I. Yalda-Mooshabad, F. J. Margetan, and R. B. Thompson Center for Nondestructive Evaluation Iowa State University Ames, Iowa 50011

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1 MONTE-CARLO SMULATON OF ULTRASONC GRAN NOSE. Yalda-Mshabad, F. J. Margetan, and R. B. Thmpsn Center fr Nndestructive Evaluatin wa State University Ames, wa 511 NTRODUCTON n ultrasnic inspectins fr small r subtle defects in metals, defect signals may be bscured by grain nise eches which arise frm the scattering f sund by the micrstructure f the metal. Mdels fr predicting micrstructural nise levels are cnsequently essential fr accurately assessing the reliability f the ultrasnic inspectins. Existing nise mdels, like the independent scatterer mdel (SM) [1), are capable f predicting nly average nise characteristics, such as the rt-mean-square (rms) nise level. Average nise levels, althugh useful, are nt sufficient fr assessing detectin reliability. One needs t knw the manner in which nise signals are distributed abut their average level. The expected peak nise level, fr example, effects the rate f "false calls", in which nise signals are mistaken fr eches frm critical defects. n this wrk, we present a Mnte-Carl methd fr Simulating time-dmain nise signals bserved in pulse/ech immersin inspectins f metal cmpnents. The methd predicts simulated time-dmain nise signals, and hence can be used t determine bth average and peak nise levels. We assume that the backscattered nise is dminated by the singlescattering f the incident beam by individual metal grains. The metal vlume is represented as an ensemble f spherical, single-crystal grains whse centers and rientatins are randmly chsen. Grain radii are determined by the nearest-neighbr distances and vlume cnservatin. The backscattered vltage signal frm each grain is calculated by treating the grain as an anistrpic scatterer in the hmgeneus average medium frmed by the ther grains. Backscattered signals frm all grains are summed t determine the ttal nise signal. Calculatins are then repeated fr many different grain ensembles t assess average and peak nise levels. Predictins f the Mnte-Carl mdel are cmpared t experiment, and t the predictins f SM. At a fixed time after the frntsurface ech, the distributin f nise vltages abut their mean value is fund t becme nrmally distributed as the density f grains increases. This is demnstrated by a series f calculatins fr a.-phase titanium specimens with different grain densities. NOSE CALCULATONS Grain nise refers t ultrasnic eches which arise frm scattering f waves by the micrstructure f a metal specimen. Fig. 1. shws the nrmal incidence inspectin f a metal specimen in pulse/ech mde in an immersin setting. We emply a time crdinate system in which the center f the frnt-surface ech appears at t=o. n this system, we designate a "time windw f interest" (TWO ) fr which ur nise calculatin will be Review f Prgress in Quantitative Nndestructive Evaluatin, Vl. 12 Edited by D.O. Thmpsn and D.E. Chimenti, Plenum Press, New Yrk,

2 --- Z grains Vltage 1! 2 3. time spatial regin f interest (a) frnt surface ech att=o... time windw f interest (b) Fig. 1. (a): Gemetry f ultrasnic inspectin. (b): Nise vltage respnse. valid. Assciated with TWO is a crrespnding "spatial regin f interest" (SRO) which enclses all metal grains that can prduce appreciable backscattered signals in the TWO. The minimum and maximum z-crdinates f the SRO are determined by the limits f the TWO and the duratin f the ultrasnic pulse. The lateral "envelpe" f the SRO is determined by the beam prfile, i.e., by the vlume in which the incident field strength is appreciable. The particulars f the pulse/ech inspectin which must be specified fr the grain nise simulatins are the time windw f interest, the transducer radius and fcal length, waterpath, and the material prperties f the metal (density, sundspeed, attenuatin, number f grains per cubic centimeter, and elastic cnstants fr single crystals). n additin ne inputs a frnt surface "reference" ech which serves t encde the gain settings f the pulser/receiver and the transducer efficiency. Using the ultrasnic measurement mdel f Thmpsn and Gray [2], the Furier transfrm f this time-dmain reference signal may be written as: Using the same citatin, the Furier transfrm f the vltage signal bserved in the nise measurement gemetry due t direct scattering by a single anistrpic grain lcated at psitin (x,y,z) in the slid is: (1 ) (2) n Eqs. (1) and (2), the hst metal is treated as an attenuative istrpic medium. The symbls v, k, p, n, and a dente the lngitudinal wave velcity, wavenumber (k = r/v), density, attenuatin cnstant, and transducer radius, respectively. Subscripts and 1 refer t water and metal, respectively and subscripts Rand S refer t the reference and nise gemetries which may have different waterpaths. is the transducer efficiency. R and T 1 are the reflectin and transmissin cefficients. C(r,x,y,z) is a measure f the incident ultrasnic field strength in the metal. O(r) accunts fr the effects f diffractin lsses in the reference signal. The waterpath (zor r ZOS) is measured utward frm the transducer face alng the central ray directin. Finally, A(r) is the scattering amplitude fr backscattered sund frm the grain in questin. The backscattering amplitude f a single grain illuminated by a lngitudinal plane wave is deduced using the Brn 172

