Evaluation of probe impedance due to thin-skin eddy-current interaction with surface cracks

Abstract

Crack detection using eddy-current nondestructive testing is often carried out at frequencies such that the skin depth of the induced current is much smaller than the crack dimensions. The induced current then flows in a thin skin at the conductor surface and at the faces of a surface crack. In the case of a crack that acts as an impenetrable barrier to electric current, the electromagnetic field at the crack surface can be represented, at an arbitrary frequency, in terms of a potential which satisfies a two-dimensional Laplace equation. The boundary conditions required in the solution of the Laplace equation have not yet been determined for the general case, but we have derived approximate boundary conditions which are applicable in the thin-skin regime. The conditions derived are valid for cracks in materials of arbitrary permeability. From the harmonic solutions of the Laplace equation, the impedance change of the excitation coil due to the defect has been calculated for cracks in aluminum and ferromagnetic steel. Comparisons between predictions and experimental measurements on rectangular slots show good agreement, thus corroborating the theory and the numerical calculations.

Thin-skin eddy-current interaction with semielliptical and epicyclic cracks

“…This means that has the same boundary conditions as the solution for a rectangular crack and can therefore be calculated in the same way [12]. Once is known, (30) is used to determine the edge condition on .…”

“…From (12), the dominant impedance term for a semicircular crack of radius and zero opening in a uniform incident magnetic field , can be written as (56) for unit coil current, where is a dimensionless normalized impedance. Note that the impedance depends on the square of the crack radius and therefore depends on its area.…”

“…Although the TM potential is determined for a crack of zero opening, the derivation of the impedance allows for the effects of a small crack gape , and the possibility that the crack is filled with a nonconducting material of relative permeability , which may differ from the permeability of the host material . The impedance is written as (11) where, for unit coil current (12) (13) and (14) The expression for , (12), can be interpreted as the sum of a volume term of order and a surface term of order . The loss of surface at the crack mouth and the additional surface at the crack base give rise to the contribution , where is the TM magnetic field tangential to the crack edge in the plane.…”

“…These calculations have been performed for long cracks of uniform depth [4] and for rectangular cracks [12]. Below, the formalism is adapted to deal with curved crack profiles.…”

“…The normal derivative at the crack mouth, given by (40) is substituted into (32) and the expansion coefficients are calculated by solving a matrix equation. To obtain the matrix equation, (32) is multiplied by , the resultant relationship integrated from to and the orthogonality properties of the sine function used to define a system of linear equations as described in [12]. A solution of the linear system is found by LU decomposition [14].…”