# Estimation of Fatigue Crack AE Emissivity Based on the Palmer-Heald Model
## 3 Results  
According to the results of the tests for each sample, two dependences were obtained--the number of AE impulses from the \(\sigma/\sigma_{y}\) ratio and from the applied load \(P\), which were used to analyze the emissivity of the material according to the Palmer-Heald and Dunegan models. The parameters of the analytical models were estimated using nonlinear approximation of empirical data by a power function and a secant function. The dependence \(N_{\Sigma}(P)\), the number of AE impulses on the applied load, was used as the initial data for the Dunegan model and the coefficients of the model \(A\) and \(n\) were determined by minimizing the objective function by the gradient method. In the Palmer-Heald model, only one parameter is determined--\(D_{a}\)--and the argument of the secant function was set based on the value of \(\sigma_{y}\) determined during mechanical tests. The actual stress \(\sigma\) was determined as the quotient of the load and the residual area of the specimen under the assumption of uniformly accelerated crack growth. Examples of the obtained experimental dependences and their approximations for each steel are presented in Figure 6.  
The results of approximation of all 70 sets of experimental data obtained for various values of the lengths of fatigue cracks show a high degree of agreement with the theoretical models of Dunegan and Palmer-Heald. For each of the three studied parameters--\(A\), \(n\), and \(D_{a}\)--the errors and the confidence interval of estimation were determined. The average errors in estimating were the following: for parameter \(D_{a}\) it was 2.3%, for parameter \(n\) it was 6.7%, and for parameter \(A\) it was \(\sim\)70%. Due to the large error in determining parameter \(A\), which can be explained by the poor conditionality of the calculations, coefficient \(D_{a}\) of the Palmer-Heald model was used to estimate parameter \(N_{\Sigma}\). The power-law indicator \(n\) of the Dunegan model was calculated to characterize the degree of danger of a crack and to verify the locally dynamic criterion for classifying AE sources in accordance with [25]. Figure 5: Diagram of the loading cycle. For each sample, the \(D_{a}\) and \(D\) parameters were calculated in the Palmer-Heald model. Figure 7 shows, as an example, the empirical distribution of parameters \(D_{a}\) and \(D\) for samples of 3-mm thick of 09G2S steel. Visually, the distributions shown in Figure 7 comply with the normal law and to verify this hypothesis, statistical tests of fit for distributions were used. The results are presented in Table 3.  
Figure 6: The number of AE impulses \(N_{\Sigma}\) depending on the current stress (\(\
icefrac{{\alpha}}{{\sigma_{y}}}\)) and load \(P\): (**a,b**) steel 09G2S, the initial length of the fatigue crack is 18.45 mm; (**c,d**) steel 45, the initial length of the fatigue crack is 20.80 mm; (**e,f**) steel 65G, the initial length of the fatigue crack is 17.81 mm. For the fit tests, two criteria were chosen: the Pearson \(\chi^{2}\) criterion, which is the most widespread for determining compliance with a certain type of distribution law and the non-parametric Anderson-Darling test, which retains the reliability in the case of limited sample sizes. The values of the test statistics for the normal distribution for the log-normal distribution and the Weibull distribution law are presented in Table 3. The \(A^{2}\) value corresponds to the Anderson-Darling test and the \(\chi^{2}\) corresponds to the Pearson test. These test statistic values are considered together with probability \(p\)-values. Since the \(p\)-values exceed 0.05, all the considered distributions can be used to represent the experimental data. However, the smallest values of the test statistics corresponding to the least deviation of empirical data from the theoretical law, and the largest \(p\)-values characterizing the greater reliability of the results are observed for the normal distribution law.