# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
## 3 Creep Properties of Coal Sample with Initial Damage  
Triaxial compression creep tests (TCCT) were conducted on initially damaged coal samples by applying incremental stresses at 40%, 50%, 60%, 70%, 80%, 85%, and 90% of their corresponding strength under different loading rates. The creep strain curves for each loading rate are presented in Figure 4.  
Figure 4 reveals that under different displacement rates, at lower applied stresses, the creep deformation stabilizes after undergoing a decelerated creep stage. As the stress level increases, the creep transitions into a steady-state creep stage. Upon reaching the final load stage, the creep enters an accelerated creep stage. When the loading rate reached \(V=0.1\) mm\(\cdot\)s\({}^{-1}\) (1000 microstrains\({}^{-1}\)), the accelerated creep failure stage exhibited instantaneous and abrupt characteristics, with a dramatic increase in deformation rate culminating in catastrophic instability (Figure 4d).  
As shown in Figure 5, the loading rate and stress level have a significant impact on the creep behavior of coal samples. Under low stress conditions, as the loading rate increases, the instantaneous strain of coal samples shows a slight downward trend. This is mainly due to the full development and propagation of internal fractures in coal samples at lower loading rates, which allows new microcracks to continuously form during the application of constant load. In the high-stress state, both extreme rates of \(V=0.002\) mm\(\cdot\)s\({}^{-1}\) (20 microstrains\({}^{-1}\)) and \(V=0.1\) mm\(\cdot\)s\({}^{-1}\) (1000 microstrains\({}^{-1}\)) experienced creep failure under the sixth-level load, leading to a significant increase in strain. Further analysis reveals that under the same loading rate, creep strain exhibits a transition from linear to nonlinear  
\begin{table}
\begin{tabular}{c c} \hline
**Loading Rates/mm\(\cdot\)s\({}^{-1}\)** & **Loading Stress (Proportion of TCS)/MPa** \\ \hline \hline \multirow{8}{*}{0.005 (50)} & 12.2 (40\%) \\  & 15.3 (50\%) \\  & 18.4 (60\%) \\  & 21.4 (70\%) \\  & 24.5 (80\%) \\  & 26.1 (85\%) \\  & 27.6 (90\%) \\ \hline \multirow{8}{*}{0.010 (1000)} & 12.0 (40\%) \\  & 15.0 (50\%) \\  & 18.0 (60\%) \\  & 21.0 (70\%) \\  & 24.0 (80\%) \\  & 25.5 (85\%) \\  & 27.0 (90\%) \\ \hline \multirow{8}{*}{0.050 (500)} & 13.7 (40\%) \\  & 17.1 (50\%) \\  & 20.6 (60\%) \\ \cline{1-1}  & 24.0 (70\%) \\ \cline{1-1}  & 27.4 (80\%) \\ \cline{1-1}  & 29.1 (85\%) \\ \cline{1-1}  & 30.8 (90\%) \\ \hline \multirow{8}{*}{0.100 (1000)} & 15.8 (40\%) \\  & 19.7 (50\%) \\ \cline{1-1}  & 23.6 (60\%) \\ \cline{1-1}  & 27.6 (70\%) \\ \cline{1-1}  & 31.5 (80\%) \\ \cline{1-1}  & 35.5 (90\%) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Graded stress table under different loading rates.  
growth as the load increases, indicating the existence of a stress threshold (24.0-31.5 MPa) that controls the creep stage. Notably, this threshold has a nonlinear correlation with the loading rate: when the loading rate increases from 0.002 mm\(\cdot\)s\({}^{-1}\) to 0.1 mm\(\cdot\)s\({}^{-1}\) (20 to 1000 microstrains\(\cdot\)s\({}^{-1}\)), the stress threshold first decreases from 24.7 MPa to 24.0 MPa and then rises back to 31.5 MPa, revealing the complex influence mechanism of the loading rate on the long-term stability of damaged coal samples.  
Figure 4: Axial strain creep properties.  
Figure 5: Diagram of the cumulative creep strain relationship under the same level of load with different loading rates.