# Analytical Modeling of Particle Scratching Process
## 3 Result and Discussion
### Normal Stress Distribution  
Normal stress distribution of the contact zone between the particle and workpiece can be obtained based on the analytical model when all parameters are determined, including nominal depth of cut \(h_{doc}\), coefficient of elastic rebounds \(k_{1}\), coefficient of plastic upheaval \(k_{2}\), radius of particle \(r\), yield strength \(\sigma_{y}\), and coefficient of strain strengthening \(n\) of the material. Shear stress distribution will be obtained based on normal stress distribution and considering the coefficient of friction. The normal stress distribution on the elastic deformation zone conforms to the Hertz contact theory, which is common, and Equation (7) provides a clear indication. The normal stress on the plastic deformation zone can be calculated according to Equations (16) and (17). All parameters have been considered to draw a series of normal stress distribution curves in the plastic deformation zone, which also express the effect of these parameters on normal stress distribution. All curves have been classified, and the parameters for each specific curve are listed in Table 1.  
The curves were drawn in three coordinate systems according to the classification of Table 1. The results are shown in Figure 4.  
As shown in Figure 4, the normal stress increases with the increase in \(t\) in the plastic deformation zone, and the rate of increase decreases gradually. The maximum normal stress appears at the separation point of the chip and the particle, as well as the shear stress. Therefore, the wear degree of separation point is the largest. As shown in Figure 4a, the normal stress increases with the increase in yield strength \(\sigma_{y}\) and the coefficient of strain strengthening \(n\). The greater yield stress, the greater normal stress under the same deformation levels, and the greater strain strengthening, the greater the growth rate of normal stress with the increase in strain, which leads to the phenomenon shown in Figure 4a. As shown in Figure 4b, the normal stress increases with the increase in rebound height \(h_{1}\), and at the starting point of curves \(P_{1}(k_{1}=0.5)\), \(P_{2}\) (\(k_{1}=0.3)\), and \(P_{3}(k_{1}=0.1)\), the values of normal stress are equal. At the end of the curves, the value decreases with the increase in the coefficient of elastic rebound \(k_{1}\); for example, the normal stress decreases from 650.6 MPa to 577.8 MPa when the coefficient of elastic rebound \(k_{1}\) increases from 0.1 to 0.5. The reason is that the increase in elastic rebound leads to an increase in the origin length of plastic deformation defined in Section 2.1, in return, leading to a decrease in maximum strain. The change in coefficient of plastic upheaval \(k_{2}\) just changes the length of normal stress distribution curves, and the value of the curves does not change; for example, curves \(P_{4}\) (ab) and \(P_{2}\) (abc) are included in curve \(P_{5}\) (abcd), because the coefficient of plastic \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Curves** & \(\sigma_{y}\)**/MPa** & \(N\) & \(h_{doc}\)/\(\mu\)**m** & \(r\)**/mm** & \(k_{1}\) & \(k_{2}\) \\ \hline \(L_{1}\) & 200 & 0.5 & 30 & 0.1 & 0.3 & 0.3 \\ \(L_{2}\) & 500 & 0.5 & 30 & 0.1 & 0.3 & 0.3 \\ \(L_{3}\) & 800 & 0.5 & 30 & 0.1 & 0.3 & 0.3 \\ \(L_{4}\) & 500 & 0.2 & 30 & 0.1 & 0.3 & 0.3 \\ \(L_{5}\) & 500 & 0.8 & 30 & 0.1 & 0.3 & 0.3 \\ \(P_{1}\) & 500 & 0.5 & 30 & 0.1 & 0.3 & 0.3 \\ \(P_{1}\) & 500 & 0.5 & 30 & 0.1 & 0.5 & 0.3 \\ \(P_{1}\) & 500 & 0.5 & 30 & 0.1 & 0.3 & 0.1 \\ \(P_{1}\) & 500 & 0.5 & 30 & 0.1 & 0.3 & 0.5 \\ \(Q_{1}\) & 500 & 0.5 & 30 & 0.1 & 0.3 & 0.3 \\ \(Q_{2}\) & 500 & 0.5 & 40 & 0.1 & 0.3 & 0.3 \\ \(Q_{3}\) & 500 & 0.5 & 50 & 0.1 & 0.3 & 0.3 \\ \(Q_{4}\) & 500 & 0.5 & 30 & 0.2 & 0.3 & 0.3 \\ \(Q_{5}\) & 500 & 0.5 & 30 & 0.3 & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of normal stress distribution curves on plastic deformation zone.