3 apprximatin [3]: (3) where p = Pgrain - Phst and C 33 = (C33 )grain - (A+21-l)hst n Eq. (3), S(k) is a frequency dependent "shape factr" determined by the size and shape f the scatterer. n the present wrk, all grains will be assumed t be spherical. Fr a sphere f a radius r, the shape factr may be derived resulting in the fllwing expressin: S(k,r) = 4rrr3 [ sin(2rk) - (2rk) cs(2rk) ]1 (2rk)3 (4) n the Mnte-Carl frmulatin, the spatial regin f interest is filled with spherical single-crystal grains, with the ttal number f grains determined by the grain density (n) and the vlume f the SRO. The grain centers, (x,y,z), are chsen randmly within this vlume. The rientatin f the principal crystalline axes f each grain are als chsen randmly frm a specified distributin. The assigned radius f each grain is prprtinal t the distance t the center f the nearest neighbring grain, with the cnstant f prprtinality chsen t cnserve vlume. The ttal nise signal is taken t be the simple sum f the backscatlered nise eches frm each grain in SRO. Examples f three such eches may be seen in Fig. 1 b. These individual eches are calculated using Eqs. (1 )-(4). Since the reference signal is assumed t be knwn, Eq. (1) can be used t determine the transducer efficiency B(w). Fr cmputatinal simplicity, the Gaussian beam mdel [4] is used t calculate C(w,x,y,z). The individual nise eches are summed in the frequency dmain, and an inverse Furier transfrm is then used t btain the time-dmain ttal nise signal. T cmplete the calculatinal algrithm, the methd fr determining p and OC33 must be specified. n the present wrk we restrict attentin t equi-axed, singlephase cubic r hexagnal crystallites. The elastic cnstants f the istrpic, macrscpic specimen are taken t be the Vigt average f the elastic cnstants f the individual crystallites. These are the mean istrpic stiffnesses "under cnstancy f strain". Average elastic prperties fr equiaxial distributins f crystals f hexagnal and cubic symmetry are available [5,6]. n particular: hexagnal symmetry (5) (A+2/l)Vigt = (3C C 12 +4C 44 ) 5, cubic symmetry (6) where the Cij dente the single crystal cnstants in a principal axis crdinate system. Fr each grain in SRO, three randmly chsen Euler angles, (jf,e,<j» [7] are used t rient the principal axes f the grain with respect t the lab crdinate system which is attached t the macrscpic specimen. n the lab system, the 33 cmpnents f grains stiffness matrix can be shwn t be: (7) 1729

4 n summary, C33 in Eq. (3) is evaluated fr each grain using Eqs. (5)-(). We als assume that the macrscpic specimen has the same density as each crystallite, i.e., p=. As an example f a Mnte-Carl calculatin, cnsider an equi-axed, a-titanium specimen with hexagnal crystallites ensnified by a is-mhz tneburst pulse using a fcused transducer having a radius f.67 cm and a fcal length f 9.64 cm. Grains with a density f 1, per cubic cm are chsen t fill ut the SRO which cntains the fcal regin and ccupies a vlume f.52 cm3. Fig 2a. shws the backscattered signal frm a single grain lcated in the prtin f the SRO nearest the transducer. The ttal nise signal frm all 52 grains in the SRO is shwn in Fig. 2b. The particular cllectin f 52 grains is referred t as "ne ensemble" f grains, and crrespnds t ne transducer psitin in a scan pattern. By repeating the calculatins fr many ensembles f grains, we can estimate sme quantities f interest such as the average and the peak nise levels that wuld be bserved during an inspectin. Tw such quantities used thrughut this paper are: the rt-mean-squared average nise level seen at time t (averaged ver ensembles); and the rati f the peak nise seen at time t fr any ensemble t the rms average nise at time t. t shuld be re-emphasized that nly thse grains are included which wuld cntribute t the nise within the TWO. Nise signals are seen utside this windw, but sme grains which culd cntribute signals at thse times have nt been included. COMPARSON WTH EXPERMENT t is illuminating t cmpare the predictins f the Mnte-Carl nise mdel t experimental nise measurements. n rder t d this, the nise mdel requires the number f grains per cubic cm, n, as an input. T estimate n, ne can analyze a micrgraph shwing the grain structure f the test specimen. The prbability that a line segment f length L placed n the pht has bth ends inside f ne grain is next determined fr the test specimen. This is then cmpared t the analytical expressin fr this 1.5., time windw f interest 1.5 _ "6 "6.5,..-.5,..- g.+-_ ; OJ \." 1)1 CO ech ech.g -1. frm frm -1. frnt f back f ,.;g:...ra"ti-n "Tg:...rai-n._._...---j time (micrsecnds) (a) time windw f interest time (micrsecnds) Fig. 2. Nise signals. (a): Single grain. (b) One ensemble f grains. (b) 173

