## 3 Finite Element Modeling (FEM) of Human-Seat Interactions
### _Material Modeling and Challenges_
#### 3.3.3 Material Modeling for Latex and Fabric Adhesive Tape  
Material Characterization of Latex Using Circular Membrane Inflation Test  
The pneumatic spring consists of a thin latex membrane, requiring accurate characterization of its mechanical properties. A circular membrane inflation test was applied, which is particularly effective for thin-film hyperelastic materials ([23; 24; 25; 26; 27; 28]). This method measures mechanical properties by analyzing the stress-strain relationship derived from the pressure-deformation behavior of spherical membranes. The experimental setup (Figure 9a) involved cutting latex samples from a tube and fixing them onto a device with a circular window, sealed with screws and rubber layers. Air was pumped into or released from the membrane to induce deformation. An optoNCDT 1402 laser sensor (Micro Epsilon, Raleigh, NC, USA) measured the displacement of the apex of the spherical cap, while an SMC PSE43 AR06 pressure sensor (SMC Corporation, Tokyo, Japan) recorded the internal pressure. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Term k** & **Relaxation Time \(\tau_{k}\)** & **Prony Coefficients \(S^{(k)}\)** \\ \hline
1 & 0.733073 & 0.0181693 \\ \hline
2 & 0.799137 & 0.0725529 \\ \hline
3 & 6.81538 & 0.0914753 \\ \hline
4 & 10.2102 & 0.0695494 \\ \hline
5 & 17.0064 & 0.00251894 \\ \hline
6 & 251.667 & 0.0251244 \\ \hline
7 & 8890.97 & 0.00165894 \\ \hline
8 & 16,063.3 & 0.14635 \\ \hline
9 & 17,171.8 & 0.000349927 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The set of coefficients from the relaxation function for the viscoelastic properties. Figure 8: The experimental stress–strain curves (green square) and the fitted curves (solid green) from the foam compression test (**a**), along with the force–displacement diagram obtained from both the experiment and simulation (**b**). The geometry of the inflated membrane is illustrated in Figure 9a, where \(p_{m}\) represents the internal pressure of the latex membrane, \(t_{m}\) is the membrane thickness, \(h_{m}\) is the height of the spherical cap, \(r_{m}\) is the radius of the spherical cap, and \(R_{w}\) is the radius of the circular window (\(R_{w}\) = 72.5 mm). Figure 9: Circular membrane inflation test set up and the height of the hemisphere cap (**a**), displacement–pressure diagram of the test (**b**) and the strain–stress relationship (**c**). The stress distribution within the spherical cap, \(\sigma_{m}\), is calculated as:  
\[\sigma_{m}=\frac{p_{m}r_{m}}{2t_{m}} \tag{4}\]  
The thickness \(t_{m}\) is given by:  
\[t_{m}=t_{m0}\lambda_{3}=\frac{t_{m0}}{\lambda^{2}} \tag{5}\]  
where \(t_{m0}\) = 0.65 mm (initial membrane thickness). From the geometric relationship, the radius \(r_{m}\) and angle \(\theta\) are expressed as:  
\[r_{m}=\frac{h_{m}}{2}+\frac{R_{w}^{2}}{2t_{m}} \tag{6}\]  
\[\theta=arcsin\left(\frac{R_{w}}{r_{m}}\right) \tag{7}\]  
The experimental results (Figure 9b) show the relationship between membrane height (\(h_{m}\)) and internal pressure (\(p_{m}\)) during the inflation process (represented by the blue line) and the deflation process (represented by the red line). The engineering strain \(\lambda\) was calculated as:  
\[\lambda=\frac{l_{m}}{l_{m0}}=\frac{r_{m}\theta}{r_{m0}\theta_{0}} \tag{8}\]  
where \(l_{m0}\) represents the initial length and \(l_{m}\) is the instantaneous length of the meridian curve of the deformed membrane, \(r_{m0}\) and \(\theta_{0}\) (\(r_{m0}\)\(\approx\) 0, \(\theta_{0}\)\(\approx\) 0) are the initial values of the radius \(r_{m}\) and angle \(\theta\), respectively. Finally, the strain-stress relationship was determined as shown in Figure 9c. The stress-strain curve was calculated based on the average values of the pressure-displacement curves obtained from the inflation and deflation processes of the circular membrane inflation test.