英漢雙語彈性力學(xué)6

1、Chapter 6 The Basic Solution ofThe Temperature Stress Problems Elasticity1第六章 溫度應(yīng)力問題的基本解法2Chapter 6 The Basic Solution of Temperature Stress Problems6-4 Solve plane problem of temperature stresses by displacement6-3 The boundary conditions of temperature filed6-2 The differential equation of heat co

2、nduction 6-1 The basic concept of temperature field and heat conduction6-5 The introducing of potential function of displacement6-6 The plane problems of thermal stresses in axisymmetric temperature fieldThe Basic Solution of The Temperature Stress Problems3溫度應(yīng)力問題的基本解法第六章 溫度應(yīng)力問題的基本解法6-4 按位移求解溫度應(yīng)力的平面

3、問題6-3 溫度場的邊界條件6-2 熱傳導(dǎo)微分方程6-1 溫度場和熱傳導(dǎo)的基本概念6-5 位移勢函數(shù)的引用6-6 軸對稱溫度場平面熱應(yīng)力問題4 When the temperature of a elastic body changes, its volume will expand or contract. If the expansion or contraction cant happen freely due to the external restrictions or internal deformation compatibility demands, additional st

4、resses will be produced in the structure. These stresses produced by temperature change are called thermal stresses, or temperature stresses. Neglecting the effects of the temperature change on the material performance, to solve the temperature stresses, we need two aspects of calculation: (1) Solve

5、 the temperature field of the elastic body by the initial conditions and boundary conditions, according to heat conduction equations. And the difference between the former temperature field and the later temperature field is the temperature change of the elastic body. (2) Solve the temperature stres

6、ses of the elastic body according to the basic equations of the elastic mechanics. This chapter will present these two aspects of calculation simply.Chapter 6 The Basic Solution of The Temperature Stress ProblemsThe Basic Solution of The Temperature Stress Problems5 當(dāng)彈性體的溫度變化時，其體積將趨于膨脹和收縮，若外部的約束或內(nèi)部的

7、變形協(xié)調(diào)要求而使膨脹或收縮不能自由發(fā)生時，結(jié)構(gòu)中就會出現(xiàn)附加的應(yīng)力。這種因溫度變化而引起的應(yīng)力稱為熱應(yīng)力，或溫度應(yīng)力。 忽略變溫對材料性能的影響，為了求得溫度應(yīng)力，需要進行兩方面的計算：（1）由問題的初始條件、邊界條件，按熱傳導(dǎo)方程求解彈性體的溫度場，而前后兩個溫度場之差就是彈性體的變溫。（2）按熱彈性力學(xué)的基本方程求解彈性體的溫度應(yīng)力。本章將對這兩方面的計算進行簡單的介紹。第六章 溫度應(yīng)力問題的基本解法溫度應(yīng)力問題的基本解法66-1 The Basic Concept of Temperature Field And Heat Conduction1.The temperature fiel

8、d: The total of the temperature at all the points in a elastic body at a certain moment, denoted by T. Unstable temperature filed or nonsteady temperature field: The temperature in the temperature field changes with time. i.e. T=T（x，y，z，t） Stable temperature filed or steady temperature field: The te

9、mperature in the temperature field is only the function of positional coordinates. i.e. T=T（x，y，z） Plane temperature field: The temperature in temperature field only changes with two positional coordinates. i.e. T=T（x，y，t）The Basic Solution of The Temperature Stress Problems76-1 溫度場和熱傳導(dǎo)的基本概念1.溫度場：在任

10、一瞬時，彈性體內(nèi)所有各點的溫度值的總體。用T表示。 不穩(wěn)定溫度場或非定常溫度場：溫度場的溫度隨時間而變化。 即 T=T（x，y，z，t） 穩(wěn)定溫度場或定常溫度場：溫度場的溫度只是位置坐標(biāo)的函數(shù)。 即 T=T（x，y，z） 平面溫度場：溫度場的溫度只隨平面內(nèi)的兩個位置坐標(biāo)而變。 即 T=T（x，y，t）溫度應(yīng)力問題的基本解法82.Isothermal surface: The surface that connects all the points with the same temperature in the temperature field at a certain moment. Ap

11、parently, the temperature doesnt changes along the isothermal surface; The changing rate is the largest along the normal direction of the isothermal surface. T+2TT+TTT-Txoy3.Temperature gradient:The vector that points to the direction in which temperature increase along the normal direction of the i

