1、1,Chapter 1 Introduction,2,第一章 緒 論,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

2、Lesson,Introduction,4,第一章 緒 論,1-1 工程力學(xué)問(wèn)題的建模,1-3 彈性力學(xué)問(wèn)題的基本假設(shè),1-4 彈性力學(xué)中的幾個(gè)基本概念,1-5 彈性力學(xué)的學(xué)習(xí)方法,1-2 彈性力學(xué)的基本內(nèi)容,習(xí)題課,緒 論,5,Introduction,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,緒 論,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,Introduction

5、,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 Modeling o

6、f the Engineering Mechanics Problem,Fig.1-1,8,緒 論,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,1-1 工程力學(xué)問(wèn)題的建模,圖1-1,9,Introduction,Material is simplified according to these hypothesises of the same kind, consecution and uniformity in each direction.,（3）Material simplification,According t

7、o 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 construction,（1）Construction

8、 Simplification,10,緒論,根據(jù)各向同性、連續(xù)、均勻等假設(shè)進(jìn)行簡(jiǎn)化。,（3）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）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）結(jié)構(gòu)簡(jiǎn)化,11,Introduction,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 mo

9、del 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,緒論,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,模型建立以后，對(duì)計(jì)算的

10、結(jié)果進(jìn)行分析整理，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）實(shí)驗(yàn)驗(yàn)證,（1）線性化,13,Introduction,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 Elasticity,1、Inves

11、tigative task,The research object of the elasticity is general and complicated shape structure piece, entity structure, plate shell etc.,2、Investigative object,14,緒論,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 彈性力學(xué)的基本內(nèi)容,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,15,Introduction,Plastici

12、ty: 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: Study statics and dynamics of the rigid body(constraint

13、 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,緒論,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：,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

14、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(wèn)題、板、殼和實(shí)體結(jié)構(gòu)等的應(yīng)力和位移分析。,17,Introduction,1-3 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 assumptio

15、n: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 body,then the elastic body obey the Hookes law-the str

16、ess 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 of the object would not change along with position c

17、oordinates change.,The basic assumption of the elasticity：,18,緒論,1-3 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,Introduction,（4）I

18、sotropy 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 the one after deformation. The small quantity in high lev

19、el 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 for some problems, then it is necessary to use another

20、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,緒論,（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

21、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(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)。,21,Introduction,1-4 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, whic

22、h 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 the stress distributed in volume of the object t

23、hat is called volumetric force,for example,gravity and inertia force.It is shown in Fig.1-2.,22,緒論,1-4 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,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 。,23,Introduction,（3）Gather degree：,（4）The component of volumetric forc

24、e：,The force of F is resolved along with the three coordinates, which will get the three components of straight intersection:,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。,

25、24,緒論,（3）集度：,（4）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：,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。,25,Introduction,2. Surface force,(3) Gather degree of surface force:the average gather degree of the surface force above :,The gather degree of the surface force at P:,(4) The components of surface force: The compone

26、nts of surface force are , , ,and then are ForceLength-2,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 force is distributed one in the surface of the object.For instance,liquid stress and contact st

27、ress . It is shown in Fig.1-3.,26,緒 論,2. 面力,P點(diǎn)所受面力的集度為：,（4）面力分量： P點(diǎn)的面力分量為 、 、，因次是力長(zhǎng)度-2。,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 。,27,Introduction,2.stress,2.Property：The same point in the object,whose stress of different cross sections is different.,1.Definition：The object

28、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 among them.The function of a part to another part among them that exp

29、resses 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.,28,緒論,（二）應(yīng)力,3.應(yīng)力集度：,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)

30、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 。,29,Introduction,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 c

31、omponent of the stress on the plane is equal in size but contrary in direction at the meaning 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 wi

32、th coordinates axis that is called the component of the stress.,Fig.1-5,30,緒 論,4.應(yīng)力分量,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。,31,Introduction,The drawing shows that the normal of the surface of the unit is y,it i

33、s called surface y.The stress that the stress component plumbs the 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：,The stress paralleling

34、 the surface of the unit is called the slicing 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.,32,緒 論,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)

35、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,圖1-6,（2）符號(hào)規(guī)定：,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 、 。,33,Introduction,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

36、 is negative athwart the positive direction of coordinates.,Fig.1-7,34,緒論,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1-7,35,Introduction,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-

37、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.,36,緒論,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1-8,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?37,Introduction,Fig.1-9,38,緒論,彈性力學(xué),材料力學(xué),

38、（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(huà)應(yīng)力圓時(shí)，應(yīng)按材料力學(xué)的符號(hào)規(guī)定。,圖1-9,39,Introduction,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 , , .,3.Strain

39、,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.,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 sh

40、own respectively by , , .,P,40,緒論,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。,（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。,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。,41,Introduction,（2）Among all points of the object have the opposite displacement, so the object produce distor

41、tion.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 displacement do not make the

42、 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.Displacement,Displacement：Whe

43、n 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 coordinates axis is posi

44、tive, negative direction along coordinates axis is negative.These three projections are called the component of the displacement of the point.,42,緒論,（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)，一般認(rèn)為是由兩種性質(zhì)的

45、位移組成：,（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,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。,43,Introduction,displacement,deformation,stress,Volumetric stress,Plane stress,Geometrical equation,Physical equation,Equation of equilibrium,Boundary condition,Fig.1-10,5.The relation of

46、 physics quantity,44,緒論,（五）各物理量之間的關(guān)系,45,Introduction,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 difficult.,Dont stand on points without measure and fix atten

47、tion 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 less, the beginners comprehension is difficult.The direct

48、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 is in order to lay the foundation for the study of the f

49、uture.,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 condition 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