# Creep Model of Weakly Cemented Soft Rock Considering Damage and Secondary Development in FLAC3D
## 3 Creep Damage Constitutive Model Considering Damage for Weakly Cemented Soft Rock
### Model Validation  
#### 3.4.1 Parameter Identification  
To validate the rationality of the proposed creep model that accounts for damage in weakly cemented soft rock, the Levenberg-Marquardt (L-M) algorithm was employed for model fitting. The L-M algorithm combines the strengths of the Gauss-Newton method and gradient descent, rendering it highly effective for handling problems with multiple parameters and complex linearization. Additionally, it promotes optimal solutions during both early and late iterations, thereby circumventing the pitfalls of converging into local optima. Based on the method described, parameter inversion was performed for the creep model, yielding rheological parameters, as shown in Table 4. The fitting results are presented in Figure 9. (Low stress levels correspond to levels 1 to 5, while high stress levels refer to level 6, at which the rock enters an accelerated creep state.) When extending the model from a one-dimensional to a three-dimensional configuration, the bulk modulus and shear modulus must adhere to the following relationship:  
\[\left\{ \begin{array}{l}G=\frac{E_{0}}{2(1+\mu)}\\ K_{0}=\frac{E_{0}}{3(1-2\mu)}\end{array} \right. \tag{16}\]  
In the equation, \(\mu\) represents [PERSON]'s ratio. For analytical convenience, the shear modulus \(K_{0}\) and bulk modulus \(G\) are unified into the elastic modulus \(E_{0}\). As shown in Table 4 and Figure 8, the proposed model precisely delineates the three stages of creep failure in weakly cemented soft rock, with a fitting accuracy exceeding 99%. Compared to the traditional Nishihara model, it adeptly captures the phenomenon of a sharp increase in creep rate and nonlinear strain-time growth during testing. Furthermore, compared to the enhanced Nishihara model without considering damage [42], it exhibits superior capability in reflecting the mechanical characteristics of the creep process during the accelerated creep stage. This results in a marked improvement in fitting accuracy, which in turn effectively mitigates the limitations of the Nishihara model in accelerated creep failure description. Consequently, the improved Nishihara model considering damage proposed in this study can accurately depict the full-stage creep characteristics of weakly cemented soft rock. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & **Confining** & **Stress** & \(G\) **or** & \multirow{2}{*}{**K/GPa**} & \multirow{2}{*}{**\(\eta^{\text{ve/(GPa\_h)}}\)**} & \multirow{2}{*}{**\(\eta^{\text{ve/(GPa\_h)}}\)**} & \multirow{2}{*}{**\(\lambda\)**} & \multirow{2}{*}{**\(\alpha\)**} & \multirow{2}{*}{**\(\mathbf{R^{2}}\)**} \\  & **Pressure** & **/MPa** & **\(\mathbf{E_{0}/GPa}\)** & & & & & & \\ \hline \multirow{8}{*}{Model in this study} & & 5.5 & 1.84 & 49.12 & 6.51 & - & - & - & 0.992 \\  & & 6.3 & 1.89 & 28.53 & 2.79 & - & - & - & 0.986 \\  & 0 & 7.1 & 1.84 & 18.57 & 2.92 & - & - & - & 0.99 \\  & & 7.9 & 1.80 & 17.71 & 2.12 & - & - & - & 0.982 \\  & & 8.7 & 14.5 & 10.32 & 1.34 & - & - & 59.35 & 0.962 \\  & & 9.5 & 1.42 & 29.66 & 5.31 & 5.51 & 0.32 & 0.02 & 0.997 \\ \cline{2-10}  & & 11.5 & 1.67 & 54.42 & 1.01 & - & - & - & 0.992 \\  & & 13.2 & 1.64 & 20.04 & 0.26 & - & - & - & 0.999 \\  & 2 & 14.9 & 1.55 & 4.83 & 0.51 & - & - & 0.34 & 0.992 \\  & & 16.6 & 1.41 & 4.21 & 0.16 & - & - & 26.44 & 0.997 \\  & & 18.3 & 1.29 & 0.95 & 0.49 & - & - & 0.196 & 0.999 \\  & & 20.0 & 1.18 & 25.68 & 0.54 & 7.08 & 0.43 & 2.46 \(\times\) 10\({}^{5}\) & 0.992 \\ \hline \multirow{2}{*}{Model in reference [42]} & 0 & 9.5 & 1.41 & \(8.2\times 10^{3}\) & 0.16 & 0.62 & 6.22 \(\times\) 10\({}^{-6}\) & - & 0.98 \\  & 2 & 20 & 1.18 & 138.52 & 0.43 & \(1.23\times 10^{11}\) & \(7.51\times 10^{-5}\) & - & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Creep parameters of weakly cemented soft rock in different models. Figure 9: Creep test and model fitting curve of weakly cemented soft rock under different conditions. (**a**) \(\sigma_{3}=0\) MPa (low stress). (**b**) \(\sigma_{3}=0\) MPa (high stress) [42].