# Effect of the Rolling Friction on the Heap Formation of Dry and Wet Coarse Discs
## 2 Materials and Methods  
### Contact Force Model  
This study is based on DEM proposed by [PERSON] and [PERSON] [4]. The Newton second law defines each particle's movement, and particle motion is the result of its interaction with neighboring particles or walls and the action of other external forces. With an appropriate time step, all forces and torques can be determined from the interactions between the particle and its immediate neighbors [23]. We selected a time step equal to 0.01 of the critical time step which depends on the normal stiffness, the damping coefficient, and the particle mass [6]. \[\Delta t_{c}=\frac{\pi}{\sqrt{\frac{S_{n}}{m}(1-z_{n}^{2})}}\]  
where \(\Delta t_{c}\) is the critical time step, \(S_{n}\) the normal stiffness, \(m\) the particle mass, and \(z_{n}\) the dimensionless damping coefficient. Particles are allowed to have small overlaps while in contact. The translational and rotational movements of particle \(i\) which is in contact with \(k\) particles at time \(t\) can be described by the following equations:  
\[m_{i}\frac{dV_{i}}{dt}=m_{i}g+\sum_{j=1}^{k}F_{ij} \tag{1}\]  
\[l_{i}\frac{d\omega_{i}}{dt}=\sum_{j=1}^{k}T_{ij} \tag{2}\]  
where \(m_{i}\) represents the particle \(i\) mass, \(\mathbf{V}_{i}\) the particle velocity, \(m_{i}g\) the gravitational force, and \(\mathbf{F}_{ij}\) the sum of interaction forces. The interaction forces include the contact forces and the viscous damping forces. Equation (2) describes the rotation caused by the interaction forces. \(l_{i}\) is the moment of inertia, \(\mathbf{\omega}_{i}\) the particle angular velocity, and \(T_{ij}\) the sum of torques which involve the torque caused by the tangential forces acting at the contact point and the rolling friction torque \(M_{i}\) which oppose the movement of the particle. Figure 1 shows the interparticle forces between two particles in contact. All these forces and the simulation results depend heavily on the contact model. Our model uses the nonlinear Hertz-Mindlin contact force model for dry particle simulation and the Hertz-Mindlin-JKR with a capillary force for the wet particle simulation [17]. The contact model has a viscous part in both normal and tangential directions. The normal force is determined according to the Hertz contact theory [24], Mindlin-Dereciewicz theory [25] is used to calculate the tangential force, and a dashpot is adopted to describe the dissipative mechanism. The Coulomb friction law limits the tangential frictional force. Because of its accuracy and efficiency, it is used by many researchers [3; 17; 26] to study the granular materials system. The normal force is divided into two parts: the repulsive force and the damping force. The repulsive force depends on the normal overlap. The damping force varies linearly with the relative velocity. Equation (3) gives the normal force acting on particle \(i\) in contact with particle \(j\):  
\[F_{n}=\frac{4E^{*}\sqrt{R^{*}}}{3}{\delta_{n}}^{3/2}-2\sqrt{\frac{5}{6}}\beta \sqrt{S_{n}m^{*}}v_{n} \tag{3}\]  
\[S_{n}=2E^{*}\sqrt{R^{*}\delta_{n}} \tag{4}\]  
Figure 1: Forces acting on particle \(i\) during contact. where \(S_{n}\) is the normal stiffness, \(E^{*}\), \(R^{*}\), and \(m^{*}\) are the equivalent Young Modulus, radius, and mass defined as:  
\[\frac{1}{E^{*}}=\frac{1-{v_{i}}^{2}}{E_{i}}+\frac{1-{v_{j}}^{2}}{E_{j}}\frac{1}{ R^{*}}=\frac{1}{R_{i}}+\frac{1}{R_{j}}\frac{1}{m^{*}}=\frac{1}{m_{i}}+\frac{1}{m_{j}}\]  
where \(E_{i}\), \(E_{j}\), \(v_{i}\), \(v_{j}\), \(R_{i}\), \(R_{j}\), \(m_{i}\), and \(m_{j}\) are [PERSON]'s modulus, Poisson ratio, radius, and mass of particles \(i\) and \(j\). \(v_{n}\) the relative normal velocity and \(\delta_{n}\) is the normal overlap (See Figure 2). For walls, the radius is considered infinite. \(\beta\) is the dimensionless damping coefficient calculated from the coefficient of restitution \(e\) given by \(\beta=\frac{\ln e}{\sqrt{\pi^{2}+ln^{2}e}}\).