Various physical and geometrical parameters can

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1 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 5 Interaction of a Monochromatic Ultraonic Beam with a Finite Length Defect at the Interface Between Two Aniotropic Layer: Kirchhoff Approximation and Fourier Repreentation Bruno Vacoin, Catherine Potel, Philippe Gatignol, and Jean-Françoi de Belleval, Member, I Abtract Thi paper preent a fat computation method to imulate the interaction between a bounded acoutic beam and a -layered aniotropic tructure with a finite defect on the internal interface. The method ue the claical Fourier decompoition of the field into plane wave, and the Kirchhoff approximation i introduced to calculate the diffuion by the defect. The validity of the approximation i etimated by comparion with the Keller Geometrical Theory of Diffraction and with reult obtained by boundary element method. The quickne of the method allow teting everal geometrical configuration (varying ident angle, thickne of the layer or the phyical nature of the defect). Thee tudie may be ued to foreee what experimental configuration would be adequate to have a chance to detect the defect. I. Introduction: The Problem Variou phyical and geometrical parameter can influence the ability to detect a defect in immered aniotropic tructure, uing ultraonic nondetructive teting (NDT). Numerical imulation now play an important role [], [] and are widely ued to conceive method and demontrate their performance. Thi paper aim to develop a rapid imulation method capable of performing parametric tudie to determine the good teting configuration (frequency, ident angle), when complex tructure (uch a compoite tructure luding a delamination defect or a weak bonding) are involved. The final aim i to integrate thi imulation method in an indutrial platform []. In thi paper, the teting etup i aumed to be contituted both of a fixed oblique emitting tranducer that can generate guided wave (uch a Lamb wave) in the tructure and of a receiver that can be moved parallel to the tructure to meaure the preure Manucript received Augut, 8; accepted June 6, 9. B. Vacoin, P. Gatignol, and J.-F. de Belleval are with Univerité de Technologie de Compiègne Laboratoire Roberval, UMR CNRS 65,Compiègne, France. B. Vacoin i alo with Univerité de Picardie Jule Verne, IUT de l Aine-Qualité, Logitique Indutrielle et Organiation, Soion-Cuffie, France. C. Potel i with Univerité du Maine Laboratoire d Acoutique de l Univerité du Maine (LAUM, UMR CNRS 66), Le Man, France ( catherine.potel@univ-leman.fr). C. Potel i alo with Fédération Acoutique du Nord-Ouet (FANO, FR CNRS ), France. Digital Object Identifier.9/TUFFC.9.7 amplitude of the re-emitted field in the external fluid; the cae of a notable dicrepancy between the field re-emitted by a healthy tructure and that by a tructure luding a defect, are particularly of interet. Therefore, a parametrical tudy i carried out to examine the influence of the ident angle, the azimuthal orientation of the ident acoutic beam, the frequency, the location, and nature of the defect. Thu, the aim of the paper i, firt, to preent a fat imulation method uing the Kirchhoff approximation to tudy the interaction of an acoutic bounded beam with an aniotropic compoite tructure luding a finite-ized delamination defect, under high-frequency approximation, no far-field approximation being made. The econd aim of the paper i then to how how thi imulation technique i uitable to work out the circumtance that are favorable to a good detection of a defect. The modeling of the propagation in aniotropic multilayered plane tructure i now available and well known [] [6], even if many problem till exit when tudying the propagation in healthy tructure, for example, complex hape and rein tranfer molding (RTM). Taking into account an infinite-ized defect between layer (bonding condition between parallel interface) i alo not a major problem [7]; the quality of the bonding can be then appreciated through the tudy of the reflection and tranmiion coefficient a a function of tiffne coefficient. The tructure involved in the literature for thi kind of modeling are either iotropic or aniotropic multilayered tructure. On the other hand, the modeling of a finite-ized defect i more complex (a ueful review of the different method ued for detecting defect, notably in compoite media, can be found in []) and often need the ue of the boundary element method (BM). It need at leat approximation, uch a the Kirchhoff approximation ued by Spie [8] to model, in the far field, the interaction of elatic plane wave with an infinite aniotropic medium in which the defect conit of traction-free phere embedded in the medium; the -D cattering i tudied uing, like many author, Green triadic function. The method ha been validated for high frequencie and weak ident angle. In the ame ituation, for variou defect, Huang et al. [9] and Schmerr [] ue the Kirchhoff approximation for plane wave and in the far field, thu validating Kirch- 885 /$5. 9 I

2 5 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 Fig.. Geometry of the problem. hoff approximation for thi application. The comparion of the Kirchhoff approximation with the BM ha been performed by Foote and Franci [] in the far field, in the cae of acoutic back-cattering from fih. The defect (the fih wimbladder) i modeled by a void, the urface of which i repreented by a meh derived from meaurement of microtomed ection, and the model are independently validated. Notably, the accuracy of the reult from the Kirchhoff approximation for high frequencie i preerved, in comparion with the BM. Another method, the ray method, can alo be ued (the pencil method [] ue the calculation of energy in ray tube): it i a high-frequency approximation of Fourier integral. Croce et al. [] ue a method that eem cloe to the method decribed in the preent paper, in the meaning that the ray technique diregard (a in the Kirchhoff approximation) a part of the cattering effect on the edge of a defect; however, thi Croce method i limited by the multiple reflection when there are too many interface. Another numerical method, the ditributed point ource method (DPSM), ha been taken up (with 67 ource and matrice to invert) and compared with experimental reult for multilayered tructure by Banerjee and Kundu [4]. More preciely, luion or cavity-type defect in a ingle aniotropic layer have been detected uing Lamb wave generated by an ultraonic beam. The detection of microcrack initiation and evolution in a fatigue ample i alo uccefully experimentally performed uing Lamb wave by Rokhlin et al. [5]. The author alo decribe the propagation of acoutic wave in aniotropic multilayered tructure. The cae of the propagation of Lamb wave in vicoelatic material i decribed by Hoten et al. [6]: reult obtained via a finite element method (FM) and a emi-analytic method are compared: the reult are in good agreement, notably for the detection of a notch in a ingle layer. All thee paper bring ignificant contribution to the problem, but, to the author knowledge, none of them implement the Kirchhoff approximation in the area of multilayered aniotropic tructure luding finite-ized interface defect, for bounded beam, with no far-field hypothei. Thu, the aim of thi paper i to tudy, uing the Kirchhoff approximation, the interaction of a monochromatic TABL I. Nomenclature of Subcript and Supercript. Notation Meaning Supercript (e) xact olution (i) Infinite defect on interface II (h) Healthy interface II (k) Kirchhoff approximation () Scattered field under the Kirchhoff approximation α Medium, α =,,, Subcript a Wave propagating toward x > b Wave propagating toward x < i Component on x -axi Incident field (in fluid medium ) ref Reflected field (in fluid medium ) tr Tranmitted field (in fluid medium ) ( For example: u ) b = cattered field propagating toward x < in h medium ; u ( ) ( h a = component on x axi of the field u ) a, i.e., the field propagating toward x > in medium for a tructure with a healthy urface II. ultraonic bounded beam with a -layer aniotropic tructure luding a finite length defect and immered in a fluid, and to obtain the reflected and tranmitted field (Fig. ). The nomenclature ued in thi paper i defined in Table I. Fluid media and above and below the tructure are emi-infinite (thee fluid media are identical but have different number for convenience). The interface plane i denoted (Ox x ), the x -axi being perpendicular to the interface (denoted I, II, and III) and the acoutic axi of the emitting tranducer (diameter a, frequency f = ω/(π)) make an angle θ with the x -axi. The tructure i made up of aniotropic layer and (thicknee denoted h and h ) perfectly bonded all along their common interface II, except on a L-length delamination-type defect. The geometry of the problem i a -D one, but the -D effect caued by the aniotropy are taken into account (in particular, the particle diplacement and tre vector may have component). II. The Method of Solution, Uing the Kirchhoff Approximation We propoe here to olve the problem of the interaction between an ident acoutic beam and a bilayered olid tructure with a finite defect by the Kirchhoff approximation. In the cope of thi tudy, the Kirchhoff approximation need to aume that the internal interface between the layer i homogeneou outide the defect and that the defect i a finite part of thi interface with another homogeneou behavior. By a homogeneou behavior, we mean that the phyical condition to be atified on both ide of a urface between media (the healthy interface or the finite defect) do not depend on the coordinate() along it. In practice, it will be aumed that the internal interface fulfill a perfect adheion between the olid, wherea the finite defect yield a delamination or a partial bonding with contant characteritic of the glue.

