# Estimation of Fatigue Crack AE Emissivity Based on the Palmer-Heald Model
## 1 Introduction
There are several fundamental research papers describing AE impulse amplitude distribution laws [15; 16; 17] while the possibility of finite element analysis makes it possible to acoustic waveguide precise calculation [18; 19; 14]. At the same time, an evaluation of the material emissivity and estimation of the number of AE impulses emitted by a defect is a problem that has not been completely solved. At present, emissivity is evaluated either qualitatively (materials are characterized by low, average or high emissivity), or approximately--by one characteristic value [20]. In a number of studies, two methods of non-destructive testing were used together to interpret and predict the \(N_{\Sigma}\) parameter. For example, the joint use of AE and magnetic testing was described in [21] and the joint use of AE and thermography was performed in [22]. The AE parameters and in particular, \(N_{\Sigma}\), are interpreted based on the known diagnostic models of magnetic testing and thermography, respectively. In this paper, to establish a relationship between the defect parameters and AE data, it is proposed to study the nature of the \(N_{\Sigma}\) change with a stress increase during the sample loading. To quantify parameter \(N_{\Sigma}\) and the nature of its change with an increasing load, it is proposed to use the coefficients of analytical Dunegan [2] and Palmer-Heald [3] models. Based on the assumption that the number of AE impulses is proportional to the volume of the plastic deformation zone of the material at the crack tip and that the accumulation rate of AE pulses is proportional to the rate of increase of the plastic deformation zone under static loading of the object, the Dunegan model can be written in the form  
\[N_{\Sigma}=AP^{n} \tag{1}\]  
where \(P\) is the applied load; \(n\) is a power indicator characterizing the stress intensity factor at the crack tip; \(A\) is a constant. The power indicator \(n\) determines the form of the dependence \(N_{\Sigma}\)(\(P\)) and \(A\) is a normalizing coefficient that ensures the correspondence of the quantitative indicators of the model and the experiment. According to the Palmer-Heald model, the number of AE impulses can be expressed as  
\[N_{\Sigma}=D\cdot a\left(sec\left(\frac{\pi}{2}\frac{\sigma}{\sigma_{y}} \right)-1\right) \tag{2}\]where \(\sigma\) is the actual stress; \(\sigma_{y}\) is theyield strength; \(a\) is half the length of the crack; \(D\) is thedimensional coefficient of the model (1/m), depending on the characteristics of the material, temperature and type of stress-strain state. The number of AE impulses recorded during loading of an object with a crack is determined by the material properties, defect characteristics and loading conditions. In the Dunegan model, the loading parameter is taken into account as the basis of the power-law dependence \(P\), the defect characteristic is taken into account using the exponent \(n\), and the coefficient \(A\) is determined by the material properties. In the Palmer-Heald model, the loading conditions are determined by the \(\sigma\)/\(\sigma_{y}\) ratio, the defect characteristics are determined by a coefficient \(a\) equal to half the crack length and the value of the coefficient \(D\) depends on the material properties. Thus, the coefficient \(D\) of the Palmer-Heald model and coefficient \(A\) of the Dunegan model can be considered as parameters that quantitatively characterize the material emissivity (as parameter that closely correlated with AE emissivity), independent of loading conditions, crack length, and stress intensity factor at its apex. This paper describes an approach to the evaluation of the quantity of AE impulses emitted of fatigue crack. The evaluation is based on of the Palmer-Heald model multiplicative parameter \(D\), which can be interpreted as an indirect evaluation of AE emissivity of the steel during the crack propagation.