## 7 Conclusions  
In this paper, 20 freeze-thaw cycles were carried out on four types of fractured sandstone with different inclination angles, followed by conventional triaxial and fatigue loading tests, combined with nuclear magnetic resonance technology to analyze the strength, deformation properties, and \(T_{2}\) spectrum distribution of the fractured sandstone, and a fatigue model was established based on the theory of internal variables in continuum mechanics. The conclusions are as follows:  
1. The strength of freeze-thawed fractured sandstone showed a decreasing trend with increasing fissure inclination, axial strain and shear damage angle increased with increasing fracture inclination, and fatigue load application enhanced the tendency of deterioration of the mechanical property parameters of fractured sandstone with increasing fissure inclination.
2. The distribution of NMR \(T_{2}\) spectra of the fractured sandstone was dominated by three peaks, and the pore distribution of the rock samples was dominated by medium pores and small pores. After 20 freeze-thaw cycles and fatigue loading, the \(T_{2}\) spectral curves of the four groups of fractured sandstone with different dips showed a small left shift, and many micropores sprouted inside the samples. The \(T_{2}\) spectral curve even showed a fourth peak after fatigue loading damage.
3. After 20 freeze-thaw cycles and fatigue loading, the \(T_{2}\) spectral area of the fractured sandstone changed significantly. The change rate of the spectral area ranged from 5.17% to 8.05% and 12.00% to 20.70%, respectively. The change amplitude of the spectral area was different, but, overall, the changing trend increased, the inclination angle of the fractured sandstone was different, and the degree of freeze-thaw-fatigue damage was also different.
4. Based on the internal variable theory of continuum mechanics, a fractured sandstone fatigue model was established with residual deformation as the internal variable. At the same time, the influence of the fracture inclination angle on the deformation modulus was taken into account. The experimental data fitted well with the theoretical model, which can well reflect the pre-fatigue mechanical behavior of fractured sandstone. Therefore, the NMR technique can be combined with the acoustic emission technique to carry out further research.  
K.Z.: Conceptualization; methodology; software; validation; formal analysis; writing--original draft; visualization. P.C.: Conceptualization; data curation; writing--review and editing; supervision; visualization. J.R.: Conceptualization; methodology; resources; supervision. Z.L.: Conceptualization; resources; data curation; supervision. K.W.: Conceptualization; software; supervision. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the National Natural Science Foundation of China (grant no. 12072259) and the Coal Union Foundation of Shaanxi Natural Science Foundation (grant no. 2019 JLP-01).  
Not applicable.  
Data Availability Statement: The data presented in this study are available on request from the corresponding author.  
This work was completed at the College of Architecture and Civil Engineering, Xi'an University of Science and technology. We thank our professors for their support and help in this project. At the same time, I would like to thank my colleagues for their continuous efforts in their work.  
Author [PERSON] was employed by the company China Construction Third Engineering Bureau Group Co., Ltd., Wuhan, China. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.