# Nonlinear Fatigue Damage of Cracked Cement Paste after Grouting Enhancement
## 3 Fatigue Test Results and Analysis  
Figure 4 shows a typical failure trajectory of the specimens. Several stages of the specimens under cyclic loading, including compaction, micro-crack initiation, stable crack propagation, and accelerated crack propagation, were observed with the increase of the number of cycles. For single crack specimens, the crack initiated in the middle of the specimen and propagated up and down around the slurry body (see Figure 4a). After the specimen was completely destroyed, the slurry body and the surrounding part of the specimen still had good integrity. It was clear that grouting reinforcement could effectively avoid the stress concentration of the crack and strengthen the specimen. The fatigue performance of the specimens after grouting was better than that of the specimens without a crack. The typical failure trajectory of the specimens of cracked specimens and non-cracked specimens was listed in Figure 4c,d. According to the test results, under the same stress level and stress ratio (\(S=0.41\), \(R=0.1\)), the average fatigue life of non-fractured specimens was 31,250 cycles. For the single-cracked specimen, crack initiation started after approximately 721 cycles and the final number of cycles to extend to the top of the specimen was approximately 21,453 cycles. For the double-cracked specimen, the crack initiation started after approximately 561 cycles and the final number of cycles to extend to the top of the specimen was approximately 19,453 cycles. At the same time, compared with the un-grouted and non-fractured specimens, the toughness of the grouted specimens was greatly improved. For the double-crack specimens, the crack initiated in the original crack tip and propagated to another crack (see Figure 4b). In some cases, the grouting body was compressed and destroyed, which indicated  
Figure 3: Fatigue loading curve. Figure 2: Specim size and grouting method. that the better grouting effect can be obtained and the grouting body has fully played its important reinforcement role. In general, double-fractured specimens have better strength and fatigue resistance than single-fissure specimens. The fatigue test results under different stress ratio and stress levels are listed in Table 2. The fatigue life of specimens slightly decreased with the low amplitude \(S_{a}\), however, after increasing the amplitude \(S_{a}\), the fatigue life was rapidly reduced. Changing the stress level \(\sigma_{\text{max}}/\sigma_{c}\) and the stress amplitude \(S_{a}\) significantly affects the fatigue process. The initial axial deformation took place before the beginning of the cycle and the axial deformation increased with the increase of the stress level. The total amount of cumulative irreversible deformation required to achieve failure will be reduced and affect the fatigue life of the specimen significantly. The S-N (stress amplitude and fatigue life relationship) curve was obtained according to the data listed in Table 2 and Figure 5. It can be seen that the stress amplitude \(S_{a}\) decreases exponentially with the increases of the fatigue life \(N_{f}\) which is similar to the research results of other scholars [20]. After data fitting, the stress-fatigue life equation can be obtained as:  
\[S_{a}=15.25522\;N_{f}^{-0.08804} \tag{1}\]  
Taking the logarithm of both sides of Equation (1), the linear relationship between the stress amplitude \(S_{a}\) and fatigue life \(N_{f}\) can be obtained as:  
\[\text{lg}S_{a}=1.183418\;-\;0.08804\;\text{lg}N_{f} \tag{2}\]  
Figure 4: Failure diagram of specimen: (**a**) single-crack after grouting; (**b**) double-crack after grouting; (**c**) cracked specimens before grouting; and (**d**) without cracks. Figure 6 shows the effect of stress ratios on fatigue life. It can be seen that the fatigue life increases greatly with the increase of the stress ratio R under the same stress level. The increase of R means the decrease of stress amplitude \(S_{a}\).