线性粘弹区

线性粘弹性LinearViscoelasticity俞炜上海交通大学流变学研究所2009年复杂流体流变学讲习班与学术研讨会2009年7月6日-11日LinearViscoelasticity线性粘弹性1.Introduction简介2.Stressrelaxation应力松弛3.BoltzmannsuperpositionprincipleBoltzmann叠加原理4.Therelaxationspectrum松弛时间谱5.Creepandcreeprecovery蠕变与蠕变回复6.Smallamplitudeoscillatoryshear小振幅振荡剪切7.Time-temperaturesuperpositionprinciple时温叠加原理2009复杂流体流变学讲习班上海交通大学流变学研究所1.Introduction简介Thistypeofbehaviorisobservedwhenthedeformationissufficientmildthatthemoleculesofapolymericmaterialaredisturbedfromtheirequilibriumconfigurationandentanglementstatetoanegligibleextent.•verysmalldeformation:小形变-totalstrainisverysmall-theearlystagesofalargerdeformation•deformationoccursveryslowly:(formaterialswithfadingmemory)非常缓慢的形变-steadysimpleshearatverylowshearratesItsprincipalutilityisasamethodforcharacterizingthemicrostructureintheirequilibriumstate表征材料平衡态的结构1.Introduction简介2009复杂流体流变学讲习班上海交通大学流变学研究所1.Introduction简介Dynamicstrainsweep动态应变扫描Dynamicstresssweep动态应力扫描*00Gσγ=1.Introduction简介Creepandrecovery蠕变与回复2009复杂流体流变学讲习班上海交通大学流变学研究所Stepstrainorstressrelaxation(阶跃应变，应力松弛)strainγt00γt0Increases0γshearstress()()00,xytGtσγγ=Shearmodulus(剪切模量)[Pa](forstepshearstrain)()()00,EtEtσεε=Tensilemodulus(拉伸模量)[Pa](forstepextensionalstrain)0γ0εrelaxationmodulus(松弛模量)linearviscoelasticity()()000lim,GtGtγγ→=2.Stressrelaxation应力松弛2.Stressrelaxation应力松弛Viscoelasticliquid粘弹性液体Viscoelasticsolid粘弹性固体00eG=⎧⎨≠⎩Equilibriummodulus(平衡模量)()0lim,etGGGtγ∞→∞==Instantaneousmodulus()000lim,tGGtγ→=Stepstrainorstressrelaxation(阶跃应变，应力松弛)relaxationmodulus(松弛模量)linearviscoelasticity()()000lim,GtGtγγ→=t0RelaxationmodulusViscoelasticsolidViscoelasticliquid2009复杂流体流变学讲习班上海交通大学流变学研究所PS2.Stressrelaxation应力松弛2.Stressrelaxation应力松弛Relaxationmodulusforpolymers聚合物的松弛模量eGgGlog(time)RelaxationmodulusG(t)(logarithmicscale)Crosslinkedmaterials(elastomer)LinearpolymersA:monodispersewithMMCB:monodispersewithMMCC:polydispersewithMMCgG0NGABCplateaumoduluslog(time)G(t)(logarithmicscale)2009复杂流体流变学讲习班上海交通大学流变学研究所0NeRTGMρ=缠结点间平均分子量2.Stressrelaxation应力松弛Relaxationmodulusforpolymers聚合物的松弛模量2.Stressrelaxation应力松弛2009复杂流体流变学讲习班上海交通大学流变学研究所Boltzmannsuperpositionprinciple(玻尔兹曼叠加原理):theadditivityofthecausesandeffects(linearity)tt2t1γγ1γ2tt1γγ1tt2γγ2Responsett1γγ1tt1τt1(t)=G(t-t1)γ1tt2γγ2tt2τt2(t)=G(t-t2)γ2t2t1γγ1γ2tt1τt1(t)=G(t-t1)γ1+G(t-t2)γ2t2ttheresponse(e.g.strain)atanytimeisdirectlyproportionaltothevalueoftheinitiatingsignal(e.g.stress).3.BoltzmannsuperpositionprincipleBoltzmann叠加原理物理意义:Thepresentstressisthesuperpositionoftheeffectofpaststrainweighedbymemoryfunction()()()ttGttdtτγ−∞′′=−∫Forasmoothstrainhistory()()()ttGtttdtτγ−∞′′′=−∫orisamathematicalconveniencewhichimpliesalltimet'priortot.Itisnotnecessaryinpractice.