# Estimation of Fatigue Crack AE Emissivity Based on the Palmer-Heald Model
## 4 Discussion  
This paper considered the approach to assessing the AE emissivity of structural steels during the propagation of a fatigue crack based on the Palmer-Heald model. From the point of view of evaluating the emissivity of the material, parameter \(D\) is informative. Parameter \(D\) characterizes the emissivity of the material and parameter \(D_{a}\) can be used to predict the number of AE impulses \(N_{\Sigma}\) during loading of a structure with a crack whose length is a priori unknown. The anisotropy of the properties of the studied steels, the inhomogeneous stress-strain state of the material at the crack tip, the influence of the shape of the crack, and the presence of additional secondary sources of AE, such as friction of the crack edges, determine the random character of parameters \(D\) and \(D_{a}\) as random variables given by the law of probability distribution. The random nature of the distribution of the parameters \(D\) and \(D_{a}\) may be due to different stress-strain states of the metal at the crack tip. Figure 9 shows that the distributions of the microhardness values (and, hence, stress and strain intensities) vary significantly from sample to sample. This is explained by the fact that in real metals and alloys, the propagation of a crack and the formation of a plastic deformation zone near its edges and apex depend not only on the absolute dimensions of the crack and loading parameters, but also on the parameters of the microstructure of the material. Among the most characteristic parameters, one should single out the grain size, the state of grain boundaries, pearlite fraction and colony size, the presence and volume fraction of non-metallic inclusions, etc. The combination of the effect of these metallurgical factors leads to differences in the shape and size of the plastic deformation zones near the crack tip. Figure 9: Photographs of metal near the crack tips with indents (areas of research are marked) and the _HV0.015_ microhardness matrices in these areas on three samples (**a–c**) of 09G2S steel. The analysis of experimental data shows that parameters \(D\) and \(D_{a}\) can be described by the normal distribution law and this hypothesis is supported by the results of the Anderson-Darling and Pearson tests. The normal distribution is not only the most reliable but also the most conservative estimation of the distribution law, since the normal distribution corresponds to the class of \"light-tailed\" distribution, and its application should not lead to an overestimation of the parameter \(N_{\Sigma}\). Using the well-known probability distribution of parameter \(D_{a}\) based on the Palmer-Heald model, we can estimate the probability of parameter \(N_{\Sigma}\). At a certain known level of the load \(\sigma\), the probability distribution of parameter \(N_{\Sigma}\) can be estimated by scaling the distribution of parameter \(D_{a}\) by a _see_\(\left[\left(\frac{\pi}{2}\frac{\partial}{\partial y}\right)-1\right]\). It is important that parameter \(D_{a}\) does not correlate with the length of the fatigue crack--the correlation coefficient between these parameters \(r_{1,D_{a}}\) is 0.22. The average values of parameters \(D\) and \(D_{a}\) have minimum values for 09G2S steel, maximum values for 65G steel and intermediate values for 45 steel. 09G2S steel has a large plasticity (see Table 2 and Figure 2) compared with steels 45 and 65G. In this regard, an increase in the parameters characterizing the emissivity of the material with an increase in the carbon content in the steel structure can be explained by the more brittle nature of the fracture of steels 45 and 65G in comparison with low-carbon steel 09G2S. It should be noted that since the amplitude discrimination threshold was set to 40 dB during the experiment, the results are quite conservative; with their help, it is possible to estimate the number of AE impulses with amplitude exceeding 40 dB.