3 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 5 The priple and the validity of the Kirchhoff approximation are firt preented in Section II-A, independently of the method employed to calculate the different acoutic field. Then ome reminder are given in Section II-B on practical tool (decompoition of a beam into plane wave which lead to Fourier integral, propagation of plane wave in aniotropic media, boundary condition for plane wave). Finally, thee tool are ued in Section II-C to implement the Kirchhoff approximation uing Fourier integral. A. Priple and Validity of the Kirchhoff Approximation ) The Priple of the Method: The Kirchhoff approximation, upercript (k), may be explained on the bai of a re-emiion priple for the (unknown) exact olution of the problem, upercript (e). a) The paive re-emiion priple: Let u denote by u α the diplacement field in the medium α (α =,,,), ee Fig.. If the exact olution u α(e) of the problem of the interaction with the bilayered tructure (luding it finite defect) were known, thi exact olution could be plit into different term in each medium. A an example, in the external fluid, we ditinguih the ident field u ( e produced by the emitter and the global reflected field u ) ref. ( e) In the fluid, only one tranmitted field u tr exit. In each olid layer of the tructure, the total field may be eparated into part: one propagating toward the reaing coordinate x (the a ubcript) and the other a( e) propagating toward the decreaing x (the b index): u a a( e and u ) b, α =,. ( e Now, let u aume that the value of the field u ) b all along the interface II are known, i.e., for x = h and for any value of the coordinate x. We may conider the halfpace x h a made up by the medium of the olid, and then calculate in thi infinite region the olution of the propagation equation that take thee known value ( e of u ) b on the plane x = h and that atifie the radiation condition toward the negative value of x. Becaue thi olution i unique, it coide with the field u () e b ( x, x ) in the whole layer. ( e Once thi field u ) b ha been determined in the layer, it may be conidered a an ident field on the upper interface I, together with the true ident field u. One then obtain a reflected field in the olid, which coide with the field u () e a ( x, x ), and a tranmitted field in the fluid, that i nothing but the true reflected field ( e u ) ref. Fig.. Priple of the Kirchhoff approximation: paive re-emiion priple. The ame argument may be followed, auming that ( e) the exact value of the field u a on the interface II are known, then calculating the field u () e a ( x, x ) in the whole layer and again the interaction of thi field with the ( e third interface III, thu getting the field u ) ( e) b and u tr. ( e Actually, the exact value of the field u ) ( e) b and u a are not known on the (only piecewie homogeneou) interface II. But if ome approximate value u (app) ( x, x = h ) may be obtained for thee field on thi interface, then, following the procedure jut decribed above, one may expect to determine approximate value u (app) ( x, x ) of the variou field and epecially of the approximate value u (app) ref of the exact reflected field. b) The Kirchhoff approximation: The Kirchhoff approximation follow the proce jut decribed at the end of Section II-A--a and ummed up in Fig.. The calculu of the approximate value of the field on the interface II need to olve preliminary problem (the olution of which are given in Section II-C-). A it ha been aid at the beginning of Section II, thee problem will be introduced under the hypothee of homogeneou behavior of the interface II, on it healthy part on the one hand, and on it defective finite part on the other hand. The firt preliminary problem concern the interaction between the ident field u and the bi-layered tructure with it internal interface II uppoed to be healthy (without any defect, ee part of Fig. ). The olution of thi problem i available for any ident plane wave, hence for any ident field that may be expreed by a ummation of plane wave. The olution of thi firt preliminary problem will be denoted u (h). The econd preliminary problem i identical to the previou one, but with the homogeneou boundary condition of the healthy interface replaced here by the homogeneou condition of the defect, all along the interface; ee part of Fig.. Thu, we may peak of an infinite defect and the olution of thi problem will be denoted u (i). The Kirchhoff approximation u () k of the exact olution u () e conit in chooing the approximate value for the field on the interface II in the following way (Fig. ): the ( k field u ) ( i b will be taken equal to the olution u ) b on the ( h defect and to the olution u ) b outide for the reflection

