## 3 Material Removal Model of Abrasive Flow  
### Normal Grinding Force of Abrasive Particle  
Since the particles have no certain geometric shape, a particle model is commonly set up with conicalness and pyramid. In this paper, the silicon carbide particle is taken as the abrasive particle, and is set as a regular hexahedron model [40; 41; 42].  
As presented in Section 2, the abrasive flow enters the constrained flow passage and forms the fully-developed turbulent state. For the movement complexity of turbulence, we select a single abrasive particle in contact with complex surface as the study subject, in which the normal grinding force is a key parameter. Based on the above hypothesis, the material removal model of single particle can be described by Figure 3.  
In this model, abrasive flow is turbulent so that the abrasive particles in the flow passage strike and cut the complex surface under the effect of normal force in vertical direction, where \(F_{N}\) is the pressure of fluid driving particle. The cutting force of the single abrasive particle can be expressed by [43; 44; 45]:  
\[F_{n}=c_{n}pS_{jc} \tag{1}\]  
where \(c_{n}\) is a constant of particle wearing, and determined by the particle shape and granulate hardness; \(p\) is the average normal stress of particles collide with surface; and \(S_{jc}\) is the projection of abrasive particle on the machined surface and can be described as  
\[S_{jc}=\sqrt{\frac{2F_{N}}{3\sqrt{3}H}} \tag{2}\]  
where \(H\) is surface hardness of workpiece. Therefore, the indentation depth is determined by the normal force and the hardness of workpiece.  
The projection area is supposed to be an isosceles right triangle, and the waist length is given by  
\[a_{y}=\sqrt{\frac{4F_{N}}{\sqrt{3}H}} \tag{3}\]  
Figure 3: Material removal model of single particle: 1, particle; 2, machined surface.  
According to micro-cutting wear mechanism [46; 47; 48], a small part of the abrasive particle is pressed into the machining surface by the normal force. Then, an abrasive particle is driven by the friction between the base fluid and abrasive particle. The furrow action is taken by abrasive particle on the machined surface, so the wear scars with trough shapes are appeared on the surface. For the turbulent abrasive flow processing, high accuracy finishing process is achieved by numerous irregular wear scars with trough shapes. In this research, the material of abrasive particle is silicon carbide, and the material of workpiece is titanium alloy. Because the hardness of silicon carbide is greater than the hardness of titanium alloy, the processing result is affected by the hardness of silicon carbide. If the slip distance of a particle is \(ds\), the removal volume of different machined surface is described as:  
\[dV=\left\{\begin{array}{cc}\frac{F_{\mathrm{in}}}{2\sqrt{H}}\cdot\frac{3R+ \delta}{3R+3-3 ay}ds,&\text{concave surface}\\ \frac{F_{\mathrm{in}}}{2\sqrt{H}}ds,&\text{plane}\\ \frac{F_{\mathrm{in}}}{2\sqrt{H}}\cdot\frac{3R-\delta}{3R-3\delta+3 ay}ds,& \text{convex surface}\end{array}\right. \tag{4}\]  
If the removal volume is maximum, the machined surface is with concave profile. Similarly, if the workpiece has a convex surface, the removal volume is minimum.