## 3 Results and Discussion
### Determination of the Crack Initiation Criterion Based on Strain
#### 3.2.2 Experimental Verification of the Double-Strain Criterion  
The evolution curve of the generalized strain of the crack tip for different crack widths, different external temperatures, and different initial moisture contents, during the cooling time, is shown in Figure 11.  
Figure 11: The evolution of generalized strain at the crack tip. (**a**) Different crack widths; (**b**) different external temperatures; (**c**) different initial moisture contents.  
From Figure 11, it can be seen that the cracks in specimens A1, A2, B1, B2, C2, C3, and C4 propagated under frost heave. Crack-tip strain of specimens A3, A4, B3, B4, C4 produces slight changes during temperature equilibrium. Taking specimen of A1 as an example, the generalized strain evolution curve in the process of frost-heave crack propagation was analyzed, as shown in Figure 12. From the figure, it can be seen that the slope of the generalized strain curve at point A begins to increase, which represents the frost-heave initiation of micro-cracking. By this time, the concrete in the micro-crack zone had become a loose discontinuous medium, as shown in Figure 13, but could still able to withstand stress. With increasing micro-crack deformation, the stress transferred by the micro-cracks decreases until it finally forms macro-cracks, as shown in Figure 13. Additionally, as showed in Figure 12, the generalized strain abruptly increases at point B, which denotes the unstable propagation of cracks [47].  
The crack-propagation processes of other specimens were analyzed using the same method that was used for specimen A1. The parameters of micro-crack initiation strain and unstable propagation strain for each specimen are shown in Table 3. From the table, it can be seen that the crack initiation strain and unstable crack propagation strain are little affected by external temperature, crack width, or the initial moisture content of the crack. The propagation strain is mainly related to material parameters. The average values of micro-crack initiation strain and unstable propagation strain were 224 \(\upmu\)\(\upvarepsilon\) and 243 \(\upmu\)\(\upvarepsilon\), respectively.  
Figure 12: The generalized strain evolution curve of the frost-heave crack propagation of specimen A1.  
Figure 13: Frost-heave propagation of cracks.  
From Table 3, it can be seen that there are differences in the crack initiation strain and unstable strain of different specimens. The reason for this is that there were differences in the processes of pouring and curing, which affects the strength of concrete specimens. However, since there was little difference between the ratio of the crack initiation strain to the unstable strain among the specimens, it can be concluded that the test data can be used to effectively analyze the whole process of crack propagation. It is worth noting that, since the crack-tip strain measured by the traditional strain gauge is the average of the strain gauge's target range, and the concrete crack strain measured here is similar to the crack-tip point strain, it is slightly larger than that obtained by traditional strain gauge tests.  
Reference [48] shows that, for concrete with \(0.062\leq\alpha\leq 0.172\), C60 concrete has an \(\alpha\) of 0.074 and a Poisson's ratio of 0.25. By substituting Equation (19), the relationship between crack-initiation strain and instability strain can be obtained, as shown in Figure 11. The theoretical and experimental values of crack initiation strain for each specimen are also depicted in Figure 11.  
From Figure 14, it can be seen that the theoretical values of crack initiation strain are basically consistent with the experimental results obtained in this study. Therefore, the double-strain model proposed in this paper can be used to evaluate the whole process of crack propagation, which avoids the need to determine the stress intensity factor at the crack tip and the fracture toughness of materials under a complex frost-heave force.