## 4 Numerical Analysis and Results
### Intender  
Three types of indenters were designed, and each was characterized by a different shape (Figure 3). The thickness (\(t\) = 8 mm) and width (\(d\) = 10 mm) were constant for all indenters.  
A cuboid measuring 90 \(\times\) 32 \(\times\) 8 mm was taken as the test specimen, which was assigned liver parameters for two different models of viscoplastic material by [PERSON] and [PERSON]. Based on a literature review of studies conducted by [PERSON] and [PERSON], among others, [PERSON]'s modulus values for liver were obtained [18; 19]. The value of this parameter depended on the source used and the gender of the patient. The results are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multicolumn{6}{c}{**Material Constants**} \\ \cline{2-7}  & \(\mu_{1}\) & \(\alpha_{1}\) & \(\mu_{2}\) & \(\alpha_{2}\) & \(\mu_{3}\) & \(\alpha_{3}\) \\ \hline
1 & 0.515 & 23.713 & 141.140 & 1.284 & 158.126 & 1.089 \\
2 & 0.330 & 25.052 & 6.471 & \(-\)0.099 & 92.001 & \(-\)0.040 \\
3 & 1.338 & 18.812 & 34.995 & \(-\)0.471 & 67.029 & 0.111 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material constants of the Ogden model.  
In order to represent ideal pressing, the two lateral planes were blocked in such a way that only translation (displacement) along the vertical plane was possible. Numerical simulations were carried out for the proposed three different penetrator shapes, which were assigned the material constants of steel in the elastic range (Table 5). Hex type elements sized at 0.5 mm were used for all components so that, on contact, the size of the elements would be the same. A demonstration of the geometry of the simulated flat penetrator is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Young’s modulus (Nava) & 0.270 & MPa \\ Young’s modulus (Yeh) & 0.640–2.0 & kPa \\ Tension stresses & 7.0–95.0 & kPa \\ Young’s modulus (Karimi) for women & 0.089–0.308 & MPa \\ Young’s modulus (Karimi) for men & 0.066–0.386 & MPa \\ \hline \hline \end{tabular}
\end{table}
Table 4: Mechanical parameters of the liver.  
Figure 4: Discretization of the system adopted for calculations.  
Figure 3: Shape and dimensions of indenters used: (**a**) WS—spherical indenter, (**b**) WP—flat indenter, (**c**) WZ—rounded indenter.  
Model scaling was performed by running a series of simulations for a duration of 8 s. An example of simulation runs for the Yeoh model is shown in Figure 5.  
Figure 5: Results for material data of Sample 1.  
The values of deformation and stress occurring on the tested surface were analyzed. Figure 6 summarizes the results of the S12 stress in MPa for the four parameters of the Yeoh model.  
From the numerical analyses carried out for various indenters, no significant differences were observed in the results obtained related to the mathematical model used. The difference between the results is 5%, which can be considered a very good fit. Larger differences were observed in the parameters of the constants that were used during the calculations. The individual results for different material constants are in the range of \(\pm\)32%, which may indicate a satisfactory fit. This may be due to the large discrepancy in the initial data of the rheological models. The above results were used in the following numerical simulations.