## 2 Materials and Methods
### Settings of EDEM Contact Model and Theoretical Basis  
The adequate selection of a contact model is the prerequisite for the effective development of the EDEM discrete element simulation model. The contact between quartz sand particles was set as rigid contact without plastic deformation. The contact between spherical particles and particles will generate normal overlap \(\xi\). The calculation formula is as follows:  
\[\xi=R_{1}+R_{2}-|r_{1}-r_{2}| \tag{1}\]  
where \(R_{1}\) and \(R_{2}\) are the radius of two spherical particles, and \(r_{1}\) and \(r_{2}\) are the center position vectors of two particles. The normal contact force \(F_{\mathrm{n}}\) and the tangential contact force \(F_{\mathrm{t}}\) generated by friction will be generated by the contact of two spherical particles, where the calculation formula of \(F_{\mathrm{n}}\) is as follows:  
\[F_{\mathrm{n}}=S_{\mathrm{n}}\xi \tag{2}\]  
where \(S_{\mathrm{n}}\) is the normal equivalent stiffness of the particle contact surface, which could be expressed as:  
\[S_{\mathrm{n}}=\frac{4 aG}{1-v} \tag{3}\]  
where \(G\) and \(v\) are the shear modulus and Poisson' s ratio of particles, respectively, and \(a\) is the radius of the particle contact surface. The calculation formula could be expressed as follows:  
\[a=\frac{3\pi(1-V)}{2n(1-\phi)}R \tag{4}\]  
where \(n\) is the number of contact surfaces of a single particle, \(\phi\) is the equivalent porosity of the medium, and \(R\) is the local mean curvature radius of the contact surface. For an ideal spherical particle, the \(R\) could be expressed as:  
\[R=\frac{R_{1}\times R_{2}}{R_{1}+R_{2}} \tag{5}\]  
Since the tangential contact force \(F_{\mathrm{t}}\) is relatively complicated, its size depends on the loading history but is not entirely determined by the relative position of particles. For example, the normal contact force increment \(\Delta F_{\mathrm{n}}\) and the tangential contact force increment \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{Quartz sand} & Shear modulus/GPa & 0.2 \\  & [PERSON]’s ratio & 0.25 \\  & Density/kg\(\cdot\)m\({}^{-3}\) & 2350 \\ \hline \multirow{3}{*}{PLA} & Shear modulus/GPa & 1.4 \\  & Poisson’s ratio & 0.35 \\  & Density/kg\(\cdot\)m\({}^{-3}\) & 1248 \\ \hline \multirow{3}{*}{Contact between quartz sand particles} & Coefficient of restitution & 0.3 \\  & Static friction factor & 0.5 \\  & Rolling friction factor & 0.01 \\ \hline \multirow{3}{*}{Contacts between quartz sand and PLA} & Coefficient of restitution & 0.2 \\  & Static friction factor & 0.5 \\ \cline{1-1}  & Rolling friction factor & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties and contact parameters. \(\Delta F_{\rm t}\) are used to describe the loading history of particles, and the tangential contact force \(F_{\rm t}\) can be calculated by the following equations:  
\[\Delta F_{\rm t}=S_{\rm t}\Delta\xi,S_{\rm t}=\frac{8 aG}{1-v} \tag{6}\]  
\[\Delta F_{\rm t}=\mu_{f}\Delta F_{\rm n} \tag{7}\]  
\[F_{\rm max}^{S}>|\Delta F_{\rm t}|=\mu_{f}\Delta F_{\rm n} \tag{8}\]  
where \(S_{\rm t}\) is the tangential equivalent stiffness, \(\Delta\xi\) is the tangential displacement increment, \(\mu_{f}\) is the friction coefficient, and \(\Delta F^{S}\)max is the static friction force. When the static friction force is greater than the tangential contact force increment, the particles do not cause micro sliding on the contact surface. Therefore, the Hertz-Mindlin no-slip contact model was selected as the contact model between quartz sand particles [46; 47]. When quartz sand particles were in contact with the test samples, since a ploughing effect from the abrasive materials existed, the test surfaces were subject to micro shearing, ploughing and cutting. Therefore, grooved grinding marks were generated. In this study, the Hertz-Mindlin with Archard Wear model was selected as the contact model between quartz particles and the surface of the test sample [48].