# Research on Residual Life Estimation Method for KMN Steel Based on Nonlinear Ultrasonic Testing
## 4 Discussion  
### Micro-Cracks Statistics  
The micro-cracks inside the specimen continue to merge during the expansion process and finally form a main crack and cause fatigue failure. The main crack is the main factor that affects the fatigue life of the material. Since only a few small areas are selected for statistics when observing the microstructure of the specimen section, the main crack (that is, the extreme value of the crack length) may not be in the observational field of view, so we must find the value of all the cracks in the dangerous area of the specimen. The extreme value can better reflect the fatigue degree of the specimen. The Gumbel distribution is the main form of the extreme value distribution. Its principle is that, when the random variable is large enough, it obeys the exponential law distribution, then the Gumbel distribution is the limit distribution for its maximum value distribution [25]. We introduce the Gumbel distribution into the process of finding the crack's extreme value, measure the largest crack size in a small range through random small specimen data, and then extrapolate to estimate the largest crack size in a large range [26; 27], and its distribution function is shown as:  
\[G(a_{\text{max}})=\exp\biggl{(}-\exp\biggl{(}-\frac{a_{\text{max}}-\lambda}{ \varphi}\biggr{)}\biggr{)}. \tag{1}\]  
Among them, \(G(a_{\text{max}})\) is the probability that the maximum crack size is less than or equal to \(a_{\text{max}}\); \(\varphi\) and \(\lambda\) are the scale parameter and the position parameter, respectively. The maximum crack size \(a_{\text{max}}\) in different volumes can be estimated by Equation (2):  
\[a_{\text{max}}=-\varphi\ln\biggl{(}-\ln\biggl{(}\frac{T-1}{T}\biggr{)}\biggr{)} +\lambda. \tag{2}\]  
Among them, \(T\) is the return period, and the calculation formula is as follows:  
\[T=V/V_{0}. \tag{3}\]  
Among them, \(V_{0}\) is the standard detection volume, and \(V\) is the control volume. \(V_{0}\) can be obtained by Equation (4):  
\[V_{0}=h\times S_{0} \tag{4}\]  
Among them, \(S_{0}\) is the standard inspection field of view area; \(h\) is the average value of the crack size. \(h\) can be obtained by Equation (5):  
\[h=\frac{\sum\limits_{i=1}^{n}a_{\text{max},i}}{n}. \tag{5}\]  
Among them, \(a_{\text{max},i}\) is the largest crack size in the standard field of view obtained from the \(i\)-th observation in \(\upmu\)m.  
The selection of the control volume is very important for the extreme value calculation. In this paper, the area where the stress is higher than \(0.9\sigma_{\text{max}}\) in the specimen is defined as the control volume. For the plate specimen in the bending fatigue test, the maximum stress \(\sigma_{\text{max}}\) of the test is located on the surface in the middle of the specimen and the internal stress in the section is inversely proportional to the distance from the specimen's surface. The closer to both ends of the specimen, the smaller the stress. When the stress is equal to \(0.9\sigma_{\text{max}}\), the \(x\)-axis coordinates are \(-l_{\text{x}}\) and \(l_{\text{x}}\), respectively, and the control volume of the specimen is:  
\[V=2l_{\text{x}}bt. \tag{6}\]  
For KMN steel, \(E=205\) GPa, \(\rho=7840\) kg\(\cdot\)m\({}^{-3}\),\(f=20\) kHz, and the specimen size are shown in Figure 8. For the plate specimens in the bending fatigue test, it can be simplified as a uniform cross-sectional beam subjected to a concentrated load in the middle. By substituting the parameters when the static fatigue specimen stress is higher than \(0.9\sigma_{\text{max}}\)into Equations (7) and (8), we can calculate that \(l_{x}\) = 16.24 mm and \(V\) = 1949 mm\({}^{3}\). By combining Equations (3)-(5), we can calculate that \(h\) = 17.27 \(\upmu\)m and \(T\) = 1.3 \(\times\) 10\({}^{7}\).