英漢雙語(yǔ)彈性力學(xué)1

1、1 2 3 Chapter 1 Introduction 1-1 The Modeling of the Engineering Mechanics Problem 1-3 The Basic Assumption of the Elasticity Problem 1-4 The Several Basic Concepts of Elasticity 1-5 The Study Method of the Elasticity 1-2 The Basic Contents of the Elasticity Exercises Lesson 4 第一章第一章 緒緒 論論 1-1 1-1 工

2、程力學(xué)問題的建模工程力學(xué)問題的建模 1-3 1-3 彈性力學(xué)問題的基本假設(shè)彈性力學(xué)問題的基本假設(shè) 1-4 1-4 彈性力學(xué)中的幾個(gè)基本概念彈性力學(xué)中的幾個(gè)基本概念 1-5 1-5 彈性力學(xué)的學(xué)習(xí)方法彈性力學(xué)的學(xué)習(xí)方法 1-2 1-2 彈性力學(xué)的基本內(nèi)容彈性力學(xué)的基本內(nèi)容 習(xí)題課習(xí)題課 5 The elasticity is a branch of the solid mechanics, the task of it is to research the elasticity objects stress, deformation and displacement due to externa

3、l force or change of temperature. The elasticity is the foundation of studying plasticity, fracture mechanics and finite element method. This course shows the mathematics modeling process of mechanics problems completely, and establishes the basic equation and boundary condition of the elasticity an

4、d proceeds to beg the solutions of some problem. The foundation of the elasticity basic equation lays a foundation for further number method. 6 彈性力學(xué)是固體力學(xué)的一個(gè)分支，研究彈性體由于外 力作用或溫度改變等原因而發(fā)生的應(yīng)力、形變和位移。 彈性力學(xué)是學(xué)習(xí)塑性力學(xué)、斷裂力學(xué)、有限元方法的 基礎(chǔ)。 本課程較為完整的表現(xiàn)了力學(xué)問題的數(shù)學(xué)建模過程， 建立了彈性力學(xué)的基本方程和邊值條件，并對(duì)一些問題進(jìn) 行了求解。彈性力學(xué)基本方程的建立為進(jìn)一步的數(shù)值方法 奠定了

5、基礎(chǔ)。 7 Through the process of establishing the mechanics model in the engineering mechanics problem, generally three parts should be simplified: Suffering Force Simplification Material Simplification Construction Simplification 1、The Modeling Process of the Engineering Mechanics Problem 1-1 The Model

6、ing of the Engineering Mechanics Problem Fig.1-1Fig.1-1 8 工程力學(xué)問題建立力 學(xué)模型的過程中，一般要 對(duì)三方面進(jìn)行簡(jiǎn)化： 受力簡(jiǎn)化 材料簡(jiǎn)化 結(jié)構(gòu)簡(jiǎn)化 一、工程力學(xué)問題的建模過程一、工程力學(xué)問題的建模過程 1-1 1-1 工程力學(xué)問題的建模工程力學(xué)問題的建模 圖圖1-11-1 9 Material is simplified according to these hypothesises of the same kind, consecution and uniformity in each direction. （3）Material

7、 simplification According to the Saint-Venants principle, the complex force system is simplified to an equivalent force system. （2）Suffering Force Simplification Such as space problem is simplified to flat surface problem and symmetry problem in axis, and entity construction is simplified to plate c

8、onstruction （1）Construction Simplification 10 根據(jù)各向同性、連續(xù)、均勻等假設(shè)進(jìn)行簡(jiǎn)化。 （3 3）材料簡(jiǎn)化）材料簡(jiǎn)化 根據(jù)圣維南原理，復(fù)雜力系簡(jiǎn)化為等效力系。 （2 2）受力簡(jiǎn)化）受力簡(jiǎn)化 如空間問題向平面問題的簡(jiǎn)化，向軸對(duì)稱問題的簡(jiǎn)化，實(shí) 體結(jié)構(gòu)向板、殼結(jié)構(gòu)的簡(jiǎn)化。 （1 1）結(jié)構(gòu)簡(jiǎn)化）結(jié)構(gòu)簡(jiǎn)化 11 Proceed to handle to the small quantity in high level. Proceed the linearization that may be linearized. 2、Advertent Problem