4 54 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 Fig.. Interaction of a plane wave (ident angle θ) with a plane defect (length L). GTD = Geometrical Theory of Diffraction. ( k) problem (ee part -a of Fig. ). Similarly, the field u a ( i) will be taken equal to the olution u a on the defect and ( h) to u a outide for the tranmiion problem (ee part -b of Fig. ). Then, the calculu of the field re-emitted in the relevant half-pace (following the procedure decribed in Section II-A--a) i performed. Thi re-emiion i then followed by the interaction with the external interface I and III, and, therefore, the approximate olution u (k) of the exact field u (e) i obtained. ) Dicuion of the Validity Domain of the Kirchhoff Approximation: The validity of the Kirchhoff approximation may be appreciated by ome phyical conideration. Strictly peaking, the ident wave on each edge of the defect produce ome diffraction effect that modify the field u (h) on the healthy ide of the interface and the field u (i) on the defect ide. Thu, the Kirchhoff approximation conit of ignoring thee diffraction effect. It hould be noted that neglecting thee diffraction effect of the ident wave i clearly reaonable if the wavelength i mall compared with the length L of the defect and if the ident direction i not too far from the normal direction to the plane of the defect. a) Comparion with the Theory of Diffraction: A comparion between the Kirchhoff approximation and the Geometrical Theory of Diffraction (GTD) of Keller [7], [8], which i alo a high-frequency method, may be carried out: becaue the GTD method take into account the diffraction of the ident wave on each edge of the defect, it i intereting to be able to etimate the error produced by the Kirchhoff approximation when neglecting thee diffraction effect. With thi in mind, we conider an infinite fluid medium (Fig. ) with a finite traight line of length L (the defect) on which the acoutic preure i aumed to be equal to zero. Fig. 4 how the profile of acoutic preure modulu calculated both by the GTD method (the olid line) and by the Kirchhoff approximation (the dotted line), for an ident monochromatic plane wave Fig. 4. Field diffracted by a zero preure defect (length L) in an infinite medium a a function of x /L (ee Fig. for the geometry and the notation). θ =, k L = 6, ζ/l = 6. Solid line: Geometrical Theory of Diffraction (GTD); dotted line: the Kirchhoff approximation; hatched region: nonvalidity zone for GTD. (angular frequency denoted ω) propagating in a fluid medium (peed of ound denoted V ). We recall that, in the GTD approach, the field i obtained by conidering ray: the pecular reflected ray on the defect urface and the ray reulting from the diffraction by the edge of the defect (Fig. ). The expreion for thee diffraction field may be found in [8]. For example, the diffraction preure field (divided by the ident preure) p j ( M ) coming from the edge j (j =,) at point M (Fig. ), can be written in the following form [omitting the term exp(iωt)] [8]. On the other hand, the Kirchhoff approximation lead to the prediction of diffraction phenomena that appear in the olution of the re-emiion problem in olid and, due to the dicontinuity of the given (approximate) boundary value on the plane II. p M j ( ) = e pk r j ( + ecoy j ) - ein q inq - co y ( ) ì é p ù expí ü - i K ( r j + ein q)+ ë ê û ú ý î 4 þ ì i - coy j in q + K r + é í O ( r ) j ( in q - y ê K j ) ë î co j j - ù ü ú ý, û þ () where θ, ψ j, and O é ëê X N ù ûú are, repectively, the ident angle, the angle between the (Ox )-axi and A j M, and a term of the Nth-order in X, and where k, K, r j, and ε are given, repectively, by k = ω/v, K = k L, r j = r j/ L, ε = for j = and ε = for j =. It can be een from () that the GTD olution p j ( M ) i not defined when θ = ψ j + π/ (the olution i ingular on the line correponding to the reflected ray at the edge of the defect), which lead to nonvalidity zone for the GTD (Fig. and 4).

5 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 55 When the plane defect conit of a lit with zero preure, the geometrical reflected field can be eaily calculated on the reflected ray on the lit. Thi field (divided by the ident preure) at point M(x,x ) i given by ( ) p M exp ik x in q ik x co q. () ref ( ) =- - + The total preure i the ummation of the geometrical reflected field p ref ( M ) and of the field which i cattered by each edge and, i.e., p M ( ) and p ( M )(Fig. ): ( ) = ( )+ ( )+ ( ) p GTD M p ref M p M p M. () For example, for an ident angle θ equal to and an adimenional frequency K = k L = 6 (Fig. 4), the comparion between the method i fairly good in the main part of the reflected field, but alo for the diffraction pattern on the left and right part of the preure profile. Thi imple cae, without any mode converion, permit to evaluate the error made on the olution, which reult from the fact that a part of the diffraction effect are omitted; thee diffraction effect are taken into account by the GTD but are not when uing Kirchhoff approximation. Similar cae can be found in the literature for defect in iotropic or aniotropic olid (ee [9] and reference contained therein). b) Comparion with reult from a boundary finite element method: The validity of the Kirchhoff approximation may be evaluated by comparion with reult obtained through a boundary finite element method. In [] and [], a Fourier tranform/boundary element hybrid method ha been introduced for the implified cae where the layer are made of the ame iotropic material. The comparion how that for all the geometrical configuration, the reult fit each other in the main part of the reflected field, wherea ome dicrepancy may appear in the ide part where diffraction effect dominate. A expected, thi dicrepancy diminihe when the length of the defect i reaed or at higher frequencie. B. The Method of Plane Wave Decompoition Before uing the Kirchhoff approximation, ome practical tool have to be decribed to obtain the different acoutic field correponding to the preliminary problem mentioned in Section II-A--b.. The ident-bounded beam ha to be decompoed into monochromatic plane wave (via Fourier tranform) and the diplacement and tre field have to be expreed in a coordinate ytem linked to the layered tructure (ee Section II-B--a). Becaue the method choen here to calculate Fourier tranform i the fat Fourier tranform algorithm, Section II-B- end with ome conideration about the ue of thi algorithm (ee Section II-B--b).. The diplacement field in an aniotropic bilayered tructure immered in a fluid have to be obtained, and thi involve writing a) the diplacement and tre field in an aniotropic medium with appropriate radiation condition (ee Section II-B-) and b) the boundary condition (luding bonding condition) at the interface eparating aniotropic media (ee Section II-B-). To be more concie, we made the choice in thi paper to treat only the reflection problem in fluid, but imilar expreion can be eaily obtained for the tranmiion problem in fluid. In addition, it hould be noted that thi cae i the claic cae of interet in NDT when the reflected field i the only acceible field that can be meaured. ) The Incident Bounded Beam in Fluid : a) Priple of the decompoition into plane wave and change of bai: The priple of the decompoition of a beam into monochromatic wave (or angular pectrum decompoition) i a well-known priple that can be applied to a calar or a vector field [] [6], baed on the linearity of the wave equation conidered. Here, the ident diplacement field u can be built, at any point M( X, X ), omitting the exp(iωt) factor, a a uperpoition of all the plane acoutic field with parameter K. It expreion can be written a u ( X ) =, X ì k ( K ) í Aˆ K i K X K ( ) exp - ( + X ) ü [ ] dk ý ò, î k þ (4) where  ( K ) i the amplitude of each plane wave and where ( K,, K ) are the component of the wave vector k of the fluid (angular frequency ω) in the coordinate ytem R = ( O, X, X, X ) linked to the emitting tranducer (Fig. ). Thee component atify the diperion relation ( ) + ( ) = = ( ) = ( ) K K k k w/ V, (5) where V i the peed of the wave propagating in the fluid. The particle diplacement u X, ( ) in the fluid, normal to the front face of the emitting tranducer, i aumed to be known and can be derived from experimental reult or analytical expreion. a Uing (4), the diplacement u ( X, ) can be written u X A K K ì (, ) = í ü ˆ exp ik X K ( ) (- ) d ý ò, î k þ (6)

