## 2 The Constitutive Model of Weathered Material  
### Improvement of [PERSON]'s Constitutive Model  
The linear elastic constitutive [23] relationship can be expressed as  
\[\sigma=\lambda\epsilon+2G\epsilon \tag{1}\]  
where \(\sigma\) stands for stress tensor, \(\epsilon\) stands for strain tensor, \(\lambda\) and \(G\) stand for the _Lame_ constants, and \(\epsilon\) stands for the volumetric strain with \(\epsilon=\epsilon_{11}+\epsilon_{22}+\epsilon_{33}\).  
The relationship among _Lame_ constants, the elastic modulus, and [PERSON]'s ratio can be expressed as  
\[\lambda=\frac{E_{t}\
u}{(1+\
u)(1-2\
u)} \tag{2}\]  
\[G=\frac{E_{t}}{2(1+\
u)} \tag{3}\]In the above equations, \(E_{t}\) is the changing tangent elastic modulus, so Equation (1) can show the nonlinear change of stress and strain.  
The _Jacobian_ matrix can be obtained by taking the partial derivative of the stress tensor to the strain tensor in Equation (1), which can be expressed as  
\[[J]=\frac{\partial\sigma}{\partial\epsilon}=\begin{bmatrix}\lambda+2G&\lambda& \lambda&0&0&0\\ \lambda&\lambda+2G&\lambda&0&0&0\\ \lambda&\lambda&\lambda+2G&0&0&0\\ 0&0&0&G&0&0\\ 0&0&0&0&G&0\\ 0&0&0&0&0&G\\ \end{bmatrix} \tag{4}\]  
where \(\lambda\) and \(G\) are both variables, the _Jacobian_ matrix is different in each incremental step.  
The strain (with 6 strain components) state of the deformed body is equivalent to one-dimensional axial strain, and the equivalent strain \(\epsilon_{e}\) can be expressed as  
\[\epsilon_{e}=\sqrt{\frac{2}{9}\Big{[}\big{(}\epsilon_{x}-\epsilon_{y}\big{)}^ {2}+\big{(}\epsilon_{y}-\epsilon_{z}\big{)}^{2}+\big{(}\epsilon_{x}-\epsilon_ {z}\big{)}^{2}+6\big{(}\gamma_{xy}{}^{2}+\gamma_{xz}{}^{2}+\gamma_{yz}{}^{2} \big{)}\Big{]}} \tag{5}\]  
Under the multi-directional stress state, [PERSON] and [PERSON] [24] put forward an orthotropic stress-strain relationship matrix. They think that after offsetting the influence of [PERSON]'s ratio, the equivalent stress-strain relationship can still be presented with Saenz's equation. However, it is necessary to carry out stress-strain tests in each direction, which greatly increases the test frequency and cost. Considering that no tensile stress [25] exists for CSGR dam in design conditions, the whole dam body is in elastic state under compression, which can be treated as an isotropic constitutive relation in the elastic stage calculation of the dam body to reduce test frequency and cost. The introduction of equivalent strain that can transform the complex multiaxial strain state into a simple uniaxial strain state is a computational scalar instead of the real strain of CSGR dams. By improving [PERSON]'s constitutive relation [14] with equivalent strain \(\epsilon_{e}\), the original axial strain \(\epsilon\) is updated to the equivalent strain \(\epsilon_{e}\), where \(E_{0}\) is the initial tangent modulus, \(E_{c}\) is the secant modulus at the peak stress \(\sigma_{c}\), and \(\epsilon_{c}\) is the peak strain. Thus, the [PERSON]'s constitutive equation suitable for CSGR with weathered materials is established, which can be expressed as  
\[\sigma=\frac{E_{0}\epsilon_{e}}{1+\big{(}\frac{E_{0}}{E_{c}}-2\big{)}\frac{ \epsilon_{e}}{\epsilon_{c}}+\big{(}\frac{\epsilon_{e}}{\epsilon_{c}}\big{)}^ {2}} \tag{6}\]  
Differentiating (6), the tangent elastic modulus (\(E_{t}\)) of Equation (6) can be expressed as  
\[E_{t}=\frac{\partial\sigma}{\partial\epsilon_{e}}=\frac{E_{0}\bigg{(}1-\big{(} \frac{\epsilon_{e}}{\epsilon_{c}}\big{)}^{2}\bigg{)}}{\bigg{[}1+\big{(}\frac{ E_{0}}{E_{c}}-2\big{)}\frac{\epsilon_{e}}{\epsilon_{c}}+\big{(}\frac{ \epsilon_{e}}{\epsilon_{c}}\big{)}^{2}\bigg{]}^{2}} \tag{7}\]