The geometry of the problem (Fig. 13)lends itself to the Schwarz-Christoffeltheory,87,88which yields a conformaltransformation to map the domain of thecrack and the adjoining half plane aboveit into a half plane. An elementarysolution for the half plane will lead to afixed potential difference across the crack.Then, an inverse transform can be appliedto produce a representation of the electricfield at the crack mouth. In this case, asuitable analytic inverse transform isapparently lacking and the mapping mustbe done numerically by using, possibly,the newton-raphson iterative technique orthe brent algorithm.89Förster90and others91have usedconformal mapping to determine themagnetic flux leakage at the crack mouth.In fact, the mapping is used widely tofind the magnetic field at the gap betweentwo pole pieces such as the field at thegap between the poles of a magneticrecording head.92In eddy currentproblems, the electric field is neededrather than the magnetic field but thesolution is essentially the same (Fig. 13).At the corners, the electric field issingular, varying in magnitude in air closeto the corner as (rcorner)–1/3, where (rcorner)is the radial distance from the apex of thecorner. This behavior is characteristic ofthe field in the vicinity of a right angledwedge.93Between the crack faces, the fieldtends to become more uniform deeperinto the crack. The magnitude of the fieldbetween the faces depends on how deepand wide the crack is. If the crack is madenarrower while the potential across thecrack remains the same, then themagnitude of the electric field increases.In the limit of closure without contact,the electric field forms a singular layer,infinitely strong, of infinitesimalthickness. It is this limiting case that willbe explored here because the singularlayer has a simple mathematicalrepresentation.Impenetrable CrackIn calculations of the field perturbationdue to a crack, it is usual and convenientto apply a boundary condition that statesthat the normal component of the currentdensity in the conductor at the crack faceis zero. Although the surface of the cracksupports a distribution of electrical chargeand the charge must get there somehow,in the quasistatic approximation thecharging current is neglected. In aconductor, the displacement current jωε0Eis neglected because it is very muchsmaller than the charge current σ0E. Evenat high eddy current test frequencies,~10 MHz, where the magnitude ofdisplacement current is greater than atlower frequencies, the ratio ε0ω·σ0–1is onthe order of 10–9for a low conductivitymetal, 0.58 MS·m–1(1 percent of theInternational Annealed Copper Standard).
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