The general problem of elastic wave propagation in multilayered anisotropic mediaAdnanH. Nayfeh Department of•4erospace Engineering andEngineering Mechanics, Universityof Cincinnati, Cincinnati, Ohio 45221 ( Received 25 February1990; revised 16September 1990; accepted 10October1990) Exact analyticaltreatmentof the interactionof harmonicelasticwaveswith n-layered anisotropic platesis presented. Each layer of the platecan possess up to aslow asmonoclinic symmetryand thus allowing resultsfor higher symmetrymaterialssuchas orthotropic, transversely isotropic, cubic,and isotropicto be obtainedasspecial cases. The waveis allowed to propagate alongan arbitraryanglefrom the normalto the plateaswell asalongany azimuthalangle.Solutions are obtained by using the transfer matrix method.According to this methodformalsolutions for eachlayerare derivedandexpressed in termsof waveamplitudes. By eliminatingtheseamplitudes the stresses and displacements on onesideof the layer are relatedto those of the otherside.By satisfying appropriate continuity conditions at interlayer interfaces a globaltransfer matrix canbe constructed whichrelates the displacements and stresses on onesideof the plate to thoseon the other. Invoking appropriate boundary conditions on the plates outerboundaries a largevarietyof importantproblems canbe solved. Of these mentionis madeof the propagation of freewaves on the plateand the propagation of waves in a periodic mediaconsisting of a periodic repetition of the plate.Confidence is the approach andresults are confirmed by comparisons with whatever is available from specialized solutions. A varietyof numericalillustrations are included. PACS numbers:43.20.Bi, 43.20.Fn, 43.30.Ma INTRODUCTION Most of the availableliterature on layeredmedia is restrictedto the study of situationswhere the individual mate- Studies of the propagation of elasticwavesin layered mediahavelongbeenof interest to researchers in thefields of geophysics, acoustics, andelectromagnetics. Applications of these studies include such technologically important areas as earthquake prediction, underground faultmapping, oil and gasexploration,architecturalnoisereduction,and the recentlyevolving concern of the analysis and design of advancedfibrousand layeredcomposite materials.Common to all of these studies isthedegree of theinteractions between the layers,which manifestthemselves in the form of reflection and transmission agents and hencegiveriseto geometric dispersion. These interactions depend, among manyfactors, upon the properties, direction of propagation, frequency and number,and natureof the interfacialconditions. Extensive review of worksonthissubject until the mid 60'shasbeenreported in the literatureasis evidenced from rial layers areisotropic. Generally speaking, for wavepropagation in suchmedia, solutions are obtainedby expressing the displacements and stresses in each layer in terms of its wavepotentialamplitudes. By satisfying appropriate inter- facial conditions, characteristic equations are constructed that involvethe amplitudes of all layers;thisconstitutes the directapproach. The degree of complication in thealgebraic manipulation of theanalysis will thusdepend uponthenumberof layers.For relatively few layersthedirectapproach is appropriate.However, as the numberof layersincreases the direct approach becomes cumbersome, and one may resort to the alternativetransfer(propagator)matrix technique introduced originally by Thomson 6 andsomewhat lateron by Haskell 7 and Gilbertand Backusfi According to this technique oneconstructs the propagation matrix for a stack of an arbitrarynumberof layersby extending the solution from onelayer to the next while satisfying the appropriate interfacialcontinuityconditions. To narrow down our discussion of problemsrelatingto the interactionof elasticwaveswith periodicmediawe note thebook by Ewing etal.1andupto theearly80'sby Brekhovskikh. 2Formore recent works onthegeneral subject of wave propagationin layered media we refer the reader to Refs.3-5 asrepresentative references. Typically a layeredmediumconsists of two or more material components attachedat their interfacein somefashion. A body made up of an arbitrary numberof different materialcomponents and whose outerboundaries are either free or supported by semi-infinite media constitutes a general layeredmedium. Often the abovedefinition is relaxedto thatRytov 9utilized thedirect approach method andderi red someanalyticalexpressions for characteristic equations of a periodicarray of two isotropiclayers.However,Rytov was only ableto present solutions for propagation eitheralongor normal to thelayers. Nayfeh mderived anexact expression for thecharacteristic equation of waves propagating normal to a periodic arrayof anarbitrarynumber of isotropic layers. includesemi-infinite solids,single-layer plates,and two semi-infinite solidsin contactas degenerate cases of layered media. Sve • extended theresults of Rytov toanyoblique incidence andderived thecharacteristic equation for theperiodic array 1521 d. Acoust.Soc. Am. 89 (4), Pt. 1, April1991 0001-4966/91/041521-11 $00.80 • 1991 Acoustical Societyof America 1!521 ownloaded 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 •7Braga andHermann•5 used thepropagator matrix methodand presented results for a