有限元法理論(一)

FiniteElementAnalysisTheory有限元法理論IntroductiontoComputationalMechanicsModelingandsimulationofmechanicsproblemsFiniteelementmethodFinitedifferencemethodMoleculardynamicsmethodBoundaryelementmethodChapter1.ElasticityandFiniteElementMethod

第一章彈性力學及有限元Theoryofelasticityisoftencalledelasticityortheoryofelasticmechanics.Itisthebranchofsolidmechanics.

彈性力學的理論簡稱為彈性理論或彈性力學。它是固體力學的一個分枝。WhatdoestheElasticitydealwith?Itdealswiththestresses,deformationsanddisplacementsinelasticsolidsproducedbyexternalforcesorchangesintemperature.

研究彈性體由于外力和溫度改變而引起的應力，形變和位移。Itanalyzesthestresses,deformationsanddisplacementsofstructuralelementswithintheelasticrangeandtherebytocheckthesufficiencyoftheirstrength,stiffnessandstability.

分析結構的應力，形變和位移，檢查是否滿足強度，剛度和穩(wěn)定性條件。TheImportantConceptinElasticity

彈性力學中的幾個重要概念ExternalForces外力Stress應力Deformation(Strain)形變(應變)Displacement位移A.externalforces外力Bodyforces體積力，體力Externalforcesortheloads,distributedoverthevolumeofthebody,arecalledbodyforces.分布在物體體內的外力叫體力:重力，慣性力Surfaceforces表面力，面力Externalforces,ortheloads,distributedoverthesurfaceofabody,arecalledsurfaceforces.分布在物體表面的外力叫面力:水壓力，接觸力B.Stress應

力Internalforces:undertheactionofexternalforces,internalforceswillbeproducedbetweenthepartsofabody.

內力：在外力作用下，物體各部分間產生相互作用的力叫內力。Stressesaretheinternalforcesactingontheperunitarea.

應力:作用在單位面積上的內力。

StressFig.應力定義圖Thenormalcomponentiscalledthenormalstress.Thetangentialcomponentiscalledtheshearingstress.

法向分量叫法向應力，切向分量叫剪應力。Thefig.ofstressnotation

坐標面上應力記號圖C.Deformation形變Bydeformationwemeanthechangeoftheshapeofabody,whichmaybeexpressedbythechangesinlengthsandanglesofitsparts.

形變--物體形狀（各部分長度和角度）的改變。TostudydeformationconditionatacertainpointP,weconsiderlinesegmentsPA,PB,PC

研究一點的變形，考慮通過P點的三個正向微段PA,PB,PC。D.Displacements位移Bydisplacement,wemeanthechangeofposition.

位置的移動叫位移。Displacementcomponentsu,v,w---theprojectionsofthedisplacementonthex,yandzaxes.

位移在坐標軸上投影叫位移分量u,v,w。Itisconsideredpositiveasitisinthepositivedirectionofthecorrespondingcoordinateaxis.

沿坐標正向的位移分量為正。Basicassumptions基本假定Thebodyiscontinuous.

物體是連續(xù)的。Thebodyisperfectlyelastic.

物體是完全彈性的。Thebodyishomogeneous.

物體是均質的。Thebodyisisotropic.

物體是各向同性的。Thedisplacementsandstrainsaresmall.

位移和應變是微小的。Thebodyiscontinuous

物體是連續(xù)的Thewholevolumeofthebodyisfilledwithcontinuousmatterwithoutanyvoid.

假定整個物體的體積都被組成這個物體的介質所充滿，不留下任何孔隙。Underthisassumption,thephysicalquantitiesinthebody,suchasstresses,strainsanddisplacements,canbeexpressedbycontinuousfunctionsofcoordinatesinthespace.

物理量（例：應力，應變，位移）能用坐標的連續(xù)函數表示。Thebodyisperfectlyelastic

物體是完全彈性的ThebodywhollyobeysHook’slawofelasticity.----Therelationsbetweenthestresscomponentsandthestraincomponentsarelinear.

物體遵守虎克定律---應力分量和應變分量是線性關系。Theelasticconstantswillbeindependentofthestressorstraincomponentsunderthisassumption.

彈性常數與應力和應變的大小無關。Thebodyishomogeneous

物體是均質的Theelasticconstantswillbeindependentofthelocationinthebody.

彈性常數與位置無關。物體由同一種材料組成。物體由多種材料組成，但每一種材料的顆粒遠小于物體，且在物體內均勻分布。Thebodyisisotropic

物體是各向同性的Theelasticconstantswillbeindependentoftheorientationofthecoordinateaxes.

彈性常數與坐標軸的方向無關。Steelstructure-------isotropic

鋼---各向同性Woodenstructure-----notisotropic

木---各向異性Thedisplacementsandstrainsaresmall

位移和應變是微小的Thedisplacementcomponentsareverysmallincomparisonwithitsoriginaldimensions.位移遠小于物體尺寸---可用變形前的尺寸代替變形后的尺寸。Thestraincomponentsandtherotationsofalllineelementsaremuchsmallerthanunity.

應變分量和轉角遠小于1---其乘積及二次冪可忽略。Fundamentalquantitiesexpressedbymatrix

基本量的矩程表示Bodyforce體力：Surfaceforce面力：Displacement位移：Stress應力：Strain應變：Fundamentalequationsexpressedbymatrix

基本方程的矩程表示Geometricalequations幾何方程Physicalequations物理方程Balanceequations平衡方程Virtualworkequations虛功方程Geometricalequations

幾何方程