## 4 Constitutive Relationship
### Model Validation  
Cretaceous rock is different from ordinary rocks. It contains a large number of pore structures. The stress-strain curve under triaxial stress had an obvious crack compression section, and a strain softening section was observed under negative temperature. The double strain Hooke constitutive model can effectively describe the stress and strain state before the yield point, but it cannot reveal these characteristics after the yield point. However, the stress-strain curve obtained by the damage constitutive model on the basis of the D-P criterion was in good agreement with the test stress-strain curve after the rock yield point. Therefore, the constitutive relationship of red sandstone can be more accurately described by establishing the constitutive model in sections. However, there is no unified standard for determining the rock yield point. The peak point of the stress-strain curve, namely, the peak stress, is generally regarded as the yield stress of rock in classical rock theory, which can not only lead to the underestimation of rock plastic deformation, but also affect the calculation of hardening parameters [43; 44; 45; 46; 47; 48; 49].  
In fact, the rock is damaged only when it reaches a certain stress state under the 3D compression process. According to previous studies, the yield stress value of sandstone under various conditions is 73-78% of the peak stress value before the peak point. For simplifying the calculations, 78% of the peak stress value before the peak point was selected as the yield point in this study. On the basis of the experimental results under different confining pressures at room temperature, the established constitutive relationship before and after the yield point was compared with the test results, as shown in Figure 14.  
It is clear from Figure 14 that the established model can competently fit the stress-strain curve under different confining pressures. However, the model did not predict the sudden drop of stress at the point of rock failure, and there was no good contact between the joints in the segmented model, especially the fitting curve when the confining pressure was 2 MPa. The difference between the theoretical values of the two models at the joint was 1.4 MPa, which was strongly related to the failure of the established damage constitutive model to better fit the elastic stage. To quantify the fitting degree of the model, the model deviation was analyzed, and the specific expressions are as follows:  
\[\delta=\frac{\sum\limits_{i=1}^{n}{|\sigma_{i}^{*}-\sigma_{i}|}}{n} \tag{19}\]  
\[\overline{\delta}=\frac{\sum\limits_{i=1}^{n}{|\sigma_{i}^{*}-\sigma_{i}|}}{ \sum\limits_{i=1}^{n}{\sigma_{i}}}\times 100 \tag{20}\]  
where \(\delta\) and \(\overline{\delta}\) are the absolute deviation and relative deviation between the modeling and experimental results, respectively, and \(\sigma_{i}\) and \(\sigma_{i}^{*}\) are the corresponding test stress and theoretical stress values, respectively (\(i\) = 1, 2, 3... ). \(n\) represents the number of data points. The model parameters obtained from the above equation are listed in Table 4.  
According to the above results, the absolute deviation of the established rock constitutive model did not exceed 1 MPa, and the relative deviation was less than 5%, which indicates that the model was able to accurately describe the stress-strain behavior of Cretaceous red sandstone under triaxial stress.  
\begin{table}
\begin{tabular}{c c c} \hline
**Confining Pressure/MPa** & **Absolute Deviation \(\delta\) /MPa** & **Relative Deviation \(\overline{\delta}\) /\%** \\ \hline
2 & 0.312 & 3.28 \\
4 & 0.526 & 3.82 \\
6 & 0.706 & 3.68 \\ \hline \end{tabular}
\end{table}
Table 4: Results of model deviation.  
Figure 14: Comparison between experimental data and model data. (**a**) W = 2 MPa; (**b**) W = 4 MPa; (**c**) W = 6 MPa.