Time-harmonic torsional waves in a composite cyliner with an imperfect interface J. R. Berger a) Division of Engineering, Colorao School of Mines, Golen, Colorao 80401 P. A. Martin b) Department of Mathematics, University of Manchester, Manchester M13 9PL, Unite Kingom S. J. McCaffery c) Division of Engineering, Colorao School of Mines, Golen, Colorao 80401 Receive 4 June 1999; accepte for publication 1 November 1999 In this paper, the propagation of time-harmonic torsional waves in composite elastic cyliners is investigate. An imperfect interface is consiere where tractions are continuous across the interface an the isplacement jump is proportional to the stress acting on the interface. A frequency equation is erive for the ro an ispersion curves of normalize frequency as a function of normalize wave number for elastic bimaterials with varying values for the interface constant F are presente. The analysis is shown to recover the ispersion curves for a bimaterial ro with a perfect wele interface (F 0), an has the correct limiting behavior for large F. It is shown that the moes, at any given frequency, are orthogonal, an it is outline how the problem of reflection of a torsional moe by a planar efect such as a circumferential crack can be treate. 000 Acoustical Society of America. S0001-4966 00 0440-7 PACS numbers: 43.0.Mv, 43.0.Gp, 43.35.Cg, 43.38.Dv ANN INTRODUCTION The motivation for this stuy comes from the application of electromagnetic acoustic transucers EMATs to the nonestructive testing of reinforce cables. We moel the cable as an infinitely long bimaterial cyliner, with a core of circular cross section surroune by a coaxial claing; the core an the claing are ifferent homogeneous isotropic elastic solis. Applications of EMATs are reviewe by Frost 1979 an by Hirao an Ogi 1997. We are intereste in the use of time-harmonic torsional waves in the composite cyliner. Johnson et al. 1994 have use EMATs to stuy staning torsional moes in a single-material circular cyliner. This is a classical problem originally stuie by Pochhammer see, for example, Achenbach, 1973, sec. 6.10, or Miklowitz, 1978, sec. 4.4. Propagation of time-harmonic torsional waves in a ro compose of two or more elastic layers has also been stuie; see Thurston s paper 1978 for a comprehensive review. Perhaps the earliest work is by Armenàkas 1965, 1967, 1971. He stuie the ispersion of harmonic waves an establishe the isplacements an stresses at the interface of each layer analytically. A frequency equation was obtaine by enforcing continuity conitions at the interface an a stress-free bounary conition on the lateral surfaces of the cylinrical ro. Charalambopoulos et al. 1998 have consiere the free a Electronic mail: jberger@mines.eu b Present aress: Department of Mathematical an Computer Sciences, Colorao School of Mines, Golen, CO 80401; electronic mail: pamartin@mines.eu c Present aress: CIRES, University of Colorao, Bouler, CO 80309. vibration of a bimaterial elastic ro of finite length. The problem was solve for time-harmonic waves using the Helmholtz ecomposition of the three-imensional elasticity equations. The interface between the layers was consiere as perfect, proviing continuity of isplacement an traction. The frequency equation for the full three-imensional ro was foun in terms of a 9 9 eterminantal equation whose roots yiel the ispersion relations for the ro. Rattanawangcharoen an Shah 199 have also consiere the layere cylinrical ro, but they stuie the problem from a more general perspective in that their formulation allowe many layers. A propagator matrix approach was use which relate the stresses an isplacements of one layer to the next. The propagator matrix was foun to implicitly generate the frequency equation for the ro. The main motivation for the paper was to arrive at an efficient computational scheme for the many-layer problem which i not rely on a homogenization metho such as integrating through the layers. In this paper, we consier the bimaterial elastic cyliner with an imperfect interface between