periodicarray of an arbitrary numberof orthotropiclayers.which facilitatesand leadsto executionease that the wave vectors of the incident and reflected waves all Refs.5•. --p at-' (1) and. FORMULATION OF THE PROBLEM Braga andHerrmann.149.Soc. Equivalently. Of these we mention:(a) dispersion With respect to the primed coordinate system(k. withsingle layeranisotropic plates.. Our solutions will be generaland includeresultspertaining to several special cases. This resulted. anotherimportant featureof our analysis concerns themanner in whichtheoblique propagationdirection isintroduced andtheway it modifies the criterionnecessary to insure periodicity. Redistribution subject to ASA license or copyright.tion angles • are specified the geometry of the platewill be that formations. If wedesignate theangle 0 (measuredfrom the normal to the interfaces) to definethe of Sveto the case of orthotropic layers. several papers werepresented on guidedwaves in laminated anisotropic platesas wellasonperiodically laminated anisotropic media. 19.. Confidence in our resultsis established by comparingwith the limited availablenumericalexamples of the special casemodels of Refs. and (c) slowness results for eitherhomogeneous or periodicmedia. 1 1 and 14. • TingandChadwick. the most comprehensive surveys of the relevantliterature.where d• isitsthickness. was motivated by theimportant observation stresses (rr•)• ofoneface of theplate k to those oftheother. •6andNayfeh et al.Gilbert•3 discussed the utility of the propagator matrix formalismof GilbertandBackus 8 to the study of wavepropagation in stratified mediaand obtained explicitexpressions for the simplecaseof a periodically layeredfluid.. I. a localcartesian coordinate (x.17 withthehelp oflinear orthogonal trans.Their work is restricted howeverto the caseof propagation alonglayer interfaces (no obliqueincidence)and also for propagation alongan axisof symmetry of eachlayer. To maintain generalitywe assumeeach layer to be arbitrarily orientedin the x•-x 2 plane. and oftheanalysis. Pt. Acoust.In order to be able to describe the relativeorientation of the layers. see http://asadl. The mostrelevant worksfor the present work are givenby propagation direction. This matrix relatesthe displacements (u. April 1991 c• = c. Theapproach used angle0 measured from the normalx3. Thus.n.we assign for eachlayerk.Thisimplies thattheirlayering is restrictedsuch that the symmetryaxesof all layers coincide. Besides the advantages gainedby the useof the linear transformation approach. Am.onceall orientaNayfeh etaL. characteristics for a multilayered plateconsisting of an arbitrary numberof arbitrarily orientedanisotropic layers. TingandChadwick •6also used thepropagator such 0< (x•)• <d (k•. For cases involving periodic anisotropic mediawe mention that Yamada and Nemat-Nasser •4 extended the results multilayered plate(theresulting medium will thus bea periodic one with respect to the individualcomponents of the plate).2o Theuse of thelinear transformation.(b) dispersion of an infinitemediumbuilt from repetition of the 1522 J.Without any lossin generalitywe shallassume a plane wave propagates in the x•-x 3 plane at an arbitrary forthereflection coefficient froma fluid-loaded arbitrarily oriented multilayered orthotropic plate.Their analysis was carriedinitially for waves traveling alongthe layeringand thentheyoutlinedhow it canbe generalized to an arbitrary directionof propagation in the sagittalplane.. 89. No.•)k with (x•)• normal to it.the plate occupies the region0 <x 3<d.)•.140. k = 1.189.It is obvious that the layeredplate constitutesthe repeatingcell of the infinite medium.x2. Thus layer k extends from thatitsoriginislocated in thebottomplane of thelayer gittalplane. liein thesame plane.to the vanishing of the determinant of a 12• 12 matrix. 1. According to this Stroh ••) andderived a characteristic equation forharmonic notationthe total thickness of the layeredplated equals to the sum of the thicknesses of its layersand hence.org/journals/doc/ASALIB-home/info/terms . 17. dueto theadded coupling of thehorizontally polarized component of the wave.e• • dia (2) by the specialized expandedmatrix form of monoclinicme- Adnan H. Vol.Schoenberg •2 utilizedthe matrix transfer method andpresented results for oblique incidence on an alternating fluid-isotropic solid medium. Hence. from the generalconstitutive relationsfor anisotropic media.)•. 4.. Nayfeh: Multilayeredanistropicmedia 1522 nloaded 01 Jul 2011 to 150. in Ref.of twoisotropic layers in theformof thevanishing determinantof an 8X8 matrix.19-21 Theanalysis was therefore conducted in a coordinate system formedby incident and reflectedplanesrather than by materialaxes..2.then this will lead to an explicitdependence of the characteristic equations upon0. were able to derive exact analytical expressions defined.Hencethe planeof eachlayerisparallelto thex•-x. and usingthe standardsummationconvention on repeatedindicesthe elasticfield equations of layer k are givenby the momentumequation -- Ox. In thispaperwe utilizecombinations of thelineartransformationapproach and the transfermatrix methodand extend the resultsof Reft 17 to the studyof the interactionof free harmonicwaveswith multilayered anisotropic media. The paperspresented include.either individuallyor collectively. their model does not account for the cou- pling between the in-planemotion (SH) and that of the samatrix approach(in conjunctionwith a formalismdeveloped for steadyplane motionsof anisotropic bodiesby Considera plate consisting of an arbitrary numbern of monoclinic layers rigidly bonded at their interfacesand stacked normalto the x3 axisof a globalorthogonal Cartesiansystem x• = (x•.In a recently heldspecial symposium onwavepropa- gation in structural composites 5. 