## 6 Conclusions  
In this paper, the uniaxial tensile test of modified polydimethylsiloxane material was carried out, and the hyperelastic constitutive model of modified polydimethylsiloxane material was established. Based on the constitutive model, a two-dimensional finite element model is established in combination with the cohesive element to simulate the tensile failure of the composite lining structure and the surrounding rock. A simplified finite element model of the composite lining is established to study the crack resistance of the modified \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Internal Water** & **Critical Thickness-to-** & **Internal Water** & **Critical Thickness-to-** \\
**Pressure (MPa)** & **Width Ratio \(\mathbf{\alpha}\)** & **Pressure (MPa)** & **Width Ratio \(\mathbf{\alpha}\)** \\ \hline
1 & 0.162 & 6 & 0.945 \\
2 & 0.313 & 7 & 1.025 \\
3 & 0.465 & 8 & 1.557 \\
4 & 0.609 & 9 & 1.926 \\
5 & 0.731 & 10 & 1.930 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Critical coating thickness-to-width ratio under different internal water pressure conditions. Figure 19: Effect of thickness-to-width ratio on the maximum true principal strain of impermeable coating under different internal water pressure. polydimethylsiloxane impermeable coating under internal water pressure. The main conclusions of this paper are as follows:  
1. The modified polydimethylsiloxane exhibits exceptional hyperelastic behavior with a true fracture elongation of 118.98%. The simplified polynomial hyperelastic constitutive model (RP-N2) accurately captures its non-linear mechanical response, achieving a correlation coefficient (R\({}^{2}\)) of 0.999 with experimental data. This behavior enables the coating to accommodate large deformations in cracked concrete linings while maintaining impermeability. Furthermore, the RP-N2 model's computational efficiency enables rapid structural assessments, maintaining 99% accuracy in strain prediction. 2. The crack width of lining structure decreases with the increase in buried depth, and the crack width gradually maintains invariance after reaching a certain buried depth. When the lateral pressure coefficient is less than 1, the maximum crack width decreases with the increase in the lateral pressure coefficient; when the lateral pressure coefficient is greater than 1, the maximum crack width increases with the increase in the lateral pressure coefficient. 3. The derived critical thickness-to-width ratios provide a quantifiable design framework. According to different internal water pressure conditions, a set of thickness-to-width ratios are suggested to determine the thickness of the modified polydimethylsiloxane impermeable coating in tunnel engineering. The proposed methodology addresses practical challenges in ultra-high-pressure tunnel design. By correlating the polydimethylsiloxane thickness with crack width and lateral pressure coefficients, engineers can optimize coating specifications for specific geological and hydraulic conditions. [PERSON] provided the original idea and wrote the original manuscript. [PERSON] provided engineering guidance and verified the feasibility of the method. [PERSON] reviewed and edited the manuscript. [PERSON] investigated and provided engineering data. [PERSON] analyzed the results of this method. All authors have read and agreed to the published version of the manuscript. This study was financially supported by the Natural Science Foundation of Sichuan Province, China (No. 2024 NSFSC0836). We acknowledge the aforementioned supports. Not applicable. The datasets used in the current study are available from the corresponding author upon reasonable request. This study was financially supported by the Natural Science Foundation of Sichuan Province, China (No. 2024 NSFSC0836). We acknowledge the aforementioned supports. Special thanks to [PERSON] for the support of this study. Authors [PERSON] and [PERSON] were employed by the company POWERCHINA Chengdu Engineering Corporation Limited.