弹性力学双语讲义(chapter1)
弹性力学讲义
yz
标轴的负方向为负。
yx y 负面：截面上的外法线 B 沿坐标轴的负方向
A
z
O
负面上的应力以沿坐标 y 轴的负方向为正，沿坐
（不考虑位置, 把应力当作均匀应力）标轴的正方向为负。
x 正应力符号规定与材力同，切应力与材力不相同。
连接前后两面中心的直线 z
ab作为矩轴，列出力矩平 衡方程，得
z
fz
F f
S
fy
f : 极限矢量,即物体在P点所受面力 的集度。方向就是F的极限方向。
fx P
fx , fy , fz：体力分量。
o
y 符号规定:
x
lim F f
V 0 S
沿坐标正方向为正，沿坐标负 方向为负。
量纲：N/m2=kg∙m/s2∙m2=kg/m∙s2
即：L-1MT-2
(4)各向同性 — 假定物体是各向同性的.
符合以上四个假定的物体，就成为理想弹性体.
(5)小变形假定 — 假定位移和形变是微小的. 它包含两个含义： ⅰ 假定应变分量 <<1. 例如：普通梁中的正应变 <<10-3 << 1,切应变 << 1;
ⅱ 假定物体的位移<<物体尺寸.
例如：梁中挠度 << 梁的高度
弹性力学在土木、水利、机械、航空等工程学科 中占有重要的地位。许多非杆件形状的结构必须用 弹性力学方法进行分析。例如，大坝，桥梁等。
§1.2 弹性力学中的几个基本概念
弹性力学的基本概念: 外力、应力、形变和位移
1. 外力：体积力和表面力，简称体力和面力
体力：分布在物体体积内的力，例如重力和惯性力。
2 yzzx
英汉双语弹性力学ppt课件
3
第二章 平面问题的基本理论
§2-1 平面应力问题与平面应变问题 §2-2 平衡微分方程 §2-3 斜面上的应力主应力 §2-4 几何方程刚体位移 §2-5 物理方程 §2-6 边界条件 §2-7 圣维南原理 §2-8 按位移求解平面问题 §2-9 按应力求解平面问题。相容方程 §2-10 常体力情况下的简化 §2-11 应力函数逆解法与半逆解法 习题课
无外力作用。
y
x
注意：平面应力问题z =0，但 问题相反。
ቤተ መጻሕፍቲ ባይዱz 0，这与平面应变
8
2.Plane strain problem Very long column bears the face force in parallel with plate face and doesn’t change
Elasticity
1
2
Chapter 2 The Basic theory of the Plane Problem
§2-1 Plane stress problem and plane strain problem
§2-2 Differential equation of equilibrium §2-3 The stress on the incline.Principal stress §2-4 Geometrical equation.The displacement of the rigid body §2-5 Physical equation §2-6 Boundary conditions §2-7 Saint-Venant’s principle §2-8 Solving the plane problem according to the displacement §2-9 Solving the plane problem according to the patible equation
弹性力学第一章
第一章 教学参考资料（一）本章的学习要求及重点1．弹性力学的研究内容，及其研究对象和研究方法，认清他们与材料力学的区别。
2．弹性力学的几个主要物理量的定义、量纲、正负方向及符号规定等，及其与材料力学相比的不同之处。
3．弹性力学的几个基本假定，及其在建立弹性力学基本方程时的应用。
（二）本章内容提要1．弹性力学的内容─弹性力学研究弹性体由于受外力作用、边界约束或温度改变等原因而发生的应力、形变和位移。
2．弹性力学中的几个基本物理量：体力—— 分布在物体体积内的力、记号为,,,x y z f f f 。
量纲为L -2MT -2，以坐标正向为正。
面力—— 分布在物体表面上的力，记号为,,,x y z f f f 。
量纲为L -2MT -2 ，以坐标正向为正。
应力—— 单位截面面积上的内力，记号x xy στ⋯⋯，量纲为L -2MT -2，以正面正向为正，负面负向为正；反之为负。
形变—— 用线应变, x y εε和切应变xy γ表示，量纲为1，线应变以伸长为正，切应变以直角减小为正。
位移—— 一点位置的移动，记号为,,u v w ，量纲为L ，以坐标正向为正。
3．弹性力学中的基本假定理想弹性体假定—连续性，完全弹性，均匀性，各向同性。
小变形假定。
4．弹性力学问题的研究方法已知：物体的边界形状，材料性质，体力,边界上的面力或约束。
求解：应力、形变和位移。
解法：在弹性体区域V 内，根据微分体上力的平衡条件，建立平衡微分方程；根据微分线段上应变和位移的几何条件，建立几何方程；根据应力和应变之间的物理条件，建立物理方程。
