ET04.pdf

# Two properties of the dipole density are worthy of

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Two properties of the dipole density are worthy of note at this point. Firstly, it tends to zero at the buried crack edge. Secondly, the derivative P ·( z ) –1 is zero at the crack mouth, z being the coordinate whose axis is normal to the surface of the conductor (Fig. 15). These properties are written as: (78) and: (79) where r e is the coordinate of an edge point and r m is the coordinate of a point at the crack mouth. For example, Eq. 80 gives the dipole density for a long straight crack of depth d in a uniform unperturbed field E 0 : 83 (80) Note that p ( z ) vanishes at z = – d and that the derivative with respect to z vanishes at z = 0 in keeping with the general properties in Eqs. 78 and 79. In addition, it is important to be aware that the electric field has a half-power singularity at the edge of an ideal crack varying locally as: 96 (81) where ρ is the perpendicular distance (meter) of a point from the edge and φ is an angle (radian) measured from the surface S 0 in a plane perpendicular to the edge. This means that, in general, the dipole density varies as: (82) near the edge. p ρ E ρ φ ρ ρ φ φ φ , ˆ cos ˆ sin ( ) × 1 2 2 p z E d z ( ) = 2 0 0 2 2 σ = p z r m 0 p r e ( ) = 0 E E t t t + = 1 0 σ p p p p A B t S = ⋅δ E E t t S A B + ( ) + = δ σ p p 0 0 lim w d p C = 0 1 0 E n n σ lim w d p C = 0 P n n 85 Modeling of Electromagnetic Testing F IGURE 16 . Integration path C 0 crosses crack at points A and B and is formed in limit as A± and B± approach surface S 0 . A A + A C 0 B + B B S 0

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The solution of the eddy current ideal crack problem has been reduced to one of finding the surface dipole density p. Thus a scalar replaces a two-component vector, the jump in the tangential electric field. Consequently fewer unknowns are needed for a numerical solution. To calculate p, it is necessary to know the continuity conditions that apply to the magnetic field at the crack surface S 0 because these conditions will be needed in the derivation of an equation from which the dipole density can be calculated. Although the details of these derivations will not be given here, it is useful to understand the continuity conditions that apply to the magnetic field at the ideal crack surface. The jump in the tangential electric field at the ideal crack is inseparable from the singular property of the electric field between the crack faces, as expressed here in terms of a current dipole layer. However, no such singular behavior occurs in the magnetic field. The truth of this can be demonstrated by following an argument like the one for the electric field but applying Stokes’ theorem to Ampère’s law rather than to the induction law, thereby forming the line integral of H around the path C 0 . Following this parallel argument, it can be deduced that the line integral of H vanishes as the closed path A A + B + B (Fig. 16) collapses onto the crack but no singular behavior of the magnetic field in the crack could lead
• Fall '19
• Wind, The Land, Magnetic Field, Dodd, Modeling of Electromagnetic Testing

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