Anisotropic properties of mechanical characteristi

arXiv:cond-mat/0607232v1[cond-mat.mtrl-sci]10Jul2006AnisotropicpropertiesofmechanicalcharacteristicsandauxeticityofcubiccrystallinemediaT.Paszkiewicz∗andS.WolskiChairofPhysics,Rzesz´owUniversityofTechnology,Al.Powsta´nc´ow.Warszawy6,PL-35-959Rzesz´owPolandAbstractExplicitexpressionsforinverseofYoung’smodulusE(n),inverseofshearmodulusG(n,m),andPoissonsratioν(n,m)forcubicmediaareconsidered.AllthesecharacteristicsofelasticmediadependoncomponentsS11,S12andS44ofthecompliancetensorS,andondirectioncosinesofmutuallyperpendicularvectorsnandmwithfourfoldsymmetryaxes.Thesecharacteristicsarestudiedforallmechanicallystablecubicmaterialsforvectorsnbelongingtotheirreduciblebodyanglesubtendedbythreecubichighsymmetrydirections[001],[111],and[110].Regionsinthestabilitytriangleofinwhichcubicelasticmaterialsarecompletelyauxetic,nonauxetic,andauxeticareestablished.Severalintermediate-valencecompoundsbelongingtotheregionofcompleteauxecityareindicated.TheextremepropertiesofE−1,G−1andνestablishedbyHayesandShuvalovareconfirmed.PACSnumbers:62.20.Dc,81.40.Jj,61.50.AhKeywords:cubicelasticmaterials,Young’sandshearmoduli,negativePoisson’sratio,auxetics∗Electronicaddress:[e-mail]opertiesofelasticmediaaredescribedbythebulk,Young’sandtheshearmodules.Poisson’sratioisanotherimportantmechanicalcharactersiticsofsuchmaterials.Withtheexceptionofbulkmodulus,generally,thesecharacteristicsareanisotropic,i.e.theydependondirectionofloadanddirectioninwhichtheresponseofamaterialismeasured.ThePoissonratioisespeciallyinteresting,becauseitcanbenegative.MaterialswhichexhibitanegativePoisson’sratioarereferredtoasauxeticmedia[1,2].Auxeticsrespondtoimposeduniaxialtensionwithlateralextensioninplaceofexpectedcontraction.LakesdescribedthesynthesisofanactualauxeticmaterialandproposedasimplemechanismunderlyingthenegativePoisson’sratio[3].Suchmediacanfindinterestingapplicationsinfuturetechnologies.TingandBarnettintroducedclassificationoftheauxeticbehaviorofanisotropiclinearmedia[4].Amongthelinearelasticmedia,aspecialattentionhasbeendevotedtocubicmedia[1,2].TurleyandSinesusedthetechniqueofrotationsbyEulerangles[5],andconsideredanisotropyofYoung’smodulus,shearmodulus,andPoisson’sratioincubicmaterials.UsingthetechniquedevelopedbyTurleyandSines,GuntonandSaundersstudiedstabilitylimitsonthePoisson’sratio,andconsideredmartensitictransformations[6].Milstein[7,8],Huangetal.[9]analysedresponsesofidealcubiccrystalstouniaxialloadings.Poisson’sratioforcubicmaterialswasstudiedbyMilsteinandHuang[10],JainandVerma[11],GuoandGoddard[12],ErdacosandRen[13],andTingandBarnett[4].Withtheexceptionofthelastpaper,theauthorswerelookingfordirectionsforwhichPoisson’sratioisnegative.IntheopinionofBaughmanetal.,negativePoissonsratiosareacommonfeatureofcubicmedia[1].Theyreferspecialytocubicmetalsinglecrystals.Ourresultssupportthisopinion[14].AnattempttostudymoreglobalpropertiesofcubicauxeticswasmadebyTokmakova[15].Todescribeelasticpropertiesofcubicmedia,sheusedthefamiliarelasticparametersC11/C44andC12/C44.Thepurposeofourpaperistostudyanisotropyofallthreemechanicalcharacteristicsofallmechanicallystableelasticmedia.Toachievethisgoal,weshallusedifferentparameters,whichwereintroducedbyEvery[16]andwhichwereusedbyusfortheunifieddescriptionofelastic,acousticandtransportpropertiesofcubicmedia[17–20].2II.ELASTICPROPERTIESOFCUBICMEDIAConsiderhomogeneousmechanicallystableanisotropicelasticmaterial.ItischaracterizedbythecomponentsSijkl(ij,k,l=1,2,3)ofthecompliancetensorS,orbythecomponentsCijkl(ij,k,l=1,2,3)ofthestiffnesstensorC.Boththesetensorshavethestructure[[V2]2][21],i.e.theyhavethecompleteVoigt’ssymmetry,e.g.Cijkl=Cji,kl=Ckl,ij.Thecomplianceandstiffnesstensorsaremutuallyinverse,i.e.S=C−1.(1)AfterWalpole[22],wedenotedegenerateeigenvaluesofCbycJ,cL,andcM.Onehas[23]cJ=C11+2C12,cL=2C44,cM=C11−C12,sJ=c−1J,sL=c−1L,sM=c−1M.(2)SimilarlysJ,sL,andsMareeigenvaluesofS.TheeigenvaluessJ,sL,andsMcanbewrittenintermsofS11,S12andS44,namely,sJ=S11+2S12,sL=S44/2,sM=S11−S12.(3)UsingtheserelationsandthesecondlineofEqs.(2),weexpressthematrixelementsSUVbyCUV(U,V=1,2,4)S11=C11+C12cJcM,S12=−C12cJcM,S44=1C44.(4)Theseexpressionsareinagreementwiththefamiliarresults[24].OnemayuseeigenvaluescJ,cL,andcMinplaceofthreeelasticconstantsC11,C12,andC44,butthemorereasonablechoiceistheuseofthreeparametersintroducedbyEvery[16],namely,s1=(C11+2C44)andtwodimensionlessparameterss2=(C11−C44)s1,s3=(C11−C12−2C44)s1.(5)Theinverserelationshold[20]C11=s1(2s2+1)/3,C12=s1(4s2−3s3−1)/3,C44=s1(1−s2)/3.(6)TheelementsofCandeigenvaluescU(U=J,L,M)areproportionaltos1.FromEq.(1),itfollowsthatSij∝s−11.Therefore,weintroducethedimensionlesscomponentsofthecompliancetensorS′ij=s1Sij.3Themechanicalstabilityisguaranteedwhenallintroducedeigenvaluesarepositive.Thisconditionyieldsfamiliarinequalities[25](A11+2A12)0,(A11−A12)0,A440,(A=C,S).(7)IntermsofEvery’sparameters,cubicmaterialisstable,ifs10ands2,s3belongtothestabilitytriangle(STforshort)intheplane(s2,s3)(cf.Fig.1)(1+2s2)|4s2−3s3−1|,(10s2−6s3)1,s21.(8)Trojstab.pngFIG.1:Triangleabcof