# Study on Preparation and Processing Properties of Mechano-Chemical Micro-Grinding Tools
## 3 Analysis of Micro-Grinding Force
### Theoretical Analysis of Grinding Force
#### 3.1.2 The Friction and Cutting Force of a Single Abrasive Grain  
The micro-grinding tool is in mechanical contact with the surface of the silicon wafer by mechanical action during the grinding process. The normal force caused by friction can be expressed as a function of the depth of cut of the abrasive [20], as shown in Equation (9).  
\[F_{nf}=\sqrt{\frac{16d_{a}^{3}\cdot r_{a}\cdot E^{*2}}{9}} \tag{9}\]  
where \(E^{*}\) is the equivalent modulus of elasticity.  
It can be further calculated to obtain  
\[F_{nf}=\frac{r_{a}^{2}}{6\big{(}(1-v_{1}^{2})/\eta E_{1}+(1-v_{2}^{2})/E_{2} \big{)}}\Bigg{(}\frac{2\pi\cdot l\cdot a_{p}}{\omega_{g}r_{g}\cdot\beta\cdot \gamma}\sqrt{1-\frac{l^{2}}{r_{g}^{2}}}\Bigg{)}^{0.6} \tag{10}\]  
where \(E_{1}\) and \(E_{2}\) are the elastic moduli of single-crystal silicon and cerium oxide, respectively, and \(v_{1}\) and \(v_{2}\) are [PERSON]'s ratios of monocrystalline silicon and cerium oxide abrasives, respectively.  
The tangential force caused by friction can be obtained by multiplying the normal force by \(\mu\) friction coefficient (the friction coefficient between the micro-grinding tool and the silicon wafer at room temperature \(\mu=0.327\)).  
For the grinding force model of a single abrasive grain, the normal grinding force is formed during the grinding process due to the generation of chips, which is proportional to the corresponding cross-sectional area. It is obtained by calculation that  
\[F_{nc}=\frac{4r_{a}^{2}\cdot k}{15}\Bigg{(}\frac{2\pi\cdot l\cdot a_{p}}{ \omega_{g}r_{g}\cdot\beta\cdot\gamma}\sqrt{1-\frac{l^{2}}{r_{g}^{2}}}\Bigg{)}^ {0.6} \tag{11}\]where \(k\) is the chip thickness coefficient, which is related to the properties of the material being processed.  
When chips are generated during the grinding process, the resulting normal and tangential forces are proportional [21]. For (100) crystal-oriented single-crystal silicon, the scale factor \(\delta\) = 0.58. Combined with the self-introduced geometric deformation coefficient, the expression of the total normal grinding force and tangential grinding force during the grinding is shown in Equations (12) and (13).  
\[F_{n}=\varepsilon\Big{(}F_{nf}+F_{nc}\Big{)}=\varepsilon\Bigg{[}\frac{r_{a}^{ 2}}{6\Big{(}\frac{1-v_{1}^{2}}{\eta E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\Big{)}}+ \frac{4r_{a}^{2}\cdot k}{15}\Bigg{]}\cdot\Bigg{(}\frac{2\pi\cdot l\cdot a_{p}} {\omega_{g}r_{g}\cdot\beta\cdot\gamma}\sqrt{1-\frac{l^{2}}{r_{g}^{2}}}\Bigg{)} ^{0.6} \tag{12}\]  
\[F_{t}=\varepsilon\Big{(}F_{tf}+F_{tc}\Big{)}=\varepsilon\Bigg{[}\frac{\mu r_{a }^{2}}{6\Big{(}\frac{1-v_{1}^{2}}{\eta E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\Big{)} }+\frac{4r_{a}^{2}\cdot k\cdot\delta}{15}\Bigg{]}\cdot\Bigg{(}\frac{2\pi\cdot l \cdot a_{p}}{\omega_{g}r_{g}\cdot\beta\cdot\gamma}\sqrt{1-\frac{l^{2}}{r_{g}^ {2}}}\Bigg{)}^{0.6} \tag{13}\]  
It can be seen from Equations (12) and (13) that the characteristics of the micro-grinding tool itself, including changes in the abrasive diameter, modulus of elasticity, abrasive volume ratio, and other factors, will affect the grinding force. At the same time, changes in grinding parameters such as speed and cutting depth will also affect the change in grinding force.