# Research on Residual Life Estimation Method for KMN Steel Based on Nonlinear Ultrasonic Testing
## 4 Discussion
### Residual Life Estimation Model  
The fatigue failure process of the impeller material can be divided into three stages, namely the fatigue crack initiation stage, the fatigue crack growing stage and the residual life stage. The corresponding fatigue cycles are \(N_{\text{i}}\), \(N_{\text{p}}\) and \(N_{\text{r}}\). The relationship between them is shown in Figure 9. Therefore, the essence of the residual life evaluation method based on nonlinear ultrasonic testing is to obtain the equivalent crack size of the impeller after service using nonlinear ultrasonic testing and micro-crack size statistics and then calculate the residual life of the impeller at this time through the fatigue crack growth formula. The main cause of blade fatigue failure is wake-induced vibration. Force analysis was conducted on the impeller, and the fatigue load can be simplified into the following models. When the blades of the semi-open impeller were impacted by the wake, the blades were subjected to vibration loads or forced vibration. When the blades of the closed impeller were impacted by the wake, the blades were subjected to a three-point bending load fixed at both ends. On the basis of the KMN bending fatigue test in this paper, two sets of KMN specimens for vibration fatigue test and tensile bending fatigue test are added. Their normalized nonlinear coefficients and the estimated values of their extreme fatigue crack are obtained and then the relationship curve between the KMN nonlinear coefficient and the equivalent crack length is obtained, as shown in Figure 10.  
Fitting the data points in the figure, we can establish the empirical formula of the impeller material KMN-I nonlinear coefficient and the equivalent crack length at the early fatigue stage (before \(10^{6}\) cycles):  
\[\beta=4.04+0.19a_{\text{e}}, \tag{13}\]  
where \(a_{\text{e}}\) is the equivalent crack length. The correlation coefficient is \(r=0.79\). After the nonlinear ultrasonic testing of the blades in service, the corresponding non-linear coefficients are obtained and substituted into the empirical formula to obtain the equivalent crack size inside the blades at this time. Assuming that the crack propagation conforms to the Paris formula, according to the fatigue assessment formula adopted in GB/T19624-2004 \"Safety Assessment of Pressure Vessels Containing Defects in Use\" [29; 30]:  
Figure 9: Schematic diagram of bending fatigue failure process and residual life (\(N_{\text{f}}\) is the fatigue cycles of the impeller after service). \[N_{\rm p}=\frac{a_{\rm N}-a_{\rm i}}{C(\Delta K)^{m}}\cdot\frac{a_{\rm i}}{a_{\rm N}} \tag{14}\]  
\[N_{\rm c}=\frac{a_{\rm c}-a_{\rm i}}{C(\Delta K)^{m}}\cdot\frac{a_{\rm i}}{a_{\rm c}} \tag{15}\]  
\[N_{\rm r}=N_{\rm c}-N_{\rm p}. \tag{16}\]  
Among them, \(a_{\rm c}\) is the critical crack size, \(a_{\rm N}\) is the crack growth size, \(a_{\rm i}\) is the crack initiation size, \(N_{\rm c}\) is the fatigue cycles when the fatigue crack propagates from \(a_{\rm i}\) to \(a_{\rm c}\), and \(N_{\rm p}\) is the fatigue cycles when the fatigue crack propagates from \(a_{\rm i}\) to \(a_{\rm N}\). Equation (16) can be used to obtain the number of fatigue cycles for the crack to propagate from \(a_{\rm N}\) to \(a_{\rm c}\); that is, the residual life \(N_{\rm r}\) in cycles of the impeller. For Cr-Mo steel, \(C\), \(m\), and \(\Delta K\) are 2.0 \(\times\) 10\({}^{-11}\), 2.5, and 3.3 MPa-\(\sqrt{\rm m}\), respectively. The fracture toughness of the impeller material KMN is \(K_{\rm 1C}\) = 172.34 MPa-\(\sqrt{\rm m}\), and the critical crack size \(a_{\rm c}\) can be calculated by Equation (17):  
\[a_{\rm c}=\frac{K_{\rm 1C}^{2}}{\pi\
u^{2}}. \tag{17}\]  
Combining the above findings, we can propose a method for evaluating the residual life of an impeller based on nonlinear ultrasonic testing. The specific steps are as follows:  
1. Perform nonlinear ultrasonic testing of the impeller in service to obtain the non-linear coefficient of the impeller at a particular point in time;
2.