## 4 Conclusions  
This study further explored the freeze-thaw damage characteristics of asphalt mixtures reinforced with eco-friendly basalt fiber through air voids, splitting tensile strength, and indirect tensile stiffness modulus tests. Based on the damage characteristics, a logistic damage model was established to quantificationally analyze the damage degree and damage grow rate of asphalt mixtures. Additionally,  
Figure 9: Comparison between predictive and experimental values for AM and BFAM based on GM (1,3) of _VA_, _STS_, and _ITSM_ tests: (**a**) Damage degree of _VA_ for AM; (**b**) Damage degree of _VA_ for BFAM; (**c**) Damage degree of _STS_ for AM; (**d**) Damage degree of _STS_ for BFAM; (**e**) Damage degree of _ITSM_ for AM; (**f**) Damage degree of _ITSM_ for BFAM.  
multi-variable grey models were established for modelling the freeze-thaw damage characteristics of asphalt mixtures. Thus, the following conclusions can be drawn:  
* Freeze-thaw cycles habr a negative effect on the volume and mechanical properties of an asphalt mixture. From air voids, splitting tensile strength, and indirect tensile stiffness modulus, the addition of basalt fiber can significantly improve the freeze-thaw resistance and mechanical performance of an asphalt mixture, leading to a reinforcement mechanism.
* The logistic damage model can quantificationally demonstrate that adding basalt fiber could significantly reduce the damage degree of asphalt mixtures by about 25%, and slow down the damage grow rate by about 45% compared to a control group.
* The ninth freeze-thaw cycle may be the turning point of damage variation of an asphalt mixture, after which the expansion and formation of air voids became slower.
* Results demonstrated that the established multi-variable grey models can accurately predict the variation trend of damage characteristics of asphalt mixtures. GM (1,3) was proven to perform better in prediction accuracy compared to GM (1,2) under the same data. GM (1,3) is an effective prediction model for reflecting the evolution of freeze-thaw damage for asphalt mixtures.
* Appropriate basalt fiber content is recommended for modifying an asphalt mixture. It is necessary to check the dispersion of basalt fibers in an asphalt mixture to ensure the dispersion of basalt fiber. Compared to ordinary asphalt mixtures, the predicted cost is mainly reliant on the basalt fibers.  
Conceptualization, W.W. and Y.C.; Methodology, W.W. and G.T.; Validation, Y.C.; Formal Analysis, G.T. and G.M.; Investigation, W.W., G.M., X.S. and S.Y.; Writing-Original Draft Preparation, W.W.; Writing-Review & Editing, Y.C.; Project Administration, G.T.; Funding Acquisition, G.T. This research was funded by [National Natural Science Foundation of Chinal grant number [51678271], Science Technology Development Program of Jilin Province] grant number [20160204008 SF], supported by Graduate Innovation Fund of Jilin University grant number [101832018C003], Transportation Science and Technology Program of Jilin Province] grant number [2018-1-9], [the Education Department's \"13 th Five-Year\" Science and Technology Program of Jilin Province] grant number [JJKH20190015 KJ]. The authors would like to appreciate anonymous reviewers for their constructive suggestions and comments to improve the quality of the paper.  
The authors declare no conflict of interest.