有限元法理論(一)

FiniteElementAnalysisTheory有限元法理論IntroductiontoComputationalMechanicsModelingandsimulationofmechanicsproblemsFiniteelementmethodFinitedifferencemethodMoleculardynamicsmethodBoundaryelementmethodChapter1.ElasticityandFiniteElementMethod

第一章彈性力學(xué)及有限元Theoryofelasticityisoftencalledelasticityortheoryofelasticmechanics.Itisthebranchofsolidmechanics.

彈性力學(xué)的理論簡(jiǎn)稱(chēng)為彈性理論或彈性力學(xué)。它是固體力學(xué)的一個(gè)分枝。WhatdoestheElasticitydealwith?Itdealswiththestresses,deformationsanddisplacementsinelasticsolidsproducedbyexternalforcesorchangesintemperature.

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

分析結(jié)構(gòu)的應(yīng)力，形變和位移，檢查是否滿(mǎn)足強(qiáng)度，剛度和穩(wěn)定性條件。TheImportantConceptinElasticity

彈性力學(xué)中的幾個(gè)重要概念ExternalForces外力Stress應(yīng)力Deformation(Strain)形變(應(yīng)變)Displacement位移A.externalforces外力Bodyforces體積力，體力Externalforcesortheloads,distributedoverthevolumeofthebody,arecalledbodyforces.分布在物體體內(nèi)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(yīng)

力Internalforces:undertheactionofexternalforces,internalforceswillbeproducedbetweenthepartsofabody.

內(nèi)力：在外力作用下，物體各部分間產(chǎn)生相互作用的力叫內(nèi)力。Stressesaretheinternalforcesactingontheperunitarea.

應(yīng)力:作用在單位面積上的內(nèi)力。

StressFig.應(yīng)力定義圖Thenormalcomponentiscalledthenormalstress.Thetangentialcomponentiscalledtheshearingstress.

法向分量叫法向應(yīng)力，切向分量叫剪應(yīng)力。Thefig.ofstressnotation

坐標(biāo)面上應(yīng)力記號(hào)圖C.Deformation形變Bydeformationwemeanthechangeoftheshapeofabody,whichmaybeexpressedbythechangesinlengthsandanglesofitsparts.

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

研究一點(diǎn)的變形，考慮通過(guò)P點(diǎn)的三個(gè)正向微段PA,PB,PC。D.Displacements位移Bydisplacement,wemeanthechangeofposition.

位置的移動(dòng)叫位移。Displacementcomponentsu,v,w---theprojectionsofthedisplacementonthex,yandzaxes.

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

沿坐標(biāo)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(zhì)的。Thebodyisisotropic.

物體是各向同性的。Thedisplacementsandstrainsaresmall.

位移和應(yīng)變是微小的。Thebodyiscontinuous

物體是連續(xù)的Thewholevolumeofthebodyisfilledwithcontinuousmatterwithoutanyvoid.

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

物理量（例：應(yīng)力，應(yīng)變，位移）能用坐標(biāo)的連續(xù)函數(shù)表示。Thebodyisperfectlyelastic

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

物體遵守虎克定律---應(yīng)力分量和應(yīng)變分量是線(xiàn)性關(guān)系。Theelasticconstantswillbeindependentofthestressorstraincomponentsunderthisassumption.

彈性常數(shù)與應(yīng)力和應(yīng)變的大小無(wú)關(guān)。Thebodyishomogeneous

物體是均質(zhì)的Theelasticconstantswillbeindependentofthelocationinthebody.

彈性常數(shù)與位置無(wú)關(guān)。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(yuǎn)小于物體，且在物體內(nèi)均勻分布。Thebodyisisotropic

物體是各向同性的Theelasticconstantswillbeindependentoftheorientationofthecoordinateaxes.

彈性常數(shù)與坐標(biāo)軸的方向無(wú)關(guān)。Steelstructure-------isotropic

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

木---各向異性Thedisplacementsandstrainsaresmall

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

應(yīng)變分量和轉(zhuǎn)角遠(yuǎn)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(yīng)力：Strain應(yīng)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