Abstract
The two-dimensional problem of a long crack in a non-magnetic, half-space conductor lying perpendicular to the flow of eddy-currents induced by a one-dimensional external current sheet is studied in the context of eddy-current non-destructive evaluation. Impedance changes due to closed, surface-breaking cracks and deep, subsurface cracks are calculated. The Wiener-Hopf technique is used to obtain an approximate solution for the magnetic field scattered by a subsurface crack and hence the impedance change. The solution is accurate to within 5% for cracks whose edges lie more than one electromagnetic skin depth (delta) below the conductor surface. For the surface-breaking crack the Wiener-Hopf method yields a high-frequency asymptotic series solution for the magnetic field. The first term corresponds to the limit in which the field perturbations by the edge and corners of the crack are decoupled. The impedance change in this limit is found in closed form. Use of the Wiener-Hopf procedure in rigorously treating the open crack problem is investigated. The opening of a deep, subsurface crack whose width is much less than delta is found to be undetectable to first order in the opening. A geometrical theory of eddy-current scattering is developed, based on the optical Geo-metrical Theory of Diffraction. The theory includes a procedure which accounts for multiple scattering of the fields between the edge of a crack and its image. The method is applied to subsurface and surface-breaking cracks, yielding solutions for a subsurface crack whose edge lies only 0.4delta below the conductor surface and for a surface-breaking crack of depth delta or more. Finally, perturbation theory is applied to the surface-breaking crack problem in the low- frequency limit, giving the impedance change for a crack of depth up to 0.4delta.