Interaction of induced currents with cracks in thin plates

The induction of eddy currents in a conducting, non-magnetic plate containing a through-the-thickness crack is considered in the limit where the plate thickness (h) is small compared with the electromagnetic skin depth (δ). The plate is represented by a current sheet and the crack by a distribution of generalized current vortices. The current density for these vortices must differ from that of conventional hydrodynamic vortices so as to be consistent with Faraday’s law for self-induction. The vortex density, and hence the induced current in the cracked plate, can be obtained by solving a one-dimensional singular integral equation involving the current density for the uncracked plate which is assumed to be known. The case where the induced currents are produced by a probe coil, as in eddy-current non destructive inspection, is investigated in detail and a simple expression is derived for ∆Z, the change in coil impedance due to the crack. The formalism is applied to the specific problem of an infinite plate containing a straight crack. Analytical results are presented for the limiting cases of long and short cracks, and a numerical method based on Gauss-Chebyshev quadrature is used for intermediate crack length. These theoretical results are compared with experimental measurements of ∆Z and the thin-plate theory is found to be in excellent agreement with experiment (to within 5% or better) for values of h/δ up to 0.4.