# Nonlinear Fatigue Damage of Cracked Cement Paste after Grouting Enhancement
## 4 Inverted S-Shaped Fatigue Damage Model  
Figure 8 shows the strain cycle-ratio curve of typical samples. Where the strain is the amount of relative deformation in the vertical compression direction which was collected during the fatigue loading process. It can be seen that the changing rules of the single-crack and double-crack samples are consistent, and the entire fatigue failure process exhibits a clear three-stage damage rule. The initial damage phase and the acceleration phase have a shorter duration, and the stability phase is relatively long. The curves of the three phases show a distinct inverted S-shaped shape, which is consistent with the results obtained in most of the literature [23; 24]. At the same cyclic ratio, the single-crack specimen has a greater strain than the double-crack specimen. The cracking strain of the resin is much larger than that of the cement, so the damage begins at the periphery of the cement away from the original crack after grouting reinforcement. The strength of the entire structure is strengthened and the anti-fatigue performance is improved after grouting enhancement. When damage mechanics is used to study the fatigue properties of cement and other solid materials under cyclic loading, the selection of the damage variables is crucial. At the same time, the damage variable needs to have a clear physical meaning, easy to measure, and the change law is consistent with the evolution of fatigue mechanical properties of materials. The common definitions of damage variables are elastic modulus, longitudinal wave velocity, energy dissipation, maximum strain, residual strain, acoustic emission, density, etc. [25; 26; 27]. According to the above collection results of strain, it can be found that the strain has all the features as the amount of damage. In this paper, we will use the maximum strain as the definition of damage, and study the damage process. According to the results of the literature study, the damage variable can be defined based on the maximum vertical deformation under cyclic loading:  
\[D=\frac{e_{max}^{n}-e_{max}^{1}}{e_{max}^{N}-e_{max}^{1}} \tag{3}\]  
Figure 8: Strain-cycle ratio curve. where \(D\) is the damage after \(n\) cycles, \(\varepsilon_{max}^{n}\) is the instantaneous maximum strain for the \(n\)-th cycle, \(\varepsilon_{max}^{1}\) is the initial fatigue maximum strain of the first cycle, and \(\varepsilon_{max}^{N}\) is the maximum strain of the last cycle. The inverted S-shaped curve has been widely used in many fields of research. The logistic model, also known as the Verhulst-Pearl model, is a model in which the S-shaped curve can be better simulated. The general expression of the Logistic model equation can be expressed as [28; 29]:  
\[y=\frac{c}{1+e^{a-bx}}. \tag{4}\]  
where \(a\), \(b\), \(c\) are related parameters affecting the shape of the curve. Applying the inverse function of the logistic model to the damage model described in this paper, a new mathematical expression of the inverted \(S\)-shaped model for fatigue damage accumulation can be expressed as:  
\[D=D_{0}-a\ln\left(\frac{b}{n}-1\right). \tag{5}\]  
where \(n\) is an independent variable, called generalized cycle, can be absolute cycle \(N\) or relative cycle \(N/N_{f}\); \(D\) is the dependent variable, represents the corresponding damage of \(N\) cycles under constant amplitude loading; \(D_{0}\) is a parameter related to the initial damage; \(a\) and \(b\) are two parameters to be determined. A perfect fitting curve of damage evolution agreeable to experimental data can be obtained by adjusting these three parameters: \(D_{0}\), \(a\), and \(b\). It can be seen from Figure 9 that the damage value \(D\) is consistent with the \(D_{0}\) when the relative cycle \(N/N_{f}\) is equal to 0.5 which is consistent with the conclusion of [29]. This means that the damage value of the curve center, the cumulative damage of the half-life, could be controlled by \(D_{0}\). Most of the value of \(D_{0}\) should be around 0.5 because it is related to the half-life damage.