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1、Finite Element Analysis Theory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 elasticity i
2、s 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 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 forces or changes in temperature. 研究弹性体
3、由于外力和温度改变而引起的应力，形 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 ，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nExternal
4、 Forces 外力 nStress 应力 nDeformation (Strain) 形变(应变) nDisplacement 位移A. external forces 外 力nBody forces 体积力，体力nExternal forces or the loads, distributed over the volume of the body, are called body forces.分布在物体体内的外力叫体力:重力，惯性力nSurface forces 表面力，面力nExternal forces, or the loads, distributed over the su
5、rface of a body, are called surface forces.分布在物体表面的外力叫面力:水压力，接触力B. Stress 应 力nInternal forces: under the action of external forces, internal forces will be produced between the parts of a body.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。nStresses are the internal forces acting on the per unit area.应力:作用在单位面积上的内力。
6、Stress Fig. 应力定义图The normal component is called the normal stress. The tangential component is called the shearing stress. 法向分量叫法向应力，切向分量叫剪应力。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 leng
7、ths and angles of its parts.形变-物体形状（各部分长度和角度）的改变 。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P点的三个正向微段 PA,PB,PC。D. Displacements 位 移nBy displacement, we mean the change of position. 位置的移动叫位移。nDisplacement components u, v, w-the projections
8、of the displacement on the x, y and z axes.位移在坐标轴上投影叫位移分量 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。Basic assumptions 基本假定nThe body is continuous. 物体是连续的。nThe body is perfectly elastic. 物体是完全弹性的。nThe body is homogeneous.
9、 物体是均质的。nThe body is isotropic. 物体是各向同性的。nThe displacements and strains are small. 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 所充满，不留下任何孔隙。nUnder this assumption, the physical quantities in the body, such
10、 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 连续函数表示。The body is perfectly elastic 物体是完全弹性的nThe body wholly obeys Hooks law of elasticity. -The relations between the stress components and the strain components are l
11、inear. 物体遵守虎克定律-应力分量和应变分量是线性 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。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。n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 小于物体，且
12、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 of the orientation of the coordinate axes. 弹性常数与坐标轴的方向无关。nSteel structure - isotropic 钢-各向同性nWooden structure-not isotropic 木-各向异性The displacements and strains are small 位移和应变是微小的nThe displacement components are very s
13、mall in comparison with its original dimensions.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。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1-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： nStres
14、s 应力：nStrain 应变： Fundamental 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 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Balance equations 平衡方程Virtual Work Equations 虚功方程n虚功原理:一个原为静止的质点系，如果约束是理想 双面定常约束，则系统继续保持静止的条件是所有 作用于该系统的主动力对作用点的虚位移所作的功 的和为零。n虚位移原理:如果在虚位移发生之前,物体处于平衡 状态,那么在虚位移发生时,外力所做虚功等于物体 的虚应变能。
