英漢雙語彈性力學8

1、Spatial ProblemChapter 8 Elasticity1第八章 空間問題2Space ProblemChapter8 Space Problem8-4 The Spherical Symmetric Problem of Space8-3 The Axially Symmetric Problem of Space8-2 The Basic Equation unde Rectangular Coordinate8-1 Introduction3空間問題第八章 空間問題8-4 空間球對稱問題8-3 空間軸對稱問題8-2 直角坐標下的基本方程8-1 概 述4In this cha

2、pter we first give out the equations of equilibrium, the geometric equations and the physical equations under rectangular coordinate for spatial problems. For the analytic solutions of spatial problems can only be obtained under peculiar boundary conditions, we discuss the axial symmetric problems a

3、nd the ball symmetric problems of space emphatically.8-1 IntroductionBall Symmetric ProblemxzyAxial Symmetric ProblemxzyPSpace Problem5 本章首先給出空間問題直角坐標下的平衡方程、幾何方程和物理方程。針對空間問題的解析解一般只能在特殊邊界條件下才可以得到，我們著重討論空間軸對稱問題和空間球對稱問題。8-1 概 述空間問題球對稱問題xzy軸對稱問題xzyP68-2 Basic Equations under Rectangular CoordinateOne. D

4、ifferential Equations of Equilibrium Consider an arbitrary point inside the body and fetch a small parallel hexahedron, which stress components on each side are shown as figure. If ab denotes the line which joins the centers of two faces of the hexahedron, then from we get Canceling terms and neglec

5、ting higher order small variables,we getSpace Problem78-2 直角坐標下的基本方程空間問題一 平衡微分方程 在物體內(nèi)任意一點 P，取圖示微小平行六面體。微小平行六面體各面上的應力分量如圖所示。 若以連接六面體前后兩面中心的直線為ab，則由 得化簡并略去高階微量，得8Similarly,we get Here we prove the relation of the equality of cross shears againfrom List the equations,cancel terms,we getThese are differ

6、ential equations of equilibrium under rectangular coordinate of spaceTwo. Geometric Equations For spatial problems, deformation components and displacement components should satisfy following geometric equations Of which the first two and the last have been obtained among plane problems, the other t

7、hree can be led out with the same method.Space Problem9空間問題同理可得這只是又一次證明了剪應力的互等關系。由 立出方程，經(jīng)約簡后得這就是空間直角坐標下的平衡微分方程。二 幾何方程 在空間問題中，形變分量與位移分量應當滿足下列 6 個幾何方程其中的第一式、第二式和第六式已在平面問題中導出，其余三式可用相同的方法導出。10Three. Physical Equations For an isotropic body, the relations between deformation components and stress compone

8、nts are as follows: These are physical equations for spatial problems. If stress components are denoted by strain components, physical equations can be written as:where：Space Problem11空間問題三 物理方程對于各向同性體，形變分量與應力分量之間的關系如下： 這就是空間問題的物理方程。 將應力分量用應變分量表示，物理方程又可表示為：其中：12Four Equations of Compatibility Differ

9、entiate the second and the third formula of geometric equations at the left.Adding these two,we get Substitute the fourth formula of geometric equations into the above equation, we get（a）Similarly（b）Space Problem13空間問題四 相容方程 將幾何方程第二式左邊對z的二階導數(shù)與第三式左邊對y的二階導數(shù)相加，得將幾何方程第四式代入，得（a）同理（b）14 Differentiate the

10、late three formulas of geometric equations separately for X,Y,Z,we get From the above equations,we getSpace Problem15空間問題 將幾何方程中的后三式分別對x、y、z求導，得并由此而得16Similarly（d） The equations of (a),(b),(c),(d)are called compatibility conditions of deformation, also known as equations of compatibility. Substituti

11、ng physical equations into the above equations,and canceling terms according to differentiate equations of equilibrium,we get the compatibility equations which are expressed with stress components:Namely（c）Space Problem17空間問題同理（d）方程(a)、(b)、(c)、(d)稱為變形協(xié)調條件，也稱相容方程。 將物理方程代入上述相容方程，并利用平衡微分方程簡化后，得用應力分量表示的

12、相容方程：即（c）18We call them Michel compatibility equations.Space Problem19空間問題稱其為密切爾相容方程。20 Among spatial problems, if the elasticity bodys geometric shape,restraint condition and any external factors are symmetrical in a certain axis(any plane which passes this axis is all symmetrical one),then all str

