## 3 Numerical Modelling
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline Density & 1549 kg/m\({}^{3}\) \\ Poisson ratio & 0.45 \\ Young modulus & 32 GPa \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Isotropic material properties of the 3D model. \begin{table}
\begin{tabular}{c c} \hline \hline Nodes & 40,131 \\ Elements & 32,640 \\ DoFs & 240,786 \\ Hexa along the thickness & 6 \\ Hexa along the width & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the 3D model. In Table 5, MAC values between corresponding modes are also reported. The last two modes of Table 5 are characterised by a MAC index lower than 70%, which is here considered as benchmark [45] for eigenvectors agreement. Consequently, in the simulation, the fifth and the sixth EMA mode shapes are not accurately reproduced in FEA. Considering the discrepancy in terms of mode order between the FEA and EMA results, the assumption of isotropic material properties is not reliable in this case study. Therefore, an anisotropic material is introduced in the Finite Element model. In Nastran, for linear structural analyses, the MAT9 definition [46] allows to assign orthotropic material properties to 3D solid elements. Since the prosthetic material is carbon fibre composite, the definition of orthotropic properties is reasonable [47; 48; 49]. The fibres confer strength along their length, meaning that the orientation and length of the fibres directly influence the behaviour of the composite material. In this case, the internal composition of the prosthetic was disclosed by Ossur company. Therefore, for simplicity, the material is not defined layer by layer, but the overall properties along the local reference frame of each element are considered. The stress-strain deformation for an orthotropic material follows Equation (4) [50]:  
\[ \begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \gamma_{23}\\ \gamma_{31}\\ \gamma_{12}\end{bmatrix} = \begin{bmatrix}\frac{1}{C_{1}}&-\frac{v_{21}}{C_{2}}& -\frac{v_{31}}{C_{3}}&0&0&0\\ -\frac{v_{12}}{C_{1}}&-\frac{v_{22}}{C_{2}}&-\frac{v_{32}}{C_{3}}&0&0&0\\ -\frac{v_{12}}{C_{1}}&-\frac{v_{22}}{C_{2}}&\frac{v_{3}}{C_{3}}&0&0&0\\ 0&0&0&\frac{1}{C_{23}}&0&0\\ 0&0&0&0&\frac{1}{C_{13}}&0\\ 0&0&0&0&0&\frac{1}{C_{12}}\end{bmatrix} \cdot \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \tau_{23}\\ \tau_{31}\\ \tau_{12}\end{bmatrix}  \tag{4}\]  
where subscripts \(1\), \(2\), and \(3\) correspond to the main directions of the local reference frame of the hexagonal elements. * \(E_{1}\), \(E_{2}\), and \(E_{3}\) are, respectively, the longitudinal and transversal Young moduli;  
Figure 8: MAC matrix, comparison between EMA and FEA results: isotropic (**a**) and orthotropic (**b**) models. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(f_{\text{FEA}}\) [Hz] & \(f_{\text{EMA}}\) [Hz] & **Mode Shape** & **MAC [\%]** \\ \hline
87.96 & 86.33 & 1 st bending \(XZ\) & 87.96 \\
401.32 & 226.49 & Torsion \(RZ/1\)st bending \(XY\) & 79.76 \\
263.94 & 275.25 & 2 nd bending \(XZ\) & 76.43 \\
499.60 & 371.77 & Torsion \(RZ/2\)nd bending \(XY\) & 95.13 \\
489.24 & 603.34 & 3 rd bending \(XZ\) & 57.74 \\
1021.20 & 735.60 & Torsion \(RZ/3\)rd bending \(XY\) & 55.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Natural frequencies and MAC values between EMA and FEA (isotropic material): mode shapes are compared following the ascending order of EMA frequencies. * \(v_{12}\) and \(v_{21}\) are the in-plane Poisson ratios;
* \(v_{31}\), \(v_{13}\), \(v_{23}\), and \(v_{32}\) are the out-of-plane Poisson ratios;
* \(G_{12}\) is the in-plane shear modulus;
* \(G_{13}\) and \(G_{13}\) are the out-of-plane shear moduli. In order to respect symmetry in the matrix, it is necessary that  
\[\frac{v_{12}}{E_{1}}=\frac{v_{21}}{E_{2}} \tag{5}\]  
\[\frac{v_{13}}{E_{1}}=\frac{v_{31}}{E_{3}} \tag{6}\]  
\[\frac{v_{23}}{E_{2}}=\frac{v_{32}}{E_{3}} \tag{7}\]  
Given the high number of parameters, some assumptions are made.