## 2 Methodology
### Fatigue Damage Evolution
#### 2.2.3 Evolution Equation under Axial Compression  
This section presents a comprehensive analysis of the fatigue behavior of concrete under axial compression from the literature [31]. The fatigue strains have been measured in the middle portion of the cylindrical specimens along the longitudinal direction, ensuring a thorough understanding of the behavior. Maximum stress level \(S_{\text{max}}\) or concrete strength \(f\) will be treated as a controlling factor in fitting the equation. The compressive specimens in the literature [31] were \(\Phi\)100 mm \(\times\) H200 mm concrete cylinders with three strength grades, 26 MPa, 52 Mpa, and 84 Mpa, respectively. These specimens were loaded under four maximum stress levels, 0.75, 0.80, 0.85, and 0.95 (75%, 80%, 85%, and 95% of the static strength), respectively, while the minimum stress level was kept constant at 0.25. The preparation of materials included a single source of Portland cement, crushed stone of maximum size 13 mm (coarse aggregates), and natural river sand (fine aggregates); the fineness moduli of fine and coarse aggregates were 2.89 and 6.08, respectively. In addition, an admixture (superplasticizer with retarder) was used in the mix for the strength of 52 Mpa and 84 Mpa to give good workability, and silica fume (powder) was used for the high-strength concrete of 84 Mpa. The mixture proportion parameters, including the amount of cement, water, aggregates, and any additives used, are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Compressive** & \multirow{2}{*}{**Cement** & **Water** & **Fine Ag-** & **Coarse Ag-** & **Admixture** & **Slump** \\
**Strength** & **(kg/m3)** & **(kg/m3)** & **(kg/m3)** & **(kg/m3)** & **(kg/m3)** & **(\%)** & **(mm)** \\ \hline
26 & 320 & 218 & 717 & 931 & / & 61 \\
52 & 450 & 156 & 698 & 1047 & 3 & 81 \\
84 & 442 & 156 & 633 & 1032 & 2 & 169 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mix proportions of the concrete used in fatigue tests under axial compression from the literature [31]. Figure 4: Family of evolution curves reflecting three-stage fatigue damage development. The damage variables were calculated by Equation (19), and the damage development trend of various specimens is shown in Figure 5. Specimens are designated as S-XX-C-XX, where S-XX indicates the maximum stress level, and C-XX specifies the strength grade. For example, S-75-C-26 means a specimen with a compressive strength of 26 Mpa and a maximum stress level \(S_{\text{max}}\) of 0.75. Damage variables of various specimens are meticulously grouped according to stress levels or material strength and are fitted to fatigue damage evolution equations following Equation (20). The equation parameters are listed in Table 2.  
The column of \(R^{2}\) in Table 2 shows that the determination coefficient of the damage evolution equation based on maximum stress level is above 0.95, whose overall correlation effect is better than that based on concrete strength. Figures 6 and 7 show the distribution of damage variables and fitted damage evolution curves under different concrete strength grades and maximum stress levels, respectively. Comparing Figures 6 and 7, the scatter between damage variables of various specimens is larger when grouped under material strength, particularly for fatigue specimens made from 26 MPa concrete. When specimens are grouped under maximum stress levels, Figure 7 shows a more satisfactory correlation between the damage evolution curve and the data points of damage variables. Therefore, with the stress level as the controlling factor, Equation (20) can ensure acceptable fitting goodness and accurately reflect the evolution of fatigue damage. The finding underscores the importance of stress levels in predicting the evolution of fatigue damage. This conclusion is consistent with the observation that stress level is the most significant factor affecting concrete's fatigue life [33].