粘弹性体发展方程中关于热力学广义力精确解

:025822724(2002)02201292051,1,2(1.,610031;2.,606-8585):,,,,,,,;,,;,,:;;:O345:AEvolutionEquationofViscoelasticMediumandExactSolutionsforGeneralizedThermodynamicForcesDAIGuang2ze1,XIAYan1,ARAKIShigetoshi2(1.SchoolofMat.ScienceandEng.,SouthwestJiaotongUniversity,Chengdu610031,China;2.Dept.ofMech.Eng.,KyotoInstituteofTechnology,Kyoto,606-8585,Japan)Abstract:Itisobviousthattheexactsolutionsoftheevolutionequation,whichisderivedbyregardingthedeformationprocessofviscoelasticmaterialsasanirreversiblethermodynamicprocess,dependverymuchonthepropertiesoftheircoefficientmatrices.Thesematricesareusuallyaffectedbythenatureofthegeneralizedthermodynamiccoordinates,whichleadsthesolutionsforthegeneralizedthermodynamiccoordinatestothat:(1)undertheconditionthatnoneutrallystableequilibriumcoordinatesexistortheyareonlyintheobservedones,theexactexplicitsolutionsareinthesameforms,furthermoretheaccelerativetermwillariseintheexplicitsolutionswhenpartsofthecoordinatesparticipateintheentropyproduction;(2)undertheconditionthatallthegeneralizedthermodynamiccoordinatesparticipateintheentropyproduction,thetermproportionaltotimewillarisewhenonlyoneneutrallystableequilibriumcoordinateexistsinthehiddenonesandthegeneralizedthermodynamiccoordinatesaregivenintheformofstepfunction;(3)undertheconditionthattheneutrallystableequilibriumcoordinateexistsinthehiddenoneswhichparticipateintheentropyproduction,theaccelerativetermwillariseintheexplicitsolutionswhenalloftheobservedcoordinatesandpartsofthehiddenonesparticipateintheentropyproductionregardlessofthenumberoftheneutrallystableequilibriumcoordinates.Keywords:dissipation;evolutionequation;viscoelasticity:2001206218:(1963-),,,.37220024JOURNALOFSOUTHWESTJIAOTONGUNIVERSITYVol.37No.2Apr.2002[1-16](EvolutionEquation)[16],,,Onsager[8,9],Laplace(aij+pbij)…qj=…qi+i€(1):i1k;ik+1n:QiqiqiQi(1)aijbij,2F=12aijqiqj0D=12bijÛqiÛqj0(2):FHelmholtz();DRayleigh,aij:bijQi1Qi,Qi3:(1)i1k,k+1n;(2)q1,q2,,qmqm+1,qm+2,,qn;21,(3):,q1,q2,,qm;,qi+1,qi+2,,qm(1),(2)(3),Qi11.1(mk)…q(1)={…q1,,…qk}…q(2)={…qk+1,,…qm,…qm+1,,…qn}(3)q1,,qk,qk+1,,qm,(1)A11,A12A21,A22+qB11,B12B21,B22…q(1)…q(2)=ŠQ(1)ŠQ(2)(4)(4)…q(2),…q(1)QiŠQ(1)=[(A11+pB11)-(A12+pB12)(A22+pB22)-1(A22+pB21)]…q(1)+[A12+pB12][A22+pB22]-1ŠQ(2)(5)031371.1.1Qi(5)Qidet(A22+pB22)0qi,,QiQiM.A.BiotR.A.Bchapery[5,6],q1,q2,,qnq1,q2,,qk,k+1,k+2,,n[12]1.1.2Qi(3),(A22+pB22),(5)(1),detaij0detaij=0(2):(m=n)LaplaceQi=6kj=i-6ns=k+1B(s)ij1t0e-s(t-s)q(1)jd()+Bij1Ûqj(1)(t)+Bij1q(1)j(t)+6ns=k+1t0Bis1e-s(t-)s()dt(6)[12](A-1)(kmn)LaplaceQi=6kj=i-6ms=k+1B(s)ij2t0e-s(t-s)q(1)jd()+Bij2Ûqj(1)(t)-Bij2qj(1)(t)+Bij2q(1)j(t)+6ms=k+1B(s)ij2t0e-s(t-)s()d+Bis2Ûs(t)+Bis2s(t)(7)[12](A-2)(6),,,(7)(2)detaij=0(2):(m=n),Laplace1312:Qi=6kj=i-6ns=k+2B(s)ij3t0e-s(t-s)q(1)j()d-Bij3t0q(1)j()d+Bij3Ûqj(1)(t)+Bij3qj(1)(t)+6ms=k+2B(s)is4t0e-s(t-)s()d+Bis4t0s()d+Bis4Û(t)Bis3s(t)(8)[12](A-3)(7),,q(1)iq(1)i(t)=q0(1)jH(t),(8)t,B(s)ij3t0q(1)j()d=B(s)ij4q0(1)itH(t),q0(1)i(6)(8);,,aijbij,(8)(kmn),LaplaceQi=6kj=i-6ms=k+2B(s)ij4t0e-s(t-s)q(1)j()d-B(s)ij4t0q(1)j()d+Bij4Ûq(1)j(t)-Bij4q(1)j(t)+Bij4q(1)j(1)(t)+6ms=k+2B(s)is4t0e-s(t-)s()d+Bis4t0s()d+Bis4Û(t)+Bis4s(t)(9)[12](A-4)(7),q(1)iq(1)i(t)=q0(1)jH(t),(9)t,B(s)ij4t0q(1)j()d=B(s)ij4q0(1)jtH(t),q0(1)i,(8),(9),(9)1.2(mk)qi,i231,Qi1.1QiQi,2,:(1),;(2),,;(3),,;(4),,;(5),,,,,,,2313720012002,(TCM):[1]BaileyRW.Creepofsteelundersimpleandcompoundstressesandtheuseofhighinitialtemperatureinsteampowerplants[A].TokyoSectionalMeetingoftheWorldPowerConference[C].Tokyo:TokyoUniversityPress,1929:108921095.[2]NortonFH.Thecreepofsteelathightemperature[M].NewYork:McGrawHill,1929.[3]FindleyWN,KhoslaG.Anequationfortensioncreepofthreeunfilledthermoplastics[J].SPEJournal,1956;(12):20225.[4]FindleyWN,PetersonDB.Predictionoflong2timecreepwithten2yearsdataonfourplasticlaminates[A].Proc.ofASTM[C].US:ASTMPress,1958;58:8412855.[5]BiotMA.Theoryofstress2strainrelationsinanisotropicviscoelasticityandrelaxationphenomena[J].J.Appl.Physics,1954;(25)11:138521391.[6]SchaperyRA.Applicationofthemodynamicstothermomechanical,fracture,andbirefringentphenomenainviscoelasticmedia[J].J.Appl.Physics,1964;(35)5:145121465.[7]SchaperyRA.Atheoryofnon2linearthermoviscoelasticitybasedonreversiblethermodynamics[A].Proc.oftheFifthU.S.NationalCongressofAppliedMechanics[C].US,1966:5112530.[8]OnsagerL.Reciprocalrelationsinirreversibleprocess[J].PhysicalReview,1931;37:4052426.[9]OnsagerL.Reciprocalrelationsinirreversibleprocess[J].PhysicalReview,1931;38:226522279.[10]DaiGZ,ArakiS,SaitoK.OnsomeconsiderationsaboutSchaperyscreepequationsforlinearviscoelasticmaterials[A].ProceedingsoftheAsian2PacificConferenceonAerospaceTechnologyandScience[C].Hangzhou:InternationalAcademicPublishers,1994:1072112.[11]DaiGZ,ArakiS,SaitoK.Someconsiderationsonstressrelaxationphenomenainviscoelasticmediumfromastandpointofirreversiblethermodynamics[J].TheoreticalandAppliedMechanics,1995;44:85293.[12]DaiGZ,ArakiS,SaitoK,etal.Exactsolutionsofevolutionequationforlinearviscoelasticmedium[J].TeoreticalandAppliedMechanics,1997;46:952108.[13]DaiGZ.Studiesonexactsoluti