Solution of the two-dimensional problem of a crack in a uniform field in eddy-current non-destructive evaluation

Abstract

A two-dimensional, time-harmonic eddy-current problem is examined in which a uniform field is perturbed by a long, surface-breaking crack in a non-magnetic, conducting half-space. The crack is assumed to be ideal in the sense that it has infinitesimal opening and yet forms a perfect barrier to the passage of electric current. A solution is sought which is accurate both at high frequencies, at which the skin depth is small compared with the crack depth, and at intermediate frequencies, where the skin depth and crack depth are of similar magnitude. The Wiener-Hopf technique is used to derive a Fredholm integral equation of the second kind for the scattered magnetic field. An approximate closed form solution of this equation is found as a series of exponentially decreasing terms. At lowest order, local solutions at the buried crack edge and at the corners where the crack meets the surface of the conductor are decoupled. Higher order terms in the series account for the coupling which occurs between the fields perturbed by the crack edge and corners.