13、esses,deformations and displacements are symmetrical in this axis. This kind of problem is called axial symmetry problem of space. The forms of elastomers of axial symmetry problem are generally divided into two kinds: cylinder or half space body. According to the characteristic of axial symmetry,we

14、 should adopt the cylindrical coordinates .if we take z axis as the axis of symmetry ,then all the stress components,strain components and displacement components will be only the function of r and z,with the coordinate have nothing to do with.8-3 Axially Symmetric Problems for SpaceSpace Problem21空

15、間問題 在空間問題中，若彈性體的幾何形狀、約束情況以及所受的外來因素，都對稱于某一軸（通過這個軸的任一平面都是對稱面），則所有的應力、形變和位移也對稱于這一軸。這種問題稱為空間軸對稱問題。 根據(jù)軸對稱的特點，應采用圓柱坐標 表示。若取對稱軸為 z 軸，則軸對稱問題的應力分量、形變分量和位移分量都將只是 r 和 z 的函數(shù)，而與 坐標無關。 軸對稱問題的彈性體的形狀一般為圓柱體或半空間體。8-3 空間軸對稱問題22One. Differential Equations of Equilibrium Consider a small element as shown in figure. For

16、axial symmetry, the elements two cylindrical planes exist only normal stresses and axial shear stresses;its two horizontal planes exist only normal stresses and radial shear stresses;its two perpendicular planes exist only round normal stresses,which are shown in figure. According to the assumption

17、of continuity, stress components of the small element s positive planes have a small increase compared with the negative ones.Attention:the increase of round normal stresses are zero at this moment. For equilibrium at radial direction and axial direction and from , canceling terms and ignoring the h

18、igh order small values,we getSpace Problem23空間問題一 平衡微分方程 取圖示微元體。由于軸對稱，在微元體的兩個圓柱面上，只有正應力和的軸向剪應力；在兩個水平面上只有正應力和徑向剪應力；在兩個垂直面上只有環(huán)向正應力，圖示。 根據(jù)連續(xù)性假設，微元體的正面相對負面其應力分量都有微小增量。注意：此時環(huán)向正應力的增量為零。 由徑向和軸向平衡，并利用 ，經(jīng)約簡并略去高階微量，得：24 These are the differential equations of equilibrium for axial symmetry problems in terms o

19、f cylindrical coordinates.Two . Geometric Equations Similar to the analysis of plane problem in term of polar coordinates, we get, the strain components caused by radial displacement are: The strain components caused by axial displacement are: From the principle of superposing, namely we get the geo

20、metric equations for spatial axial symmetry problems:Space Problem25空間問題 這就是軸對稱問題的柱坐標平衡微分方程。二 幾何方程 通過與平面問題及極坐標中同樣的分析，可見，由徑向位移引起的形變分量為：由軸向位移引起的形變分量為： 由疊加原理，即得空間軸對稱問題的幾何方程：26Three. Physical Equations Because the cylindrical coordinates are orthogonal coordinates as the rectangular ones, we can get the

21、 physical equations directly from Hookes law: If stress components are expressed with strain components, the above equations can be written as:Where :Space Problem27空間問題三 物理方程 由于圓柱坐標，是和直角坐標一樣的正交坐標，所以可直接根據(jù)虎克定律得物理方程： 應力分量用形變分量表示的物理方程：其中：28Four. Solution of Axial Symmetry Problems Substitute the geomet

22、ric equations into the physical equations which stress components are expressed with strain components, we get the elastic equations:Where : Substitute the above equations into the differential equations of equilibrium, and use the notation:We get These are known as basic differential equations for

23、solving the spatial axial symmetry problems in terms of displacement components. Obviously, the displacement components in above equations are functions coordinates r and z,they cant be solved directly. So we introduce the following method: Space Problem29空間問題四 軸對稱問題的求解 將幾何方程代入應力分量用應變分量表示的物理方程，得彈性方程

24、：其中： 再將彈性方程代入平衡微分方程，并記：得到這就是按位移求解空間軸對稱問題所需要的基本微分方程。 顯然，上述基本微分方程中的位移分量是坐標r、z 的函數(shù)，不可能直接求解，為此介紹下列方法：30Five. Displacement Tendency Function For simplicity, ignoring the body force, the basic differential equations in term of displacement components can be simplified as: Supposing now the displacement ha

