## 4 Behavior of the Ski during a Carving Turn  
To study the behavior of the ski during a downhill turn, the ski-snow interaction had to be modeled. For doing this, the [PERSON] [19] model of snow was used. It is based on experimental compression tests on big frozen snow cylindrical samples. [PERSON] came up with a relation that described the snow as a piece-wise linear function as shown in Figure 16. The snow was characterized by an initial linear behavior in which the elastic constant assumed a value that varied between \(\text{E}_{\text{tan}}=115\) (old snow) and \(\text{E}_{\text{tan}}=160\) MPa (artificial snow). For higher strains, the behavior could be still described with a linear relation in which elastic module assumed a value between \(\text{E}_{\text{res}}=9\) and \(\text{E}_{\text{res}}=35\) MPa.  
The previously developed and validated solid model of the ski was improved by adding the interaction with the snow. The ski was tested for different roll angles and loads. The roll angle describes the rotation of the ski along its longitudinal axis with respect to the ground. Table 4 shows the studied combinations.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Roll Angle [\({}^{\circ}\)]** & **Force Applied [kg]** & **Maximum Deflection [mm]** & **Maximum Snow Penetration [mm]** \\ \hline
10 & 50 & 2.85 & 8.26 \\
10 & 100 & 2.88 & 8.29 \\
10 & 200 & 2.94 & 8.35 \\
30 & 50 & 9.00 & 12.82 \\
30 & 100 & 9.07 & 12.89 \\
30 & 200 & 9.15  
Simulations were performed with a non-linear solver using a large strain formulation. The snow was modeled using a von [PERSON] elastoplastic law with linear isotropic hardening. The contact was modeled as discrete. The models of contact and friction were drawn up from already discretized quantities, i.e., displacements and nodal forces. The problem of contact/friction was solved by uncoupling it from the problem of the equilibrium of structure.  
Figure 17 shows the deformed ski under load. Figures 18 and 19 show the deformed shape of the ski along the symmetry plane for several levels of load and different roll angles. These results were fundamental to understand the downhill behavior of the ski: The combination of its carving, roll angle, and load applied determined the turning angle (assuming a conductive turn was done, i.e., without sliding between the ski and the snow). The actual stiffness of the ski was fundamental in this regard: A stiffer ski would require a much higher load to ensure the same turning angle (at equal carving and roll angle). Therefore, skis having the same geometry should have different stiffnesses according to the weight of the athlete. Numerical simulations could be an effective tool for designing the perfect ski.  
Figure 20 shows the relation between the roll angle and the turning radius. The relation is well described by a decreasing power function.  
\[r_{turn}=1529.5\cdot\vartheta_{roll}^{-1.129} \tag{3}\]  
The applied load seemed to have a moderate effect on the ski deflection compared to the roll angle. With a roll angle of 45\({}^{\circ}\), in the hypothesis of the conduction turn, the ski showed a turning radius of about 22 m. The geometrical rounding radius of the ski was 40 m. For lower roll angles (30\({}^{\circ}\) and 10\({}^{\circ}\)), the rounding radius results equaled to 34 and 111 m, respectively. At a roll angle of 60\({}^{\circ}\), the turning radius decreased to 14 m.  
Figure 21 shows the von Mises stress in the cross-section of the ski for different roll angles and loads. Differently from what was observed during the three-point-bending and torsion-bending tests, the interaction with the snow prevented big deformations and the maximum stress level results were far below the yielding for each layer.  
Figure 19: Deflection of the ski for different roll angles and 200 kg.  
Figure 18: Deflection of the ski for different roll angles and loads.