5 prbability which is predicted by the Mnte-Carl calculains as described belw. One begins by determining the size distributin functin f the spherical grains. f there are n grains per unit vlume, the prbability that a given grain has a radius between rand r+dr is fund t be: p(r) dr = 41tnr2 exp(-41tnr3/3) dr (9) Then, the prbability that a line segment f length L has bth ends inside f ne grain may be calculated, with the result: P(L) = exp(-1tnl3/6) - (1tnL3/6)1/3 r(2/3, 21tnL3/3) (1) where r dentes the incmplete gamma functin. T test this prcedure, P(L) was estimated frm a micrgraph f an equi-axed cpper specimen (cubic crystallites). By cmparing with the experimental P(L) values fr varius grain densities f the spherical grain mdel, as shwn in Fig. 3., we bserved that a density f 1, grains per cubic cm was a reasnable chice. The nise mdel als requires an effective ultrasnic attenuatin as an input. This was estimated in tw ways: the traditinal analysis f multiple back surface eches resulted in an attenuatin f ex =.1f2.5 nepers per cm fr 1 MHz<f<B MHz; and the analysis f backscattered nise as a functin f depth resulted in an attenuatin f ex =.5f1.2 nepers per cm fr 2 MHz<f<6 MHz. Bth attenuatins were used in subsequent mdel nise calculatins. The specimen was ensnified by a 5-MHz bradband pulse using a fcused transducer having a radius f.636 cm and a fcal length f 7.2 cm. Backscattered nise eches were btained at 1 transducer psitins. The measured rms average nise level is shwn in Fig. 4a., and cmpared with mdel predictins fr 1 ensembles fr each f the tw attenuatin functins. n this case, the SRO brackets the fcal zne which is centered r , :::J - n PLl frm mdel n = 1, PLl frm mdel n = 1, PLl frm micrgraph PLl frm mdel n = 1,,..j-_..,...-.,...;;::"",,;.;,;;;a;.;;;.;::.::... _... ="""'..2.4 L (cm).6. Fig. 3. The prbability distributin functin f a line segment placed n micrstructure. 1731

6 cm beneath the frnt surface. Fig. 4b. shws the rati f the peak nise t the rms nise seen in the experiment and predicted by the mdel. Measured and predicted nise characteristics are seen t,be in gd agreement. Nte that the predicted abslute nise signals are abut 7 db belw the input frnt surface reference signal. and n adjustable parameters are invlved in the nise calculatins. MCM: a=o.1f2.5 Z"' 1.5 -r-; :--,., g '.5 - E. +--'-r-,...-; : l ' c::: E ' c::: Q) a. time (micrsecnds) :-- --., 4 : 1 +-_ r time (micrsecnds) MCM: a=o.5ft.2 : time (micrsecnds) (a) U: time (micrsecnds) (b) Experiment time (micrsecnds) :: time (micrsecnds) Fig. 4. Nise signals frm Cpper. (a): RMS nise levels. (b): Peak t rms nise levels. MODEL APPLCATONS One use f the Mnte-Carl mdel is t test ther single-scattering mdels and techniques fr nise signal analysis. The independent scatterer mdel (SM). fr example. predicts the rms average nise level frm the measurement system parameters and the micrstructural quantity n 1/21A rms which is knwn as the "figure-f-merit" (FOM) [1.]. n the fllwing example. we predict backscattered nise frm an a-phase titanium specimen ensnified with a 15-MHz tneburst frm a fcused transducer (a=.67 cm and F=9.65 cm). The rms nise levels near the fcal zne are predicted fr seven different grain densities fr bth the Mnte-Carl and the independent scatterer mdels as shwn in Fig. 5. The Mnte-Carl results emply 5 ensembles fr each density. Althugh. the average nise levels are seen in Fig. 5. t increase with increasing grain density. the rms nise level is expected t drp with further increase in grain density at sme pint. The Mnte-Carl nise mdel may als be used t determine hw nise vltages are distributed abut their mean value. We have determined the prbability distributin f nise vltages fr the afrementined a-titanium inspectin. This was dne by cnstructing a histgram f all nise vltages seen fr all ensembles in a shrt-duratin time windw. Results fr a lw and a high density f grains are shwn in Figs. 6a. and 6b. 1732

