ELASTIC WAVE SCATTERING FROM ROUGH SURFACES AND CRACKS

Transcription

1 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES AND CRACKS S. Ayter and B.A. Auld Edward L. Ginzton Laboratory Stanford Univerity Stanford, CA INTRODUCTION The cattering from rough urface and crack in the high frequency regime i analyzed via a cattering formula baed on the reciprocity.relation. Scattering from the mooth crack i invetigated firt.to rederive the "flah point" concept by Fourier tranform method. Baed on thi analyi, an inverion procedure i propoed for obtaining the characteritic function of the crack, which, for the cae of rough crack, give information about the roughne a well a the dimenion and hape of the crack. The theory i applicable to both 2-D and 3-D cattering problem, a well a urface wave cattering from urface breaking crack. Elatodynamic ray theory predict that cattering from crack can be decribed in term of dicrete ource point on the contour of the crack. 1 Thee point are generally called the "flah point", and their poition depend on the tranmitter and receiver location a well a the crack hape. For intance, for 2-D cattering problem, (or for deep urface breaking crack under urface wave excitation), the two edge of the crack act a the flah point. The theoretical derivation for bulk crack involve application of the Kirchhoff approximation for the total field in the repreentation theorem, converting the urface integral into a line integral via "integration by part," and alp lying the method of tationary phae to obtain the flah point. In thi tudy we ue a cattering formula baed on the reciprocity relation,2,3 with the Kirchhoff approximation and dedu~e the flah point by Fourier tranform method. For the cae of mooth crack, both method yield the ame reult. In the firt method, becaue of the nature of the integrand, 611

2 612 S. A YTER AND B. A. AULD the econd term in the "integration by part" tep vanihe, and the firt term, after ome manipulation give the flah point. However, for rough crack, the econd term no longer vanihe, and the effect of the roughne appear mainly there. Owing to the complicated nature of thi econd term, one cannot readily conclude about the general nature of the rough urface cattering. In our cae, the method for the mooth crack can be extended to the rough crack cattering without much difficulty, and the effect can be analyzed via imple Fourier tranform relation. SCATTERING FORMULA AND ITS APPLICATION TO SCATTERING FROM CRACKS A cattering formula due to Auld,2 and Kin0 3 i ued to calculate the cattering coefficient. Thi formula ha variou advantage over the repreentation theorem. Firt of all, it give the tranducer-to-tranducer cattering coefficient in term of the tranducer terminal voltage, which are the actual meaured quantitie in an experiment. Secondly, intead of the Green' function, it ue the actual tranducer' far field which are eentially plane wave at the flaw. Thi i epecially important when the Green' function i either unknown or very difficult to calculate. Thirdly, due to it relative implicity, the mathematical tep are more tractable, and thi allow more phyical inight into the problem. Finally, the formulation i not retricted by the tranducer type, or the excitation method o long a the field pattern due to the excitation can be etimated. The formula expree the change in the cattering coefficient, of 21 ' due to the preence of the crack a, 1 4(P P )1/2 1 2 f. F (1) where ubcript 1 and 2 indicate the particle velocity and tre field ditribution excited by tranducer 1 and 2 repectively, which are driven by input power PI and P2 ' and the prime denote the preence of the crack. The crack urface i deignated by S,with the normal pointing toward the flaw being uf (ee Fig. 1). One hould note that the integral in Eq. (1) i a cloed urface integral, including both face of the crack. =, A For open crack..!. Tl' n = 0 on the front and back urface. For cloed crack, T 1 n i continuou acro the crack, Then

3 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 613 SURFACE SF ENCLOSING VOLUME VF ~--+- SCATTERER pl.c' SOLUTION I (SCATTERER PRESENT) SOLUTION 2 (SCATTERER ABSENT) Fig. 1. Definition of the term ued in the cattering formula. Eq. (1) can be rewritten a 1 / or 21 f:,v' T. n ds (2) 4(P P )1/ front face where f:,v{ i the jump in the particle velocity ditribution acro the crack. Preliminary Conideration and Aumption The general cattering geometry i outlined in Fig. 2. It i aumed that the crack lie in the x-y plane and i in the far field of the tranducer. Since the crack width of interet, 2c, i in the order of everal wavelength, in the vicinity of the crack, the unperturbed field can be approximated by plane wave, but the ampli-

