## 4 Numerical Simulations of Frost-Heave Failure of the Slope Rock Mass
### Numerical Model of a Slope with an Ice Crack  
The numerical model of the slope, as shown in Figure 10, is 180 m high and 200 m wide, with 3,600,000 elements. The slope model's left and lower sides are adiabatic boundaries that constrain displacement in the horizontal and vertical directions, respectively, and the slope surface is a free and heat-transfer boundary. The precast crack f3 is 40 m \(\times\) 1 m in dimension and is filled with ice. The model's initial temperature is 0 \({}^{\circ}\)C, with a temperature load increment of \(-1\)\({}^{\circ}\)C at the heat transfer boundary. The numerical simulation is divided into 20 steps, each lasting 1 h. The thermodynamic and physical and mechanical parameters of the rock mass and ice are shown in Tables 3 and 4, respectively. To study the stress distribution of a heterogeneous rock mass under the action of frost-heave force, the calculation element in RFPA2D-thermal is endowed with heterogeneous characteristics. The parameters of each calculation element are assumed to follow the Weibull distribution. The homogeneity parameter U controls the parameter's nonuniformity; the smaller the U value, the stronger the material's nonuniformity. After multiple trial calculations, the final crack growth is comparable with the actual scenario when the homogeneity is U = 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Materials** & \begin{tabular}{c} **Thermal Capacity/(m\({}^{3}\).\({}^{\circ}\)C)** \\ **\(\times\) 10\({}^{6}\)** \\ \end{tabular} & \begin{tabular}{c} **Thermal** \\ **Conductivity/(m.\({}^{\circ}\)C)** \\ \end{tabular} & \
\begin{tabular}{c} **Coefficient of Thermal** \\ **Expansion/\({}^{\circ}\)C X 10\({}^{-5}\)** \\ \end{tabular} &
\begin{tabular}{c} **Ratio of Compression** \\ **Strength to Tensile** \\ \end{tabular} \\ \hline Rock mass & 2.3 & 2.67 & 0.54 & 15 \\ Ice & 2.1 & 2.2 & \(-150\) & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The thermodynamic parameters of the numerical model. The Analysis of the Stress and Crack Propagation Law of the Slope Rock Mass under Negative Temperature  
In view of the lack of relevant physical and mechanical parameters of class II rock mass and considering that the deep rock mass is less affected by temperature, class II and III temporarily take the same parameters shown in Table 1. The physical and mechanical parameters of quartz schist suffering 25 indoor freeze-thaw cycles are determined as the parameters for class IV. The physical and mechanical parameters of the filling medium in the weak interlayer f2 are obtained from the field test, as shown in Table 5.  
Figure 11 shows the slice stress distribution curve and the minimum principal stress distribution diagram at each typical time. The slices are cut along the left, middle, and right sides of f3, with line A, line B, and line C as the corresponding cutting lines. Because the calculation elements behind element 70 are so far from f3 and the influence of frost expansion is so minor, each drawing only presents the distribution curve of the minimum \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Materials**} & **Unit Weight** & **Elastic Modulus** & **Compressive Strength** & **Poisson’s Ratio** & **Internal Friction Angle/\({}^{\circ}\)** \\  & **Weight/kN/m\({}^{\text{3}}\)** & **Modulus/GPa** & **Strength/MPa** & **Poisson’s Ratio** & **In general Friction Angle/\({}^{\circ}\)** \\ \hline Class II rock mass & 27.10 & 16.31 & 47.16 & 0.35 & 67.8 \\ Class III rock mass & 27.10 & 16.31 & 47.16 & 0.35 & 67.8 \\ Class IV rock mass & 27.00 & 12.30 & 39.13 & 0.38 & 65.7 \\ f2 & 24.00 & 1.5 & 6 & 0.4 & 30 \\ Ice & 9.17 & 0.6 & 6 & 0.35 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The physical and mechanical parameters of the numerical model. Figure 10: The numerical model for frost-heave deterioration of rock slope. II: Class II rock mass; III: class III rock mass: IV: class IV rock mass. principal stress \(\sigma_{3}\) for the first 70 elements.