the core an the claing. We o this because it is unrealistic to assume a perfectly bone wele interface for our intene application to reinforce cables. We moel the imperfect interface using a linear moification to the stanar perfect-interface conitions, allowing some slippage. The interface conitions involve a single imensionless parameter F. We stuy the effect of varying F on the ispersion relations. Note that the results for a perfectly bone interface can be recovere by setting F 0. EMATs can be use to excite propagating moes with a specifie axial wavelength, where is etermine by the physical spacing between the magnets of alternating polarity. 1161 J. Acoust. Soc. Am. 107 (3), March 000 0001-4966/000/107(3)/1161/7/$17.00 000 Acoustical Society of America 1161
One then ajusts the frequency until one of the propagating torsional moes is excite. When such a moe interacts with a efect in the composite cyliner, other allowable moes at the frequency, but with various wavelengths, will be stimulate; evanescent moes ecaying exponentially with istance from the efect will also be present, in general. We show that the torsional moes at a given frequency are orthogonal, extening a proof ue to Gregory 1983. We also iscuss the evanescent moes an their computation. Finally, we outline how our knowlege of the moal structure for the composite cyliner can be use to moel the problem of reflection of a torsional moe by a thin efect in a cross-sectional plane. The EMAT system can only receive waves with the same wavelength as the incient moe, so that some information at the excitation frequency is lost; but the experiment can be repeate at other moal frequencies. I. FORMULATION Let (r,,z) be cylinrical polar coorinates. We consier the infinite isotropic elastic bimaterial cyliner shown in Fig. 1. The cyliner consists of a soli core, r a, surroune by an annular claing, a r b; the core an claing are mae of materials 1 an, respectively. Material m has Lamé mouli m an m, m 1,. The analysis presente here generally follows Armenàkas 1965. In general, the isplacement fiel u (u,v,w) in each portion of the bimaterial can be written using the Lamé scalar potential an vector potential ( r,, z ); see, for example, Achenbach, 1973, sec..13. We are intereste in torsional waves, for which the only non zero isplacement component is the tangential isplacement v, an v itself is require to be inepenent of. Hence, the only potential neee is the z-component of the vector potential, z, say. In terms of, we have v r. The only nontrivial stress components are an r v r v r z v z. FIG. 1. Geometry of the bimaterial cyliner. The potential satisfies 1 3 1 c t, where is the Laplace operator an c is the shear wave spee. For waves propagating in the positive z-irection, the appropriate solution of 4 can be written as r,z,t r e i kz t, where i is 1, k an are real, an solves 1 r r r /c k 0. This is Bessel s equation of orer zero. Its solutions epen on the sign of k c. Thus, efine Z n J n, W n Y n, an q /c k if k c, 7 an Z n 1 n I n, W n K n, an q k /c if k c, where J n an Y n are Bessel functions an I n an K n are moifie Bessel functions. The factor ( 1) n will allow a unifie treatment for all frequencies. The behavior of the solution as q 0 will be examine in some etail later; for now we assume that q 0 ( k c ). So, the appropriate solution of 6 is r q AZ 0 qr b BW 0 qr, where A an B are arbitrary constants, an the factors q an b have been introuce for later convenience, implying that A an B are imensionless; recall that b is the outer raius. The isplacement fiel obtaine by substituting 5 an 9 in 1 is v q 1 AZ 1 qr qb BW 1 qr e i kz t, 4 5 6 8 9 10 as Z 0 (x) Z 1 (x) an W 0 (x) W 1 (x). Note that I 0 I 1. From, we obtain for the stress, r AZ qr qb BW qr e i kz t, 11 as Z 1 (x) x 1 Z 1 (x) Z (x) an W 1 (x) x 1 W 1 (x) W (x). Let us now use the expressions above, using subscripts 1 an to inicate quantities in the core an claing, respectively. Thus, from 10, the isplacement in the claing is v q 1 A Z 1 q r q b B W 1 q r e i kz t. 1 For the core, the solution for v 1 must be boune at the origin so we have v 1 q 1 1 A 1 Z 1 q 1 r e i kz t. In these expressions, q j is efine by q j k j k if k j k, k k j if k j k j 1,,, 13 14 where k j /c j. Note that the wave number, k, is the same in the expressions for q 1 an q ; this observation gives a 116 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 116
