## 2 Numerical Modeling and Analysis
### Analysis Results  
The analysis described above allows for the study of the displacements in the three axes over the entire surface of the brace, the developing stresses in the brace, which are critical for optimizing the required brace thickness, as well as the contact pressures (Figure 7) developing between the brace and the body, which are needed to evaluate the patient's comfort. The results are illustrated in the following figures:  
Regarding the displacements in the \(X\)-axis, a maximum displacement of 14.28 mm is observed on the left rear side of the brace, which is not localized but extends almost uniformly over the entire side of the rear face of the spinal brace (Figure 8a). On the \(Y\)-axis the maximum displacement is 26.53 mm (blue color indication), as illustrated in Figure 8b. This movement is localized in a small area on the upper right side of the rear face of the brace. Finally, on the \(Z\)-axis the maximum displacement is 16.21 mm, identified by a red colored indicator on the lower right side of the rear face of the brace (Figure 8c). A maximum von Mises stress equal to 18.15 MPa is detected, which is locally reflected by a red colored indication at the critical areas of the spinal brace without exceeding the tensile strength of the material, which is in the order of 30-40 MPa. Stresses also develop on the front face of the brace, ranging from 1.30 MPa to 10.60 MPa. Figure 7: Contact pressures on the surfaces representing the human body (First numerical model). Furthermore, analyses of the brace for different values of the modulus of elasticity of the material have been conducted, with values of (a) E = 2340 MPa, (b) E = 1520 MPa, and (c) E = 320 MPa. In Figure 10 the developing stresses of the brace for various values of the brace cross-section thickness are shown. The maximum values of observed von Mises stresses and displacements are listed in Tables 3 and 4, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Section Thickness**} & \multicolumn{3}{c}{**Displacements**} \\ \cline{2-4}  & \(X\)**-Axis** & \(Y\)**-Axis** & \(Z\)**-Axis** \\ \hline t = 5 mm & 13.65 mm & 25.81 mm & 14.74 mm \\ t = 4 mm & 14.28 mm & 26.53 mm & 16.21 mm \\ t = 3 mm & 15.11 mm & 27.41 mm & 18.01 mm \\ t = 2 mm & 16.32 mm & 27.64 mm & 20.35 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Max displacements in the three axes for various values of the brace cross-section thickness. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{1}{c}{**Max von Mises Stress**} & \multicolumn{1}{c}{**Modulus of Elasticity E**} \\ \hline S\({}_{\text{max}}\) = 40.36 MPa & E = 2340 MPa \\ \hline S\({}_{\text{max}}\) = 26.91 MPa & E = 1520 MPa \\ \hline S\({}_{\text{max}}\) =18.15 MPa & E = 1000 MPa \\ \hline S\({}_{\text{max}}\) = 6.17 MPa & E = 320 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Max von Mises stresses for various values of modulus of elasticity of the brace material. Figure 9: Von Mises stress results (MPa) for brace with thickness: (a) t = 5 mm, (b) t = 4 mm, (c) t = 3 mm, (d) t = 2 mm. Interesting conclusions can be drawn regarding the effect of the thickness of the cross-section of the brace and the modulus of elasticity of its material. It is observed that as the thickness increases, a local increase in stresses is observed. As can be observed in Figure 9a, for t = 5 mm the maximum stress is 20.29 MPa, for t = 4 mm it is 18.15 MPa, for t = 3 mm 16.60 MPa, and for t = 2 mm 14.06 MPa. This can be interpreted by considering that for thicker, hence stiffer, braces, a local concentration is observed, while an overall decrease in stresses in the whole brace takes place. According to Table 2, the maximum displacements in the \(X\)-, \(Y\)-, and \(Z\)-axes decrease as the thickness increases. For thickness t = 5 mm the maximum displacement is 13.65 mm in the \(X\)-axis, 25.81 mm in the \(Y\)-axis and 14.74 mm in the \(Z\)-axis.