在弹性体边界S 上，根据面力条件，建立应力边界条件，根据约束条件，建立位移边界条件。
然后在边界条件下，求解区域内的微分方程，得出应力、形变和位移。
（三）弹性力学的发展简史与其他任何学科一样，从这门力学的发展史中，我们可以看出人们认识自然的不断深化的过程：从简单到复杂，从粗糙到精确，从错误到正确的演变历史。
弹性力学 第1章绪论
如果除上述基本假设以外，还引用某 些补充的假设，例如对于薄板（或薄壳）， 引用补充的几何假设，即直线素假设，这 样的弹性理论也可称为应用弹性理论。
弹性力学的主要对象和基本内容 弹性力学是研究非杆状弹性体(例如板、壳、 挡土墙、堤坝和地基等实体结构)在外力作用下或 由于温度改变等原因所产生的应力、应变和位移。
钱伟长(1912.10.9-2010.7.30)
钱伟长，著名力学家、应用数学家、教育家和 社会活动家。是我国近代力学的奠基人之一。 兼长应用数学、物理学、中文信息学，著述甚 丰。特别在弹性力学、变分原理、摄动方法等领域 有重要成就。早年提出的薄板薄壳非线性内禀统一 理论对欧美的固体力学和理性力学有过重大的影响。 创办了我国第一个力学研究室，筹建了中国科学院 力学研究所和自动化研究所。长期从事高等教育领 导工作，为培养我国科学技术人才作出重要贡献。 社会活动十分活跃，积极推动了祖国的统一大业。
弹性力学的任务 分析各种结构物或其构件在弹性阶段的应力和位移
校核它们是否具有所需的强度和刚度
寻求或改进它们的计算方法
材料力学与弹性力学的区别 在材料力学中研究杆状构件，除了从静力学、 几何学、物理学三方面进行分折以外，大多还需 要引用一些关于构件的应变状态或应力分布的假 定，这就大大简化了数学推演。但是，得出的解 答有时是近似的。在弹性力学中研究杆状构件一 般都不引进那些假定。因此，得出的结果就比较 精确，其解可以用来校核材料力学所得出的近似 解答。
弹性力学的基本假设与材料力学完全相 同，但是在研究方法上有较大的差别，主要 体现在
研究对象：材料力学研究的主要是杆件；而弹性 力学研究的是块、板、壳等复杂结构。 研究方法：材料力学主要是借助一些平面假设， 在构件分析中简化了数学推导，或者说舍弃了数学 严格性，但在保证精度的前提下为工程计算提供了 简便算法；而弹性力学则是数学严格的。故有时本 学科亦称为弹性结构的数学理论。
弹性力学双语版-西安交通大学幻灯片PPT
将几何方程第四式代入，得
2y
z2
2y2z
yz
yz
（a）
同理
2z
x2
2x
z2
2 zx
zx
2x
y2
2y
x2
2
xy
xy
（b）
14
Differentiate the late three formulas of geometric equations separately for X,Y,Z,we get
2 z 2 y y 2 z y 3 v z 2 z 3 w y 2 y 2 z v z w y
Substitute the fourth formula of geometric equations into the above equation, we get
并由此而得
xxyzyzxzxyx2y2uz 22 u22x
yzx yz
16
Namely
x xyz yzx zx y2 y 2zx
（c）
Similarly
zyzyxzyxxzxyyzyxzyxz22xz22yxzy
（d）
The equations of (a),(b),(c),(d)are called compatibility conditions of deformation, also known as equations of compatibility.
components and stress components are as follows:
x
1 E
x
y
z
yz
1 G
yz
y
1 E
y
z
弹性力学第一章
•The analysis in elasticity shows that the stresses are by no means uniform, but are concentrated near the hole.
•No assumption, that a plane section of the beam remains plane after bending, is made in Elasticity.
弹性力学 第一章
19
•A prismatical tension member with a small hole
弹性力学 第一章
7
Comparison among the three courses in solid mechanics
固体力学三门学科的比较
• Three branches have the same purpose and do differ from one another both in objects studied and the methods of analysis used.