9、 in Modeling Process After the model is established, proceed to analyse and neaten to the result of the computation, and return the actual problem and proceed the verification. Generally and mostly it is proceeded through experiment. （2）Experiment Verification （1）Linearization 12 對(duì)高階小量進(jìn)行處理，能進(jìn)行線性化的，進(jìn)

10、行線性化。 二、建模過程中注意的問題二、建模過程中注意的問題 模型建立以后，對(duì)計(jì)算的結(jié)果進(jìn)行分析整理，返回實(shí)際問 題進(jìn)行驗(yàn)證，一般主要通過實(shí)驗(yàn)進(jìn)行。 （2 2）實(shí)驗(yàn)驗(yàn)證）實(shí)驗(yàn)驗(yàn)證 （1 1）線性化）線性化 13 The elasticity is a branch of the solid mechanics, the task of it is to research the elasticity objects stress, deformation and displacement due to external force or change of temperature. 1-2 T

11、he Basic Contents of the Elasticity 1、Investigative task The research object of the elasticity is general and complicated shape structure piece, entity structure, plate shell etc. 2、Investigative object 14 彈性力學(xué)是固體力學(xué)的一個(gè)分支，研究彈性體由于受 外力作用或由于溫度改變等原因而發(fā)生的應(yīng)力、形變和位 移。 1-2 1-2 彈性力學(xué)的基本內(nèi)容彈性力學(xué)的基本內(nèi)容 一、研究任務(wù)一、研究任務(wù) 彈

12、性力學(xué)的研究對(duì)象為一般及復(fù)雜形狀的構(gòu)件、實(shí)體 結(jié)構(gòu)、板殼等。 二、研究對(duì)象二、研究對(duì)象 15 Plasticity: plasticity analysis and design of the structure. 3、The relation about the other course： Material mechanics: research stress and displacement of the bar structure piece that is pulled, pressed, sheared, bent or turned. Theoretical mechanics:

13、Study statics and dynamics of the rigid body(constraint force,velocity,acceleration). Structual mechanics: research internal force and displacement of the bar structure. Elasticity: stress and displacement analysis of general plane problem, plate, shell and entity structure etc. 16 塑性力學(xué)：結(jié)構(gòu)的塑性分析、設(shè)計(jì)；

14、三、與其他學(xué)科的關(guān)系：三、與其他學(xué)科的關(guān)系： 材料力學(xué)：研究桿狀構(gòu)件在拉、壓、剪、彎、扭狀態(tài) 下的應(yīng)力和位移； 理論力學(xué)：研究剛體的靜、動(dòng)力學(xué)（約束力、速度、 加速度）。 結(jié)構(gòu)力學(xué)：研究桿系結(jié)構(gòu)的內(nèi)力與位移； 彈性力學(xué)：一般平面問題、板、殼和實(shí)體結(jié)構(gòu)等的 應(yīng)力和位移分析。 17 1-3 The Basic Assumption of the Elasticity In elasticity, doing some necessary assumptions under the premise that can satisfy the practical needing precision an

15、d making the problem solved. （1）Consecution assumption:Some physics measures inside the object, for example stress, strain and displacement etc.whose variety regulation may be denoted by continuous function in coordinate. （2）Ideal elasticity assumption: supposing that the object is a ideal elastic b

16、ody,then the elastic body obey the Hookes law-the stress becomes the direct proportation with homologous deformation.And the elasticity constant doesnt change along with the variety of stress and deformation. （3）Even assumption:supposing the object be constituted by the same material, the elasticity

17、 of the object would not change along with position coordinates change. The basic assumption of the elasticity： 18 1-3 1-3 彈性力學(xué)的基本假設(shè)彈性力學(xué)的基本假設(shè) 在彈性力學(xué)中，在滿足實(shí)用所需精度的前提下做一些 必要的假設(shè)，使問題得以求解。 （1）連續(xù)性假設(shè)：這樣物體內(nèi)的一些物理量，例如 應(yīng)力、應(yīng)變和位移等可用坐標(biāo)的連續(xù)函數(shù)表示它們的變 化規(guī)律。 （2）完全彈性假設(shè)：假定物體為完全彈性體，則 服從虎克定律-應(yīng)力和相應(yīng)的形變成正比，彈性常數(shù) 不隨應(yīng)力或形變的大小而變化。 （3）

