# Nonlinear Fatigue Damage of Cracked Cement Paste after Grouting Enhancement
## 5 Conclusions  
Currently, the understanding on the fatigue properties of grouting reinforcement materials is still limited and its fatigue failure mechanism needs further investigation. In this study, the evolution of the fatigue damage was characterized by an analytical inverted S-shaped model according to the recorded maximum vertical deformation during the loading test. To reveal the mechanism of fatigue damage of rock-like materials after grouting enhancement, the uniaxial compression test under cyclic loading was carried out. The effect of the upper limit stress level, the stress ratio, and the stress amplitude on the fatigue life was studied. From the above discussion, some important conclusions can be drawn:  
1. The cracks initiated in the middle of the specimen and propagated up and down around the slurry body. The grouting reinforcement could effectively avoid the stress concentration of the crack and made the specimen more complete and strong. The fatigue life decreases slowly, first, and then decreases rapidly with the increase of the stress amplitude. The upper limit stress level and the stress amplitude are two significant factors to the fatigue life.
2. The exponential function of the stress amplitude and the fatigue life was obtained through the S-N curve and the experimental data. There was also a double logarithmic linear relationship between the stress amplitude and the fatigue life.  
Figure 10: Fitting result of the inverted S-curve.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \(\mathbf{D_{0}}\) & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{R}\) \\ \hline S-2 & 0.5712 & 0.0815 & 1.005 & 0.9911 \\ S-4 & 0.5349 & 0.1369 & 1.032 & 0.9946 \\ D-1 & 0.5163 & 0.1037 & 1.01 & 0.9951 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter fitting results.  
3. The damage evolution is phased into three stages and the three stages can be observed from the tested specimens, which are influenced by the loading parameters (such as the upper limit stress level, the stress ratio, and the stress amplitude).
4. The fatigue damage evolution curve was acquired by the curve-fitting method through the experimental data, which is based on the Logistic model. An inverted S-shaped curve was obtained and used to describe the damage evolution. The fitting result was consistent with the experimental data.  
This study exclusively focused on the fatigue damage of grouting-enhanced rock-like materials under constant amplitude fatigue condition. For the future work, the fatigue performance under variable amplitude loading will be investigated. In addition, the fatigue damage model for rock-like materials with different types of grouting materials will be studied in the future.  
[PERSON] and [PERSON] designed the scheme, tested the data and wrote the paper; [PERSON] and [PERSON] offered constructive suggestions for the fatigue test design; [PERSON] and [PERSON] also helped to proofread the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number [51404212]; the Foundation for University Key Teacher by the He'nan Educational Committee, grant number [2016 GGJS-002]; the program of China Scholarships Council; grant number [201608410176]; the third author acknowledges the financial support from the China Scholarship Council, grant number [201406370141].  
The authors declare no conflict of interest.