AuthortowhomcorrespondenceshouldbeaddressedEmaillakesengrwisceduJournalofVol39No1820050021998305181645 ID: 380438
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CompositeswithInclusionsofNegativeBulkModulus:ExtremeDampingandNegativePoissonsRatioY.C.WR.S.LMST-8,MSG755,LosAlamosNationalLaboratory,LosAlamos,NM87545,USADepartmentofEngineeringPhysics,EngineeringMechanicsProgramBiomedicalEngineeringDepartment,MaterialsScienceProgramandRheologyResearchCenter,UniversityofWisconsin-Madison147EngineeringResearchBuilding1500EngineeringDrive,Madison,WI53706-1609,USA(ReceivedJune21,2004)(AcceptedNovember8,2004)Theeffectofanegativebulkmodulusphaseinelasticcompositesisstudied.Negativebulkmodulusisshowntobepossibleinselectedunitcells.Inisotropicsolids,canbeattainedwhennegativePoissonsratioissufficientlysmall,belowthestabilitylimit(forstresscontrol)1.Suchmaterials,ifusedasinclusions,arepredictedtobestablewithrespecttothebandformation,eveniftheyarelarge.CompositeswithsphericalinclusionsofnegativebulkmoduliareshowntoexhibitnegativePoissonsratioandanomaliesincompositebulkmodulusandYoungsmodulus(andinthecorrespondingmechanicaldamping)butnotintheshearmodulus.KEYWORDS:stability,viscoelasticity,negativePoissonsratio,negativemodulus.INTRODUCTIONORMOSTELASTICsystems,stiffnessispositivei.e.,adeformedobjectexperiencesaforceinthesamedirectionasthedeformation.Negativestiffnessispossibleinthesystems,suchasprestrainedobjectsincludingpostbuckledelements,whichcontainstoredenergy[1].Heterogeneoussystemswithoneconstituentofnegativestiffnessareofinterestsincetheyarepredictedtogiverisetoextremeoveralldampingandstiffness[2],highviscoelasticdamping,andnegativeaxialstiffnesswasobserved[3]incompliantsystemscontainingpostbuckledtubes.Highviscoelasticdampinghasalsobeenobservedinmetalmatrix(Sn)compositescontainingVOparticulateinclusions[4]whichundergoaferroelasticphasetransformation.Ferroelasticsareofinterestinthiscontextsincethey *Authortowhomcorrespondenceshouldbeaddressed.E-mail:lakes@engr.wisc.eduJournalof,Vol.39,No.18/20050021-9983/05/181645 13$10.00/0DOI:10.1177/00219983050511122005SagePublications ),thisallowsnegativeYoungsmodulusandbulkmodulusspecifically Thesecondcondition(2b)forstrongellipticityentailstheconstrainedtensorialmodulusbepositive,�0.Stronglyellipticmaterialsarestablewithrespecttoformationofbands.Displacement-typeboundaryvalueproblemshaveuniquesolutionsif[12]theelasticconstantsareintherangeforstrongellipticity.Thisrangeisconsiderablylessrestrictivethanthatforthetractioncondition.ThismeansthatablockofmaterialconstrainedatthesurfacecanhaveanegativebulkmodulusorYoungsmodulusandbestablewithrespecttobothglobaldeformationandbandformation.Wecanhavewithinstrongellipticityif,issufficientlylargerthansince[13]. 23¼þ 63 43¼þ2 43¼C1111 astheLameconstants.TherangeofelasticmodulicorrespondingtostabilityandinstabilityundervariousconditionsisshowninthemapinFigure1.Therepresentationoftheupperrightquadrant,aswellasthetermdilationalisduetoMilton[10].Thestippledregioncor-respondsto1hencetoaswellasTheshadedregioncorrespondstoorhence,afailureofstrongellipticity. Figure1.MapofelasticmaterialpropertiescorrespondingtodifferentvaluesofbulkmodulusKandshearmodulusG,allowingnegativevalues.CompositeswithInclusionsofNegativeBulkModulus NEGATIVEMODULINegativeSpringConstantNegativestiffnessisknowntooccurinthecontextofpostbuckledstructuresandobjects[14].Stiffnessofalumpedsystemreferstotheratioofforcetodeflection.Acolumnwhichhasbeenbuckledintoan-shapedconfigurationisunstableandtendstosnapthrough.Bypressinglaterallyonthecolumn,onecancauseittosnapthrough.Iftheload deformationcharacteristicisstudiedinthedisplacementcontrol(whichentailsahardconstraint),negativestiffnessisobservedexperimentally.Negativestiffnessalsooccursinsingle-cellmodelsoffoammaterials.Suchmodelswereobservedexperimentallytoexhibitacompressiveforce deformationrelationwhichisnotmonotonic[15].Inwardbulgeofcellribsgivesrisetoageometricnonlinearity.NegativeShearModulusPrestrainedlatticecellstheoreticallycangiverisetonegativePoissonsratioandevennegativeshearmodulus[16].Suchlatticestructureswereoriginallyexaminedinastudyofgeneralizedcontinuummechanics[17].Negativeshearmodulusisalsoinferredfromthebehaviorofferroelasticandferroelectricmaterialsinthevicinityofphasetransitions.Astemperatureisloweredtothetransformationtemperature,theshearmodulussoftenstozero(orasclosetozeroastheexperimentcanresolve).Below,bandsordomainsforminthematerial,asanticipatedbythecontinuumtheoryandillustratedinthemapinFigure1.Singledomaincrystalsarepossible,basedonacompetitionbetweenthesurfaceenergyandtheenergyassociatedwithdomainboundaries.NegativeBulkModulusEXISTENCESincenegativebulkmodulusmaterialsarenotsubjecttoinstabilityassociatedwithdomainformation,itisnaturaltoaskwhethersuchmaterialsexist.Incrystallinematerials,softeningofthebulkmodulus(analogoustosofteningoftheshearmodulusinferroelastics)hasbeenobservedinYbInCu[18]crystalsatatemperatureof67K.Cerium[19]exhibitsavolumetricphasetransformation[20]inresponsetolowtemperature(116Kat1atm)orhighpressure(7670atmat298K).Thereisa15%volumechangewithnochangeincrystalstructure.Suchtransitionshavebeeninterpretedinthecontextofnegativecompressibility[21]hence,negativebulkmodulusConsiderinthisveinthetemperature