The General Problem of Elastic Wave Propagation in Multilayered Anisotropic Media

8/13/2019 The General Problem of Elastic Wave Propagation in Multilayered Anisotropic Media

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The general problem of elastic wave propagation in multilayeredanisotropic mediaAdnanH. NayfehDepartment f 4erospacengineeringndEngineering echanics, niversityof incinnati, incinnati, hio45221( Received 5 February1990; evised 6September 990; ccepted 0October1990)Exact analytical reatmentof the interactionof harmonicelasticwaveswith n-layeredanisotropic lates s presented. ach ayer of the plate can possessp to as ow as monoclinicsymmetryand thus allowing results or higher symmetrymaterialssuchas orthotropic,transverselysotropic,cubic,and isotropic o be obtainedas special ases. he wave s allowedto propagate longan arbitraryangle rom the normal o the plateas well as alonganyazimuthalangle.Solutions re obtained y using he transfermatrix method.According o thismethod ormalsolutionsor each ayerare derivedand expressedn termsof waveamplitudes.By eliminating heseamplitudes he stressesnd displacementsn onesideof the layer arerelated o thoseof the otherside.By satisfying ppropriate ontinuity onditions t interlayerinterfaces global ransfermatrix canbe constructed hichrelates he displacementsndstressesn one sideof the plate to thoseon the other. nvoking appropriateboundaryconditions n the platesouterboundaries largevarietyof importantproblems anbe solved.Of thesemention s madeof the propagation f free waveson the plateand the propagation fwavesn a periodicmediaconsistingf a periodic epetition f the plate.Confidences theapproach nd results re confirmed y comparisons ith whatever s available rom specializedsolutions.A varietyof numerical llustrations re included.PACS numbers: 43.20.Bi, 43.20.Fn, 43.30.Ma

INTRODUCTION

Studiesof the propagation f elasticwaves n layeredmediahave ongbeenof interest o researchersn the fieldsofgeophysics,coustics,ndelectromagnetics.pplications fthese tudiesnclude uch echnologicallymportant reas searthquakerediction, ndergroundaultmapping, il andgas exploration,architecturalnoise eduction,and the re-centlyevolving oncernof the analysis nd designof ad-vanced ibrousand layeredcompositematerials.Commonto all of these tudiess he degree f the nteractions etweenthe layers,which manifest hemselvesn the form of reflec-tion and transmission gents nd hencegiverise o geomet-ric dispersion.hese nteractions epend, mongmany ac-tors, upon the properties, direction of propagation,frequency nd number,and nature of the interfacialcondi-tions.Extensiveeviewof workson thissubject ntil the mid60'shasbeen eported n the literatureas s evidencedromthebook y Ewing t al.1andup to theearly80'sby Brek-hovskikh.For more ecentworks n hegeneral ubjectfwave propagation n layered media we refer the reader toRefs.3-5 asrepresentativeeferences.Typically a layeredmediumconsists f two or more ma-terial components ttachedat their interface n some ash-ion. A body made up of an arbitrary numberof differentmaterialcomponents nd whoseouterboundaries re eitherfree or supportedby semi-infinitemedia constitutes gen-eral layered medium. Often the abovedefinition s relaxed toinclude semi-infinite olids,single-layer lates,and twosemi-infinite olids n contactas degenerate ases f layeredmedia.

Most of the available iterature on layered media is re-stricted to the study of situationswhere the individual mate-rial layers re sotropic.Generally peaking,or wavepropa-gation n suchmedia, solutions re obtainedby expressingthe displacements nd stressesn each layer in terms of itswave potentialamplitudes.By satisfying ppropriate nter-facial conditions, haracteristicquations re constructedthat involve he amplitudes f all layers; his constituteshedirectapproach. he degree f complicationn thealgebraicmanipulation f theanalysiswill thusdepend pon he num-berof layers.For relatively ew ayers he directapproachsappropriate.However, as the numberof layers ncreaseshedirect approachbecomes umbersome, nd one may resortto the alternative ransfer propagator)matrix techniqueintroduced riginally y Thomsonand somewhatateronby Haskell and Gilbert and BackusfiAccordingo thistechnique neconstructshe propagationmatrix for a stackof an arbitrary numberof layersby extending he solutionfrom one ayer to the next while satisfyinghe appropriateinterfacialcontinuityconditions.To narrow down our discussion f problems elating tothe interactionof elasticwaveswith periodicmedia we notethatRytov utilizedhedirect pproach ethod ndderi edsomeanalyticalexpressionsor characteristic quations f aperiodicarray of two isotropic ayers.However,Rytov wasonly able o present olutionsor propagation itheralongornormalo the ayers. ayfehmderived n exact xpressionfor the characteristic quationof wavespropagating ormalto a periodic rrayof an arbitrarynumberof sotropicayers.Sve extendedhe esults f Rytov o anyobliquencidenceandderived hecharacteristicquationor theperiodic rray

1521 d. Acoust.Soc. Am. 89 (4), Pt. 1, April 1991 0001-4966/91/041521-11 $00.80 1991 AcousticalSocietyof America 1 521

ded 01 Jul 2011 to 150.140.149.189. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.


