SEMI-ELLIPTICAL SURFACE FLAW EC INTERACTION AND INVERSION: THEORY. B. A. Auld and s. Jefferies

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1 SEMI-ELLIPTICAL SUFACE FLAW EC INTEACTION AND INVESION: THEOY B. A. Auld nd s. Jefferies Edwrd L. Ginzton Lbortory Stnford University Stnford, CA 9435 J, C. Moulder nd J, C. Gerlitz Ntionl Bureu of Stndrds Boulder, CO 833 INTODUCTION In 1984 the concept of the flw profile ( plot of the mgnitude nd phse of ~Z long the length of the flw) ws introduced s tool for inverting eddy current mesurements tken on surfce breking flws interrogted with sptilly nonuniform probe field [1,2]. Comprison of direct mesurements of pproximtely rectngulr EDM notches with theoreticl clcultions for rectngulr flws showed stisfctory greement. Inversion procedures developed for sizing flw from its flw profile dt were lso found to give resonble results. During the pst yer the gol hs been to develop probe-flw interction theory for semi-ellipticl flws in nonuniform interrogting field. This is gin collbortive effort between Stnford nd the Ntionl Bureu of Stndrds t Boulder, The theory, nd some comprison with experiment, is given in this pper nd detils of the experiments re described in compnion pper [3]. THE FLAW POFILE CONCEPT AND INVESION Figure 1 illustrtes the mesurement of the flw profile for flw whose surfce length is much greter thn the men dimeter 2r of the eddy current probe. The concentric circles t the left represent the eddy currents produced when the coil, which is centered on the plne of the surfce breking flw, is t position x on the x-xis shown in the figure. The flw profile is tken by scnning the probe long the x-xis nd recording ~Z s function of x In blind test where the orienttion of the flw is not known priori, the flw profile must be obtined by tking rster scn nd extrcting the profile from the rster response, s described in [1]. For long flw the profile hs the chrcteristic shpe shown in the figure, nd it hs been shown [1] tht the length 2c of the flw cn be obtined from either the spcing of the shoulders or the outer slope intercepts of the profile. Once the length hs been determined the depth nd the opening ~u of the flw my be found by constructing McFetridge Chrt for the flw (Fig. 2). 383

2 POBE SCAN D I ECT I O~ / ' EDDY / /-::_-..., '\ CUENT I I /' -,', \ I VOTEX I { ~ l CENTEED \ \ \.':::/,' / AT x \ '...,, /' ' "" 6u X 6Z(Xo) FLAW AMPLITUDE POFILE Fig. 1. Flw mplitude profile for n verge coil dimeter 2r > flw length 2c. c: N <l lj.. w ::>... z (.!) <t.1 ::E ALUMINUM '.51 r' PHASE OF 6Z (deg) Fig. 2. Mgnitude versus phse of ~Z t the flw profile center for rectngulr f lw (luminum 661, 8 =.51", r =.32"). (2c = 2, normlized to r ). This is plot of mgnitude-versus-phse of ~Z t the center of the profile, clculted for given 2c nd different vlues of the dimensions nd ~u. By plotting the experimentlly mesured mplitude nd phse of ~Z on the chrt one cn find the inverted vlues of nd \ u. POTENTIAL FOMULATION OF THE ~z FOMULA The bsic 6.Z formul for surfce breking f l w is 6.Z 1 I (E H.. - E.. H ) dxdy I2 y X y X mouth where the xy plne is the surfce of the substrte nd the z-xis is directed into the substrte. Orienttion of the flw with respect to the (1) 384

