## 3 Novel Creep Constitutive Model for the Host Rocks of the Weilai Tunnel
### Development of the Proposed Creep Constitutive Model
#### 3.3.3 Effect of Pore Water Pressure: Second Variant of the Proposed Model
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(p\) & \(\sigma_{1}\)\(-\sigma_{3}\) & \(G_{1}\) & \(G_{p}\) & \(G_{2}\) & \(\eta_{2}\) (GPa.h) & \(\eta_{3}\) & \(\alpha\) & \(n\) & \(k\) & \(t_{f}\) (h) & \(\mathbf{R^{2}}\) \\ (MPa) & (MPa) & (GPa) & (GPa) & (GPa) & (GPa) & \(\eta_{2}\) (GPa.h) & \(\alpha\) & \(n\) & \(k\) & \(t_{f}\) (h) & \(\mathbf{R^{2}}\) \\ \hline
0.05 & 11.99 & 2.67 & 3.76 & 5.26 & 15.41 & & 1.021 & & & & 0.9917 \\  & 14.99 & 2.30 & 10.47 & 6.70 & 31.17 & & 0.082 & & & & 0.9877 \\  & 17.99 & 1.93 & 10.77 & 10.99 & 407.90 & & 0.020 & & & & 0.9873 \\  & 20.97 & 2.02 & 25.36 & 15.90 & 83.81 & & 0.687 & & & & 0.9889 \\  & 23.98 & 5.16 & 0.81 & 7.33 & 446.58 & 0.69 & 0.782 & 11.64 & 0.024 & 552 & 0.9914 \\
0.10 & 11.99 & 2.60 & 3.65 & 5.11 & 36.57 & & 0.975 & & & & 0.9921 \\  & 14.99 & 2.23 & 10.17 & 6.51 & 74.00 & & 0.077 & & & & 0.9981 \\  & 17.99 & 1.87 & 10.47 & 10.68 & 968.77 & & 0.018 & & & & 0.9823 \\  & 20.97 & 1.97 & 24.64 & 15.44 & 198.06 & & 0.754 & & & & 0.9789 \\  & 23.98 & 2.00 & 0.79 & 7.12 & 1060.63 & 1.65 & 0.783 & 9.72 & 0.035 & 489 & 0.9911 \\
0.15 & 11.99 & 2.70 & 3.52 & 5.04 & 13.59 & & 0.942 & & & & 0.9918 \\  & 14.99 & 2.33 & 9.79 & 6.42 & 27.51 & & 0.075 & & & & 0.9845 \\  & 17.99 & 1.95 & 10.06 & 10.53 & 358.01 & & 0.018 & & & & 0.9872 \\  & 20.97 & 2.05 & 23.72 & 15.63 & 73.96 & & 0.741 & & & & 0.9886 \\  & 23.98 & 2.08 & 0.76 & 7.20 & 394.11 & 0.63 & 0.771 & 10.85 & 0.041 & 447 & 0.9929 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model parameters affected by pore water pressure (\(p\)) in the loading states. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(p\) & \(\sigma_{1}\)\(-\sigma_{3}\) & \(G_{p}\) (GPa) & \(G_{2}\) (GPa) & \(\eta_{2}\) (GPa.h) & \(\alpha\) & \(R^{2}\) \\ (MPa) & (MPa) & (GPa) & (GPa) & (GPa) & (GPa) & (GPa) & (GPa) & (GPa) \\ \hline
0.05 & 11.99 & 2.01 & 0.014 & 0.0312 & 1.62 & 0.9911 \\  & 14.99 & 4.23 & 0.08 & 0.0071 & 27.87 & 0.9869 \\  & 17.99 & 9.34 & 0.28 & 0.0202 & 28.15 & 0.9881 \\  & 20.97 & 3.02 & 0.19 & 0.0112 & 12.63 & 0.9874 \\
0.10 & 11.99 & 2.33 & 0.018 & 0.0605 & 1.73 & 0.9928 \\  & 14.99 & 4.23 & 0.105 & 0.0137 & 27.98 & 0.9891 \\  & 17.99 & 4.93 & 0.37 & 0.03 Figure 16: Creep strain of sandstone affected by pore water pressure (the deviatoric stress is 17.99 MPa). Figure 15: Creep strain of sandstone affected by pore water pressure (the deviatoric stress is 11.99 MPa). Figure 17 shows that, even under low pore water pressure, with the effects of pore water pressure, the creep of sandstone develops more rapidly than in dry conditions. Indeed, with the influence of pore water pressure, the tertiary creep triggers earlier, after only 40 h. However, in dry conditions, tertiary creep happens after nearly 60 h. In the context of long-term stability of the deep soft rock tunnel, the effects of the pore water pressure must be strongly addressed. Similar to the dry states, considering different damage index values, the evolution of the viscoplastic strain of the host rocks in wet conditions (\(p=0.10\) MPa ) and the damage variable are examined, and the results are displayed in Figures 18 and 19. This also gives an expansion on the understanding of the model (second variant). Figures 18 and 19 reveal the significance of rock damage effects in the proposed creep constitutive model. Figure 17: Creep strain of sandstone affected by pore water pressure (the deviatoric stress is 23.98 MPa). Figure 18: Viscoplastic strain of the model considering different damage index values in wet states.