# Study of Rock Crack Extension under Liquid Nitrogen Low-Temperature Fracturing
## 3 Feasibility Study on Low-Temperature Frozen Shale Using a Numerical Simulation Method  
Numerical simulations are carried out using RFPA, a software that not only takes into account the non-homogeneity of a rock mass at micro and macro levels, but also captures the key behavior of rock deformation and damage [27]. Taking the numerical calculation model of the water-ice single crack of [PERSON] et al. [28] as an example, the thermal coupling property of RFPA2D-Thermal was used to analyze the frost heaving deformation and fracture process. The model was 0.6 m (length) \(\times\) 0.6 m (with), the cross-section length of the crack was 0.04 m, and the center width of the short axis of the crack was 0.004 m, which was a plane strain state and ensured that the model size was 15 times the crack length. When quadrilateral elements were used, the size was refined to 0.001 m, and the compression-tension ratio was 10:1. The initial temperature of the rock mass was 0 \({}^{\circ}\)C, and the AB edge on the crack surface was cooled to a temperature of \(-\)20 \({}^{\circ}\)C. The three edges were adiabatically treated, AB, with CD constraint Y direction displacement and BC constraint X direction displacement. The specific boundary conditions are shown in Figure 9. The thermodynamic parameters are shown in Table 2.  
Figure 8: Crack propagation change during the cryogenic thermal shock experiments (sample 1). (**a**) Front; (**b**) sides. Considering the effect of water migration during frost heaving, the equivalent thermal expansion coefficient method used for numerical analysis and the equivalent thermal expansion coefficient were introduced \(\alpha\)[28]:  
\[(1-\alpha\Delta T)^{3}=\Big{(}1+\beta u^{T}\Big{)}(1-\xi), \tag{1}\]  
where \(\beta\) is the phase change volume expansion coefficient of the water-ice under free state (0.09); the crack water begins to phase change from 0 \({}^{\circ}\)C, that is, the phase change point \(T_{f}\)= 0 \({}^{\circ}\)C; the freezing temperature is \(-\)20 \({}^{\circ}\)C, so \(\Delta T=-\)20 \({}^{\circ}\)C; \(u^{T}\) = 1; and \(\xi\) represents the proportional relationship between the water transfer flux and the original volumetric water. The curve of the equivalent thermal expansion coefficient with the water transfer flux ratio was obtained by substituting Formula (1) into the numerical value, as shown in Figure 10.  
Figure 11 is the minimum principal stress cloud near the crack when the water transfer flux ratio \(\xi\) is 0.06. In RFPA2D-Thermal, the compressive stress was set to be positive and the tensile stress was set to be negative. An obvious tension concentration could be observed at the crack tip, and the maximum tensile stress even reached 9.84 MPa, which was less than the tensile strength of 10 MPa. The local unit failure was only caused by the heterogeneity of the rock material. The ice on the crack wall and inside the crack was mainly compressed. Given that the crack shape was non-curved and slightly changed in the numerical calculation, the stress distribution in the crack ice was not uniform. However, the stress value was in a constant state, and the overall difference was small. The tensile stress at the crack tip = 0 continued to increase with the decrease in the freezing temperature, and \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Medium**} \\ \cline{2-3}
**Parameter Name** & **Rock** & **Ice** \\ \hline Compressive strength (MPa) & 100 & 20 \\ Uniformity of compressive strength & 2 & 30 \\ Poisson ratio & 0.37 & 0.35 \\ Elastic modulus (MPa) & 10,000 & 600 \\ Homogeneity of elastic modulus & 2 & 20 \\ Density (kg\(\cdot\)m\({}^{-3}\)) & 2410 & 917 \\ Angle of internal friction (\({}^{\circ}\)) & 35 & 20 \\ Coefficient of thermal conductivity (W\(\cdot\)m\({}^{-1,\circ}\)C\({}^{-1}\)) & 2.67 & 2.2 \\ Thermal capacity (\(\cdot\)m\({}^{-3,\circ}\)C\({}^{-1}\times 10^{6}\)) & 1.9 & 1.7 \\ Thermal expansion coefficient (\({}^{\circ}\)C\({}^{-1}\times 10^{-5}\)) & 0.3 & - \\ \hline \hline \end{tabular}
\end{table}