15-17 in order to construct a transfer matrix for each layer k. theorientation of thek th layerin thexi space canbedescribed bya rotationof an angle&• between(x'•)• and x•. 17 was introduced in their earlier works that dealt In this sectionwe follow the analytical procedureof waves in periodically anisotropic media. 1.x3)asillustrated in Fig._ planewhichisalso chosento coincidewith the bottom surfaceof the layered plate. (3) yields the known constitutiverelations: wherewe usedthe contracting subscript notations 1• 11.. (5) c'..c•..2 . andu• are the components of stress.•.ifi.13. respectively.Vol.andc.• o o which. 0.•(ij.2.4-•23.h 0 0 0 C. e}•.. i.org/journals/doc/ASALIB-home/info/terms. respectively.5' = 2e• (withi%j) defines theengineering shear straincomponents...X• X2 ! 2 3 FIG. c. 4 n-2 n-I n Since rr•.•o'. April 1991 "•/ /c.89. (6) Adrian H.l = 1.. if applied to Eq. = fi.Am.anddisplacement. o 0 G6 / e. 2-•22..i....k. In Eq. 3-•33.•. Pt.4.1.1.•3 0 0 C56 C_• 6 0 e'• e•2 whereri 0isthe cosines ofthe angle between x[and xj..fi. 1.. Herea. andp' and c•k.Yi2 (3) riO= --sin& 0 cos& 0 ..... / o0 .140.. No. 5-. and6-•12 to relate c•jkt to C•. = fi.• ck c.e.j ..a• aretensors andsince weareconducting ouranalysis in theglobal xi coordinate. arethematerial density and elastic constants. 6)..189.jfio•&. Y. (4a) eop = 15'o•. strain. (4b) (•) 'Ctl.... see http://asadl.cl. 1523 J..theytransform according to •r•3 rr .2 C. Redistribution subject to ASA license or copyright. respectively. Soc. .s Ct12 C52 C53 C53 C.15'p•e.149. thetransformation tensor fie reduces to 0 0 0 Ci• C• 0 e•3 I.= c..3 0 C•3 0 0 0 C. Nayfeh: Multilayered artisttopic media 1523 ownloaded 01 Jul 2011 to 150.o[s . o 0 cq . (3).3 and p.q.Modelgeometry. (x[) k to x. anyorthogonaltransformation of theprimed to thenonprimed coordinates.5.. (2) through therelation of Eq.Fora rotation of angle Oin thex{ -x5plane. Acoust. April1991 AdnanH.23 ] T_+ }k (9b) (16b) K33 = C55 sin 219--pc 2+ C33 a2. _p -axj at (7) II.C45)a sin 0...6. k.qandC •q entries are listedin AppendixA. •'"A• (18) (19) 1524 J. 13. (3) will remainzero in (6). and yieldsthe sixth-degree polynomial equation A•.K•:(Ctq )Ke3(a •) ' Wq --K•:(aq )K33(aq ) --Ke3(aq )Ki3(• q ) 1524 Kll(aq)K23(øsq) --K13(aq)KI2(aq) (12) K•(aq)Ke3(aq) -K•e(aq)K•3(aq) (13) alently (xg)• and at any azimuthalangle4. eliminating the commonamplitudesU.. (10) relating a to c.//2. and demands the vanishing of the determinantin Eq.W• and V6 = V•. respectively. q = 1.its transformed matrix will resemble that ofmonoclinic forms to CombiningEqs. It allowsthe waveto be incidenton the layer at an arbitrary angle 0 from the normal x 3 or equiv- For each a•.• .. for examples.j 8 2u. K•3 = (C•3 + C55)a sin 0. and usingsuperposition. The existenceof nontrivial solutionsfor Ul. notice the nonvanishing of the transversedisplacement component u2in Eq..43aregiven in AppendixB.O.. Soc. (12) and (13) with the stress-strain relations(6). Equation (10) admitssix solutions for a (having the properties) whereX• is the 6 X 6 square matrixof Eq. wecanuse therelations (9) and express the displacementratios V•=U2•/U. (7) resultsin a system of three coupledequations for the displacements u•. m. and u3. Nayfeh: Multilayered anistropic media nloaded 01 Jul 2011 to 150. (6) into Eq. Thischoice of solutions leadsto the threecoupledequations K.4P-.. Furthermore. . I thenfor an angleof incidence 0. q = 1. ANALYSIS Substituting from Eq..42. respectively. (15) Noticethat the specific relations in the entries of the square matrix of Eq.41 a4 d-A2 a2 d-/13= 0. In terms of the rotat- ed coordinatesystemx. andstresses at (x•)•.. is symmetric. (8).where thetransformation relations between theC. namelyKin.2. (3) is particularized to orthotropic media. (14). (8) where where • is the wave number. and Eq =e igctqx'. the momentum equation trans- -- &r. = d (•). Pt. No.x• = d.n.. we can finally relate the displacements andstresses at thetopof thelayered plate. and U3 definesthe variablescolumn specialized to the upper and lower surfaces of the layer.V.ct) ].asin Fig.3. via the transfermatrix multiplication A =A. to those at its bottom.and. In fact.4.K. = X•D•X • •.. see http://asadl.n = 1.Am.Vol. we propose a solutionfor the displacements u• in the form (Ul._ resultingin P+ =.. and V and W are ratiosof the displacement amplitudes of u2 and u3 to that of u •...•.. w is thecircularfrequency.U2. can be seenby inspection of the ratios (12) and (13) in conjunction with the restrictions ( 11). U2. where (16a) K22 = C66 sin 219 -. K23= (C36d. (a) U. If we now identify the planeof incidence to be the (14) x •x3. wherethecoefficients A •.33. a is still an unknown parameter.89..O. (9a).