# Empirical Approach to Defect Detection Probability by Acoustic Emission Testing
## 4 Results and Discussion  
### AE Signal Simulation Results  
A priori determination of AE source parameters is one of the main results of this study. According to the proposed model, the emissivity of fatigue crack may be estimated using the multiplicative coefficient \(D_{a}\) of the Palmer-Heald model (1). For fractures at the stage of stable growth, it corresponds to the normal distribution law. Table 2 shows mean values and standard deviations of parameter \(D_{a}\), obtained on the basis of empirical data for the studied steel grades with respect to the wall thickness of the investigated object. According to the distribution of \(D_{a}\), it is possible to estimate  
the probability that a certain number of AE signals will be recorded when an object with fatigue crack is loaded. Figure 7 shows the density distribution of the AE hits quantity \(N_{\Sigma}\) for steels 09G2S and 45 at different maximum values of mechanical stress in the loading cycle. A high emissivity of steel 45 is noted as compared to steel 09G2S; in addition, an increase in the maximum mechanical stress \(\sigma_{max}\) from \(0.5\sigma_{y}\) to \(0.7\sigma_{y}\) leads to an increase in \(N_{\Sigma}\) by on average three times. When analyzing the amplitude distribution, we have found that for fatigue cracks in 09G2S and 45 steels at the stage of stable growth, amplitude distribution follows the Weibull distribution law. Moreover, the distribution of the parameter \(\lambda\) corresponds to the normal distribution law with parameters \(\mu\) = 5.5 and \(\sigma\) = 1.13 for steel 09G2S, \(\mu\) = 7.76 and \(\sigma\) = 0.69--for steel 45. The parameter \(k\) is characterized by a small scatter and for both steels can be represented using the normal distribution with the parameters \(\mu\) = 1.14 and \(\sigma\) = 0.095. Figure 8a shows the Weibull distribution functions for the mean value of the scale parameter \(\overline{\lambda}=5,51\) within the confidence interval of the parameter \(k\) with a confidence level of 0.95. Figure 8b shows the Weibull distribution functions for the mean value of the parameter \(k\) within the confidence interval (3.25, 7.77) of the parameter \(\lambda\) with the same confidence value of 0.95. These dependencies show that the scatter of the shape parameter has a significantly smaller effect on the distribution type than the spread of the scale parameter. The obtained values of AE emissivity and parameters of the impulse amplitude distribution correspond to the data presented in [29; 30; 31]. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Steel Grade**} & \multicolumn{2}{c}{\(D_{a}\)} \\ \cline{2-3}  & **Mean Value** & **Standard Deviation** \\ \hline
09G2S & \(51h\) & \(25h\) \\
45 & \(107h\) & \(35.7h\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters \(D_{a}\) distributions for steels under study. Figure 7: Probability density function \(N_{\Sigma}\) for steels 09G2S (**a**) and 45 (**b**). Examples of AE signal waveforms obtained as a result of modeling are shown in Figures 9 and 10. Figure 9a-c shows the signals calculated in accordance with the method described in paragraph 3.2.1, for the distances between the AE source and AE sensor of 2, 5 and 7 m, respectively. Figure 10 shows more realistic data: modeled signals are shown against the background of noise caused by the pump during hydraulic loading. The correctness of AE signal simulation and their correspondence to the experimental data are demonstrated in [26]. The AE waveforms shown in Figures 9 and 10 are the components of the signals \(s_{1}\) (formula 3) and \(s_{2}\) (formula 4), which are used to determine the location of the AE  
Figure 8: Weibull distribution function with the mean value of the parameter \(\lambda\) within the confidence interval of the parameter \(k\) (**a**) and with the mean value of the parameter \(k\) within the confidence interval of the parameter \(\lambda\) (**b**).