连续介质力学(固体)-24-30

连续介质力学（固体）ContinuumMechanicsMechanicsofContinuaMechanicsofContinuousMedia第24-30讲赵亚溥(ZhaoYa-Pu)中国科学院力学研究所非线性力学国家重点实验室2010年秋季复习11::22iklmiklmFuuuu事实上，Landau&Lifshitz教材中的(10.1)式可表达为如下张量形式：11::22iklmiklmRvvvv则(34.3)式可可以表示为：作业：Landau&Lifshitz有关丝状液晶的耗散函数如下，请理解：PhysicistJamesClarkMaxwellcalledFourier'sbookagreatmathematicalpoem.Fourierisamathematicalpoem.__LordKelvin第二十四讲：Thermalconductivity第二十四讲：Thermalconductivity作业：Furtherreading第二十五讲：微细尺度传热第二十六讲：Theabsorptionofsoundsinsolids声波吸收：在实际传声介质中声能传播的途中逐渐转变成热，从而出现随距离而逐渐衰减的现象。耗散过程。声波吸收的原因：很多，如：介质的粘性、热传导，介质的微观动力学过程引起的弛豫效应等。声波吸收的例子：礼堂、高速公路的吸声板，最大限度地增加声波的吸收，减少声波反射。声波的经典吸收理论：粘滞性及热传导两部分吸收组成。粘滞吸收：当声波通过介质时，介质质点因相对运动而产生内摩擦，也即粘滞作用，导致声的吸收。热传导吸收：因声波传播基本上是绝热的，当介质中有声波通过时，介质产生压缩和膨胀的交替变化，压缩区温度升高，膨胀区温度降低，之间形成温度梯度，引起热传导。该过程是不可逆的，因此产生声能的耗散，称为热传导吸收。第二十六讲：Theabsorptionofsoundsinsolids发展简史概述：C.G.Stokes在1845年导出由粘滞性引起的流体中声波吸收公式，其吸收系数除了与粘滞系数成正比外，还与声波频率的二次方成正比。G.R.Kirchhoff于1868年提出了由热传导引起的声波吸收理论，其吸收系数除了与介质的热导率成正比外，还与声波的频率成二次方关系。以上两部分统称为声波的经典吸收理论。固体中声波的吸收研究开展的稍迟一些，20世纪30年代末起才出现这方面的测量，吸收机制比流体复杂的多。大量测量发现，几乎所有的气体都与经典吸收理论有偏差。1920年，AlbertEinstein从声波散射来确定缔合气体的反应率，从而促进了对气体分子热弛豫吸收理论的广泛研究。弛豫吸收：由介质分子的微观内过程引起，主要机制有：（1）分子热弛豫吸收；（2）化学弛豫；（3）结构弛豫；（4）多种弛豫等。第二十七讲：Deformationwithchangeoftemperature第二十八讲：DiffusioninsolidsWiki:Theconceptofdiffusionemergedfromphysicalsciences.Theparadigmaticexampleswereheatdiffusion,moleculardiffusionandBrownianmotion.TheirmathematicaldescriptionwaselaboratedbyJosephFourierin1822,AdolfFickin1855,andbyAlbertEinsteinandMarianSmoluchowskiin1905/06,respectively.Diffusion:flowofmatterHeat:flowofthermalenergyElectriccurrent:flowofelectricchargeDifferentkindsofflowsinmaterialHeat:flowofthermalenergyFourier’slawofheatconduction(1811)xTqq:heatflux(Jm-2s-1)xTGradient?TemperaturegradientThermalconductivityJosephFourier(1768-1830)Electriccurrent:flowofchargeOhm’slawofelectricalconduction(1827)xVEjj:chargeflux(Cm-2s-1),currentdensityxVGradient?Electricpotentialgradient,electricfieldEelectricalconductivityGeorgSimonOhm(1787-1854)Diffusion:flowofmassFick’sfirstlawofdiffusionin1855xcDjj:massflux(kgm-2s-1,molesm-2s-1)xcGradient?concentrationgradient,kgm-4D:Diffusivity,m2s-11829-1901DiffusionMassflowprocessbywhichspecieschangetheirpositionrelativetotheirneighboursDrivenbythermalenergyandagradientThermalenergy→thermalvibrations→AtomicjumpsConcentration/chemicalpotentialElectricGradientMagneticStressFlux(J)(restricteddefinition)→Flow/area/time[Atoms/m2/s]AssumethatonlyBismovingintoAAssumesteadystateconditions→Jf(x,t)(Noaccumulationofmatter)Fick’sfirstlawdxdcDAdtdnNo.ofatomscrossingareaAperunittimeCross-sectionalareaConcentrationgradientMattertransportisdowntheconcentrationgradientDiffusioncoefficient/diffusivityAFlowdirectionAsafirstapproximationassumeDf(t)dxdcDAdtdngradientionconcentrattimeareaatomsJ//dxdcJdxdcDJdxdcDdtdnAJ1Fick’sfirstlawDiffusivity(D)→f(A,B,T)D=f(c)Df(c)C1C2Steadystatediffusionx→Concentration→DiffusionSteadystateJf(x,t)Non-steadystateJ=f(x,t)D=f(c)D=f(c)Df(c)Df(c)Fick’ssecondlawJxJx+xxxxxJJonAccumulatixxJJJonAccumulatixxxxJJJxtcxxJsmAtomsmsmAtoms23.1xxJxtcccDtxxFick’sfirstlawccDtxxDf(x)22ccDtx22ccDtxRHSisthecurvatureofthecvsxcurvex→c→x→c→+vecurvaturec↑ast↑vecurvaturec↓ast↑LHSisthechangeisconcentrationwithtime22ccDtx(,)Erf2xcxtABDtSolutionwith2constantsdeterminedfromBoundaryConditionsandInitialCondition202ErfexpuduErf()=1Erf(-)=-1Erf(0)=0Erf(-x)=-Erf(x)u→Exp(u2)→0AreaGaussianerrorfunctionABApplicationsbasedonFick’sIIlawx→Concentration→Cavg↑tt10|c(x,t1)t2t1|c(x,t1)t=0|c(x,0)A&Bweldedtogetherandheatedtohightemperature(keptconstant→T0)Fluxf(x)|tf(t)|xNon-steadystateIfD=f(c)c(+x,t)c(-x,t)i.e.asymmetryabouty-axisC(+x,0)=C1C(x,0)=C2C1C2A=(C1+C2)/2B=(C2–C1)/2DeterminationofDiffusivityB0QkTDDeTemperaturedependenceofdiffusivityArrheniustypeApproximateformulafordepthofpenetrationDtxBoltzmanndistributionfunctionD0:pre-exponentialfactor(slightlydependsontemperature)Q:temperatureindependentATOMICMODELSOFDIFFUSIONInterstitialMechanism•Self-diffusion:Inanelementalsolid,atomsalsomigrate.LabelsomeatomsAftersometimeABCDVacancyMechanismDirectInterchangeandRingInterstitialDiffusion1212HmAtT0KvibrationoftheatomsprovidestheenergytoovercometheenergybarrierHm(enthalpyofmotion)→frequencyofvibrations,’→numberofsuccessfuljumps/timemB'HkTe12Vacantsitec=atoms/volumec=1/3concentrationgradientdc/dx=(1/3)/=1/4Flux=Noofatoms/area/time=’/area=’/2242'')/(dxdcJDmB2HkTDe20DkTQeDD0OncomparisonwithSubstitutionalDiffusionProbabilityforajump(probabilitythatthesiteisvacant).(probabilitythattheatomhassufficientenergy)Hm→enthalpyofmotionofatom’→frequencyofsuccessfuljumpsfmBB'HHkTkTeefmB'HHkTefmB2HHkTDeAsderivedforinterstitialdiffusion422''JDdcdxElementHfHmHf+HmQAu9780177174Ag957917