英漢雙語(yǔ)彈性力學(xué)1

1、1 23Chapter 1 Introduction1-1 The Modeling of the Engineering Mechanics Problem1-3 The Basic Assumption of the Elasticity Problem1-4 The Several Basic Concepts of Elasticity1-5 The Study Method of the Elasticity1-2 The Basic Contents of the ElasticityExercises Lesson4第一章第一章 緒緒 論論1-1 1-1 工程力學(xué)問(wèn)題的建模工程力

2、學(xué)問(wèn)題的建模1-3 1-3 彈性力學(xué)問(wèn)題的基本假設(shè)彈性力學(xué)問(wèn)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 external force or change

3、 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 and proceeds to beg

4、 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(wèn)題的數(shù)學(xué)建模過(guò)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(wèn)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)值方法奠定了基礎(chǔ)。7 Through the proces

5、s of establishing the mechanics model in the engineering mechanics problem, generally three parts should be simplified:Suffering Force Simplification Material SimplificationConstruction Simplification1、The Modeling Process of the Engineering Mechanics Problem1-1 The Modeling of the Engineering Mecha

6、nics ProblemFig.1-1Fig.1-18 工程力學(xué)問(wèn)題建立力學(xué)模型的過(guò)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(wèn)題的建模過(guò)程一、工程力學(xué)問(wèn)題的建模過(guò)程1-1 1-1 工程力學(xué)問(wèn)題的建模工程力學(xué)問(wèn)題的建模圖圖1-11-19 Material is simplified according to these hypothesises of the same kind, consecution and uniformity in each direction. （3）Material simplification According to the Saint-

7、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 construction（1）Construction Simplificatio

8、n10 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(wèn)題向平面問(wèn)題的簡(jiǎn)化，向軸對(duì)稱問(wèn)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 in Modeling Process After the model is establi

9、shed, 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）Linearization12 對(duì)高階小量進(jìn)行處理，能進(jìn)行線性化的，進(jìn)行線性化。二、建模過(guò)程中注意的問(wèn)題二、建模過(guò)程中注意的問(wèn)題 模型建立以后，對(duì)計(jì)算的結(jié)果進(jìn)行分析整理，

10、返回實(shí)際問(wèn)題進(jìn)行驗(yàn)證，一般主要通過(guò)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 The Basic Contents of the Elasticity1、Investigative task

11、 The research object of the elasticity is general and complicated shape structure piece, entity structure, plate shell etc.2、Investigative object14 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: plast

12、icity 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: Study statics and dynamics of the rigid body(constraint force,vel

13、ocity,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)；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)

14、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(wèn)題、板、殼和實(shí)體結(jié)構(gòu)等的應(yīng)力和位移分析。171-3 The Basic Assumption of the Elasticity In elasticity, doing some necessary assumptions under the premise that can satisfy the practical needing precision and making the problem solved. （1）Consecution assumption:Some physics measur

15、es 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 body,then the elastic body obey the Hookes law-the stress becomes the direc

16、t 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 of the object would not change along with position coordinates change. Th

17、e basic assumption of the elasticity：181-3 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(wèn)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）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 ela

18、stic 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 the one after deformation. The small quantity in high level may be ignored when inve

19、stigating 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 for some problems, then it is necessary to use another brief method.But it is stil

20、l 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

21、不少問(wèn)題是適用的，但對(duì)于一些問(wèn)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)。211-4 The Several Basic Concepts of the Elasticity It can be divided into the stress of volume and plane according to the different distribution of the external function, which are called volumetric force and surface forc

22、e respectively. （2）Property：volumetric force is different from the different location in a point;the volumetric force is continuous in distribution.1.External stress1.Volumetric force（1）Definition：It is the stress distributed in volume of the object that is called volumetric force,for example,gravit

23、y and inertia force.It is shown in Fig.1-2.QzxyVOPXYQFFig.1-2Z221-4 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1-21-2zxyVOPXYQFZ23（3）Gather degree： （4）The component of volumetric force： The force

24、 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 force at point P.Plus sign and negative sign are separately determined by the direction of components,and then isForceLength-3。The av

25、erage gather degree of volumetric force:VQThe gather degree of volumetric force at point P:VQFVlim0QThe direction of F is the limited one of24（3 3）集度：）集度： （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：VQP點(diǎn)所受體力的集度為：VQFVlim0FQ的方向就是 的極限方向。252. Surface for

26、ceSSQ(3) Gather degree of surface force:the average gather degree of the surface force above :The gather degree of the surface force at P:SQFS0lim(4) The components of surface force: The components of surface force are , , ,and then are ForceLength-2XYZxyzPSXYZFQFig.1-3(2) Property:In general, surfa

27、ce force is the function of located coordinates at point in the surface of the object.(1) Definition:surface force is distributed one in the surface of the object.For instance,liquid stress and contact stress .It is shown in Fig.1-3.Q262. 2. 面面力力S上面力的平均集度為：SQ（3）面力集度：P點(diǎn)所受面力的集度為：SQFS0lim（4）面力分量： P點(diǎn)的面力

