## 3 Results and Discussion  
### Logistic Damage Model of Asphalt Mixtures under Freeze-Thaw Cycles  
To illustrate the effects of eco-friendly basalt fiber on the performance of an asphalt mixture, a series of experiments was carried out for two kinds of asphalt mixtures, i.e., the control group (AM, asphalt mixtures without basalt fiber) and test group (BFAM, asphalt mixture with basalt fiber). The freeze-thaw damage can be characterized by air voids (_VA_) for the volume parameter, splitting tensile strength (_STS_), and indirect tensile stiffness modulus (_ITSM_) for mechanical parameters. Therefore, the damage degrees of _VA_, _STS_, and _ITSM_ were adopted as the evaluating indicators of asphalt mixture under various freeze-thaw cycles. The _VA_, _STS_, and _ITSM_ of AM and BFAM varying with freeze-thaw cycles are plotted in Figure 5a,c,e, and the corresponding damage degree results are shown in Figure 5b,d,f, in which the logistic models are expressed in curves with uncertain data. Based on the logistic damage model theory in the Section 2.4.1, the logistic damage models of _VA_, _STS_, and _ITSM_ for AM and BFAM could be established and listed in Table 2. These logistic damage models could be proven effective, due to the higher correlation coefficients R\({}^{2}\) above 0.97 and Adj. R-Squared values in Table 2 represent the fitting parameter accuracy. As plotted in Figure 5a,c,e, it can be clearly seen that the _VA_ results of asphalt mixtures gradually increased with the increasing of the freeze-thaw cycles, whereas the _STS_ and _ITSM_ results presented a decreasing trend. Simultaneously, Figure 5b,d,f illustrate that asphalt mixtures were gradually damaged with freeze-thaw cycles. Furthermore, the variation trend of the damage degree of the asphalt mixtures increased rapidly at first and then more slowly. Under the action of freeze-thaw cycles, the internal structure of asphalt mixtures was damaged due to volume expansion and temperature stress. The adhesion capability between asphalt and aggregates became weaker and weaker under the continuous action of freeze-thaw cycles. Before 9 freeze-thaw cycles, the air voids in asphalt mixture first extended and then, adjacent air voids were coalesced, leading to the significant variation trend. However, the expansion and formation of air voids became slow after 9 freeze-thaw cycles. In addition, by comparative analysis of control and test groups, the addition of basalt fiber can significantly improve the _STS_ and _ITSM_, and decline the _VA_ of asphalt mixture. Meanwhile, the damage degree of eco-friendly basalt fiber modified asphalt mixture has been as well greatly reduced. This may be owing to the fact that the basalt fiber formed a spatial networking structure, playing the role of reinforcement and toughening. Due to the poor absorption of water, but good adhesiveness with asphalt for basalt fiber, basalt fiber can also limit shrinkage cracking due to temperature stress under freeze-thaw cycles. From Table 2, by comparative analysis of the damage characteristics (i.e., _VA_, _STS_, and _ITSM_) of two kinds of asphalt mixtures, i.e., ordinary asphalt mixture (AM) and basalt fiber modified asphalt mixture (BFAM), it clearly shows that the model parameter \"\(A_{2}\)\" values of the logistic damage model of AM are always more than those of BFAM. Furthermore, the model parameter \"\(x_{0}\)\" values of AM are higher than those of BFAM at all times. The variations of the model parameters between those asphalt mixtures quantificationally demonstrate that the addition of basalt fiber into an asphalt mixture can significantly reduce the damage degree of asphalt mixtures and slow down the damage grow rate.