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26420058JournalofQingdaoUniversityofScienceandTechnologyVol.26No.4Aug.2005:167226987(2005)0420331205(,266061):JMAlexander,,22,,:;;;;:TG386.43;TG394;O344.7:ATheoreticalAnalysisofPlasticWrinklinginDielessThin2walledTubeInternalHighPressureFormingTIANZhong2ke(Chinese2GermanTechnologicalFaculty,QingdaoUniversityofScienceandTechnology,Qingdao266061,China)Abstract:Basedontheaxisymmetriccollapsemechanismmodel,proposedbyJ.M.Al2exanderforthethin2walledtubeunderaxialcompression,thispaperintroducesazoningmodeltoanalyzetheendplasticwrinklingcharacteristicofthedielessthin2walledtubeinternalhighpressureforming.Accordingtothethinshellplasticstabilitytheory,basedontheplasticinstabilitycoefficientsandenergyrule,thetheoreticalforcalculatingcriticalloadsofplasticwrinklingontheendsofthehydroformedtubeisestablishedun2dertheconditionofsystempotentialenergyminimization.Keywords:thin2walledtube;internalhighpressureforming;plasticwrinkling;zoningmodel;criticalloads,,,;,()[1],,,,,,[2]JMAlexander:2005203208:(1970),,.26,,22,,1/[3,4]JMAlexander,,3[5],,(1l),Alexander,l(1):1+lEpDY0-tD23l2=0(1),Ep,Y0,D,t1AlexanderFig.1AxisymmetriccollapsemechanismmodelproposedbyJMAlexander,2(2),;,,,2Fig.2Zoningmodelforendplasticwrinklinganalysisofdielessthin2walledtubeinternalhighpressureforming2,[6]FRShanley[7],,,[8]Prandtl2Reuss,dijdeijdpij,dij=deij+dpijHooke,2deij=dsij2G+(1-2)ijd0E2334:E,G,,dsij,ijKronecker,d0Lvy2Mises,dpij=sijdd,d=h(J2)dJ2,h(J2)J2,dJ2J2dij=deij+dpij=dsij2G+(1-2)ijd0E+sijd=1+Edsij+(1-2)Eijd0+sijh(J2)dJ2(2)FRShanley,(2),dij=[aij]dij(3)[aij],i,j=1,2,,6(3)dij=[bij]dij(4)[bij],i,j=1,2,,6Kirchhoff2Love,(z=0)dx=1E(dx-dy)+h(J2)dJ2(x-0)dy=1E(dy-dx)+h(J2)dJ2(y-0)dxy=2(1+)Edxy+2h(J2)dJ2xy:a11=1E+1Ex-12y2a12=-E+1Ex-12yy-12za16=3Ex-12yxya22=1E+1Ey-12x2a26=3Ey-12xxya66=2(1+)E+9E2xy1E=12i1E0t-1E,i,E0t=dxdxxx,E0t,,()(),,,,,52dx5y2+52dy5x2-52dxy5x5y=52w5x5y2-52w5x252w5y2-kx52w5y2-ky52w5x2Donnell[9],w,kxx,kyy(x,y),dx=525y2,dy=525x2,dxy=-525x5ya22545x4-2a26545x35y+(a66+2a12)545x25y2-2a16545x5y3+a11545y4=52w5x5y2-52w5x252w5y2-kx52w5y2-ky52w5x23,,xy,w=Asin(mx/L)cos(ny/R)[10]A,mn,LR,kx=0,33326ky=1/R52w5x5y2-52w5x252w5y2-1R52w5x2=A2222[cos(2x)-cos(2y)]+A2Rsin(x)cos(y)=g1cos(2x)+g2(2y)+g3sin(x)cos(y)+0y2x2+0x2y2g1=A2222a22(2)4,g2=-A2222a11(2)4,g3=A2R[a224+(a66+2a12)22+a114],0x0y1)Ub=12L02R0(dMxdx+dMydy+2dMxydxy)dxdy=t348A2(b114+2b1222+b224+4b6622)RLUm=t2L02R0(dxdx+dydy+dxydxy)dxdy=t4[a11(32g22+g23)4+2a12g2322+a22(32g21+g23)4+a66g2322]RL+t[a11(0x)2+2a120x0y+a22(0y)2]R2)W=0xtL02R0-125w5x2dxdy+0ytL02R0-125w5y2dxdy=-t4A2(0x2+0y2)RL3)=Ub+Um-WddA=0,t212[b114+2(b12+2b66)22+b224]+0x2+0y2+4R2[a114+(2a12+a66)22+a224]=0(5)4)n=0,=0,()m=1,=/L,L=2l,l(1)(5)t212b114+0x2+1R2a22=0(6)(6)Mises(0x)2-0x0y+(0y)2=Y20,0x0y0y=P0iD/2t,0x=F0a/Dt,:F0aP0i(P0i,F0i),PiFa:,,;,,3[1][11],3KSG800Fig.3ExperimentalDataCurvesofInternalHighPressureFormingofKSG8004334:KSG800(1),Pi=2512MPa,Fa=50125kN)4KSG8001KSG800Table1TestdataofinitialyieldloadsofKSG800kPi/MPaFa,n/kNFa,e/kN1.027.5100.032.272.021.5134.581.824KSG800Fig.4ThereactionbetweeninitialyieldloadsofKSG800andpositiondistributionofcriticalplasticwrinklingload4,4,,,[12],,,,,:(Universit¾tPaderborn)LUF(LehrstuhlfrUmformendeundSpanendeFertigungstechnik)[1].[J].,2005,26(3):242245.[2],,,.[J].,2001,8(4),3035.[3]SRGuillow,GLu,RHGrzebieta.Quasi2staticaxialcom2pressionofthin2walledcircularaluminiumtubes[J].Interna2tionalJournalofMechanicalSciences,2001,43:21032123.[4]GLu,RenjieMao.Astudyoftheplasticbucklingofaxiallycompressedcylindricalshellswithathick2shelltheory[J].InternationalJournalofMechanicalSciences,2001,43:23192330.[5]JMAlexander.Anapproximateanalysisofthecollapseofthincylindricalshellsunderaxialloading[J].QuarterlyJour2nalofMechanicsandAppliedMathematics,1960,13:1015.[6]ZPBazant.Structuralstability[J].InternationalJournalofSolidsandStructures,2000,37:5567.[7]FRShanley.InelasticColumnTheory[J].JournalofAero2nauticalScience,1947,14:261268.[8].[M].:,1979.[9]LHDonnell,CCWan.Effectofimperfectionsonbucklingofthincylindersandcolumnsunderaxialcompression[J].JournalofAppliedMechanics,1950,17:73.[10],,.[M].:,1994.[11],.DASYLab[J].,2002,21(3):437438.[12],.[M].:,1983.533