## 2 Materials and Methods
### Ski
#### 2.3.2 Solid Model  
The same geometrical simplifications of the shell model were also used for the 3D simulations (symmetry for the three-point-bending model and cutting of tail and tip for both models). Moreover, additional assumptions have been made in order to obtain all the 9 constants necessary for the proper description of the materials in the space. According to [21], materials with unidirectional long fibers have the same elastic module in each radial direction (perpendicular to the reinforcing fibers). The characterization tests were performed on thin laminates and it was not possible to characterize the remaining out of plane properties. The shear module G and the Poisson's ratio \(\
u\) founded in the xy plane were used also for the other 2 directions (zx and yz). While in the 3D modeling approach some additional hypothesis were introduced for the material properties, from a geometrical point of view the 3D model (Figures 5 and 6) better reproduced the real shape of the ski, as it also included the lateral protective layer that was neglected in the shell approach. Figure 4: Shell models used for the three-point-bending (left) and torsion-bending (right) simulations. Figure 5: Isometric view of the solid model used for the three-point-bending test. Figure 11: Material 290 tensile tests. Figure 10: Material 24 tensile tests. Figure 9: Material 207 tensile tests. Table 1 reports the in-plane parameters for each material. Table 2 reports the maximum displacement in the midpoint of the ski when bended under a load of 120 N in the three-point-bending tests. Measured values were compared with those obtained from the two different FEM analyses (with shell and solid elements). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{9}{c}{**Material**} \\ \cline{2-10}  & **9** & **24** & **139** & **207** & **290** & **135** & **31** & **86** & **217** \\ \hline \(\mathbf{E_{x}}\)**[MPa]** & 550 & 35,000 & 800 & 850 & 20,000 & 4000 & 2000 & 13,700 & 35,000 \\ \(\mathbf{E_{y}}\)**[MPa]** & 567 & 8700 & 750 & 897 & 13,700 & 569 & 127 & 420 & 12,436 \\ \(\mathbf{v_{xy}}\)**[-]** & 0.45 & 0.23 & 0.45 & 0.42 & 0.18 & 0.39 & 0.39 & 0.40 & 0.56 \\ \(\mathbf{v_{yx}}\)**[-]** & 0.43 & 0.35 & 0.38 & 0.30 & 0.12 & 0.39 & 0.39 & 0.40 & 0.48 \\ \(\mathbf{G_{xy}}\)**[MPa]** & 776 & 35,010 & 1097 & 1291 & 29,125 & 3009 & 1184 & 10,086 & 25,327 \\ \(\mathbf{\sigma_{x}}\)**[MPa]** & 11 & 152 & 10 & 6 & 110 & 98 & 87 & - & - \\ \(\mathbf{\sigma_{y}}\)**[MPa]** & 10 & 9 & 13 & 6 & 57 & 22 & 21 & - & - \\
**Thickness [mm]** & 0.35 & 0.35 & 0.50 & 1.10 & 1.20 & 0.55 & 0.55 & 0-5.45 & 0.5 \\ \hline \hline \end{tabular}  
* The sub-indexes x and y refer to the longitudinal and axial ski direction, respectively.  
\end{table} 
Table 1: Material elastic properties. Figure 12: Material 31 tensile tests. Figure 13: Material 135 tensile tests. Table 3 reports the results of the torsion-bending tests (40N @ 0.45m) and the related numerical predictions. Figure 14 shows the normal stresses in the thickness of the ski (on the symmetry plane and the position where the load was applied--0 corresponds to the bottom part) during the three-point-bending test. Figure 15 refers to the bending-torsion test. Both tests were repeated increasing the load up from the first yielding. The most loaded layer was the n. 217, in which the stress reached a value of about 32 MPa both in the three-point-bending and in the bending-torsion test. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Displacement [mm]** & **Simulation Time * [min]** & **Error [\%]** \\ \hline Experimental & 40 & & \\ Shell & 41 & 35 & 2.5\% \\ Solid & 44 & 60 & 10.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the maximum deflection of the ski in the three-point bending test: Finite element analysis (FEM) vs. experiments. Figure 14: Solid model results—three-point-bending test (normal and [PERSON] stresses in the section). Figure 15: Solid model results—torsion-bending test (normal and [PERSON] stresses in the section).