有限元法理论(一)

FiniteElementAnalysisTheory有限元法理论IntroductiontoComputationalMechanicsModelingandsimulationofmechanicsproblemsFiniteelementmethodFinitedifferencemethodMoleculardynamicsmethodBoundaryelementmethodChapter1.ElasticityandFiniteElementMethod

第一章弹性力学及有限元Theoryofelasticityisoftencalledelasticityortheoryofelasticmechanics.Itisthebranchofsolidmechanics.

弹性力学的理论简称为弹性理论或弹性力学。它是固体力学的一个分枝。WhatdoestheElasticitydealwith?Itdealswiththestresses,deformationsanddisplacementsinelasticsolidsproducedbyexternalforcesorchangesintemperature.

研究弹性体由于外力和温度改变而引起的应力，形变和位移。Itanalyzesthestresses,deformationsanddisplacementsofstructuralelementswithintheelasticrangeandtherebytocheckthesufficiencyoftheirstrength,stiffnessandstability.

分析结构的应力，形变和位移，检查是否满足强度，刚度和稳定性条件。TheImportantConceptinElasticity

弹性力学中的几个重要概念ExternalForces外力Stress应力Deformation(Strain)形变(应变)Displacement位移A.externalforces外力Bodyforces体积力，体力Externalforcesortheloads,distributedoverthevolumeofthebody,arecalledbodyforces.分布在物体体内的外力叫体力:重力，惯性力Surfaceforces表面力，面力Externalforces,ortheloads,distributedoverthesurfaceofabody,arecalledsurfaceforces.分布在物体表面的外力叫面力:水压力，接触力B.Stress应

力Internalforces:undertheactionofexternalforces,internalforceswillbeproducedbetweenthepartsofabody.

内力：在外力作用下，物体各部分间产生相互作用的力叫内力。Stressesaretheinternalforcesactingontheperunitarea.

应力:作用在单位面积上的内力。

StressFig.应力定义图Thenormalcomponentiscalledthenormalstress.Thetangentialcomponentiscalledtheshearingstress.

法向分量叫法向应力，切向分量叫剪应力。Thefig.ofstressnotation

坐标面上应力记号图C.Deformation形变Bydeformationwemeanthechangeoftheshapeofabody,whichmaybeexpressedbythechangesinlengthsandanglesofitsparts.

形变--物体形状（各部分长度和角度）的改变。TostudydeformationconditionatacertainpointP,weconsiderlinesegmentsPA,PB,PC

研究一点的变形，考虑通过P点的三个正向微段PA,PB,PC。D.Displacements位移Bydisplacement,wemeanthechangeofposition.

位置的移动叫位移。Displacementcomponentsu,v,w---theprojectionsofthedisplacementonthex,yandzaxes.

位移在坐标轴上投影叫位移分量u,v,w。Itisconsideredpositiveasitisinthepositivedirectionofthecorrespondingcoordinateaxis.

沿坐标正向的位移分量为正。Basicassumptions基本假定Thebodyiscontinuous.

物体是连续的。Thebodyisperfectlyelastic.

物体是完全弹性的。Thebodyishomogeneous.

物体是均质的。Thebodyisisotropic.

物体是各向同性的。Thedisplacementsandstrainsaresmall.

位移和应变是微小的。Thebodyiscontinuous

物体是连续的Thewholevolumeofthebodyisfilledwithcontinuousmatterwithoutanyvoid.

假定整个物体的体积都被组成这个物体的介质所充满，不留下任何孔隙。Underthisassumption,thephysicalquantitiesinthebody,suchasstresses,strainsanddisplacements,canbeexpressedbycontinuousfunctionsofcoordinatesinthespace.

物理量（例：应力，应变，位移）能用坐标的连续函数表示。Thebodyisperfectlyelastic

物体是完全弹性的ThebodywhollyobeysHook’slawofelasticity.----Therelationsbetweenthestresscomponentsandthestraincomponentsarelinear.

物体遵守虎克定律---应力分量和应变分量是线性关系。Theelasticconstantswillbeindependentofthestressorstraincomponentsunderthisassumption.

弹性常数与应力和应变的大小无关。Thebodyishomogeneous

物体是均质的Theelasticconstantswillbeindependentofthelocationinthebody.

弹性常数与位置无关。物体由同一种材料组成。物体由多种材料组成，但每一种材料的颗粒远小于物体，且在物体内均匀分布。Thebodyisisotropic

物体是各向同性的Theelasticconstantswillbeindependentoftheorientationofthecoordinateaxes.

弹性常数与坐标轴的方向无关。Steelstructure-------isotropic

钢---各向同性Woodenstructure-----notisotropic

木---各向异性Thedisplacementsandstrainsaresmall

位移和应变是微小的Thedisplacementcomponentsareverysmallincomparisonwithitsoriginaldimensions.位移远小于物体尺寸---可用变形前的尺寸代替变形后的尺寸。Thestraincomponentsandtherotationsofalllineelementsaremuchsmallerthanunity.

应变分量和转角远小于1---其乘积及二次幂可忽略。Fundamentalquantitiesexpressedbymatrix

基本量的矩程表示Bodyforce体力：Surfaceforce面力：Displacement位移：Stress应力：Strain应变：Fundamentalequationsexpressedbymatrix

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

几何方程