= K . (17) a 6 d-.U3)•--( 1. = 0 to those at (x•).org/journals/doc/ASALIB-home/info/terms .6 and getting P[ = A•P F ..q and Wq= U3q/U. By applyingthe above procedure for eachlayerfollowed by invoking thecontinuity of the displacement and stress components (16b) at the layer interfaces. Equation(14) can be usedto relatethe displacements Kll = Cll sin 219 --pc2d-C5515g 2.149.Wq sin19) + C•CtqVq]. althoughsolutions (8) are explicitlyindependent of x2. Redistribution subject to ASA license or copyright..189.O. A½oust.2 . media.. k=1. c is the phase velocity ( = co/•).. pff = {[Ul. K•2 = Cl6sin 2 0 + C45 a2.W) U exp[i•(x• sin O+ ax 3 -. evenif the matrix of Eq.. U.. = O. u2. Notice that no matterwhat rotationalangle.6.6 is used.2 . = + wqsin0) + D3q = i• [C4s(aq d.q as Vq =K•3(aq )K:•(aq ).U3. (14) andD• is a 6 X 6 diagonal matrixwhose entries are The matrix •/• constitutes the transfermatrix for the monoclinic layer k.pc 2 + C44 a2.. we finally write the formal solutions for the displacements and stresses in the expanded matrixform.. (9a) wherethe summationconvention is implied. suchas W: = -. 1..the zero entriesin Eq.x> = 0.140.2 . an implicitdependence iscontained in thetransformation. Noticethat. D•q= i•(Cl3sin0 + C36 sinOVq + C33a qWq ).... Thisis done by specializing (14) to these twolocations. since Eq..•r-•I) = 0.In our subsequent analysis we choose to arrangethesesix eigenvalues as a•.. then it holdsfor a single layerin particular [see Eq. 1/or3. 1.• areequal totheeigenvalues Eq. (24) (:25) = det(X•X•')det(Dk)= det(Dk). det(A•) = 1. x3 = 0. so are 22= 1/21. Vol. The characteristic (i) det(A) = det(A. '461 `462 263 (27) det(X.6 and d is the total thickness...2 . Since the entries of D• are made up of pairs that are inverseof each other. n. if A. 6 and1/aq constitute thesame set. we have shownthat the globaltransfermatrix correctlyreduces to the corresponding matrix of the singlematerialplate when all layer properties are the same. and 25 are eigenvaluesof A k.crI)= O.Bysubstituting from (17) into (18) and carryingthe inverse era we get 241`442 243 `451 `45 .j . which togetherwith the relation(17). we nowindicatethat suchpropertis are alsocharacteristics of the transfer matrices of theindividual layers.• is 1/•r. (16) ]. To showthis.6 format. No.2 . using thefactthat the products of the two equalrank square matrices M.4 ff • havethesame setof eigenvalues. III. Am. (20) By inspecting (23b) and (23c). (v) A very important consequence of the abovelisted propertiesis the resultingrelationsthat exist betweenthe invarientserA.. In fact.as. 22 bycyclic permutation. we can rewrite the relation (23a) as det(X•D•X5 IX2D2X f •'"X.A• andD• also havethe sameeigenvalues. and compare the resulting expressions. Now.n.. As a consequence of ( a). (19) that lead to the characteristic of.4.. see http://asadl. CombinaAdnanH. 6) ofA• aregiven bythediagonalelements of Dk. we conclude that (23c) can be obtained from (23b) by merelyinvertingthe diagonal matrices D• andthe eigenvalue a. thus.by employing ( 11). canease theexecution of theanalysis and leadto simple analytical representation of theresults. Equation(19) will nowbeused to present solutions for a variety of situations.... Redistribution subject to ASA license or copyright.Sec.arethedisplacement andstress columnvectors at thetop.. we consider a single cell conclude that: medium.let usassume that theeigenvalue of A is•r. In the first. (c) The results of (b). •"'X2D2X2 •X•D. = X• and D = D..rrI) = 0.respectively. PROPERTIES OF THE TRANSFER MATRIX M•M2•M2M •) have thesame eigenvalues.453 = 0. 22 Since Dk isdiagonal.Thus. Nayfeb:Multilayered anistropic media 1525 ownloaded 01 Jul 2011 to 150. (b) As a consequence of their similarity. X ff • -.eiga•o• o (28) which is consistentwith the formal solution (8).. (23c) The globaltransfermatrix A has severalproperties which. X• andthesixvalues aq arethesame for every k. (23a) (23b) det(X. (19) holdsfor any number of layersn..D. With thiswe nowconcentrate on listingand discussing properties of the individualtransfermatrixA k(a) det (Ak) = 1.. q = 1. 4..where nowP * andP . isseen to beunity. Accordingly./L 3...D. Thus weconclude thattheeigenvalues %. l/a... 89. and the fact that the eigenvalues eigenvalues •rqandinvarients I• of A. I•=1. then it is obvious thatD• andD ff • havethesame entries (eigenval- ues).x3 = d. andlower....•X•D c •X • •' "X.2. Pt. Here we generalizethe classicalFloquct periodicity condition to require XD ff •X. _ • "'D• Thisconclusion canalso bearrivedat by notingthatAk and D• are similar andhence their determinants are equal.org/journals/doc/ASALIB-home/info/terms.2 .. we conclude the symmetricrelations Is=I.X • • .if exploited. itsdeterminant isequal to theproduct of itsentries which. impliesthat nesses). q = 1.2 . leadto theconclusion that.M2 and M2M• (although 1525 J.. Before weproceed to listanddiscuss these properties. •.. (21) wherewe usedthe fact that hereX. q = 1.)det(A•_. we hypothesize that any general propertyof Ak is alsoa propertyof A. I4=12. Now. It follows thenthattheeigenvalue ofA . (26) ofAff • aretheinverse of thecorresponding eigenvalues of Ak.140. namely a free n-layeredplate. • . ofthetotal plate.. equationfor sucha situationis obtainedby choosing 0 := 0 (22) and invoking the stress-free upper and bottom surfacesin equation (ii) Theeigenvalues ofA .. ). theglobal matrix A collapses to det(Ak) = det(X• )det(XF •)det(D• ) A: XtDX • t. write it in terms of both the eigenvalues (say 2q. Acoust. This means that the sixpossible isa diagonal matrix whose entries aregiven byexp(i•ctqd).