## 3 Results and Discussion
### Determination of the Crack Initiation Criterion Based on Strain
#### 3.2.1 Double-Strain Criterion of Crack Propagation
\tag{7}\]  
where \(\alpha\) is the ratio of the tensile and compressive strength of the concrete materials; \(\sigma_{s}\) is the ultimate stress of the material; \([\sigma_{t}]\) is the allowable stress for the material; and \(b\) is a parameter used to reflect the influence of external force on the failure of the material, which is given as follows:  
\[b=\dfrac{(1+a)\tau_{0}-[\sigma_{t}]}{[\sigma_{t}]-\tau_{0}} \tag{8}\]  
where \(\tau_{0}\) is the shear strength of the material, and the range of parameter \(b\) is \(0\leq b\leq 1\). In the process of solving the boundary equation of the micro-crack generation zone at the crack tip under plane strain, since \(\sigma_{3}\
eq 0\), it is necessary to judge the magnitude relationship between the two principal stresses \(\sigma_{2}\) and \(\sigma_{3}\). When \(\sigma_{2}=\sigma_{3}\) in Equation (7), Equation (9) can be obtained as follows:  
\[\theta_{0}=2\alpha rc\sin(1-2v) \tag{9}\]  
When Equation (6) is equal to Equation (7), we can obtain \(\theta_{1}\) as follows:  
\[\theta_{1}=2\alpha rc\sin\dfrac{(1-2v)\alpha}{2+\alpha} \tag{10}\]  
\(\theta_{0}>\theta_{1}\) can be obtained by comparing Equation (9) with Equation (10). When \(0\leq\theta\leq\theta_{1}\), \(\sigma_{1}>\sigma_{2}>\sigma_{3}\), and \(\sigma_{2}\geq\dfrac{\sigma_{1}+\alpha\sigma_{3}}{1+\alpha}\), by introducing Equation (5) into Equation (7), the boundary expression of the crack initiation zone at the crack tip can be obtained as follows:  
\[r_{CU}=\dfrac{K_{I}^{2}}{2\pi\sigma_{s}}\cos^{2}\dfrac{\theta}{2}\bigg{(}1-2 \alpha v+\dfrac{1-b}{1+b}\sin\dfrac{\theta}{2}\bigg{)}^{2}\ (0\leq\theta\leq\theta_{1}) \tag{11}\]  
Figure 9 indicates that cracks propagate along the \(x\)-axis. Letting \(\theta=0^{\circ}\) and \(K_{I}=K_{IC}\) in Equation (11), the critical length of the micro-crack initiation zone at the crack tip on the \(x\)-axis can be obtained as follows:  
\[r_{CU}=\dfrac{K_{IC}^{2}}{2\pi\sigma_{s}^{2}}(1-2\alpha v)^{2} \tag{12}\]  
where \(K_{IC}\) is the fracture toughness of the material and \(r_{cu0}\) is the critical length of micro-crack initiation at the crack tip when \(\theta=0^{\circ}\). The critical length of the micro-crack initiation zone at the crack tip will be determined by the equivalent tensile strain theory. According to the principle of material mechanics for plane strain problems, the generalized [PERSON]'s law can be expressed as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=\frac{1}{E}[\sigma_{1}-v(\sigma_{2}+ \sigma_{3})]\\ \varepsilon_{2}=\frac{1}{E}[\sigma_{2}-v(\sigma_{3}+\sigma_{1})]\\ \varepsilon_{3}=0\end{array} \right. \tag{13}\]  
where \(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\) are the principal strains and \(E\) is the [PERSON]'s modulus of elasticity of the material. By introducing Equation (5) into Equation (13), the principal strain field at the crack tip under the plane strain problem can be obtained:  
\[\left\{ \begin{array}{l}\varepsilon_{1}=\frac{\Big{(}1-v^{2}\Big{)}K_{I}}{E \sqrt{2\pi v}}\cos\frac{\theta}{2}\Big{(}\frac{1-2v}{1-v}+\frac{1}{1-v}\sin \frac{\theta}{2}\Big{)}\\ \varepsilon_{2}=\frac{\Big{(}1-v^{2}\Big{)}K_{I}}{E\sqrt{2\pi v}}\cos\frac{ \theta}{2}\Big{(}\frac{1-2v}{1-v}-\frac{1}{1-v}\sin\frac{\theta}{2}\Big{)}\\ \varepsilon_{3}=0\end{array} \right. \tag{14}\]  
Figure 9 shows that the frost-heave crack propagation belongs to mode I. In this study, only the tension state of the concrete cracks is discussed, so \(\varepsilon_{1}\geq 0\), \(\varepsilon_{2}\geq 0\), and \(\theta\leq\theta_{1}\). \[ \begin{array}{l}\varepsilon_{e}=\sqrt{\left\langle\varepsilon_{1}\right\rangle ^{2}+\left\langle\varepsilon_{2}\right\rangle^{2}}\\ =\frac{\Big{(}1-v^{2}\Big{)}K_{I}}{E\sqrt{\pi v}}\cos\frac{\theta}{2}\sqrt{ \Big{(}\frac{1-2v}{1-v}\Big{)}^{2}+\Big{(}\frac{1}{1-v}\Big{)}^{2}\sin^{2} \frac{\theta}{2}}\end{array}  \tag{15}\]  
where \(\varepsilon_{e}\) is the equivalent strain and \(<\) > is a function which expresses the following meanings:  
\[\left\langle x\right\rangle=\left\{