25、s tendency, we use displacement tendency function to denote the displacement components: Thus we get: Substitute with the basic differential equations which ignoring the body force,we get:Namely Space Problem31空間問題五 位移勢函數(shù) 為簡單起見，不計體力。位移分量的基本微分方程簡化為： 現(xiàn)在假設位移是有勢的，把位移分量用位移勢函數(shù) 表示為：從而有代入不計體力的基本微分方程，得即32 is

26、 a mediation function. The solving representations of stress components from displacement tendency function are: If only , we get .Namely So for an axial symmetry problem, if we find a suitable mediation function ,from which the displacement components and stress components satisfy the boundary cond

27、itions, then we get the correct solution of the problem. In order to solve axial symmetry problems, Lame introduces a displacement function Attention: not all the displacement functions of spatial problems have tendency. But if they have, the volumetric strain .Six Lame Displacement FunctionDefine W

28、here Space Problem33空間問題取 ，則 。即 為調和函數(shù)，由位移勢函數(shù)求應力分量的表達式為： 為求解軸對稱問題，拉甫引用一個位移函數(shù) 這樣，對于一個軸對稱問題，如果找到適當?shù)恼{和函數(shù) ，使得由此給出的位移分量和應力分量能夠滿足邊界條件，就得到該問題的正確解答。注：并不是所有問題中的位移函數(shù)都是有勢的。若位移勢函數(shù)有勢，則體積應變 。六 拉甫位移函數(shù)令其中34 Substitute the above functions into the basic differential functions which in the absence of body force, we ge

29、t: Namely is a repeated mediation function, we call it Lame displacement function. The representations of stress components from this function are: So for an axial symmetry problem, if we find a suitable repeated mediation function ,from which the displacement components and stress components satisf

30、y the boundary conditions, then we get the correct solution of the problem.Space Problem35空間問題 將上式代入不計體力位移分量的基本微分方程，可見：即 是重調和函數(shù)，稱為拉甫位移函數(shù)。由拉甫位移函數(shù)求應力分量的表達式為： 可見，對于一個軸對稱問題，只須找到恰當?shù)闹卣{和的拉甫位移函數(shù) ，使得該位移函數(shù)給出的位移分量和應力分量能夠滿足邊界條件，就得到該問題的正確解答。36Seven Example: half space body which is under the action of outward dr

31、awn concentrated forces in the boundaryConsider a half space body, which body forces are ignored. It receives outward drawn concentrated forces in the boundary, as shown in figure. Please solve its stresses and displacements. Solution:choose the coordinate system as fig. Through the dimensional anal

32、ysis, Lames displacement function is positive one order power of length coordinate of which F multiplies R、z、. After preliminary calculation, we set displacement function as: According to the relations of displacement components and stress components and displacement function:xzyPRzSpace Problem37空間

33、問題七 舉例：半空間體在邊界上受法向集中力 設有半空間體，體力不計，在其邊界上受有法向集中力，如圖所示。試求其應力與位移。解：取坐標系如圖。通過量綱分析，拉甫位移函數(shù)應是F乘以R、z、等長度坐標的正一次冪，試算后，設位移函數(shù)為根據(jù)位移分量和應力分量與位移函數(shù)的關系：xzyPRz38We can obtain the displacement components and the stress componentsSpace Problem39空間問題可以求得位移分量和應力分量40The boundary conditions are(a)(b)According to the Saint-Ve

34、nants Principle,we have(c)The boundary condition (a) is satisfied. From boundary condition (b),we get(d)From condition (c), we get(e)Solving in terms of (d) and (e),we getSpace Problem41空間問題邊界條件是(a)(b)根據(jù)圣維南原理，有(c)邊界條件(a)是滿足的。由邊界條件(b)得(d)由條件(c)得(e)由(d)及(e)二式的聯(lián)立求解，得42Substitute the obtained A1 and A2

35、into the forgoing representations, we get Space Problem43空間問題將得出的A1及A2回代，得44 Among spatial problems, if the elasticity bodys geometric shape,restraint condition and any external factors are symmetrical in a certain point (any plane which passes this point is all symmetrical one),then all stresses,st

36、rains and displacements are symmetrical in this point. This kind of problem is called spherically symmetry problem of space. According to the characteristic of spherically symmetry, we should adopt the spherical coordinates .if we take elasticity bodys symmetrical point as the coordinates origin , t