relation between q 1 an q, which we will iscuss later. With reference to Fig. 1, we now consier bounary an interface conitions on the isplacement fiel given by 1 an 13. At the outer surface, we have the traction-free bounary conition r 0 at r b. 15 We consier the interface conitions in the following section. A. Interface conitions A variety of conitions may be taken on the interface r a in orer to represent imperfect interface conitions. A review of interface conitions for elastic wave problems has been presente by Martin 199. For most moels, the isplacement u an traction t on one sie of the interface are assume to be linearly relate to the isplacement u an traction t on the other sie of the interface. For example, the moel of Rokhlin an Wang 1991, originally erive for plane interfaces, takes interface conitions of the form t Gu Bt, u Ft Au, where A, B, F, an G are 3 3 matrices, an the square brackets inicate a jump in the quantity across the interface; for example, if the interface is at r a, we have u u u u a, u a,, 16 suppressing the epenence on z an t. If the coupling term G can be neglecte, an furthermore if A an B are set equal to zero, we recover the moel of Jones an Whittier 1967 for a flexibly bone interface, t 0, u Ft, 17 18 where F is a constant iagonal matrix. For simplicity, we will use the Jones Whittier moel for the analysis presente here. For thin, elastic interfacial layers, the elements of F have been relate to the thickness an elastic constants of the layer by, for example, Jones an Whittier 1967, Mal an Xu 1989, an Pilarski an Rose 1988. For torsional waves, u reuces to a scalar for the tangential isplacement v an t reuces to a scalar for the tangential shear stress r. The interface conitions are then an r a r a v a/ 1 F r a, 19 0 where v v (a ) v 1 (a ) an F is a imensionless scalar. Our goal is to investigate solutions which satisfy 15, 19, an 0 as the interface parameter F is varie. We note that if F 0, the perfect interface conitions of continuity of traction an isplacement are recovere. II. FREQUENCY EQUATION FOR THE ROD We now present the etails for the set of equations which will etermine the ispersion relations in the bimaterial ro. Substituting the isplacement fiel of 1 in the bounary conition, 15, yiels A Z q b q b B W q b 0. 1 Following the Jones-Whittier moel, we have for continuity of traction across the interface, from 1, 13, an 19, 1 / A 1 Z q 1 a A Z q a q b B W q a 0. Note that neither of these equations changes in the case of the perfectly bone interface. The isplacement jump across the interface is given by 0. We then have q 1 b 1 A 1 Z 1 q 1 a Fq 1 az q 1 a q b 1 A Z 1 q a q bb W 1 q a 0. 3 Equations 1 3 provie three equations in the three unknown constants A 1, A, an B. In matrix form, the system of equations is Db 0, 4 where the elements of the nonsymmetric matrix D are obtaine irectly from 1 3 an b (A 1,A,B ) T. For a nontrivial solution we then require et D 0. 5 This is the frequency equation for the ro. The quantity et D seems to epen on only five imensionless parameters, namely q 1 b, q b, a/b, 1 /, an F; 6 in particular, the ensity ratio or, equivalently, c 1 /c oes not appear explicitly. However, this is illusory: we have to know how to choose Z n (J n or ( 1) n I n?) an W n (Y n or K n? in each material, an these choices epen on the relative sizes of k, k 1, an k, information that we cannot extract from a knowlege of 6 alone. Thus we procee as follows. Assume that we are given values for a/b, 1 /, F, an c /c 1 k 1 /k, 7 say. Choose a value for the axial wave number kb. We now seek values of k b, say, so that 5 is satisfie. Note that k 1 b k b, an then q 1 b an q b are efine by 14, with