有限元法理論(一)

單擊此處編輯母版標(biāo)題樣式,單擊此處編輯母版文本樣式,第二級(jí),第三級(jí),第四級(jí),第五級(jí),*,Finite Element Analysis Theory,有限元法理論,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 彈性力學(xué)及有限元,Theory of elasticity is often called elasticity or theory of elastic mechanics.It is the branch of solid mechanics.,彈性力學(xué)的理論簡稱為彈性理論或彈性力學(xué)它是固體力學(xué)的一個(gè)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.,研究彈性體由于外力和溫度改變而引起的應(yīng)力，形變和位移。

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.,分析結(jié)構(gòu)的應(yīng)力，形變和位移，檢查是否滿足強(qiáng)度，剛度和穩(wěn)定性條件,The Important Concept in Elasticity,彈性力學(xué)中的幾個(gè)重要概念,External Forces,外力,Stress,應(yīng)力,Deformation(Strain),形變,(,應(yīng)變,),Displacement,位移,A.external forces,外 力,Body forces,體積力，體力,External forces or the loads,distributed over the volume of the body,are called body forces.,分布在物體體內(nèi)的外力叫體力:重力，慣性力,Surface forces,表面力，面力,External 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:水壓力，接觸力,B.Stress 應(yīng),力,Internal forces:under the action of external forces,internal forces will be produced between the parts of a body.,內(nèi)力：在外力作用下，,物體各部分間產(chǎn)生相互作用的力叫內(nèi)力,。

Stresses are the internal forces acting on the per unit area.,應(yīng)力:作用在單位面積上的內(nèi)力,Stress Fig.應(yīng)力定義圖,The normal component is called the normal stress.,The tangential component is called the shearing stress.,法向分量叫法向應(yīng)力，切向分量叫剪應(yīng)力,The fig.of stress notation,坐標(biāo)面上應(yīng)力記號(hào)圖,C.Deformation 形 變,By 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-物體形狀（各部分長度和角度）的改變,To study deformation condition at a certain point P,we consider line segments PA,PB,PC,研究一點(diǎn)的變形，考慮通過P點(diǎn)的三個(gè)正向微段PA,PB,PC,。

D.Displacements 位 移,By displacement,we mean the change of position.,位置的移動(dòng)叫位移Displacement components u,v,w-the projections of the displacement on the x,y and z axes.,位移在坐標(biāo)軸上投影叫位移分量 u,v,w,It is considered positive as it is in the positive direction of the corresponding coordinate axis.,沿坐標(biāo)正向的位移分量為正Basic assumptions 基本假定,The body is continuous.,物體是連續(xù)的The body is perfectly elastic.,物體是完全彈性的The body is homogeneous.,物體是均質(zhì)的The body is isotropic.,物體是各向同性的The displacements and strains are small.,位移和應(yīng)變是微小的。

The body is continuous,物體是連續(xù)的,The whole volume of the body is filled with continuous matter without any void.,假定整個(gè)物體的體積都被組成這個(gè)物體的介質(zhì)所充滿，不留下任何孔隙Under 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(yīng)力，應(yīng)變，位移）能用坐標(biāo)的連續(xù)函數(shù)表示The body is perfectly elastic,物體是完全彈性的,The 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-應(yīng)力分量和應(yīng)變分量是線性關(guān)系。

The elastic constants will be independent of the stress or strain components under this assumption.,彈性常數(shù)與應(yīng)力和應(yīng)變的大小無關(guān)The body is homogeneous,物體是均質(zhì)的,The elastic constants will be independent of the location in the body.,彈性常數(shù)與位置無關(guān)物體由同一種材料組成物體由多種材料組成，但每一種材料的顆粒遠(yuǎn)小于物體，且在物體內(nèi)均勻分布The body is isotropic,物體是各向同性的,The elastic constants will be independent of the orientation of the coordinate axes.,彈性常數(shù)與坐標(biāo)軸的方向無關(guān)Steel structure-isotropic,鋼-各向同性,Wooden structure-not isotropic,木-各向異性,The displacements and strains are small,位移和應(yīng)變是微小的,The displacement components are very small in comparison with its original dimensions.,位移遠(yuǎn)小于物體尺寸-可用變形前的尺寸代替變形后的尺寸。

The strain components and the rotations of all line elements are much smaller than unity.,應(yīng)變分量和轉(zhuǎn)角遠(yuǎn)小于1-其乘積及二次冪可忽略Fundamental quantities expressed by matrix基本量的矩程表示,Body force 體力：,Surface force 面力：,Displacement 位移：,Stress 應(yīng)力：,Strain 應(yīng)變：,Fundamental equations expressed by matrix基本方程的矩程表示,Geometrical equations 幾何方程,Physical equations 物理方程,Balance equations 平衡方程,Virtual work equations 虛功方程,Geometrical equations,幾何方程,應(yīng)變分量與位移分量的幾何關(guān)系,變形協(xié)調(diào)方程,Physical equations,物理方程,Balance equations,平衡方程,Virtual Work Equations,虛功方程,虛功原理:一個(gè)原為靜止的質(zhì)點(diǎn)系，如果約束是理想雙面定常約束，則系統(tǒng)繼續(xù)保持靜止的條件是所有作用于該系統(tǒng)的主動(dòng)力對作用點(diǎn)的虛位移所作的功的和為零。

虛位移原理:如果在虛位移發(fā)生之前,物體處于平衡狀態(tài),那么在虛位移發(fā)生時(shí),外力所做虛功等于物體的虛應(yīng)變能,。