6 56 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 which permit, by mean of an invere Fourier tranform, obtaining the angular pectrum U, and thu the amplitude  K ( ): K U K A K ( ) = ˆ ( ) k ó = u X ik X (, ) exp( ) d õ p ô X. To tudy the interaction of each monochromatic plane wave with the -layered tructure, a change of coordinate ytem i neceary, the new one R = ( Ox,, x, x ) being linked to the tructure (Fig. ). The tranlation from the origin O to the new one O reult in a change of phae exp( - i O O) = exp ( - k ik Z ), and the rotation angle θ around the x -axi (correponding to the ident angle of the acoutic beam) reult in a Jacobian J ( k ) uch that (7) æ k ö dk = co q + inq dk = J k d k, èç k ø ( ) (8) where k and k are the component of the wave number k on the x - and x -axe (K and K are function of k ). The amplitude  ( k ) of each ident plane wave, when referenced at the plane x = (interface I) can thu be expreed a a function of it amplitude A ˆ K, when referenced at the plane X =, by ( ) and ( ) = ( ) (- ) ( ) (9) ˆ ˆ A k A K exp ik Z J k, which lead to the following expreion of the ident particle diplacement in the fluid, in the coordinate ytem R: u ( x, x ) = ì k Aˆ ( k ) í ü ò ( k ) expé- i k ( x + k x ) ù dký î k ëê ûú, () þ where P k k / k i the polarization vector of the ident wave in fluid. = ( ) b) Ue of the fat Fourier tranform algorithm: The method ued to calculate the Fourier tranform and invere Fourier tranform i the fat Fourier tranform (FFT). Thi algorithm impoe a contant tep ampling. However, due to the rotation angle θ around the x -axi (correponding to the ident angle of the acoutic beam) between the plane linked to the emitting tranducer and the plane linked to the interface of the layered tructure, a contant tep ampling along X -axi (and thu for K ) lead to a noncontant tep ampling along the x -axi (and thu for k ), a it can be een in the expreion of the Jacobian J k ( ) in (8). Two method can be ued to obtain the amplitude Â( k ) [given by (9)] of each ident plane wave (referenced at the plane x = ) for a contant tep ampling for k. The firt one conit in calculating the angular pectrum U ( K ) [given by (7)], then the Jacobian J ( k ), and finally the amplitude A ˆ ( k ), with a contant tep ampling for K (which give a noncontant tep ampling for k ). An interpolation of the amplitude  ( k ) i then provided to obtain new amplitude A ˆ ( k ), correponding to a contant tep ampling for k. The econd method conit in interpolating the angular pectrum U K ' ( ) ( ) to obtain new value U K correponding to a noncontant tep ampling in K ', but correponding to a contant tep ampling in k. The Jacobian J ( k ) and the amplitude  ( k ) are then calculated with thi contant tep ampling in k. It ha been hown [5] that thi econd method give more accurate reult than the firt one. In other word,. U ( K ) i calculated from (9) and uing a contant tep ampling in K,. U K ', ( ) i interpolated to obtain U K ( ) uing the noncontant tep ampling in K ' which give a contant tep ampling in k. c) Subequent ue of the decompoition method: The method of decompoition of a beam into plane wave that ha jut been decribed in Section II-B--a i alo ued to calculate the field re-emitted by the finite-ized defect, following the paive re-emiion priple decribed in Section II-A--a. Thi calculation require that the reemiion data on the interface II admit a Fourier integral, which i enured by introducing a cattering problem in Section II-C for which the data are zero outide of the defect. ) Plane Wave in Aniotropic Media: Generally peaking, the interaction between an oblique ident monochromatic plane wave propagating in the plane (Ox x ) and an aniotropic layered tructure generate 6 plane wave [numbered by (η)] in each layer, with different velocitie. The propagation equation in each layer ue the ame form a the one developed by Rokhlin et al. [7], [8] and completed by Ribeiro et al. [9] with the help of the inhomogeneou waveform [4], [4]. It i ueful to introduce the lowne vector (η) m of the wave (η), defined by [4], [4] () () () m = h n/ h = h k/ w, V () where (η) n, (η) k and (η) V are repectively the propagation direction vector, the wave vector and the velocity of the

7 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 57 wave (η) in the conidered layer. It hould be noted that, due to the boundary condition at each interface (I, II, or III) which lead to Snell-Decarte law, the projection m and m of the lowne vector of all the wave on x and x -axi are the ame (with here m = ). In each aniotropic layer, the diplacement vector (η) U of each plane wave (η) ha the following form ( ) x x t = a é- i k x + k x - wt ù ëê ûú = a P exp é ëê - iw ( m x + m x - t) ù ûú, () () () () U(, ;) h h Pexp h where (η) a and (η) P are, repectively, the amplitude and the polarization vector of the wave (η), m, and (η) m are the component on axe x and x, repectively, of the lowne vector (η) m. Uing Hooke law in aniotropic olid media (and ummation convention on repeated indexe), the tree can be expreed a function of diplacement by Tij = c ijk U x k, () where the c ijkl are the elatic contant of the layer. The tre vector (η) T (aociated with the normal e x to the interface) of each plane wave (η) ha the following form T e. = T i x i (4) Following the priple of the Kirchhoff approximation explained in Section II-A-, it i neceary to identify in each layer the wave that propagate (or decreae) in the direction x > (denoted a ) and thoe that propagate (or decreae) in the direction x < (denoted b ). For propagative wave, thi identification mut be done by uing an energetic criterion [44] [46], baed on the ign of the normal power flux given by Synge [47]: 4 ( ) (5) * * F =- iw - TjU j + TjU j, where X* i the complex conjugate of X. For inhomogeneou wave, a decreaing condition mut be applied, uing the ign of the imaginary part m " of the rd component (η) m of the lowne vector of the wave (η). Subequently, for convenience, the indexe η =,, will refer to b wave that propagate (or decreae) in the direction x <, and η = 4,5,6 will refer to a wave that propagate (or decreae) in the direction x >. The energetic and radiation criteria can thu be ummarized a follow: ì F <, propagative wave, field b, h =,, í î m " >, inhomogeneou wave, (6a) ì F >, propagative wave, fielda, h = 456,, í î m " <, inhomogeneou wave. (6b) ) Plane Wave Tranmitted Through Interface (Boundary Condition, Including Bonding Condition, at Interface II): In cae of a perfect (rigid) bonding between olid media and at the interface II, the boundary condition conit in the continuity of the diplacement and tre vector (aociated with the normal e x to the interface), i.e., U ( x, x = h ;) t = U ( x, x = h ;), t (7a) and T ( x, x = h ;) t = T ( x, x = h ;), t " x, x = h, " t. (7b) Condition (7) are thoe involved in the preliminary problem olved for a healthy interface II (ee Section II- A--b and Section III-C-) In cae of a total delamination at interface II, the boundary condition conit of etting the tre vector to zero, i.e., T ( x, x = h ;) t =, x = h," t, (8a) and T ( x, x = h ;) t =, x = h, " t, (8b) for all the value of x correponding to point on the defect. In cae of intermediate (elatic) bonding between olid media and at the interface II (which i the cae of the defect conidered here), the boundary condition choen here depend on thoe introduced by Pilarki et al. [7] and widely ued from that time. In a bidimenional cae, thee bonding condition conit in chematizing the bonding by a uniform ditribution of pring without ma working under traction-compreion and hear deformation [48], [49]. Uing the tre vector aociated with the normal e x to the interface and defined by (4), a linear relation between the tre vector and a hift of diplacement ΔU can thu be written uch that T ( x, x = h ;) t = T ( x, x = h ;), t x = h, " t, (9a) U ( x, x = h ;) t ¹ U ( x, x = h ;), t x = h, " t, (9b) T ( x, x = h ;) t = KD U( x, x = h ;), t x = h, " t, (9c) for all the value of x correponding to point on the defect, where DU = U - U () and K i a ( ) matrix of tiffne coefficient. In practice, the coupling between traction/compreion and hear effect can be ignored, o that the matrix K can be conidered a a diagonal matrix. It hould be noted that the poitive ign of the coefficient of the matrix K are traightforwardly aociated with the direction of the nor-

