有限元法理论(一)

1、Finite Element Analysis Theory有限元法理论有限元法理论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 弹性力学及有限元 Theory of elasti

2、city is often called elasticity or theory of elastic mechanics. It is the branch of solid mechanics. 弹性力学的理论简称为弹性理论或弹性力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。学。它是固体力学的一个分枝。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. 研究弹性体由于外力和温度改变而引起的应力，形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。变和位移。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. 分析结构的应力，形变和位移，检查是否满足强度，分析结

4、构的应力，形变和位移，检查是否满足强度，刚度和稳定性条件刚度和稳定性条件。The Important Concept in Elasticity弹性力学中的几个重要概念n External Forces 外力外力n Stress 应力应力n Deformation (Strain) 形变形变(应变应变)n Displacement 位移位移A. external forces 外 力nBody forces 体积力，体力体积力，体力nExternal forces or the loads, distributed over the volume of the body, are called

5、 body forces. 分布在物体体内的外力叫体力分布在物体体内的外力叫体力: :重力，惯性力重力，惯性力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. 分布在物体表面的外力叫面力分布在物体表面的外力叫面力: :水压力，接触力水压力，接触力B. Stress 应 力nInternal forces: under the action of external forces, interna

6、l forces will be produced between the parts of a body. 内力：在外力作用下，内力：在外力作用下，物体各部分间产生相互作用物体各部分间产生相互作用的力叫内力的力叫内力。nStresses are the internal forces acting on the per unit area. 应力应力: :作用在单位面积上的内力作用在单位面积上的内力。 Stress Fig. 应力定义图The normal component is called the normal stress.The tangential component is ca

7、lled the shearing stress. 法向分量叫法向应力，切向分量叫剪应力法向分量叫法向应力，切向分量叫剪应力。0limAQSA The fig. of stress notation 坐标面上应力记号图C. Deformation 形 变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. 形变形变-物体形状（各部分长度和角度）的改变物体形状（各部分长度和角度）的改变。

8、nTo study deformation condition at a certain point P, we consider line segments PA, PB, PC 研究一点的变形，考虑通过研究一点的变形，考虑通过P P点的三个正向微段点的三个正向微段PA,PB,PCPA,PB,PC。D. Displacements 位 移nBy displacement, we mean the change of position. 位置的移动叫位移。位置的移动叫位移。nDisplacement components u, v, w-the projections of the displ

9、acement on the x, y and z axes. 位移在坐标轴上投影叫位移分量位移在坐标轴上投影叫位移分量 u, v, w u, v, w 。nIt is considered positive as it is in the positive direction of the corresponding coordinate axis. 沿坐标正向的位移分量为正。沿坐标正向的位移分量为正。Basic assumptions 基本假定nThe body is continuous. 物体是连续的。物体是连续的。nThe body is perfectly elastic. 物体是

10、完全弹性的。物体是完全弹性的。nThe body is homogeneous. 物体是均质的。物体是均质的。nThe body is isotropic. 物体是各向同性的。物体是各向同性的。nThe displacements and strains are small. 位移和应变是微小的。位移和应变是微小的。The body is continuous 物体是连续的nThe whole volume of the body is filled with continuous matter without any void. 假定整个物体的体积都被组成这个物体的介质所假定整个物体的体积都

11、被组成这个物体的介质所充满，不留下任何孔隙。充满，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：应力，应变，位移）能用坐标的连物理量（例：应力，应变，位移）能用坐标的连续函数表示。续函数表示。The body is perfectly elastic 物体是完全弹性

12、的nThe body wholly obeys Hooks law of elasticity. -The relations between the stress components and the strain components are linear. 物体遵守虎克定律物体遵守虎克定律-应力分量和应变分量是线性应力分量和应变分量是线性关系。关系。nThe elastic constants will be independent of the stress or strain components under this assumption. 弹性常数与应力和应变的大小无关。弹性常数

13、与应力和应变的大小无关。The body is homogeneous 物体是均质的nThe elastic constants will be independent of the location in the body. 弹性常数与位置无关。弹性常数与位置无关。n物体由同一种材料组成。物体由同一种材料组成。n物体由多种材料组成，但每一种材料的颗粒远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。小于物体，且在物体内均匀分布。The body is isotropic 物体是各向同性的nThe elastic constants will be independent

14、 of the orientation of the coordinate axes. 弹性常数与坐标轴的方向无关。弹性常数与坐标轴的方向无关。nSteel structure - isotropic 钢钢-各向同性各向同性nWooden structure-not isotropic 木木-各向异性各向异性The displacements and strains are small位移和应变是微小的nThe displacement components are very small in comparison with its original dimensions. 位移远小于物体尺寸

15、位移远小于物体尺寸-可用变形前的尺寸代替变形可用变形前的尺寸代替变形后的尺寸。后的尺寸。nThe strain components and the rotations of all line elements are much smaller than unity. 应变分量和转角远小于应变分量和转角远小于1-1-其乘积及二次幂可忽略。其乘积及二次幂可忽略。Fundamental quantities expressed by matrix基本量的矩程表示nBody force 体力：nSurface force 面力： nDisplacement 位移： nStress 应力：nStrai

16、n 应变： Txyzxyyzzx Txyzxyyzzx Tu v w Tpx y zFundamental equations expressed by matrix基本方程的矩程表示nGeometrical equations 几何方程 nPhysical equations 物理方程nBalance equations 平衡方程nVirtual work equations 虚功方程Geometrical equations几何方程 xyzxyyzzxuxvywzuvyxvwzywuxz222222222222222222222yxyxyzxyxzxyyzzyyzxyzxxzxzyzxyzxzyxx yy zxxyzzyy zz xyxyzxzz xx yxxyz 应变分量与位移分量的几何关系变形协调方程Physical equations 物理方程D1D100011100011000(1)1 200(1)(1