# 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  
### Determination of Damage Variable and Damage Threshold  
[PERSON] et al. [44] highlighted that creep under low stress is predominantly attributable to the closure of internal cracks and the compaction of the rock matrix, while creep under high stress results in the generation of numerous microcracks and even macroscopic fractures. The formation of internal microcracks compromises the rock's internal structure, thereby inducing alterations in various rheological parameters. To evaluate these changes, a damage variable can be introduced to quantify the impairment of rheological parameters. Building on this premise, [PERSON] established the evolution equation of damage over time as follows [13]:  
\[D(t)=\frac{E_{0}-E_{\infty}}{E_{0}}(1-e^{-\alpha t}) \tag{1}\]  
where \(D(t)\)--damage function; \(E_{\infty}\)--final elastic modulus of the rock mass; \(E_{0}\)--initial elastic modulus of the undisturbed rock mass; \(\alpha\)--elastic modulus decay parameter; and \(t\)--time.  
[PERSON] et al. [45] proposed, based on their research, that the evolution of model parameters follows the same pattern as the elastic modulus, with parameters changing over time. The evolution equations for the model parameters are outlined as follows:  
\[\left\{\begin{array}{l}Z(t)=Z_{0}[1-D(t)]=Z_{0}[1-\frac{E_{0}-E_{\infty}}{E_ {0}}(1-e^{-\alpha t})]\\ D(t)=\frac{E_{0}-E_{\infty}}{E_{0}}(1-e^{-\alpha t})\end{array}\right. \tag{2}\]  
where \(Z(t)\) represents the parameter damage function. Total strain-time curves of weakly cemented mudstone specimens under step-loading conditions, as shown in Figures 5a and 6a, indicate that at the fifth stress level, the volumetric strain trend transitions from compression to expansion. According to Table 3, under uniaxial conditions, the instantaneous elastic modulus of weakly cemented soft rock demonstrates no notable attenuation at the first to fourth stress levels. However, at the fifth stress level (\(\sigma\) = 8.7 MPa), the instantaneous elastic modulus exhibits a noticeable attenuation. Under 2 MPa confining pressure, the instantaneous elastic modulus presents significant attenuation at the third stress level (\(\sigma\) = 14.9 MPa), with attenuation rates reaching 26.3% and 4.93%, respectively.  
According to Figure 7 ([PERSON] et al. [46]), the rock test curve can be divided into the following stages: OA is the crack closure stage, AB is the elastic stage, BC is the stable crack propagation stage, CD is the accelerated crack propagation stage, and DE is the post-peak failure stage. Referring to the research of [PERSON] et al. [46], during the accelerated crack propagation stage, the elastic modulus of rock specimens undergoes substantial changes. It can be concluded that weakly cemented mudstone enters the accelerated crack propagation stage at this point. The rapid development of internal cracks leads to the swift attenuation of the instantaneous elastic modulus, thus inducing the volumetric strain to transition from compression to dilation. Therefore, this study considers the significant attenuation of the instantaneous elastic modulus as the damage threshold, marking the onset of rock damage.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\sigma_{3}=0\) MPa} & \multicolumn{2}{c}{Stress level/MPa} & 5.5 & 6.3 & 7.1 & 7.9 & 8.7 & 9.5 \\ \cline{2-7}  & Modulus of instantaneous elasticity/GPa & 1.84 & 1.89 & 1.84 & 1.81 & 1.45 & 1.42 \\ \hline \multirow{2}{*}{\(\sigma_{3}=2\) MPa} & \multicolumn{2}{c}{Stress level/MPa} & 11.5 & 13.2 & 14.9 & 16.6 & 18.3 & 20 \\ \cline{2-7}  & Modulus of instantaneous elasticity/GPa & 2.03 & 2.01 & 1.93 & 1.78 & 1.69 & 1.56 \\ \hline \end{tabular}
\end{table}
Table 3: Instantaneous elastic modulus at different stress levels.