TEAM Workshop Problem No. 15
Rectangular Slot in a Thick Plate

A Problem in Non-Destructive Evaluation



Introduction

Inversion of eddy-current data and the reconstruction of flaws is the preeminent problem in electromagnetic nondesstuctive evaluation (NDE). This places a premium on developing good forward models for computing field-flaw interactions, because all inversion algorithms must, of necessity, rely on such calculations. There has evolved in recent years several sophisticated computational models for the forward problem [1-4], but these models differ significantly in their theoretical and numerical approaches. For exemple, [1-3] use a volume-integral approach that incorporates fast Fourier transforms with conjugate gradients to solve the resulting linear system of equations, whereas [4] uses finite-elements.
Because of this diversity of theoretical-computational approaches, it has become clear that there is now a great need to present experimental data from benchmark problems, whose purpose is to not only validate individual models and codes, but to also allow comparisons between competing models and codes. In this paper we present two such problems for the calculation of impedance change, DZ. These problems,
    1. Rectangular slot in a thick plate (900 Hz)
    2. Rectangular slot in a thick plate (7 kHz),
have the common feature of being based on practical eddy-current testing techniques, and of utilizing simple geometries. Additional problems of this genre, including cracks in a thin plate, cracks in a double plate system, and cracks in a thin plate with a tangent coil, are collected together in [6]; further details of each experiment can be found in the references cited in this paper.


Problem No. 1

The experimental arrangement is shown schematically in Figure 1. Here, a circular air-cored coil is scanned, parallel to the x-axis, along the length of a rectangular slot in an aluminium alloy plate. Both the frequenty and th ecoil lift-off are fixed, and DZ is measured as a function of coil-center position. The parameters for this test experiment are listed in Table 1. This problem is completely described in [7], and is also included in [5]. Solutions appear in [3, 8], where a volume-integral equation is used. Preliminary calculations for this case were first reported by Dunbar [9].


Problem No. 2

The rectangular slot geometry for this problem is identical to that of Problem No. 1 (see Fig.1). The experimental arrangements uses a larger coil, at a higher frequenty (see Table 2 for the parameters). The skin depth at this frequenty is one-fifth of the slot depth, which makes this problem differ from No. 1 by nearing the thin-skin limit. Theoretical calculations for thid problem have been published [8].


Objective

The objective is to compute the change in the inductance and resistance of the driving-point impedance of the coil (compared to its value over an unflawed portion of the plate) as a function of coil position, and compare the computed results to the experimental results tabulated in Table 3. This is to be done for each problem. In addition, the computed and experimental results are to be compared by plotting the magnitude and phase of each versus coil-center position. Plot the magnitude and phase (in degrees) on separate graphs, for each test. The magnitude and phase are given by:

|DZ| = ((DXL)² + (DR)² ))½

Arg(DZ) = arctg (DXL / DR)

where DXL = wDL.



Figure 1: Schematic configuration for the measurement of DZ due to a surface breaking slot.


Table 1 - Parameters of Test Experiment No. 1 (see Fig. 1)

Table 2 - Parameters of Test Experiment No. 2 (see Figure 1)

Table 3 - Experimental Results for Problem 1 and 2


References


  1. J. R. Bowler, L. D. Sabbagh, and H. A. Sabbagh, "A Theoretical and Computational Model of Eddy-Current Probes Incorporating Volume Integral and Conjugate Gradient Methods", IEEE Trans. Magnetics, Vol. 25, No. 3, May 1989, pp. 2650-2664.
  2. H. A. Sabbagh, L. D. Sabbagh, and J. R. Bowler, "A Volume-Integral Code for Eddy-Current Nondestructive Evaluation", COMPEL-The International Journal for Computation and Mathematics in Electrical and Engineering, Vol. 9 (1990), Supplement A, pp. 67-70.
  3. J. R. Bowler, S. A. Jenkins, L. D. Sabbagh, and H. A. Sabbagh, "Eddy-Current Probe Impedance Due to a Volumetric Flaw", J. Applied Physics, Vol. 70, No. 3, 1 August 1991, pp. 1107-1114.
  4. W. Lord and R. Palanisamy, in G. Birnbaum and G. Free, eds., "Eddy-Current Characterization of Materials and Structures", ASTM STP722 (American Society for Testing and Materials, Philadelphia, 1981), pp. 5-81.
  5. H. A. Sabbagh, ed., "The ACES Collection of Canonical Problems: Set1", The Applied Computational Electromagnetics Society, Spring 1990, pp. 3-8.
  6. Applied Computational Elecromagnetics Society Newsletter, Volume 6, No. 1, March 1991, pp. 17-34.
  7. S. K. Burke, "A Benchmark Problem for Computation of DZ in Eddy-Current Nondestructive Evaluation (NDE)", J. Nondestructive Evaluation, Vol. 7, Nos. 1/2, 1968, pp. 35-41.
  8. D. McA. McKirdy, J. Nondestructive Evaluation, Vol. 8 (1989), pp. 45-51.
  9. W. Scott Dunbar, J. Nondestructive Evaluation, Vol. 7 (1988), pp. 43-53.