It became evident in the late 1980s that the

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It became evident in the late 1980s that the differences between the London group’s model and Auld’s approach ought to be reconcilable in a unified theory that would be valid for arbitrary permeability. In seeking the unified approach, the perturbation in the magnetic field at the crack mouth was taken into account by Lewis, Michael, Lugg and Collins, 104,105 who derived a boundary condition using a flux conservation argument applied to a region around the opening. The resulting theory is applicable to materials of arbitrary relative permeability and corroborates the unfolding model in the high permeability limit. Alternative Formulations A more formal approach to obtaining the unified theory is to start with a technique valid at an arbitrary frequency and specialize it systematically for the thin penetration regime. A suitable 89 Modeling of Electromagnetic Testing F IGURE 18. Variation with probe position for coil whose axis is in plane of semielliptical simulated crack in aluminum: (a) resistance; (b) reactance. Calculations were performed at 250 Hz and by using 32 × 16 grid. See Harrison and Burke for details of coil parameters and simulated crack. 76 (a) Resistance ( ) 0.5 0 –0.5 –1 –1.5 –2 –2.5 –3 –50 –40 –30 –20 –10 0 10 20 30 40 (–2.0) (–1.6) (–1.2) (–0.8) (–0.4) (0.4) (0.8) (1.2) (1.6) Probe position, mm (in.) (b) 2 1.5 1 0.5 0 –0.5 Legend = theoretical plot for 32 × 16 cells = observations Reactance ( ) –50 –40 –30 –20 –10 0 10 20 30 40 (–2.0) (–1.6) (–1.2) (–0.8) (–0.4) (0.4) (0.8) (1.2) (1.6) Probe position, mm (in.)
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formulation for this strategy is one where the electromagnetic field in the conductor is expressed in terms of transverse electric and transverse magnetic hertz potentials, 106 ψ and ψ ´ respectively. Then, the electric and magnetic fields take the forms: (92) and: (93) where z < 0 and where the preferred direction ^ x is normal to the crack plane. In a half-space problem formulated using hertz potentials, it is usual to choose the preferred direction as the normal to the interface between the air and the conductor. In this way the two potentials are decoupled at the interface. Although the present choice of preferred direction leads to coupled interface conditions, the chosen modes are decoupled at the crack surface. In fact, the transverse electric mode does not interact directly with an ideal crack at all. Instead, it is perturbed indirectly through its coupling with the transverse magnetic mode at the surface of the conductor. Because direct transverse electric interaction with the crack is absent, the transverse electric potential and its gradients are continuous at the ideal crack plane. In contrast, the transverse magnetic potential is subject to a direct interaction of the crack with the field and therefore exhibits a discontinuity at the crack. To examine the discontinuity of the transverse magnetic hertz potential, it is necessary to reconsider the properties of the electromagnetic field at the crack.
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  • Fall '19
  • Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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