# Mechanical Properties and Damage Evolution of Heated Granite Subjected to Liquid Nitrogen Cooling
## 4 Discussion on Damage Evolution
### Model Validation  
The statistical damage constitutive model (Equation (8)) has been widely adopted [32; 33; 34]. This model has been proved to describe the stress-strain relationships of rock [32; 33]. For different types of rocks, [PERSON] et al. [34] found that this model fitted well with hard rock with few voids, but it did not fit well with rock with high voids. This is because the current model cannot describe the void deformation during the initial compression stage. Granite is one kind of hard rock with few voids. Due to the thermal treatment, some cracks are induced within granite, which results in an obvious initial crack closure in the experimental stress-strain curves (Figure 6). Notably, the initial crack closure in the experimental stress-strain curves contains two forms of strain: the void deformation and elastic deformation of the rock matrix. It is assumed that the initial void deformation does not affect the damage evolution of the rock. The focus of our study is to analyze the damage evolution of granite. Thus, the void deformation is not considered in the established theoretical curve; the void deformation is subtracted from the experimental curve. The experimental and theoretical stress-strain curves for heated granite after different cooling treatments are shown in Figure 6. The theoretical curves are in a good agreement with the experimental  
Figure 5: Effect of liquid nitrogen cooling on deformability: (**a**) [PERSON]’s modulus, (**b**) reduction rate of Young’s modulus between air and liquid nitrogen cooling, (**c**) [PERSON]’s ratio, and (**d**) increase rate of Poisson’s ratio between air and liquid nitrogen cooling. curves. On the one hand, the theoretical curves can accurately show the peak stress and its corresponding strain and can correctly reveal the effect of the temperature and cooling treatments on the stress-strain relationship. On the other hand, the theoretical curve can correctly describe the transformation from linear to nonlinear in the stress-strain curves, which successfully reflects the whole failure process: elasticity, damage, failure, and strain softening. The theoretical curves are in a good agreement with the experimental curves before the peak stress, which correctly reflects the damage behavior before the peak stress. The post-peak theoretical curves have a similar trend to the experimental curves, which can reveal the strain softening post-peak. Figure 6: _Cont._  
The reason for the difference between the theoretical and experimental curves in that the post-peak is analyzed. Firstly, the granite used in the experiment is brittle and more likely to have sudden unstable failure after the peak stress, resulting in a large slope of the post-peak curve for certain specimens. The theoretical curves describe the cumulative effect of matrix damage and represent the ideal mode of failure, which leads to a smaller slope of the post-peak theoretical curves (Figure 6e,h). Secondly, sliding deformation occurs between the splitting cracks within uniaxial specimens before the failure (Figure 6b,e,h), however, the sliding deformation is not considered in the theoretical curves. This leads to a higher strain of failure than the theoretical value. The parameters of the theoretical curve are determined by the tested strain, and a higher tested strain results in a slower slope of post-peak theoretical curves. In order to improve the description accuracy of the uniaxial post-peak curve by the theoretical model, the peak strains of certain specimens are reduced and then used to calculate the theoretical curves. Specimens without a significant sliding deformation need not do so, and their tested strain can be used to directly determine the theoretical curves. The distribution parameters \(m\) and \(f_{0}\) of the theoretical model depend on the measured data.