# Estimation of Fatigue Crack AE Emissivity Based on the Palmer-Heald Model
## 3 Results
Table 4 shows the mean values and standard deviations (STD) of the \(D\) and \(D_{a}\) parameters for samples with a thickness of 3 mm and 5 mm. The ratio of the same type parameters corresponding to different sample thicknesses turned out to be approximately equal to the thickness ratio (0.6). \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Thickness, mm**} & \multicolumn{2}{c}{\(D_{a}\)} & \multicolumn{2}{c}{_D, 1/mm_} \\ \cline{2-5}  & **Average Value** & **STD** & **Average Value** & **STD** \\ \hline
3 & 144.1 & 68.8 & 15.9 & 6.8 \\
5 & 256.9 & 140.54 & 26.8 & 14.30 \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of \(D\) and \(D_{a}\) distributions for various thickness. Figure 7: Empirical distributions of parameter \(D_{a}\) (**a**) and parameter \(D\) (**b**) included in the Palmer–Heald model. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Distribution**} & \multicolumn{2}{c}{**Anderson-Darling Test**} & \multicolumn{2}{c}{**Pearson \(\chi^{2}\) Test**} \\ \cline{2-5}  & \(A^{2}\) & \(p\)-Value & \(\chi^{2}\) & \(p\)-Value \\ \cline{2-5}  & \multicolumn{4}{c}{_D-Parameter_} \\ \hline Normal & 0.21 & 0.98 & 1.12 & 0.57 \\ Log-Normal & 0.63 & 0.61 & 3.12 & 0.21 \\ Weibull & 0.31 & 0.93 & 1.78 & 0.41 \\ \hline \multicolumn{5}{c}{_D\({}_{a}\)_-parameter_} \\ \hline Normal & 0.25 & 0.96 & 1.88 & 0.39 \\ Log-Normal & 0.49 & 0.75 & 2.62 & 0.27 \\ Weibull & 0.56 & 0.68 & 3.32 & 0.19 \\ \hline \end{tabular}
\end{table} 
Table 3: Statistical tests of fit for distributions results. The average values and standard deviations of the parameters \(D\) and \(D_{a}\) for all the steel grades are shown in Table 5. Given the linear dependence of the distribution of \(D\) and \(D_{a}\) parameters on the thickness of the sample, for the case of a metal stress-strain state constancy, the average values and standard deviations (STD) can be estimated as a function of sample thickness \(t\). The power exponent \(n\) of the Dunegan model has a more deterministic character. Its value correlates with both the crack length and the stress intensity factor at the crack tip. The dependence of the exponent \(n\) on the crack length is shown in Figure 8.  
Despite the high value of the correlation coefficient \(r\) of the length of the fatigue crack and the parameter \(n\) (\(r\) = 0.85) for small sizes of the defect, there is a significant scatter in the exponent \(n\), which can be explained by the inhomogeneous microstructure of the material, as well as the difference in the nature of crack propagation. Figure 9 shows the results of a study of the deformed metal volume directly at the crack tip for several samples of 09G2S steel. Zones with different microhardness exceeding the hardness level of undeformed metal (_180... 240 HV0.015_) are also conventionally marked with colors on the matrices in the figure: yellow--areas with a hardness of _250...260 HV0.015_; orange--_261...299 HV0.015_; red--_300...330 HV0.015_; maroon--above _330 HV0.015_; in blue--places where hardness is not defined. The division into such ranges is nominal, the purpose of which is to demonstrate the presence of plastically deformed zones with a rough estimate of the degree of deformation. Figure 8: Dependence of the power exponent \(n\) of the Dunegan model on the crack length. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Steel**} & \multicolumn{2}{c}{\(D_{a}\)} & \multicolumn{2}{c}{\(D\), 1/mm} \\ \cline{2-5}  & **Average Value** & **STD** & **Average Value** & **STD** \\ \hline
09G2S & 51 t & 25 t & 5.3 t & 2.6 t \\
45 & 107 t & 35.7 t & 7.8 t & 3.1 t \\
65G & 289 t & 64.8 t & 15.3 t & 4.2 t \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of \(D\) and \(D_{a}\) distributions for steels under study.