4 614 S. A YTER AND B. A. AULD z +8 -c CRACKJ c x TRANSDUCER 2 kr TRANSDUCER I Fig. 2. General cattering geometry. In 2-D kt and kr are in the x-z plane, but in e-d they are not necearily o. tude mut be corrected for the diffraction effect, The unrer- + turbed field are therefore characterized by A(w) ~zp[j(wt-ko k r)] where a = T,or R correponding to tranmitter and receiver a repectively, ko = w/v i the wavenumber, and k i the unit vector in the propagation direction. The A(w) fgctor ummarize the effect of diffraction, for 2-D A(w) "" { ~ w for 3-D (3) In all cae, by the term cattering data, we mean the variation of the cattering coefficient in frequency, with the tranmitter and receiver poition fixed, It i alo aumed that in the 2-D cae, the propagation vector ~ and kr are in the x-z plane, but in the 3-D cae they are not necearily o, A further aumption i that the crack location i known, i.e., the tranducer are arranged 50 that the crack i near the center of their main lobe.

5 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 615 For the perturbed field ditribution, we ue the Kirchhoff approximation, namely, taking for the field on the front urface thoe that would be preent if the crack were a pecular reflector, and on the back face auming the field to be zero. Then, for fixed tranducer poition, Eq. (2) reduce to of 21 "" I front face We next introduce y(x,y), the characteritic function of the crack, (4) (x,y) on crack face; y(x,y) = (5) Thi reduce Eq. (4) to l oofoo elewhere -j k it :i: or 21 "" A 2 y(x,y) e 0 dx dy (6) _00-00 ~ A 1\ -+where k = (k T + k R ), and the direction of k will be referred to a+th~ "can direction~. Note that for backcattering (k T = k R ), and k i parallel to k T SCATTERING FROM SMOOTH CRACKS 2-D Cae For the two-dimenional cae, or deep urface breaking crack, the characteritic function i a function only of x Uing Eq. (3), the counterpart of Eq. (6) for the 2-D cae i found to be I csr 21 (W) "" W y(x) ~ A From k w/v and the definition o -jk k x x e 0 dx (7) (8)

6 616 S. AYTER AND B. A. AULD where V i the phae velocity of the wave of illumination, one readily ee that or 21 (00) - oo '{Y(T)} (9) where VT ) Y(T) Y ( ~ A k x and r{ } i the Fourier tranform operation from the T into the 00 domain. Uing the time differentiation property Fourier tranformation,4 Eq. (9) can be expreed a, (10) domain of or 21 (00) - : f ;T Y(T) I (11) Equation (11) tate that (d/dt)y(t) and or 21 (oo) are Fourier tranform pair. Therefore, in the time domain with pule operation two pule of oppoite phae are oberved at the receiving tranducer, with a time difference (ee Fig. 3) 2c k -7- " x/v (12) correponding to the time delay between the ray that hit the edge of the crack and return to the receiver. Therefore, the ignal appear a if it i coming from the edge (flah point). Figure 3 chematically illutrate the argument. 3-D Cae For the three-dimenional cae, Eq. (6) can be interpreted in term of the two-dimenional Fourier tranform of the characteritic function y(x,y) from the x-y domain into the k,k domain, where x y k x k y!e. (k ) V... x!e. (k) " V y and Since the tranducer poitio~ are fixed, the compone~t o~ the projection of the can vector k on the crack plane, k x and t. yare fixed quantitie. Therefore, a the frequgncy change, tre cattering coefficient can the Fourier domain along the line (13) k y k Y k k. x x (14)

7 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 617 y(x) -c c x T' -2c Fig. 3. Scattering mechanim for the 2-D cae. The ame argument applie to the Rayleigh wave cattering from deep urface breaking crack. Rotating the coordinate in the x-y plane via the coordinate tranformation T [(k x)x + (k y)y]/v + (15) ~ A u [-(k y) + (k x)y]/v one can rewrite Eq. (6) a,

