Using a two dimensional finite element code, the response of a U-core eddy current probe was computed for a subsurface flaw in a stainless steel medium. Next, using a three dimensional scattering model, the change in coil impedance was calculated for the same situation. From a comparison of these two results, it was concluded that the two dimensional finite element code overestimates the eddy current sensor response for the practical problem at hand by a factor of 10. This agreed well with the result obtained using an approximate technique described in this paper to estimate the true response from two dimensional calculations. Application of such desensitization factor should allow the two dimensional calculations to be effectively used in design studies.</p

Palanisamy, R.

Thompson, Donald

Thompson, R.Bruce

Digital Repository @ Iowa State University (ISU)

ESTIMATES OF EDDY CURRENT RESPONSE TO SUBSURFACE CRACKS FROM 2-D FINITE ELEMENT CODE PREDICTIONS ABSTRACT R. Palanisamy, R. B. Thompson and D. O. Thompson Ames Laboratory, USDOE Iowa State University Ames, IA 50011 Using a two dimensional finite element code, the response of a U-core eddy current probe was computed for a subsurface flaw in a stainless steel medium. Next, using a three dimensional scattering model, the change in coil impedance was calculated for the same situation. From a comparison of these two results, it was concluded that the two dimensional finite element code overestimates the eddy current sensor response for the practical problem at hand by a factor of 10. This agreed well with the result obtained using an approximate technique described in this paper to estimate the true response from two dimensional calcula-tions. Application of such desensitization factor should allow the two dimensional calculations to be effectively used in design studies. INTRODUCTION The selection or development of a reliable eddy current inspec-tion system for a specified flaw requires accurate determination of the sensor characteristic. Traditionally, this is achieved theoretically by predicting the coil response (i.e., the change in impedance, ~Z) to cracks in the given test specimen. In the last few years, one of the popular techniques employed in such analysis is the two-dimensional finite element model. This numeri-cal method calculates ~Z2D in ohms for unit length of the coil since the current source, conducting medium and the flaw are assumed to be infinitely long in the third direction. The physical signifi-cance of the 'infinitely long' flaw must be interpreted with some caution. 569 570 R. PALANISAMY ET AL. In two dimensional analysis the eddy currents flow perpendi-cular to the plane of the figure, so that they are obstructed by the crack area. However, in this analysis, the currents cannot flow together behind the crack as they would in a three dimensional calculation. Hence, it appears likely that these calculations overestimate somewhat the probe sensitivity values (~Z/Z). It is useful to know the error associated with such predictions. This paper describes an approximate technique to estimate the true response of an eddy current sensor (~Z3D) to a finite subsur-face crack from the results of two dimensional calculations (~Z2D)' Both ~Z2D and ~Z3D were theoretically computed for a specific case using finite element and scattering models, respectively. The results are found to be in good agreement with the estimate. 2-D FINITE ELEMENT ANALYSIS In the present study, a horseshoe ferrite core eddy current probe is used to detect subsurface flaws in a semi-infinite stainless steel medium. This situation is similar to the sleeved bolt hole inspection problem reported in references 1 and 2. In the present analysis, we deal with a single conducting medium as opposed to the layered medium in the sleeved bolt hole geometry. The U-core eddy current probe and the cross-section of the test geometry are depicted in Fig. 1. (a) I Coi 1 current R. = mean length Ferrite core w core thickness fy-aXiS Coil Stainless steel Fig. 1. a) U-core eddy current probe. (b) Ferrite core b) Cross-section of U-core probe over a semi-infinite conducting (stainless steel) medium. ESTIMATES OF EDDY CURRENT RESPONSE TO SUBSURFACE CRACKS 571 The material of the semi-infinite medium is assumed to be stainless steel (nonma!netic, and the electrical conductivity = 1.494 x 106 (ohm-m)- ». The relative permeability and electrical conductivity of ferrite core are 5000 and 1 x 10-6 (ohm-m)-l, respectively. The flaw (0.03" wide and 0.015" deep) is assumed to be located at a depth of 0.05" from the surface. The excitation frequency is 40 kHz. Finite element solution of quasi-static magnetic fields in materials, and the prediction of eddy current probe impedance using this technique are presented in references 3 and 4, respectively. As the U-core probe slides over the slab, maximum impedance change (~Z) occurs when the probe is directly above the flaw which is located at a depth of 0.05" below the surface. 