弹性力学课件三一

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,weget

ThecompatibilityequationforstrainThecompatibilityequationforstrainmustbesatisfiedbythestraincomponentsx,yandxytoensuretheexistenceofsingle-valuedcontinuousfunctionsuandvconnectedwiththestraincomponentsbythegeometricalequations.若所选的x、y和xy不满足这个方程，那么，由几何方程中的任意两个所求出的位移分量，将不满足第三个方程例如选x=0，y=0，xy=cxy

不满足相容方程由此应变求位移第三个方程不能满足，所求u，v不存在Byusingphysicalequations,thecompatibilityequationcanbetransformedintoarelationbetweenthestresscomponents.

Substitutionofthephysicalequationsinto

Foraplanestressproblem

Totransformthisequationintoadifferentformmoresuitableforuse,weeliminatetheterminvolvingxy

byusingthedifferentialequationsofequilibrium.

Differentiatingthefirstequationwithrespecttoxandthesecondwithrespecttoy,addingthemupandnotingthatxy=yx,wegetSubstitutingthisintothecompatibilityequationandperformingsomesimplification,weobtainthecompatibilityequationintermsofstresses.Foraplanestrainproblem,anequationsimilartoaboveequationmaybeobtainedsimplybyTheresultis

Thus,1.Inthesolutionofaplaneproblemintermsofstresses,thestresscomponentsmustsatisfythedifferentialequationsofequilibriumandthecompatibilityequation.

Besides,theymustsatisfythestressboundaryconditions.2.Sincethedisplacementboundaryconditionscanbeexpressedneitherintermsofstresscomponentsnorintermsoftheirderivativeswithrespecttothecoordinates,displacementboundaryproblemsandmixedboundaryproblems

cannotbesolvedintermsofstresses.

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