## 3 Results and Discussion
### Analysis of Numerical Simulation Results
#### 3.2.3 Cyclic Loading and Unloading Stress-Strain Analysis  
The numerical simulation and experimentally measured stress-strain curves are shown below. As seen in Figure 18, the spacing of the experimentally measured hysteresis loops showed two stages of 'dense-sparse'. With the increase in freeze-thaw cycles, the spacing of the hysteresis loops became more and more sparse. On the other hand, the simulated hysteresis loops show equal spacing, with no noticeable 'dense-sparse' phase. At the early loading stage, the peak values of the experimental curves are almost on the same curve (Peak-to-peak envelope), gradually shifting in the direction of strain increase. Many scholars [35; 36; 37; 38; 39; 40] have found that the envelope curve coincides with the uniaxial compression curve of the sandstone specimen. At the later loading stage, the curve deviates significantly from the uniaxial compression stress-strain curve. This phenomenon indicates  
Figure 17: Specim Contact Force diagram. (**a**) 20, (**b**) 40, (**c**) 60, (**d**) 80.  
that cyclic loading accumulates internal damage in the specimen, and irreversible strain damage occurs. Figure 18: _Cont._  
The loading and unloading methods used in the numerical simulation are not the same as the experiments, which are loaded by triangular waves, while sinusoidal waves load the numerical simulation. Figure 18 shows that the cumulative plastic strain is different between the simulation and the measurement under the same number of loading and unloading cycles, mainly because the numerical model does not have a compacting phase, and enters the elastic strain phase directly. For this reason, we define a new parameter, Cumulative Strain Difference (\(\epsilon_{cs}\)), to evaluate the strain difference between simulation and experiment. It is calculated from Equation (8). \[\varepsilon_{c}=\frac{\left|\frac{\varepsilon_{ms}-\varepsilon_{cs}}{\sigma_{ ms}-\sigma_{cs}}\right|}{\left(\frac{\varepsilon_{d}}{\sigma_{a}}\right)} \tag{8}\]  
where \(\varepsilon_{cs}\) is the strain generated by a loading in the simulation process, [%];  
\(\varepsilon_{ms}\) is the strain generated by the corresponding loading in the measured process, [%];  
\(\sigma_{cs}\) is the peak stress in the simulation process with the corresponding loading, [MPa];  
\(\sigma_{ms}\) is the peak stress in the measured process with the corresponding loading, [MPa];  
\(\varepsilon_{a}\) is the strain difference between measured and simulated during parameter calibration, [%];  
\(\sigma_{a}\) is the measured peak stress during parameter calibration, [MPa]. As an illustration, consider the stress-strain data of unfreeze-thaw rock samples (see Table 5 for details). Among them, the specimen was damaged at cycle level 13, and the strain values were not informative, so only the first 12 cycle levels were calculated. The specific calculations are shown in Table 5.  
Figure 18: Comparison of stress–strain curves with different freeze–thaw times. (**a**) 0 freeze–thaw (simulation), (**b**) 0 freeze–thaw (measurement), (**c**) 20 freeze–thaw (simulation), (**d**) 20 freeze–thaw (measurement), (**e**) 40 freeze–thaw (simulation), (**f**) 40 freeze–thaw (measurement), (**g**) 60 freeze–thaw (simulation), (**h**) 60 freeze–thaw (measurement), (**i**) 80 freeze–thaw (simulation), (**j**) 80 freeze–thaw (measurement). The results obtained from the calculations in Table 5 are plotted in Figure 19. The variation of the Cumulative Strain Difference for different numbers of freeze-thaw cycles is also shown in Figure 19.  
From the results, the maximum value of Cumulative Strain Difference is 37.10%, which occurs in cycle level 1, and the minimum value is 7.00%, which occurs in cycle level 10. The average Cumulative Strain Difference was calculated to be 20.57%. From the past studies [41; 42; 43; 44], the strain of numerical simulation of uniaxial compression is about 25% lower than the actual measurement on average. So, using a sinusoidal wave in the paper is feasible.