Elasticity: 弹性力学
1. plates and shells 板，壳 2.blocks: 块体 e.g. dams,foundations 坝，基础
3.analyze bar element precisely 对杆件作精确分析
弹性力学 第一章
2010122 弹性力学(中英文)(2011)
天津大学《弹性力学》课程教学大纲课程编号：2010122 课程名称：弹性力学学时：96 学分： 6学时分配：授课：96 上机：0 实验： 0 实践： 0 实践（周） 0授课学院：机械工程学院适用专业：工程力学先修课程：高等数学，材料力学，张量分析和场论一、课程的性质与目的弹性力学是固体力学学科的分支。
该课程是研究和分析工程结构和材料强度和学习《有限元法》、《塑性力学》、《断裂力学》等后续课程的理论基础。
课程的基本任务是研究弹性体在外载荷作用下，物体内部产生的位移、变形和应力分布规律，为解决工程结构和材料的强度、刚度和稳定性等问题提供解决思路和方法。
二、教学基本要求要求学生对应力、应变等基本概念有较深入的理解，掌握弹性力学解决问题的思路和方法。
能够系统地掌握弹性力学的基本理论、边值问题的提法和求解、弹性力学平面问题、柱形杆的扭转和能量原理，了解空间问题、复变函数解法、热应力和弹性波等。
三、教学内容弹性力学I1．绪论1．1弹性力学的任务、内容和研究方法1．2弹性力学的发展简史和工程应用1．3弹性力学的基本假设和载荷分类2．应力理论2．1内力和应力2．2斜面应力公式2．3应力分量转换公式2．4主应力，应力不变量2．5最大剪应力，八面体剪应力2．6应力偏量2．7应力平衡微分方程2．8正交曲线坐标系中的平衡方程3．应变理论3．1位移和应变3．2小应变张量3．3刚体转动3．4应变协调方程3．5位移单值条件3．6由应变求位移3．7正交曲线坐标系中的几何方程4．本构关系4．1广义胡克定律4．2应变能和应变余能4．3热弹性本构关系4．4应变能正定性5．弹性理论的微分提法、解法及一般原理5．1弹性力学问题的微分提法5．2位移解法5．3应力解法5．4应力函数解法5．5迭加原理5．6解的唯一性原理5．7圣维南原理6．柱形杆问题6．1问题的提法，单拉和纯弯情况6．2柱形杆的自由扭转6．3反逆法与半逆法，扭转问题解例6．4薄膜比拟6．5较复杂的扭转问题6．6柱形杆的一般弯曲7．平面问题7．1平面问题及其分类7．2平面问题的基本解法7．3应力函数的性质7．4直角坐标解例7．5极坐标中的平面问题7．6轴对称问题7．7非轴对称问题7．8关于解和解法的讨论弹性力学II8．复变函数解法8．1平面问题的复格式8．2单连域中复势的确定程度8．3多连域中复势的多值性8．4级数解法8．5保角变换解法8．6柯西积分公式的应用9．空间问题9．1齐次拉梅－纳维方程的一般解9．2非齐次拉梅－纳维方程的解9．3位移的势函数分解9．4空间轴对称问题9．5半空间问题9．6接触问题10．能量原理10．1基本概念和术语10．2可能功原理，功的互等定理10．3虚功原理和余虚功原理10．4最小势能原理和最小余能原理10．5弹性力学变分问题的欧拉方程10．6弹性力学变分问题的直接解法（一）10．7可变边界条件，卡氏定理10．8广义变分原理10．9弹性力学变分问题的直接解法（二）11．热应力11．1热传导基本概念11．2热弹性基本方程11．3热应力问题简例及不产生热应力的条件11．4基本方程的求解11．5平面热应力问题12．弹性波的传播12．1杆中的弹性波12．2无限介质中的弹性波12．3球面波12．4平面波12．5平面波的发射与折射12．6平面波在自由界面处的反射，瑞利波12．7勒夫波四、学时分配五、评价与考核方式平时成绩（出勤、作业等）20%，期末考试成绩80%。
弹性力学基础-中英
The actual point of yield is often difficult to identify. A number of techniques are used to locateσy. The tangent method <or knee method> locates the yield strength at the intersection of the elastic slope and the initial portion of the plastic region <not reliably>. The preferred method is the percentage offset method where yield strength is obtained by drawing a line parallel to the initial elastic region data at 0.2% strain <0.002> offset. Where this line intersects the stress-strain curve then becomes known as the 0.2% yield strength.
Plastic means permanent!
