# Performance of Stone Masking Asphalt Mixtures Fabricated by Different Compaction Methods
## 3 Results
### Fatigue Performance Comparison
#### 3.5.2 Fatigue Equation  
Substituting the values of the regression coefficients in Table 17 into Equation (3), we obtained the equivalent fatigue lives of the samples designed by different methods and corresponding to different stress ratios at different failure probabilities. The results are presented in Table 18. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Design Methods**} & \multirow{2}{*}{**Coefficient Type**} & \multicolumn{6}{c}{**Coefficients of the Weibull Distribution Equation Under the Following Stress Ratios S**} \\ \cline{3-7}  & & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** \\ \hline \multirow{4}{*}{Marshall} & \(m_{0}\) & 7.0395 & 4.8676 & 2.743 & 3.9622 & 2.7147 \\  & \(\ln t\) & 70.125 & 41.525 & 21.27 & 27.611 & 17.04 \\  & \(R^{2}\) & 0.9699 & 0.9978 & 0.9881 & 0.9741 & 0.9706 \\ \hline \multirow{4}{*}{SGC} & \(m_{0}\) & 6.9635 & 4.4745 & 2.7435 & 3.7427 & 4.7643 \\  & \(\ln t\) & 70.358 & 38.579 & 21.825 & 26.925 & 30.408 \\  & \(R^{2}\) & 0.9953 & 0.9852 & 0.9921 & 0.9961 & 0.9806 \\ \hline \multirow{4}{*}{VVTM} & \(m_{0}\) & 6.8605 & 4.2347 & 2.6284 & 4.574 & 4.1766 \\  & \(\ln t\) & 69.72 & 37.154 & 21.09 & 33.143 & 27.369 \\ \cline{1-1}  & \(R^{2}\) & 0.9936 & 0.9873 & 0.9632 & 0.9704 & 0.9716 \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Fatigue test results after fitting the data to the Weibull distribution. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Design Methods**} & \multirow{2}{*}{**\(P\) (\%)**} & \multicolumn{6}{c}{**Equivalent Fatigue Life (\(\overline{N}\)) under the Following Stress Ratio S**} \\ \cline{3-7}  & & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** \\ \hline \multirow{4}{*}{Marshall} & 5 & 13,901 & 2754 & 790 & 502 & 178 \\  & 10 & 15,398 & 3193 & 1026 & 602 & 232 \\  & 20 & 17,130 & 3725 & 1349 & 728 & 306 \\  & 30 & 18,310 & 4101 & 1601 & 819 & 364 \\  & 40 & 19,268 & 4416 & 1825 & 897 & 416 \\  & 50 & 20,122 & 4701 & 2040 & 969 & 465 \\ \hline \multirow{4}{*}{SGC} & 5 & 15,951 & 2859 & 965 & 602 & 317 \\  & 10 & 17,688 & 3358 & 1255 & 730 & 369 \\  & 20 & 19,701 & 3971 & 1650 & 892 & 432 \\  & 30 & 21,074 & 4410 & 1957 & 1011 & 476 \\  & 40 & 22,189 & 4778 & 2231 & 1113 & 514 \\  & 50 & 23,183 & 5116 & 2494 & 1207 & 548 \\ \hline \multirow{4}{*}{VVTM} & 5 & 16,808 & 3204 & 986 & 733 & 344 \\  & 10 & 18,667 & 3798 & 1297 & 857 & 409 \\ \cline{1-1}  & 20 & 20,825 & 4535 & 1725 & 1010 & 490 \\ \cline{1-1}  & 30 & 22,298 & 5066 & 2062 & 1119 & 548 \\ \cline{1-1}  & 40 & 23,497 & 5514 & 2365 & 1211 & 597 \\ \cline{1-1}  & 50 & 24,566 & 5926 & 2656 & 1294 & 642 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Equivalent fatigue lives of the specimens at different failure probabilities. On a log-log scale, the fatigue life \(N\) of the asphalt mixture and the stress ratio \(S\) are linearly related as:  
\[\ln N=a-b\ln S, \tag{4}\]  
where \(a\) and \(b\) are the regression coefficients. The coefficients \(a\) and \(b\) in the fatigue equation define the intercept and slope of the straight line, respectively. The fatigue life \(N\) is an increasing function of \(a\), whereas \(b\) reflects the sensitivity of the fatigue life to changes in the stress ratio. In other words, as \(a\) increases or \(b\) decreases, the fatigue performance improves. The parameters and correlation coefficients of the fatigue equation are shown in Table 19.  
Considering that fatigue lines are sometimes expressed in the form \(S\)= f(\(N\)), the corresponding fatigue curves were constructed at failure probabilities of 5% and 50% (seeing Figure 8). At the same failure probability, the coefficients \(a\) and \(b\) were larger and smaller, respectively, in the VVTM-designed SMA-13 mixture than in the Marshall-designed and SGC-designed mixtures. This result demonstrates the higher fatigue performance of the VVTM-designed mixture than of the other mixtures. Figure 8: Fatigue curves of the SMA-13 asphalt mixtures designed by the three methods.