# Research on Mechanical Properties and Damage Constitutive Model of Water-Bearing Coal
## 5 Validation of the Constitutive Model  
The constitutive model developed in Section 4 is validated using the experimental data collected in Section 3. [PERSON] et al. [32] assumed that the strength of the coal micro-element body obeyed the distribution of the power function, and on this basis, the two-stage damage constitutive model of water-bearing coal was deduced. The model of [PERSON] et al. [32] was used to calculate the test results in this article, and the two models' fitting results were compared and analyzed. The fitting results of the two models are shown in Figures 9-12. Figure 9: The uniaxial compression test data points and theoretical model curve of coal samples in test group 1: **(a)** the author’s model; **(b)** the comparison model. In the study literature on the coal (rock) constitutive model, there are few indications to quantitatively assess the fitting result of the constitutive model. To scientifically evaluate the fitting effect of the damage constitutive model, the following four statistical indicators are introduced in this paper:  
Figure 11: The uniaxial compression test data points and theoretical model curve of coal samples in test group 3: **(a)** the author’s model; **(b)** the comparison model. Figure 12: The uniaxial compression test data points and theoretical model curve of coal samples in test group 4: **(a)** the author’s model; **(b)** the comparison model. Figure 10: The uniaxial compression test data points and theoretical model curve of coal samples in test group 2: **(a)** the author’s model; **(b)** the comparison model. Mean absolute error (MAE), which is defined as: \[\text{MAE}(y,\hat{g})=\frac{1}{l}\sum_{i=1}^{l}(|y_{i}-f(x_{i})|)\] (30)
2. Mean square error (MSE), which is defined as: \[\text{MSE}(y,\hat{g})=\frac{1}{l}\sum_{i=1}^{l}(y_{i}-f(x_{i}))^{2}\] (31)
3. Root mean square error (RMSE), which is defined as: \[\text{RMSE}(y,\hat{g})=\sqrt{\frac{1}{l}\sum_{i=1}^{l}(y_{i}-f(x_{i}))^{2}}\] (32)
4. Decision coefficient (R\({}^{2}\)), which is defined as: \[\text{R}^{2}=1-\frac{\text{MSE}(y,\hat{g})}{\frac{1}{l}\sum_{i=1}^{l}(y_{i}- \overline{y})^{2}}\] (33)  
In Equations (30)-(33): \(y\) is the parameter to be estimated, it is the strength of the coal sample in this paper, and \(\hat{g}\) is the estimator of \(y\); \(l\) is the number of statistical samples; \(y_{i}\) is the true value of the test data; \(f(x_{i})\) is the predicted value of the damage constitutive model; \(\overline{y}\) is the average amount of the test data. The MAE, MSE, and RMSE can reflect the deviation between the experimental amount and the theoretical amount of the model; the smaller the value is, the higher the fitting degree of the model is, and when the value is zero, the experimental value is equal to the theoretical value, and the damage constitutive model is the perfect model. The value of R\({}^{2}\) can reflect the degree of approximation of the theoretical value of the model to the real test value; the larger the value is, the better the model fitting effect is. The fitting effect evaluation indexes (MAE, MSE, RMSE, and R\({}^{2}\)) corresponding to the three-stage damage constitutive model in this paper and [PERSON] et al. 's [32] two-stage damage constitutive model are shown in Tables 2-5.