7 T- 9 7 ci 6 ' 5 c -:::J c 4 3 <: en 2 E : a n n time (micrsecnds) time (micrsecnds) Fig. 5. RMS nise levels (a): Mnte-Carl Mdel. (b): ndependent Scatterer Mdel. b respectively. The Gaussian prbability functin is als pltted using the bserved mean and the variance f the vltage distributins in each case. Fr small-grained specimens, i.e., n = 1, grains per cubic cm f Fig. 6b., nise vltages are fund t be distributed in a Gaussian manner. Hwever, the distributin can be very nn-gaussian fr large-grained specimens, i.e., fr n = 1 grains per cubic cm f Fig. 6a. The rati f the peak nise t rms nise averaged ver a time windw f interest can be used t track the apprach t Gaussian behavir. Fig. 7 shws the Mnte-Carl mdel predictins f this average quantity fr different grain densities (1 ensembles f grains at each density) fr the tneburst equiaxial a-titanium inspectin. The expected value f this rati fr a Gaussian distributin is shwn by the slid line in Fig. 7. We bserve that the rati f peak nise t rms nise appraches the expected Gaussian level frm abve as the grain density increases. ii:" '" is :. '" e a. Ql g ( a.1 "... MCM :: - Gaussian ",,.,,..1 Nise Vltage (Vlts) a.:- <n is. 15. '" e Cl. Ql, g g b MCM -Gaussian..3 Nise Vltage (VHs) Fig. 6. Vltage distributins. (a): n 1 grain/cm3. (b): n = 1, grains/cm3 1733

8 f... E MCM calculatins 1 ensembles /' equiaxial a-titanium 15 MHz tneburst / Expected value fr Gaussian Distributin T"" : 6 g M T"" M n, grains/cm 3 T"" M Fig. 7. Average peal< nise t rms nise. T"" The Mnte-Carl nise mdel can als be used t prvide wavefrms fr defect signals in the presence f nise. Fr example, simulated hard-alpha inclusins can be added t the cluster f grains. The predicted wavefrms can then be used t test signal prcessing techniques fr defect detectin. The mdel can als be extended t include the effects f texture and grain elngatin and may be integrated int a predictive detectability mdel fr flaws in metals. ACKNOWLEDGMENT This wrk was spnsred by the Center fr Advanced Nndestructive Evaluatin, perated by the Ames Labratry, USDOE, fr the Air Frce Wright Labratry/Materials Directrate under Cntract N. W-745-ENG-2 with wa State University. REFERENCES 1. F. J. Margetan, T. A. Gray, and R. B. Thmpsn, Review f Prgress in ONDE. 1B. D. O. Thmpsn and D. E. Chimenti, Eds., (Plenum Press, New Yrk, 1991), p R. B. Thmpsn and T. A. Gray, J. Acust. Sc. Am. 74(4): (193). 3. J. E. Gubernatis, E. Dmany, J. A. Krumhansel, and M. Huberman, Labratry f Atmic and Slid State Physics and Material Science Center. (Crnell University, thaca, New Yrk, 1976), Reprt # R. B. Thmpsn and E. F. Lpes, J f Nndestr. Eval. 4(2): (194). 5. Yan Li and R. B. Thmpsn, J. ADD. Phys. 67(5): (199). 6. R. B. Thmpsn, J. F. Smith, S. S. Lee, and G. C. Jhnsn, Metal. Trans. 2A: (199). 7. B. A. Auld, (Wiley, New Yrk, 1973), pp F. J. Margetan and R. B. Thmpsn, Review f Prgress in ONDE. 11 B. D. O. Thmpsn and D. E. Chimenti, Eds., (Plenum Press, New Yrk, 1992), p