# Research on Mechanical Properties and Damage Constitutive Model of Water-Bearing Coal
## 4 Damage Constitutive Model
### Sectional Damage Constitutive Model  
Referring to the research results of Lu [42], the one-dimensional constitutive relation of water-bearing coal in the fracture compaction stage can be defined as:  
\[\sigma=\sigma_{a}\bigg{(}\frac{\epsilon}{\epsilon_{a}}\bigg{)}^{2} \tag{22}\]  
In Equation (22): \(\sigma_{a}\) and \(\epsilon_{a}\) are the maximal stress and strain in the fracture compaction stage of water-bearing coal under uniaxial compression.  
Figure 3 shows that the stress-strain curve of water-bearing coal during the pore compaction stage is nearly horizontal, so the one-dimensional constitutive model in this stage can be defined as:  
\[\sigma=\sigma_{a}+\frac{\sigma_{b}-\sigma_{a}}{\epsilon_{b}-\epsilon_{a}}( \epsilon-\epsilon_{a}) \tag{23}\]  
In Equation (23): \(\sigma_{b}\) and \(\epsilon_{b}\) are the maximal stress and strain in the pore compaction stage of water-bearing coal under uniaxial compression.  
From Equations (13), (15) and (21), the one-dimensional constitutive model of water-bearing coal in elastic deformation, plastic deformation, and post-peak failure stages can be obtained:  
\[\sigma=\sigma_{b}+E_{\omega}(\epsilon-\epsilon_{b})\exp\left\{-\left[\frac{ \left(\alpha+\frac{1}{\sqrt{3}}\right)E_{\omega}\epsilon}{n}\right]^{m}\right\} \tag{24}\]  
At the peak point (\(\epsilon_{c}\), \(\sigma_{c}\)), the uniaxial compression stress-strain curve meets the boundary requirements: (1) \(\epsilon=\epsilon_{c}\), \(\sigma=\sigma_{c}\); (2) \(\epsilon=\epsilon_{c}\), \(\frac{\partial\sigma}{\partial\epsilon}=0\). Substituting the boundary conditions into Equation (24), the expression of \(m\) and \(n\) can be obtained after solving:  
\[m=\frac{\epsilon_{c}}{R(\epsilon_{b}-\epsilon_{c})} \tag{25}\]  
\[n=\bigg{(}\alpha+\frac{1}{\sqrt{3}}\bigg{)}E_{\omega}\epsilon_{c}(-R)^{-\frac {1}{m}} \tag{26}\]For the specific stress-strain curve, \(R\) is a constant in Equations (25) and (26), and its calculation equation is as follows:  
\[R=\ln\frac{\sigma_{c}-\sigma_{b}}{E_{\omega}(\varepsilon_{c}-\varepsilon_{b})} \tag{27}\]  
The expressions of \(m\) and \(n\), namely Equations (25) and (26), are substituted into Equation (24), and the subsequent equation can be derived:  
\[\sigma=\sigma_{b}+E_{\omega}(\varepsilon-\varepsilon_{b})\exp\left[R\bigg{(} \frac{\varepsilon}{\varepsilon_{c}}\bigg{)}^{\frac{\varepsilon_{c}}{R( \varepsilon_{b}-\varepsilon_{c})}}\right] \tag{28}\]  
By combining Equations (21), (22) and (26), the segmented damage constitutive model of water-bearing coal can be obtained:  
\[\sigma=\left\{\begin{aligned} \sigma_{a}\bigg{(}\frac{ \varepsilon}{\varepsilon_{a}}\bigg{)}^{2}&&(\varepsilon< \varepsilon_{a})\\ \sigma_{a}+\frac{\sigma_{b}-\sigma_{a}}{\varepsilon_{b}- \varepsilon_{a}}(\varepsilon-\varepsilon_{a})&&(\varepsilon _{a}\leq\varepsilon<\varepsilon_{b})\\ \sigma_{b}+E_{\omega}(\varepsilon-\varepsilon_{b})\exp\left[R \bigg{(}\frac{\varepsilon}{\varepsilon_{c}}\bigg{)}^{\frac{\varepsilon_{c }}{R(\varepsilon_{b}-\varepsilon_{c})}}\right](\varepsilon\geq\varepsilon_{ b})\end{aligned}\right. \tag{29}\]