H_{\text{bw}}\delta_{\text{mg}}\Big{[}1-(1-\lambda_{\text{mg}})\frac{\ln\delta_{ \text{h}}-\ln\delta_{\text{mg}}}{\ln\delta_{\text{h}}-\ln\delta_{0}}\Big{]}\\ A_{\text{ts}}=\pi R_{\text{m}}\delta_{\text{mg}}\big{[}1+\psi(\delta_{\text{mg}} )\big{]}\\ \psi(\delta_{\text{mg}})=-2\Big{(}\frac{\delta_{\text{mg}}-\delta_{0}}{\delta_ {\text{h}}-\delta_{0}}\Big{)}^{3}+3\Big{(}\frac{\delta_{\text{mg}}-\delta_{0}} {\delta_{\text{h}}-\delta_{0}}\Big{)}^{3}\\ F_{\text{tsp}}=H_{\text{bw}}-H_{\text{bw}}(1-\lambda_{\text{mg}})\frac{\ln \delta_{\text{h}}-\ln\delta_{\text{mg}}}{\ln\delta_{\text{h}}-\ln\delta_{0}}\\ \delta_{\text{mg}}=\delta_{\text{mgt}}+\delta_{\text{mgg}}\end{array} \right. \tag{5}\]  
where \(F_{\text{tsp}}\) is the average polishing pressure during the elastic-plastic contact deformation stage, \(A_{\text{ts}}\) is the contact area between the abrasive and workpiece, and \(\lambda_{\text{mg}}\) is the average polishing pressure coefficient between the abrasive and workpiece. The force between the abrasive particle and the workpiece can be expressed by Equation (6),  
\[F_{\text{mg}}=F_{\text{mgs}}+F_{\text{ts}}+F_{\text{mgt}} \tag{6}\]  
where \(F_{\text{mg}}\) is the force between the abrasive particle and workpiece, \(F_{\text{mgt}}\) is the total elastic force between the abrasive particle and workpiece, \(F_{\text{ts}}\) is the total elastic-plastic force between the abrasive particle and workpiece, and \(F_{\text{mgs}}\) is the total plastic force between the abrasive particle and workpiece. The elastic contact deformation between the flap wheel substrate and the workpiece is shown in Figure 4. The total force between the flap wheel and the workpiece can be expressed by Equation (7),  
\[F_{\text{P}}=F_{\text{bg}}+F_{\text{mg}} \tag{7}\]  
where \(F_{\text{P}}\) is the total force between the flap wheel and workpiece, and \(F_{\text{bg}}\) is the elastic force between the flap wheel substrate and workpiece. According to the Hertz elastic contact theory, the elastic force and deformation between the flap wheel substrate and the workpiece can be expressed by Equation (8),  
\[\left\{ \begin{array}{l}F_{\text{bg}}=\frac{4}{3}E_{\text{bg}}\sqrt{R_{\text{ bg}}}\sqrt{\delta_{\text{bg}}^{3}}\\ \\ \frac{1}{E_{\text{mbt}}}=\left(\frac{1-\
u_{\text{mt}}^{2}}{E_{\text{mt}}}+\frac {1-\
u_{\text{mt}}^{2}}{E_{\text{bt}}}\right)\\ \\ \frac{1}{R_{\text{bg}}}=\frac{1}{R_{\text{bg}}}\pm\frac{1}{R_{\text{h}}}\end{array} \right. \tag{8}\]  
where \(E_{\text{mbt}}\) is the equivalent elastic modulus of the flap wheel and workpiece, \(R_{\text{bg}}\) is the equivalent radius of the flap wheel and workpiece, \(R_{\text{b}}\) is flap wheel radius, and \(R_{\text{jq}}\) is the curvature radius of the workpiece at the contact point. Figure 4: Elastic contact deformation between flap wheel substrate and workpiece.