## 4 Conclusions  
A direct tension test was conducted on two kinds of asphalt mixture (AC-13 and SMA-13) with five degrees of aging (UA, STOA, LTOA2d, LTOA5d and LTOA8d) at different test temperatures (10 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, 30 \({}^{\circ}\)C, 40 \({}^{\circ}\)C). The objective is to comprehensively investigate  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & \multicolumn{2}{c}{**Fitting Formula**} & \(\mathbf{R^{2}}\) \\ \hline AC-13 & c = 0.999 – 0.017d + 0.003T + 0.003d\({}^{2}\) – 9.590\(\times 10^{-5}\)T\({}^{2}\) – 0.002 dT & 0.95 \\ SMA-13 & c = 0.976 + 0.038d + 7.414\(\times 10^{-4}\)T – 0.003 + 4.200\(\times 10^{-7}\)T\({}^{2}\) – 0.002 dT & 0.94 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Non-linear surface fitting result.  
Figure 15: Non-linear surface fitting graph: (**a**) AC-13; (**b**) SMA-13.  
the effect of aging degree/test temperature/gradation on the cracking resistance of the asphalt mixture. The main findings of this paper are as follows:  
1. For both AC-13 and SMA-13, the tensile strength gradually ascends as the degree of aging progresses further, but descends as test temperature rises. Additionally, AC-13 has a higer tensile strength than SMA-13. The ultimate strain increases as the degree of aging degree progresses further, but decreases as the test temperature increases. SMA-13 can produce a higher ultimate strain compared with AC-13.
2. For both AC-13 and SMA-13, the pre-peak strain energy density increases as the degree of aging progresses further, but goes down as the test temperature increases. SMA-13 can produce a higher ultimate strain than AC-13. STOA conditions can boost the pre-peak and post-peak strain energy density, while LTOA conditions can reduce them. Additionally, the pre-peak and post-peak strain energy density both exhibit a tendency to increase first and then decrease, with their maximum occurring at 20 \({}^{\circ}\)C. Moreover, the superiority of SMA-13 lies in the post-peak strain energy density rather than the pre-peak strain energy density.
3. In the discussed region of aging degree and test temperature, SMA-13 has a higher CRI and aging coefficient value than AC-13 by 3-69%; this means that SMA-13 is endowed with a better cracking resistance whether it is under aging conditions or not.
4. STOA can enlarge the aging coefficient of the asphalt mixture, which means that STOA can increase cracking resistance even under aging conditions. Additionally, this positive effect can be weakened by an elevation in test temperature. LTOA can lower the aging coefficient, which indicates that LTOA can weaken the cracking resistance and the lowering effect can be boosted by elevations in the test temperature
5. The relationship between aging coefficient and the two factors (aging degree and test temperature) can be reflected by the functional expression of the non-linear surface (R\({}^{2}\) = 0.95 for AC-13 and R\({}^{2}\) = 0.94 for SMA-13).  
Conceptualization, S.Y.; data curation, S.Y. and K.L.; formal analysis, S.Y.; funding acquisition, Z.Z.; investigation, S.Y.; methodology, S.Y.; resources, K.L.; supervision, Z.Z.; validation, K.L.; writing--original draft, S.Y.; writing--review and editing, S.Y. and K.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Science and Technology Planning Project of Transportation of Guangdong Province, grant number 1912-0002.  
Data Availability Statement: Data sharing not applicable.  
The authors declare no conflict of interest.