Plastic deformation---it is irreversible or permanent.
O
A
B
C
D
E
elastic region
yield strength 屈服应力 屈服强度
plastic region
ultimate tensile strength 抗拉强度
弹性力学1
∆l x
p
εx =
∆l x lx
同理： 同理：
εy =
∆l y ly
,
εz =
∆l z lz
弹性力学（第一章） 弹性力学（第一章）
2、剪应变 、 各线段之间直角的改变，用弧度表示。 γ 各线段之间直角的改变，用弧度表示。(γyz、 γzx、 γxy)。 。 剪应变以直角变小时为正，变大时为负。 剪应变以直角变小时为正，变大时为负。
2、物体任意点的应力状态 、
σz C τzx z σy τyz P σx τzy A 0 x σz τyx τxy τxz τzy σx τxz τxy τzx τyx B τyz σy
PA = ∆x PB = ∆y PC = ∆z
y
（1）正负号规定： ）正负号规定： 正面应力沿坐标轴正方向为正，负面应力沿坐标轴负方向为正。 正面应力沿坐标轴正方向为正，负面应力沿坐标轴负方向为正。
弹性力学（第一章） 弹性力学（第一章）
课程的教学基本要求
1．理解面力、体力、应力、应变和位移的基本概念并熟 ．理解面力、体力、应力、 悉相应的记号等规定。 悉相应的记号等规定。 2．理解弹性力学中的基本假定，熟悉弹性力学三大类基 ．理解弹性力学中的基本假定， 本方程（ 平衡方程、几何方程、物理方程） 本方程 （ 平衡方程 、 几何方程 、 物理方程 ） ， 了解按应力 求解和按位移求解基本方程的思路和步骤， 求解和按位移求解基本方程的思路和步骤 ， 掌握平面应力 问题和平面应变问题的特点。 问题和平面应变问题的特点。 3．能正确写出边界条件，能正确应用圣维南原理。 ．能正确写出边界条件，能正确应用圣维南原理。 4．理解平面问题的逆解法和半逆解法的基本思路。熟悉 ．理解平面问题的逆解法和半逆解法的基本思路。 矩形梁纯弯曲的解和简支梁受均布荷载的解。 矩形梁纯弯曲的解和简支梁受均布荷载的解 。 了解斜截面 的应力和应变。 的应力和应变。
弹性力学第1章
2Deformation:Displacements and Strains We begin development of the basicfield equations of elasticity theory byfirst investigating thekinematics of material deformation.As a result of applied loadings,elastic solids will changeshape or deform,and these deformations can be quantified by knowing the displacements ofmaterial points in the body.The continuum hypothesis establishes a displacementfield at allpoints within the elastic ing appropriate geometry,particular measures of deformationcan be constructed leading to the development of the strain tensor.As expected,the straincomponents are related to the displacementfield.The purpose of this chapter is to introduce thebasic definitions of displacement and strain,establish relations between these twofieldquantities,andfinally investigate requirements to ensure single-valued,continuous displace-mentfields.As appropriate for linear elasticity,these kinematical results are developed underthe conditions of small deformation theory.Developments in this chapter lead to two funda-mental sets offield equations:the strain-displacement relations and the compatibility equa-tions.Furtherfield equation development,including internal force and stress distribution,equilibrium and elastic constitutive behavior,occurs in subsequent chapters.2.1General DeformationsUnder the application of external loading,elastic solids deform.A simple two-dimensionalcantilever beam example is shown in Figure2-1.The undeformed configuration is taken withthe rectangular beam in the vertical position,and the end loading displaces material points tothe deformed shape as shown.As is typical in most problems,the deformation varies frompoint to point and is thus said to be nonhomogenous.A superimposed square mesh is shown inthe two configurations,and this indicates how elements within the material deform locally.It isapparent that elements within the mesh undergo extensional and shearing deformation.Anelastic solid is said to be deformed or strained when the relative displacements between pointsin the body are changed.This is in contrast to rigid-body motion where the distance betweenpoints remains the same.In order to quantify deformation,consider the general example shown in Figure2-2.In the undeformed configuration,we identify two neighboring material points P o and P connected withthe relative position vector r as shown.Through a general deformation,these points are mappedto locations P0and P0in the deformed configuration.Forfinite or large deformation theory,theo27undeformed and deformed configurations can be significantly different,and a distinction between these two configurations must be maintained leading to Lagrangian and Eulerian descriptions;see,for example,Malvern(1969)or Chandrasekharaiah and Debnath(1994). However,since we are developing linear elasticity,which uses only small deformation theory, the distinction between undeformed and deformed configurations can be dropped.Using Cartesian coordinates,define the displacement vectors of points P o and P to be u o and u,respectively.Since P and P o are neighboring points,we can use a Taylor series expansion around point P o to express the components of u asu¼u oþ@u@xr xþ@u@yr yþ@u@zr zv¼v oþ@v@xr xþ@v@yr yþ@v@zr zw¼w oþ@w@xr xþ@w@yr yþ@w@zr z(2:1:1)FIGURE2-1Two-dimensional deformation example.