# Analytical Modeling of Particle Scratching Process
## 2 Physical Model and Analytical Model of Scratching Process
### Analytical Model  
Forces applied to the particle mainly come from material compressive stress and friction. The resultant forces in the \(Z\) direction are 0 due to the symmetrical particle geometry. There are two work-material deformation zones, namely, the elastic deformation zone _F-HEIG_ and the plastic deformation zone _HEI-JBK_, which are mentioned in Section 2.1. The forces applied on the particle are obtained by integrating the compressive stress and friction stress on the surface of zone _F-HEIG_ and zone _HEI-JBK_, respectively. As shown in Figure 3a, compressive stress and shear stress produced by friction are applied to the abrasive surface. \(h_{doc}\) is the nominal depth of cut, \(h_{1}\) is the rebound height, and \(h_{2}\) is the wedge height of side flow. \(h_{3}\) is the actual depth of the groove. h\({}_{3}\) is expressed as:  
\[h_{3}=h_{doc}-h_{1} \tag{3}\]  
In order to calculate the forces applied on the particle, the \(t\)-axis is built whose origin is the \(F\) point and the positive direction is the same as the direction of the \(Y\)-axis. Figure 3: Analytical model two-body abrasive wear. (**a**) Side view. (**b**) Top view. As shown in Figure 3b, in elastic deformation zone _F-HEIG_, the element of area dS1' can be expressed as:  
\[dS_{1}^{{}^{\prime}}=r^{{}^{\prime}}\cdot dS\cdot\left( \begin{matrix}dt\\ \sin\theta\end{matrix} \right) \tag{4}\]  
where \(r^{\prime}\) is the section-radius of the element in zone _F-HEIG_, which is perpendicular to the _Y_-axis, \(r^{\prime}=r\cdot sin\theta\). \(\theta\) is the included angle between line \(OO_{1}\) and the line connecting point \(O\) and an arbitrary element in zone _F-HEIG_. In element area \(dS_{1}\), the force produced in the \(X\) direction and \(Y\) direction are expressed as:  
\[dF_{X_{F-HEIG}}=4\int_{0}^{\pi/2}\left(\tau_{1}\cdot\cos\delta\cdot dS_{1}^{{}^ {\prime}}\right)d\delta \tag{5}\]  
\[dF_{Y_{F-HEIG}}=4\int_{0}^{\pi/2}\left(\sigma_{1}\cdot\cos\theta\cdot dS_{1}^{{}^ {\prime}}\right)d\delta \tag{6}\]  
where \(\sigma_{1}\) is the pressed stress of an arbitrary element in zone _F-HEIG_, and \(\sigma_{1}\) can be expressed as:  
\[\sigma_{1}=\sigma_{Y}\cdot\frac{arccoa(1-h_{1}/r)-arccoa(1-t/r)}{arccoa(1-h_{1 }/r)} \tag{7}\]  
\(\tau_{1}\) is the shear stress of an arbitrary element in zone _F-HEIG_, and \(\tau_{1}\) can be expressed as:  
\[\left\{ \begin{array}{l}\tau_{1}=\mu\sigma_{1},\mu\sigma_{1}<\tau_{\max}( \text{sliding zone})\\ \tau_{1}=\tau_{\max},\mu\sigma_{1}\geq\tau_{\max}(\text{sticking zone})\end{array} \right. \tag{8}\]  
\(cos\theta\) and \(sin\theta\) can be expressed as:  
\[\cos\theta=\frac{(r-t)}{r} \tag{9}\]  
\[\sin\theta=\sqrt{r^{2}-\left(r-t\right)^{2}_{r}} \tag{10}\]  
In zone _F-HEIG_, the force produced in the \(X\) direction and \(Y\) direction are expressed as:  
\[F_{X_{F-HEIG}}=\int_{0}^{h_{1}}dF_{X_{F-HEIG}} \tag{11}\]  
\[F_{Y_{F-HEIG}}=\int_{0}^{h_{1}}dF_{Y_{F-HEIG}} \tag{12}\]  
In plastic deformation zone _HEI-JBK_, the element of area \(dS_{2}\)' can be expressed as:  
\[dS_{2}^{{}^{\prime}}=r^{{}^{\prime\prime}}\cdot d\psi\cdot\left( \begin{matrix} dt\\ \sin\gamma\end{matrix} \right) \tag{13}\]  
where \(r^{\prime\prime}\) is the section-radius of the element in zone _F-HEIG_ which is perpendicular to the _Y_-axis, \(r^{\prime\prime}=r\cdot sin\gamma\), and \(\gamma\) is the included angle between line \(OO_{1}\) and the line connecting point \(O\) and an arbitrary element in zone _HEI-JBK_.