18、均勻性假設(shè)：假定物體由同一材料組成，這 樣物體的彈性不隨位置坐標(biāo)而變化。 彈性力學(xué)的基本假設(shè)為： 19 （4）Isotropy assumption: The elastic properties of one point in object are the same in every direction. （5）Assumption of small deformation: supposing displacement and deformation is very small.Then using the dimension before deformation instead of t

19、he one after deformation. The small quantity in high level may be ignored when investigating strain and displacement of the object.Which is very important to the linearization of the equation. The assumptions above are suitable for many problems in engineering, but they exist errors much differently

20、 for some problems, then it is necessary to use another brief method.But it is still the same for the basic theories of many concepts.The elasticity is the foundation of the subjects of learning plasticity,fracture mechanics and finite element method and etc. 20 （4）各向同性假設(shè)：物體內(nèi)一點(diǎn)的彈性性質(zhì)在所有 各個(gè)方向都相同。 （5）小

21、變形假設(shè)：假定位移和形變是微小的。這樣， 可以用變形前的尺寸代替變形后的尺寸，在考察物體的 應(yīng)變和位移時(shí)，可以略去高階小量，這對(duì)于方程的線性 化十分重要。 以上的假設(shè)對(duì)于工程中不少問題是適用的，但對(duì)于 一些問題的誤差太大，就必須用另外的簡(jiǎn)化方案，但許 多概念基本理論仍然是共同的，彈性力學(xué)是學(xué)習(xí)塑性力 學(xué)、斷裂力學(xué)、有限元方法等學(xué)科的基礎(chǔ)。 21 1-4 The Several Basic Concepts of the Elasticity It can be divided into the stress of volume and plane according to the differ

22、ent distribution of the external function, which are called volumetric force and surface force respectively. （2）Property：volumetric force is different from the different location in a point;the volumetric force is continuous in distribution. 1.External stress 1.Volumetric force （1）Definition：It is t

23、he stress distributed in volume of the object that is called volumetric force,for example,gravity and inertia force.It is shown in Fig.1-2.Q z x y V O P X Y Q F Fig.1-2 Z 22 1-4 1-4 彈性力學(xué)中的幾個(gè)基本概念彈性力學(xué)中的幾個(gè)基本概念 按照外力作用的不同分布方式，可分為體積力和表面力， 分別簡(jiǎn)稱體力和面力。 （2）性質(zhì)：體力隨點(diǎn)的位置不同 而不同；體力是連續(xù)分布的。 （一）外力（一）外力 1.體力 （1）定義：所謂體力

24、是分布在物 體體積內(nèi)的力，如重力和慣性力。 如圖12所示 。Q 圖圖1-21-2 z x y V O P X Y Q F Z 23 （3）Gather degree： （4）The component of volumetric force： The force of F is resolved along with the three coordinates, which will get the three components of straight intersection: kZjYiXF X,Y,Z are called the components of volumetric f

25、orce at point P.Plus sign and negative sign are separately determined by the direction of components,and then isForceLength-3。 The average gather degree of volumetric force: V Q The gather degree of volumetric force at point P: V Q F V lim 0 QThe direction of F is the limited one of 24 （3 3）集度：）集度：

26、（4 4）體力分量：）體力分量： 將F沿三個(gè)坐標(biāo)軸分解，可得到三個(gè)正交的分力： kZjYiXF X、Y、Z稱為物體在P點(diǎn)的體力分量，正負(fù)號(hào)視分 力指向而定，因次是力長(zhǎng)度-3。 體力的平均集度為： V Q P點(diǎn)所受體力的集度為： V Q F V lim 0 F Q的方向就是 的極限方向。 25 2. Surface force S S Q (3) Gather degree of surface force:the average gather degree of the surface force above : The gather degree of the surface force a

27、t P: S Q F S 0 lim (4) The components of surface force: The components of surface force are , , ,and then are ForceLength-2 XYZ x y z P S X Y Z F Q Fig.1-3 (2) Property:In general, surface force is the function of located coordinates at point in the surface of the object. (1) Definition:surface forc

