# A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate
## 6 Conclusions  
The creep characteristics of damaged coal samples were investigated through stepwise loading creep tests, and a new integer-order viscoelastic-plastic creep model was established. The conclusions are as follows:  
(1) Coal samples with identical initial damage were prepared using cyclic displacement-controlled loading-unloading protocols via an MTS servo-controlled testing system. As the number of cycles increases, the cumulative plastic deformation increases and the degree of damage escalates, ultimately yielding coal specimens with an initial damage variable value of 0.3 for this test. (2) The damaged coal samples exhibit distinct rheological behavior, progressing through three characteristic creep stages: decelerated creep, steady-state creep, and accelerated creep. Compared with intact specimens, coal samples subjected to higher loading rates demonstrate significantly shorter duration and higher deformation rates in the accelerated creep phase, highlighting the critical influence of loading rate on the creep behavior of coal. (3) Under the condition of maintaining a constant initial damage variable value, when the loading rate is progressively increased from 0.005 mm\(\cdot\)s\({}^{-1}\) to 0.1 mm\(\cdot\)s\({}^{-1}\) (50 to 100 microstrains\(\cdot\)s\({}^{-1}\)), the experimental results demonstrate that such rate enhancement significantly reduces the creep duration of coal samples. Notably, the damage rate during the accelerated creep stage exhibits a sharp upward trend, which provides crucial insights for investigating the rate effects in creep deformation mechanisms. (4) An integer-order viscoelastic-plastic creep model incorporating the initial damage effects of coal samples has been developed, with parameters successfully identified for \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(V\)/mm\(\cdot\)s\({}^{-1}\) & \(\sigma_{1}\)-\(\sigma_{3}\) & \(K\)/GPa & \(G_{0}\)/GPa & \(G_{1}\)/GPa & \(\eta_{1}\)/GPa\(\cdot\)h & \(\eta_{2}\)/GPa\(\cdot\)h & \(\eta_{c}\)/GPa\(\cdot\)h & \(\alpha\) \\ \hline \multirow{9}{*}{0.005} & 12.2 & 5.226 & 1.0014 & 7.8705 & 3.2220 & - & - & - \\  & 15.3 & 5.2464 & 1.0140 & 11.5109 & 8.2670 & - & - & - \\  & 18.4 & 0.9280 & 1.8270 & 10.5516 & 4.9341 & - & - & - \\  & 21.4 & 0.4296 & 1.6345 & 9.4184 & 5.1057 & 1.9520 & - & - \\  & 24.5 & 0.4370 & 1.8047 & 10.0072 & 2.1598 & 7.1888 & - & - \\  & 26.1 & 0.5314 & 3.5944 & 7.3451 & 5.5411 & 6.5304 & 3.2011 & 76.8 \\  & 27.6 & 0.3781 & 1.6321 & 8.4501 & 5.8510 & 0.5313 & 3.2145 & 45.4 \\ \hline \multirow{9}{*}{0.01} & 12.0 & 6.8729 & 1.0953 & 13.1982 & 2.3849 & - & - & - \\  & 15.0 & 0.9642 & 2.0594 & 10.4216 & 3.5060 & - & - & - \\  & 18.0 & 0.5760 & 5.9033 & 10.2757 & 4.4911 & - & - & - \\  & 21.0 & 0.6740 & 3.1974 & 10.8972 & 8.3033 & - & - & - \\  & 24.0 & 12.830 & 1.1541 & 8.4180 & 3.8084 & 0.3305 & - & - \\  & 25.5 & 7.9332 & 1.0650 & 7.7545 & 3.7684 & 3.5917 & 2.0055 & 56.3 \\  & 27.0 & 8.6911 & 1.0220 & 8.8841 & 3.2509 & 3.9112 & 2.7123 & 71.0 \\ \hline \multirow{9}{*}{0.05} & 13.7 & 0.7721 & 3.4427 & 9.8614 & 1.4964 & - & - & - \\  & 17.1 & 0.6101 & 5.7079 & 7.9588 & 2.9583 & - & - & - \\  & 20.6 & 1.3977 & 1.7510 & 8.8411 & 2.7169 & - & - & - \\  & 24.0 & 0.5584 & 6.4483 & 7.1948 & 3.4242 & 1.6842 & - & - \\  & 27.4 & 4.8800 & 1.2038 & 7.8248 & 4.2479 & 2.1687 & - & - \\  & 29.1 & 0.4242 & 3.2182 & 12.3183 & 5.4993 & 7.4177 & 3.0738 & 46.0 \\  & 30.8 & 0.4579 & 3.3866 & 4.8155 & 1.3154 & 1.9944 & 3.2081 & 8.2 \\ \hline \multirow{9}{*}{0.1} & 15.8 & 0.6634 & 11.2076 & 21.1316 & 3.4952 & - & - & - \\  & 19.7 & 7.4510 & 1.3789 & 17.5394 & 6.2641 & - & - & - \\  & 23.6 & 0.6675 & 6.8392 & 21.8978 & 7.2299 & - & - & - \\  & 27.6 & 1.2806 & 2.2003 & 14.7146 & 12.3827 & 0.4193 & - & - \\  & 31.5 & 3.0127 & 1.3061 & 12.4878 & 4.2972 & 4.8699 & - & - \\  & 35.5 & 4.1327 & 1.4932 & 12.944 & 1.5625 & 1.2069 & 8.4631 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values in nonlinear integer-order creep models with different loading rates. specimens under varying loading rates.