A PARAMETRIC FINITE ELEMENT STUDY OF A GENERIC CRACK SIZING

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1 A PARAMTRIC FIIT LMT STUDY OF A GRIC CRACK SIZIG PROBLM M. G. Wismer, P.L. Levin, and R. Ludwig Department f lectrical ngineering Wrcester Plytechnic Institute Wrcester, Massachusetts ITRODUCTIO lbis paper discusses the applicatin f a numerical analysis technique t study eddy-cu1tent transducers. Prbe design and perating parameters are detennined by the cnstraints f excitatin current and the material being tested. Because the depth f penetratin f the induced eddy-cu1tent is related t the excitatin frequency as well as the permeability and cnductivity f the specimen, the perating frequency may be selected fr a certain depth f reslutin. Hwever, subsurface field depth is als a functin f relative prbe size and lift-ff. Althugh there are n knwn clsed frm slutins relating prbe size and lift-ff t penetratin, apprximate analytical relatinships have been develped by Mttl[3]. The prbe design prcess can be facilitated by a numerical mdel. The impedance signature f a prbe that is scanned ver a surface flaw will depend upn the relative size f the sensr, the drive frequency, lift-ff psitin and material parameters f the specimen. umerical mdels can help determine signal sensitivity t each parameter. Fr DT purpses it is desirable t ptimize the apprpriate values f cil characteristics that wuld increase the chances f flaw detectin within a given material. The results f this paper were generated using the finite element methd develped and applied t specific DT prblems by Palanisarny[l]. Because the flexibility f this apprach permits the mdeling f an arbitrary 2-D input prbe it can be extended t study the effect f misaligned r tilted cils. These cases have imprtant practical implicatins since scanning prbes are ften nt perfectly aligned with the specimen. The study f impedance changes in cils due t flaws is extensive and includes analytical, numerical and experimental techniques. Many f the analytical frmulatins cnsider the relatinship between the radius f a relatively small flaw and the radius f a circular cil suspended abve the material. Fr these cases the vlume flaw thery develped by Auld is incrprated[ 4]. Ddd et al[s] cmpared these analytic results with experimental fr abslute and differential axisymmetric prbes. umerically, axisymmetric prbes signals were predicted fr an arbitrary 2-D crack pening by L. D. Sabbagh[6]. 1bis finite element cde is capable f analyzing axisymmetric and 2-D prblems. A generic single frequency 2-D gemetry fr an abslute air-cre transducer is cnsidered as shwn in Figure 1. The analyzed specimen is assumed t be a cnducting half-space where the bundaries in the x and y directins terminate at infinity. Revie.y 0/ Prgress in Quantitalive ndestructive valuatin, Vl. lob dited by D.O. Thmpsn and D.. Chimenti. Plenum Press. ew Yrk,

2 L +J -J () $-+ +-r-+ Fig crss sectin f air cre cil ver a half-space. FORMULATIO The electrmagnetic field prblem can be frmulated in terms f the magnetic vectr ptential A where, fr 2-0 gemetries, nly the z cmpnent is nn-zer. The gverning equatin fr sinusidal field quantities takes n the frm [1] (1) where ~ 0', c are, respectively, material permeability and cnductivity and drive frequency. lz is the impressed current density supplied t the cil. quatin (1) is slved numerically by finite elements using the Galerkin weighted residual methd. The steps invlved are part f standard prcedures which include casting the equatin in a weak frm by frming an inner prduct with a plynmial weighting functin <Ili (2) where < > indicates integratin ver the regin f interest The field quantity Az is apprximated by unknwn cefficients Aj multiplying basis functins <Ilj which are chsen frm the same family as the weighting functins (3) Here the indices j dente ndes thrughut the slutin dmain n. Therefre the cntinuus representatin (2) is transfrmed.int a discrete system f equatins (4) 1938

3 where Aj is understd t be the ndal value f Az and +j is chsen t be a linear basis functin ver triangular elements. Integrating (4) by parts the fmal numerical fnnulatin includes nly first derivatives and knwn bundary cnditins n Az [ <- 1 V~ V~ - J. CO at!. t!.. > ] A-= -<J t!.. >+- 1 i _t!.. daz J.1 dl '1'1 'I'J '1'1 'I'J J Z'l'1 J.1 I d '1'1 (5) The circular abslute cil with current density Iz is mdeled in 2-D by tw sets f elements as shwn in Figure 2a. The same cil can be apprximated using ndal currents Iz at the center nde as in Figure 2b where Iz = Iz L\: and L\: equals the cil crss sectinal area. The ndal inputs are represented by delta functins, a(), and (5) becmes (6) where h is the lift-ff, r is the mean cil width and Xl is the psitin f the psitive current The impedance calculatins fr the single wire lp f 2b requires the flux thrugh the tw wires and crrespnds t the results fr a finite cil. Vltage is defined by [1] where (7) (8a) and (8b) <lli>=jz<'i>=~=ii!:ac= element area -1 I ~I. 1 1 Fig 2a.Cil mdeled with 16 elements each with current density l z I i I Xl Iz=lz!:ac!:ac = cil area, I I Fig 2b.Cil mdeled with ndal values at the cil center. 1939