3 x-xis nd the y-xis is s shown in Fig. 1. Therefore, the mgnetic field of the probe hs only n x component t the position of the flw, since the probe hs circulr symmetry. Integrtion is over the surfce opening, or mouth, of the flw. The unprimed fields re those t the surfce in the bsence of the flw nd the primed fields in the presence of the flw. An pproximtion is mde by ssuming tht the mgnetic field in the mouth of the flw is the sme s the originl field t the sme point on the unflwed surfce (H~ =H). X X In the theory of [1] the mgnetic field inside the volume of the flw ws obtined, for rectngulr flws, by expnding in seprted vrible solutions to Lplce's eqution. This pproch, bsed on the qusisttic pproximtion tht is vlid for ll eddy current testing, enbles one to evlute the primed electric field in Eq. (1) [4]. For semi-ellipticl flws this pproch is not convenient nd it is more prcticl to solve the interior mgnetosttic problem numericlly by the finite difference method. This suggestion ws mde by the nondestructive testing group t the London Center for Mrine Technology, University College London [5]. To implement finite difference evlution of ~Z it is desirble to reformulte Eq. (1) in terms of the mgnetosttic potentil. Substituting the grdient of for H, using Mxwell's equtions, nd simplifying with vrious vector identities reduces Eq. (1) to the form ~z iw~o f (f~ ~ ~- } ~ ~~) dxdy I th { <3 z J t (3 z J mou (2) where the primed nd unprimed quntities hve the sme significnce s in Eq. (1). Further lgebr converts this to M ~ I 2 ( ~/x) 2 dx + ~ f J <V/.Q,) 2 d.q, I2' mouth tip 2 ( i + i. Mu) ~ ~/z dxdy I 2 o2 {outh (3) where KL nd KT re the Khn coefficients defined in [4], nd the vrible ".Q," in the second term is mesure of length long the crck tip. The substrte conductivity is nd its skin depth is o. FINITE DIFFEENCE ANALYSIS The problem is to solve for the mgnetosttic potentil in the interior of the flw ( ~ in Eq. (3)), with Dirichlet boundry conditions t the mouth of the flw (i.e., the unperturbed interrogting potentil), nd Neumnn boundry conditions t the curved tip of the flw (i.e., the norml derivtive is zero). As in the 1984 clcultions, the interior mgnetic field is ssumed to lie entirely in the xz plne, so tht the potentil problem is two-dimensionl. After converting the Lplce eqution nd boundry conditions to finite difference form the set of difference equtions is solved by itertion, strting with some pproximte solution such s the solution for rectngulr flw. The number of mesh points used is typiclly 19 long the mouth (x direction) nd 9 in the depth (z direction). The mximum number of itertions is 1. Once ~ hs been found nd the derivtives of Eq. (3) evluted, the problem is solved. There re no tedious numericl summtions of nlytic seprted vrible solutions to be crried out s in the "nlyticl" solution for rectngulr flws. Becuse 385

4 of this, the finite difference pproch does not require significntly more numericl computtion thn the "nlyticl" pproch, nd it hs the importnt dvntge of pplying to flws of ny shpe. The progrm ws tested nd debugged by reclculting the rectngulr flws nlyzed in [1]. UNIFOM INTEOGATING FIELD The first complete nlysis of probe-flw interction ws rectngulr flw in uniform interrogting field [4], where it is shown tht the probe response cn be expressed s L'lZ 2 ch CJI c (1 + i) 21 + i cl'lu \ with function of w only, nd zo zl functions of /c only. ' Here, Ho is the uniform interrogting field nd the Z's re shpe prmeters depending only on the spect rtio /c. These prmeters hve been reclculted for rectngulr flws, using the finite difference progrm, nd hve lso been evluted for semi-ellipticl flws. The results, used in [6], re listed in Tble 1. ectngulr Tble 1. Uniform Field Prmeters Semi-Ellipticl /c zo zl /c zo zl o (4) NONUNIFOM INTEOGATING FIELD Finite difference clcultions hve been mde for number of flws produced by the Ntionl Bureu of Stndrds [3] nd Mrtin Mriett [7], using the sme ir-core probe s in 1984 [1]. Figure 3 shows computed mgnitude nd phse profiles for rectngulr nd semi-ellipticl ftigue crck models. The curves re generlly similr except for the fct tht the semi-ellipticl geometry hs more sloping shoulders, s expected from the curvture of the flw ends. Another feture to note is tht the slope intercepts of the two mgnitude curves re essentilly the sme. This mens tht for prcticl purposes the rectngulr length inversion curves [1] cn lso be used for semi-ellipticl flws. Figure 4 shows McFetridge chrt for semi-ellipticl flw with the sme prmeters s the rectngulr flw of Fig. 2. Here, there is clerly difference between the two cses, so tht the shpe of the flw must be tken into ccount in inverting nd L'lu. Figures 5 nd 6 show comprisons of Stnford clcultions with NBS mesurements for n EDM notch nd ftigue crck. More detiled discussion of these nd other results is given in [3]. One point should, however, be emphsized. There is very significnt difference between the experimentl curves tken with two nominlly identicl ir-core probes. 386