•rI) = 0. substituting from (17) into (18) and recognizing that X •-_•t• X• = I (identity) fori = 1. To this end. April1991 p + = p ... if we expandthe characteristic equation der (A -. and 1/as (iv) In thedegenerate case where all layers aremadeup of the samematerial (but not necessarily haveequalthick- This propertycanbe easilyprovenby employing the wellknownresultthat thedeterminant of theproduct isequalto the productof the determinants.( c) andthe definition( 18) we The result16= 1 alsoconfirmsthe fact that det(A) = 1..189.4&and. andthus A canberepresented by Ak for k = 1. A second important situationis that of a periodicmedium consisting of a repetitionof the unit cell (plate).D. By inspection we seethat these eigenvaluesconsistof three pairs with the entriesof each pair beingthe inverseof eachother. and 26 = 1/25.24= 1/23..D F •X . (iii) The property described under(ii) canonlyimply thatthe% consists of three pairs where theentries of each pair are the inverse of eachother..2.149.q= 1. we note that for eachlayerk the axes(x•) &and (xfi) &coincide with the azimuthalangles• = 0 ø and • = 90 ø.. on the nineconstants describing the orthotropiccase. (27) and (30).i: . (3) by noting the additionalconditionsimposedby symmetry.-. canonlyholdin thelimit. This resultmeans that theSH wavemotions uncouple from those of the sagittalplane. (32). ).This is due to the fact that couplingof the SH wavefieldequations with those of the sagittalplanewavewill persist...Thus.whose x'• axis is normal to the plane of isotropy. (27) and (30) applyif the wavepropagates alongdirections other than theseprincipalaxes.•= Notice that a• and a 6 of Eq. 22 by V.3=C•2. Pt. 1..q = 1.C66 sin • 0)/C44].13/2 = 0. To thisendsubstituting fromEq.cubic. C.189. Acoust.. strictlyspeaking.Soc. we alsorecognize that results for all symmetryclasses higher than orthotropic (which include transversely isotropic.(C•3+ C5•) 2sin e0.. PROPAGATION SYMMETRY ALONG AXES OF ROTATIONAL (30) In terms ofthe individual entriesA 0ofA theinvariantslq are ( -. If x. (31) Returningto the caseoforthotropicsymmetry.•A(i•.3 C•2 Ch C•3 (34) det (A -. (10) leads. Nayfeh: Multilayeredanistropicmedia 1526 nloaded 01 Jul 2011 to 150. oncethe characteristic equations for propagation eitheroff or alongprincipalaxesare derivedfor orthotropic symmetry. If thewavehappens to propagate alongan inplane axisof sym- whereA (il.. the corresponding results for the highersymmetrymaterials can be obtainedby merely applying the appropriate restrictions on their properties. No.i2. Vol. C26.s = Equation(29) canalsobeexpanded andwrittenin termsof theinvariants I..3./•3' (35) aS•. with (36) A = C33C55 . for propagationalong axesof rotational symmetry. 89.• = i• < i''' <i. IV. of A which. (32) The corresponding transfermatrix for a stackof 0ø.[ --B-'I.tionsof (26) and (19) yieldsthe characteristic equation C.i•.transversely isotropic. . the analysis reduces to a two-dimensional andhencesimpler one.al' a4 • -. uncoupling of these equations occurs. April 1991 Adnan H. Here..4 thedisplacements andstress amplitudes reduce to 1526 J. under restricted conditions.i.140.pc2)C33 -3(C$5 sin 20 -pce)Css -.Ie•acø•ø)= 0..andC4• alsovanish. Redistribution subject to ASA license or copyright. As waspointed outin anearlier work.Recognizing that theseclasses of materials are different from monoclinic materials in that eachpossess two orthogonal principalaxesin the planes of the layers.For suchsituations the appropriate transfer matrix doesnot containthis nonexistent coupling. (27) and (30)..can be easilyobtainedfrom Eq.i.org/journals/doc/ASALIB-home/info/terms .90 ø orthotropic layershave beenrecentlyderivedin order to studythe reflection from a fluid-loaded laminated plateof unidirectional fibrous composites? For completeness we here outline the analysisleading to the derivation of the transfermatrixfor propagation alongaxesof symmetry. HIGHER SYMMETRY MATERIALS metryof layerk (namely for •& = 0 øor 90 ø)thenEqs.• ) isthe rnthdeterminant formedfrom the rowsi -• ivi 2..• and columnsj = i•.C•3 = 2C.149..Eqs. For this reason.for • = 0..andisotropic)are contained asspecial cases of the corresponding solutions for the orthotropiccase. the matrixelements C•6..3 •. (36) correspond to the (SH) motion while thoseof Eq.-. 4. the presence of cubic symmetry requiresthe further restrictions As for the sagittalplanemotionwe noticethat for each c%. C22 C.namely..•4 the presentgeneralcaseand hencewill not be pursued further. Nayfehrecently studied the interaction of the (SH) elastic waves withmultilayered media e4 asaprelude to (33) C•2 -.ie. This simplification of theelastic stiffness matrixhasimplications for theanalysis commencing at Eq.1)r•I. oneneeds only to assurethat further appropriatereductions in the number of nonzeroelasticconstants are exploitedin Eqs. Thisis of course dueto the presence of thesuperfluous coupling[as implied by (10) ] betweenthe equations describing the horizontally polarizedwave (SH) and thosebelongingto the The results of Eqs. C•3=C•2. (27) and (30) obtained for themonoclinic material case can.respectively.As a consequence Eq.i. andx• are chosen to coincide with the in-plane principalaxesfor orthotropicsymmetrythen C'• = C• = C• = C•'• = 0. hold for highersymmetryonessuchas orthotropic.