the associate selections of Z n an W n ictate by 7 an 8. In fact, the relations between q 1, q, k 1, k, k, an are complicate, because they epen on the relative magnitues of k, k 1, an k ; there are four cases, as summarize in Table I. In this table, the secon column specifies the four cases in terms of the shear wave spees of the two materials these are material constants an the axial wave spee c a /k. A similar table was given by Kleczewski an Parnes 1987 in their stuy of torsional moes when the claing is unboune (b in our notation. In orer to compare with Armenàkas 1965 for F 0, we have etermine the ispersion curves of normalize frequency, 1163 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1163
TABLE I. Relations between q 1 an q, in which c /c 1 k 1 /k an c a /k. Wave numbers Wave spees q 1 q Relation between q 1 an q k k 1 k k k k 1 c 1 c c a k 1 k k k q 1 q k (1 ) c c 1 c a k k k 1 c 1 c a c k 1 k k k q 1 q k (1 ) k 1 k k c c a c 1 k k 1 k k q 1 q k (1 ) k 1 k k c a c 1 c k k 1 k k k k 1 k c a c c 1 q 1 q k (1 ) k b a / b a / c, as a function of normalize axial wave number, k b a /, for a given value of the interface parameter F. We note that the frequency equation etermine here cannot be written in terms of a single argument such as qa, which can be one in the case of a ro mae from a single material. As such, we stuy numerical solutions to 5 in the next section for values of a/b, F, an the elastic constants. III. DISPERSION CURVES WITH VARYING INTERFACE CONDITIONS To benchmark the analysis presente here, we first present results which can be irectly compare with Armenàkas 1965 in the case of a perfect interface, F 0. We take a/b 0.5, 1 / 10, an c 1 /c 1.83 so that the ensity ratio, 1 / 3. We show the ispersion curves of frequency,, versus wave number,, for F 0, F 1, F 10, an F 100 for the secon moe in Fig. an the thir FIG.. Dispersion curves for the secon moe in the bimaterial cyliner. FIG. 3. Dispersion curves for the thir moe in the bimaterial cyliner. moe in Fig. 3. The first moe will be analyze in a subsequent section. The ispersion curve for F 0 agrees exactly with the analysis of Armenàkas 1965. As the interface parameter is increase, note the ecrease in, especially at the smaller values of. At higher values of, the loss of perfect continuity at the interface has a reuce effect. One feature of note in Fig. 3 is the curve for F 100, which exhibits a corner at 0.8. This is not a numerical artifact: the figure was prouce using very small increments in. Similar behavior was foun for other large values of F. A secon way of visualizing the behavior of the ispersion curves as the interface parameter is varie is illustrate in Fig. 4. In the figure, we show results for the secon moe an plot frequency,, versus the interface parameter, F, as the wave number is varie. As expecte, we see a much greater effect on by F at the smaller wave numbers. This suggests a possible measurement approach for etermining F where is fixe, is measure, an then F is etermine from the figure. The curves shown in Fig. 4 appear to be approaching asymptotic values for large F. With reference to the interface conitions given by 0, we see that in the limit as F we recover the bounary conition r a 0. This is the appropriate bounary conition for the outer bounary of a soli ro of raius a an for the inner bounary conition for a hollow tube with inner raius a. Inthe case of F, the frequency equation given by 5 reflects this change in bounary conition, where Z q 1 a g q a,q b 0, 8 1164 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1164