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of two sotropicayersn the ormof thevanishing etermi-nant of an 8X8 matrix.Schoenberg2 utilized he matrixtransfermethod ndpresentedesultsor obliquencidenceon an alternating luid-isotropic olid medium.Gilbert 3discussedhe utility of the propagatormatrix formalismofGilbertandBackusto the studyof wavepropagationnstratifiedmediaand obtained xplicitexpressionsor thesimplecaseof a periodically ayered luid.For casesnvolvingperiodic nisotropicmediawe men-tion that Yamada and Nemat-Nasser 4 extended the resultsof Sve o the case f orthotropic ayers.This resulted, ue otheadded oupling f thehorizontally olarized omponentof the wave, o the vanishing f the determinant f a 12 12matrix. n a recently eldspecial ymposiumnwavepropa-gation n structural omposites, several aperswerepre-sented n guidedwaves n laminatedanisotropic latesaswellason periodicallyaminated nisotropic edia.The pa-perspresentednclude,either ndividuallyor collectively,the most comprehensiveurveysof the relevant iterature.The most elevantworks or the presentwork are givenbyBraga ndHerrmann, TingandChadwick,6andNayfehet al.7Braga ndHermann5sedhepropagator atrixmethodand presentedesults or a periodicarray of an arbi-trary numberof orthotropic ayers.Their work is restrictedhowever o the caseof propagation long ayer interfaces(no oblique ncidence)and also for propagation long anaxisof symmetry f each ayer.This mplies hat their ayer-ing is restrictedsuch that the symmetryaxesof all layerscoincide. Thus, their model does not account for the cou-pling between he in-planemotion (SH) and that of the sa-gittalplane. ingandChadwick6alsousedhepropagatormatrix approach (in conjunctionwith a formalism devel-oped for steadyplane motionsof anisotropic odiesbyStroh ) andderived characteristicquationorharmonicwaves n periodically nisotropicmedia.Their analysiswascarried nitially for waves ravelingalong he layeringandthentheyoutlinedhow t canbe generalizedo an arbitrarydirectionof propagationn the sagittalplane.Nayfeh taL,17with hehelp f inear rthogonalrans-formations,ere ble o derive xact nalyticalxpressionsfor thereflectionoefficientroma fluid-loadedrbitrarilyoriented ultilayeredrthotropiclate. heapproachsedin Ref. 17 was introduced in their earlier works that dealtwithsingleayeranisotropiclates. 9,2ohe useof the ineartransformation,which facilitatesand leads o executioneaseoftheanalysis, asmotivated y the mportant bservationthat the wave vectors of the incident and reflected waves alllie n thesame lane. 7.19-21heanalysis as hereforeon-ducted n a coordinate ystemormedby incident nd re-flectedplanes ather than by material axes.In thispaperwe utilize combinations f the inear rans-formationapproach nd the transfermatrix methodand ex-tend the resultsof Reft 17 to the study of the interactionoffree harmonicwaveswith multilayeredanisotropicmedia.Our solutionswill be generaland include esultspertainingto several pecial ases. f thesewe mention: a) dispersioncharacteristicsor a multilayered lateconsistingf an arbi-trary numberof arbitrarily orientedanisotropicayers; b)dispersion f an infinitemediumbuilt from repetitionof the

multilayered late the resultingmediumwill thusbea peri-odic one with respect o the individualcomponentsf theplate); and (c) slownessesults or eitherhomogeneousrperiodicmedia. t is obvious hat the layeredplate consti-tutes the repeatingcell of the infinite medium.Besideshe advantages ainedby the useof the lineartransformationpproach, nother mportant eatureof ouranalysis oncernshemanner n which heoblique ropaga-tion direction s ntroduced nd he way t modifieshe crite-rion necessaryo insureperiodicity.f wedesignateheangle0 (measured rom the normal to the interfaces) to define hepropagation irection, hen this will lead to an explicitde-pendence f the characteristic quations pon0. Confidencein our results s established y comparingwith the limitedavailablenumericalexamples f the special asemodelsofRefs. 1 1 and 14.