(27) and (30).into Appendix A andinspecting theresulting entries leads to theconclusion that for propagation along rotationalsymmetryaxes. to thesimpler solution fortheaq al. givenin Ref. B = (C• sin 20 -.. C. (9b). we now derive results for orthotropicsymmetry.Or6 = [(pC 2-.I• cos[2•d cos0 ] + 12cosiedcos0 ] . (29) C•4 = C.C36. The implication hereisthat for the present problem this results in simplified versions of Eqs. on the other hand.. also sagittalplane. Am.4zIC) I/2]/eA. afteralgebraic manipulation reduce to cos[3•dcos0 ] . (35) correspond to the sagittal planewaves.. C= (C• sin • 0 --pc•)(C• sin • O--pc•). el whichdeals withsingle orthotropic andmonoclinic plates. and cubic. Of greatest importanceisthefactthatK•3 andKa3in Eq. Thus. for off-principal-axis propagation.2. whichparticularizes the constitutive relations (3) to orthotropic media.(Be.In fact. /•2 •--. Finally. (9b) will vanish. see http://asadl. (37) Resultsfor material possessing transverse isotropy. demonstrate a frequencydependent "dispersive" character of the wavefronts. April1991 Adnan H. the curveswill thus constitutethe wave frontsfor an effective homogenized mediumwhose propertiesare volumefractionweighted properties of the individual layers. (10) andfor a fixedO. DISCUSSIONS cos[•'dcosg ] +I•/2 AND NUMERICAL ILLUSTRATION In this section. The proper. (43) describes thevariation ofthethree phase of the 6X6 matrixA of (18). (40) and D• is againthe discuss the casein which all layersare the same.3.howfunctions of theproduct of frequency andunitcellthickness. tiesof a representative orthotropic materialthat we usedin (38) our calculations are givenin GPa by C h = 128. formalsolutions for propagation along anaxis of symmetry of an orthotropic material aregiven by (40a) where ingappropriate rotational angles.4 are similar. we demonstrate the variations of phase velocityc withangle of incidence 0 for specified frequencies Fand orientation angles ½. In the first. For the layered mediacase. Thus. q---1.. for thefreewaves ona single layered plateandon theperiodFor an isotropic material. dictates that the characteristic Eq. Pt. For the anisotropic ease. for a fixedfrequency.This is expected to undoubtedly result in a description of thebehavior of single homogeneous anisotropic materials. these Utilizing`4'.4 fortheaq defined in Eq. however.I•Vq sint9).For Fd = 0 MHz ram. C•=18.. (35). Acoust.2. for specified angles of incidence 0. andgeometric stacking arespecified.(43} and(44) gives theroots c = cL andc = CTyielding twoconcenttic spherical wavefrontcurves asis expected.0 ø.combination of Eqs. the threesolutions will be coupledresulting in nonspherical wavefronts. we now E• = e g•". U•3. 1. Conventional dispersion curves in the formsof variawith frequency for specified orientation angles.by varying the of layers. (30). Soc.140. This is done. anditsconstituents (layers)areassigned volume fractionsaddingto unity.together with the factthat for thiscase D and. front curves. we present phase velocitydispersion curvesplottedas using eitherEq.4. (42) and cos[2•dcos0 ] --I• Vl.Vol. sin 2 0C•Ctq • (Cl3+ C•)Ctq sin0 Diq= i•( Ci3sin0 + C33ctq Wq ).for a ingit with.org/journals/doc/ASALIB-home/info/terms.quasilongitudinal andtwoquasishear motions. For thisspecialized single medium case. (41). Thus.thisiseffectively a formof demonstration of the dependence of wavefront (inverseof slowness } curves frequency in a discrete manner. Thus. Nayfeh: Multilayered anistropic media 1527 ownloaded 01 Jul 2011 to 150.. structed namelyFd. (40b) Onceagain. for examples a combinationof 0 ø. (43) which alsospecializes the formal solution(8) to the one appropriate for thesingle homogeneous medium. (10) uncouic media for propagation along an axisof symmetry of each plesandgives layerare given. q=1.4of themonoclinic case) areidentical withthose variable O. thecorresponding characteristic equations curves will be independent of frequency. To showthe extentof generalityin the results. Eq. (44) A• A3•=0 # (41) =0.Withoutanylossin generality the thickness d of the representative unit cell is keptconstant.for example. In the sections of wave velocities with wavenumber can also be conond.. by 4X4diagonal matrix whose entries are given ase g"•t'".we present our numerical results in two categories.189. This can be doneby specializing Eq. With reftransfer matrixis againconstructed by multiplication of the erence to Eq. Here.2. U12. Dzq ---i•Css(ct q q.Eq.Am.3. see http://asadl. we can constructwave front curvesand hence. As wasdiscussed earlier. q = 1. we illustratethe analyticalresults(27). Ch=6. they are by no means exhaustive of the varietyof the phenomenology containedin the analysis. 