frequency,, for the thir moe in the composite ro correspon to the secon moe frequencies in the hollow tube. Some iscussion on the interface parameter F is perhaps in orer. As mentione in the Introuction, the application motivating the analysis presente here is the nonestructive evaluation of reinforce cables. Typically these cables are fabricate with a steel core surroune by an aluminum claing. Because of the unerlying wire-rope structure of the core an the claing, the interface conitions are imperfect. Our approach here is to treat the interface parameter in a phenomenological manner to account for the imperfect interface. As such, we o not stipulate a strict physical interpretation to the numerical value of F, nor o we attempt to relate the value of F to elastic constants. IV. FIRST TORSIONAL MODE Armenàkas 1965 note that the first torsional moe is not properly escribe by the solution of the Bessel equation, unlike the higher torsional moes analyze above. Therefore, special consieration of the first torsional moe is necessary, an this is carrie out next. For example, when q 0, there can be a nonispersive moe propagating in the claing of the ro with ispersive moes in the core. Suppose that k c, so that q 0 an 6 reuces to FIG. 4. Normalize frequency vs interface parameter at fixe wave number in the bimaterial cyliner for the secon moe. g q a,q b Z q a W q b Z q b W q a. The frequency equation 8 has two sets of solutions, one given by Z (q 1 a) 0 an the other given by g(q a,q b) 0. The first of these is the frequency equation for a soli ro of raius a, whereas the secon is the frequency equation for a hollow tube of inner raius a an outer raius b. So, the imperfect interface formulation has the expecte behavior for large values of F. Numerically, one must be somewhat careful in hanling the limit as F an check if the relations between q 1 an q given in Table I still hol. We present some typical results in Table II where we have use F 1 10 10. Note in the table that we report values of normalize frequency as we have throughout this paper, even for the soli ro moes in material 1 reporte in the table. From the table, we see that the asymptotic values of frequency,, for the secon moe in the composite ro, are the nonispersive first moes (q 1 a 0) in the soli ro. Also, the asymptotic values of TABLE II. Frequencies for the secon an thir moes in the composite ro with F 1 10 10, an corresponing frequencies for the first moe in a soli ro an the secon moe in a hollow tube. secon moe thir moe, hollow tube secon moe, soli ro first moe 0. 0.365 1.86 1.86 0.365 0.4 0.730 1.33 1.33 0.730 0.6 1.095 1.405 1.405 1.095 r r 0, with general solution r A B log r. This gives a solution for v proportional to B/r, which cannot be the general solution as it involves only one arbitrary constant, B. As we are intereste in rather than, we return to 6 ; ifferentiation with respect to r gives r 1 r r r /c k 0, a secon-orer orinary ifferential equation for. When k c, the general solution of this equation is r Ar B/r. Hence, in imensionless form, we have an v r,z,t Ar Bb /r e i kz t r r,z,t B b/r e i kz t. 9 30 Note that the expressions 9 an 30 can also be obtaine by taking the limit q 0 in 10 an 11 apart from some numerical factors which can be absorbe into A an B. This accounts for the various q-factors in 10 : they lea to meaningful boune expressions for small q. Let us assume that q 0. The outer bounary conition 15 implies that B 0, whence r 0 in the claing an v A re i kz t. 31 Within the core, v 1 is given by 13. For continuity of tractions across r a, 19 gives 1165 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1165
A 1 Z q 1 a 0, whereas the imperfect-interface conition 0 gives A 1 Z 1 q 1 a q 1 aa 0. 3 33 Now, for nontrivial solutions, we require k k 1, so that 3 gives q 1 a j,s, the s-th zero of J (x). Then A 1 is arbitrary with A given by 33 with Z 1 J 1. It is interesting to note that the interface parameter F oes not enter into any of the equations, so that the ispersion curves for the first torsional moe when q 0 are ientical, regarless of whether the interface is perfect or imperfect. This is consistent with the fact that since q 0, a nonispersive moe is propagating in the claing an 3 is simply the frequency equation for the ispersive moes in the core. Alternatively, let us assume that q 1 0; for boune isplacements in the core, we obtain v 1 r,z,t A 1 re i kz t. Within the claing, v is given by 1, so that the outer bounary conition gives 1. The interface conition 19 gives A Z q