## 5 Conclusions  
Since water and load are two important factors affecting the mechanical properties of unsaturated soil, the damage variable evolution laws of unsaturated soil under the coupling effect of moistening and stress fields were explored. Based on the fractional calculus theory and the effective stress principle of unsaturated soil, a constitutive model of unsaturated soil was established. The proposed model can provide a scientific basis for the deformation and instability of unsaturated soil. The sensitivity of the main parameters of the proposed model was discussed, and the following conclusions were reached.  
1. Based on the classical soil mechanics principle, unsaturated soil mechanics and VG model, the mathematical formula for effective stress of unsaturated soil expressed by saturation is established in this paper. On this basis, the mathematical formula of mechanical shear strength of unsaturated soil, considering the coupling of moistening and stress fields, is established by Schrefler's shear strength formula.
2. A fractional-order strain-hardening model based on a conventional unsaturated triaxial shear test is established by using fractional-order calculus theory. Based on the principle of damage mechanics, the coupling damage variable of unsaturated soil under the influence of stress and water content is established. Finally, a fractional-order strain-hardening model of unsaturated soil under the coupling of stress and moistening fields is established.
3. The experimental data of unsaturated loess with strain-hardening characteristics are used to verify the proposed model. Results show that the model can describe the whole process of strain-hardening of unsaturated soil under the coupling of stress and moistening fields.
4. Through discussions on parameters of the proposed model, it is found that there is an initial threshold as unsaturated soil being damaged in coupling of the stress and moistening fields. The initial threshold value increases with the increase in confining pressure and decreases with the increase in matric suction. The hardening ability of unsaturated soil increases with the increase in matric suction or confining pressure. The overall mechanical properties of unsaturated soils improve nonlinearly with the increase of matric suction or confining pressure.  
The constitutive model proposed in this study was a discussion of the mechanical properties of unsaturated soil under the coupling effect of the moistening and stress fields, which could theoretically describe the stress-strain process. However, due to the lack of complete physical experimental data, it was difficult to discuss parameters \(\delta\) and \(\beta\) in the validation process. Further validation of the proposed model is required with experimental results of multiple groups of unsaturated soil.  
Conceptualization, H.Z.; Methodology, H.Z.; Resources, H.Z. and P.W.; Writing--original draft, P.W.; Writing--review & editing, H.Z.; Supervision, P.W.; Funding acquisition, H.Z. and P.W. All authors have read and agreed to the published version of the manuscript.  
This research was financially supported by Sichuan Huaxi Group Co., Ltd. (No. HXXX2019/015, No. HXXX2019/019, No. HXXX2018/030), the Open Fund of Sichuan Provincial Engineering Research Center of City Solid Waste Energy and Building Materials Conversion and Utilization Technology (No. GF2022 ZC009), and the Open Fund of Sichuan Engineering Research Center for Mechanical Properties and Engineering Technology of Unsaturated Soils (No. SC-FBHT2022-04).  
Not applicable.  
**Data Availability Statement:** Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. All data shown in the figures and tables can be provided on request.  
The authors declare no conflict of interest.