28、e is distributed one in the surface of the object.For instance,liquid stress and contact stress . It is shown in Fig.1-3. Q 26 2. 2. 面面 力力 S 上面力的平均集度為： S Q （3）面力集度： P點(diǎn)所受面力的集度為： S Q F S 0 lim （4）面力分量： P點(diǎn)的面力分量為 、 、， 因次是力長(zhǎng)度-2。 XYZ x y z P S X Y Z F Q 圖1-3 （2）性質(zhì)：面力一般是物體表面點(diǎn)的位置坐標(biāo)的函數(shù)。 （1）定義：分布在物體表面上的力。如流體壓

29、力和接觸力 。如圖13所示 。 Q 27 2.stress 3.Stress gather degree： The average gather degree of internal force above : A Q A The stress at point P: A Q s A 0 lim -Positive stress-Shearing strength And then areForceLength-2 The stress component at point P is , . 2.Property：The same point in the object,whose stres

30、s of different cross sections is different. 1.Definition：The object bears the external force function.Additional internal force is produced among every cross sections of the object interior.For displaying these internal forces,we use a cross section to cut the object open, and then take out a part a

31、mong them.The function of a part to another part among them that expresses for internal force, which are resultant force of distributed forces that distribute on the cross section.When the area of cross section incline to the zero,the distributed force on the cross section is shown as Fig.1-4s. x y

32、z A B P o A s Q n m Fig.1-4 28 （二）應(yīng)力（二）應(yīng)力 3.應(yīng)力集度： 上的內(nèi)力的平均集度為： A Q A P點(diǎn)的應(yīng)力為： A Q s A 0 lim -正應(yīng)力-切應(yīng)力 因次是力長(zhǎng)度-2。 P點(diǎn)的應(yīng)力分量為 、 2.性質(zhì)：在物體內(nèi)的同一點(diǎn)，不同截面上的應(yīng)力是不同的。 1.定義：物體承受外力作用，物體內(nèi)部各截面之間產(chǎn)生附加內(nèi) 力，為了顯示出這些內(nèi)力，我們用一截面截開物體，并取出其 中一部分，其中一部分對(duì)另一部分的作用，表現(xiàn)為內(nèi)力，它們 是分布在截面上分布力的合力。當(dāng)截面面積趨于零時(shí)截面上的 分布力。如圖14所示 。s x y z A B P o A s Q n m 圖

33、圖1-41-4 29 4.The component of the stress Stress is relevant with not only the position of point but also the direction of the cross section. It is not a general vector but is two rank tensor. The component of the stress on the plane is equal in size but contrary in direction at the meaning of omitti

34、ng the small quantity in high level. (1) For analyzing the state of one point,one small positive parallel hexahedron is taken out from the point.The component of the stress of each section along with coordinates axis that is called the component of the stress. x y z o Fig.1-5 A B C P 30 4.4.應(yīng)力分量應(yīng)力分量

35、 應(yīng)力不僅和點(diǎn)的位置有關(guān)，和截面的方位也有關(guān)，不是 一般的矢量，而是二階張量。 相對(duì)平面上的應(yīng)力分量在略 去高階小量的意義上大小相等， 方向相反。 （1）為了分析一點(diǎn)的應(yīng)力狀 態(tài)，在這一點(diǎn)從物體內(nèi)取出一個(gè) 微小的正平行六面體，各面上的 應(yīng)力沿坐標(biāo)軸的分量稱為應(yīng)力分 量。 x y z o 圖圖1-51-5 A B C P 31 The drawing shows that the normal of the surface of the unit is y,it is called surface y.The stress that the stress component plumbs the

36、 surface of the unit is called the positive stress. The positive stress is recorded y,the positive direction along y axis is positive,whose suffix means the direction along coordinates axis. （2）Symbol provision： y x y z o yx Fig.1-6 yz The stress paralleling the surface of the unit is called the sli

37、cing stress,which is showed by , and whose the first suffix y means the flat surface of the place and the second suffix x,z mean respectively along the direction of the coordinates axis. , is showed in Fig.1-6. yx yx yz yz 32 圖示單元體面的法線為y,稱 為y面，應(yīng)力分量垂直于單元體 面的應(yīng)力稱為正應(yīng)力。 正應(yīng)力記為y,沿y軸的正 向?yàn)檎?其下標(biāo)表示所沿坐標(biāo)軸 的方向。 y