4 with ex and ey representing unit vectrs. xplicitly evaluating the integral (Sa) alng discrete ndes yields with 1 (VxA)' ds =- r'ix. a~ dxdz s J X( ax (9a) (9b) where it is assumed that I is a characteristic length f the cil and xn is the psitin f the negative wire (ie. xn = xi + r). By numerically calculating these integrals the impedance per unit length is related t the ndal values f Az by (10) te that An and A I have ppsite signs s that the magnitudes f their real and imaginary parts add tgether as required t mdel an abslute cil. In cntrast, the impedance f a differential cil is prprtinal t the sum An + A I, thus detecting the difference in their magnitudes. TIlls implies that a differential cil is mre sensitive t an impedance change than an abslute cil. ddy-current prbes measure a relative change in impedance and are usually calibrated with respect t an unflawed specimen prir t scanning. The nnnalized impedance is btained by dividing the signal frm the flawed half-space by the signal frm the flawless half-space. The magnitude and phase f the impedance is recrded at every psitin and pltted versus the center pint f the prbe. Fr this analysis the discretized regin used t mdel the half-space has a length f 26 mm and a height f 3.75 mm with a simulated crack width f 1.33 mm. The ttal height f the discretized regin is 15 mm. These dimensins are chsen s that the uter bundary which is Dirichlet and set at Az = 0 will nt affect the slutin. T study the effects f prbe sensitivity the crack width remains cnstant and the cil width r, lift-ff h, penneability fmaterialll and drive frequency are varied ne at a time. TILTD COIL FORMULA no dal currents are als used t simulate the tilted cil but the delta functins are shifted frm the riginal psitins t accunt fr the tilt angle. The tilted sensr can be cmpletely defined by its center lift-ff, prbe width and angle f inclinatin ~ with respect t the material surface as shwn in Figure 3. The psitive and negative input currents have different lift-ff heights hl and h2 and are separated alng the x-axis by a distance d such that (1la) ~ =h+!. sincj> 2 d =rc~ (l1b) (lie) 1940

5 ~ hff1~ I{~~ 7.~ 13 :.. d ~ x- Fig 3. 2-D representatin f a tilted sensr with an angle f inclinatin,.. Fig 4. Tilted cil mdeled with ncxlal values. An analytic slutin fr an inclined prbe ver a balf-space is btained by superimpsing the results f tw single wires with apprpriate lift-ffs and x-axis psitin. The single wire slutin was wrked ut be Stll [2] and can be superimpsed s that fr y.::; 0 in Figure 3 the result is Az(x,y) = -I.ldl 1"" -khl Iz e 1t 0 J.1+Y e YY cs(kx) eik- "" ~ I.ldl Iz r -k e eyy cs(k[x-d]) elk 1t J J.1+Y (12) Upn making the apprpriate substitutins (lla-c) equatin (12) takes the fnn J.lJ.1 i 00 e -k( b- ~SiD' ) Ar.(x,y) = - Iz k ey'l cs(kx) elk- 1t 0 J.1+Y J.lJ.1 e -k( 100 bt rid + ) -Iz k eyy cs(k[x-rcs~]) elk (13) 1t 0 J.1+Y The cmparisn f the finite element mdel with the analytical predictins prvided by (13) is in excellent agreement fr x = O. Fr the numerically tilted prbe the input currents are again delta functins but this time their psitin, in general, will nt cincide with a nde as illustrated in Figure 4. Rewriting (6) fr the shifted currents results in 1941

6 + Iz< B(y-h 2 ) B(x-[xl+d]) 4>i > +.!.1 0:"lAz4>i dl (14) J.1 I ad The integratin f the surce tenns n the right hand side leads t ndal cunents weighted by the relative area cvered by the input nde and the ther tw ndes f the element. Thus the current is redistributed t the three surrunding ndes t simulate an input at any pint within an element. This is a cnsequence f the weighted residual fnnulatin. The impedance f the tilted cil is calculated based n (10) where the values f An and At are fund at the crdinates f the input lcatins by an interplatin f the basis functins [7]. These impedances are nnnalized by the signals f the flat cil ver a half-space. SIMULA 1D RSULTS Impedance magnitude and phase are pltted fr different relative prbe sizes in Figure 6. As the cil width increases frm half f the crack width t five times the flaw pening, the magnitude f the signal als is amplified as mre flux passes thrugh a wider cil. The space between the maxima are clser as the cil width decreases, hwever, making smaller prbes mre sensitive t the crack edges. Figure loa shws this peak separatin as a functin f nrmalized prbe size thus indicating flaw detectin capability. Figure 7 shws the effect f driving frequency n signal strength. The magnitude increases with frequency because f the smaller skin depth in the material. The higher frequency restricts the fields t the surface thus increasing the signal strength at the measurement lcatin. Penneability als affects the depth f penetratin f the s that the 1.02 gl.d2 ]1.01 ~ g I. 01 I.DO. ".. Ii:,." I - cii/ericeo.s I ~ h - cil!.c ':1:-\ -- cilzerice 2.0 I 1\ :.\ ---- cil!.crtl-. O,:.1'\ cui'ctti.2 I ~ : I ~,, : I,, I'i i --~.. "II ftl, : ~ I I I ;~,( It " I, I"" 1 l' i, ~,. ;, ~ : I \, \., \ " \ :L- O '!zf.!!"'_""'i'j'.!!!"'_"'ito.oo'!!'"""'_s!i:.!!'"""':!o"l. "!""""':!"l"!""""!i~o.oo psitin (mm) Fig S. umerical impedance signatures (magnitude) fr different relative prbe sizes ~ =ti 'j ~IO.OO e ~ c: ~ 1.00 fr /1 ~ 2.00 CII Q ~""""""""""''''''''I"""'''''''''"T''...,,,...,..,......, O width/crck width Fig 6. Prbe sensitivity due t relative prbe width. 1942

