DOC

Stress field of orthotropic cylinder subjected to axial compression

By Judith Young,2014-02-18 23:33
11 views 0
Stress field of orthotropic cylinder subjected to axial compressionof,to,field,axial

    Stress field of orthotropic cylinder subjected

    to axial compression

    App1.Math.Mech.Eng1.Ed.31(3),305316(2010)

    DOI10.1007/s104830100304z

    ?ShanghaiUniversityandSpringer-Verlag

    BerlinHeidelberg2010

    AppliedMathematics

    andMechanics

    (EnglishEdition)

    Stressfieldoforthotropiccylindersubjectedtoaxialcompression

    WeizhouZHONG(钟卫洲),,ShunchengSONG(宋顺成),GangCHEN(陈刚).,

    Xi.chengHUANG(黄西成),PengHUANG(黄鹏)

    (1.SchoolofMechanicsandEngineering,SouthwestJiaotongUniversity, Chengdu610031,P.R.China;

    2.InstituteofStructuralMechanics,ChinaAcademyofEngineeringPhysics Mianyang621900,SichuanProvince,P.R.China)

    (ContributedbyShunchengSONG)

    AbstractBased0ntheconstancyhypothesisofmaterialvolume,thecircumferential andradialstressesofacylinderspecimenareanalyzedwhenthecylinderissubjecttoa loadingalongtheaxialdirection.Thecircumferentialandradialstressdistributionisa powerfunctionofradiusparameterwhentheconstitutiverelationofspecimenmaterial isorthotropic.Thestressdistributionisaquadraticfunctionofradiusparameterfor transverselyisotropicmateria1.Alongthecylinderaxialline.thecircumferentialand radialstressesaremaximumandequaltoeachother.Inthecircumferenceboundary surface.theradialstressiszeroandthecircumferentialstressvalueisminima1.The failuretheoryofmaximumtensilecircumferentialstrainisappliedtocalculatethecritical axialloading.Thecircumferenceboundary.1ayerfailurecriterionoforthotropiccylinders

isdescribedwiththeHillTsaistrengththeory.Theobtainedstrengththeoryisrelated

    toaxialstressandmechanicalpropertiesofspecimenmaterialandtothespecimenaxial

    deformationstrainrateandthechangerateofstrainrate.

    Keywordsorthotropic,axialcompression,axialsymmetry,stressdistribution,strain rate

    ChineseLibraryClassification0344.3

    2000MathematicsSubjectClassification74K10,74D05

    1Introduction

    Recently,manyanisotropicmaterialsareappliedwidelyinindustryandintheciviland weaponfields.Foranisotropicmaterials,basicmechanicalpropertiessuchaselasticmodulus,

    Poissonratio.andtensionandcompressionstrengthesvaryindifferentmaterialorientations. Therefore,theanalysisonanisotropicmaterialstructuresisquitecomplicated.Foranisotropic

    materials.thestudyontheconstitutiverelationandfailurecriterionissignificantinevaluat

    ingstructurereliability.Manyresearchershavedonemuchstudyontheconstitutiverelation andthenumericalsimulationmethodsoforthotropicmaterials.Cazacueta1.lJintroduceda

    ReceivedNov.5,2009/RevisedJan.11,2010

    ProjectsupportedbytheNationalNaturalScienceFoudationofChina(No.50874095)andtheNa-

    tionalBasicResearchProgramofChina(973Program)

    CorrespondingauthorWeizhouZHONG,Ph.D.,E-mail:wz_zhong@sina.com

    306WeizhouZHONG,ShunchengSONG,GangCHEN,Xi

    chengHUANG,andPengHUANG

    macroscopicorthotropicyieldcriterionthatcandescribeboththeanisotropyofmaterialsand theyieldingasymmetrybetweentensionandcompression.Theyieldfunctionisexpressedin termsoftheprincipalvaluesofthestressdeviator.Plunketteta1.[21proposedayieldfunction describinganisotropicbehaviorsoftexturedmetals.Theproposedanisotropicyieldfunction hasgreataccuracyboththetensileandcompressiveanisotropyinyieldstresses.Basedon theenergyprincipleofbucklingandthemixedhardeningconstitutiverelationoforthotropic

materials.ZengandFudeducedtheformulaofelastoplasticcriticalstressesandcalculated

    criticalstressesofsimply.supportedorthotropiccircularcylindricalshellsunderaxialcompres

    sion.AbdAUaandFarhan[4Jdevelopedthesolutionofthenon-homogeneousorthotropicelastic

    cylinderforplanestrainproblems.Analyticalexpressionsfordisplacementandstresseswereob

    tained.Basedontheelasto

    plasticmechanicsandthecontinuumdamagetheory,TianandFuJ

    proposedayieldcriterionthatrelatestothesphericaltensorofstressanddiscussedanelasto plasticbucklingproblemofanorthotropicthinplate.Jefferyeta1.Jsimulatedbiologicalsoft tissuebytheorthotropichyperelasticconstitutivelaw.RomashchenkoandTarasovskaya[Jcar

    riedoutnumericalstudiesonthedynamicbehaviorofmultilayerthick-walledcylindricalshells

    withdifferentspiralreinforcementstructures.RedekoplSJmadeuseofdifierentialquadrature

    methodstopredictthebucklingcharacteristicsofanorthotropicshellofrevolutionofarbitrary

    meridiansubjectedtoanormalpressure.GrigorenkoandRozhok[9JadoptedadiscreteFourier

    seriesapproachtoanalyzethestressstateoforthotropicandtransverselyisotropicelliptical hollowcylinders.Manyresearchers[10-141havedonemuchstudyonthecomputationmethods

    oforthotropicmaterialstructuresaswel1.

