## 6 Numerical Validation  
The numerical validation of the mathematical model in this section was conducted using MATLAB; R201b. It permitted the identification of the impact of the \(11\) parameters on the \(PTF\). The evaluation of the real value of [PERSON]'s modulus allowed for a more realistic numerical simulation. The composite material properties used in the experimental validation were obtained using specimens according to ISO 527-4 [21]. According to ISO 527-4, specimens of rectangular plastic molds were prepared. The prepared mixture of carbon composite material was then molded inside. When the curing process finished, the specimens were separated from the molds and fixed on a universal testing machine. The rectangular part was fixed on both sides of the machine's jaw. The upper jaw moves upward. The applied force to move the jaw and the distance allow the calculation the  
Figure 7: Force \(F\) and torque \(T\) applied on the intermediate. Pressure created at the interference fit is represented by \(P\).  
stresses generated in the specimen. The machine's software generates the Young's modulus, the strain, and the stress at fracture, which were set as a limit in the numerical simulation of the PTF. The properties of the 10% carbon composite are given in Table 1.  
As shown in Figure 8, a numerical simulation was conducted to determine the effect of the nominal diameter \(d\) and the composite thickness \(e=R_{02}-R_{l2}\) on the \(PTF\) value that the EPATH assembly can support. The intermediate's material was steel. For this purpose, the von Mises stresses were calculated to set the maximum stresses that the composite material can support. If the von Mises value is lower than the maximum stress the composite material can support, then the parameters governing the interference fit can be considered acceptable. The acceptable range of \(PTF\) is marked inside the red area. It can be noticed that the thickness of the composite \(e\) and the nominal diameter \(d\) play a major role. It is obvious that, when \(e\) increases, \(PTF\) increases. Also, at the same nominal diameter, \(PTF\) becomes constant after a certain value of \(e\). This allowed for the optimization of the design of the robotic part and avoided over-dimensioning.  
The numerical simulation of the von Mises stresses and the Push Through Force for the 10% volume fraction coefficient of the carbon particle is plotted in \((a,b)\) for steel intermediates, in \((c,d)\) for brass intermediates, and in \((e,f)\) for aluminum intermediates is and shown in Figure 9. The limit line of the admissible von Mises stresses was drawn based on the values of the maximum stresses found in the traction test (Table 1). It can be noticed that the Push Through Force (\(PTF\)) is higher when the interference fit value (\(\delta_{r}\)) is higher and the nominal diameter (\(d\)) is lower. A combination of \((\delta_{r},d)\), which creates stresses above the limit line, causes plastic deformation of the composite material or even breaks it. Therefore, the choice of \((\delta_{r},d)\) was made from the acceptable area in the graph.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Characteristic** & **Value** & **Unit** \\ \hline \(v_{f}\) & 10 & \% \\ Stress at fracture & 28 & MPa \\ Theoretical Young’s modulus & 2.6 & GPa \\ Experimental Young’s modulus & 2.66 & GPa \\ Young’s modulus deviation & 2.3 & \% \\ Strain at fracture & 0.6 & \% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Carbon composite characterization and traction test results.  
Figure 8: \(PTF\) represented as a function of the nominal diameter \(d\) and the thickness of the composite material \(e\) for a 10% volume fraction coefficient and 0.04 mm radial interference fit. The intermediate used for the simulation was made of steel. The acceptable range of \(PTF\) is marked inside the red area.