弹性力学课件32

2.9SOLUTIONOFPLANEPROBLEMINTERMSOFDISPLACEMENTForthesolutionofanelasticityproblem,wecanproceedinthreedifferentways:1.Takethedisplacementcomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequations

andboundaryconditions

containingthedisplacementcomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythegeometricalequationsandthenthestresscomponentsbythephysicalequations.

2.Takethestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindthestraincomponentsbythephysicalequationsandthenthedisplacementcomponentsbythegeometricequation.3.Takesomeofthedisplacementcomponentsandalsosomeofthestresscomponentsasthebasicunknownfunctions,formulateasystemofdifferentialequationsandboundaryconditionscontainingthestresscomponentsonly,solvefortheseunknownfunctionsandtherebyfindtheotherunknownfunctions.Nowweproceedtoformulatethedifferentialequationsandboundaryconditionsforsolutionofaplaneproblemintermsofdisplacements.Thegeometricequations：

1.Thedifferentialequations

Thephysicalequations(planestressproblem)SubstitutionofgeometricEotheseequationsNow,usingtheserelationsinequilibriumequationsThedifferentialequationsforthesolutionoftheproblemintermsofdisplacements.

2.Boundaryconditions

lx+mxy=Xmy+lxy=YStressboundaryconditionsofaplaneproblemWeobtainthestressboundaryconditionsoftheproblemintermsofdisplacements：Tosumup,Thedisplacecomponentsu(x,y),v(x,y)inaplanestressproblemmustsatisfythroughoutthebodyconsideredandalsosatisfyonthesurfaceofthebody.Foraplanestrainproblem,itisnecessaryinaboveequations.2.10SOLUTIONOFPLANEPROBLEMINTERMSOFSTRESSESThetwodifferentialequationsofequilibriumcontainthestresscomponentsonlyandmaybeusedfortheirsolution.Thethirddifferentialequationcanbeobtainedfromthegeometricalandphysicalequationsbyeliminatingthedisplacementcomponentstherein.ThegeometricalequationsofaplaneproblemareAddingthesecondderivativeofxwithrespecttoyandthesecondderivativeofywithrespecttox,wegetThecompatibilityequationforstrainThecompatibilityequationforstrainmustbesatisfiedbythestraincomponentsx,yandxytoensuretheexistenceofsingle-valuedcontinuousfunctionsuandvconnectedwiththestraincomponentsbythegeometricalequations.若所选选的x、y和xy不满足足这个个方程程，那那么，，由几几何方方程中中的任任意两两个所所求出出的位位移分分量，，将不不满足足第三三个方方程例如选x=0，y=0，xy=cxy

不满足相容方程由此应应变求求位移移第三个个方程程不能能满足足，所所求u，v不存在在Byusingphysicalequations,thecompatibilityequationcanbetransformedintoarelationbetweenthestresscomponents.Substitutionofthephysicalequationsinto

