## 4 Determination of Duncan-Chang Model Parameters
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**S (\%)** & **Parameters** & **FTC = 0** & **FTC = 1** & **FTC = 5** & **FTC = 10** & **FTC = 30** & **FTC = 60** & **FTC = 120** \\ \hline \multirow{2}{*}{0.0} & K & 102.03 & 87.10 & 73.45 & 67.45 & 51.88 & 53.52 & 45.61 \\  & N & 0.34 & 0.37 & 0.48 & 0.47 & 0.52 & 0.54 & 0.53 \\ \hline \multirow{2}{*}{0.5} & K & 87.30 & 81.85 & 69.50 & 63.68 & 51.88 & 51.44 & 44.40 \\  & N & 0.37 & 0.37 & 0.50 & 0.49 & 0.52 & 0.49 & 0.51 \\ \hline \multirow{2}{*}{1.0} & K & 74.56 & 72.28 & 66.37 & 59.16 & 51.29 & 49.75 & 41.60 \\  & N & 0.44 & 0.44 & 0.52 & 0.56 & 0.48 & 0.48 & 0.56 \\ \hline \multirow{2}{*}{2.0} & K & 74.66 & 69.18 & 62.95 & 55.46 & 45.81 & 45.64 & 40.28 \\  & N & 0.42 & 0.47 & 0.55 & 0.59 & 0.52 & 0.49 & 0.55 \\ \hline \multirow{2}{*}{3.0} & K & 81.58 & 71.12 & 64.57 & 57.54 & 48.08 & 47.29 & 41.03 \\  & N & 0.41 & 0.45 & 0.56 & 0.61 & 0.57 & 0.46 & 0.55 \\ \hline \end{tabular}
\end{table} 
Table 9: Values of material constants K and N.  
Figure 6: Variation of average \(\rm R_{f}\) of different confining pressure with FTCs. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**S (\%)** & **Parameters** & **FTC = 0** & **FTC = 1** & **FTC = 5** & **FTC = 10** & **FTC = 30** & **FTC = 60** & **FTC = 120** \\ \hline \multirow{2}{*}{0.0} & K & 102.03 & 87.10 & 73.45 & 67.45 & 51.88 & 53.52 & 45.61 \\  & N & 0.34 & 0.37 & 0.48 & 0.47 & 0.52 & 0.54 & 0.53 \\ \hline \multirow{2}{*}{0.5} & K & 87.30 & 81.85 & 69.50 & 63.68 & 51.88 & 51.44 & 44.40 \\  & N & 0.37 & 0.37 & 0.50 & 0.49 & 0.52 & 0.49 & 0.51 \\ \hline \multirow{2}{*}{1.0} & K & 74.56 & 72.28 & 66.37 & 59.16 & 51.29 & 49.75 & 41.60 \\  & N & 0.44 & 0.44 & 0.52 & 0.56 & 0.48 & 0.48 & 0.56 \\ \hline \multirow{2}{*}{2.0} & K & 74.66 & 69.18 & 62.95 & 55.46 & 45.81 & 45.64 & 40.28 \\  & N & 0.42 & 0.47 & 0.55 & 0.59 & 0.52 & 0.49 & 0.55 \\ \hline \multirow{2}{*}{3.0} & K & 81.58 & 71.12 & 64.57 & 57.54 & 48.08 & 47.29 & 41.03 \\  & N & 0.41 & 0.45 & 0.56 & 0.61 & 0.57 & 0.46 & 0.55 \\ \hline \end{tabular}
\end{table} 
Table 8: Parameters of rational function that could reflect the relationship between \(\rm R_{f}\) and FTC under different S.  
With the increase of FTC, the change rule of material constant K rapidly decreased below 30 cycles, and then slowly decreased after 60 cycles. The regulation of N increased below 10 cycles, and then decreased between 10 and 60 cycles; it then increased after 120 cycles. The regression relation of K-FTC and N-FTC could be fitted by the exponential function; parameters of this function are shown in Table 10. And the fitting result is shown in Figure 8.  
Stress level \(\mathrm{S_{L}}\) is defined as the ratio of actual principal-stress difference (\(\sigma_{1}-\sigma_{3}\)) to failure principal-stress difference (\(\sigma_{1}-\sigma_{3}\))/:  
\[\mathrm{S_{L}}=\frac{(\sigma_{1}-\sigma_{3})}{(\sigma_{1}-\sigma_{3})_{f}}, \tag{10}\]  
By substituting Equations (5), (6), (9) and (10) into Equation (4), we can get the following formula:  
\[\mathrm{E_{t}}=\mathrm{K}\mathrm{P_{a}}\bigg{(}\frac{\sigma_{3}}{\mathrm{P_{a} }}\bigg{)}^{\mathrm{N}}\bigg{[}1-\frac{\mathrm{R}_{f}(1-\sin\Phi)(\sigma_{1}- \sigma_{3})}{2\cos\Phi+2\sigma_{3}\sin\Phi}\bigg{]}^{2}. \tag{11}\]  
Simplified:  
\[\mathrm{E_{t}}=\mathrm{E_{f}}\Big{(}1-\mathrm{R}_{f}\mathrm{S_{L}}\Big{)}^{2}. \tag{12}\]