## 4 Long-Term Settlement Prediction Analysis
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Unit Weight kg/m3** & \begin{tabular}{c} **Elasticity Modulus** \\ **/MPa** \\ \end{tabular} & **Poisson Ratio** & \
\begin{tabular}{c} **Cohesion** \\ **/kPa** \\ \end{tabular} &
\begin{tabular}{c} **Internal Friction Angle** \\ \(f^{\circ}\) \\ \end{tabular} \\ \hline Rock & 2600 & 800 & 0.28 & 300 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rock physical parameters. \[t=\frac{q}{2}\left[1+\frac{1}{k}-\left(1-\frac{1}{k}\right)\left(\frac{r}{q}\right) ^{3}\right] \tag{4}\]  
where \(t\) is the deviatoric stress parameter, \(q\) is the Mises stress, \(r\) is the deviatoric stress invariant, \(\beta\) is the tilt angle of the yield surface in the p-t stress space and is related to the friction angle \(\varphi\), and \(k\) is the ratio of the strength from triaxial tensile tests to the strength from triaxial compression tests, reflecting the influence of principal stresses on yield. When \(k=1\) and \(t=q\), the yield surface is the [PERSON] circle in the \(\pi\)-plane. To ensure that the yield surface is convex, it is required that \(0.778\leq k\leq 1.0\). \(d\) is the intercept of the yield surface on the t-axis in the p-t stress space. It represents another form of cohesion and is related to the input hardening parameters. It can be determined as follows:  
\[d=\left\{ \begin{array}{l}\left(1-\frac{1}{3}tan\beta\right)\sigma_{c}\\ \left(\frac{1}{k}+\frac{1}{3}tan\beta\right)\sigma_{t}\\ \frac{\sqrt{3}}{2}\left(1+\frac{1}{k}\right)\tau\end{array} \right.\]  
In a linear model, the expression for the plastic flow potential \(G\) is given by Equation (5). \[G=t-ptan\psi \tag{5}\]  
In this equation, \(\psi\) represents the dilatancy angle in the p-t stress space. When \(\psi=0^{\circ}\), it indicates that the volume of the plastic deformable material remains unchanged, whereas when \(\psi>0^{\circ}\), the material exhibits dilatancy effects. When using the associated flow rule, \(\psi\) equals \(\beta\), and when the value of k is 1, the linear Drucker-Prager model degenerates into the classical Drucker-Prager model [22; 23]. Therefore, we can combine the time-hardening process with the Drucker-Prager yield-failure criterion and its associated flow rule to jointly formulate the time-hardening and Drucker-Prager plastic-coupled creep model of the soil [24; 25]. The values of the relevant parameters are shown in Tables 4 and 5, where \(\sigma_{0}\) represents the axial load from the compressive creep characteristic test, and \(A\), \(n\) and \(m\) are material parameters for the time-hardening creep law. \(R^{2}\) represents the coefficient of determination obtained from the fit. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Compaction Density** & \(\sigma_{0}\)**/kPa** & \(A\) & \(n\) & \(m\) & \(R^{2}\) \\ \hline \multirow{4}{*}{85\%} & 50 & \(1\times 10^{-8}\) & 1.21398 & \(-0.72969\) & 0.98891 \\  & 100 & \(1\times 10^{-8}\) & 1.26747 & \(-0.82022\) & 0.95937 \\  & 200 & \(1\times 10^{-8}\) & 1.23795 & \(-0.92080\) & 0.88839 \\  & 400 & \(1\times 10^{-8}\) & 1.22657 & \(-0.92554\) & 0.84951 \\  & 50 & \(1\times 10^{-8}\) & 1.11915 & \(-0.82803\) & 0.92200 \\
90\% & 100 & \(1\times 10^{-8}\) & 1.15649 & \(-0.75232\) & 0.92676 \\  & 200 & \(1\times 10^{-8}\) & 1.15450 & \(-0.87983\) & 0.93797 \\  & 400 & \(1\times 10^{-8}\) & 1.13348 & \(-0.91752\) & 0.89378 \\  & 50 & \(1\times 10^{-8}\) & 1.10226 & \(-0.88783\) & 0.81795 \\  & 100 & \(1\times 10^{-8}\) & 1.13559 & \(-0.74755\) & 0.96733 \\
95\% & 200 & \(1\times 10^{-8}\) & 1.14587 & \(-0.81434\) & 0.95700 \\  & 400 & \(1\times 10^{-8}\) & 1.11372 & \(-0.90938\) & 0.90810 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Time-hardening creep model parameters (different degrees of compaction).