8 58 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 mal e x, which point out from medium toward medium. Condition (9) are aumed to be thoe on the finiteized defect on interface II. To olve the preliminary problem for an infinite defect on interface II (ee Section II-A- and Section II-C-), thee condition (9) are extended all along interface II. Once the Kirchhoff approximation priple i explained and the practical tool for calculating the acoutic field uing plane wave are given, the following Section II-C aim at preenting the implementation of the Kirchhoff approximation method (uch a it ha been decribed in Section II-A) uing Fourier integral: the preliminary problem have to be olved (Section II-C-) and the cattering problem i then introduced (Section II-C-). C. Implementation of the Kirchhoff Approximation Uing Plane Wave Decompoition To be more concie, we made the choice in thi paper to treat only the reflection problem in fluid, but imilar expreion can be eaily obtained for the tranmiion problem in fluid. The parametric tudy in Section III deal only with thi reflected field. A explained in Section II-A--b, the Kirchhoff approximation need to olve preliminary problem: the firt one concern the interaction between the ident field u and the bilayered tructure with it internal interface II uppoed to be healthy (without any defect); the econd preliminary problem i identical to the previou one, but with an infinite defect at interface II; thi i the aim of Section II-C- to explicitly treat thee preliminary problem, firt for plane wave and econd for bounded beam. Finally, a mentioned in Section II-B--c, a cattering problem ha to be introduced (ee Section II-C-), to enure that the re-emiion data on the interface II would admit a Fourier integral. ) Solution of Preliminary Problem: a) Solution of preliminary problem for plane wave: Thee preliminary problem mut be firt olved for each monochromatic plane wave contituting the identbounded beam, the amplitude of each plane wave being equal to  ( k ) [ee (9), Section II-B--a]. For each problem, 4 calar equation derived from the boundary condition mut be written, uing notation defined in Section II-B-, a follow.. Four equation coming from the continuity of the component on x -axi of the diplacement vector U and from the continuity of the tre vector T at interface I (fluid /olid ): U ( x, x = ;) t = U ( x, x = ;), t " x, x =, " t, T (a) ( x, x = ;) t = T ( x, x = ;), t " x, x =, " t. (b). Four imilar equation at interface III (olid /fluid ). Six equation derived from the boundary condition (9) at interface II (olid /olid ), whether thi interface i healthy or conit of an infinite defect Fourteen unknown amplitude correpond to thee 4 boundary condition: the amplitude a ref of the wave reflected in fluid, the amplitude a a of the wave propagating or decreaing in each layer α, α =, (6 per layer), and the amplitude a tr of the wave tranmitted in fluid. The preliminary problem for plane wave can thu be olved: the problem (*) with a healthy interface II [(*) = (h)] or an infinite defect [(*) = (i)] on that interface. In particular, the olution for plane wave U α(*) of the problem (*) can be plit, in each layer α (α =,), into the olution U a a(*) and U b a(*) correponding to the wave propagating or decreaing toward x > (or x < ), repectively: (*) (*) (*) U = Ua + U b, a =,, (a) a a a where [omitting the exp(iωt) factor and uing ()] and U U 6 a(*) ( h) a(*) a = å Ua 4 6 å a(*)( h) a ( h) = a P exp é ëê - i k x + k 4 a =,, a(*) a(*) b = å U b å a ( x ) a(*)( h) a ( h) = a P exp é ëê - i kx + k a =,, ( a x ) ù ûú, ù ûú, (b) (c) uing criteria (6) (ee Section II-B-). It hould be noted that the amplitude (η) a α(*) are function of amplitude  ( k ) of each ident plane wave contituting the ident-bounded beam. Therefore, it could be convenient to write them in the form ˆ, ( ) = ( ) () a(*) ( h) a(*) a k A k T where (η) T α(*) are amplitude ratio. b) Solution of preliminary problem for bounded beam: Finally, the field correponding to the ident-bounded beam can be obtained by uperpoing the field obtained

9 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 59 in Section II-C--a for plane wave, uing the uperpoition priple a explained in Section II-B-. In particular, the field u α(*) in layer α (α =,) of the problem (*) can be written uch that where u () i a cattering term that mut be calculated by the above-decribed procedure. For thi cattering problem, the (approximate) value of the field on the interface II are: where (*) (*) u = ua + ub (*), (*) = or (*) = (), i a a a (4a) ì () ub ( x, x = h ) = í î ub ( x, x = h ) = u outide the defect, (7a) () ( i b ) - u b ( h) on the defect, (7b) and 6 a(*) ( ) = å ò { ( ) a(*) ( ( h) a ) P 4 u a x, x Aˆ k T k a exp( -i k x ) exp -ik x d k, (4b) u b x, x Aˆ k T k ( )} a(*) ( ) = å ò { ( ) a(*) ( ( h) a ) P a exp( -i k x ) exp -ik x d k. (4c) ( )} q. (4b) and (4c) will be ued in Section II-C, when olving the cattering problem (7). It i convenient to write here the expreion of the reflected diplacement field, for the problem with a healthy interface II [(*) = (h)] or an infinite defect on interface II [(*) = (i)]: u ò ( ) = ( ) + (*) (*) ref ref ref { ( ) x, x a k P exp ik x exp( -i kx ) d k }, (5) where P ref i the polarization vector of the reflected wave in fluid. q. (5) will be ued ubequently in Section II- C- to calculate the preure P (*) ref in fluid [ee (4)]. ) The Aociated Scattering Problem: Once the approximate value u (k) have been obtained on the interface II, by conidering the olution u (h), and u (i) of the preliminary problem mentioned in Section II-A--b (the olution of which i given in Section II-C-), we need to olve the radiation problem in the (virtual) half-pace x h and x h. To calculate the olution of thee radiation problem, a plane wave decompoition of the field may be introduced by the way of a patial Fourier tranform along the line x = h. Thi Fourier tranform will be eaily defined if an aociated cattering problem i introduced, conidering the field generated by the defect a a perturbation of the field of the healthy tructure. Thu, the approximate global olution may be ought in the form of a um: () k (), u = u + u (6) which permit u to obtain u () b ( x, x ) in layer (ee Section II-A--a, the paive re-emiion priple). Similar expreion can alo be written for the approximate field ( u ) a. Once the correponding cattered field i calculated in the half pace x h, by plane wave decompoition, one may conider it interaction with the interface I, getting ( ) the reflected and tranmitted cattered field u a ( and u ) ref. Note that in the cattering problem, the ident field u no longer enter into the interaction with the interface I. Finally, the expected total approximate reflected field i obtained by the following expreion: () k ref ref ref (). u = u + u (8) a) The cattering data on the interface II: The olution of the preliminary problem (h) and (i) have been obtained in Section II-C-. It i now poible to apply the Kirchhoff approximation method (ee Section II-A-) to obtain the approximate olution u (k). It hould be noted that, for the cattering problem (7), the field u () b ( x, x = h ) and u () a ( x, x = h ) are perfectly known on the interface II, and that they contitute re-emiion data for the reflection problem and the tranmiion problem, repectively. Subtituting (4c) for x = h in the cattering problem () lead to expreion for the field u b ( x, x = h ) on interface II (with the aim of determining the reflected field in fluid ): u b () ( x, x = h ) =, outide the defect, (9a) () u b ( x, x = h ) å ò { i h Aˆ k é () () ëê T k T k = ( ) ( )- ( ) ( ) - P exp -i k h exp ikx d k, on the defect, ( ) } ù ûú (9b) which involve the difference of the amplitude ratio () i T ( k ) for the infinite defect problem and the amplitude ratio h h () () T k ( ) for the healthy interface II. Simi-