I. FORMULATION OF THE PROBLEM

Considera plate consisting f an arbitrary numbern ofmonoclinic ayers rigidly bonded at their interfacesandstackednormal o the x3 axisof a globalorthogonalCarte-siansystem = (x,x2,x3)as llustratedn Fig. 1.Hence heplaneof each ayer s parallel o thex-x,_planewhich s alsochosen o coincidewith the bottom surfaceof the layeredplate. To maintain generalitywe assumeeach layer to bearbitrarily oriented n the x-x 2 plane. n order to be able todescribehe relativeorientationof the layers,we assign oreach ayerk, k = 1,2 ....n, a localcartesian oordinate x,)ksuch hat tsorigin s ocatedn thebottomplaneof the ayerwith (x) normal to it. Thus layer k extends from0< (x)


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2

3

4

n-2

n-In

XX2

FIG. 1.Modelgeometry.

r3

'Ctl, 0;2 C3 0 0Ct12 C52 C53 0 0 C56C;3 C53 C.3 0 0 C_

0 0 0 C,h C;s 00 0 0 Ci C 0

c', ck c; o o

e'e2e3

I.Yi2(3)

wherewe used he contracting ubscript otations 11,2-22, 3-33,4-23, 5-.13, and6-12 to relate jktoC, ij.k,l = 1.2.3 ndp,q- 1,2.... ). Herea,i,,o[sanduare the componentsf stress, train,and displacement,e-spectively,ndp'andck,are hematerial ensityndelasticconstants,espectively.n Eq. (3), Y,5' 2e (with %j) de-fines heengineeringhear traincomponents.

Sincer, e},andc,5.are ensorsndsincewearecon-ducting uranalysisn theglobal i coordinate,nyorthog-onal ransformationf theprimed o thenonprimedoordi-nates,.e., (x[) k to x,., hey ransform ccordingorr ... = fi,,,ifi,,o',, (4a)eop 15'o,15'pe,, (4b)c .... = fi,,,,fi,,jfio&,c.,, ()

whereri s he osinesf he ngleetween[ and , respec-tively.For a rotation f angleO n thex{ -x5 plane, he rans-formationensorie reducesoriO= --sin& cos& , (5)

0 0which, f appliedo Eq. (2) throughherelation f Eq. (3)yields the known constitutiverelations:

"/ /c,= ,;cl, o 0 G6 e,; o 0 cq ,/ o 0 (6)1523 J. Acoust. oc.Am.,Vol.89, No.4, Pt.1, April 991 Adrian . Nayfeh:Multilayeredrtisttopic edia 1523

ded 01 Jul 2011 to 150.140.149.189. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.


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wherehe ransformationelationsetweenheC,qandC qentries re isted n AppendixA. Notice that no matter whatrotationalangle 6 is used, he zero entries n Eq. (3) willremainzero n (6). In fact, even f the matrix of Eq. (3) isparticularizedo orthotropicmedia, ts transformedmatrixwill resemble that ofmonoclinic media. In terms of the rotat-ed coordinatesystemx,, the momentum equation trans-forms to

&r;j 8 2u;-- _p -- (7)axj atII. ANALYSIS

Substituting rom Eq. (6) into Eq. (7) results n a sys-tem of three coupledequations or the displacements , u2,and u3. If we now identify the plane of incidence o be thex x3,as n Fig. I then or an angleof incidence , we proposea solution or the displacements in the form(Ul,//2,U3) --- 1,V,W) U exp[i(x sin O+ ax 3 -- ct) ],

(8)where is the wave number, c is the phase velocity( = co/), w is the circular requency, is still an unknownparameter, nd V and W are ratiosof the displacementm-plitudes f u2 and u3 to that of u , respectively. otice that,althoughsolutions 8) are explicitly ndependent f x 2, animplicitdependences containedn the transformation. ur-thermore, notice the nonvanishingof the transversedis-placement omponent 2 n Eq. (8). This choice f solutionsleads o the threecoupledequationsK,,, (a) U, = O, m,n = 1,2,3, (9a)

where he summationconvention s mplied,K,,, is symmet-ric, namelyKin,, = K ..... andKll = Cll sin 19-pc2 d-C5515g,K2 = Cl6sin 0 + C452,K3 = (C3 + C55)a sin 0,K22= C66 in 19- pc + C442,K23= (C36d- C45)a sin 0,K33 C55 in 19--pc + C332. (9b)The existenceof nontrivial solutions or Ul, U2, and U3

demands he vanishingof the determinant n Eq. (9a), andyields he sixth-degree olynomial quationa 6 d- 41 4 d-A2 2 d-/13= 0, (10)

relating to c, where he coefficients , 42,and 43aregivenin Appendix B. Equation (10) admitssix solutions or a(having the properties)

For each , q = 1,2....6, wecanuse herelations9) andexpress the displacement atios V=U2/U;q andWq= U3q/U;q sKll(aq)K23(sq)-K13(aq)KI2(aq)12)VqK3(aqK:(aq- K:(CtqKe3(aK(aq)Ke3(aq)Ke(aq)K3(aq)13)Wq-K:(aqK33(aq--Ke3(aqKi3(

CombiningEqs. (12) and (13) with the stress-strainrelations 6), and usingsuperposition, e finally write theformal solutions or the displacementsnd stressesn theexpandedmatrix orm,

(14)

whereEq= e gctqx',

Dq= i(Cl3sin0 + C36 inOVq C33aWq,= + wqsin0) +

D3q i [C4s(aq - Wqsin19) CCtqVq],q = 1,2....6. (15)

