## 3 Verifications
### Discussion on Concrete Parameters Affecting Fatigue Degradation
#### 3.3.1 Loading Frequency  
Taking the axial compression fatigue test as an example, we have analyzed the effect of frequency on fatigue damage evolution by referring to the fatigue deformation behavior of concrete specimens under different combinations of test conditions in pieces of literature [39; 40]. The main experimental condition combinations are shown in Table 3, where the first four rows of \(S_{\text{max}}=0.85\) are from the literature [40], and the rest are from the literature [39]. Similar to the discussion in Section 2.2.3, the damage variables are then summarized according to frequency and maximum stress level, and the damage evolution equation is fitted, as shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Maximum Stress** & **Stress Ratio** & **Frequency (Hz)** & **Number of Specimens** \\ \hline
0.85 & 0.3 & 1/16 & 2 \\
0.85 & 0.3 & 1⁄4 & 2 \\
0.85 & 0.3 & 1 & 2 \\
0.85 & 0.3 & 4 & 2 \\
0.80 & 0.1 & 2 & 4 \\
0.80 & 0.2 & 2 & 4 \\
0.80 & 0.2 & 3 & 4 \\
0.75 & 0.1 & 2 & 4 \\
0.75 & 0.1 & 3 & 4 \\
0.70 & 0.1 & 3 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Loading conditions of fatigue tests under axial compression from the literature [39; 40]. The maximum stress level, stress ratio (ratio between minimum and maximum fatigue stress), and loading frequency vary.  
As shown in Table 4, the damage evolution trends of different specimens are more consistent under the same stress level conditions, where the fitting goodness of damage evolution reaches 0.97 at a lower stress level of \(S_{\text{max}}\) = 0.75. It is worth noting that the damage evolution appears more discrete at \(S_{\text{max}}\) = 0.85 compared to the stress level \(S_{\text{max}}\) = 0.75. This phenomenon may be because, at higher stress levels of \(S_{\text{max}}\geq\) 0.80 [24], the frequency dramatically affects the number of failure cycles, therefore affecting the evolution of fatigue deformation, whereas, at lower \(S_{\text{max}}\), the frequency has little effect on the evolution of fatigue deformation [41].  
For a detailed analysis of the effect of loading frequency on fatigue behavior, damage evolution curves are fitted for different loading frequencies at the same stress level, as presented in Figure 19. In the figure, \((S_{\text{max}},\,R,f)\)-x represents the test results of the x-th specimen under test conditions \((S_{\text{max}},\,R,f)\). For example, (0.80, 0.1, 2)-1 represents the test results of the first specimen under the test conditions of a maximum stress level of 0.80, stress ratio of 0.1, and loading frequency of 2 Hz.  
It is illustrated that the differences in damage variable evolution at different frequencies are not significant for lower stress levels, which is consistent with the previous findings. As shown by the curves in Figure 19 for higher stress levels, the extent of damage development decreases with increasing loading frequency, leading to relatively high theoretical fatigue residual strength at high loading frequency.  
Scholars have pointed out that for lower stress levels \(S_{\text{max}}\), the influence of frequency can be ignored, and the impact will increase as the stress level increases. Therefore, fitting the damage evolution equation based on stress levels does not need to consider the influence of frequency at lower stress levels. However, applying the damage evolution equation to analyze strength attenuation at higher stress levels should also consider the influence of different loading frequencies on the results.