7 percent change in signal magnitude increases with J.lr as seen in Figure 8. Overall the relative impedance is inversely prprtinal t J.Ir because f the bundary cnditins enfrced at the interface. The effect f tilt angle n impedance measurements is clarified by Figure 9. Signal strength decreases as the prbe is mre inclined since fewer nrmal flux lines pass thrugh the cil. As expected the peaks in Figure 9 are different crrespnding t the tw lift-ffs hi and h2. Bth maxima are discernible up t an angle f 30 degrees. Finally Figure 10 is a quantitative indicatin f prbe sensitivity due t prbe size and tilt angle. It illustrates the effects f wider prbes and larger tilt angles in terms f flaw imaging capability. Future investigatins are required t investigate tilt angles between 10 and 20 degrees, t explain the effects f phase angle and t study issues such as crack depth and varying cnductivity > 1.02 ] 1.02 " " > freq_10khz - freq-50khz.--- freq-l00khz freq-150khz 1.01 g'1.00 " ~0.99 "- 0.9S 0> ".f Jjr J,Ar jjr _ 'f,..."""""""""'~"""""""""''''''''''m-m"'''''''''''''''''"ttttm-m""ttl psitin (mm) Fig 7. umerical impedance signatures (magnitude) fr different drive frequencies. 0.9S fn""""""""m-m""""'''''''''''"'t'"~''"'"'!'"ottt"'''''''''''''''''''''''' psitin (mm) Fig 8. umerical impedance signatures (magnitude) fr different material pcrmcabilities. COCLUSIOS This paper discusses in tw dimensins hw numerical analysis techniques can be used t ptimize design parameters f a scanning single frequency eddy-current prbe. The influence f drive frequency, cil size and material parameters n impedance signatures is clearly illustrated and can be chsen fr imprved flaw detectin. In additin prbe misalignment with respect t the surface f the specimen are investigated by calibrating the numerical mdel against an analytical slutin f a tilted cil ver a cnducting half-space. Subsequent results f a tilted scanning prbe ver a surface-breaking crack exemplify hw the tilt angle affects the signal respnse. 1943

8 _ U ;g D,~,t.OO"""""'_"'ID'".OO"""""'_lI"'.OO"""''''''''D.''''''''''''''''lI''''!.OO''''''''~ID.OO - D ~...,...,...'"'"...'"'"... -ll O. li.oo "'0 Cl> 0.11 "'0 0\ li.oo '0.00 D.73~..,...,......,...,...,... -'.00 -'0.00 -li.oo psitin (mm) Fig 9. umerical impedance signatures (magnitude) fr different cil tilt angles s= C' ~1.05 t , iii 0.90., 'U0.85 ~ c: 0> r."""'"ttttt"""'"'i'tt'i..".,...."""'...,.,"""'... x tilt angle (degrees) Fig 10. Signal strength versus tilt angle. RFRCS 1. R. Palanisamy, "Finite lement Mdeling f ddy-current ndestructive Testing Phenmena," Ph.D Dissertatin, Department f lectrical ngineering, Clrad State University, Frt Cllins, D. L. Stll, The Analysis f ddy-currents, Oxfrd, ngland, Oarendn Press. 3. Z. Mttl, "The Quantitative Relatins Between True and Standard Depth f Penetratin fr Air-cred Prbe Cils in ddy-current Testing," DI Internatinal,.2, pp 11-18, February B. A. Auld, F. Muennemann and D. K. Winslw, "ddy-current Prbe Respnse t Open and Clsed Surface Flaws," J. ndestr val, Vl. 2, pp. 1-21, C. V. Ddd, C. D. Cx and W.. Deeds, "xperimental Verificatin f ddy-current Flaw Thery," Rev. PrfU. QUantitative ndestructive val., Vl. 4A, pp L. D. Sabbagh and Harld A. Sabbagh, "A Cmputer Mdel f ddy-current Prbe-Crack Interactin," Rev. Pr~. Ouantitative ndestructive val., Vl. 9A, pp , B. Becker, G. F. Carey and 1. T. Oden, Finite lements An Intrductin, Vl. I, nglewd Cliffs, ew Jersey, Prentice-Hall Inc.,