## 4 Unconfined Compressive Resilient Modulus Test
### Establishment of the Fatigue Model Based on the True Stress Ratio
The calculated results are presented in Table 6. According to the test setting parameters in Table 6, the four-point bending fatigue tests were carried out as shown in Figure 12b, and the test results are presented in Table 6.  
The fatigue life of the cement-treated aggregate base material was fitted by S-N fatigue equation with a nominal stress ratio and true stress ratio, respectively. The fitting results are shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\
u\) & \(\sigma\) & \(S_{fp}\) & \(t_{1}\)(\%) & \(t_{2}\)(\%) & \multicolumn{6}{c}{**Fatigue Life of Parallel Tests N\({}_{f}\)**} & \multirow{2}{*}{**Average N\({}_{f}\)**} \\ \cline{4-9} (MPa/s) & (MPa/s) & & & & & **1** & **2** & **3** & **4** \\ \hline
5 & 0.25 & 1.11 & 0.19 & 0.23 & 12,687,941 & 12,619,781 & 12,809,765 & 12,753,580 & 12,717,767 \\
13 & 0.65 & 1.30 & 0.50 & 0.50 & 2283 & 2251 & 2237 & 2189 & 2239 \\
20 & 1.00 & 1.63 & 0.77 & 0.61 & 567 & 596 & 582 & 603 & 587 \\
30 & 1.50 & 2.43 & 1.15 & 0.62 & 302 & 317 & 320 & 309 & 312 \\
40 & 2.00 & 2.61 & 1.54 & 0.77 & 30 & 37 & 33 & 36 & 34 \\
50 & 2.50 & 2.72 & 1.69 & 0.91 & 6 & 8 & 10 & 8 & 8 \\ \hline \hline \end{tabular}  
* where \(\
u\) is loading rate; \(\sigma\) is stress level; \(S_{f}\) is the flexural strength value; \(t_{1}\) is the nominal stress ratio, \(t_{1}=\sigma/1.3\) MPa, 1.3 MPa is the standard strength; and \(t_{2}\) is the true stress ratio, \(t_{2}=\sigma/S_{\varphi}\).  
\end{table} 
Table 6: Tests results of flexural fatigue under different stress levels. Figure 12: Fatigue test (**a**) Loading waveform (**b**) Flexural test. Figure 13: Fatigue curve fitted by nominal stress ratio and true stress ratio. It can be observed from Figure 13 that:  
1 When stress levels of 1.5 MPa, 2.0 MPa and 2.5 MPa were exerted on the specimens during the four-point bending fatigue tests, the nominal stress ratios calculated by the standard strength 1.30 MPa were larger than 1. For these stress ratios, the fatigue life should theoretically be less than 1. However, their actual average fatigue lives were 312, 34 and 8, respectively, as shown in Table 6. Meanwhile, when the nominal stress ratio was 1, the actual fatigue life was 247, as shown in Figure 13, which means that the fatigue curve will not go through the point of (1, 1). The reason for this is that the strength of the cement-treated aggregate base material increases with the loading rates. However, the fatigue equation characterized by the nominal stress ratio could not consider it. 2 When the true stress ratios were adopted in the _S-N_ fatigue equation to reveal the characteristics of the cement-treated aggregate base material, the discrepancy mentioned above was eliminated. Based on the test and fitting results, the fatigue curve characterized by the true stress ratio will pass through the point of (1, 1), as shown in Figure 13. The parameter \(\mathrm{k}^{\prime}\) of the fatigue equation \(\mathrm{N}_{f}=\mathrm{k}^{\prime}(1/t)^{n^{\prime}}\) was 1, so the equation can be expressed as \(\mathrm{N}_{f}=(1/t)^{n^{\prime}}\). This indicates that the fatigue life \(\mathrm{N}_{f}\) will be 1 when the stress ratio \(t\) is 1, which means the fatigue equation curve based on the true stress ratio can be extended to the strength failure point of (1, 1), where both the true stress ratio and the fatigue life are 1. The \(\mathrm{N}_{f}=1\) represents the strength failure where the fatigue life is 1. Therefore, based on the test and fitting of the results, the fatigue curve reveals the internal relationship between the strength failure and the fatigue failure. The fatigue equation based on the true stress ratio is more rational than the fatigue equation of the nominal stress ratio. The fatigue test results can be extended in both directions of the fatigue curve. However, if the fatigue curve characterized by the nominal stress extends in the two directions, it will lead to a larger deviation. The greater the scope of the extension, the greater the deviation.