a q b B W q a 0; the frequency equation is then obtaine by combining this equation with 1 : it is the same equation as for a hollow cylinrical tube. The other interface conition, 0, then etermines A 1 as A 1 q a 1 A Z 1 q a q b /a B W 1 q a. Again, these equations o not involve F. V. EVANESCENT MODES So far, we have only consiere propagating torsional moes. However, cyliners can also support evanescent moes, which ecay exponentially with z. Such moes can be constructe by writing r,z,t r e kz i t, where solves 1 r whence r r /c k 0, r q AJ 0 qr b BY 0 qr with q ( /c) k. Then, proceeing exactly as before, we arrive at the frequency equation 5 in which Z n an W n are to be replace by J n an Y n, respectively. VI. DISCUSSION ON MODE ORTHOGONALITY We have constructe various torsional moes for the composite cyliner in the general form u r,,z,t Re U r, e i kz t. In our computations, we have fixe the axial wave number k an then calculate the frequencies of the allowable moes. This is convenient for comparisons with Armenàkas 1965 an it is appropriate for the application to EMATs; these can be use to excite propagating moes of a specifie axial wavelength. However, once such a moe has been excite, we are intereste in stuying its reflection by efects in the cyliner. This is most conveniently one by specifying the frequency an then etermining all the allowable moes at that frequency. With this in min, we write a typical moe as u n r,,z,t Re U n r, e i k n z t, where the wave number k (n) nee not be real. These moes are biorthogonal. To be more explicit, enote the stresses corresponing to u (n) by n r,,z,t Re S n r, e i k n z t. Then, if k (n) k (m), we have U m z S n zz S m rz U n r S m z U n rr 0, A 34 where A is the cross section of the composite cyliner. This relation can be prove by a simple extension of the proof given by Gregory 1983. One applies the elastic reciprocal theorem twice, once in the core an once in the claing, an then as the results; the interface conitions imply that the contributions from integrating over the two sies of the interface cancel. In fact, 34 hols for all moes in composite cyliners of any cross section, an with any number of imperfect cylinrical interfaces. For our problem, with torsional moes given by v n r,z,t Re V n r e i k n z t, Eq. 34 reuces to b V m r V n r r r 0, 0 m n, 35 so that torsional moes are actually orthogonal. This orthogonality relation is useful when the reflection of a torsional moe by certain efects is examine. For example, we may consier a bimaterial cyliner with a planar break crack perpenicular to the cyliner s axis, giving an iealize moel of a amage cable. Specifically, we partition the cross-section A into a broken part A b an an unbroken part A u, so that A A b A u. The bounaries of A b an A u are concentric circles; for example, we might take A u to be the circle 0 r c, with A b as the annulus c r b, so that the cable is circumferentially cracke. Then, if a torsional moe is incient on the efect, the reflecte an transmitte fiels can be written as moal sums. This is a stanar approach for planar obstacles in waveguies. In the context of torsional waves, it has been use recently by Engan 1998 to analyze the effect of a step-change in raius of homogeneous circular cyliners. For the present problem, application of the bounary conitions at the efect plane leas to a system of equations for the reflection an transmission coefficients; of particular interest are the reflecte an transmitte moes with the same wavelength as the incient moe, because these are the only moes that can be etecte by the EMAT. Again, in a stanar way, one can erive integral equations an/or variational expressions for the reflection an transmis- 1166 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1166
