It became evident in the late 1980sthat the differences between the Londongroup’s model and Auld’s approach oughtto be reconcilable in a unified theory thatwould be valid for arbitrary permeability.In seeking the unified approach, theperturbation in the magnetic field at thecrack mouth was taken into account byLewis, Michael, Lugg and Collins,104,105who derived a boundary condition usinga flux conservation argument applied to aregion around the opening. The resultingtheory is applicable to materials ofarbitrary relative permeability andcorroborates the unfolding model in thehigh permeability limit.Alternative FormulationsA more formal approach to obtaining theunified theory is to start with a techniquevalid at an arbitrary frequency andspecialize it systematically for the thinpenetration regime. A suitable89Modeling of Electromagnetic TestingFIGURE18.Variation with probe position for coil whose axis is in plane of semiellipticalsimulated crack in aluminum: (a) resistance; (b) reactance. Calculations were performed at250 Hz and by using 32 ×16 grid. See Harrison and Burke for details of coil parameters andsimulated crack.76(a)Resistance (Ω)0.50–0.5–1–1.5–2–2.5–3–50–40–30–20–10010203040(–2.0)(–1.6)(–1.2)(–0.8)(–0.4)(0.4)(0.8)(1.2)(1.6)Probe position, mm (in.)(b)21.510.50–0.5Legend= theoretical plot for 32 ×16 cells= observationsReactance (Ω)–50–40–30–20–10010203040(–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 wherethe electromagnetic field in the conductoris expressed in terms of transverse electricand transverse magnetic hertzpotentials,106ψand ψ´ respectively. Then,the electric and magnetic fields take theforms:(92)and:(93)where z< 0 and where the preferreddirection ^xis normal to the crack plane.In a half-space problem formulatedusing hertz potentials, it is usual tochoose the preferred direction as thenormal to the interface between the airand the conductor. In this way the twopotentials are decoupled at the interface.Although the present choice of preferreddirection leads to coupled interfaceconditions, the chosen modes aredecoupled at the crack surface. In fact, thetransverse electric mode does not interactdirectly with an ideal crack at all. Instead,it is perturbed indirectly through itscoupling with the transverse magneticmode at the surface of the conductor.Because direct transverse electricinteraction with the crack is absent, thetransverse electric potential and itsgradients are continuous at the ideal crackplane. In contrast, the transversemagnetic potential is subject to a directinteraction of the crack with the field andtherefore exhibits a discontinuity at thecrack.To examine the discontinuity of thetransverse magnetic hertz potential, it isnecessary to reconsider the properties ofthe electromagnetic field at the crack.
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