12、sothermal surface. It is denoted by T, and its value is denoted by , where n is the normal direction of the isothermal surface. The components of temperature gradient at each coordinate areThe Basic Solution of The Temperature Stress Problems9溫度應(yīng)力問題的基本解法2.等溫面：在任一瞬時，連接溫度場內(nèi)溫度相同各點的曲面。顯然，沿著等溫面，溫度不變；沿著等溫

13、面的法線方向，溫度的變化率最大。T+2TT+TTT-Txoy3.溫度梯度：沿等溫面的法線方向，指向溫度增大方向的矢量。用T表示，其大小用 表示。其中n為等溫面的法線方向。溫度梯度在各坐標(biāo)軸的分量為10Define to be the unit vector in normal direction of the isothermal surface, pointing to the temperature increasing direction.T（1）4.Thermal flux speed: The quantity of heat flowing through the area S o

14、n the isothermal surface in unit time, denoted by .The Basic Solution of The Temperature Stress Problems11溫度應(yīng)力問題的基本解法取 為等溫面法線方向且指向增溫方向的單位矢量，則有T（1）4.熱流速度：在單位時間內(nèi)通過等溫面面積S 的熱量。用 表示。12Its value is: (2) Thermal flux density: The thermal flux speed flowing through unit area on the isothermal surface, denot

15、ed by . Then we have5.The basic theorem of heat transfer: The thermal flux density is in direct proportion to the temperature gradient and in the reverse direction of it. i.e.(3)The Basic Solution of The Temperature Stress Problems is called the coefficient of the heat transfer. Equations (1), (2) a

16、nd (3) lead to13熱流密度：通過等溫面單位面積的熱流速度。用 表示，則有 溫度應(yīng)力問題的基本解法其大小為(2)稱為導(dǎo)熱系數(shù)。由（1）、（2）、（3）式得5.熱傳導(dǎo)基本定理：熱流密度與溫度梯度成正比而方向相反。即 (3)T14We can see that the coefficient of the heat transfer means “the thermal flux speed through unit area of the isothermal surface per unit temperature gradient”. From equations (1) and

17、 (3), we can see that the value of the thermal flux densityThe projections of the thermal flux density on axes: The Basic Solution of The Temperature Stress ProblemsIt is obvious that the component of thermal flux density in any direction is equal to the coefficient of heat transfer multiplied by th

18、e descending rate of the temperature in this direction.15溫度應(yīng)力問題的基本解法由（1）和（3）可見，熱流密度的大小可見，導(dǎo)熱系數(shù)表示“在單位溫度梯度下通過等溫面單位面積的熱流速度”。熱流密度在坐標(biāo)軸上的投影可見：熱流密度在任一方向的分量，等于導(dǎo)熱系數(shù)乘以溫度在該方向的遞減率。16 The principle of heat quantity equilibrium: Within any period of time, the heat quantity accumulated in any minute part of the obj

19、ect equals the heat quantity conducted into this minute part plus the heat quantity supplied by internal heat source.6-2 The Differential Equation of Heat ConductionxyzThe Basic Solution of The Temperature Stress Problems Take a minute hexahedron dxdydz as shown in the above figure. Suppose that the

20、 temperature of this hexahedron rises from T to . The heat quantity accumulated by temperature is , where is the density of the object, C is the heat quantity needed when the temperature of the object with a unit mass rise one degreespecific thermal capability.17 熱量平衡原理：在任意一段時間內(nèi)，物體的任一微小部分所積蓄的熱量，等于傳入

21、該微小部分的熱量加上內(nèi)部熱源所供給的熱量。6-2 熱傳導(dǎo)微分方程xyz 取圖示微小六面體dxdydz。假定該六面體的溫度在dt時間內(nèi)由T 升高到 。由溫度所積蓄的熱量是 ，其中 是物體的密度，C 是單位質(zhì)量的物體升高一度時所需的熱量比熱容。溫度應(yīng)力問題的基本解法18 Within the same period of time dt, the heat quantity qxdydzdt is conducted into the hexahedron from left, and the heat quantity is conducted out the hexahedron throug

22、h right. Hence, the net heat quantity conducted into isThe Basic Solution of The Temperature Stress Problems Introduce into it . We can see thatThe net heat quantity conducted into it from left and right is:The net heat quantity conducted into it from top and bottom is:The net heat quantity conducte

23、d into it from front and back is: Hence, the total net heat quantity conducted into the hexahedron is:which can be abbreviated as:19溫度應(yīng)力問題的基本解法 在同一段時間dt內(nèi)，由六面體左面?zhèn)魅霟崃縬xdydzdt，由右面?zhèn)鞒鰺崃?。因此，傳入的凈熱量為將 代入可見：由左右兩面?zhèn)魅氲膬魺崃繛橛缮舷聝擅鎮(zhèn)魅氲膬魺崃繛橛汕昂髢擅鎮(zhèn)魅氲膬魺崃繛椋阂虼?，傳入六面體的總凈熱量為：簡記為：20 Suppose that there is a positive heat reso

