To standard depth of penetration δ the geometry of

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The geometry of the problem (Fig. 13) lends itself to the Schwarz-Christoffel theory, 87,88 which yields a conformal transformation to map the domain of the crack and the adjoining half plane above it into a half plane. An elementary solution for the half plane will lead to a fixed potential difference across the crack. Then, an inverse transform can be applied to produce a representation of the electric field at the crack mouth. In this case, a suitable analytic inverse transform is apparently lacking and the mapping must be done numerically by using, possibly, the newton-raphson iterative technique or the brent algorithm. 89 Förster 90 and others 91 have used conformal mapping to determine the magnetic flux leakage at the crack mouth. In fact, the mapping is used widely to find the magnetic field at the gap between two pole pieces such as the field at the gap between the poles of a magnetic recording head. 92 In eddy current problems, the electric field is needed rather than the magnetic field but the solution is essentially the same (Fig. 13). At the corners, the electric field is singular, varying in magnitude in air close to the corner as ( r corner ) –1/3 , where ( r corner ) is the radial distance from the apex of the corner. This behavior is characteristic of the field in the vicinity of a right angled wedge. 93 Between the crack faces, the field tends to become more uniform deeper into the crack. The magnitude of the field between the faces depends on how deep and wide the crack is. If the crack is made narrower while the potential across the crack remains the same, then the magnitude of the electric field increases. In the limit of closure without contact, the electric field forms a singular layer, infinitely strong, of infinitesimal thickness. It is this limiting case that will be explored here because the singular layer has a simple mathematical representation. Impenetrable Crack In calculations of the field perturbation due to a crack, it is usual and convenient to apply a boundary condition that states that the normal component of the current density in the conductor at the crack face is zero. Although the surface of the crack supports a distribution of electrical charge and the charge must get there somehow, in the quasistatic approximation the charging current is neglected. In a conductor, the displacement current j ω ε 0 E is neglected because it is very much smaller than the charge current σ 0 E . Even at high eddy current test frequencies, ~10 MHz, where the magnitude of displacement current is greater than at lower frequencies, the ratio ε 0 ω · σ 0 –1 is on the order of 10 –9 for a low conductivity metal, 0.58 MS·m –1 (1 percent of the International Annealed Copper Standard).
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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