Notice hat the specific elationsn the entriesof the squarematrix of Eq. (14), suchas W: = -- W and V6 = V, forexamples, an be seenby inspection f the ratios (12) and(13) in conjunctionwith the restrictions 11 .Equation (14) can be used o relate he displacementsandstressest (x). = 0 to those t (x), = d (). This sdoneby specializing14) to these wo ocations, liminatingthe commonamplitudesU, ..... U,6 and getting

P[ = AP F , k=1,2 .....n, (16a)where

pff = {[Ul,U2,U3,O.33,O.3,O.23T_+k (16b)defines he variablescolumn specialized o the upper andlower surfaces f the layer, k, respectively, nd

A. = XDX , (17)whereX is the 6 X 6 squarematrixof Eq. (14) andD is a6 X 6 diagonalmatrixwhose ntries reThe matrix / constituteshe transfermatrix for themonoclinicayer k. It allows he wave o be ncidenton thelayer at an arbitrary angle 0 from the normal x 3 or equiv-alently (xg) and at any azimuthalangle4. By applying heabove rocedureor each ayer ollowed y nvoking hecon-tinuity of the displacementnd stress omponents16b) atthe layer interfaces,we can finally relate the displacementsandstressest the top of the ayeredplate,x = d, to those tits bottom,x> = 0, via the transfermatrix multiplicationA =A.,,_ '"A (18)

resulting nP+ =.4P-, (19)

1524 J. Aoust. oc.Am.,Vol.89, No. 4, Pt. 1, April1991 AdnanH. Nayfeh:Multilayered nistropic edia 1524


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where owP * andP - are hedisplacementndstressol-umnvectorst the op,x3 = d, and ower, 3 = 0, of the otalplate,respectively.III. PROPERTIES OF THE TRANSFER MATRIX

The global ransfermatrix A has severalpropertieswhich, f exploited, aneaseheexecution f theanalysis ndlead o simple nalytical epresentationf the results.Beforeweproceedo listanddiscusshese roperties, e now ndi-cate hat suchpropertis re alsocharacteristicsf the trans-fer matrices f the ndividual ayers. n fact,sinceEq. (19)holds or any numberof layersn, then t holds or a singlelayer n particular seeEq. (16) ], and husA canbe repre-sented y Ak for k = 1,2 ....n. Accordingly,we hypothesizethat any generalpropertyof Ak is alsoa propertyof A.With thiswe now concentrate n listingand discussingproperties f the ndividual ransfermatrixA k-

(a) det (Ak) = 1. (20)This propertycanbe easilyprovenby employinghe well-known esult hat thedeterminant f the product s equal othe productof the determinants,which togetherwith therelation 17), implies hat

det(Ak = det(X )det(XF )det(D )= det(XX')det(Dk)= det(Dk). (21)

Thisconclusionanalsobearrivedat by noting hatAk andD are similarand hence heir determinantsre equal.2SinceDk isdiagonal,tsdeterminantsequal o theproductof itsentrieswhich,by employing 11 , is seen o beunity.(b) As a consequencef their similarity,A and D alsohave he sameeigenvalues.his means hat the six possibleeigenvaluessay q,q= 1,2....6) ofA aregiven y hediag-onal elements f Dk. By inspection e see hat these igen-valuesconsistof three pairs with the entriesof each pairbeing he inverseof each other. thus, if A,/L 3, and 25 areeigenvaluesof A k, so are 22= 1/21,24= 1/23, and26 = 1/25.(c) The resultsof (b), and the fact that the eigenvaluesofAff are he nversef thecorrespondingigenvaluesfAk, lead o theconclusionhat 4&and 4 ff have hesamesetof eigenvalues.As a consequencef ( a )- ( c ) and the definition 18 weconclude that:

(i) det(A) = det(A,)det(A_, )... det(A) = 1.(22)(ii) TheeigenvaluesfA - areequal o theeigenvaluesof.4. To show his, et usassumehat the eigenvalue f A is r.It followshen hat heeigenvaluefA - is 1/r.By substi-tuting from (17) into (18) and carrying he inverse ra we

getdet(X.D.. "'X2D2X2 XD,X - rI) = 0, (23a)det(X,D F X ,- XD c X ' "X.,

XD ff X,, - r-I) = 0. (23b)Now, using he fact hat the products f the two equal anksquare matrices M,M2 and M2M (although

MM2 M2M have hesame igenvalues,2bycyclic er-mutation, we can rewrite the relation (23a) asdet(XDX5 IX2D2X '"X,,D. X ff -- crI)= O.

