# Study on Preparation and Processing Properties of Mechano-Chemical Micro-Grinding Tools
## 3 Analysis of Micro-Grinding Force
### Theoretical Analysis of Grinding Force
#### 3.1.1. Grinding Depth Model of the Abrasive  
When the micro-grinding tool processes, the scratches on the surface of the workpiece can reflect the motion state of the abrasive, and the volume of the removed single-crystal silicon can be expressed by the product of the length of the abrasive mark and the average cutting area of the abrasive. When grinding, the silicon wafer is in a stationary state, and the grinding width is the radius of the soft abrasive, that is, the center of the abrasive disc moves on the edge of the silicon wafer. To obtain the length of the wear mark, the coordinate system shown in Figure 5 is established with \(O\) point as the origin of the coordinate system, \(OD\) on the geometric center line of the silicon wafer, \(O^{\prime}\) as the geometric center of the micro-grinding tool, and the apex \(A\) in the contact zone between the micro-grinding tool and the silicon wafer is assumed to be \(l\) away from \(AO\), and let \(l\) be the variable.  
The geometric relationship can be deduced from the relationship between the rotation angle of the micro-grinding tool \(\alpha\) and \(l\), as shown in Equation (4).  
\[\alpha=arccos\sqrt{1-\left(\frac{l}{r_{g}}\right)^{2}} \tag{4}\]  
where \(r_{g}\) is the radius of the micro-grinding tool, and \(l\) is the distance between the contact area's vertex and the silicon wafer's centerline.  
The abrasive mark length \(L(l)\) can be expressed as the product of the micro-grinding tool's angle and the micro-grinding tool's radius, as shown in Equation (5).  
\[L(l)=r_{g}\cdot\alpha=r_{g}\cdot arccos\sqrt{1-\left(\frac{l}{R_{g}}\right)^{2}} \tag{5}\]  
Figure 4: Schematic diagram of abrasive grain stress.  
Figure 5: Schematic diagram of material removal.  
The instantaneous amount of material removed \(dV\) from the geometric centerline \(l\) of the silicon wafer is shown in Equation (6).  
\[dV=S\cdot L(l)\cdot\beta\cdot N \tag{6}\]  
where \(L(l)\) is the length of the wear mark at the geometric centerline \(l\) of the silicon wafer, \(S\) is the material removal area of a single abrasive grain, \(\beta\) is the repetitive cutting coefficient of the abrasive grain, and the value is 0.66 [19]; \(N\) is the number of effective abrasive grains from the center of the end face geometry of the micro-grinding tool.  
The instantaneous removal volume of the material can be calculated as shown in Equation (7).  
\[dV=r_{g}\cdot\gamma\cdot\left[\frac{2\big{(}r_{a}-z_{g}\big{)}}{r_{a}}\right]^ {25}\cdot\left(1-\frac{l^{2}}{r_{g}^{2}}\right)^{-0.5}\cdot\beta dl \tag{7}\]  
Among them, \(\gamma\) is the volume ratio of the abrasive, and \(r_{a}-z_{g}\) is the maximum grinding depth of the abrasive.  
The average depth of cut can be further calculated as shown in Equation (8).  
\[d_{a}=\frac{1}{2}\big{(}r_{a}-z_{g}\big{)}=\frac{r_{a}}{4}\Bigg{(}\frac{2\pi \cdot l\cdot a_{p}}{\omega_{g}r_{g}\cdot\beta\cdot\gamma}\sqrt{1-\frac{l^{2}} {r_{g}^{2}}}\Bigg{)}^{0.4} \tag{8}\]  
where \(\omega_{g}\) is the speed of the abrasive, and \(a_{p}\) is the depth of cut for a given micro-grinding tool.  
As can be seen from the results of Equation (8), factors such as the size of the micro-grinding tool and the size of the abrasive grain will impact the material removal volume of the micro-grinding tool. It can also be seen that even under the condition of a macroscopic depth of cut, a change in the grinding position causes a change in the depth of the cut.