Influence of a circular crack on the electromagnetic field in a conductor
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INFLUENCE OF A CIRCULAR CRACK ON THE ELECTROMAGNETIC FIELD IN A CONDUCTOR Z. T. Nazarchuk,1 Ya. P. Kulynych,1, 2 and O. S. Koval’1
UDC 537.874
The problem of scattering of low-frequency electromagnetic fields on circular cracks is reduced by the method of small parameter to a system of hypersingular integral equations of the Newton-potential type. For the solution of these equations, we propose to use the method of orthogonal polynomials of two variables. It is shown that the electric dipole whose moment is proportional to the normal component of the sounding field is an electrodynamic analog of the crack. Keywords: hypersingular integral equation, method of small parameter, diffraction, electromagnetic field, crack.
The reliable data on the defects and, in particular, on cracks, i.e., the most dangerous defects, are required for the development of contemporary methods of diagnostics of the service life and durability of engineering structures. The nondestructive testing methods can be regarded as one of the tools that can be used for the accumulation of these data. For the development of the theoretical foundations of eddy-current nondestructive testing, it is necessary to study and develop the methods aimed at the solution of the three-dimensional problems of scattering of electromagnetic fields by cracks in conducting materials. In the major part of research works dealing with the theory of eddy-current nondestructive testing, the problems of interaction of eddy currents with local bulk defects are reduced to weakly singular integral equations over the volume (or over the surface) of the defect. The available numerical algorithms used for the solution of these equations are unsuitable for the boundary integral equations obtained for cracks, which belong to the class of two-dimensional hypersingular integral equations. The development and justification of analytic methods aimed the solution of these equations is connected with significant mathematical difficulties. This is why it is necessary to create new constructive algorithms for their numerical solution. In what follows, we present the results of investigation of the problem of scattering of electromagnetic fields on a crack in a conducting material under the condition that the size of the defect is small as compared with the depth of penetration of the electromagnetic field. For this purpose, we use one of the efficient approximate methods, namely, the method of small parameter. Basic Assumptions and Statement of the Problem Consider a material with conductivity containing a circular crack of radius a . The magnetic permeability of the crack and material is equal to μ 0 . As a unit of length, we take the radius a and choose a system of Cartesian coordinates Ox1 x 2 x 3 whose origin is located at the center of the crack. Assume that the crack lies in the plane 1 2
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected].
Translated from Fizyko-Khimichna Mekhanika Mate
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