# Estimation of Fatigue Crack AE Emissivity Based on the Palmer-Heald Model
## 1 Introduction  
The acoustic emission (AE) method is successfully used to detect defects in items and structures for various purposes. Fatigue crack is one of the most widespread and dangerous defects which could be successfully detected with help of the AE method. One of the most complex and relevant directions in the development of the AE-diagnostic methodology is the establishment of a relationship between the parameters of a fatigue crack and the AE impulses parameters [1]. The strong relationship between the microstructure of materials and AE parameters complicates the construction of a fatigue crack propagation model. In the theory of AE diagnostics, a number of analytical and phenomenological models have been developed that connect the crack parameters and AE data during the static loading, e.g., the Dunegan model [2], the Palmer-Heald model [3], and the Holt and Goddard model [4]. In the case of propagation of a fatigue crack under cyclic loading, an equation similar to the Paris-Erdogan law has been proposed [5; 6]. The disadvantage of existing models is their parametric uncertainty--the functional relationship between the mechanical characteristics of the material and the AE parameters is established with accuracy to coefficients depending on the material properties, structure features and operating conditions. The parametric uncertainty of the models does not allow a quantitative estimation of the parameters of AE impulses induced by the crack. The active development of technology for scanning electron and laser confocal microscopy [7] and the improvement of testing and measuring equipment contribute to the development of a phenomenological approach in acoustic emission. Previous studies [8; 9; 10] included analyses of AE data during fatigue crack propagation with identification of the fatigue crack propagation stage and classification of various AE sources that appeared during the crack growth. The disadvantage of anempirical approach is the difficulty of data summarizing and processing as well as the need, in many experiments, to substantiate and confirm the empirical relationships. The complex analytical and phenomenological approach to the fracture process, which was proposed in [11], allowed the design of a theoretically confirmed fracture process model which allows the identification of various stages of material fracture based on the parameters of AE impulses. This study represents a study of the AE signature of fatigue crack, focused on the improvement of industrial applications of the AE testing method. The objective of this study was to assess the emissivity of structural steels during the development of fatigue cracks in order to clarify the probability of fatigue defect detection during acoustic emission testing. Acoustic emission testing is the only nondestructive testing method for which the assessment of the reliability of defect detection is not implemented in the field practice. Currently, there are only a few papers devoted to evaluation of the reliability of AE testing [12; 13; 14]. To assess the reliability of defect detection during AE monitoring, a quantitative diagnostic model should be formed that allows the determination of not just the functional dependence, but also the quantitative values of AE parameters induced by a defect of a certain type in response to a specific loading effect with a certain probability. One of the main parameters influencing the probability of defect detection by the AE method is emissivity, the integral characteristic of which is \(N_{\Sigma}\)--the number of impulses emitted during the loading of the sample. The uncertainty of the quantity of AE impulses is one of the main reasons of defect detection probability evaluation complexity. To assess the probability of defect detection by the AE method, it is necessary to specify the function of the AE source, including the estimation of emissivity, the estimation of the law of distribution of amplitudes and also the function of the acoustic waveguide.