## 3 Results and Discussion
### Determination of the Crack Initiation Criterion Based on Strain
#### 3.2.1 Double-Strain Criterion of Crack Propagation  
Comparing Figure 9a-c, it can be seen that frost-heave crack propagation is in accordance with the characteristics of a mode I crack propagation. Additionally, the crack width is much smaller than the specimen length. Therefore, the specimen was simplified by a plane model, as shown in Figure 10. There is an edge crack with a length of \(a\) on the half plane, and an approximately uniformly distributed frost-heave force \(q\) was assumed for the cracks. The model has ox symmetry. Therefore, only the first quadrant was studied. Figure 9: Crack propagation. (**a**) Crack propagation on specimen surface; (**b**) generalized strain distribution on specimen surface; (**c**) AE events location on specimen surface. Before the micro-cracks start to propagate, the stress field at the crack tip can be analyzed according to the linear elastic fracture mechanics, and the principal stress field distribution at the crack tip is obtained. The boundary conditions of the model are as follows:  
\[x=0,\ y\geq 0\text{: }\sigma_{x}=0,\ \tau_{xy}=0 \tag{2a}\] \[y=0,\ 0\leq x\leq a\text{: }\sigma_{y}=-q,\ \tau_{xy}=0\] (2b) \[y=0,\ a\leq x\leq\infty\text{: }u_{y}=0,\ \tau_{xy}=0 \tag{2c}\]  
Since \(x^{2}+y^{2}\rightarrow\infty\), all stress components tend to zero. The stress intensity factor at the crack tip [44] is expressed as follows:  
\[K_{I}=1.1215q\sqrt{\pi a} \tag{3}\]  
For a type \(I\) crack, the stress field near the tip is as follows:  
\[\left\{ \begin{array}{l}\sigma_{x}=\dfrac{K_{I}}{\sqrt{2\pi r}}\cos\dfrac{ \theta}{2}\Big{(}1-\sin\dfrac{\theta}{2}\sin\dfrac{3\theta}{2}\Big{)}\\ \sigma_{y}=\dfrac{K_{I}}{\sqrt{2\pi r}}\cos\dfrac{\theta}{2}\Big{(}1+\sin \dfrac{\theta}{2}\sin\dfrac{3\theta}{2}\Big{)}\\ \tau_{x}=\dfrac{K_{I}}{\sqrt{2\pi r}}\cos\dfrac{\theta}{2}\sin\dfrac{\theta}{2 }\cos\dfrac{3\theta}{2}\end{array} \right. \tag{4}\]  
where \(r\) and \(\theta\) are the polar coordinates; \(\sigma_{x}\), \(\sigma_{y}\), and \(\tau_{x}\) are the stress components near the crack tip; and \(K_{I}\) is the stress intensity factor at the crack tip. The principal stress near the crack tip was obtained using the following equations:  
\[\left\{ \begin{array}{l}\sigma_{1}=\dfrac{K_{I}}{\sqrt{2\pi r}}\cos\dfrac{ \theta}{2}\Big{(}1+\sin\dfrac{\theta}{2}\Big{)}\\ \sigma_{2}=\dfrac{K_{I}}{\sqrt{2\pi r}}\cos\dfrac{\theta}{2}\Big{(}1-\sin \dfrac{\theta}{2}\Big{)}\\ \sigma_{3}=\dfrac{2 vK_{I}}{\sqrt{2\pi r}}\cos\dfrac{\theta}{2}\end{array} \right. \tag{5}\]  
where \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) are the principal stresses and \(v\) is Poisson's ratio. Using the solution of the elastic stress field at the crack tip, the critical length of the micro-crack initiation zone which is at the crack tip under unified strength as the yield criterion is analyzed. The unified strength theory of concrete [45] takes into account the difference between the tensile and compressive properties of materials and the influence of intermediate principal stress. Therefore,  
Figure 10: Frost-heave force model of crack. in this study, the unified strength theory of concrete was taken as the failure criterion of concrete. The mathematical expression of the unified strength theory [46] is as follows:  
\[\left\{ \begin{array}{l}F=\sigma_{1}-\dfrac{\alpha}{1+b}(b\sigma_{2}+\sigma_{3} )=[\sigma_{t}]=\sigma_{s}\\ \sigma_{2}\leq\dfrac{\sigma_{1}+\alpha\sigma_{3}}{1+\alpha}\end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}F^{\prime}=\dfrac{1}{1+b}(\sigma_{1}+b\sigma_{2})- \alpha\sigma_{3}=[\sigma_{t}]=\sigma_{s}\\ \sigma_{2}\geq\dfrac{\sigma_{1}+\alpha\sigma_{3}}{1+\alpha}\end{array} \right.