## 3 Numerical Results
### Quasi-Static Rolling
#### 3.2.2 Extension by Considering the Impact of the Friction Coefficient  
The Coulomb friction is now taken into account using the regularization method presented in Section 2. The friction coefficient impact on the quasi-static rolling is investigated.  
Figure 9 shows the displacement evolution in the contact zone for various values of \(\mu\). We observe that increasing \(\mu\) favors the stick regime. The longitudinal displacement remains constant for \(\mu\geq 0.3\) and the lateral displacement variation decreases as the coefficient of friction increases. Indeed, the tangential pressure needed to switch to the slip regime increases knowing that the structure is subjected to the same deflection. The lateral displacement maximum has moved towards the exit of the contact patch when the friction is added. Furthermore, the lateral displacement at the entry of the contact patch is no longer similar to the one at the exit of the contact patch. The vertical displacement is more or less the same since the friction is mainly acting on the tangential components.  
Figure 10 shows the normal contact pressure distribution at different stages of the contact. Similar to the frictionless contact, the contact pressure maximum is located at the entrance (resp. exit) for the the nodes located at the entry (resp. exit) of the block. Unlike the frictionless case, the distribution at the exit is different than the distribution at the entry since the friction law is characterized by a time-dependent path and the mechanical system is no longer conservative. Increasing \(\mu\) leads also to the saturation of the normal contact pressure.  
Figure 8: Time evolution of the normal contact pressure (top), the total contact force (middle) and the frequency spectrum of the contact force variation (bottom). Node selection for the contact pressure analysis: block entry \(\theta_{e}\)(—), block center \(\theta_{c}\)(—), block exit \(\theta_{s}\)(—).  
Figure 11 shows the tangential contact pressure is increasing with respect to \(\mu\) unlike the normal contact pressure where a saturation effect is noticed. The tangential contact pressure has the same evolution as the normal pressure for the nodes located at the middle of the block \(\theta=\theta_{c}\) and at the exit of the block \(\theta=\theta_{s}\); however, an increase in the norm of the tangential pressure of the leading edge at the exit of the contact patch is observed. The differences observed between the tangential pressure for various values of \(\mu\) is explained by the change of the slope in the stick regime and the change of the tangential pressure value in the slip regime.  
Figure 12 shows the total contact force variation. As for the frictionless case, the total lateral force \(F_{y}\) is null since the plane \((y=0)\) is a symmetry plane of the structure. The variation of the vertical load is a little bit greater as the coefficient of friction \(\mu\) increases until reaching an invariant evolution. In fact, \(F_{z}\) remains the same for \(\mu=(0.3-0.5-1.0)\)  
Figure 10: Time evolution of the normal contact pressure in the contact patch (\(y=\text{width}/2\), \(r=R_{e}\)) for frictionless contact (—), \(\mu=0.1\) (—), \(\mu=0.3\) (—), \(\mu=0.5\) (—) and \(\mu=1.0\) (—) (\(\theta=\theta_{e}\) (**left**); \(\theta=\theta_{e}\) (**middle**); \(\theta=\theta_{s}\) (**right**)).  
Figure 9: Time evolution of the quasi-static displacement in the contact patch (\(y=\text{width}/2\) and \(r=R_{e}\)) for frictionless contact (—), \(\mu=0.1\) (—), \(\mu=0.3\) (—), \(\mu=0.5\) (—) and \(\mu=1.0\) (—).  
This is coherent with the saturation phenomena observed for the normal contact pressure. The total longitudinal contact force amplitude decreases as we increase \(\mu\). High frequency oscillations appear for larger values of \(\mu\). These oscillations represent a consequence of the tangent pressure variation described in Figure 11.