## 4 Discussion of Test Results  
### SCB Test Results  
#### 4.1.1 Definition of the Three Stages within the Fatigue Curve  
As Figure 5 shows, the fatigue cracking process of ECC can be divided into three stages: the crack initiation stage, the crack stable propagation stage and the accelerated destruction stage.  
The slope variation of the curve shown in Figure 5 was used to distinguish the cracking stages. However, due to the dynamic load of 10 Hz during the test, the curve presented a fluctuating state. Thus, a three regression model was used to fit the original data, as Figure 6 shows, and then the slope of the fatigue curve could be calculated easily. As Figure 6 demonstrates, the first segment with a loading cycle between 0-6000, with an decreasing slope of the displacement-cycle curve, was fitted by four-parameter logistic rodboard model, as Equation 1 shows; the second segment with loading cycle interval 6000-14,000, with a steady slope of the displacement-cycle curve was fitted by linear model, as Equation 2 shows; and the third segment with a loading cycle between 14,000-21,000, with an increasing slope of the displacement-cycle curve was fitted by a three-parameter exponential model, as Equation (3) shows. The slope of each model at different cycles could be calculated by Equations (4)-(6), respectively. The fitting results of the three regression models are shown in Table 1. Because the second  
Figure 4: Universal test machine (UTM) loading cooperated with digital image correlation (DIC).  
Figure 5: Three stages of ECC fatigue cracking process during the SCB test under stress ratio 0.5.  
segment was fitted by a linear model, the scope was constant and equal to \(b\). By solving equation \({y_{1}}^{\prime}(x)=b\), the critical loading cycles \(x_{1}\) between stage I and stage II could be determined. Similarly, by solving \({y_{3}}^{\prime}(x)=b\), the critical loading cycles \(x_{2}\) between stage II and stage III could be calculated. The loading cycles of each fatigue cracking stage could be obtained as shown in Table 2.  
\[y_{1}(x)=c_{1}+\frac{d_{1}-c_{1}}{1+\left(\frac{x}{b_{1}}\right)^{a_{1}}} \tag{1}\]  
\[y_{2}(x)=a_{2}+b_{2}\cdot x \tag{2}\]  
\[y_{3}(x)=a_{3}+b_{3}\cdot e^{\epsilon_{3}x} \tag{3}\]  
\[K_{1}={y_{1}}^{\prime}(x)=\frac{a_{1}\times(c_{1}-d_{1})\times\left(\frac{x}{ b_{1}}\right)^{(a_{1}-1)}}{b_{1}\times\left(\left(\frac{x}{b_{1}}\right)^{a_{1}}+1 \right)^{2}} \tag{4}\]  
\[K_{2}={y_{2}}^{\prime}(x)=b_{2} \tag{5}\]  
\[K_{3}={y_{3}}^{\prime}(x)=b_{3}\cdot c_{3}\cdot e^{\epsilon_{3}x} \tag{6}\]  
where, \(x\) is loading cycles, \(y\) is vertical displacement. \(a_{1}\), \(b_{1}\), \(c_{1}\), \(d_{1}\), \(a_{2}\), \(b_{2}\), \(a_{3}\), \(b_{3}\), \(c_{3}\) are model regression fitting parameters, \(K_{1}\), \(K_{2}\) and \(K_{3}\) are the slope of the three regression models respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Segment**} & \multicolumn{4}{c}{**Regression Fitting Parameters**} \\ \cline{2-5}  & \(a\) & \(b\) & \(c\) & \(d\) \\ \hline
1 & \(-0.881\) & \(3.817\) & \(-0.043\) & \(0.1411\) \\
2 & \(0.135\) & \(6.933\times 10^{-7}\) & – & – \\
3 & \(0.146\) & \(4.149\times 10^{-8}\) & \(6.403\times 10^{-4}\) & – \\ \hline \end{tabular}
\end{table}
Table 1: Regression fitting parameters of three-stage equations.  
Figure 6: Three-stage fitting method for distinguishing fatigue cracking stages of ECC.