## 6 Conclusions  
In this paper, an unloading-damage test was carried out on a yellow sandstone sample with 60% unloading quantity under three types of confining pressure, and freeze-thaw mechanical tests were performed on the resulting unloaded-damaged samples. A freeze-thaw strength decay model based on the average change in porosity before and after freeze-thawing and the ability to consider the effect of unloading residual confining pressure was established. The model parameters were identified and verified by fitting the test data, and the following conclusions were finally obtained. 1. In the process of unloading sandstone under different initial confining pressures, the evolution process of the amount of unloading on the deformation response of the rock sample is largely the same. Unloading quantities of 60% and 90% were used as the dividing points to create three stages: the slow section, uniform section, and accelerated section. A 60% unloading quantity is the dividing point for the opening and continuous expansion of microfractures induced by unloading inside the rock sample. The uniaxial strength of the unloaded-damaged samples under different initial confining pressures with 60% unloading is roughly the same, which is smaller than the uniaxial strength of the original sample, reflecting the feasibility and reasonableness of the abovementioned unloading quantity as the classification criterion of unloading damage and the unloading-damage test;
2. The initial confining pressure of the three groups of unloaded-damaged samples is different, while the change in porosity with the number of freeze-thaw cycles and the value of change after freezing-thawing are constant across different initial confining pressures. This indicates that the degree and trend of freeze-thaw deterioration of unloaded-damaged samples with 60% unloading are largely the same, independent of the initial envelope pressure, so the average change in porosity can be used to measure the freeze-thaw deterioration of the unloaded-damaged samples;  
\begin{table} \begin{tabular}{c} **Tested** \\ **Values** \\ \end{tabular}  \\ \hline
0.19 & 107.672 & 107.17 & 122.725 & 122.22 & 143.628 & 143.32 & 35.568 & 34.32 \\
0.31 & 105.827 & 105.29 & 121.059 & 120.51 & 142.279 & 142.35 & 34.574 & 33.70 \\
0.47 & 103.416 & 103.61 & 118.873 & 118.453. The peak strength damage rate of the unloaded-damaged samples after freeze-thaw reloading (both uniaxial and triaxial) increases with the increase in the number of freeze-thaw cycles. The unloading residual pressure value inhibits the increase; a larger value of unloading residual pressure leads to a more obvious degree of inhibition;
4. Using the extended exponential attenuation model, the decay constant was modified by fitting the experimental data. The value of unloading residual confining pressure was introduced as the decay parameter and the freeze-thaw strength decay model was established with the easily determined average change in porosity. The values calculated by the model are largely consistent with the tested values. It is verified that the proposed model can reflect the decay of freeze-thaw strength using the average change in porosity after freeze-thaw cycles and can also consider the effect of unloading residual confining pressure. This reveals the micromechanical behavior induced by microstructure damage after freeze-thaw cycles of unloaded-damaged samples and predicts the decay in freeze-thaw strength. This provides some guidance for freeze-thaw hazard prediction of excavation and unloaded rock in cold regions. Conceptualization, X.C., W.D., L.C., B.M. and S.G.; Methodology, X.C., W.D., L.C., B.M. and H.J. ; Validation, B.M. and S.G.; Formal analysis, L.C., B.M., H.J. and W.W.; Investigation, W.W.; Data curation, W.D. ; Writing original draft, W.D. and S.G.; Writing review & editing, X.C., L.C., B.M., S.G. and W.W.; Supervision, X.C. ; Project administration, X.C., W.D. and H.J. ; Funding acquisition, X.C. All authors have read and agreed to the published version of the manuscript.