24、urce to supply heat inside the object, which supply heat quantity W per unit volume in unit time. Then the heat quantity that supplied by this heat resource during time dt is Wdxdydzdt.According to the principle of heat quantity equilibrium,which can be simplified as: LetThis is the differential equ

25、ations of heat transfer.ThusThe Basic Solution of The Temperature Stress Problems21 假定物體內(nèi)部有正熱源供熱，在單位時間、單位體積供熱為W，則該熱源在時間dt內(nèi)所供熱量為Wdxdydzdt。 根據(jù)熱量平衡原理得：溫度應(yīng)力問題的基本解法化簡后得：記則這就是熱傳導(dǎo)微分方程。226-3 The Boundary Conditions of Temperature Filed To solve the differential equation, and sequentially solve the temperatu

26、re filed, the temperature of the object at initial moment must be known, i.e. the so-called initial condition. At the same time, the rule of heat exchange between the object surface and the surrounding medium after the initial moment must be also known, i.e. the so-called boundary conditions. The in

27、itial condition and the boundary conditions are called by a joint name of the initial value conditions. Initial condition: Boundary conditions are divided into four kinds of forms: The first kind of boundary condition: The temperature at any point on the object surface is known at all moments , i.e.

28、 where Ts is the surface temperature of the object.The Basic Solution of The Temperature Stress Problems236-3 溫度場的邊值條件 初始條件： 邊界條件分四種形式： 第一類邊界條件 已知物體表面上任意一點在所有瞬時的溫度，即 其中Ts 是物體表面溫度。 溫度應(yīng)力問題的基本解法 為了能夠求解熱傳導(dǎo)微分方程，從而求得溫度場，必須已知物體在初瞬時的溫度，即所謂初始條件；同時還必須已知初瞬時以后物體表面與周圍介質(zhì)之間熱交換的規(guī)律，即所謂邊界條件。初始條件和邊界條件合稱為初值條件。24 The se

29、cond kind of boundary condition: The normal thermal flux density at any point on the object surface is known, i.e. where the subscript s means “surface”, and n means “normal”. The third kind of boundary condition: The heat release situation of convection at any point on the boundary of the object is

30、 known at all moments. According to the convection theorem of heat quantity, the thermal flux density transmitting from the object surface to the surrounding medium per unit time is in direct proportion to the temperature difference between them, i.e. Where Te is the temperature of the surrounding m

31、edium; is called the coefficient of the heat release of convection, or heat coefficient for short. The forth kind of boundary condition: It is known that the two objects contact completely, and exchange heat through the form of heat conduction, i.e.The Basic Solution of The Temperature Stress Proble

32、ms25 第三類邊界條件 已知物體邊界上任意一點在所有瞬時的運流（對流）放熱情況。按照熱量的運流定理，在單位時間內(nèi)從物體表面?zhèn)飨蛑車橘|(zhì)的熱流密度，是和兩者的溫差成正比的，即 溫度應(yīng)力問題的基本解法其中Te是周圍介質(zhì)的溫度； 稱為運流放熱系數(shù)，或簡稱熱系數(shù)。 第四類邊界條件 已知兩物體完全接觸，并以熱傳導(dǎo)方式進行熱交換。即 第二類邊界條件 已知物體表面上任意一點的法向熱流密度，即 其中角碼 s 表示“表面”，角碼n 表示法向。266-4 Solve Plane Problem of Temperature Stress by Displacement Suppose the temperatu

33、re change of every point in the elastic body is T. For an isotropic body, if there is no constricts, then the minute length at every point of the elastic body will generate normal strain , (where is the coefficient of expansion of the elastic body). Thus, the components of strain at every point of t

34、he elastic body are However, because the elastic body is restricted by the external restrictions and mutual restrictions among each section in the object, the above-mentioned deformations can not happen freely. Then the stress is produced, i.e. the so-called temperature stress.This temperature stres

35、s will result in additional strain due to the elasticity of the object, as expressed by Hookes law. Therefore, the components of the total strain of the elastic body areThe Basic Solution of The Temperature Stress Problems Suppose the temperature change of every point in the elastic body is T. For a

36、n isotropic body, if there is no constricts, then the minute length at every point of the elastic body will generate normal strain , (where is the coefficient of expansion of the elastic body). Thus, the components of strain at every point of the elastic body are However, because the elastic body is