8 618 S. AYTER AND B. A. AULD (16) where V V -+ A -+ A..r(" u) Y ( --(k. x, - k. yu),-- Ik 12 Ik 12 Defining y(,) a the integral of..r("u) along the direction normal to the can direction, i.e., (17) f.r.(" u)du -co (18) co one can interpret Eq. (16) a (19) The procedure i outlined chematically in Fig. 4. The limit of y(,) are defined at the point where the tangent line to the contour of y("u) i in the u direction, and the lope of y(,) i dicontinuou at thoe point. A a reult, the econd derivative how a correponding impulive behavior. A in the 2-D cae, thee are the flah point time domain becaue of Eq. (19). INVERSION SCHEME The flah point are baically the reult of the diffraction mechanim. Therefore if one take the effect of diffraction from the cattering data, one can obtain directly the characteritic function of the crack. In mathematical form, 2-D cae 3-D cae (20) Thi inverion procedure give the crack dimenion, a well a the information about the crack hape for the 3-D cae. For intance, with a ingle meaurement it i poible to determine both dimenion of the crack, provided that the crack orientation i known (ee Fig. 4).

9 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 619 RECEIVER AT T CD Y(TI{r(T.VldV -CD y'cti T T T Fig. 4. Scattering mechanim for the 3-D cae. Tne weakne of the procedure lie in the fact that the Fourier inverion mechanim require the data over the whole w-domain. When the data i bandlimited, harp variation in the actual characteritic function will caue ocillation in the inverted data (Gibb phenomenon).4 However, thi problem i equipment related, and with wider band tranducer it effect can be reduced. The inherent problem of the inverion method i that it i valid only in the high frequency regime. Therefore, even if the tranducer can excite low frequencie efficiently, the data at low frequencie

10 620 S. AYTER AND B. A. AULD may not agree with the theoretical expectation, hence one may have to reject that data, or analyze it via low frequency theorie. ROUGH CRACKS Relevance to Fracture Mechanic A a crack grow under cyclic loading, it exhibit different roughne characteritic in different growth regime. A typical plot of growth rate v effective tre intenity i hown in Fig. 5. In Regime I, the crack growth i via non-continuum mechanim, and mainly affected by the microtructure. For thi reaon, face roughening i the dominant character. In Regime II, the microtructure ha little influence on crack growth and the crack propagate in continuum. In thi regime, the crack tip i continuouly blunted and rehaped. Therefore, although the crack face are moother, the crack tip grow irregularly.5 REGIME I REGIME II REGIME m 10-2r ~,_ ,., U >- u " Ē E z 't:> "- 0." THRESHOLD t.ko 10-~~------~~ ~ LOG t.k Fig. 5. Crack growth rate, da/dn, a a function of the tre intenity range, AK.

11 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 621 The tre intenity factor of a rough crack differ from that for a mooth crack due to tip irregularitie, non-flatne of the urface, and branching of the crack. 6 For thi reaon, the dimenion of a crack, without any knowledge of it roughne characteritic, are not ufficient to predict failure. Another apect of the rough crack problem i crack cloure, due to either the urface roughne or the oxide debri formed inide the crack during growth. When the crack i cloed at certain pot, it will affect the growth rate of the crack becaue of the reduction of the effective tre intenity range (ee Fig. 6). Hence, being able to determine the cloure point i a important a the roughne information itelf. Modeling of Rough Crack To model the roughne, we firt ued the perturbation analyi firt ued by Brekhovkikh to calculate the Rayleigh wave attenuation due to urface roughne. 7 Thi analyi involve repreenting the boundary condition on a tre-free rough urface a an equivalent boundary condition on a mooth urface. With thi method, it ha K(V mm I II III NO OXIDE-INDUCED ROUGHNESS-INDUCED CLOSURE CLOSURE CLOSURE 6Keff=Kmax-Kmin 6Keff=Kmax-KCI 6Keff=Kmax-KCI K max N ro K Fig. 6. Crack cloure and it effect on the tre intenity range.

