# Research on Mechanical Properties and Damage Constitutive Model of Water-Bearing Coal
## 5 Validation of the Constitutive Model
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Index**} & \multicolumn{2}{c}{\(\omega=\) 0\%} & \multicolumn{2}{c}{\(\omega=\) 3.31\%} & \multicolumn{2}{c}{\(\omega=\) 7.06\%} & \multicolumn{2}{c}{\(\omega=\) 10.34\%} & \multicolumn{2}{c}{\(\omega=\) 13.14\%} \\ \cline{2-11}  & \begin{tabular}{c} **The** \\ **Authors’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Comparison** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Authors’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Methods’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} & \begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} & \
\begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} &
\begin{tabular}{c} **The** \\ **Models’** \\ **Model** \\ \end{tabular} \\ \hline MAE & 0.507 & 0.476 & 0.683 & 0.894 & 0.057 & 0.682 & 0.595 & 0.665 & 1.059 & 0.580 \\ MSE & 0.429 & 0.618 & 0.869 & 1.681 & 0.008 & 0.825 & 0.775 & 0.666 & 1.916 & 0.668 \\ RMSE & 0.655 & 0.786 & 0.932 & 1.296 & 0.092 & 0.908 & 0.881 & 0.816 & 1.384 & 0.817 \\ R\({}^{2}\) & 0.89 & 0.84 & 0.94 & 0.88 & 0.99 & 0.87 & 0.81 & 0.83 & 0.51 & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The fitting effect evaluation index values of the two models based on the experimental data in test group 1. From the fitting results in Figures 9-12, it can be noticed that the three-stage constitutive model curve derived in this paper is basically identical to the experimental curve at each loading rate. By observing the values of each evaluation index in Tables 2-5, it can be found that the maximum MAE, MSE, and RMSE corresponding to the model in this paper are 1.059, 4.838, and 2.200, respectively, and the maximum R\({}^{2}\) is 0.99, indicating that the model has a high fitting degree and good applicability. [PERSON] et al. 's [32] model can accurately describe the changing trend of the uniaxial compression total stress-strain curve of water-bearing coal, but there are many inconsistent points. The MAE, MSE, and RMSE values of this model are generally larger than those of this model, and the R\({}^{2}\) values are mostly smaller than those of this model. The fitting effect of the traditional continuous damage constitutive model in the fracture compaction stage of the specimen is worse than that in other stages, and there is a certain deviation between the theoretical and experimental value. [PERSON] et al. [32] put forward a two-stage damage constitutive model by improving the continuous damage constitutive model, which has a good matching effect to the full stress-strain curve of common small porosity and low permeability coal. The three-stage damage constitutive model suggested in this study is more universal and can fit the full stress-strain curve of large porosity and high permeability coal wells. [PERSON] et al. 's [32] model is a special form of this model; when \(\sigma_{b}=\sigma_{a}\) and \(\varepsilon_{b}=\varepsilon_{a}\), the model in this paper degenerates into [PERSON] et al. 's [32] model.