(23c)By inspecting 23b) and (23c), we conclude hat (23c) canbe obtained rom (23b) by merely nverting he diagonalmatricesD and the eigenvalue . Since he entriesof D aremade up of pairs that are inverseof each other, then it isobvioushatD andD ff have hesame ntrieseigenval-ues). husweconcludehat heeigenvalues, q = 1,2....6and 1/aq constitutehesame et.(iii) The property escribednder ii) canonly mplythat he% consistsf three airswhereheentries f eachpair are the nverse f eachother. n our subsequentnalysiswe choose o arrange hesesix eigenvaluess a, l/a, ,1/or3,as, and 1/as(iv) In thedegenerateasewhere ll layers remadeupof the samematerial (but not necessarilyaveequal hick-nesses), and hesixvalues q are hesameor every .Now, substitutingrom (17) into (18) and recognizinghatX -_t = I (identity) or = 1,2....n, theglobalmatrixAcollapseso

A: XtDX t, (24)wherewe used he fact that hereX,, = X and

D = D.D,, _ "'D (:25)isa diagonal atrixwhosentriesregiven yexp(ictqd),q = 1,2 ....6 and d is the total thickness.Thus, we haveshown hat the global ransfermatrix correctly educesothe correspondingmatrix of the singlematerial plate whenall layer properties re the same.

(v) A very important consequencef the above istedproperties s the resulting elations hat exist between heinvarientserA. To this end, if we expand he characteristicequation der (A -- rrI) = 0, write it in terms of both theeigenvaluesrqand nvarients of A, q = 1,2....6 format,and compare the resulting expressions,we conclude thesymmetric elationsIs=I, I4=12, I=1. (26)

The result16 = 1 alsoconfirms he fact that det(A) = 1.Equation 19) will nowbe used o present olutionsora variety of situations.n the first, we consider singlecellmedium, namely a free n-layeredplate. The characteristicequation or sucha situation s obtainedby choosing := 0and invoking the stress-free pper and bottom surfaces nEq. (19) that lead to the characteristic quation

241`442 43`451 45 .453 0. (27)'461 `462 263A second mportant situation s that of a periodicmediumconsisting f a repetitionof the unit cell (plate). Here wegeneralize he classicalFloquct periodicity condition to re-quire

p + = p - eigao (28)which is consistentwith the formal solution (8). Combina-

1525 J. Acoust.Sec. Am., Vol. 89, No. 4, Pt. 1, April 1991 AdnanH. Nayfeb:Multilayered nistropicmedia 1525ded 01 Jul 2011 to 150.140.149.189. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.


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tionsof (26) and (19) yields he characteristic quationdet (A -- Ie ac)= 0. (29)

Equation 29) canalsobeexpanded ndwritten n termsofthe nvariants. of A which, fteralgebraic anipulationreduce tocos[3dcos0 ] - I cos 2d cos0 ] + 12cosiedcos0 ]

- 13/2 = 0. (30)In terms fthe ndividualntriesAofA theinvariantslqregiven n Ref. 22 by

( -- 1)rI. = i < i'''


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Wqpea-C, in0- CCtq (38)(Cl3+ C)Ctqsin0Diq= i( Ci3sin0 + C33ctqq ,Dzq ---Css(ctq- IVq in 9). (39)

Accordingly,ormalsolutionsor propagationlong n axisof symmetry f an orthotropicmaterial regivenby

(40a)where

E = eg", q --- ,2,3,4. (40b)Onceagain,Eq. (40) canbe used o relate hedisplace-mentsnd tressestx}kl = 0 to hosetx}k)= d (n) orsucha restricteddirectionof propagation. his can be donebyspecializingq. (40) to x3n 0 and o x}l= d (nandeliminatinghecommon mplitude olumn f Ul, U12,U3,and U4 esultingn an equation imilar o (27) with X& nowgivenby the 4X4 matrix of Eq. (40) and D is again he4X4diagonalatrix hosentriesregivenseg"t'",q = 1,2,3,4or heaq definedn Eq. (35). Henceheglobaltransfermatrix s againconstructed y multiplication f theindividualmaterial transfermatrices.The propertiesof theresulting X 4 matrix (here eferred o asA' to avoidconfus-ing t with 4 of themonoclinicase)are denticalwith hoseof the 6X6 matrixA of (18), except or the fact that`4 hasonly two pairsof eigenvaluesather than three.Utilizing`4 , thecorrespondingharacteristicquationsfor the reewaves n a singleayeredplateandon theperiod-ic media or propagationlong n axisof symmetry f eachlayerare given, espectively,yA A3=0 (41)

andcos[2dcos0 ] --I cos['dcosg ] +I/2 =0. (42)Vl. DISCUSSIONS AND NUMERICAL ILLUSTRATION

