# Optimization of Shape Design of Grommet through Analysis of Physical Properties of EPDM Materials
## 3 Nonlinear Analysis Using FEM
### Nonlinear Analysis by Setting Material Constants  
Unlike metals, rubber retains its elasticity even under large strain. As rubber has hyper-elastic properties that exhibit nonlinearity between load and strain, it is important to understand its nonlinear properties [15]. We conducted uniaxial and biaxial tensile tests of EPDM rubber to obtain stress-strain rate information. Then, we determined the nonlinear material constants required for finite element analysis. The uniaxial and biaxial tensile tests were performed using an EPDM 50 material to obtain stress-strain data, as shown in Figure 6. We determined the material constants required for nonlinear analysis. The tensile test was performed using the KS standard dumbbell-type three test [13]. The Mullins effect is observed in EPDM materials, such as rubber, in which the initial molecular structure is rearranged upon repeated loads [16; 17; 18; 19]. As shown in Figure 6a,b, as strain range gradually increases, if a strain larger than the previously applied strain is received, a certain permanent strain occurs and strain does not become zero, even if stress is zero. In addition, while the gauge distance of a specimen increases in the repeated loading process, the cross-sectional area decreases. The nonlinear material constants for finite element analysis were obtained through the curve fitting of the stress-strain data obtained in the uniaxial and biaxial tensile tests. The relationship between stress and strain was determined to obtain the final nonlinear material constants considering the change in the cross-sectional area under the repeated loading of rubber, as shown in Figure 7.  
The Ogden model was used for nonlinear analysis. An Ogden model is a hyper elastic material model that can be used for predicting the nonlinear stress-strain behavior of materials such as rubber or polymer. Ogden model was introduced by [PERSON] in 1972, and the strain energy density function for an Ogden material is as follows (3)  
\[\mathrm{w}=\sum_{k=1}^{n}\mu_{k}\left(\frac{\lambda_{1}^{a_{k}}+\lambda_{2}^{ a_{k}}+\lambda_{3}^{a_{k}}-3}{a_{k}}\right) \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Level** & **a (mm)** & **b (mm)** & **c (mm)** & **d (mm)** & **e (\({}^{\circ}\)C)** \\ \hline
1 & 4 & 10 & 1 & 5 & 135 \\
2 & 4.5 & 11 & 1.5 & 6 & 145 \\
3 & 5 & 12 & 2 & 7 & 155 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters according to shape. Figure 5: Grommet shape parameter settings. where w: Strain energy density; \(\mu_{k}\), \(a_{k}\): Ogden constants; and \(\lambda\): Stretch ratio; \(n=3\). Table 7 shows the coefficients of the Ogden model with a strain range of 100%, which is a material model used to describe nonlinear material constants. The ABAQUS commercial software was used [20; 21; 22]. Figure 8 shows an image of the analysis results. Figure 8a shows the initial state before analysis and (b) shows graphically one of the various analysis results. The maximum value of insertion force and separation force was confirmed through the analysis results. The results for insertion force and separation force were obtained for 32 conditions through the nonlinear analysis, as shown in Table 8. ANOVA was performed for insertion force and separation force according to shape design parameters. Results showed that all variables except d were significant among the variables that affected insertion force, while shape parameters b and e affected separation force. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{6}{c}{**Ogden Model 3 rd Constant Values**} \\ \cline{2-7}  & \(\mu\)**1** & \(\mu\)**2** & \(\mu\)**3** & \(\alpha\)**1** & \(\alpha\)**2** & \(\alpha\)**3** \\ \hline EPDM50 (100\%) & 3.557 & 8.004 & 4.550 \(\times\) 10\({}^{-1}\) & 2.000 \(\times\) 10\({}^{-2}\) & 4.000 \(\times\) 10\({}^{-3}\) & 2.381 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Ogden model 3 rd constant values. Figure 6: Experimental stress–strain curves for compound EPDM.