Analytical modeling for the plate with a flat-bottom hole based on the reflection and transmission theory in pulsed eddy current testing

Abstract

Ferromagnetic structures such as pipes or vessels are widely used in petroleum, chemical and power generation industries. Periodical nondestructive testing (NDT) is vital for continued safe operation. As a NDT technology, pulsed eddy current testing (PECT) technology which is excited by a square-wave pulse rather than a sinusoidal waveform has been widely used for its advantages of non-contact and acquisition of information at various depths in one excitation process. In PECT, the analytical modeling is important because it gives a better understanding of the signal and benefits the inverse process of PECT in extracting information of structures. The foundation of theoretical model of PECT is the Dodd-Deeds model presented by Dodd and Deeds in 1968, Theodoulidis and Kriezis represented the integral solution of Dodd-Deeds model in the form of series by using the truncated region eigenfunction expansion (TREE) method. Using the Dodd-Deeds model and the TREE method, other analytical modelings have been solved. However, most modelings assume that the wall thinning of the specimen is uniform, and the analytical solution only contains the variables in the z direction (the direction perpendicular to the surface of the specimen), such as the thickness of the specimen. With the rapid development of PECT, problems such as the footprint of the probe, the quantitative analysis of local wall thinning also need to be solved. These problems are related to the variable in the r direction (the direction parallel to the surface of the specimen), so the analytical modelings mentioned above are not available any more. To solve these problems, the analytical modeling of the plate with a flat-bottom hole is proposed. Considering the fact that the boundary condition in the analytical modeling of the plate with a flat-bottom hole is complicated, the assumption that the transverse wave number and the longitudinal wave number in the layer where the flat-bottom hole located are the same is made in this paper, and the transverse wave number is set to be only related to the structure in the r direction. Firstly, the expressions of magnetic vector potential in all the layers are obtained by using the reflection and refraction theory of electromagnetic wave. Then the analytical solution is solved based on the extended Cheng's matrix method by introducing the construction coefficient Wn. Finally, the 16MnR specimen with the flat bottom holes is conducted as an example, and experiments are carried out. The good agreement between results calculated by the analytical model and the experimental results measured verifies the developed analytical model.