37、hen all the stress components,strain components and displacement components will be only the function of radial coordinate r,with the other two coordinates have nothing to do with. Obviously, spherically symmetric problems can only exist in hollow or solid round spheroid. 8-4 Spherically Symmetric P

38、roblem For SpaceSpace Problem45空間問題 在空間問題中，如果彈性體的幾何形狀、約束情況以及所受的外來因素，都對稱于某一點（通過這一點的任意平面都是對稱面），則所有的應力、形變和位移也對稱于這一點。這種問題稱為空間球對稱問題。 根據(jù)球對稱的特點，應采用球坐標 表示。若以彈性體的對稱點為坐標原點 ，則球對稱問題的應力分量、形變分量和位移分量都將只是徑向坐標 r 的函數(shù)，而與其余兩個坐標無關。 顯然，球對稱問題只可能發(fā)生于空心或實心的圓球體中。8-4 空間球對稱問題46One. Differential Equations of Equilibrium For symm

39、etry, the small element only has radial volume force . From radial equilibrium, and considering , Neglecting the higher order small variables, we get the differential equations of equilibrium for spherically symmetric problems: Fetch a small element. Fetch a small hexahedron from the elastomer. It i

40、s formed by two pellet faces, which distance is ,and two pairs of radial planes, which angle is respectively. For spherical symmetry, each plane only has normal stress. Its stress situations are shown in fig.Space Problem47空間問題一 平衡微分方程 取微元體。用相距 的兩個圓球面和兩兩互成 角的兩對徑向平面，從彈性體割取一個微小六面體。由于球對稱，各面上只有正應力，其應力情況

41、如圖所示。 由于對稱性，微元體只有徑向體積力 。由徑向平衡，并考慮到 ，再略去高階微量，即得球對稱問題的平衡微分方程：48Two Geometric Equations For symmetry, it can only exist radial displacement ; for the same reason, it can only exist radial normal strain and tangent normal strain , it cant exist shear strain along the coordinate direction. The geometric

42、equations for spherically symmetric problems are:Three Physical Equations The physical equations for spherically symmetric problems can directly be led out from Hookes law If stress components are expressed with strain components, we getSpace Problem49空間問題二 幾何方程 由于對稱，只可能發(fā)生徑向位移 ；又由于對稱，只可能發(fā)生徑向正應變 及切向正

43、應變 ，不可能發(fā)生坐標方向的剪應變。球對稱問題的幾何方程為：三 物理方程 球對稱問題的物理方程可直接根據(jù)虎克定律得來：將應力用應變表示為：50Four. The Basic Differential Equation in Terms of Displacement Substitute the geometric equations into the physical equations, we get the elastic equations:Substitute the above equations into the differential equations of equilib

44、rium,we get This is known as the basic differential equations for solving the spherically symmetric problems in terms of displacement.Space Problem51空間問題四 位移法求解的基本微分方程 將幾何方程代入物理方程，得彈性方程再代入平衡微分方程，得這就是按位移求解球對稱問題時所需要用的基本微分方程。52Example: a hollow pellet which is under action of the even distributed press

45、ure consider a hollow pellet. Its interior radius is a, the exterior is b, the inner pressure is qa, outer pressure is qb. At the absence of body force, please find its stresses and displacements.Its solution isAnd the stress components are:Solution :for ignoring the body force, the differential equ

46、ation for spherically symmetric problems can be simplified as xzySpace ProblemFive 53空間問題五 舉例：空心圓球受均布壓力 設有空心圓球，內(nèi)半徑為a，外半徑為b，內(nèi)壓為qa，外壓為qb，體力不計，試求其應力及位移。其解為得應力分量解： 由于體力不計，球對稱問題的微分方程簡化為xzy54Substitute the boundary conditions into the above formulas, we getAnd then we get the radial displacement of the pr

47、oblem: The stress expressions are:Space Problem55空間問題將邊界條件代入上式解得于是得問題的徑向位移應力表達式56Exercise 8.1 suppose there is a equal section pole with arbitrary shape. its density is ,with its upper end hung and lower end free, which is shown as fig. Try to prove the stress components be suitable for any condition.zySolution : the stress components are:The body force components are:Space Problem57空間問題練習8.1 設有任意形狀的等截面桿，密度為 ，上端懸掛,下端自由，如圖所示。試證明應力分量能滿足所有一切條件。zy解： 已知應力分量為體力分量為58One The Inspection of Differential Equations of EquilibriumObviously they are satisfied.Two. The Inspection o