38、 x y z o yx 圖圖1-61-6 （2 2）符號(hào)規(guī)定：）符號(hào)規(guī)定： yz yz 平行于單元體面的應(yīng)力稱為 切應(yīng)力，用 、 表示，其第 一下標(biāo)y表示所在的平面，第二下 標(biāo)x、z分別表示沿坐標(biāo)軸的方向。 如圖16所示的 、 。 yx yx yz yz 33 The components of the stress on other x,z positive surface is shown in Fig.1- 7. The stress on positive surface is positive along the positive direction of coordinates,

39、and is negative athwart the positive direction of coordinates. Fig.1-7 xy 34 其它x、z正面上的 應(yīng)力分量的表示如圖1 7所示。 凡正面上的應(yīng)力沿坐 標(biāo)正向?yàn)檎孀鴺?biāo)正向 為負(fù)。 圖圖1-71-7 35 The stress that paralleling the surface of the unit is shown likeyx、yz in Fig.1-8,which is positive along the negative direction of x axis and z axis.Fig.1-8Fi

40、g.1-8 Fig.1-8 shows that the normal of the surface of the unit is the negative direction of y, the positive stress is recorded ,which is positive along the negative direction of y axis. y 36 平行于單元體面 的應(yīng)力如圖示的yx、 yz，沿x軸、z軸的 負(fù)向?yàn)檎?圖圖1-81-8 圖18所示單元體 面的法線為y的負(fù)向，正 應(yīng)力記為 ,沿y軸負(fù)向 為正。 y 37 Elasticity （3）Noticing

41、 the slicing stress sign of elasticity is distinguishing with material mechanics.Generally the slicing stress is positive in elasticity, such as the Fig.1- 9shows.However, the sign closing both sides is different in material mechanics. Material Mechanics We should draw stress circle according to the

42、 sign provision of material mechanics. Fig.1-9Fig.1-9 38 彈性力學(xué)彈性力學(xué) 材料力學(xué)材料力學(xué) （3）注意彈性力學(xué) 切應(yīng)力符號(hào)和材料力學(xué) 是有區(qū)別的，圖19中， 彈性力學(xué)里，切應(yīng)力都 為正，而材料力學(xué)中相 鄰兩面的的符號(hào)是不同 的。 在畫應(yīng)力圓時(shí)，應(yīng) 按材料力學(xué)的符號(hào)規(guī)定。 圖圖1-91-9 39 2.Shearing stress：In Fig.1-5, the orthogonal changes of the line segment PA,PB,PC are means with the radian, which is calle

43、d shearing stress.Shearing stress is shown respectively by , , . yz zx xy 3.Strain The strain is the changes of the shape.The strain of the object may come down to the changes of the length and the angle. x y 1.Positive strain：In Fig.1-5 the flex of the line segment PA,PB,PC per-unit length, namely

44、unit flexible or opposite flex, which is called positive strain, positive strain is shown respectively by , , . z P Fig.1-5Fig.1-5 A B C P 40 2.切應(yīng)變：圖1-5中線段 PA、PB、PC之間的直角的改 變，用弧度表示，稱為切應(yīng)變。 分別用 、 、 表示。 yz zx xy （三）形變（應(yīng)變）（三）形變（應(yīng)變） 形變就是形狀的改變。物體的形變可以歸結(jié)為長(zhǎng)度的改 變和角度的改變。 x y 1.正應(yīng)變：圖1-5中線段PA、PB、PC每單位長(zhǎng)度的伸 縮，即單位伸

45、縮或相對(duì)伸縮，稱為正應(yīng)變。分別用 、 、 表示。 z P 圖圖1-51-5 A B C P 41 （2）Among all points of the object have the opposite displacement, so the object produce distortion.In the elasticity, studying primarily the displacement that the distortion of the object causes. （1）The displacement that the movement of the whole obje

46、ct like a rigid body proceeding causes, generally which includes the translation and the rotation. In this way the displacement do not make the opposite distance of the shape and particle of the object change.(Object only contain outside effect but have no inside effect) 1. When the positions of all

47、 point of the object change, thinking generally it is constituted by the displacement of two kinds of property： 4.Displacement Displacement：When the object transform, the amount of changes of the all point position calls the displacement. 2.The expressing method of the displacement The displacement