Foraplanestressproblem

Totransformthisequationintoadifferentformmoresuitableforuse,weeliminatetheterminvolvingxybyusingthedifferentialequationsofequilibrium.Differentiatingthefirstequationwithrespecttoxandthesecondwithrespecttoy,addingthemupandnotingthatxy=yx,wegetSubstitutingthisintothecompatibilityequationandperformingsomesimplification,weobtainthecompatibilityequationintermsofstresses.Foraplanestrainproblem,anequationsimilartoaboveequationmaybeobtainedsimplybyTheresultisThus,1.Inthesolutionofaplaneproblemintermsofstresses,thestresscomponentsmustsatisfythedifferentialequationsofequilibriumandthecompatibilityequation.Besides,theymustsatisfythestressboundaryconditions.2.Sincethedisplacementboundaryconditionscanbeexpressedneitherintermsofstresscomponentsnorintermsoftheirderivativeswithrespecttothecoordinates,displacementboundaryproblemsandmixedboundaryproblemscannotbesolvedintermsofstresses.Inthesolutionofelasticityproblems,itisnecessarytodistinguishbetweensimplyconnectedbodiesandmultiplyconnectedones.Abodyissaidtobesimplyconnectedifanarbitraryclosedcurvelyinginthebodycanbeshrunktoapoint,bycontinuouscontraction,withoutpassingoutsideitsboundaries.Solidblocksandhollowspheresareexamplesofsimplyconnectedbodies.Otherwise,thebodywillbesaidtobemultiplyconnected.Ringsandhollowcylindersareexamplesofmultiplyconnectedones.Inthecaseofmultiplyconnectedbody,theremightbesomearbitraryfunctionsleadingtomulti-valueddisplacements,whichareimpossibleinacontinuousbody.Then,wehavetoconsidertheconditionofsingle-valueddisplacementstodeterminethestresses.Inplaneproblems,however,wemayalsobrieflydefineasimplyconnectedbodyasonewithonlyonecontinuousboundaryandamultiplyconnectedbodyasonewithtwoormoreboundaries.2.11CASEOFCONSTANTBODYFORCESInmanyengineeringproblems,thebodyforcesareconstant,I.e.,thecomponentsXandYdonotvarywithcoordinatesthroughoutthevolumeofthewholebody.（thegravityforces,theinertiaforces）Ontheconditionofconstantbodyforces,thecompatibilityequationswillreducetothehomogeneousdifferentialequationNow,thedifferentialequationsofequilibriumandthestressboundaryconditions,aswellasthecompatibilityequations,donotcontainanyelasticconstantandarethesameforbothkindsofplaneproblems.Hence,inastressboundaryproblemforasimplyconnectedbodywithacertainboundaryandsubjectedtocertainexternalforces,thestresscomponentsx,y,xywillbeindependentoftheelasticpropertiesofthebodyandhavethesamedistributioninbothplanestressconditionandplanestraincondition.Thisconclusionisveryhelpfulintheexperimentalanalysisofthestressesinastructureoritselements.（1）可将将某种材料料，某种状状态下所求求的应力分分量的结论论用于其他他材料或其其他状态（（边界条件件，外荷载载相同）（2）Wemayuseamodelinplanestresscondition(athinslice)insteadofoneinplanestraincondition(alongcylindricalbody).（1）Wemayuseanymodelmaterialconvenientforstressmeasurementinsteadofthestructurematerialonwhichthemeasurementmightbeimpossible.(3)Beside,inthecaseofastressboundaryproblemforsimplyconnectedbodiessubjectedtoconstantbodyforces,astressanalysisfortheactionofthebodyforcesmaybeconvertedtotheanalysisfortheactionofsurfaceforces.Thestresscomponentsx,y,xyaredeterminedbythedifferentialequations（a）（b）andtheboundaryconditionsl(x)s+m(xy)s=Xm(y)s+l(xy)s=YNow,wesetx=’x-Xx,y=’y-Yy,xy=’xyandproceedtofindthedifferentialequationsandboundaryconditionswhichmustbesatisfiedby’x,’y,’xy.(c)Substitutetheminto(a),(b)and(c)andobtainl(’x)s+m(’xy)s=X+lXxm(’y)s+l(’xy)s=Y+mYyWeseethatthedifferentialequationsandboundaryconditionstobesatisfiedby’x,’y,’xymustbethesameasthoseinaproblemwithzerobodyforcesandwithsurfaceforcecomponentsXandYincreasedbylXxandmYy,respectively.Thisconclusionsuggestsaprocessforthesolutionofx,y,xy:NeglectthebodyforcesandapplyfictitioussurfaceforcecomponentsX”=lXxandY”=mYyinadditiontotheoriginalsurfaceforces;Solveforthestresscomponents,’x,’y,’xy,byappropriatemethods;Findx=’x-Xx,y=’y-Yy,xy=’xy2.12AIRY’’SSTRESSFUNCTION.INVERSEMETHODANDSEMI-INVERSEMETHODisnonhomogeneousand,therefore,itsgeneralsolutionmaybeexpressedasthesumofaparticularsolutionandthegeneralsolutionofthehomogeneoussystemWhenbodyforcesareconstant,theparticularsolutionmaybetakenasx=-Xx，y=-Yy，xy=0orx=0，y=0，xy=-Xy-Yxorx=-Xx-Yy，y=-Xx-Yy，xy=0Whichsatisfyequations(1)(2)Accordingtodifferentialcalculus,for(1),thereexistsacertainfunctionA(x，y)sothat：RewriteSimilarly,(2)ensurestheexistenceofanotherfunctionB(x，y)sothat：Sincexy=yx,wehaveWhichensurestheexistenceofstillanotherfunction(x,y)sothatWeobtainthegeneralsolutionofhomogeneousequations：Now,thesuperpositionofthegeneralsolutionwiththeparticularsolutionyieldsthefollowingcompletesolution:Thefunction(x,y)isknownasthestressfunctionforplaneproblems,ortheAiry’sstressfunction.Inorderforthestresscomponentstosatisfythecompatibilityequationaswell,thestressfunctionmustsatisfyacertainequation.or

2(Xx)=

2(Yy)=0intheconditionofconstantbodyforceorbesimplywrittenas：Whenbodyforcesarenotconsidered,thesolutionwillreducetoThus：inthesolutionofplaneproblemsintermsofstresses,whenthebodyforcesareconstant,itisonlytosolveforthestressfunctionfromthesingledifferentialequationandthenfindthestresscomponentsbyButthesestresscomponentsmustsatisfythestressboundarycondition.Inthecaseofmultiplyconnectedbodies,theconditionofsingle-valueddisplacementsmustbeinspectedinaddition.Tosolvethepartialdifferentialequationsofelasticitytogetherwiththegivenboundaryconditions,thedirectmethodofsolutionisusuallyimpossible.Wehavetousetheinversemethodorthesemi-inversemethod.Intheinversemethod,somefunctionssatisfyingthedifferentialequationsaretakenandexaminedtoseewhatboundaryconditionsthesefunctionswillsatisfyandtherebytoknowwhatproblemstheycansolve.InthecaseofsolutionbyAiry’sstressfunction,weselectsomefunctionsatisfyingthecompatibilityequation,findthestresscomponents,andthenfindthesurfaceforcecomponents.Inthisway,weidentifytheproblemwhichthestressfunctioncansolve.Inthesemi-inversemethod,weassumethesolutionforthestressesordisplacementsinagivenproblem,basedontheloadingconditionandboundaryconditionsoftheproblem,andthenproceedtoshowthatallthedifferentialequationsandboundaryconditionsaresatisfied.InthecaseofsolutionbyAiry’sstressfunction,wemakeassumptionsregardingthestresscomponents,findthecorrespondingstressfunctionandproceedtoshowthatthestresscomponentsderivedfromthisstressfunctioncansatisfyalltheboundaryconditions.Ifsomeoftheboundaryconditionsarenotsatisfied,thenwehavetomodifytheassumptionsmade.本章小结1、平面问题平面应力问题和平面应变问题2、平面问题题的基本未知知量：x、y、xy、x、y、xy、u、v3、平面问题题的基本方程程和边界条件件：平衡微分方程：几何方程：物理方程：平面应力问题：平面应变问题题边界界条条件件：：位位移移边边界界条条件件和和应应力力边边界界条条件件lx+mxy=Xmy+lxy=Y应力边界条件4、、平平面面问问题题的的求求解解方方法法：：按按位位移移求求解解；；按按应应力力求求解解按位位移移求求解解，，要要求求位位移移满满足足拉拉密密方方程程，，在在边边界