5 MAGNITUDE OF 6Z (fi) vs xo.2r ~ s-.159] LENGTH, 7.5 OMAL I ZED DEPTH,.64 TOr'.32"' OPE I G'. PHASE OF 6Z (deg) vs x 9r s '.159] LENGTH' 7.5 NOMALIZED DEPTH,.64 TOr'.32'" OPENING '. 5 ~--~-5~------~o~------~5----~ xo/7 Fig. 3. Computed f tigue crck (~u = ) model f or rectngulr nd s emi-elli pticl shpes (2c = 7.5 normlized to r, /c =.171, /6 = 4). () Mgnitude, (b) Phse. 387

6 This points up the extreme importnce of eddy current probe chrcteriztion in quntittive NDE. 3 N <l LL w McFETIDGE CHAT (2c/'f=2. SEMI-ELLIPTICAL) z (!) ~.1 ALUMINUM PHASE OF t:.z (degl Fig. 4. Mgnitude versus phse of ~Z t the flw profile center for semi-ellipticl flw (Aluminum 661, o =.51", F =.32".) (2c = 2, normlized to I). INVESION This section constitutes brief review of the experimentl dt discussed in [3] nd their inversion. The flw shpes re identified in Tble 2 (SE =semi-ellipticl EDM, =rectngulr EDM, F =ftigue). Figure 7 compres inverted flw length with ctul flw length. In Fig. 8 experimentl dt points for semi-ellipticl EDM notches re plotted on theoreticl McFetridge chrt, nd Fig. 9 does the sme for rectngulr EDM notches. Finlly, Tble 2 compres inverted nd ctul vlues of depth nd opening ~u. This dt ws ll tken with n ir-core probe hving Y =.32". It should be noted tht the error in depth inversion is significntly greter for the deeper flws. The reson is cler from Figs. 8 nd 9, where it cn be seen tht the curves become more compressed for lrger vlues of /o. Since ll curves on the McFetridge chrt re for the sme o, or frequency, the solution to this problem is to repet the mesurement t lower frequency. 388

7 MAGNITUDE OF AZ (!l) vs xo.3r ~ s '.159) LENGTH, 4.55 NOMALIZED.25 DEPTH,.87 TOr'.32" OPE lng PHASE OF AZ (degl vs xo ~ , ~THEOY s '.159} LENGTH 4.55 O_MAL I ZE~, DEPTH,.87 TO r.32 OPENING.23 Fig. 5, Comprison of theory nd experiment for EDM notch NBS 3-A, 2c = 4.55, =.87, ~u =.23 normlized to Y ; =.51". Semi-ellipticl shpe. () Mgnitude. (b) Phse, 389

8 MAGNITUDE OF llz (n) vs ~o , ALUM INUM PHASE OF llz (deg) vs x ~--~--~-. 7 ALUMINUM 8'.21} LE GTH ' 5.34 NOMALIZED DEPTH ' I I 3 TO r '.32" OPENING ' 4 L--_~ ~ L-----~ L-_J ~o lr Fig. 6. Comprison of theory nd experiment for Stnford ftigue crck (Otto Buck, ockwell, 1979), 2c = 5.34, = 1.13, ~u =.; o =.64". () Mgnitude. (b) Phse. 39