37、 restricted by the external restrictions and mutual restrictions among each section in the object, the above-mentioned deformations can not happen freely. Then the stress is produced, i.e. the so-called temperature stress.This temperature stress will result in additional strain due to the elasticity

38、 of the object, as expressed by Hookes law. Therefore, the components of the total strain of the elastic body are276-4 按位移求解溫度應(yīng)力的平面問題 設(shè)彈性體內(nèi)各點的溫變?yōu)門。對于各向同性體，若不受約束，則彈性體內(nèi)各點的微小長度，都將產(chǎn)生正應(yīng)變 （ 是彈性體的膨脹系數(shù)），這樣，彈性體內(nèi)各點的形變分量為溫度應(yīng)力問題的基本解法 但是，由于彈性體所受的外在約束以及體內(nèi)各部分之間的相互約束，上述形變并不能自由發(fā)生，于是就產(chǎn)生了應(yīng)力，即所謂溫度應(yīng)力。這個溫度應(yīng)力又將由于物體的彈性而引起附

39、加的形變，如虎克定理所示。因此，彈性體總的形變分量是：28The Basic Solution of The Temperature Stress Problems For the temperature change problems of plane stress, the above equations are simplified as They are the physical equations of thermal elastic mechanics of the problems of plane stress . 29對于平面應(yīng)力的變溫問題，上式簡化為溫度應(yīng)力問題的基本解法這

40、就是平面應(yīng)力問題熱彈性力學(xué)的物理方程。30 Express the components of stress by the components of strain and the temperature change T, then the physical equations becomeThe Basic Solution of The Temperature Stress Problems The geometric equations still are Introducing the geometric equations into the physical equations y

41、ields the components of stress which are expressed by the components of displacement and temperature change T 31溫度應(yīng)力問題的基本解法將應(yīng)力分量用形變分量和變溫T表示的物理方程為：幾何方程仍然為：將幾何方程代入物理方程，得用位移分量和變溫T 表示的應(yīng)力分量32 Introducing the above equations into the differential equations of equilibrium ignoring body forces The Basic Sol

42、ution of The Temperature Stress Problems33將上式代入不計體力的平衡微分方程溫度應(yīng)力問題的基本解法34and simplifying yelds（1） These are the differential equations solving the problems of plane stress of temperature stress by displacement. In the same way, introducing the components of the stresses into stress boundary conditions

43、 without surface forceThe Basic Solution of The Temperature Stress Problems35簡化得：這就是按位移求解溫度應(yīng)力平面應(yīng)力問題的微分方程。 同理，將應(yīng)力分量代入無面力的應(yīng)力邊界條件溫度應(yīng)力問題的基本解法（1）36（2）and simplifying yields These are the stress boundary conditions to solve plane stress problems of temperature stress by displacement. The boundary conditio

44、ns of displacement still are Compare equations (1),(2) with the equations (1),(2) in 2-8, chapter 2. We can see that the components X and Y of the body forces are displaced byThe Basic Solution of The Temperature Stress Problems37溫度應(yīng)力問題的基本解法簡化后得：這是按位移求解溫度應(yīng)力平面應(yīng)力問題的應(yīng)力邊界條件。 位移邊界條件仍然為： 將式(1)、(2)與第二章2-8中

45、式(1)、(2)對比，可見（2）38While the components and of the surface forces are displaced by For plane strain problems of temperature stress, it is only needed that in the plane stress problems of temperature stress）（is displaced bymaammmm+-1112EEis displaced byis displaced byThen the corresponding equations u

46、nder the conditions of plane strain are obtained. The Basic Solution of The Temperature Stress Problems39代替了體力分量 X 及 Y ，而：則得到在平面應(yīng)變條件下的相應(yīng)方程。代替了面力分量 及 。 對于溫度應(yīng)力的平面應(yīng)變問題，只須將溫度應(yīng)力平面應(yīng)力問題的溫度應(yīng)力問題的基本解法406-5 The introduction of displacement potential function From last section we know that when solving the prob

47、lems of temperature stress by displacement under the situation of plane stress, we must let the components of displacement u and v satisfy the differential equationsAnd the boundary conditions of displacement and stress must be satisfied also on boundaries. We should do it by two steps when solving

48、the problems: (1) Figure out an arbitrary group of particular solution of the above differential equations.It need only satisfy the differential equations, but not always satisfy the boundary conditions. (2) Figure out a group of supplementary solution of the differential equations ignoring temperat

