## 4 Evaluation of Damage Degree  
The different indexes analyzed above indicate different degrees of rock damage. Damage variable assessment is an important analysis to measure the irreversible deterioration degree of rock materials, which can quantitatively evaluate the degree of rock damage. The choice of expression method should be combined with the difference and complexity of the physical properties of materials. At present, scholars mainly quantitatively evaluate the deterioration degree of stone heritage from the non-destructive and micro-level, such as physical properties, mechanical properties, and mineral chemical composition [24; 25]. For example, [PERSON] et al. [26] and [PERSON] et al. [27] evaluate the weathering of tombstones and stupas according to surface hardness. [PERSON] et al. [28] and [PERSON] et al. [29] used wave velocity to analyze the internal deterioration of the Yungang Grottoes and the Beijing Xihuang Temple to provide a theoretical basis for formulating appropriate protection measures. Compared with the non-dimensional evaluation, the comparison between indicators tends to be a qualitative evaluation. [PERSON] et al. [30] and [PERSON] et al. [31] studied several advances in the theoretical study of rock damage mechanics and believed that rock damage degree could be defined by wave velocity variation. [PERSON] et al. [32] and [PERSON] et al. [33] proposed evaluating the grade of rock deterioration through strength change and proposed a decay prediction model for sandstone mechanical indexes under freeze-thaw action. This study refers to the above research to define the damage factors and evaluate the degree of weathering of samples. In addition, [PERSON] et al. [34] developed three models for predicting the degree of flint thermal response damage based on density, wave velocity and porosity. The results showed that the prediction model similar to the neural network was more accurate than MLR and ANFIS, and the damage factor based on porosity was more effective than that based on elastic modulus. To explore the degree of damage to sandstone caused by different freeze-thaw modes, two damage factors, strength damage factor \(D_{\sigma}\) and wave velocity damage factor \(D_{\
u}\), were introduced to quantitatively analyze the damage degree of rock. The calculation methods are shown in Equations (6) and (7), and the results are shown in Table 3.  
\[D_{\sigma}=1-\frac{\sigma_{1}}{\sigma_{0}} \tag{6}\]  
Figure 17: Correlation curve between surface hardness and UCS of rock sample. \[D_{v}=1-\left(\frac{v_{1}}{v_{0}}\right)^{2} \tag{7}\]  
The change in damage factors of rock samples under different freeze-thaw modes is shown in Figure 18. It can be seen that under the effect of different freeze-thaw modes, the strength damage factors of rock samples show a trend of rapid growth at first and then slow growth. Among them, the damaging effect of submerged freeze-thaw is the strongest, followed by periodic saturated freeze-thaw, and capillary freeze-thaw is relatively small. The strength decline of rock samples is the most significant in the first 20 freeze-thaw cycles, and the damage factor accounts for about 60% of the total damage degree, which reflects the deep damage to rock in the initial freeze-thaw stage. At the 80 th of freeze-thaw cycle, the damage factor of the submerged freeze-thaw rock sample is as high as 0.526, and that of capillary freeze-thaw and periodic saturated rock sample is 0.299 and 0.451, respectively. Wave velocity damage factor also relatively accurately reflects the degree of rock damage. Compared with immersion freeze-thaw, the wave velocity damage factors of rock under capillary freeze-thaw and periodic saturation freeze-thaw are low, mainly in the process of the experimental sample immersed in water entering the inside rock, and are more likely to cause greater damage to the rock. The moisture content of the rock itself and its microenvironment cannot be ignored.