## 4 Numerical Analysis and Results
### Head Attachments  
The stress-strain relationship of hyperelastic materials, such as silicone rubber, is usually defined as nonlinear, elastic, and incompressible [34]. In this study, a static compressive load test was performed for Smooth-On Ecoflex 00-20, which is the silicone rubber used in the experiment. The resulting curve in the stress-strain graph was nonlinear, indicating that the stiffness of this silicone material increases with the applied load. In order to correlate the stress level with the change in stiffness of silicone, the stress-strain relationship obtained from the test was fitted to a quadratic curve, which can be described by Equation (8). The stresses at each load increment in the experiment were used to calculate the corresponding strains using Equation (8). \[\sigma=147.7\varepsilon^{2}+57.14\varepsilon+0.3583 \tag{8}\]  
Here, \(\sigma\) is stress and \(\varepsilon\) is strain. The resulting strain values were used to obtain the corresponding stiffness as the slope of the square curve at each stress level. Table 9 shows the stress and strain values at each load increment. Figure 11 shows this relationship for Smooth-On Ecoflex 00-20 in graphic form. Figure 12 shows the two developed geometric shapes of the attachments: cylindrical and conical. To facilitate further analysis, the designations E_15 and D_0 are used. The distinctive feature of the proposed geometries was an indentation in the form of a ring at  
Figure 11: Stress–strain relation for Smooth-On Ecoflex 00–20 under static compressive loading. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Stress \(\sigma\) [kPa]** & **Strain \(\varepsilon\) [\%]** \\ \hline \multirow{4}{*}{Ecoflex 00–20} & 1.2875 & 0.01563 \\  & 2.5375 & 0.03498 \\ \cline{1-1}  & 3.7875 & 0.05281 \\ \cline{1-1}  & 5.0375 & 0.06943 \\ \cline{1-1}  & 6.2875 & 0.08506 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Material constants of the synthetic materials used to make the attachments [34]. the top of the base for the measuring head of the FibroScan(r) device. The elements were discretized at the 0.5 mm level using Tetra-type elements. The assigned values of material constants were taken from previous in-house studies and from the manufacturer's website. Numerical analysis was performed for two geometric shapes of attachments and three types of material: Ecoflex 00-20, Medical Gel #0, and Agar Progel at forcing frequencies ranging from 1.0-5.0 MHz. Figures 13 and 14 show the different simulation time steps for wavefront propagation at 3 MHz using a cylindrical and a conical attachment made of Ecoflex 00-20. Table 10 shows a summary of these results for Ecoflex 00-20. Tables 11 and 12 show the results of material Medical Gel #0 and Agar Progel, sequentially. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{3}{c}{**Ecoflex 00–20 Material Attachment**} \\ \multicolumn{3}{c}{**Cylindrical**} & \multicolumn{2}{c}{**Conical**} \\ \hline
**Inductor Frequency** & **Wave Velocity** & **Inductor Frequency** & **Wave Velocity** \\
**[MHz]** & **[m/s]** & **[MHz]** & **[m/s]** \\ \hline
1.0 & 843 & 1.0 & 1192 \\
1.5 & 906 & 1.5 & 1201 \\
2.0 & 969 & 2.0 & 1236 \\
2.5 & 992 & 2.5 & 1276 \\
3.0 & 1050 & 3.0 & 1320 \\
3.5 & 1082 & 3.5 & 1345 \\
4.0 & 1132 & 4.0 & 1367 \\
4.5 & 1183 & 4.5 & 1371 \\
5.0 & 1243 & 5.0 & 1399 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Summary of results for cylindrical and conical heads made of Ecoflex 00–20 material. A mechanical wave generated using a cylindrical head at a frequency of 3 MHz gradually propagates in the medium, and its amplitude decreases with time (e.g., for the first time step of 0.00016 s, the maximum displacement is 0.11 mm, and for the last 0.001 s, it has a value of 0.03 mm). It propagates along the axis of the cylinder. At 0.00016 s, the wave is at a relatively early stage of propagation. It moves along the head of the cylinder, propagating through the medium as a longitudinal wave.