−∞()()()0ttGttdtτγ′′=−∫Tensorialforms()()()tijijtGttdtτγ−∞′′=−∫()()()tijijtGtttdtτγ−∞′′′=−∫()()()ttmtttdtτγ−∞′′′=−−∫()()(memoryfunction)Gttmttt′∂−′−=′∂AlllinearviscoelasticbehaviorisgovernedbytheBoltzmannsuperpositionprinciplebasedonthesinglematerialfunctionG(t).3.BoltzmannsuperpositionprincipleBoltzmann叠加原理Boltzmannsuperpositionprinciple(玻尔兹曼叠加原理):theadditivityofthecausesandeffects(linearity)2009复杂流体流变学讲习班上海交通大学流变学研究所Forasmoothstrainhistoryforstepextensionalstrain()()()tijijtGttdtτγ−∞′′=−∫()00020000000ijtεγεε⎡⎤⎢⎥≥=−⎢⎥⎢⎥−⎣⎦extensionalstress()112203EGtσττε=−=Young’smodulus()()()03EtEtGtσε==InfinitesimalstrainLinearviscoelasticmaterials3.BoltzmannsuperpositionprincipleBoltzmann叠加原理Boltzmannsuperpositionprinciple(玻尔兹曼叠加原理):theadditivityofthecausesandeffects(linearity)Forasmoothstrainhistory()()()tijijtGtttdtτγ−∞′′′=−∫forsteadysimpleshear()00000000ijtγγγ⎡⎤⎢⎥≥=⎢⎥⎢⎥⎣⎦shearstress()()()()12210tttGttdtGsdsττγγ−∞∞′′==−=∫∫zeroshearviscosity()0120Gsdsητγ∞==∫normalstressdifferences120NN==Thenormalstressdifferenceswillbezeroinanysimplesheardeformation,whetherornotitissteadyintime,aslongasthelineartheoryisvalid,i.e.,aslongasthedeformationissufficientlysmallorslow.3.BoltzmannsuperpositionprincipleBoltzmann叠加原理Boltzmannsuperpositionprinciple(玻尔兹曼叠加原理):theadditivityofthecausesandeffects(linearity)2009复杂流体流变学讲习班上海交通大学流变学研究所3.BoltzmannsuperpositionprincipleBoltzmann叠加原理Generaldifferentialequationoflinearviscoelasticity线性粘弹性的广义微分形式β0istheonlynon-zeroparameterβ1istheonlynon-zeroparameterβ0andβ1aretheonlynon-zeroparameters(Elasticsolid)(Newtonianfluid)(Kelvinmodel)α1andβ1aretheonlynon-zeroparameters(Maxwellmodel)3.BoltzmannsuperpositionprincipleBoltzmann叠加原理2009复杂流体流变学讲习班上海交通大学流变学研究所4.Therelaxationspectrum松弛时间谱Empiricalequationsfortherelaxationmodulus松弛模量的经验方程()()00exptGtσγλ=−inastepstrainexperiment()()0expGtGtλ=−shearrelaxationmodulus()()()0exptijijtGtttdtτλγ−∞′′′=−−⎡⎤⎣⎦∫linearconstitutiveequationThegeneralizedMaxwellmodel广义Maxwell模型……..()()()exptijkkijktGtttdtτλγ−∞′′′=−−⎡⎤⎣⎦∑∫()()1expNkkkGtGttλ=′=−−⎡⎤⎣⎦∑initialmodulusrelaxationtime()()()exptijkkijktGtttdtτλγ−∞′′′=−−⎡⎤⎣⎦∑∫()()1expNkkkGtGtλ==−∑¾Usually5~10modesaresufficienttofitexperimentaldatareasonablywell¾Suchaseriesofvaluesiscalleda“discretespectrum”ofamaterial离散谱¾ThebehaviorofG(t)atsufficientlylongtimes(terminalzone)willbedominatedbyGi-λipa