6. but mostimportantlyonthewave number • or frequency F (more precisely on the parameter Fd).2. Here we mentionthat further 1527 J. . (29) admitsthe solution a•=cos•0. the situation is muchmore complicated dueto thedependence of thephase velocities. notonlyon theindividual layerproperties.25. (40) to x3•n• = 0 andto x}•l= d (n•and eliminating thecommon amplitude column of Ul•.respectively.60 ølayupconstitutes a fourlayered cellwhereas a combination of 0 0. different layerscan be constructed from this chosen materialby assign- Accordingly. except for the fact that`4 ' has velocities with the incidentangleandhence constitute wave only two pairsof eigenvalues ratherthan three. Thischoice is not restrictiveandhastheadvantage of saving space by not havingto list differentmaterialproperties. However.4.theglobaltransfer matrixfor sucha situation collapses to the form givenin (24}. (39) C[s = 12.149.60 ø. C ]• = 7.and0 øcells defines a single homogeneous material. Usingthismatrix. Once the number where CLandCr arethelongitudinal andshear wavespeeds in themedium respectively.90 ø.Wq =pea --C•.sin • 0. az3.. C•6 = 8.0 ø. (29) or (30). however. Redistribution subject to ASA license or copyright.js . C•=72.sin • 19.ever. for fixed valuesof 8. Hence theglobal ct•= ea/c•-.Eq. and U•4resulting in anequation similarto (27) with X&now givenby the 4X4 matrix of Eq.89. (40) canbeused to relatethedisplace- ments and stresses atx}kl= 0 tothose atx}k)= d (n) forsuch a restricted directionof propagation. C•=5. theirproperties.andp= 2g/cm •. and (42) with a limited selection of numerical examples. C•3=32.5 = c2/ear -. While thecases we present herearecertainly typi- cal.the results (43) admit individualmaterial transfermatrices. No.The properties of the threerootsfor the phasevelocityc corresponding to one resulting 4 X 4 matrix (herereferred to asA' to avoidconfus.and -. 45 ø. Variation of phasevelocityc with wavenumber • for angleof incidence0 = 45 ø.Am. 0 t I 2 I 3 I 4 I 5 I 6 I 7 I 8 FIG.45 ø) layup. No. At the zerowave-number limit.Sample examples. using a (60 ø. April1991 AdnanH. oneof these seems to beassociated with a rapid change in the slopeof the quasilongitudinal mode.45 ø).2. (27). 6(c) and to a lesser degree in 6(b) are due to tropyof the (0 ø. It isobvious thatthe largest valuecorresponds to thequasilongitudinal mode.little change isseen to take placein thesevalues. eachfigure displays threevalues of wavespeeds corresponding to one quasilongitudinal andtwo quasishear. 2-5. 4 especially at Fd = 4 layups. --60 ø) layup periodic medium as a representative case. To furthershowthe versatility of the analyses we also generate. see http://asadl. -.As • increases.0 ø. and (0 ø. Nayfeh:Multilayered anistropic media 1528 nloaded 01 Jul 2011 to 150. usingthe characteristic Eq.in Figs.The complicated features playedin Fig.90 ø). 2 with (60 ø. -. SameasFig. 89. 1.90 ø. -.org/journals/doc/ASALIB-home/info/terms .confidence in our analyticaland computational procedure was established by reproducing the numericalresultsof Refs. theisotropiclikebehavior (suggestedbythe closeness of thetwo quasitransverse modes of Figs. Notice in contrast that the "clean" behavior dis- plane where Kl = c sin0and K 2 = CCOS 0.0 ø. Pt. for the selectedvalues MHz mm broughtaboutby the presence ß ''' Fd = 0. 2 with (0 ø 90 ø. 3.Soc. 1528 J.At I I I I I I I I 0 i 2 3 4.60 ø) layup.4 and 5 as compared withthose of Fig.90 ø).0 ø. isworthcommenting upon. (60 ø.189. They correspond to (0 ø.5 6 7 relativelylow values of thewavenumber. Redistribution subject to ASA license or copyright. Furthermore.Vol. Acoust.45 ø. Note •4 once again thatall of these plates have thesame thickness d. In these figures thephase velocity c isgiven by km/s andthe wave number •' by mm.45 ø. 7 for freewaveson a finitethickness multi- C•4 •4 I' I I I I I I 2 3 • 5 6 7 8 I FIG. 4. and 4 MHz mm. 6(a)-(c) we depict. wave front curvesin the K•-K 2 modes.60 ø) multivaluedbehaviorshownin Fig.149.0 ø -. .45 ø. FIG. These curves demonstrate the inverse of the slowness curves asfunctions of frequency andhence displayanddemonstrate wavefrontdispersion behavior. -.90 ø) layup. -. 2.140. 0 = 45 ø. 3).75 ø.45 ø) and (60 ø. 6 and 9 that constitutespecialcasesof the present work. 15 ø. whichdemonstrate thedependence of suchcurveson the number of layers and their orientations..60 ø. 30 ø. the dispersion curves of Fig.•. periodicmedia layup configurations. respectively. 6(a) reflects the variations of effective wave speed c (namelyat Fd = 0) with the angleof incidence. aregiven fortherepresentative angle.(0 ø. (0 ø. Closeexamination of these figures reveals several interesting features. SameasFig. It is consistent with the static predictionof the quasi-isoshownin Fig. 4. other higherordermodes appear. of the higher-order In Fig.90 ø.60 ø). GONC/U$1ON •4 3 IL O I I 2 I 3 I ß I 5 I • I ? I 8 We have derivedanalyticalexpressions that are easily adaptableto numerical illustrationsof the interactionof elasticwaveswith multilayeredanisotropic media.layered plate consisting of ( --60 ø.15ø. 0 ø. 4.00 ø) layup.'" (a) K• FIG. -. Wavefrontcurves for fixedfrequenciesare alsoincluded to demonstrate their dispersive behaviors.'7 ¾11.(c) Same as (a) repeatedat Fd = 4 MHz mm. No. . 9 KI 1529 J. 0 ø. 5. 60 ø) layup. FIG. Vol.149.(b) Same as (a) repeated at Fd = 2 MHz ram. Resultsin the formsof dispersion curves are givenfor several representative layering. K2 K• K1 ..189.75 ø. The curvesdisplayed on this figureare typical of free wavesin anisotropic plates.org/journals/doc/ASALIB-home/info/terms. IO KI '" ""'61.140.Wavesare allowed to propagate along arbitrary directionsin both azimuthal as well as incidence planes. Nayfeb:Multilayered anistropic meclla 1529 ownloaded 01 Jul 2011 to 150. 