# Constitutive Characteristics of Rock Damage under Freeze-Thaw Cycles
## 5 Discussion
### Damage Percentage Analysis of Rocks under Different FTCs  
Based on the verified cumulative damage constitutive model describing the decay of rock shear strength, combined with Equation (36) for calculating rock damage percentage, the changes in rock damage mechanical properties under different freeze-thaw cycles can be obtained, as shown in Figure 8.  
Figure 8: Damage percentage of rocks under different FTCs: (**a**) Confining pressure = 0; (**b**) Confining pressure = 2 MPa; (**c**) Confining pressure = 5 MPa; (**d**) Confining pressure = 10 MPa.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number of FTCs**} & **Confining** & **Extreme Stress** & **Modulus of Elasticity** \\  & **Pressure/MPa** & **/MPa** & **/GPa** \\ \hline
10 & 0 & 4.23 & 0.867 \\
10 & 2 & 14.5 & 1.295 \\
10 & 5 & 19.6 & 1.452 \\
10 & 10 & 24.3 & 1.565 \\
20 & 0 & 3.8 & 0.761 \\
20 & 2 & 12.7 & 1.156 \\
20 & 5 & 18.9 & 1.289 \\
20 & 10 & 23.5 & 1 325 \\
30 & 0 & 3.74 & 0.529 \\
30 & 2 & 11.3 & 0.890 \\
30 & 5 & 18.1 & 1.066 \\
30 & 10 & 22.9 & 1.240 \\
40 & 0 & 3.3 & 0.515 \\
40 & 2 & 10.5 & 0.710 \\
40 & 5 & 17.2 & 0.917 \\
40 & 10 & 21.2 & 0.932 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter values of cumulative damage model of rock shear strength under different normal stresses and FTCs.  
(1) It can be seen from Figure 8 that, under the same number of freeze-thaw cycles, the initial damage percentage of rocks gradually increases with the increase in freeze-thaw cycles. In addition, it can be seen from Figure 8a that with the increase in freeze-thaw cycles, the strain value required for the sliding zone soil to reach the damage variable of 1 under the same normal stress load also increases, i.e., as shown in Figure 8c, when the rock passes 10, 20, 30 and 40 times and the freeze-thaw cycle reaches the failure value 1, its shear strain is 52%, 68%, 75% and 81%, respectively. A larger number of freeze-thaw cycles can lead to a larger shear displacement of the rock, and the change is obvious. Under the same number of freeze-thaw cycles, a larger number of freeze-thaw cycles will cause greater damage to the rock material.  
(2) It can be seen from Figure 8b-d that, with the increase in the number of freeze-thaw cycles, the change in rock damage value tends to be gentle under different freeze-thaw cycles. At the same time, when the peak value of the curve describing the change of damage value reaches 1, the required final strain value gradually increases, indicating that more cracks appear in the rock with the increase in the number of freeze-thaw cycles. With the increase in the number of freeze-thaw cycles, cracks appear continuously in the rock, and obvious plastic zones appear along with them, making the rock produce plastic hardening characteristics under the accumulation of freeze-thaw cycles. Meanwhile, with the increase in the number of freeze-thaw cycles, the rock is damaged to different degrees. With the increase in freeze-thaw cycles, the greater the damage degree of the rock, the more damaged parts in the rock. The energy required to produce further damage is also greater, so the ductile failure characteristics of the rock are becoming more and more obvious.