# Mechanical Behavior of Weathered Granite Exposed to Water
## 4 Modeling
### Model for Predicting Strain Softening
According to the above expression, without considering the elastoplastic coupling, Equation (2) can be obtained:  
\[\epsilon_{1}^{p}=\epsilon_{1}-\frac{\sigma_{1}}{E_{0}} \tag{2}\]  
where \(\epsilon_{1}^{p}\) is the plastic internal variable, \(\epsilon_{1}\) represents the axial strain, and \(E_{0}\) indicates the elastic modulus before the peak. Equation (2) can yield the value of the plastic internal variable at the peak residual stress, and the strain-softening parameter can be determined by this method, as expressed in Equation (3):  
\[\epsilon_{1}^{p}=\epsilon_{1r}-\frac{\sigma_{1r}}{E_{0}} \tag{3}\]  
where \(\epsilon_{1r}^{p}\) is the maximum plastic strain, \(\epsilon_{1r}\) indicates the residual strain of the rock mass, and \(\sigma_{1r}\) denotes the residual stress on the rock mass. Equations (4) and (5) define the internal friction angle and the cohesive force, respectively:  
\[\varphi=\left\{ \begin{array}{l}\varphi_{0}+\frac{\epsilon_{1r}^{p}}{\epsilon _{1r}-\frac{\sigma_{1r}}{E_{0}}},0\leq\epsilon_{1}^{p}\leq\epsilon_{1r}-\frac {\sigma_{1r}}{E_{0}}\\ \varphi_{r},\epsilon_{1}^{p}>\epsilon_{1r}-\frac{\sigma_{1r}}{E_{0}}\end{array}  \right\} \tag{4}\]  
Figure 8: The stress–strain curve. \[c=\left\{ \begin{array}{l}c_{0}+\frac{\varepsilon_{1}^{P}}{\varepsilon_{1r}-\frac{ \sigma_{1r}}{E_{0}}},0\leq\varepsilon_{1}^{P}\leq\varepsilon_{1r}-\frac{ \sigma_{1r}}{E_{0}}\\ c_{r},\varepsilon_{1}^{P}>\varepsilon_{1r}-\frac{\sigma_{1r}}{E_{0}}\end{array} \right\} \tag{5}\]  
According to the Mohr-Coulomb strength criterion, after the stress on the rock mass reaches the peak value, the stress state satisfies Equation (6):  
\[\sigma_{1}=\frac{2c\cos\varphi}{1-\sin\varphi}+\sigma_{3}\frac{1+\sin\varphi} {1-\sin\varphi} \tag{6}\]  
where \(\sigma_{3}\) represents the confining pressure on the rock. For the strain softening, substituting Equation (2) into Equations (4) and (5), respectively, yields Equations (7) and (8):  
\[\varphi=\varphi_{0}+\frac{\varepsilon_{1}-\frac{\sigma_{1}}{E_{0}}}{ \varepsilon_{1r}-\frac{\sigma_{1r}}{E_{0}}}(\varphi_{r}-\varphi_{0}) \tag{7}\]  
\[c=c_{0}+\frac{\varepsilon_{1}-\frac{\sigma_{1}}{E_{0}}}{\varepsilon_{1r}- \frac{\sigma_{1r}}{E_{0}}}(c_{r}-c_{0}) \tag{8}\]  
Taking the partial derivative of Equations (7) and (8) with respect to the principal axial strain yields Equations (9) and (10), respectively:  
\[\frac{\partial\varphi}{\partial\varepsilon_{1}}=\frac{1-\frac{\varepsilon_{P }}{E_{0}}}{\varepsilon_{1r}-\frac{\sigma_{1r}}{E_{0}}}(\varphi_{r}-\varphi_{0}) \tag{9}\]  
\[\frac{\partial c}{\partial\varepsilon_{1}}=\frac{1-\frac{\varepsilon_{P}}{E_{0 }}}{\varepsilon_{1r}-\frac{\sigma_{1r}}{E_{0}}}(c_{r}-c_{0}) \tag{10}\]  
where \(E_{p}\) is the deformation modulus of the veathered rock in the softening stage after reaching the peak stress. Taking the partial derivative of Equation (6) with respect to the principal axial strain yields Equation (11):  
\[\frac{\partial\varphi}{\partial\varepsilon_{1}}=\frac{2\cos\varphi\frac{ \partial c}{\partial\varepsilon_{1}}}{1-\sin\varphi}+\frac{[2\sigma_{3}\cos \varphi+2c(1-\sin\varphi)]\frac{\partial\varphi}{\partial\varepsilon_{1}}}{ \left(1-\sin\varphi\right)^{2}} \tag{11}\]  
Finally, the deformation modulus of the veathered rock in the softening stage after reaching the peak stress can be obtained by substituting Equations (9) and (10) into Equation (11):  
\[E_{p}=\frac{2\cos\varphi(1-\sin\varphi)E_{0}(c_{r}-c_{0})+2[\sigma_{3}\cos \varphi+c(1-\sin\varphi)]E_{0}(\varphi_{r}-\varphi_{0})}{\left(1-\sin\varphi \right)^{2}(E_{0}\varepsilon_{1r}-\sigma_{1r})+2\cos\varphi(1-\sin\varphi)(c_ {r}-c_{0})+2[\sigma_{3}\cos\varphi+c(1-\sin\varphi)](\varphi_{r}-\varphi_{0})} \tag{12}\]