1. These include thecases of propagation of freeharmonicwaves in a multilayered platesand in periodic media constructed from a repetitionof the layeredplate. 2 with (0 ø. 69. Redistribution subject to ASA license or copyright.js .A plate consisting of an arbitrarynumberof layerseachpossessing aslow asmonoclinic symmetryischosen asa representative cell of the medium. (a) Wavertonicurves for Fd = 0 MHz mm and a (60 *.30 ø. April1991 AdnanH.Soc.60 ø. Acoust. see http://asadl.. Characteristic equations for a varietyof physical systemsarediscussed. SameasFig..60 ø) layup.45 ø. Am. 6. Pt. 6036 + 2C. Am.1C33044 -.3 G6+ 2C.6)S2G '-+ C•2G4.Css .6 )52G2. Gilbert and G. "Seismic Wavesin StratifiedMedia--ll. C36= (C•3 -. 326-332 (1966).2C. Appl. C23: C•3G2q3 C26• (C.C•4S2. 73-101 (1987). •F." Geephys. 1957). Cambridge. A. Ewing.)pc 2sin 40 + (Cll + •55+ C66)P 2c4 sin2 0 --p3c6]/A. Press.M."Acoustical Properties era Thinly Laminated Media. C.(c."J. No.G. Backus.o )GS3. J. April1991 AdrianH.C.GsG•.2C$.C.2C. Ting(Eds.2C'•2-. 17 (1953). AMD-Vol. Seismol. see http://asadl." J.org/journals/doc/ASALIB-home/info/terms .UK. 68-80 (1956). Sec. SA. '•B.6)SG3.C•6G.H. Thomson..20. Brekhovskikh. + c.(C .2C.89. 1983).. Phys. "Transmission of Elastic Waves Through a Stratified Solidmedium. "S. 1. 7. C22: c. 60 ø) layupfree plate. • 0 I I 2 I 3 I 4 I 5 I 6 I 7 I C4s= ( C •4 -.C . M. ACKNOWLEDGMENT This work hasbeensupported by AFOSR.6)S2G q.C-52 q-2C... P. N."Time-Harmonic Waves Propagation Normalto the LayersoœMulti-layered PeriodicMedia. L..605 ." Bull. Waves in LayeredMedia (Academic.6C36Css)sin 40 . Ci6 • ( C iI -.• .140. ElasticWaves in Layered Media(McGraw-Hill.G4).3.C•3 . Frazer.. New York. Acoust. Mech. R. New York. Accoust. :L.189. 677-682 ( 1971). 1988). 90. Sec.6)GS3 q.C '•2 -.%.T.6) S2G2q-C•2(S4q. Cii •. Mech. andF.033 + C. 2 C66= (C•I + C•2 -.c•Gs) sine 0 . C33• C. "Propagator Matricesin ElasticWaveand Vibration Problems. 91. (10) are givenby AI •--[ (0. Wave Propagation inStructural Composites (American Societyof MechanicalEngineers.G•Css .C3605 + C. Mal andT. Appl. Nayfeh. Sec.C•3 044 q-2013036045 -. Combination of the transformation matrix (5) with the constitutiverelations (3) yields the following transformed properties: APPENDIX B The variouscoefficients ofEq.iC4s + C . 1966).C'16(84q-G4).C•3X2. New York.C'•3)SG.20.149.C•2 + 2C•. 8 C44= C•402 q.Vol." Phys.i q-C•2 -4C•.20.s 4 + 2(ci: + 2C•. Rytov.C..+ 2C•. 89 (1950). Nayfeh:Multilayered anistropic media 1530 nloaded 01 Jul 2011 to 150. FIG.21.(C..6033045 q-C3305s066 -.. Sve. APPENDIX A whereG = cos& and S = sin •.605 • • 36 • • 2 2 a3= [ (c. N.3C."Geophysics 31. Variation of phase velocity c withwave number •' for a (60 ø.G4-. with •W..20•30506 . Kennett. Fryer and L. 2. Ci3 • C•3G2q. 42.Go+ c• + G. 38.C]6C55)sin 2O A2= [ (CIi03306 -.. Elastodynamic Eigensolutions for SomeAnisotropic Systems. 40. •N. Pt. •C. Am. E.2 -. Redistribution subject to ASA license or copyright.)." J.iC•6-. C55= C.C • )SG.W.2C•. 43. "The Dispersion o• Surface Waves in Multilayered Media. G4 q-C•2S4 q-2(C '•2 Ci2 • (C. M.. mA. J.C'•2. 4.SeismicWavePropagation in StratifiedMedia (CambridgeU.0 ø. + 2C. 1530 J.3044055 q-013025 -."Time-Harmonic Waves Travelling Obliquely in a Periodically Laminated Medium. T.5G 2q. . Appl. Haskell. S. Astron.C•55 2.6)SG 3 + (C '•2. 92-96 (1974).K. •'W. Jardel:sky. Pt. 24A. SOc. andD. pp. 2•A. H. "Experimental Ultrasonic Reflection •4M. Mech. B. andGuided WavePropagation in Fibrous Composite Laminates." Philos.NJ." J. pp. Chimenti andA. "Ultrasonic reflection andguided wave "A. 56.js . 625-649 (1958). Appl. Redistribution subject to ASA license or copyright." J. "Ultrasonic Wave Reflection from 1531 J. 5." Wave Motion 6. Stroh. see http://asadl. April 1991 Adrian Fl_ klayfeh: Multilayerod anistropic media 1531 ownloaded 01 Jul 2011 to 150. Chadwick.Vol. H.4." J. E."to appearin J. E. Acoust. 87. "Dislocations and Cracks inAnisotropic Elasticity.. E. in Ref. 863 (1988). 881 (1990). Appl. Soc. "Harmonic Waves withArbitrary Propagation Directionin layeredOrthotropic ElasticComposites. W. Media.69-80. Nayfeh. Chimenti." in Ref.H. Acoast. •3K. Nayfeb. Acoust.•2M. Z007 (1989).189.89. Nayfeb andD. •r. 531-542( 1981 ). Liquid-Loaded Orthotropic Plates withPdpplications to Fibrous Composites. H. Nayfeb andD. E.Gilberg. T. 1409-1415(1990).Yamada andS. Am. Chimenti. '•Theoretical Wave Propagation in Multilayered Orthotropic Media. pp.Braga and G.1. 1968). M. Schoenberg. mA. 3. 88. 73. 137-162 (1983). Mech. 29-38. TingandP. 303-320 (1984). Am.Matrix Theory (Prentice-Hall." in Ref. "Plane Waves inAnisotropic layered Composites. Chimenti. Herrmann. "FreeWavePropagation in Plates of GeneralAnisotropic Media. No. 17-28. 5. "Harmonic Waves in Periodically 23D. Layered Anisotropic Elastic Composites. 55. 22j. 53-68. Composite Mater.Soc. E. "Wave Propagation inAlternating Solid and Fluid-Layers. Nemat-Nasser. Chimenti andA. "A Propagator Matrix Method forPeriodically Stratified 2øD. Nayfeb.H. T. 15.Am. pp." J.149. N. Franklin. E.Am.org/journals/doc/ASALIB-home/info/terms. N."in Ref. Nayfeh. •A. C. Englewood Cliffs. Taylor." J. •SA. propagation in biaxiallylaminatedcomposite plates.140. Mag.5. Acoust. "ThePropagation of Horizontally Polarized Shear Waves in MultilayeredAnisotropic Media. 5. SOc. H.
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