## 4 Determination of Duncan-Chang Model Parameters  
As shown in Figure 1, relational curves between principal-stress deviation (\(\sigma_{1}-\sigma_{3}\)) and axial strain \(\varepsilon_{1}\) of samples are hyperbolic. Therefore, the Duncan-Chang model was suitably applied to analyze the relationship between sample stress and strain [13]. The relationship between model parameters and FTC is discussed below. [PERSON] et al. (1963) [13] analyzed the stress-strain curve of soil morphology on the basis of a large number of triaxial shear-test data, and suggested that the hyperbolic function was suitable for simulating the stress-strain relationship of soils:  
\[\sigma_{1}-\sigma_{3}=\frac{\varepsilon_{1}}{\text{a}+\text{b}\varepsilon_{1}}, \tag{3}\]  
where a and b are parameters that could be determined by conventional triaxial shear test. Since \(d\sigma_{2}=d\sigma_{3}=0\) in the triaxial test, tangent elastic modulus \(\text{E}_{\text{t}}\) could be obtained by taking the derivative of Equation (3) and eliminating axial strain \(\varepsilon_{1}\):  
\[\text{E}_{\text{t}}=\frac{1}{\text{a}}[1-\text{b}(\sigma_{1}-\sigma_{3})]= \text{E}_{\text{i}}\left[1-\frac{(\sigma_{1}-\sigma_{3})}{(\sigma_{1}-\sigma_{ 3})_{ult}I}\right], \tag{4}\]  
where \(\text{E}_{\text{i}}\) is the initial elastic modulus, and \((\sigma_{1}-\sigma_{3})_{ult}\) is the ultimate principal-stress deviation. Ultimate principal-stress deviation is difficult to be directly obtained through the test, but failure principal-stress difference is easy to determine. Therefore, in order to obtain the ultimate principal-stress deviation, the failure ratio was defined as follows:  
\[\text{R}_{f}=\frac{(\sigma_{1}-\sigma_{3})_{f}}{(\sigma_{1}-\sigma_{3})_{ult}}. \tag{5}\]  
According to Mohr-Coulomb strength theory:  
\[(\sigma_{1}-\sigma_{3})_{f}=\frac{2\text{c}\cos\Phi+2\sigma_{3}\sin\Phi}{1- \sin\Phi}. \tag{6}\]  
We transformed the coordinate system of Equation (3) to get the following equation:  
\[\frac{\varepsilon_{1}}{\sigma_{1}-\sigma_{3}}=\text{a}+\text{b}\varepsilon_{1}. \tag{7}\]  
According to Equation (7), triaxial test data are transformed into another coordinate, and curves of \(\varepsilon_{1}/(\sigma_{1}-\sigma_{3})\) and \(\varepsilon_{1}\) were drawn in Figure 4. It can be seen from Figure 4 that there was a first-order linear correlation between stress and strain in the \(\varepsilon_{1}/(\sigma_{1}-\sigma_{3})-\varepsilon_{1}\) coordinate. The stress-strain curve of the sample conformed to the hyperbolic function. The y-intercept of the fitting line was parameter a, and the slope was parameter b, as shown in Table 6. The reciprocal of parameters a and b are \(\text{E}_{\text{i}}\) and \((\sigma_{1}-\sigma_{3})_{ult}\), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**S (\%)**} & \multicolumn{4}{c}{**c (kPa)-FTC**} & \multicolumn{4}{c}{\(\varphi\) **(\({}^{\circ}\))-FTC**} \\ \cline{2-11}  & \(\mathbf{A_{1}}\) & \(\mathbf{B_{1}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{D_{1}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{A_{2}}\) & \(\mathbf{B_{2}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{D_{2}}\) & \(\mathbf{R^{2}}\) \\ \hline
0.0 & 13.58 & 1.75 & 1.32 & 0.021 & 0.993 & 55.02 & \(-\)1.69 & 46.16 & \(-\)0.0046 & 0.983 \\
0.5 & 1.66 & 0.91 & 1.66 & 0.029 & 0.962 & 52.09 & \(-\)1.52 & 44.63 & \(-\)0.0054 & 0.982 \\
1.0 & 12.47 & 1.04 & 2.21 & 0.033 & 0.962 & 52.00 & \(-\)1.53 & 44.03 & \(-\)0.0073 & 0.985 \\
2.0 & 10.80 & 2.14 & 2.85 & 0.026 & 0.985 & 52.62 & \(-\)1.64 & 43.04 & \(-\)0.0076 & 0.988 \\
3.0 & 10.72 & 2.93 & 3.15 & 0.026 & 0.986 & 52.07 & \(-\)2.02 & 42.91 & \(-\)0.0088 & 0.988 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of exponential and Fourier functions that can reflect relationship between c, \(\varphi\), and FTC under different S.  
In Figure 5, under different confining pressures, the variation tendencies of initial tangent modulus \(\rm E_{i}\) and FTC were similar, decreasing rapidly below 30 cycles, and then decreasing slowly after 30 cycles.