(Undeformed)(Deformed) FIGURE2-2General deformation between two neighboring points.28FOUNDATIONS AND ELEMENTARY APPLICATIONSNote that the higher-order terms of the expansion have been dropped since the components of r are small.The change in the relative position vector r can be written asD r¼r0Àr¼uÀu o(2:1:2) and using(2.1.1)givesD r x¼@u@xr xþ@u@yr yþ@u@zr zD r y¼@v@xr xþ@v@yr yþ@v@zr zD r z¼@w@xr xþ@w@yr yþ@w@zr z(2:1:3)or in index notationD r i¼u i,j r j(2:1:4) The tensor u i,j is called the displacement gradient tensor,and may be written out asu i,j¼@u@x@u@y@u@z@v@x@v@y@v@z@w@x@w@y@w@z2666666437777775(2:1:5)From relation(1.2.10),this tensor can be decomposed into symmetric and antisymmetric parts asu i,j¼e ijþ!ij(2:1:6) wheree ij¼12(u i,jþu j,i)!ij¼12(u i,jÀu j,i)(2:1:7)The tensor e ij is called the strain tensor,while!ij is referred to as the rotation tensor.Relations (2.1.4)and(2.1.6)thus imply that for small deformation theory,the change in the relative position vector between neighboring points can be expressed in terms of a sum of strain and rotation bining relations(2.1.2),(2.1.4),and(2.1.6),and choosing r i¼dx i, we can also write the general result in the formu i¼u o iþe ij dx jþ!ij dx j(2:1:8) Because we are considering a general displacementfield,these results include both strain deformation and rigid-body motion.Recall from Exercise1-14that a dual vector!i canDeformation:Displacements and Strains29be associated with the rotation tensor such that !i ¼À1=2e ijk !jk .Using this definition,it is found that!1¼!32¼12@u 3@x 2À@u 2@x 3 !2¼!13¼12@u 1@x 3À@u 3@x 1 !3¼!21¼12@u 2@x 1À@u 1@x 2 (2:1:9)which can be expressed collectively in vector format as v ¼(1=2)(r Âu ).As is shown in the next section,these components represent rigid-body rotation of material elements about the coordinate axes.These general results indicate that the strain deformation is related to the strain tensor e ij ,which in turn is a related to the displacement gradients.We next pursue a more geometric approach and determine specific connections between the strain tensor components and geometric deformation of material elements.2.2Geometric Construction of Small Deformation TheoryAlthough the previous section developed general relations for small deformation theory,we now wish to establish a more geometrical interpretation of these results.Typically,elasticity variables and equations are field quantities defined at each point in the material continuum.However,particular field equations are often developed by first investigating the behavior of infinitesimal elements (with coordinate boundaries),and then a limiting process is invoked that allows the element to shrink to a point.Thus,consider the common deformational behavior of a rectangular element as shown in Figure 2-3.The usual types of motion include rigid-body rotation and extensional and shearing deformations as illustrated.Rigid-body motion does not contribute to the strain field,and thus also does not affect the stresses.We therefore focus our study primarily on the extensional and shearing deformation.Figure 2-4illustrates the two-dimensional deformation of a rectangular element with original dimensions dx by dy .After deformation,the element takes a rhombus form as shown in the dotted outline.The displacements of various corner reference points areindicated(Rigid Body Rotation)(Undeformed Element)(Horizontal Extension)(Vertical Extension)(Shearing Deformation)FIGURE 2-3Typical deformations of a rectangular element.30FOUNDATIONS AND ELEMENTARY APPLICATIONSin the figure.Reference point A is taken at location (x,y ),and the displacement components of this point are thus u (x,y )and v (x,y ).The corresponding displacements of point B are u (x þdx ,y )and v (x þdx ,y ),and the displacements of the other corner points are defined in an analogous manner.According to small deformation theory,u (x þdx ,y )%u (x ,y )þ(@u =@x )dx ,with similar expansions for all other terms.The normal or extensional strain component in a direction n is defined as the change in length per unit length of fibers oriented in