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At the crack face because these are not defined

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at the crack face, because these are not defined. Instead, the boundary conditions at an arbitrary frequency (Eqs. 78 and 79) are imposed on p . In finding a solution using the moment technique using N equations for N unknowns, a reduction in the unknowns needed to approximate p can be made because the prior knowledge derived from Eqs. 78 and 79 restricts its behavior at the perimeter of the crack. This technique releases some degrees of freedom that can be used to represent ψ ( r ± ) as a solution of the laplace equation on the crack face. By management of the unknown coefficients in this way, a solution can be found that agrees with experiment. 108 91 Modeling of Electromagnetic Testing F IGURE 19. Inductance and resistance variation with probe position for coil whose axis is in plane of semielliptical artificial crack in aluminum: (a) inductance plot; (b) resistance plot. Theory (solid line) is compared with experimental results (points) acquired at 50 kHz. See Harrison and Burke for details of coil parameters and simulated crack. 76 (a) Inductance (mH) 2.5 2 1.5 1 0.5 0 –20 –15 –10 –5 0 5 10 15 20 (–0.8) (–0.6) (–0.4) (–0.2) (0.2) (0.4) (0.6) (0.8) Probe position, mm (in.) (b) Resistance ( ) 200 150 100 50 0 –20 –15 –10 –5 0 5 10 15 20 (–0.8) (–0.6) (–0.4) (–0.2) (0.2) (0.4) (0.6) (0.8) Probe position, mm (in.) Legend = theoretical plot for 32 × 16 cells = observations
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Thin Penetration Regime As Auld has shown, a suitable boundary condition for formulating a well defined laplace problem on S 0 in the thin penetration regime can be derived from the magnetic field at the crack mouth. The transverse magnetic component of the magnetic field in the Y direction can be written: (99) where ψ (c) is the perturbed potential (volt) due to the crack. As it stands, Eq. 99 cannot be used immediately as a boundary condition because the perturbed field at the mouth is not known in advance . Auld got around this problem by neglecting the perturbation of the magnetic field at the crack mouth, a reasonable approximation because it can be small for nonferromagnetic materials. Taking the field perturbation into account increases the complexity of the problem 107 but improves the accuracy of the results for nonferrous alloys and gives results valid for ferromagnetic materials. 109 Results of impedance predictions 110 and measurements for a semielliptical artificial crack are shown in Fig. 19. The experimental data are taken from a series of measurements made at 16 frequencies. 76 For a comparison with thin penetration theory, results at the highest frequency (50 kHz) are shown. Calculations were performed with conformal mapping. 110 At this frequency, the depth of the simulated crack, 8.61 mm (0.339 in.), is more than 18 times the standard depth of penetration, 0.47 mm (0.019 in.). Note that the theory underpredicts the resistive component by about 10 percent.
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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