# An Improved Cutting Force Model for Ultrasonically Assisted Grinding of Hard and Brittle Materials
## 2 Model Development
### Cutting Force Analysis  
Due to the introduction of the axial ultrasonic vibration, the diamond grains in the end face of the tool are vibrated as perpendicular to the workpiece surface, and those in the lateral face are parallel to the workpiece surface, while those in the transition fillet surface penetrate into the workpiece surface at a certain angle. Therefore, the abrasive grains at different locations of the tool result in different cutting forces because of their different  
Figure 3: Trajectory of an abrasive grain in UAG.  
Figure 2: Illustration of ultrasonically assisted grinding process.  
ways of interacting with the workpiece. In this study, assumptions and simplifications are made as follows:  
* All abrasive grains are rigid indenters with an octahedral shape. The semi-angle between the two opposite edges of an abrasive grain is \(\beta\), and the length of the edges is \(b\), as shown in Figure 4;
* The material is ideally removed by brittle fracture;
* All the diamond grains in the tool surface have the same protrusion height and are uniformly distributed.  
In consideration of different cutting depths, the process of axial ultrasonically assisted grinding contains two cases: case one is \(a_{p}>r_{c}\), as shown in Figure 5, and the other is \(a_{p}<r_{c}\), as shown in Figure 6 (where \(a_{p}\) is cutting depth, \(r_{c}\) is the radius of the transition filled). In the literature, the diamond grains in the end face of the tool generate the intermittent hammering actions primarily in the axial direction [12; 15]. The diamond grains in the lateral face of the tool generate a continuous contact with sinusoidal trajectory primarily in the feed direction [26; 27; 28]. The cutting state of the abrasive grains in the transition filled surface is more complex and may not only consist of the intermittent hammering actions but also the continuous contact actions.  
Figure 4: A diamond grain simplified as an octahedron.  
Figure 5: Cutting force analysis of \(a_{p}>r_{c}\).  
Decomposition can be made on the cutting forces generated by the abrasive grains in the different areas of the tool in both the axial and feed directions. For the case of \(a_{p}>r_{c}\), it is obtained that \(F_{a}=F_{a1}+F_{a2}\) and \(F_{f}=F_{f1}+F_{f2}\), as shown in Figure 5. For the case of \(a_{p}<r_{c}\), \(F_{a}=F_{a1}+F_{a2}\) and \(F_{f}=F_{f2}\) are found, as shown in Figure 6. The parameters in the above equations are defined as:  
\(F_{a}\) is the axial force of the tool;  
\(F_{c}\) is the cutting force of the diamond grains on the transition fillet surface of the tool;  
\(F_{a1}\) is the axial force of the diamond grains on the end face of the tool;  
\(F_{a2}\) is the component of \(F_{c}\) in the axial direction;  
\(F_{f}\) is the feed force of the tool;  
\(F_{f1}\) is the feed force of the diamond grains in the lateral face of the tool;  
\(F_{f2}\) is the component of \(F_{c}\) in the feed direction.  
Through the above analysis, the cutting force in UAG can be expressed as  
\[\left\{\begin{array}{c}F_{a}=F_{a1}+F_{a2}\\ F_{f}=\tau F_{f1}+F_{f2}\end{array}\right.,\text{ if }a_{p}>r_{c},\tau=1; \text{ if }a_{p}<r_{c},\tau=0. \tag{4}\]  
Since the radial force of the tool is too small to have any impact effect, it was not considered in this study. For the cutting force model, the indentation theory on brittle materials is used, as illustrated in Figure 7. \(C_{L}\) and \(C_{h}\) are the lateral crack length and height, respectively, and are given by [32; 33; 34]:  
\[C_{L}=C_{c}\left(\frac{1}{\tan\beta}\right)^{5/12}\left(\frac{E^{3/4}}{H_{V} \cdot K_{IC}\cdot\left(1-\
u^{2}\right)^{1/2}}\right)^{1/2}F_{e1}{}^{5/8} \tag{5}\]  
\[C_{h}=C_{b}\left(\frac{1}{\tan\beta}\right)^{1/3}\frac{E^{1/2}}{H_{V}}F_{e1}{} ^{1/2} \tag{6}\]  
where \(C_{c}\) and \(C_{b}\) are dimensionless constants; \(E\) is the elastic modulus; \(H_{V}\) is the hardness; \(K_{IC}\) is the fracture toughness; \(\
u\) is the Poisson's ratio; and \(\delta\) is the maximum penetration depth.  
Figure 6: Cutting force analysis of \(a_{p}<r_{c}\).