## 2 Constitutive Relation of Modified Polydimethylsiloxane Material
### Identification of Constitutive Model  
The stress-strain relationship of hyperelastic materials is expressed by the strain potential energy function:  
\[S=\frac{\partial U}{\partial E} \tag{6}\]  
where \(S\) is the second Piola-Kirchhoff stress, \(E\) is the Green strain, and \(U\) is the strain potential energy function.  
There are mainly nine kinds of strain potential energy function forms in ABAQUS: Arruda-Boyce form, Marlow form, Mooney-Rivlin form, neo-Hookean form, Ogden form, Polynomina (polynomial) form, Reduced Polynomial (simplified polynomial) form, Yeoh form, and VanderWaals form. Users can provide material test data, and the ABAQUS  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Nominated Stress-Strain** & **True Stress-Strain** \\ \hline Fracture strength (MPa) & 2.71 & 8.90 \\ Fracture elongation (\%) & 228.69 & 118.98 \\ Initial elastic modulus (MPa) & 0.97 & 1.37 \\ Elastic modulus of secant (MPa) & 1.11 & 7.36 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties obtained by two analysis methods.  
Figure 3: Stress-strain curve. (**a**) Nominal stress-strain curve. (**b**) True stress-strain curve.  
software (2019) can match the optimal strain potential energy. According to the test data of the direct tensile test of the modified polydimethylsiloxane membrane, the optimal strain potential energy function form is the reduced polynomial form. The form of the reduced polynominal strain energy potential is as follows:  
\[U=\sum_{i+j=1}^{N}C_{i0}\big{(}\overline{I}_{1}-3\big{)}^{i}+\sum_{i=1}^{N}D_{i }^{-1}\Big{(}I^{el}-1\Big{)}^{2i} \tag{7}\]  
where \(U\) is the strain energy per unit of reference volume, \(N\) is a material parameter, \(C_{i0}\) and \(D_{i}\) are temperature-dependent material parameters, and \(\overline{I}_{1}\) is the first deviatoric strain invariant defined as follows:  
\[\overline{I}_{1}=\overline{\lambda}_{1}^{2}+\overline{\lambda}_{2}^{2}+ \overline{\lambda}_{3}^{2} \tag{8}\]  
where the deviatoric stretches \(\overline{\lambda}_{i}=\lambda_{i}/\sqrt[3]{J}\), \(J\) is the total volume ratio, and \(\lambda_{i}\) denote the principal stretches. \(J^{el}\) is the elastic volume ratio as defined as follows:  
\[J^{el}=\frac{J}{J^{th}} \tag{9}\]  
where \(J^{th}\) is the thermal volume ratio, defined as follows:  
\[J^{th}=\left(1+\varepsilon^{th}\right)^{3} \tag{10}\]  
where \(\varepsilon^{th}\) is the linear thermal expansion strain that is obtained from the temperature and the isotropic thermal expansion coefficient.  
The initial shear modulus and bulk modulus are given by the following equations:  
\[\mu_{0}=2C_{10} \tag{11}\]  
\[K_{0}=\frac{2}{D_{1}} \tag{12}\]  
After the ABAQUS software evaluation, the reduced polynominal form with a strain potential energy order of \(N=2\) is determined as the optimal strain potential energy form, referred to as RP-N2, and the relevant parameters are automatically fitted, which are \(C_{10}=361376.7\), \(C_{20}=4428.9\), \(D_{1}=0\), \(D_{2}=0\). The constitutive relation of the modified polydimethylsiloxane in the form of RP-N2 strain potential energy is shown in Figure 4.  
Figure 4: RP-N2 hyperelastic constitutive relation.