In this section,we illustrate he analytical esults 27),(30), (41), and (42) with a limited selection f numericalexamples.While thecases e present erearecertainly ypi-cal, they are by no means xhaustive f the varietyof thephenomenology ontained n the analysis.Once the numberof layers, heirproperties,ndgeometric tacking respeci-fied,we present ur numerical esults n two categories.nthe first, we demonstratehe variations f phasevelocitycwith angleof incidence for specifiedrequenciesand ori-entation ngles; his seffectively formof demonstrationof the dependence f wave ront (inverseof slowness curveswith frequencyor specified rientation ngles.n the sec-ond, we present hase elocitydispersion urvesplottedasfunctions f theproduct f frequencyndunitcell hickness,namelyFd, for specified ngles f incidence . The proper-

tiesof a representativerthotropicmaterial hat we used nour calculations re given n GPa by C h = 128, C ] = 7,Ch=6, C=72, C=5, C3=32, C=18,C[s = 12.25, 6 = 8,andp= 2 g/cm . Here, ifferentay-ers can be constructedrom this chosenmaterial by assign-ing appropriateotational ngles. his choice s not restric-tive and has he advantage f saving pace y not having olist differentmaterialproperties. hus, for examples com-binationof 0 ,90 ,60 ,and -- 60 ayupconstitutes four-layered ell whereas combination f 0 0,0 ,0 ,and0 cellsdefines single omogeneousaterial.Withoutany oss ngenerality he thickness of the representativenit cell iskeptconstant, nd ts constituentslayers)are assignedol-ume fractionsadding o unity.To show he extentof generality n the results,we nowdiscusshe case n which all layersare the same. This isexpectedo undoubtedlyesultn a descriptionf thebehav-ior of single omogeneousnisotropic aterials. s wasdis-cussed arlier, he global ransfermatrix for sucha situationcollapseso the form given n (24}. Using hismatrix, o-getherwith the fact that for thiscaseD and 4 are similar,dictates hat the characteristic q. (29) admits he solution

a=cos 0, q=1,2 .... , (43)which alsospecializeshe formal solution 8) to the oneappropriateor thesingle omogeneousedium.With ref-erence o Eq. (10) and or a fixedO, he results 43) admitthree roots or the phasevelocityc correspondingo onequasilongitudinalnd wo quasishear otions. hus, or avariable ,Eq. (43) describeshevariation f he hree hasevelocities ith the ncidentangleand hence onstitutewavefront curves. or thisspecializedinglemedium ase, hesecurveswill be ndependent f frequency, owever.For an sotropicmaterial, or example, q. (10) uncou-plesandgives

ct= ea/c - sin 0; a z = c2/ear- sin 19, (44).5whereCLandCr are he ongitudinalndshearwavespeedsin themedium espectively.hus,combinationf Eqs. 43}and 44) giveshe oots = cL andc = CTyieldingwocon-centtic pherical ave rontcurves s s expected.or theanisotropic ase,however, he threesolutions ill be cou-pled esultingn nonsphericalave ronts.For the ayeredmediacase, he situations muchmorecomplicatedue o the dependencef thephase elocities,not onlyon the ndividualayerproperties,ut most mpor-tantlyon hewave umber or frequency (morepreciselyon the parameter d). However, or a fixed requency, ecan constructwave front curvesand hence,by varying thefrequencyn a discretemanner, emonstratefrequency-dependentdispersive"haracter f the wave ronts.ForFd = 0 MHz ram, the curveswill thus constitute he wavefronts or an effective omogenized ediumwhoseproper-tiesare volume ractionweighted roperties f the individ-ual layers.Conventionalispersionurvesn the formsof varia-tions of wave velocities with wavenumber can also be con-structed sing itherEq. (29) or (30). This s done,how-ever, for fixed valuesof 8. Here we mention hat further

1527 J. Acoust. oc.Am.,Vol.89, No.4, Pt. 1, April1991 AdnanH. Nayfeh:Multilayerednistropic edia 1527aded 01 Jul 2011 to 150.140.149.189. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.j


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confidencen our analyticaland computational rocedurewas established y reproducinghe numerical esultsofRefs. 6 and 9 that constitutespecialcasesof the presentwork.Sample xamples, hichdemonstratehedependenceof such curveson the number of layers and their orienta-tions, regiven or he epresentativengle, = 45 , n Figs.2-5. They correspondo (0 ,90 ); (0 ,90 ,45 , -- 45 );(60 ,0 , -- 60 ); and (0 , 15 , 30 ,45 , 60 , 75 , 90 ),periodicmedia layup configurations,espectively. oteonce gain hatall of these lates ave hesamehickness.In theseigureshephase elocity sgiven y km/s and hewave umber' by mm .Closeexamination f these igures eveals everal nter-esting eatures. t the zerowave-numberimit, each iguredisplayshreevaluesof wavespeeds orrespondingo onequasilongitudinalnd wo quasishear.t is obvioushat thelargest aluecorrespondso thequasilongitudinalode.Atrelatively ow values f the wavenumber, ittle changes seento take place n thesevalues.As increases, ther higher-ordermodes ppear; neof these eemso beassociateditha rapid changen the slopeof the quasilongitudinal ode.Furthermore, heisotropiclikebehaviorsuggestedbytheclosenessf the two quasitransverse odesof Figs.4 and 5 ascompared ith thoseof Fig. 3 ), isworthcommenting pon.It is consistentwith the static predictionof the quasi-iso-tropy of the (0 ,90 ,45 , -- 45 ) and (60 ,0 , -- 60 )layups.In Fig. 6(a)-(c) we depict, for the selectedvaluesFd = 0,2, and 4 MHz mm, wave front curves n the K-K 2planewhereKl = c sin 0and K 2 = CCOS, using (60 ,0 ,--60 ) layup periodic medium as a representative ase.These curves demonstrate the inverse of the slowness curvesas unctions f frequency ndhencedisplayanddemonstratewavefrontdispersion ehavior.The complicated eatures