10 6 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 larly, the field u () a ( x, x = h ) can be obtained, with the view of olving the tranmiion problem. b) Reolution of the cattering problem: Let u conider the problem of the re-emiion toward x <, to obtain ( ) the reflected cattered field u ref in fluid and thu, uing ( k) (7), the Kirchhoff approximation u ref of the exact olution. With the radiation condition toward x <, the field () u b x, x ( ) i expreed in olid in the form u () b () () x x A () (, ) = ˆ å ò { h ( k ) h exp -i k x P ( ) - ( ik x ) k} exp d, () () () where the amplitude h Aˆ ( k ) are going to be determined from re-emiion data (9), uing exactly the ame method a that explained in Section II-B- for the ident beam. A u () b ( x, x = h ) i known on interface II, an invere Fourier tranform of () lead to the vector equation ˆ () å A ( k ) P exp( -i k h ) () = u b ( x, h) exp + p ò ( ik x ) x d, () which contitute a linear ytem of equation for the () () unknown amplitude h Aˆ ( k ), η =,,. It hould be noted that the integrand of the right hand term of () i null outide the defect, becaue of (9a). Then, uing (), () the whole field u b ( x, x ) i known in layer. Similar expreion can be obtained for the tranmitted ( ) cattered field u tr in fluid. c) Interaction of the cattered field with interface I: Now, following the re-emiion priple (ee Section II-A--a), () the field u b ( x, x ) behave like an ident field on the interface I at x =, and the problem of the interaction of each plane wave contituting thi ident beam with the interface I ha to be olved. () Uing (), thi ident beam u b ( x, x = ) at x = can be written a (omitting exp(iωt) factor) u x, x = ò U b k exp( ikx ) d k, () ( ) = ( ) - b () () where U () b () () k A () ( ) = ˆ å h ( k ) h i the angular pectrum (plane wave). P () ( The interaction of thee ident wave U ) b with interface I generate reflected wave U a (with unknown ( ) amplitude (η) a () (η = 4,5,6)) in medium and a tranmitted wave U () ( (with an unknown amplitude a ref ) ) in fluid. Thu, the whole field U () in olid at x = i made ( up of the ident b-type wave U ) b propagating (or decreaing) toward x < and of the reflected a-type ( ) wave U a propagating (or decreaing) toward x > : U ( k ) = U ( k )+ U ( k ), (4) () () () a b where U 6 () a () () k a () ( ) = å h ( k ) h 4 P. (5) Finally, the writing of the boundary condition for plane wave at interface I (x = ) lead to 4 equation baed on the continuity of the component on the x -axi of the diplacement vector U and from the continuity of the tre vector T, which i aociated with the normal e x to the interface, ee (4), at interface I (olid /fluid ): () () U ( x, x = ;) t = U ( x, x = ;), t (6a) and T () () ( x, x = ;) t = T ( x, x = ;), t " x, x =, " t. (6b) It hould be noted that, for thi cattering problem, the ident field i no longer involved in thee equation. Four unknown amplitude correpond to thee 4 boundary condition: the amplitude a ref ) of the wave tranmit- ( ted in fluid and the amplitude (η) a () (η = 4,5,6) of the wave reflected in olid. Thee unknown amplitude can now be determined by olving the linear ytem of (6). ) Solution of the Reflection Problem: Total Reflected Field in Fluid : The cattered diplacement field in the fluid () i uch that u ò ( ) = ( ) () () ref x, x a ref k Pref { exp ( + ik x ) exp -i ( kx )}d k. (7) Finally, the whole reflected field in fluid baed on the Kirchhoff approximation i given by (8) in Section II-C- : () k ref ref ref (), u = u + u (8) ( where u h ) ref i given by (6) with (*) = (h), and the whole field in fluid can be written a () k () k ref. (9) u = u + u

11 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 6 TABL II. Material Characteritic of the Unidirectional Carbon poxy Medium. latic Contant (GPa) uch that 6th-Order Axi i Parallel to the x -Axi [5]. c c c c c 44 ρ (kg/m ) The reflected preure P (*) ref in fluid, correponding to the problem with a healthy interface II [(*) = (h)], an infinite defect on interface II [(*) = (i)] or a finite defect [(*) = (k)] on interface II under the Kirchhoff approximation i then given by where (*) ó (*) ref õ { ref ( ) = ( ) P x, x iwr V ô a k exp(+ ik x (*) = ( k),( h),( i), () k ref ref ref (). ) exp( -ik x )} d k, (4a) a = a + a (4b) III. Reult for Carbon/poxy Structure Some reult obtained on carbon/epoxy tructure (ee Table II for the elatic contant and the denity) are preented in thi ection, firt for a normal idence (θ = ) and variou thicknee (ee Section III-A), then when a Lamb mode i excited in the perfectly bonded tructure (healthy interface II, ee Section III-B-). In the latter cae, the influence of the aniotropy (Section III-B-), of the length and location of the defect (Section III-C and III-D), and of the nature of the bonding (Section III-) are conidered. A far a carbon/epoxy tructure are concerned, when the carbon fiber make an angle ψ with the x -axi, the layer i called a ψ-layer. A an example, a /9 tructure i made up of the layer with fiber parallel to the x -axi and of the layer with fiber perpendicular to x -axi. In the cae conidered in thi ection, the layer have the ame thickne (h = h ). The ident beam i a Gauian beam and the particle diplacement u X, ( ) in the fluid, normal to the front face of the emitting tranducer (ee Section II-B--a) ha the following form é ù u X U X a (, ) = exp -( / ). ëê ûú (4) (*) The figure preent the reflected preure moduli P ref (given by (4) and normalized by the ident preure) in a plane parallel to the interface (ζ/a = ) a a function of x /a (ee Fig. ): a thin olid line i ued for an healthy interface II (*) = (h), a dotted line for an infinite defect on interface II (*) = (i), and a thick olid line for a finiteized defect on interface II (*) = (k). The defect i taken to be a full delamination in Section III-A, III-B, III-C, and III-D, i.e., the tre vector i equal to zero on the defect, wherea the cae of an intermediate (elatic) bonding between olid media and at the interface II i conidered in Section III-. The frequency i f = MHz and the diameter of the tranducer i a = mm; ubequently, all the length are expreed a multiple of the radiu a. A. Cae of Normal Incidence When θ = (normal idence) and when d = (defect localized jut in front of the tranducer), the hape and the amplitude of the reflected field for an infinite [(*) = (i)] or for a finite defect [(*) = (k)] on interface II are very imilar (ome diffraction effect can jut be oberved for the finite defect cae), a can be een when comparing Fig. 5(a) and (b). For the purpoe of nondetructive teting, the amplitude of the reflected field for a healthy interface II and for a defect on interface II hould be quite different, to provide a good detection of the defect. Thi i the cae for Fig. 5(b) when the product f.h =. MHz mm but thi i no longer the cae when the product f h = MHz mm (ee Fig. 6): the reflection coefficient in water for plane wave ident on the healthy tructure ocillate periodically between and.8 a a function of the product frequency by thickne, and thu, if thi product i uch that the reflection coefficient i cloe to.8, the difference between the reflected field for a healthy tructure and for a tructure with a defect (maximum amplitude equal to for a large defect in the cae of a delaminating) i too mall to provide a good detection of the defect. A a conequence, it eem intereting to tudy the cae of oblique idence, and epecially when the ident beam may generate a Lamb wave into the tructure for a given couple (ident angle, frequency). B. Cae of a Lamb Mode Propagating in the Structure A far a propagation of Lamb mode in the tructure i concerned, the mot intereting cae for detecting a defect correpond to the generation of a Lamb mode in the healthy tructure (ee Section III-B-. and III-B-). When the ize of the defect i long enough, the generation of Lamb mode between the upper interface and the defect can alo enable the detection and thu be conidered. To be more concie, thee lat reult are not reported here. ) Cae of a Single Iotropic Layer: The firt reult for Lamb mode (Fig. 7) concern the cae of identical media contituting layer and, which amount to a defect located in a ingle -mm-thick layer. The fiber are parallel to the x -axi (one ingle 9 layer or a 9 /9 tructure), which amount to an iotropic ymmetry in the agittal plane. The lination of the axi of the ident