12 622 S. A YTER AND B. A. AULD been found that8 the contribution of the rough urface alone, (or) will be proportional to r (or) - -jk (k T + k R ) ~(x,y) r 0 z where ~(x,y) i the roughne function. (21) The econd method conidered i an extenion of Kirchhoff boundary condition to the rough urface, decribed in the literature a "the tangent plane approximation".9 In thi cae one take the total field a the um of incident wave plu the pecularly reflected wave from the local tangent plane to the rough urface. When the tranducer angle are around normal incidence with repect to the rough urface, mode coupling i negligible provided that the roughne amplitude i le than wavelength. The reult of calar diffraction theory will then apply and one find, () -jk (kt+kr) ~(x,y) or - or 0 e 0 z ~ or(o) [1 - jko (k T + k R ) z~(x,y) +... ] (22) in agreement with the reult of perturbation analyi. The perturbation analyi i valid for roughne amplitude 1~(x,Y)1 «1 and for it lope max a = x,y The tangent plane method i alo valid for mall variation in lope, but can tolerate larger roughne amplitude. Another approach being conidered i to model the roughne a a uperpoition of mall, known geometry, catterer with a random ditribution. In thi cae, each catterer i mall compared to the wavelength, hence the field diturbance around each catterer can be viualized via quai-tatic field theory. Thi kind of model permit fater variation for the roughne function ince the diturbance due to each catterer i tightly localized, and the adjacent catterer do not interact to firt order. With thi kind of model, the reult i alo the ame a that given by Eq. (22). With the above model for an open crack, the characteritic function for a rough crack can be een to be of the form y(x,y) -jk k.z~(x,y) o S e -+- " ~ y(x,y) [1 - jk k -zt(x,y) +... ] o

13 ELASTIC WAVE SCATTERING FROM ROUGH SURFACES/CRACKS 623 If the crack i cloed at certain pot, the characteritic function will how a jump at the cloure point. For intance, if the face contact rigidly, i.e., iv{ = 0, then Y r = 0 at thoe point. INVERSION OF ROUGH CRACKS Unlike the mooth crack cae, the characteritic function of rough crack i frequency dependent, and in general the Fourier -+ h inverion algorithm doe not apply. However when k z r;(x,y) «1, the characteritic function can be approximated a, Y (x,y,w) r -+ h ~ y(x,y) - jwy(x,y) k z r;(x,y)/v The inverion algorithm then give the um of the mooth crack characteritic function, and the roughne contribution, which i proportional to the derivative of the roughne function, i.e., for the 2-D cae, and it + v h Z for the 3-D cae. Hence, for the 3-D cae one meaure the roughne function in the can direction. In the flah point repreentation of cattering from a rough crack, the econd (2-D cae) and third (3-D cae) derivative of the roughne function are uperimpoed on the regular flah point. Hence, any harp variation in the roughne function will appear a econdary flah point in the cattering data. REFERENCES 1. J.D. Achenbach, K. Wiwanathan, and A. Norri, "An Inverion Integral for Crack-Scattering Data," Wave Motion, 1:299 (1979) 2. B.A. Auld" "General Electromechanical Reciprocity Relation Applied to the Calculation of Elatic Wave Scattering Coefficient," Wave ::1otion, 1:3 (1979). 3. G.S. Kino, "The Application of Reciprocity Theory to the Scattering of Acoutic Wave by Flaw," J. Appl. Phy. 49:3190 (1978).

14 624 S. AYTER AND B. A. AULD 4. A. Papouli, "The Fourier Integral and it Application," McGraw Hill, New York (1962). 5. R.O. Ritchie, S. Sureh, D.M. Park, E.K. Tchegg, and S.E. Stanzl, Reearch on Fatigue Threhold, in: "Fatigue Threhold," Proceeding of the InternationalSympoium, Stockholm, June 1981, J. BMcklund, A.F. Blom, and C.J. Beever, ed., EMAS Publ. Ltd., Warley, U.K. (1981). 6. M.L. Kachanov, "A Microcrack Model of Rock Inelaticity, Part II: Propagation of Microcrack," Mechanic of Material, 1:29 (1982) 7. L.M. Brekhovkikh, "Propagation of Surface Rayleigh Wave along the Uneven Boundary of an Elatic Body," Sov. Phy.-Acou. 5:288 (1959) 8. S. Ayter and B.A. Auld, "Reonance and Crack Roughne Effect in Surface Breaking Crack," Proceeding of DARPA/AFWAL Review of Progre in Quantitative NDE, La Jolla, July 1980, (1981). 9. P. Beckman and A. Spizzichino, "The Scattering of Electromagnetic Wave from Rough Surface," Pergamon Pre (1963). ACKNOWLEDGEMENT Thi work wa ponored by the Center for Advanced Nondetructive Evaluation, operated by the Ame Laboratory, USDOE, for the Air Force Wright Aeronautical Laboratorie/Material Laboratory and the Defene Advanced Reearch Project Agency under Contract No. W ENG-82 with Iowa State Univerity.