# Effect of the Rolling Friction on the Heap Formation of Dry and Wet Coarse Discs
## 2 Materials and Methods
### Rolling Friction  
Rolling friction torque is added to the present model to study the heap formation of a soil sample. Rolling friction is a dimensionless coefficient defined as the force that resists the movement of a particle rolling on a surface [29]. In some cases, it can be taken as the angle of internal friction [30]. In this case, the granular material should have a uniform particle size, density, and moisture content. Some researchers consider the rolling friction as a coefficient with a unit of length [31]. When a spherical particle is in rotation on a surface, the contact between it and the surface generates a force \(R_{w}\) that does not pass through the sphere's center. The eccentricity between the direction of the force and the vertical line passing through the particle center is considered the rolling friction coefficient (Figure 3). However, it can be decomposed into two parts: a length parameter and a dimensionless parameter [22]. For non-spherical particles, according to [23; 32; 33], it is the eccentricity \(\varepsilon\) between the line of action of the normal contact force and the centroid of the particle (Figure 4). For the study of the heap formation of spherical or circular particles, rolling friction is of particular interest to obtain good simulation results. [PERSON] et al. (2018) proved that if the rolling friction is ignored or set to zero, even with a maximum sliding friction coefficient, the model cannot correctly predict repose angle. Instead, it produces a lower angle of repose than the one given by the experiments. It is one reason we need to implement a rolling friction torque in contact models to study heap formation. Another primary reason is the difficulty in DEM simulations to model the real shape of the granular material. With rolling friction, the effect of the particle shape is well simulated. In fact, when a particle is rolling, energy dissipation comes mainly from its shape effect, even though a little part is due to micro-sliding, visco-elasticity, plasticity, and surface adhesion (2018). Rolling friction simulates the particle shape by opposing the rolling and dissipating the rolling energy of the particle as its natural shape would do. Thus, it is a good alternative to give us realistic results with simple particle shape by introducing a \"shape like\" behavior to a particle (2018). However, rolling friction torque is always resistant, while the torque induced by particle shape can either resist the movement or favor it. The particle shape's simplicity further reduces the computer resource consumption that is very important when dealing with thousands of particles. [PERSON] et al. (2018) classify the rolling friction models into four groups called model A, model B, model C, and model D. In this study, only models A, B, and C will be investigated. Model A, or directional constant torque model, applies a resistant torque to both particles in contact. That torque is always against the relative rolling direction. [PERSON] (2018) used that model to study the formation of a sand pile. According to it, the rolling resistance torque is given by:  
\[T_{r}=-\mu_{r}R_{r}|F_{n}|\frac{\omega_{rel}}{|\omega_{rel}|} \tag{11}\]  
Figure 4: Forces acting on particle \(i\) during contact. Figure 3: Rolling friction mechanism on a circular particle. where \(\mu_{r}\) is the coefficient of rolling resistance, \(R_{r}\) the effective rolling radius is given by Equation (12), \(F_{n}\) is the contact normal force, and \(\omega_{rel}\) the relative angular velocity between the particles in contact. \[R_{r}=\frac{r_{t}r_{f}}{r_{i}+r_{j}} \tag{12}\]  
Model B, also called the viscous model, applies a resistant torque given by:  
\[T_{r}=-\mu_{r}R_{r}|F_{n}|\big{(}\omega_{i}r_{i}-\omega_{j}r_{j}\big{)} \tag{13}\]  
This rolling resistance torque is proportional to the translational velocity, as we can notice in the formula. In model C (elastic-plastic spring-dashpot models), the rolling friction is divided into two parts: a spring torque \(T_{r}^{k}\) and a viscous damping torque \(T_{r}^{d}\).