48、of random point in object, whose projection of u,v,w in the axis of x,y,z are used to mean it.Positive direction along coordinates axis is positive, negative direction along coordinates axis is negative.These three projections are called the component of the displacement of the point. 42 （2）物體的各點(diǎn)間有相

49、對(duì)位移，因而物體產(chǎn)生了變形。 彈性力學(xué)中主要研究物體由變形而引起的位移。 （1）整個(gè)物體象一個(gè)剛體一樣進(jìn)行的運(yùn)動(dòng)所引起的位移， 一般包括平移和轉(zhuǎn)動(dòng)。這樣位移并不使物體的形狀、質(zhì)點(diǎn)間 的相對(duì)距離發(fā)生變化。（物體只有外效應(yīng)而無內(nèi)效應(yīng)）。 1.當(dāng)物體各點(diǎn)發(fā)生位置改變時(shí)，一般認(rèn)為是由 兩種性質(zhì)的位移組成： （四）位移（四）位移 位移：物體變形時(shí)，各點(diǎn)位置的改變量稱為位移。 2.位移的表示方法 x yz uvw 物體內(nèi)任意一點(diǎn)的位移，用它在 、 、 軸上的投 影 、 、 來表示，以沿坐標(biāo)軸正向?yàn)檎刈鴺?biāo)軸負(fù)向?yàn)?負(fù)。這三個(gè)投影稱為該點(diǎn)的位移分量。 43 displacement deformation

50、 stress Volumetric stress Plane stress Geometrical equation Physical equationEquation of equilibrium Boundary condition Fig.1-10Fig.1-10 5.The relation of physics quantity 44 位移位移 形變形變 應(yīng)力應(yīng)力 體力體力 面力面力 幾何方程幾何方程物理方程物理方程平衡方程平衡方程 邊界條件邊界條件 圖圖1-101-10 （五）各物理量之間的關(guān)系（五）各物理量之間的關(guān)系 45 The deducing of the formula

51、 of the elasticity is more complicated.The meaning of the formula is not clear and definite,and the formula is not easy to memorize, so the beginner will feel difficult. Dont stand on points without measure and fix attention on the main process of the deduce.The deduce and the memory of the formula

52、had better to pass the matrix form. 1-5 The study method of the elasticity Because the basic equation is a system of partial differential equation and contact is less, the beginners comprehension is difficult.The direct solution of the system of partial differential equation is very difficult,only w

53、hen the boundary condition is simple, the solution may be solved.Most problems need to be solved through the numerical method, so the meaning of the basic equation is in order to lay the foundation for the study of the future. In deduce process,Be good at utilizing the small deformation to omit the

54、small quantity in high level. In boundary condition,distinguishing the primary boundary and the subordinate boundary.In the subordinate boundary,using the condition of the equivalent force system to instead according to the Saint-Venant principle. At the rear of each chapter,attaching some exercises

55、.And these exercises may deepen the comprehension of the concept and the method. 46 彈性力學(xué)的公式推導(dǎo)比較繁復(fù)，公式的意義不明確，不 便記憶，因此初學(xué)者會(huì)感到困難。 在學(xué)習(xí)中，不要過分拘泥于細(xì)節(jié)，應(yīng)著眼于推導(dǎo)的主要過 程，公式的推導(dǎo)和記憶，最好通過矩陣形式和張量。 1-5 1-5 彈性力學(xué)的學(xué)習(xí)方法彈性力學(xué)的學(xué)習(xí)方法 由于基本方程是偏微分方程組，接觸較少，理解有困難。 偏微分方程組的直接求解是十分困難的，只有在邊界條件比 較簡(jiǎn)單時(shí)，可以解出，大多需要通過數(shù)值方法求解，因此基 本方程的意義很大程度上是為將來的學(xué)習(xí)打基礎(chǔ)。 在推導(dǎo)過程中，善于利用小變形略去高階小量，在邊界條 件中，要分清主要邊界和次要邊界，在次要邊界上根據(jù)圣維 南原理，用等效力系的條件進(jìn)行替代。 在每章的最后，附有一些習(xí)題，通過練習(xí)，加深對(duì)概念和 方法的理解。 47 Introductionexercise lesson exercise 1What is the research object and content of the elasticity?What are the different and the sim