有限元法理論(一)

1、Finite Element Analysis Theory有限元法理論有限元法理論Introduction to Computational MechanicsnModeling and simulation of mechanics problemsnFinite element methodnFinite difference methodnMolecular dynamics methodnBoundary element methodChapter 1. Elasticity and Finite Element Method第一章 彈性力學及有限元 Theory of elasti

2、city is often called elasticity or theory of elastic mechanics. It is the branch of solid mechanics. 彈性力學的理論簡稱為彈性理論或彈性力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。學。它是固體力學的一個分枝。What does the Elasticity deal with?nIt deals with the stresses, deformations and displacements in elastic solids produced by external f

3、orces or changes in temperature. 研究彈性體由于外力和溫度改變而引起的應力，形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。變和位移。nIt analyzes the stresses, deformations and displacements of structural elements within the elastic range and thereby to check the sufficiency of their strength, stiffness and stability. 分析結構的應力，形變和位移，檢查是否滿足強度，分析結

4、構的應力，形變和位移，檢查是否滿足強度，剛度和穩(wěn)定性條件剛度和穩(wěn)定性條件。The Important Concept in Elasticity彈性力學中的幾個重要概念n External Forces 外力外力n Stress 應力應力n Deformation (Strain) 形變形變(應變應變)n Displacement 位移位移A. external forces 外 力nBody forces 體積力，體力體積力，體力nExternal forces or the loads, distributed over the volume of the body, are called

5、 body forces. 分布在物體體內的外力叫體力分布在物體體內的外力叫體力: :重力，慣性力重力，慣性力nSurface forces 表面力，面力表面力，面力nExternal forces, or the loads, distributed over the surface of a body, are called surface forces. 分布在物體表面的外力叫面力分布在物體表面的外力叫面力: :水壓力，接觸力水壓力，接觸力B. Stress 應 力nInternal forces: under the action of external forces, interna

6、l forces will be produced between the parts of a body. 內力：在外力作用下，內力：在外力作用下，物體各部分間產(chǎn)生相互作用物體各部分間產(chǎn)生相互作用的力叫內力的力叫內力。nStresses are the internal forces acting on the per unit area. 應力應力: :作用在單位面積上的內力作用在單位面積上的內力。 Stress Fig. 應力定義圖The normal component is called the normal stress.The tangential component is ca

7、lled the shearing stress. 法向分量叫法向應力，切向分量叫剪應力法向分量叫法向應力，切向分量叫剪應力。0limAQSA The fig. of stress notation 坐標面上應力記號圖C. Deformation 形 變nBy deformation we mean the change of the shape of a body, which may be expressed by the changes in lengths and angles of its parts. 形變形變-物體形狀（各部分長度和角度）的改變物體形狀（各部分長度和角度）的改變。

8、nTo study deformation condition at a certain point P, we consider line segments PA, PB, PC 研究一點的變形，考慮通過研究一點的變形，考慮通過P P點的三個正向微段點的三個正向微段PA,PB,PCPA,PB,PC。D. Displacements 位 移nBy displacement, we mean the change of position. 位置的移動叫位移。位置的移動叫位移。nDisplacement components u, v, w-the projections of the displ

9、acement on the x, y and z axes. 位移在坐標軸上投影叫位移分量位移在坐標軸上投影叫位移分量 u, v, w u, v, w 。nIt is considered positive as it is in the positive direction of the corresponding coordinate axis. 沿坐標正向的位移分量為正。沿坐標正向的位移分量為正。Basic assumptions 基本假定nThe body is continuous. 物體是連續(xù)的。物體是連續(xù)的。nThe body is perfectly elastic. 物體是

10、完全彈性的。物體是完全彈性的。nThe body is homogeneous. 物體是均質的。物體是均質的。nThe body is isotropic. 物體是各向同性的。物體是各向同性的。nThe displacements and strains are small. 位移和應變是微小的。位移和應變是微小的。The body is continuous 物體是連續(xù)的nThe whole volume of the body is filled with continuous matter without any void. 假定整個物體的體積都被組成這個物體的介質所假定整個物體的體積都

11、被組成這個物體的介質所充滿，不留下任何孔隙。充滿，不留下任何孔隙。nUnder this assumption, the physical quantities in the body, such as stresses, strains and displacements, can be expressed by continuous functions of coordinates in the space. 物理量（例：應力，應變，位移）能用坐標的連物理量（例：應力，應變，位移）能用坐標的連續(xù)函數(shù)表示。續(xù)函數(shù)表示。The body is perfectly elastic 物體是完全彈性

12、的nThe body wholly obeys Hooks law of elasticity. -The relations between the stress components and the strain components are linear. 物體遵守虎克定律物體遵守虎克定律-應力分量和應變分量是線性應力分量和應變分量是線性關系。關系。nThe elastic constants will be independent of the stress or strain components under this assumption. 彈性常數(shù)與應力和應變的大小無關。彈性常數(shù)

13、與應力和應變的大小無關。The body is homogeneous 物體是均質的nThe elastic constants will be independent of the location in the body. 彈性常數(shù)與位置無關。彈性常數(shù)與位置無關。n物體由同一種材料組成。物體由同一種材料組成。n物體由多種材料組成，但每一種材料的顆粒遠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。小于物體，且在物體內均勻分布。The body is isotropic 物體是各向同性的nThe elastic constants will be independent

14、 of the orientation of the coordinate axes. 彈性常數(shù)與坐標軸的方向無關。彈性常數(shù)與坐標軸的方向無關。nSteel structure - isotropic 鋼鋼-各向同性各向同性nWooden structure-not isotropic 木木-各向異性各向異性The displacements and strains are small位移和應變是微小的nThe displacement components are very small in comparison with its original dimensions. 位移遠小于物體尺寸

15、位移遠小于物體尺寸-可用變形前的尺寸代替變形可用變形前的尺寸代替變形后的尺寸。后的尺寸。nThe strain components and the rotations of all line elements are much smaller than unity. 應變分量和轉角遠小于應變分量和轉角遠小于1-1-其乘積及二次冪可忽略。其乘積及二次冪可忽略。Fundamental quantities expressed by matrix基本量的矩程表示nBody force 體力：nSurface force 面力： nDisplacement 位移： nStress 應力：nStrai

16、n 應變： Txyzxyyzzx Txyzxyyzzx Tu v w Tpx y zFundamental equations expressed by matrix基本方程的矩程表示nGeometrical equations 幾何方程 nPhysical equations 物理方程nBalance equations 平衡方程nVirtual work equations 虛功方程Geometrical equations幾何方程 xyzxyyzzxuxvywzuvyxvwzywuxz222222222222222222222yxyxyzxyxzxyyzzyyzxyzxxzxzyzxyzxzyxx yy zxxyzzyy zz xyxyzxzz xx yxxyz 應變分量與位移分量的幾何關系變形協(xié)調方程Physical equations 物理方程D1D100011100011000(1)1 200(1)(1