    Fororthotropicmaterialswithtensionstrengththatisnotconsistentwithcompression strength.thedamagemodeisnotcrushingalongtheloadingdirectionortheinclinedshear

    failureinthequasistaticandHopkinsondynamiccompressionexperiments.Thedominant damageoforthotropicspecimenisthetensionfailurealongthespecimenedge.Forexample, whenthecompressionloadingisalongthegrainofwood,thecircumferentia1tensionfailure

    occursManypapers[15l9Jfocusonthewoodstressandstraindistributionsunderthegrain 1oading,theuniaxialstressstatefailure,andthemechanicalbehaviorsofmulti

    axesloading

    laminatedstructure.FOrconcreteandrockmaterialswithtensionstrengththatismuchless thancompressionstrength,therearfacespallingorradialcircumferentialdamagegeneratesun

    dertensilestress.XuandZhangI20Jtookthedamagevariableasaweightedfactortocalculate theequivalentshearstrengthparametersofanyspacialsectioninaweightedfashionwiththe shearstrengthparametersofintactrockandjoints.Theshearstrengthparametersofrockmass werefittedasorthotropicequivalentshearstrengthparametersbyuseofthespacialequivalent shearstrengthparameters.Recently,muchstudyisonstructureexperimentsandnumerical simulationsoforthotropicmaterials.Theresearchonthedynamicalmechanicalpropertiesof orthotropicmaterialsisrarelydone.Inordertoknowthefailuremechanismsoforthotropic materials.itisnecessarytoanalyzestressdistributionsunderstaticanddynamicloadings. Basedontheconstancyhypothesisofmaterialvolume.thecircumferentialandradialstresses ofacylinderspecimenunderaxial1oadingareanalyzedinthework.Thecircumferentialand radia1stressdistributionexpressionsareobtainedfororthotropicandtransverselyisotropicma-

    terials.Accordingtothefailurecriterionofmaximumtensilestrainthecriticalaxialloading i8obtained.Thecircumferenceboundary

    layerfailurebehaviorisdescribedwiththeHillTsai

    strengththeory.Thefailureexpressionisrelatedtothestrainrateofspecimenaxialdeforma

    tion.Thefailurebehaviorsofwood.concrete.androckcylinderssubjectedtoaxialcompression

    loadingarethendiscussed.

    2Relationofaxialloadingversuscircumferentialstrain

    Giventhematerialvolumeasconstant,thetheoreticalanalysisonthecompressionbehavior ofacylinderspecimensubjectedtoaxialloadingisdone.Ifthefrictioncoefficientofloading Stressfieldoforthotropiccylindersubjectedtoaxialcompression307

    (a)Beforecompressionfb)Aftercompression

    Fig.1Axialcompressionmodelofcylinder Theaxialstrainzandthecircumferentialstraincanbewrittenas T

    hend

    Ezn,

    

    1.

    27rRend1Rend

     nn

    (1)

    C

    ,

    Ordingtothematerialvolume'sconstancyhypothesis,thefollowingequationcanbe

    oerlved:

    Equation(3)canbesimplifiedas

    0=7rRdh.d

    ,,Rend,/2h0

    hend

    (3)

    AccordingtoEqs.(1)(3),therelationbetweentheaxialstrainzandthecircumferentia1 strainCOcanbewrittenas

    2eo+=0(4)

    Equation(4)indicatesthattheaxialcompressionstrain'Sabso1utevalueistwiceofthe

    circumferentialstrain'Sonewhenthematerialvolumeisconstant.

    Iftheloadingis1e8sthan

    thespecimenaxialcompression'8elasticlimitvalue,theaxialstraincanbeexDres8eda8

    P

    EaTcR2

    end

    '

(5)

    whereistheaxialelasticmodulus.

    Accordingtof4),thefollowingequationcanbederived JF)=22dlnRend

    =

    27rd

    Equation(6)expressestherelationofaxialloadingversuscircumferentia1sirain.Ifthecircu

    m

    ferentialfailurestrainissmall,thecircumferentialtensionfailureoccurswhenthespecimenis

    

    ?