2 Taking advantage of the symmetry about y-axis (Fig. lb), finite element analysis was performed only for one half of the region. Contours of constant magnetic vector potential values (absolute) are plotted in Fig. 2. The results of this two dimensional analysis are discussed later in this article. Fig. 2. Contours of constant magnetic vector potential (absolute) at 40 kHz. 572 R. PALANISAMY ET Al. 3-D FIELD SCATTERING THEORY The change in eddy current sensor impedance is found from the incident and scattered fields of the flaw by using the recipro-city theorem. 5 For a void flaw in a linear, homogeneous, isotropic, conducting medium with free space permitivity and permeability, the change AZ in sensor impedance is given by where, a I electrical conductivity of the medium current at the sensor terminals E = electric field without the flaw electric field with the flaw E' vf dv volume of the flaw differential volume element. (1) It is apparent that, to compute AZ, it is necessary to know E and E' within the boundaries of the flaw. 6 ,7 The strategy for computing these electric fields is to approximate the incident field in the vicinity of the flaw by its constant plus linearly varying components. The respective scattered fields are then approximated for an ellipsoidal flaw by the dipole and quadrupole field solutions to the static form of Maxwell's equations. This quasi-static approximation is good when the spatial variations of the fields in the vicinity of the flaw take place over a distance small compared to a skin depth (i.e., the flaw size does not exceed a skin depth). Making use of these approximations, Kincaid6 ,7 has derived expressions to calculate ~z for surface and subsurface flaws. As a special case, let us consider a single component incident field which has constant direction and amplitude over a semi-infinite conductor surface. In Fig. 3, this field is chosen to be in the z direction with constant amplitude Eo' Within the conductor, the field equation along the positive x direction is E z = (2) For this situation, Kincaid7 has obtained an expression for AZ due to an ellipsoidal void flaw at a distance d from the surface of the conductor (Fig. 3). AZ 8 b2(Eo)2 -(l+j~ 1+' ~ (~)2 Ll = yra I e 15 { J 45 15 } (3) ESTIMATES OF EDDY CURRENT RESPONSE TO SUBSURFACE CRACKS Flaw location x 573 Fig. 3. Single component (EZ) uniform incident field over a semi-infinite medium with a subsurface flaw. where, <5 skin depth in conductor a, b major and minor axes of ellipsoidal void flaw (a and b are parallel to x and y axes, respectively, and b/a > 1/2). ESTIMATE OF Z3D FROM Z2D Figure 4 illustrates in greater detail the assumptions made in the interpretation of the ~Z/Z calculations obtained using the 2-D finite element code. Since the currents flow parallel to the infinite dimension of the crack, no ~z will be predicted if the width, w, is set equal to zero. The best simulation of the 3-D crack case appears to be obtained when the width wand depth d of the 2-D crack are set equal to the length 2a and depth c of the 3-D crack. As shown in Fig. 4., the disturbance of the current flow in the 2-D claculation is comparable to that in the plane of the crack in the 3-D case. However, since the crack induced current disruption is less in other planes, one would expect the 2-D calculation to provide a systematic overestimate of ~Z/Z. A rough estimate of the error which this produces follows. The inductance (L3D) of a coil is given by L3D = ~2 fff ~H2dxdydz (4) 574 R. PALANISAMY ET AL. (a) (b) Fig. 4. a) Lines of current flow in two dimensional calculation. b) Lines of current flow in three dimensional calculation. where I, Hand p are current in the coil, magnetic field and magnetic permeability, respectively. In 2-D numerical computation the impedance is obtained as 'ohms/unit length'. That is 1 J -2 L2D = I2 J pH dxdy (5) where H is the magnetic field in the 2-D analysis. The flaw induced fractional change in inductance can be ex-pressed as where, or, H = H(x,y,z), flaw free H field H~ = Hl(X,y,z), H field with flaw 2JffH ~Hdxdydz o 2 fffH dxdydz o where H = H -HI and H/H «1. o 0 (6) (7) ESTIMATES OF EDDY CURRENT RESPONSE TO SUBSURFACE CRACKS 575 Let t and w (Fig. 1) be the mean distance between the pole faces, and the core thickness, respectively. Suppose that Ho(x,y) and ~H(x,y) in the vicinity of the flaw are known from the results of the finite element calculation. Then, when the probe is centered over the flaw, one expects that, near the flaw position, H (x,y,z) ~ H (x,y)f(w/t) o 0 (8) where f(w/t) is a factor, accounting for field decrease due to finite length effects, which varies from zero when wIt = 0 to unity when w/t~. Similarly, one expects that near the flaw position, ~H(x,y,z) ~ ~H(x,y)f(w/t)g(z/a) (9) where, the factor g(z/a) accounts for the field variation in the z direction. That is, at the flaw where z = 0, g(O) = 1, and when z increases, g(z/a)~ as the fields return to their flaw free values. Finally, it is assumed that the L3D is related to the L2D by the expression (10) where, h(w/t) is a factor accounting for finite length effects which varies from zero when wIt = 0 to unity when w/t~. Substi-tution of Eqs. (8)-(10) into Eq. (7) yields the desired result eo ~ 2 ] feo g(z/a)dz (~L) ~ f (wIt) - (~L) L 3D w w t L 2D (-)h(-) t t (11) The term inside the square brackets describes the finite length effects of the coil and is expected to be slowly varying when ~t. Note, for example, that both the numerator and denominator increase with wIt and these variations would tend to cancel one another. Thus, for the purpose of this rough estimate, this factor is set equal to unity. If one further assumes that when a~c, the spatial extent of the field perturbation by a crack is on the order of the crack radius, then r g(z/a)dz _00 ~ 2a (12) and one obtains the final result 576 R. PALANISAMY ET Al. (13) RESULTS AND CONCLUSION Taking a = 0.015" and = 0.16" as predicted by the finite element calculations,2 one obtains(AL/L)3D ~ 0.2(~L/L)2D. From the 2-D finite element prediction, (~L/L)2D = 1.2 x 10-3 . There-fore, (~L/L)3D = 2.4 x 10-4 . Using the scattering model, ~Z3D was calculated for an oval shaped crack (major and minor axes were assumed as 0.03" and 0.015", respectively) at a depth of 0.05" from the surface. That is, ~Z3D = (.17 x 10-5)/12 ohms, where I is the drive current required to produce a current density of 106 amps/m2 in the coil whereas the 2D finite element code predicted ~Z2D = (0.425 x 1O-4}/I2 ohms/m. Assuming that W=0.16" (= 0.004 meter), these results imply 0.17 x 10-5/1 2 0.1 (4) (0.425xlO-4/I2}xO.004 By way of comparison, the order of magnitude calculations leading to the estimate in Eq. (13) can be recast in the form 05) 2 which has the value 0.2f for the case under consideration. The two are considered to be in excellent agreement, particularly when one would expect f to be somewhat less than unity for a coil in which t = w. Hence, it is concluded that the 2-D finite element calculations overestimate the sensitivity by a factor of 10 for the practical problem at hand. Application of this desensitization factor should allow those calculations to be effectively used in design studies in which the properties of the materials are varied. ACKNOWLEDGEMENT This work was sponsored by the Center for Advanced Nondestruc-tive Evaluation, operated by the Ames Laboratory, USDOE, for the Naval Sea Systems Command and the Defense Advanced Research Projects Agency under Contract No. W-7405-ENG-82 with Iowa State University. ESTIMATES OF EDDY CURRENT RESPONSE TO SUBSURFACE CRACKS 577 REFERENCES 1. R. Palanisamy and K. M. Lakin, "Development of an eddy current inspection technique for sleeved engine disk bolt holes", Review of Progress in Quantitative NDE, University of Cali-fornia, San Diego, La Jolla, CA, August 1-6, 1982. 2. R. Palanisamy, G. L. Burkhardt, R. E. Beissner and D. o. Thompson, "Eddy current detection of subsurface cracks in engine disk bolt holes", presented at the Review of Progress in Quantitative NDE, University of California, Santa Cruz, CA, August 7-12, 1983. 3. M. V. K. Chari, "Finite-element solution of the eddy-current problem in magnetic structures", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-93, No.1, January-February, 1974, pp. 62-72. 4. W. Lord and R. Palanisamy, "Development of theoretical models for NDT eddy current phenomena", Ed. by George Birnbaum and George Free, ASTM Pub. Code No. (PCN)04-722000-22, 1981, pp. 5-21. 5. Auld, B. A., "Quantitative modeling of flaw responses in eddy current testing", Fourth Monthly Report, Electric Power Research Institute Contract No. RPI395-3, Feb. 1979. 6. T. G. Kincaid, K. Fong and M. V. K. Chari, "Progress in solving the 3-dimensional inversion problem for eddy current NDE", Proceedings of the ARPA/AFML Review of Progress in Quantita-tive NDE, Rockwell International Report to Air Force Materials Laboratory, 1981, pp. 463-468. 7. T. G. Kincaid, personal communication. 

Estimates of Eddy Current Response to Subsurface Cracks from 2-D Finite Element Code Predictions

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Estimates of Eddy Current Response to Subsurface Cracks from 2-D Finite Element Code Predictions

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