## 5 Coupled Effects of the Freezing Temperature and Number of Cycles  
The peak compressive strength and elastic modulus of the rock materials can be utilized to characterize the extent of the freeze-thaw damage [30]. Previous discussions have thoroughly explored the effects of freezing temperature (\(T\)) and the number of freeze-thaw cycles (\(N\)) on the mechanical properties and failure morphology of coal-measure sandstone. The experimental results indicate that the damage deterioration effects of the freezing temperature (\(T\)) and cycle number (\(N\)) on coal-measure sandstone are interrelated. Figure 11a,b present the variation surfaces of the average peak compressive strength and the elastic modulus of the specimens as a function of the freeze-thaw cycle number and the freezing temperature, while Figure 12a,b show their fitted surfaces, with the fitting functions (surface equations) as follows:  
\[\sigma_{c}=0.0092N^{2}+0.0137T^{2}-0.8727N+0.5911T+0.0008 NT+28.7881,\ \mathrm{R}^{2}=0.9309. \tag{1}\]  
\[E=0.0009N^{2}+0.0001T^{2}-0.059N+0.0192T+0.0003 NT+2.4009,\ \mathrm{R}^{2}=0.9438 \tag{2}\]  
Figure 11: The mean values of the peak compressive strength and elastic modulus of the specimens at different freezing temperatures and number of freeze-thaw cycles coupling effect. (**a**) Peak compressive strength; (**b**) elastic modulus.  
Figure 12: The fitting surface of the peak compressive strength and elastic modulus of the specimens at different freezing temperatures and number of freeze-thaw cycles coupling effect. (**a**) Peak compressive strength; (**b**) elastic modulus.  
From Figure 11 and Equations (1) and (2), the following can be observed: 1 Both \(\sigma_{c}\) and \(E\) decrease with a reduction in the freezing temperature (\(T\)) and an increase in the cycle number (\(N\)), indicating an enhancement in the damage deterioration effect on coal-measure sandstone. 2 By examining the shape of the surface concerning the variations in freezing temperature (\(T\)) and cycle number (\(N\)), and comparing the coefficients of \(T^{2}\), \(T\) with \(N^{2}\), and \(N\) in Equations (1) and (2), it is evident that the effect of the freeze-thaw cycle number (\(N\)) on the deterioration of coal-measure sandstone is stronger than that of the freezing temperature (\(T\)). 3 The coefficient of the \(NT\) term in Equations (1) and (2) reflects the coupling effect between the freeze-thaw cycle number (\(N\)) and the freezing temperature (\(T\)). Notably, since \(T\) is negative, the \(NT\) term is consistently negative, indicating that an increase in the freeze-thaw cycle number (\(N\)) and a decrease in the freezing temperature (\(T\)) enhance the deterioration effect in coal-measure sandstone.  
Figure 13 compares the predicted values of the peak compressive strength and elastic modulus of coal-measure sandstone under different freeze-thaw cycle numbers and freezing temperatures with the experimental values (the slope of the line in the figure is 1), further demonstrating the rationality of the fitting function for the peak compressive strength of coal-measure sandstone. This surface equation can provide a significant reference for the strength assessment of coal-measure sandstone in cold-region open-pit mines. However, the above results are limited to experimental studies on coal-measure sandstone, and when the properties of the rock change, Equations (1) and (2) need to be revised.