sion coefficients; see, for example, Schwinger an Saxon 1968 for a etaile iscussion on relate scattering problems. VII. CONCLUSIONS We have presente an analysis for torsional waves propagating in a bimaterial ro with imperfect interface conitions. To the authors knowlege, the effect of imperfect interface conitions on ispersive wave motion has not been stuie for ros. We fin the expecte behavior for waves in the ro when we take the interface parameter F 0 an F. When F 0 we fin that the frequency ecreases with increasing F at a given wave number in the ispersion relations. This effect was shown to be more pronounce at small wave numbers. The propagation of nonispersive moes in the claing was also investigate, an the frequency equation for ispersive moes in the core was recovere. We also showe that, at any given frequency, the moes are orthogonal. This fact can be exploite in the solution of a scattering problem, where an incient torsional moe interacts with an annular efect in the bimaterial ro. The moel evelope here shoul be useful in analyzing nonestructive evaluation measurements in reinforce cables where perfect interface conitions may not exist. ACKNOWLEDGMENTS Two of us J.R.B. an S.J.M. gratefully acknowlege the support receive from the Center for Avance Control of Energy an Power Systems, a National Science Founation Inustry/University Cooperative Research Center, at the Colorao School of Mines. J.R.B. also acknowleges the aitional support provie by the Engineering an Physical Sciences Research Council as a visiting fellow in the Department of Mathematics at the University of Manchester. Achenbach, J. D. 1973. Wave Propagation in Elastic Solis North- Hollan, New York. Armenàkas, A. E. 1965. Torsional waves in composite ros, J. Acoust. Soc. Am. 38, 439 446. Armenàkas, A. E. 1967. Propagation of harmonic waves in composite circular cylinrical shells. I: Theoretical investigation, AIAA J. 5, 740 744. Armenàkas, A. E. 1971. Propagation of harmonic waves in composite circular cylinrical shells. Part II: Numerical analysis, AIAA J. 9, 599 605. Charalambopoulos, A., Fotiais, D. I., an Massalas, C. V. 1998. Free vibrations of a ouble layere elastic isotropic cylinrical ro, Int. J. Eng. Sci. 36, 711 731. Engan, H. E. 1998. Torsional wave scattering from a iameter step in a ro, J. Acoust. Soc. Am. 104, 015 04. Frost, H. M. 1979. Electromagnetic-ultrasoun transucers: Principles, practice, an applications, in Physical Acoustics, eite by W. P. Mason an R. N. Thurston Acaemic, New York, Vol. 14, pp. 179 75. Gregory, R. D. 1983. A note on bi-orthogonality relations for elastic cyliners of general cross section, J. Elast. 13, 351 355. Hirao, M., an Ogi, H. 1997. Electromagnetic acoustic resonance an materials characterization, Ultrasonics 35, 413 41. Johnson, W., Aul, B. A., an Alers, G. A. 1994. Spectroscopy of resonant torsional moes in cylinrical ros using electromagnetic-acoustic transuction, J. Acoust. Soc. Am. 95, 1413 1418. Jones, J. P., an Whittier, J. S. 1967. Waves at a flexibly bone interface, J. Appl. Mech. 34, 905 909. Kleczewski, D., an Parnes, R. 1987. Torsional ispersion relations in a raially ual elastic meium, J. Acoust. Soc. Am. 81, 30 36. Mal, A. K., an Xu, P. C. 1989. Elastic waves in layere meia with interface features, in Elastic Wave Propagation, eite by M. F. McCarthy an M. A. Hayes North-Hollan, Amsteram, pp. 67 73. Martin, P. A. 199. Bounary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers, J. Nonestruct. Eval. 11, 167 174. Miklowitz, J. 1978. The Theory of Elastic Waves an Waveguies North- Hollan, Amsteram. Pilarski, A., an Rose, J. L. 1988. A transverse-wave ultrasonic obliqueincience technique for interfacial weakness etection in ahesive bons, J. Appl. Phys. 63, 300 307. Rattanawangcharoen, N., an Shah, A. H. 199. Guie waves in laminate isotropic circular cyliner, Comput. Mech. 10, 97 105. Rokhlin, S. I., an Wang, Y. J. 1991. Analysis of bounary conitions for elastic wave interaction with an interface between two solis, J. Acoust. Soc. Am. 89, 503 515. Schwinger, J., an Saxon, D. S. 1968. Discontinuities in Waveguies Goron & Breach, New York. Thurston, R. N. 1978. Elastic waves in ros an cla ros, J. Acoust. Soc. Am. 64, 1 37. 1167 J. Acoust. Soc. Am., Vol. 107, No. 3, March 000 Berger et al.: Torsional waves in composite cyliners 1167