49、ure change T ,which can satisfy the boundary conditions after being superposed with the particular solution.The Basic Solution of The Temperature Stress Problems416-5 位移勢函數(shù)的引用 由上一節(jié)知：在平面應(yīng)力的情況下按位移求解溫度應(yīng)力問題時，須使位移分量u 和v 滿足微分方程：并在邊界上滿足位移邊界條件和應(yīng)力邊界條件。實際求解時，宜分兩步進行：（1）求出上述微分的任意一組特解，它只需滿足微分方程，而不一定要滿足邊界條件。（2）不計

50、變溫T，求出微分方程的一組補充解，使它和特解疊加以后，能滿足邊界條件。溫度應(yīng)力問題的基本解法42 Introduce into a function , and take the particular solution of displacement as The function is called the potential function of displacement. Introducing and into the differential equations instead of u and v respectively and simplifying yields: Beca

51、use and are both constants, so when let satisfy the differential equations. So and can be a group of particular solution of the differential equations.The Basic Solution of The Temperature Stress ProblemsIntroducing .and into the expression of the components of stress expressed by the components of

52、displacement and the temperature change T43溫度應(yīng)力問題的基本解法 引用一個函數(shù) ，將位移特解取為：函數(shù) 稱為位移勢函數(shù)。以 和 分別作為u和v代入微分方程，簡化后得：由于 和 都是常量，所以?。簳r， 滿足微分方程。因此 ， 可以作為微分方程的一組特解。將以及代入位移分量和變溫T表示的應(yīng)力分量表達(dá)式44yields the components of stress of corresponding particular solutions of displacement The Basic Solution of The Temperature Str

53、ess Problems45溫度應(yīng)力問題的基本解法可得相應(yīng)位移特解的應(yīng)力分量是：46 Suppose and are the supplementary solution of displacement. Then . and must satisfy the homogeneous differential equationsThe Basic Solution of The Temperature Stress ProblemsThe components of stress corresponding to the supplementary solution of displaceme

54、nt are (Notice that the temperature change is ignored, i.e. T=0.)47 設(shè) ， 為位移的補充解，則 ， 需滿足齊次微分方程：相應(yīng)于位移補充解的應(yīng)力分量為（注意不計變溫，即T=0）：溫度應(yīng)力問題的基本解法48 Thus the components of the total displacement are: They should satisfy the boundary conditions of displacement. The components of the total stress are: . They shoul

55、d satisfy the boundary conditions of the stress. In the problems of the stress boundary (no boundary condition of displacement), the components of stress corresponding to the supplementary solution of displacement can be expressed by stress function directly, i.e.The Basic Solution of The Temperatur

56、e Stress Problemsin which the stress function can be chosen according to the request of the boundary conditions of stress. In the case of plane strain, for the above-mentioned equations,）（is displaced bymaammmm+-1112EEis displaced byis displaced by49總的應(yīng)力分量是：需滿足應(yīng)力邊界條件。在應(yīng)力邊界問題中（沒有位移邊界條件），可以把相應(yīng)于位移補充解的應(yīng)

57、力分量直接用應(yīng)力函數(shù)來表示，即其中的應(yīng)力函數(shù) 可以按照應(yīng)力邊界條件的要求來選取。溫度應(yīng)力問題的基本解法 在平面應(yīng)變條件下，將上述各方程中的這樣總的位移分量是：需滿足位移邊界條件50Solution: The differential equation that the potential function of displacement need satisfy isLetIntroducing it into the above equation yields:Example 1: The temperature change of the rectangular thin plate sh

58、own in the figure is:where T0 is a constant. If ab, evaluate the temperature stress.yoaabbxThe Basic Solution of The Temperature Stress ProblemsComparing the coefficient of the two sides yields:51溫度應(yīng)力問題的基本解法例1：圖示矩形薄板中發(fā)生如下的變溫：其中的T0 是常量。若 ，試求其溫度應(yīng)力。xyoaabb解：位移勢函數(shù) 所應(yīng)滿足的微分方程為比較兩邊系數(shù)，得代入上式，得取52Substituting

59、 A and B back yields the potential function of displacement:So the components of stress corresponding to the particular solution of displacement are: To obtain the supplementary solutions, let and we can arrive at the needed components of stress corresponding to the supplementary solutions of displa

60、cement The Basic Solution of The Temperature Stress ProblemsTherefore, the total components of stress are:The boundary conditions require:53將A，B回代，得位移勢函數(shù)于是相應(yīng)于位移特解的應(yīng)力分量為 為求補充解，取 可得所需要的相應(yīng)于位移補充解的應(yīng)力分量：溫度應(yīng)力問題的基本解法因此，總的應(yīng)力分量為邊界條件要求54It is obvious that the last three conditions are satisfied, while the firs