## 3 Results and Discussion
### Four-Point Bending Fatigue Test
#### 3.3.5 Fatigue Cumulative Damage Regulation Based on Fatigue Resistivity  
The resistivity corresponding to the maximum fatigue strain \(\varepsilon_{max}\) is called the fatigue resistivity \(\rho\). The change curve of fatigue resistivity \(\rho\) with a cyclic ratio (the ratio of the cycle number to the cycle life) is called the fatigue resistivity evolution curve. The \(\rho\)--\(\eta\)/\(N\) curve is shown in Figure 11. It was not difficult to find that \(\varepsilon_{max}\) and \(\rho\) had the same three-stage change trend with the increase of the cycle ratio in the fatigue process. In the first stage, \(\eta\)/\(N\) was approximately 0.1, and \(\rho\) increased rapidly with the increase of cycle ratio, but the growth rate gradually slowed down. In the second stage, \(\eta\)/\(N\) was approximately 0.1-0.9, and \(\rho\) increased steadily and had a linear relationship with the cycle ratio; its growth rate was basically a fixed value. In the third stage, \(\eta\)/\(N\) was approximately 0.9, and \(\rho\) increased sharply until the specimen was destroyed. As shown in Figure 11a, the stress level was low, and the fatigue resistivity changed more obviously with the increase of temperature than it did in Figure 11b,c. With the increase of stress level, the influence of temperature on fatigue life decreased gradually. It can be seen that, in the damage stability stage and Stage II of linear development in the entire process of fatigue damage, the resistivity was approximately linear with the fatigue damage \(D\); the fatigue resistivity and the damage amount corresponded to a one-to-one relationship, as shown in the BC section of Figure 12.  
Figure 11: Fatigue resistivity evolution curve. (**a**) \(\rho\)-\(D\) curve at stress level 0.6; (**b**) \(\rho\)-\(D\) curve at stress level 0.7; (**c**) \(\rho\)-\(D\) curve at stress level 0.8. The fatigue resistivity is the same as \(\varepsilon_{max}\), which can be used to characterize the fatigue cumulative damage process of CNFPUC materials. Because of this, the damage variables defined by the fatigue resistivity \(\rho\) are proposed:  
\[D=\left\{ \begin{aligned} 0\left(\rho^{p}\leq\rho_{0}^{p}\right) \\ \frac{\rho^{p}-\rho_{0}^{p}}{\rho_{c}^{p}-\rho_{0}^{p}}\Big{(}\rho_{0}^{p} \leq\rho^{p}\leq\rho_{c}^{p}\Big{)}\end{aligned} \right. \tag{8}\]  
where \(\rho_{0}^{p}\) is the resistivity corresponding to the initial damage of the specimen and corresponding to the end of Stage I of the \(\rho^{p}-n/N\) curve; \(\rho_{c}^{p}\) is the ultimate resistivity in the final fatigue failure of the specimen, and \(\rho^{p}\) is the instantaneous resistivity. It can be seen that, when the damage began, \(\rho^{p}=\rho_{0}^{p}\), \(D=0\), \(\rho_{0}^{p}\) might have been called the damage threshold resistivity. When damage developed to an unstable fracture, \(\rho_{c}^{p}\) usually adopted the resistivity at the critical cycle ratio \(N_{cr}\)/\(N\) of 0.9, thus \(\rho^{p}=\rho_{c}^{p}\), \(D=1\). Therefore, the damage variable \(D\) could meet the requirements of the generalized damage variable, and the physical meaning of the definition was clear; the response of each parameter was remarkable and easy to measure and calculate. The means of \(\rho_{0}^{p}\) and \(\rho_{c}^{p}\) are shown in Tables 12 and 13. It can be seen from the tables that \(\rho_{0}^{p}\) and \(\rho_{c}^{p}\) increased with the increase of stress level S and the increase of temperature. There was a linear relationship between the damage variables and the fatigue resistivity. The bending fatigue damage of CNFPUC materials at different ambient temperatures and stress levels could be obtained by numerical substitution (8) in Tables 12 and 13. The cumulative damage of the CNFPUC could be quantitatively evaluated in engineering practice. Damage threshold resistivity or ultimate resistivity could be used as a standard for initial damage or failure.