    培‰

    曲眦l喜】

    

    n

    m

    m

    m

    ?啪舭基

    ?f=

    q;

    

    308WeizhouZHONG,Shun-chengSONG,GangCHEN,Xi

    chengHUANG,andPengHUANG

    loadedalongtheaxialdirection.AcCordingto(2)and(6),thecriticalaxialloadingPcanbe

    writtenas(7)forthecaseofcircumferentialtensionfailure: P27rR3e.gO,

    whereg0isthematerialcircumferentialtensionfailurestrain 3Responseandstressdistributionunderaxialcompressionloading 3.1Dimensionvarianceandradialacceleration

    Thespecimenradiusisrandtheaxiallengthishinthecompressionprocess,asshown inFig.1.Takingthematerialvolumeconstancyintoaccount,wecanobtainthefollowing equation:

    dV=d(zrrh)=27rrhdr+7rr.d=0(8)

    Equation(8)canbesimplifiedas2hdr:-rdh.

    Asthestructureandloadingareaxisymmetric,thecorrespondingmechanicalresponse shouldbeaxisymmetric.Thecircumferentialvelocityandaccelerationareequaltozero.The radialvelocitycanbewrittenas

    TheradialaccelerationiS

    drrdh

    2hdt

    n=:一旦

    2hdt()(一去)一一J一一/

    Accordingto(9),(10)canbesimplifiedas

    37'/,,.rd2h

    IJ

    AsE=dh/hand=dh/hdt,(11)canbewrittenas

    rz

    26c

    rd2h

    2hdt2

    Accordingto(1),therelationoftheinitialheightandtheendheightisshownas hend=hoe

    Accordingto(3)and(13),thefollowingrelationcanbegained

    whichcanbesimplifiedas

    7rR0=7rRdh0e,

    Red=R0e-gz/

    (10)

    (13)

    Equation(15)istherelationbetweentheinitialradiusandtheendradius.Theendradiusis

relatedtoe一;/2.

    T

    l1

    Stressfieldoforthotropiccylindersubjectedtoaxialcompression309 3.2Radialandcircumferentialstressdistributions

    Fortheaxisymmetricstructureandloadingproblems,thestressdistributionisaxisymmetric

    Therefore,theradialandcircumferentialstressesareonlythefunctionsofradiusparameter

    TheshearstressesTrOandTOrareequaltozero.ThesketchisshowninFig.2. oe+doe

    Fig.2Sketchofplaneinfinitesimalanalysis

    Accordingtotheelasticmechanicstheory[.thedeformationconsistentequationis E=0(16)

    Assumethatthestressfunctionis=to-randthedensityisP. Thestresscomponentsare

    =,=

    ctr

    +r

    r'

    Thephysicsequationsoftheaxisymmetricprobleminthecylindricalcoordinatesystemare

    .

    (torO0"rzeoz

    印一一,(19)

    where,Ee,andEzarethematerialradial,circumferential,andaxialelasticmoduli.Ac

    cordingto(12)and(17)(19),(16)canbewrittenas

    1d

    .//1.,1d1

    Eodr2l\\EeErEe)rdrErr2

    +

    1

    )

    Fororthotropicmaterial,therelationofengineeringelasticconstantsis

    

    /]ro

    

    ErEe.

    Accordingto(21),(20)canbesimplifiedas

    1d2T

    +

    警一1+11dgz/1:.

    Thegeneralsolutionto(22)isshownas r一瓜+rB+EoEr(vzO-Vzr)Crz

    (20)

    (22)

    r

    (笠一.1diz)Eo42dt)(23)9\'

    

    

    t}1

    I

    310Wei-zhouZHONG,Shun-chengSONG,GangCHEN,XichengHUANG,andPengHUANG wheretheparametersAandBareintegralconstants.Accordingto(17)and(23),thestress

    expressionscanbewrittenas :(1+,)+r,1B

    .(日一b'z)

    '(Eo)Ez

    (3+)

9ErEe(2)

    p:,(1+Eov~)A+,晡:1B

    +!!

    .(Ee,}

    (Ee+

    ;3Erver)(生一.1dizI42dt29\

    (24)

    (25)

    Foracylinderspecimen,thestressvalueiSlimitedalongtheaxisofsymmetry;since1+ ,,//>1,theintegra1constantAshouldbeequaltozero;theradialstressequalszeroin

    thecircumferencesurface;theintegralconstantBisexpressedas fes/21

    +

    p(1diz)().一一

    AccordingtotheexpressionsofparametersAandB,(24)and(25)canbewrittenas

    [(R0e-~o/2)卜一r]

    +P

    (学一2dt][e-~=/2).一一r一:

    EeEr(t,zel,zr,=

    ?—?(R.e.)卜一r]

    (26)

    (27)

    +p

    (譬一1di,)[(Roe-~-J2).r-1

    

    r

    ].(28)9ErEol'

    Equations(27)and(28)indicatethatcircumferentialandradialstressdistributionsare

    powerfunctionsofradiusparameterfororthotropiccylindersunderaxial1oading.Thestress

Report this document

For any questions or suggestions please email
cust-service@docsford.com