4

I I I I I I I I0 i 2 3 4. .5 6 7

FIG. 3. Sameas Fig. 2 with (0 90 ,45 , -- 45 ) layup.

shown n Fig. 6(c) and to a lesserdegree n 6(b) are due tomultivaluedbehaviorshown n Fig. 4 especially t Fd = 4MHz mm broughtaboutby the presence f the higher-ordermodes. Notice in contrast that the "clean" behavior dis-played n Fig. 6(a) reflects he variationsof effectivewavespeed (namely at Fd = 0) with the angleof incidence.To furthershow he versatility f the analyses e alsogenerate, sing he characteristic q. (27), the dispersioncurves f Fig. 7 for freewaveson a finite hickness ulti-

C4

I' I I I I II 2 3 5 6 7 8

FIG. 2. Variation of phasevelocityc with wavenumber for angleof inci-dence0 = 45 ; 0 ,90 ) layup.

4

I I I I I I I I0 t 2 3 4 5 6 7 8

FIG. 4. Sameas Fig. 2 with (60 ,0 -- 60 ) layup.1528 J. Acoust.Soc. Am.,Vol. 89, No. 4, Pt. 1, April1991 AdnanH. Nayfeh:Multilayered nistropicmedia 1528


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4

3

IL I I I I I I IO I 2 3 5 ? 8

FIG. 5. SameasFig. 2 with (0 , 15,30 ,45 ,60 ,75 ,00 ) layup.

layered plate consisting f ( --60 , 0 , 60 ) layup. Thecurvesdisplayedon this figureare typical of free waves nanisotropiclates.'711.GONC/U$1ON

We have derivedanalyticalexpressionshat are easilyadaptable o numerical illustrationsof the interactionofelasticwaveswith multilayeredanisotropicmedia. A plateconsisting f an arbitrarynumberof layerseachpossessingas ow asmonoclinic ymmetry s chosen sa representativecell of the medium.Waves are allowed to propagate longarbitrary directions n both azimuthal as well as incidenceplanes.Characteristic quationsor a varietyof physical ys-temsarediscussed.hese nclude hecases f propagation ffreeharmonicwaves n a multilayeredplatesand n periodicmedia constructed rom a repetitionof the layered plate.Results n the formsof dispersion urves re given or severalrepresentativeayering.Wave ront curves or fixed requen-ciesare also ncluded o demonstrateheir dispersive eha-viors.

K2

'" "'61 ..'"a)K1

K

, KI9

K

, KIIO

FIG. 6. (a) Wavertonicurves or Fd = 0 MHz mm and a (60 *, 0 , -- 60 )layup. b) Same s (a) repeatedt Fd = 2 MHz ram. c) Same s (a) re-peatedat Fd = 4 MHz mm.

1529 J. Acoust.Soc. Am., Vol. 69, No. 4, Pt. 1, April 1991 AdnanH. Nayfeb:Multilayered nistropicmeclla 1529aded 01 Jul 2011 to 150.140.149.189. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.j


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3

I I I I I I I0 I 2 3 4 5 6 7 8

FIG. 7. Variation f phase elocity withwavenumber' for a (60 ,0 ,60 ) layup ree plate.

ACKNOWLEDGMENTThis work hasbeensupported y AFOSR.

APPENDIX ACombination of the transformation matrix (5) with the

constitutive elations (3) yields the following transformedproperties:

Cii -Ci2 Ci3 Ci6

C22:C23:C26

C33C36=C4s=C44=C55=C66=

C ;, G 4 q- C2S4 q-2(C '2+ 2C,652G2,(C;i q- C2 -4C,6) S2G2q- C2(S4q- G4),C3G2q- C3X2,( C iI -- C '2 - 2C$,6)SG+ (C '2 C2 + 2C,oGS3,c;,s 4 + 2(ci: + 2C,6)S2G- + C2G4,C3G2q-(C; - C'2 2C;6)GS3q- (C ;2 -- C-52 - 2C;6 SG3,C;3,(C3 -- C'3 SG,( C 4 -- C )SG,C402 q- C55 2,C;5G 2 q- C4S2,(CI + C2 -- 2C'2 - 2C,6)S2Gq- C'16(84q-G4),

where G = cos& and S = sin .