12 6 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 It i important to notice that thi interference effect reduce the amplitude of the firt maximum (here equal to.6 for the normalized preure), wherea for the ame ( ident angle, the maximum of the reflected preure Pref i ) on the tructure with the infinite defect indicated by the dotted line on Fig. 7(a) ha it value near.9. A expected, the reflected preure Pref k) for a finite-ize defect fol- ( ( low the curve Pref h) outide and far from the defect, a ( hown in Fig. 7(b), wherea it follow the curve Pref i) in the ( defect region. Between thee region, the preure Pref k ) preent ome cattering effect due to the edge of the defect. A a conequence, the amplitude maximum of the reflected preure on the tructure with the finite defect i here about.5 time greater than the amplitude maximum of the reflected preure on the healthy tructure, and we may conclude that thi configuration i well uited for a good detection of the defect. Fig. 5. Reflected preure P ref (*) for a 9 /9 carbon/epoxy tructure, h = h =.5 mm, full delaminating. Thin olid line (*) = (h), dotted line (*) = (i), and thick olid line (*) = (k). Incident angle θ =, frequency f = MHz. ζ/a =, d =, L/a =. beam i uch that a Lamb mode may be generated in the healthy tructure, namely, the antiymmetric mode A in the cae of Fig. 7. The generation of a Lamb mode may be ( recognized by looking at the reflected preure Pref h) in the fluid (the thin olid line on Fig. 7) along the cut line at the ditance ζ from the interface I (ee Fig. ). Indeed, a null value of that preure can be oberved between maxima. The firt maximum, on the left, correpond to the pecular reflected field on the firt interface. The econd maximum, on the right, i followed by a low decreaing of the preure. Thi correpond to the re-emiion of the generalized Lamb wave with the well-known aociated leaky wave. Becaue the pecular reflected field and the Lamb re-emiion field are in phae oppoition, thi explain the null value of the total preure in the location where the field counterbalance each other. ) Influence of Aniotropy: When the fiber in layer are perpendicular to thoe in layer (a 9 / tructure), the tructure i aniotropic. Starting with the preceding configuration (Section III-B-), by a progreive rotation around the x -axi of thi econd layer, one may follow continuouly the Lamb mode A to obtain a guided mode for thi nonymmetric tructure, which i found to be generated at an idence of.75. The correponding reflected preure are hown on Fig. 8. The thin line preent the trong trough characteritic of the generation of a guided mode in the healthy tructure; a the dotted line for the tructure with the infinite defect i very imilar to that of Fig. 7(a), it ha not been reported on Fig. 8, for clarity. The diffraction part of the thick line, for the finite defect, are reinforced, compared with that of Fig. 7, due to a greater change of media when paing from layer to layer. A imilar cae i preented in Fig. 9, for a /9 tructure. For the ame ident angle, the ame guided mode i generated in the tructure. For thee cae, the ratio between the maxima of preure for the healthy tructure and for the tructure with the finite defect i about.4. We may conclude that the generation of thi guided mode lead to a good detection of the defect. The Fig. preent the cae of a fully aniotropic tructure ( / ). The guided mode may be generated for a.5 idence. One oberve in thi cae that the leaky wave preent everal maxima that correpond to ucceive reflection in the tructure. The diffraction effect, for the finite defect, are important in thi cae. Again, thi configuration i uitable for good detection of the defect. C. Influence of the Length of the Defect The aim of thi ection i to how the influence of the length L of the defect on it detectability. The configuration conidered for the 9 / tructure i that of Fig. 8

13 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 6 Fig. 6. Reflected preure P (*) ref for a 9 /9 carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =, frequency f = MHz. ζ/a =, d =, L/a =. for L/a =. Fig. and correpond, repectively, to a length of the defect equal to a and to a/ (a i the diameter of the emitter, ee Section I). A expected, the thick olid curve correponding to the ( reflected preure Pref k) (finite-length defect) i cloer to the dotted curve correponding to the reflected preure ( Pref i) in Fig. (L/a = ) than in Fig. 8 (L/a = ), but ome diffraction effect till exit. In both cae, the defect i well een but there i more diffraction (epecially on the left edge of the defect) when L/a = than for a larger defect. Thi can be explained by the fact that, the greater the length of the defect, the le prominent i the diffraction phenomenon due to the dicontinuity of the given (approximate) boundary value on the plane II (ee Section II-A-). On the other hand, when the length of the defect i much maller than the radiu of the emitter (Fig., ( L/a =.), the reflected preure Pref k) follow the thin ( olid curve correponding to the reflected preure Pref h ) for the healthy tructure. Although the preence of the defect generate much diffraction, a can be een in the ( thick olid curve for the preure Pref k), if thi curve were moothed, it would be much too cloe to the curve for the healthy tructure for there to be good detection of the defect. D. Influence of the Location of the Defect Fig. and 4 correpond to the ame configuration a that of Fig. 8, except that the location of the defect i changed along the interface II: d = for Fig. 8, d = a for Fig., and d = a for Fig. 4 (ee Fig. for the geometry and definition of d). For clarity, the dotted line correponding to an infinite defect i not reported on Fig. and 4 (it i the ame a that of Fig. ). Fig. 7. Reflected preure P ref (*) for a 9 /9 carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h), dotted line (*) = (i), and thick olid line (*) = (k). Incident angle θ =.78, frequency f = MHz, propagation of an A Lamb mode. ζ/a =, d =, L/a =. When d = a, the defect i lightly viible, but the diffraction originating from the right edge of the defect can be ditinguihed. When d =, the defect i clearly ( viible, and the center of the reflected field Pref k) coide ( partially with the field Pref i). The diffraction effect originating from the right edge of the defect are well viible. ( When d = a, the reflected field Pref k) coide partially ( with the field Pref h) for the healthy tructure, with diffraction effect coming from the left edge of the defect. However, the detectability of the defect i not o clear. For larger value of d, an artifact of the Kirchhoff approximation would appear. Indeed, the multiple reflection between the upper interface and the infinite defect would lead to value of the re-emiion data much greater than