pressurephasediagramoftin[22].Tinundergoesatransformationfromtetragonal(whiteor-Sn)todiamondstructure(gray-Sn)witha26%volumeincrease.Thetransformationundertemperaturecontrolisveryslow[23].Hydrostaticcompressionfavorsthehighdensitywhitephase.Therefore,compressionofgraytinunderstresscontrolwillcauseasnapthrougheffect.Inprinciple,ifthiscompressionweredoneunderdisplacementcontrol,anegativebulkmoduluswouldoccur.Ingeneral,atransformationtemperaturewhichdecreaseswithincreasingpressurewillfavorasnapthrougheffect,providedthelowtemperaturephaseislessdensethanthehighdensityphase.Also,analysis[24]ofanIsingmodelofalatticepredicts0nearthecriticaltemperature.1648Y.C.WANGANDR.S.L sufficientlylarge,thebulkmodulusbecomesnegativeandPoissonsratioisless1.Thediagonalstiffness,determinedbytheisotropycondition(5),remainspositivethroughouttherangeofinFigure3.Insummary,negativebulkmodulusmaybeinferredfromthebehaviorofcertainmaterialsinthevicinityofphasetransformationsaswellasfromalatticemodelofacrystal.STABILITYWhileitisknownthatanobjectwithnegativebulkmoduluscanbestableifconstrainedonallitsboundaries,itisofinterestinthecontextofpotentialexperimentswhetheritmightbestableunderpartialconstraintasinatension/compressiontest.Wethereforeconsiderstabilityofnegativebulkmodulusinthecontextofabodyunderdisplacementconstraintalongtwoparallelfaces,asisdoneinameasurementofYoungsmodulusindisplacementcontrol.Inthiscase,thelateralsurfacesareunderaspecifiedstressofzero.Severalcasesoftransversedeformationareconsideredinthecontextofstabilityundercontrolledtransversestress.ConsidertheelementaryisotropicformforHookeslawwithasstrainandasstress. 1E 1E Theassumeddisplacementconstraintentails0.ThenForCase1,letthestressonthelateralsurfacesbeequal:.Then 1E2 Thecompliancefortransversetwo-dimensionalbulkdeformationisthen " Forstabilitywithrespecttotransversedeformation,thiscompliancemustbepositive,so1andThisisthenormalrangeforstabilityofabulkelasticsolidwithallsurfacesfree.CompositeswithInclusionsofNegativeBulkModulus Heresubscript1indicatestheinclusionandsubscript2indicatesthematrix;sentsthecorrespondingvolumefraction.CompositeswhichattaintheHashin Shtrikmanformulaeforbulkmodulushaveamorphologyinwhichthecompositeisfilledwithcoatedspheresofdifferentsizesbutidenticalratioofspheresizetocoatingthickness.Theattainmentisexactforthebulkmodulus[27]andapproximatefortheshearmodulus.Hierarchicallaminates[28]exactlyattaintheshearmodulusformula.TheHashin Shtrikmanformulaerepresentboundsonthebehaviorofanyisotropiccomposite,withthetacitassumptionthatthemodulusofeachphaseispositive.Inthepresentwork,themoduliareallowedtobecomplex,givingrisetoviscoelasticity.Specifically,inwhich.Themechanicaldampingistanwithasthephaseanglebetweenthestressandstrainsinusoids.Similarly,thebulkmodulusbecomesacomplexquantity*.Forviscoelasticcomposites,thedynamicelastic-viscoelasticcorrespondenceprinciple[29,30]isappliedtothesolutionsforelasticconstantsofcompositessothatallelasticconstantsbecomecomplexquantities[31].Figures4 7showtheresultsofthecompositeanalysis.AsshowninFigures4and5,thereisnoeffectduetothebulkmodulusoftheinclusionsonthecompositeshearmodulus;however,thereisastrongeffectonboththecompositebulkmodulusandYoungsmodulus.InclusionsofnegativebulkmodulusgiverisetoareducedYoungsmodulus,ortoanextremehighvalueofYoungsmodulus,dependingontheinclusionofbulkmodulusvalueasshowninFigure4.Alsoshownisagiantpeakintheextensionalmechanicaldampingtan,muchlargerthantheassumedmatrixdampingtanThisbehaviorissimilartothatofcompositeswithinclusionsofnegativeshearmodulus[2].Thedifferenceisthatif,theinclusionsmustbesufficientlysmalltosuppressthe Figure4.CompositeYoungsmodulusE(solidsquares,shortdashcurve),mechanicaldampingtan(triangles,solidcurve)andshearmodulusG(opensquares,longdashcurve)vsinclusionbulkmodulus(allowingnegativevalues)inadiluteHashinShtrikmancompositewith5%byvolumeinclusions.MatrixpropertiesareG19.2GPa,tan0.02,K41.6GPa,0.3,tanCompositeswithInclusionsofNegativeBulkModulus 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