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1、090801109劉維龍090801117劉帥192—194頁(yè)原文Thus,foranaxisymmertricproblem,ifwesucceedinfindingaharmonicfunctionΨ(r,z>sothatthedisplacementcomponents(9.4.8>andthestresscomponents(9.4.10>satisfyalltheboundaryconditions,therightsolutionisobtained.b5E2RGbCAPWhenadisp
2、lacementpotentialexists,thevolumestraineisconstantthroughoutthebodyconcerned,asshownbyEqs.(9.4.3>and(9.4.4>.Sincesuchcircumstancesrarelyoccur,fewproblemscanbethoroughlysolvedbyusingdisplacementpotentialonly.Hence,formanypracticalproblems,wehavetoresortt
3、omorecomplicateddisplacementfunctions.Sometimes,however,adisplacementpotentialcanbeusedasasupplementarydisplacementfunctiontoreducethemathematicaloperations.p1EanqFDPw9.5LOVE’SANDGALERKIN’SDISPLACEMENT’FUNCTIONSForthesolutionofaxisymmertricspatialproble
4、ms,A.E.H.Loveintroducedadisplacementfunction(r,z>andexpressedthedisplacementcomponentsas:DXDiTa9E3d(9.5.1>Where.ThevolumestraingivenbyEq.(8.8.6>becomesRTCrpUDGiT(9.5.2>WithEqs.(9.5.1>and99.5.2>,thefristofEqs.(9.4.6>isidenticallysatisfiedandthesecondequa
5、tionrequires,whichshowsthat?mustbeabiharmonicfunction.SubstitutingEqs.(9.5.1>and(9.5.2>intoEqs.99.1.4>andnotingthat,wehave5PCzVD7HxA8/8(9.5.3>Thus,anaxisymmertricproblemcanbesolvedifwesucceedinfindingaproperbiharmonicfunction.sothatthedisplacementcompon
6、entsgivenbyEqs.(9.5.1>andthestresscomponentsgivenbyEqs.(9.5.3>satisfytheboundaryconditions.jLBHrnAILgForthesolutionsofspatialproblemsingeneral,B.GalerkingeneralizedtheLove’sdisplacementfunctionandexpressedthedisplacementcomponentsintermsofthreedisplacem
7、entfunctions(x,y,z>,(x,y,z>,(x,y,z>asxHAQX74J0X(9.5.1>Where.ThevolumestraingivenbyEqs.(9.1.2>becomesLDAYtRyKfE(9.5.5>SubstitutingEqs.(9.5.4>and(9.5.5>intoEqs.(9.4.1>.weseethatthethreedisplacementfunctionsmustsatisfythatis,theymustbebiharmonicfunctions.S
8、ubstitutionofEqs.(9.5.4>and(9.5.5>intoEqs.(9.1.1>yieldsZzz6ZB2Ltk8/8(9.5.6>(9.5.7>Thus,aspatialproblemingeneralcanbesolvedifwesucceedinfindingthreebiharmonicfunctions,,,sothatthedisplacementcomponentsgivenbyEqs.(9.5.4>andthestres