14 64 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 (*) Fig. 8. Reflected preure P ref for a 9 / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. Fig.. Reflected preure P (*) ref for a / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.5, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. (*) Fig. 9. Reflected preure P ref for a /9 carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. they ought to be. To avoid thi phenomenon, the Kirchhoff approximation mut be implemented in an iterative procedure [5].. Influence of the Nature of the Bonding The ection aim to tudy the influence of the ( ) matrix K of tiffne coefficient, defined in Section II-B-, and involved in boundary condition (9). The reult preented jut above correpond to a total delaminating, i.e., to a null matrix K. From a numerical point of view, it appear that a total delaminating correpond to numerical Fig.. Reflected preure P (*) ref for a 9 / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h), dotted line (*) = (i), and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. value up to nearly K ii = N m, ee Fig. 5 (Fig. 8 correpond to K ii = 5 N m ). When intermediate (elatic) bonding between the olid media and at the interface II i conidered, it can be een that the defect can till be detected. A expected, the greater the coefficient of the matrix K (the embedding condition correpond to infinite coefficient), the le the defect can be detected (ee Fig. 6 for K ii = N m ) and the ( weaker the reflected field Pref k). From K ii = 4 N m, the bonding amount to a perfect embedding and the curve

15 vacoin et al.: interaction of a monochromatic ultraonic beam with a finite length defect 65 Fig.. Reflected preure P (*) ref for a 9 / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h), dotted line (*) = (i), and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =.. Fig. 4. Reflected preure P (*) ref for a 9 / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d = a, L/a =. Fig.. Reflected preure P (*) ref for a 9 / carbon/epoxy tructure, h = h = 5 mm, full delaminating. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d = a, L/a =. Fig. 5. Reflected preure P ref (*) for a 9 / carbon/epoxy tructure, h = h = 5 mm. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. Coefficient K ii = N m. ( for Pref k) ( and Pref h) are uperimpoed. A a conequence, it hould be noted that there i a very limited range of value of the defect tiffne coefficient for which ultraound will be enitive to the preence of a bonding defect. IV. Concluion Thi paper aimed at providing a emi-analytical model for the interaction of a monochromatic ultraonic bounded beam with a -layered aniotropic tructure luding a finite defect on the internal interface. Delamination or partial bonding condition have been aumed on the defect. The cattering of the acoutic field by the defect ha been calculated uing the Kirchhoff approximation. Plane wave decompoition of the field, uing patial Fourier tranform, have been ued to olve the problem of croing the variou interface of the tructure (or eventually of a multilayered tructure in the cae of a general tratified medium) The interaction with the defect ha been calculated by introducing a cattered field, which correpond

16 66 I Tranaction on Ultraonic, Ferroelectric, and Frequency Control, vol. 56, no., October 9 Fig. 6. Reflected preure P ref (*) for a 9 / carbon/epoxy tructure, h = h = 5 mm. Thin olid line (*) = (h) and thick olid line (*) = (k). Incident angle θ =.75, frequency f = MHz, propagation of a Lamb mode. ζ/a =, d =, L/a =. Coefficient K ii = N.m. to the perturbation due to the defect, compared with the olution for the healthy tructure. Several reult have been preented for the cae of a carbon/epoxy compoite when the fiber in the layer are parallel or perpendicular. Firt, it i een that normal idence i not alway the bet configuration to detect clearly the defect. Intead, an ident angle uch that a Lamb mode or guided mode can be generated in the healthy tructure may be more uitable. 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Levent and J. F. de Belleval Some reult of ultraonic beam model in aniotropic medium uing plane wave decompoition, Acut. Acta Acut., vol. 8, uppl., pp. S46, 996. [5] M. Cataing, Propagation ultraonore dan le milieux tratifié plan contitué de matériaux aborbant et orthotrope, Ph.D. diertation, Univ. Bordeaux I, 99. [5] N. Bedrici, Ph. Gatignol, and C. Potel, An iterative method for the interaction between a bounded beam and an interface defect in olid, under Kirchhoff approximation, Acut. Acta Acut., vol. 95, no., pp. 89, 9. Bruno Vacoin i holder of the agrégation in the field of mechanic e 987. He teache quality management and nondetructive teting method at Univerity Picardie Jule Verne in Soion, France. In 8, he received a Ph.D. degree in mechanical engineering from the Univerity of Technology of Compiègne (UTC), Compiègne, France, and collaborate with thi Univerity in the Laboratoire Roberval. Hi reearch i in the propagation of ultraonic beam in aniotropic multilayered media, luding delaminating defect-type. Catherine Potel received a Ph.D. degree in mechanical engineering from the Univerity of Technology of Compiègne (UTC), Compiègne, France, in 994. She wa lecturer at the Univerity of Amien, France, from 996 to and ha been a profeor of phyical acoutic and of mechanic at the Univerity of Le Man, France, in the Laboratoire d Acoutique de l Univerité du Maine e. Her reearch i in ultraonic for nondetructive evaluation and material characterization, with pecial interet in propagation in aniotropic multilayered media uch a compoite and in rough plate. Philippe Gatignol i holder of the agrégation in the field of mathematic, and he received a Ph.D. degree from the Univerity of Pari VI in 978. He wa lecturer at the Univerity of Pari VI from 96 to 98. He wa profeor of phyical acoutic at the Univerity of Technology of Compiègne (UTC), Compiègne, France, from 98 to 7 and i now meritu Profeor. He wa project leader at the Scientific Direction of the Science for ngineering Department at the National Scientific Reearch Center (CNRS) and at the French Minitry of Reearch from 986 to 99 and from 998 to, repectively. He wa alo reponible for the training of Ph.D. tudent in mechanic at UTC until 4. Hi reearch i in phyical acoutic and ultraonic for nondetructive evaluation. Jean-Françoi de Belleval (M 87) wa born in Mareille, France, in 944. He received the diploma of engineering at the cole Polytechnique, Pari, France, and a Ph.D. degree from the Univerity of Pari VI in 974 on the ubject of the relationhip between the acoutic field of a hot jet and it turbulence and infrared emiion. He wa profeor of phyical acoutic at the Univerity of Technology of Compiègne (UTC), Compiègne, France, from 976 to 4 and i now meritu Profeor. He wa Director of the Laboratoire Roberval, a reearch unit aociated with CNRS from 99 to. Previouly, he worked at ONRA (French Aeropace Reearch Center). Hi reearch i in ultraonic for nondetructive evaluation and material characterization, with pecial interet in propagation in aniotropic multilayered media uch a compoite.