## 3 Nonlinear Creep Model of AS
### Determination of Creep Parameters
#### 3.3.2 Determination of the Remaining Creep Parameters  
When \(\sigma_{C}<\sigma_{s_{1}}\), the parameters to be fitted include \(K,\ G_{0},\ G_{1}\) and \(\eta_{1}\); when \(\sigma_{s_{1}}\leq\sigma_{C}<\sigma_{s_{2}}\), the parameters to be fitted are \(K,\ G_{0},\ G_{1},\ \eta_{1}\) and \(\eta_{2}\); when \(\sigma_{C}\geq\sigma_{s_{2}}\), the creep parameters to be fitted are \(K,\ G_{0},\ G_{1},\ \eta_{1},\ \eta_{2}\) and the rheological index \(n\). In this study, the least squares method was used to fit the parameters. First, the fitting function was determined. \[\varepsilon=f(t,K,G_{0},G_{1},\eta_{1},\eta_{2},\eta_{c},n) \tag{21}\]  
Use \(b_{1}\) to \(b_{7}\) to represent the approximate values of parameters \(K,\ G_{0},\ G_{1},\ \eta_{1},\ \eta_{2},\ \eta_{c}\), and \(n\) and define:  
\[\left\{ \begin{array}{c}\delta_{1}=K-b_{1},\delta_{2}=G_{0}-b_{2},\delta_{3}= G_{1}-b_{3},\delta_{4}=\eta_{1}-b_{4}\\ \delta_{5}=\eta_{2}-b_{5},\delta_{6}=\eta_{c}-b_{6},\delta_{7}=n-b_{7}\end{array} \right. \tag{22}\]  
By expanding the Taylor series and extracting only the linear part, we can obtain:  
\[\varepsilon=f(t,b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7})+\sum_{j=1}^{7}f^{ \prime}_{Bj}(t,b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7})\delta_{j} \tag{23}\]  
Then, the following equations can be obtained:  
\[f^{\prime}_{B1}=\frac{\partial f}{\partial K^{\prime}},f^{\prime}_{B2}=\frac {\partial f}{\partial G_{0}},f^{\prime}_{B3}=\frac{\partial f}{\partial G_{1} },f^{\prime}_{B4}=\frac{\partial f}{\partial\eta_{1}},f^{\prime}_{B5}=\frac {\partial f}{\partial\eta_{2}},f^{\prime}_{B6}=\frac{\partial f}{\partial \eta_{c}},f^{\prime}_{B7}=\frac{\partial f}{\partial n} \tag{24}\]  
According to Equation (24), the objective function of the least squares fitting problem is obtained:  
\[Q=\sum{[\varepsilon_{j}-f(t_{j},K,G_{0},G_{1},\eta_{1},\eta_{2},\eta_{c},n)]^ {2}}-\sum{f_{Bj}(t,K,G_{0},G_{1},\eta_{1},\eta_{2},\eta_{c},n)\delta_{j}} \tag{25}\]  
If  
\[\frac{\partial Q}{\partial\delta_{k}}=0,(k=1,2,3,4,5,6,7) \tag{26}\]  
Then  
\[\sum_{j=1}^{7}a_{kj}\delta_{j}=C_{k},(k=1,2,3,4,5,6,7) \tag{27}\]  
where \(a_{kj}\) and \(C_{k}\) in Equation (27) are defined as follows:  
\[\left\{ \begin{array}{c}a_{kj}=\sum{f^{\prime}_{Bk}(t_{j},b_{1},b_{2},b_{3},b _{4},b_{5},b_{6},b_{7})}\cdot{f^{\prime}_{Bj}(t_{j},b_{1},b_{2},b_{3},b_{4},b _{5},b_{6},b_{7})}k,j=1,2,\cdots,7\\ C_{k}=\sum{[\varepsilon_{j}-f(t_{j},b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7}) ]}\cdot{f^{\prime}_{Bk}(t_{j},b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7})}k=1,2, \cdots,7\end{array} \right. \tag{28}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**C (MPa)** & \(\sigma_{\text{re}}\) (MPa)/\(\delta\) & \(\sigma_{\text{s}_{1}}\) (MPa) & \(\sigma_{\text{ds}}\) (MPa)/\(\delta\) & \(\sigma_{\text{s}_{2}}\) (MPa) \\ \hline
3 & 30.3/71.29\% & 24.24 & 33.7/79.29\% & 28.65 \\
6 & 35.0/73.92\% & 28.00 & 39.3/83.00\% & 33.41 \\
9 & 38.5/72.99\% & 30.80 & 42.6/80.76\% & 36.21 \\
**12** & 41.8/72.86\% & 33.44 & 45.9/80.01\% & 39.02 \\
**15** & 46.8/74.38\% & 37.44 & 50.1/79.62\% & 42.59 \\
**18** & 52.5/78.01\% & 42.00 & 56.8/84.40\% & 48.28 \\ \hline \hline \end{tabular} Note: \(\sigma_{\text{re}}\) represents the stress when the volume changed from compressive to expansive, \(\sigma_{\text{ds}}\) represents the stress level in the volume expansion stage.  
\end{table} 
Table 3: The creep parameters \(\sigma_{s_{1}}\) and \(\sigma_{s_{2}}\) under different confining pressures. By solving Equation (28), the \(\delta_{j}\) values can be obtained, which yield the approximate values of the creep parameters.