## 4 Determination of Duncan-Chang Model Parameters
The regression relation of \(\rm E_{i}\) to FTC could be fitted by the exponential function, and fitting parameters are shown in Table 7.  
The failure principal-stress deviator \((\sigma_{1}-\sigma_{3})_{\text{{\it{eff}}}}\) and limiting principal-stress deviator \((\sigma_{1}-\sigma_{3})_{\text{{\it{ult}}}}\) were determined. According to Equation (5), fracture ratio \(\text{R}_{\text{{\it{f}}}}\) could be obtained. As shown in Figure 6, \(\text{R}_{\text{{\it{f}}}}\) increased after the first cycle, and then decreased between five and 60 cycles. After 120 cycles, \(\text{R}_{\text{{\it{f}}}}\) obviously increased. The following rational function was used for curve fitting:  
\[y=\left(\text{A}_{3}*x^{3}+\text{B}_{3}*x^{2}+\text{C}_{3}*x+\text{D}\right)/ \left(x+\text{E}_{3}\right), \tag{8}\]  
where \(x\) is the FTC number that the samples underwent, and A\({}_{3}\), B\({}_{3}\), C\({}_{3}\), D\({}_{3}\), and E\({}_{3}\) are fitting parameters. The fitting result is shown in Figure 6, and parameter values are shown in Table 8. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\sigma_{3}\) (kPa) & **S (\%)** & \(\text{A}_{1}\) & \(\text{B}_{1}\) & \(\text{C}_{1}\) & \(\text{D}_{1}\) & \(\text{R}^{2}\) \\ \hline \multirow{6}{*}{100} & 0.0 & 4.09 & \(-\)0.19 & 6.02 & \(-\)0.0022 & 0.99996 \\  & 0.5 & 3.07 & \(-\)0.13 & 5.73 & \(-\)0.0019 & 0.99999 \\  & 1.0 & 2.07 & \(-\)0.12 & 5.63 & \(-\)0.0023 & 1.00000 \\  & 2.0 & 2.75 & \(-\)0.11 & 4.83 & \(-\)0.0012 & 0.99999 \\  & 3.0 & 2.80 & \(-\)0.13 & 5.17 & \(-\)0.0017 & 0.99999 \\ \hline \multirow{6}{*}{200} & 0.0 & 4.54 & \(-\)0.13 & 7.97 & \(-\)0.0011 & 0.99993 \\  & 0.5 & 3.72 & \(-\)0.14 & 7.80 & \(-\)0.0016 & 0.99998 \\  & 1.0 & 2.43 & \(-\)0.07 & 7.60 & \(-\)0.0017 & 0.99998 \\  & 2.0 & 2.89 & \(-\)0.07 & 6.99 & \(-\)0.0015 & 0.99999 \\  & 3.0 & 3.00 & \(-\)0.16 & 7.97 & \(-\)0.0025 & 0.99994 \\ \hline \multirow{6}{*}{300} & 0.0 & 4.36 & \(-\)0.14 & 10.45 & \(-\)0.0019 & 0.99991 \\  & 0.5 & 3.82 & \(-\)0.05 & 9.27 & \(-\)0.0013 & 0.99998 \\ \cline{1-1}  & 1.0 & 4.61 & \(-\)0.04 & 7.96 & \(-\)0.00001 & 0.99996 \\ \cline{1-1}  & 2.0 & 6.29 & \(-\)0.04 & 6.12 & 0.0018 & 0.99995 \\ \cline{1-1}  & 3.0 & 10.35 & \(-\)0.02 & 2.33 & 0.0082 & 0.99995 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of exponential functions that can reflect relationship between E\({}_{\text{{\it{i}}}}\) and FTC under different \(\sigma_{3}\) and S.  
According to [PERSON]'s suggestion, initial tangent modulus \(\rm E_{i}\) is related to \(\rm\sigma_{3}\), and its empirical relationship is as follows:  
\[\rm E_{i}=KP_{a}\Big{(}\frac{\rm\sigma_{3}}{P_{a}}\Big{)}^{N}, \tag{9}\]  
where K and N are material constant and \(\rm P_{a}=101.4\) kPa is the standard atmospheric pressure. We drew the diagram of \(\rm log_{10}(\rm\sigma_{3}/P_{a})\) and \(\rm log_{10}\) (\(\rm E_{i}/P_{a}\)), as shown in Figure 7. According to the relationship between \(\rm log_{10}(\rm\sigma_{3}/P_{a})\) and \(\rm log_{10}(\rm E_{i}/P_{a})\), the y-intercept and slope of the fitting line were \(\rm log_{10}(\rm K)\) and N, respectively, so material constants K and N were obtained, as shown in Table 9.