## 3 Results
### Relationship between Weathering Index and UCS  
As a reference index, the quantitative relationship between UCS and the non-destructive index can be analyzed, and then the corresponding physical parameters can be measured  
Figure 14: UCS and loss rate of rock under freeze–thaw action, the solid lines show the trend of strength value change, and the dotted lines reflect the variation characteristics of strength loss rate. Figure 13: Surface hardness of rock samples under different test conditions. by non-destructive testing to evaluate the overall deterioration of stone cultural relics. In general, it is not possible to sample stone heritage sites for the determination of strength parameters for conservation purposes. Research on the internal relationship between indexes, especially in evaluating strength parameters based on non-destructive indexes and establishing their functional relationship, can better evaluate the deterioration degree of cultural heritages and evaluate the trend of change. [PERSON] et al. [18] established a quantitative relationship with compressive strength based on dry density, porosity, and surface hardness to predict rock mechanical parameters. [PERSON] et al. [19] took three types of rocks as research objects and took wave velocity as an independent variable to explore its internal relationship with strength, density, and other indicators. [PERSON] et al. [20], [PERSON] et al. [21], and [PERSON] et al. [22] compared the laboratory test results with the field test data and believed that the rock strength value could be inferred based on the nondestructive index. In this study, nondestructive indexes were taken as independent variables and strength parameters were taken as dependent variables to establish functional relationships among indexes. In a certain range, the more serious the internal deterioration of rock, the more significant the pore development, the smaller the ultrasonic wave velocity, and the lower the UCS. Wave velocities and UCS of each series of rock samples were fitted, as shown in Figure 15. There is an exponential relationship between wave velocity and compressive strength:  
\[\text{UCS}=ae^{(V_{p}/b)}+c \tag{2}\]  
where, \(a\), \(b\) and \(c\) are constants, and \(V_{p}\) is wave velocity. Pore porosity is an index reflecting the size and number of rock fissures. Surface hardness is one of the main indexes reflecting the damage to rock surface structure. In this paper, the variation trend of the two characterization indexes and UCS is analyzed, and the concrete relationship between the microstructure and mechanical properties of rock is discussed. The porosity of open holes was fitted to the measured points of compressive strength, as shown in Figure 16.  
Figure 15: Correlation curve between wave velocity and UCS of rock samples. The curves in Figure 13 show that there is a linear correlation between open porosity and UCS of rock samples under freeze-thaw action, and its variation trend is in line with the functional relationship of Equation (3):  
\[\text{UCS}=a-b\times\varphi \tag{3}\]  
where \(a\) and \(b\) are constants, and \(\varphi\) is the open porosity. Although the porosity and UCS of rock samples in different freeze-thaw environments are linearly fitted, the decrease rate of porosity is different due to the different uniformity of rock porosity changes caused by different freeze-thaw modes. Due to the advantages of a wide measuring range, arbitrary direction, and ease of carrying and operating, surface hardness is widely used in in situ testing [18; 23]. The relationships between surface hardness and compressive strength of rock samples were fitted, as shown in Figure 17. Linear and exponential fitting can be carried out between surface hardness and compressive strength of yellow sandstone, as shown in Equations (4) and (5). \[\text{UCS}=a\times R+b \tag{4}\]  
\[\text{UCS}=ae^{(R/b)}+c \tag{5}\]  
where \(a\), \(b\), and \(c\) are constants, and \(R\) is the value of surface hardness.