Asymptotic analysis of the dispersion relation of

1Asymptoticanalysisofthedispersionrelationofanincompressibleelasticlayerofuniformthickness.W.Hussain*DepartmentofMathematics,SchoolofArtsandSciences,LahoreUniversityofManagementSciences(LUMS),OppositeSector‘U’,D.H.A.,LahoreCantt.54792,Pakistan.*E-mailaddress:[e-mail](W.Hussain)Tel:[phone](10lines);fax:[phone]AbstractThispaperisconcernedwithanasymptoticanalysisofthedispersionrelationforwavepropagationinanelasticlayerofuniformthickness.Thelayerissubjecttoanunderlyingsimplesheardeformationaccompaniedbyanarbitraryuniformhydrostaticstress.Inrespectofageneralformofincompressible,isotropicelasticstrain-energyfunction,thedispersionequationforinfinitesimalwavesisobtainedasymptoticallyforincrementaltractionboundaryconditionsonthefacesofthelayer.Twosetsofwavespeedsasafunctionofwavenumber,layerthicknessandmaterialparameterareobtainedandthenumericalresultsareillustratedfortwodifferentclassesofstrain-energyfunctions.Foraparticularclassofstrain-energyfunctions,itisshownthatwavespeedsdon’tdependupontheamountofshearandthematerialparameters.Formaterialsnotinthisspecialclassvelocitiesdodependuponthesheardeformation.Keywords:Asymptoticanalysis,non-linearelasticity,dispersionrelation,waves.MSC(2000):74B15,74B20.1.IntroductionThestudyofinfinitesimalwavespropagatinginfinitelydeformedelasticmaterialwasinitiatedinaseriesofpapersbyBiot,summarizedinhismonograph[1],andbyHayesandRivlin[2],whostudiedsurfacewavesinahalf-spacesubjecttopurehomogeneousstrain.DowaikhandOgden[3,4]forincompressibleandcompressibleisotropicelasticmaterialsrespectively,haveinvestigatedsurfacewavesanddeformationsofahalf-space,againforafinitedeformationcorrespondingtopurehomogeneousstrain.In[11,12]thereflectionofinfinitesimalplanewavesfromtheplaneboundaryofahalf-spacesubjecttopurehomogeneousstrainisworkedout.DiscussionofFlexuralwavesinincompressiblepre-stressedelasticcompositesandsmallamplitudevibrationsofpre-stressedlaminatesisdonebyRogersonandSandifordin[13]and[14]respectively.Ineachofthepaperscitedabove,theconsideredfinitedeformationisapurehomogeneousstrainsothattheorientationoftheprincipalaxesofstrainisfixed.Theeffectofprincipalaxisorientationofwavesanddeformationsinaplateorhalf-spacehasbeenexemplifiedinthecaseofsimpleshearinRefs.[5,6].Morerecently,HussainandOgden[7]didtheproblemofreflectionandtransmissionofplanewaves,inwhichthetwohalf-spacesconsistofthesameincompressibleisotropicelasticmaterialbutaresubjecttoequalandoppositesimpleshears(seealsotherelatedpaper[8]).2In[6]ConnorandOgdenexaminedthepropagationofdispersivewavesinalayer,offinitethickness,whichissubjecttoaplanedeformationconsistingofsimpleshearparalleltothefacesofthelayer.Asaconsequencedispersionequationwasderivedandanalyzednon-asymptoticallyforincrementalboundaryconditions.Inthispapertheanalysisin[6]isextendedandthedispersionrelationisanalyzedasymptoticallyandthenumericalresultsforthelargewavevelocitiesandlowwavenumbersareobtained.TherequirednotationsandequationsaresummarizedinSection2.InSection3thebriefderivationofthedispersionequationfrom[6]isgivenrelevanttotheincrementaltractionboundaryconditions.InSection4theasymptoticexpansionofthedispersionequationisderived,andbyusingtheasymptoticsolutionsofthepropagationcondition,twosetsofvelocitiesfordifferentmodesasafunctionofwavenumber,layerthicknessandmaterialparameterareobtained.GraphicalresultsareprovidedinSection5inordertoillustratethedependenceofthewavevelocitiesonthewavenumberandlayerthicknessforthetwocategoriesofstrainenergyfunctions.Thefeaturesoftheresultsaredescribedforthetwoclassesofstrain-energyfunctionsinSections5.1and5.2.Unlike[6],forthespecialclassofstrain-energyfunctions,thewavevelocitiesareindependentofthesimplesheardeformation.Formaterialsnotinthisspecialclass,ontheotherhand,itisshownthatthewavevelocitiesdodependupontheamountofshear.Inbothclassesofstrain-energyfunctions,thewavevelocitiesareindependentofthehydrostaticstress,whichisdifferentfromthepreviouswork[6].2.BasicequationsLetXandx,respectively,bethepositionvectorsofatypicalmaterialparticleintheundeformed(reference)anddeformedconfigurationsB0andBofthematerial.Wewritethedeformation,csay,intheformx=c(X),XÎB0.(2.1)ThedeformationgradientAisdefinedbyA=Gradc,(2.2)wherethegradientoperatorGradiswithrespectto0B.4ForanisochoricdeformationwehaveAdet.1321=≡lll(2.8)Foranisotropicelasticmaterialwithstrain-energyfunctionWperunitvolume,Wdependssymmetricallyon.,,321lllWetakethematerialtobeincompressible,sothatEq.(2.8)holdsidentically.TheassociatedprincipalCauchystresses321,,sssarethengivenby,pWiii-∂∂=lls},3,2,1{∈i(2.9)wherepisthearbitraryhydrostaticstressassociatedwiththeincompressibilityconstraint.LetSdenotethenominalstresstensor.Then,intheabsenceofbodyforces,theequilibriumofthebodyisgovernedbytheequationDivS=0,(2.10)whereDivisthedivergenceoperatorin.0BForahomogeneousincompressibleelasticmaterialwehave.1--∂∂=AASpW(2.11)Forconveniencewetakee1=E1,e2=E2,e3=E3.Ifthedeformationisconfinedto(1,2)-plane,wehavev(3)=e(3),andwemaysetl3=1,sothatEq.(2.8)reducesto.121=ll(2.12)Letfdenotetheori