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1、Finite Element Analysis Theory,有限元法理论,Introduction to Computational Mechanics,Modeling and simulation of mechanics problems Finite element method Finite difference method Molecular dynamics method Boundary element method,Chapter 1. Elasticity and Finite Element Method 第一章 弹性力学及有限元,Theory of elastici
2、ty 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学。它是固体力学的一个分枝。,What does the Elasticity deal with?,It deals with the stresses, deformations and displacements in elastic solids produced by external forces or changes in temperature.
3、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。 It 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,Exte
4、rnal Forces 外力 Stress 应力 Deformation (Strain) 形变(应变) Displacement 位移,A. external forces 外 力,Body forces 体积力，体力 External 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 Surface forces 表面力，面力 External forces, or the loads, distributed over the
5、 surface 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,Internal 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。 Stresses are the internal forces acting on the per unit area. 应力:作用在单位面积
6、上的内力。,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,By deformation we mean the change of the shape of a body, which may be expressed by the chang
7、es in lengths and angles of its parts. 形变-物体形状（各部分长度和角度）的改变。 To 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,By displacement, we mean the change of position. 位置的移动叫位移。 Displacement components u, v, w-the pr
8、ojections 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 。 It 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,The body is continuous. 物体是连续的。 The body is perfectly elastic. 物体是完全弹性的。 The body is
9、homogeneous. 物体是均质的。 The body is isotropic. 物体是各向同性的。 The displacements and strains are small. 位移和应变是微小的。,The body is continuous 物体是连续的,The 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。 Under this assumption, the physical quantities in t
10、he 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）能用坐标的连续函数表示。,The body is perfectly elastic 物体是完全弹性的,The body wholly obeys Hooks law of elasticity. -The relations between the stress components and the strain co
11、mponents are linear. 物体遵守虎克定律-应力分量和应变分量是线性关系。 The 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,The elastic constants will be independent of the location in the body. 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
12、一种材料的颗粒远小于物体，且在物体内均匀分布。,The body is isotropic 物体是各向同性的,The 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。 Steel structure - isotropic 钢-各向同性 Wooden structure-not isotropic 木-各向异性,The displacements and strains are small 位移和应变是微小的,The displacement compo
13、nents are very small 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。 The 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,Body force 体力： Surface force 面力： Displace
14、ment 位移： Stress 应力： Strain 应变： ,Fundamental equations expressed by matrix 基本方程的矩程表示,Geometrical equations 几何方程 Physical equations 物理方程 Balance equations 平衡方程 Virtual 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,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。 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。,
