压电热弹性材料四边简支层合板的精确解

254Vol.25No.420084Apr.2008ENGINEERINGMECHANICS2302006-09-292007-04-03(07JCYBJC02100)(1971)(E-mail:[e-mail])*(1978)(E-mail:[e-mail])(1968)(E-mail:[e-mail]).1000-4750(2008)04-0230-06*(300300)--O326;TB33AEXACTIONSOLUTIONSFORSIMPLYSUPPORTEDRECTANGULARLAMINATEDPIEZOTHERMOELASTICPLATESLIUYan-hong,*LIJia-yu,QINGGuang-hui(AeronauticalEngineeringCollege,CivilAviationUniversityofChina,Tianjin300300,China)Abstract:Baseduponpiezothermoelasticmaterialgoverningequations,thehomogeneousstateequationforthemechanical,electric,thermalcouplingproblemofpiezothermoelasticmaterialsisderivedbymeansofreformingtheconstitutiveequationsofpiezothermoelasticmaterialsaswellascombiningthepiezothermoelasticmaterialthermalequilibriumequations.Thehomogeneousstateequationcanbesolvedindependentlythroughthepreciseintegrationmethod.Thetreatmentsimplifiesthesolutionprogramsoftenusedtosolvesecondorderdifferentialequationonthethermalequilibriumandgradientrelationship,anddecreasesgreatlytheworkloadofnumericalcomputation.Keywords:piezothermoelasticmaterials;laminatedplates;exactsolution;preciseintegrationmethod;homogeneousstateequationMindlin[1]--Nowacki[2]Iesan[3]Tauchert[4]Jonnalagadda[5]Tauchert[6]--[7]Kapuria[8]231VelBatra[9][1011][12]1TT−−σ=cεeEλTD=eε+κE+γ(1)σ(ijσ,,,,ijxyz=)ε(ijε,,,,ijxyz=)λ-(iλ,1,2,3,4,5,6i=)c(ijcijjicc=,1,2,3,4,5,6ij=)e(ije,1,2,3i=1,2,3,4,5,6j=)E(iE,,ixyz=)D(iD,,ixyz=)κ(ijκijjiκκ=,1,2,3ij=)γ(iγ1,2,3i=)TΤ-/,/,xxyzyyxzuwvzvuzwεαεβεβεα=,=+∂∂=,=∂∂+/zzxywzvuεεαβ=∂∂,=+(2),,uvw/xα=∂∂/yβ=∂∂φ,,xyEEαφβφ=−=−/zEzφ=−∂∂(3)=−pk∆(4)p(ip,,ixyz=)k(,jjiijikkk=,1,2,3ij=),iT=∆(,,ixyz=),00xyxxxzxyyyyzxyzxyzσσσσσσ∂∂∂++=∂∂∂∂∂∂++=∂∂∂0yzxzzzxyzσσσ∂∂∂++=∂∂∂(5)0yxzDDDxyz∂∂∂++=∂∂∂(6)0yxzpppxyz∂∂∂++=∂∂∂(7)(4)(1)T0T⎡⎤⎡⎤−⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥−⎢⎥⎣⎦⎣⎦⎣⎦εσceoλEDeκoγ∆pook(8)(2)(4)(8)z(5)(7)[12]d/dz=RHR(9)T[,,,,,,,,,]xzyzzzzzDpuvwTσσσφ=RHT[]H=AB,16172191117272210127849500000000000000000000000000000000000AAsssAAsssAssAαααβββαβαβ−⎡⎤⎢⎥−⎢⎥⎢⎥=−−⎢⎥⎢⎥⎢⎥⎣⎦A,17891023111224234456000000000000000000000000000000000sssssssssssssssααββαβαβ−⎡⎤⎢⎥−⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎣⎦B161316Assαβ22=−−,171416Assαβ=−(+),27A=2321516ssβα22−−,171849Assαβ22=−−,1950Asα2=−−20sβ2is(124i=)2[8]0(0,)xxwvTxaσφ======0(0,)yywuTybσφ======(10)(10),()((),())cos()sin(),(,)((),())sin()cos(),(,)((),())sin()sin(),(,)((),())sin()sin(),(,)((),mnmnxzxzmnmnyzmnmnmnzzzzzzmnmnmnzzmnmnmnyzuzuzxyvzvzxyDzDzxypwpzwzxyTzTσσζησσζησσζηζηφφ∞∞∞∞∞∞∞∞=====∑∑∑∑∑∑∑∑())sin()sin()mnmnzxyζη∞∞∑∑(11)π/,π/manbζη==(11)(9)mn−d()/d()mnmnzzz=