APPENDIX BThe variouscoefficients fEq. (10) are givenby

A ---(0,1C33044- C3 44 -2013036045- 20,3044055-013025- 20,6033045-C3305s066- C]6C55)sinO2 2A2= [ (C 03306 - C, C 6 - 2C, C3605 C, GCss C, C4s C ,3G6 + 2C,,C,60362C,3C,605

- 2030506 - C6G, + 2C,6C36Css)sin0 - (C,,033 C,,G4 -- C3 - 2C,,Css 20,605 36

a3= [ (c,,GsG- cGs) sine - (c, ,%. + c,,Go+ c + G.G.)pc2sin 0+ (Cll + 55 + C66)Pc4 in2 --p3c6]/A,with

W. M. Ewing,W. S. Jardel:sky,nd F. Press, lasticWavesn LayeredMedia(McGraw-Hill, New York, 1957).:L. M. Brekhovskikh,Wavesn LayeredMedia (Academic,New York,1966).'B. L. N. Kennett,SeismicWavePropagationn StratifiedMedia (Cam-bridgeU. P., Cambridge,UK, 1983).40. J. Fryer and L. N. Frazer, "SeismicWaves n StratifiedMedia--ll.Elastodynamic igensolutionsor SomeAnisotropic ystems," eephys.J. R. Astron. Sec. 91, 73-101 (1987).SA.K. Mal andT. C. T. Ting Eds.), Wave ropagationnStructural om-posites American Societyof MechanicalEngineers,New York, 1988),AMD-Vol. 90.

'W.T. Thomson,Transmissionf ElasticWaves hroughStratifiedSolid medium,"J. Appl. Phys.21, 89 (1950).N. A. Haskell, The Dispersion SurfaceWaves n MultilayeredMe-dia," Bull. Seismol. Sec. Am. 43, 17 (1953).F. Gilbert and G. E. Backus, PropagatorMatrices n ElasticWave andVibration Problems," Geophysics 1, 326-332 (1966)."S.M. Rytov,"Acoustical roperties ra Thinly LaminatedMedia," Phys.Accoust. 2, 68-80 (1956).mA.H. Nayfeh, Time-HarmonicWavesPropagation ormal o the Lay-ersoMulti-layered eriodicMedia," J. Appl. Mech. 42, 92-96 (1974).C. Sve,"Time-HarmonicWavesTravellingObliquelyn a PeriodicallyLaminated Medium," J. Appl. Mech. 38, 677-682 ( 1971 .

1530 J. Acoust.Sec. Am.,Vol. 89, No. 4, Pt. 1, April1991 AdrianH. Nayfeh:Multilayered nistropicmedia 1530


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2M.Schoenberg,Wave ropagationnAlternatingolid ndFluid-Lay-ers," Wave Motion 6, 303-320 (1984).3K.E.Gilberg,A PropagatoratrixMethodorPeriodicallytratifiedMedia," J. Acoust. SOc.Am. 73, 137-162 (1983).4M.YamadandS. Nemat-Nasser,Harmonic aves ithArbitraryPropagation irection n layeredOrthotropicElasticComposites,".Composite ater, 15, 531-542 1981 .A. M. B.BragandG. Herrmann,PlaneWavesnAnisotropicayeredComposites,"n Ref. 5, pp.69-80.r. C. T. TingandP. Chadwick,Harmonic avesn PeriodicallyLayeredAnisotropic lasticComposites,n Ref. 5, pp. 53-68."A. H. Nayfeb, . W. Taylor, ndD. E. Chimenti,Theoretical avePropagationn Multilayered rthotropic edia," n Ref.5, pp. 17-28.SA. . Stroh,DislocationsndCracksnAnisotropiclasticity,"hilos.Mag. 3, 625-649 (1958).mA.H. Nayfeb ndD. E. Chimenti,Ultrasonic aveReflectionrom

Liquid-Loaded rthotropic lateswith Pdpplicationso FibrousCompo-sites," . Appl. Mech. 55, 863 (1988).2D.E. Chimenti ndA. H. Nayfeb, Experimentalltrasonic eflectionandGuidedWavePropagationn FibrousCompositeaminates,"n Ref.5, pp. 29-38.2A.H. Nayfeb ndD. E. Chimenti,FreeWavePropagationn Plates fGeneralAnisotropicMedia," J. Appl. Mech. 56, 881 (1990).22j. N. Franklin,Matrix Theory Prentice-Hall, nglewood liffs,NJ,1968).23D. . ChimentindA. H. Nayfeh, Ultrasoniceflectionndguided avepropagationn biaxially aminatedcomposite lates," o appear n J.Acoast. SOc. Am. 87, 1409-1415(1990).24A.H. Nayfeh, ThePropagationf Horizontally olarized hearWavesin MultilayeredAnisotropicMedia," J. Acoust.Soc. Am. 88, Z007(1989).

1531 J. Acoust. oc.Am.,Vol.89, No.4, Pt. 1, April 991 Adrian l_ layfeh: ultilayerodnistropic edia 1531

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The General Problem of Elastic Wave Propagation in Multilayered Anisotropic Media