28、分量為 、 、，因次是力長(zhǎng)度-2。XYZxyzPSXYZFQ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。如流體壓力和接觸力 。如圖13所示 。Q272.stress3.Stress gather degree：The average gather degree of internal force above :AQAThe stress at point P:AQsA0lim-Positive stress-Shearing strengthAnd then areForceLength-2The stress component at poin

29、t P is , .2.Property：The same point in the object,whose stress 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 cros

30、s section to cut the object open, and then take out a part among 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 distr

31、ibuted force on the cross section is shown as Fig.1-4s.xyzABPoAsQnmFig.1-428（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：AQAP點(diǎn)的應(yīng)力為：AQsA0lim-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(kāi)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 。sxy

32、zABPoAsQnm圖圖1-41-4294.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

33、of omitting 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. xyzoFig.1-5ABCP304.4.應(yīng)力分量應(yīng)力分量

34、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。xyzo圖圖1-51-5ABCP31 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 surface of the un

35、it 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：yxyzoyxFig.1-6yz The stress paralleling the surface of the unit is called the slicing stress,which is showed

36、 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.yxyxyzyz32 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。yxyzoyx圖圖1-61-6（2 2）符號(hào)規(guī)定：）符號(hào)規(guī)定：yzyz 平行

37、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 、 。yxyxyzyz33 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, and is negative athwart the positive direction of coordi

38、nates. Fig.1-7xy34 其它x、z正面上的應(yīng)力分量的表示如圖17所示。 凡正面上的應(yīng)力沿坐標(biāo)正向?yàn)檎?，逆坐?biāo)正向?yàn)樨?fù)。圖圖1-71-735 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.1-8Fig.1-8 Fig.1-8 shows that the normal of the surface of the unit i

39、s the negative direction of y, the positive stress is recorded ,which is positive along the negative direction of y axis.y36 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D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?yàn)檎?。y37Elasticity （3）Noticing the slicing stress sign of elasticity is distinguishing with material mecha

40、nics.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. MaterialMechanics We should draw stress circle according to the sign provision of material mechanics.Fig.1-9Fig.1-938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

41、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(huà)應(yīng)力圓時(shí)，應(yīng)按材料力學(xué)的符號(hào)規(guī)定。圖圖1-91-939 2.Shearing stress：In Fig.1-5, the orthogonal changes of the line segment PA,PB,PC are means with the radian, which is called shearing stress.Shearing stress is shown respectively by , , .yzzxxy3.Strain The strain i

42、s the changes of the shape.The strain of the object may come down to the changes of the length and the angle.xy 1.Positive strain：In Fig.1-5 the flex of the line segment PA,PB,PC per-unit length, namely unit flexible or opposite flex, which is called positive strain, positive strain is shown respect

43、ively by , , .zPFig.1-5Fig.1-5ABCP40 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。yzzxxy（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。xy 1.正應(yīng)變：圖1-5中線段PA、PB、PC每單位長(zhǎng)度的伸縮，即單位伸縮或相對(duì)伸縮，稱為正應(yīng)變。分別用 、 、 表示。zP圖圖1-51-5ABCP41 （2）Among all points of the object have the opposite displacement, so the ob

44、ject 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 object like a rigid body proceeding causes, generally which includes the translation and the rotation. In this way the displacem

45、ent 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 point of the object change, thinking generally it is constituted by the displacement of two kinds of property：4.Displacemen

46、tDisplacement：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 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 coordina

47、tes 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(wú)內(nèi)效應(yīng)）。 1.當(dāng)物體各點(diǎn)發(fā)生位置改變時(shí)

48、，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xyzuvw 物體內(nèi)任意一點(diǎn)的位移，用它在 、 、 軸上的投影 、 、 來(lái)表示，以沿坐標(biāo)軸正向?yàn)檎?，沿坐?biāo)軸負(fù)向?yàn)樨?fù)。這三個(gè)投影稱為該點(diǎn)的位移分量。43displacementdeformationstressVolumetric stressPlane stressGeometrical equationPhysical equationEquation of equilibriumBoundary conditionFig.1-10Fig.1-105.The relation

49、 of physics quantity44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 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 dif

50、ficult. Dont stand on points without measure and fix attention on the main process of the deduce.The deduce and the memory of the formula 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 l

51、ess, the beginners comprehension is difficult.The direct solution of the system of partial differential equation is very difficult,only when 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

52、 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 small quantity in high level. In boundary condition,distinguishing the primary boundary and the subordinate boundary.In the subordinate boundary,using the conditio

53、n of the equivalent force system to instead according to the Saint-Venant principle. At the rear of each chapter,attaching some exercises.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(guò)分拘泥于細(xì)節(jié)，應(yīng)著眼于推導(dǎo)的主要過(guò)程，公式的推導(dǎo)和記憶，最好通過(guò)

54、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(guò)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(lái)的學(xué)習(xí)打基礎(chǔ)。 在推導(dǎo)過(guò)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(guò)練習(xí)，加深對(duì)概念和方法的理解。47Introductionexercise lessonexercise 1What is the research object and content of the elasticity?What are the different and the similar comparing with mat