RHR(12)()[(),(),(),(),(),mnmnmnmnmnmnxzyzzzzzzzzzDzpzσσσ=RT(),(),(),(),()],mnmnmnmnmnuzvzwzzTzφT[],=HAB91116172110121727227849500000000000,000000000000000000000000ssAAsssAAsssAAζζζηηηζηζη⎡⎤−⎢⎥−⎢⎥⎢⎥=⎢⎥−⎢⎥⎢⎥⎣⎦A1789102311122423445600000000000000000,0000000000000000sssssssssssssssζζηηζηζη−⎡⎤⎢⎥−⎢⎥⎢⎥=−−⎢⎥−−⎢⎥⎢⎥⎣⎦B161316Assζη22=+171416Assζη=(+)2715Asη2=+16sζ2491718Assζη22=+,501920Assζη22=+(12)()e()[0,]mnzmnzzzh⋅=,∈HRR(13)n(13)jzh=j1()()()mnmnjjjjjhhh−=RHR(14)()()ejjhzjh⋅=HH,()jhHjjzj()jhH1()(0)mnmnnh=RΠR(15)11()()()nnhhh−=ΠHHH1(0)mnR(0z=)3[8]7([0/0/90/0/0/90/0p]PZT-5A)1mh=50abh==PZT-5A0.1phh=Gr/Epeh=0.9/6h1sin()sin(),0,zzbzzttbtbxyTTσζησφφ======2sin()sin(),0bzztzzbtbtTxyTζησσφφ======(tb)1mn==11Table1Resultandcomparison12W1bU1bW2bP2b−641.340−19.825−8.0614482.247[8]−641.342−20.044−8.0528481.211W1b=w(a/2,b/2,0)/K1U1b=u(0,b/2,0)/K1W2b=10×w(a/2,b/2,0)/K2P2b=pz(a/2,b/2,0)/K2K1K2K1=a/h/10.3×109K2=a/h×22.5×10-61[8]5[90p/0/90/0/0p]PZT-5A工程力学233Gr/Ep。层合板厚度=10mmH，长宽50abH==，压电层厚度均为0.05phH=，各弹性层厚度均为0.9/3ehH=。工况为：sin()sin(),0zzttzzbtbbTxyTσζησφφ======。选取1mn==进行求解，层板上表面、下表面广义位移和应力值见表2。表2层板上下表面广义位移和应力值Table2Generalizedstressanddisplacementvaluesoflaminatedplatetop/bottomsurfaceutvtwtDztpzt-5.378×10-7-6.698×10-7-1.274×10-51.222×10-470.722ubvbwbDzbpzb2.583×10-71.293×10-7-1.266×10-51.218×10-470.201沿厚度方向，各广义应力和位移的变化趋势分别见图1―图10。-2500-1600-700200110020000.00.10.20.30.40.50.60.70.80.91.0σxzz/H图1xzσ沿板厚度方向的分布Fig.1Distributionofxzσthroughthicknessoftheplate-1000-2006001400220030000.00.10.20.30.40.50.60.70.80.91.0σyzz/H图2yzσ沿板厚度方向的分布Fig.2Distributionofyzσthroughthicknessoftheplate-55152535450.00.10.20.30.40.50.60.70.80.91.0σzzz/H图3zzσ沿板厚度方向的分布Fig.3Distributionofzzσthroughthicknessoftheplate012243648600.00.10.20.30.40.50.60.70.80.91.0Φz/H图4φ沿板厚度方向的分布Fig.4Distributionofφthroughthicknessoftheplate0.00.20.40.60.81.00.00.10.20.30.40.50.60.70.80.91.0Tz/H图5T沿板厚度方向的分布Fig.5DistributionofTthroughthicknessoftheplateφ相对厚度z/H相对厚度z/H相对厚度z/H相对厚度z/H相对厚度z/H剪应力σxz剪应力σyz应力σzz电势φ温度增量T234工程力学-6.0-4.2-2.4-0.61.23.00.00.10.20.30.40.50.60.70.80.91.0u(x10-7)z/H图6u沿板厚度方向的分布Fig.6Distributionofuthroughthicknessoftheplate-8-6-4-2020.00.10.20.30.40.50.60.70.80.91.0v(x10-7)z/H图7v沿板厚度方向的分布Fig.7Distributionofvthroughthicknessoftheplate-1.275-1.273-1.271-1.269-1.267-1.2650.00.10.20.30.40.50.60.70.80.91.0w(x10-5)z/H图8w沿厚度方向的分布Fig.8Distributionofwthroughthicknessoftheplate12.18012.18812.19612.20012.21212.2200.00.10.20.30.40.50.60.70.80.91.0Dz(x10-5)z/H图9zD沿板厚度方向的分布Fig.9DistributionofzDthroughthicknessoftheplate70.2070.3170.4170.5270.6270.730.00.10.20.30.40.50.