9 1 LENGTH INVESION COELATION MATIN STANFOD MAIETTA' FATIGUE: 15 MMIOC 13 Sl 8 16 MMIOB MM9C MM9B w 19 MM4C MM4B : I NS83A w 21 MM2C 2 NS838 > 3 N583C ~ NSB3 ~4 I ,2 I~ 15 <.> (\J NSB4A 6 SB48 7 S84C 8 NS84 9 NSB5A -2 1 SB5B II esse 12 N8S clf (ACTUAL) Fig. 7. Length inversion correltion plot. McFETIDGE CHAT (2clr = 5. SEMI -ELLIPTICAL) ~~~= o~.75o~o~-v- --~o-., ~s=~l , o.4 N <I ::;.3 w E o.2 z l:) <l ~.1 ( 'r ~ r6 rs r4 / "!J ALUMINUM 66 1 s =.51 SEM! ELLIPTICAL NB53A N8538 o NBS3C NBS3 PHASE OF 6Z (deg) Fig. 8. Depth nd opening inversion correltion plot for semi-ellipticl flws (2c is nominlly 5, normlized to r ). 391

10 McFETIDGE CHAT (2c/r 5. ECTANGULA).5 So.4 N <l lj...3 w Cl z.2 ;::) ~ (!) <t ~.1 ~u.5 ~ / r 6 8 ALUMINUM '.51 ECTANGULA o SS4A NBS4B t>. BS4C o BS5B NBS5C NBS5 6 PHASE Fig. 9. Depth nd opening inversion correltion plot for rectngulr flws (2c is nominlly 5, no rmlized to r ). 14 Flw Tble 2. Flw depth nd opening inversions Shpe Prmeter Actul Inverted % Error NBS 3A NBS 3B NBS 3C NBS 3D NBS 4A NBS 4B NBS 4C NBS 4D NBS 5A NBS 5B Stnford 1 Stnford 2 SE SE SE SE F F.277".23" % -23% -16% -19% +46% -46% +42% -42% -.7% -31% -18% -11% -14% -1% +34% -41% +7.3% -52% % -38% % % - 392

11 SUMMAY Previously reported flw profile nd inversion clcultions for rectngulr flws [1) hve been extended to semi-ellipticl flws. The flw profile mgnitude curves show significnt differences, mking it possible to distinguish between the two shpes. For length inversion there is little difference between the slope intercept curves for the two shpes. (This is physiclly resonble becuse the exterior slope of the profile is controlled by the interction of the probe field with the end of the flw, regrdless of its internl shpe.) A consequence of this is tht the length of semi-ellipticl flw cn be obtined from the rectngulr inversion curves. For depth nd opening inversion, however, it is necessry to use McFetridge chrt for the correct shpe (rectngulr or semi-ellipticl). EFEENCES 1. B. A. Auld, G. McFetridge, M. izit, nd S. Jefferies, eview of Progress in Quntittive Nondestructive Evlution 4, D.. Thompson nd D. E. Chimenti, Eds., (Plenum Press, New York, )984), p J. Moulder nd J. C. Gerlitz; B. A. Auld, M. izit, s. Jefferies, nd G. McFetridge, eview of Progress in Quntittive Nondestructive Evlution 4, D.. Thompson nd D. E. Chimenti, Eds., (Plenum Press, New York, 1984), p J. Moulder nd J. C. Gerlitz; B. A. Auld nd s. Jefferies, "Semiellipticl surfce flw EC interction nd inversion: experiment," these proceedings. 4. B. A. Auld, F. Muennemnn, nd M. izit, Nondestructive Testing, Vol. 7,. Shrpe, Ed. (Acdemic Press, London, 1984), p Collins, privte communiction, July E. Smith, "Appliction of uniform field eddy current technique to 3-D EDM notches nd ftigue crcks," these proceedings. 7. W. D. ummel, B. K. Christner, nd S. J. Mullen, "The influence of clibrtion nd cceptnce criteri levels on crck detection nd discrimintion by eddy current techniques," these proceedings. 393