the n -direction.Normal strain is positive if fibers increase in length and negative if the fiber is shortened.In Figure 2-4,the normal strain in the x direction can thus be defined bye x ¼A 0B 0ÀAB From the geometry in Figure 2-4,A 0B 0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx þ@u @x dx 2þ@v @x dx 2s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2@u @x þ@u @x 2þ@v @x 2dx s %1þ@u @xdx where,consistent with small deformation theory,we have dropped the higher-order ing these results and the fact that AB ¼dx ,the normal strain in the x -direction reduces toe x ¼@u@x (2:2:1)In similar fashion,the normal strain in the y -direction becomese y ¼@v@y (2:2:2)A second type of strain is shearing deformation,which involves angles changes (see Figure 2-3).Shear strain is defined as the change in angle between two originally orthogonalx FIGURE 2-4Two-dimensional geometric strain deformation.Deformation:Displacements and Strains 31directions in the continuum material.This definition is actually referred to as the engineering shear strain.Theory of elasticity applications generally use a tensor formalism that requires a shear strain definition corresponding to one-half the angle change between orthogonal axes; see previous relation(2:1:7)1.Measured in radians,shear strain is positive if the right angle between the positive directions of the two axes decreases.Thus,the sign of the shear strain depends on the coordinate system.In Figure2-4,the engineering shear strain with respect to the x-and y-directions can be defined asg xy¼p2ÀffC0A0B0¼aþbFor small deformations,a%tan a and b%tan b,and the shear strain can then be expressed asg xy¼@v@xdxdxþ@u@xdxþ@u@ydydyþ@v@ydy¼@u@yþ@v@x(2:2:3)where we have again neglected higher-order terms in the displacement gradients.Note that each derivative term is positive if lines AB and AC rotate inward as shown in thefigure.By simple interchange of x and y and u and v,it is apparent that g xy¼g yx.By considering similar behaviors in the y-z and x-z planes,these results can be easily extended to the general three-dimensional case,giving the results:e x¼@u@x,e y¼@v@y,e z¼@w@zg xy¼@u@yþ@v@x,g yz¼@v@zþ@w@y,g zx¼@w@xþ@u@z(2:2:4)Thus,we define three normal and three shearing strain components leading to a total of six independent components that completely describe small deformation theory.This set of equations is normally referred to as the strain-displacement relations.However,these results are written in terms of the engineering strain components,and tensorial elasticity theory prefers to use the strain tensor e ij defined by(2:1:7)1.This represents only a minor change because the normal strains are identical and shearing strains differ by a factor of one-half;for example,e11¼e x¼e x and e12¼e xy¼1=2g xy,and so forth.Therefore,using the strain tensor e ij,the strain-displacement relations can be expressed in component form ase x¼@u@x,e y¼@v@y,e z¼@w@ze xy¼1@uþ@v,e yz¼1@vþ@w,e zx¼1@wþ@u(2:2:5)Using the more compact tensor notation,these relations are written ase ij¼12(u i,jþu j,i)(2:2:6)32FOUNDATIONS AND ELEMENTARY APPLICATIONSwhile in direct vector/matrix notation as the form reads:e¼12r uþ(r u)TÂÃ(2:2:7)where e is the strain matrix and r u is the displacement gradient matrix and(r u)T is its transpose.The strain is a symmetric second-order tensor(e ij¼e ji)and is commonly written in matrix format:e¼[e]¼e x e xy e xze xy e y e yze xz e yz e z2435(2:2:8)Before we conclude this geometric presentation,consider the rigid-body rotation of our two-dimensional element in the x-y plane,as shown in Figure2-5.If the element is rotated through a small rigid-body angular displacement about the z-axis,using the bottom element edge,the rotation angle is determined as@v=@x,while using the left edge,the angle is given byÀ@u=@y. These two expressions are of course the same;that is,@v=@x¼À@u=@y and note that this would imply e xy¼0.The rotation can then be expressed as!z¼[(@v=@x)À(@u=@y)]=2, which matches with the expression given earlier in(2:1:9)3.The other components of rotation follow in an analogous manner.Relations for the constant rotation!z can be integrated to give the result:u*¼u oÀ!z yv*¼v oþ!z x(2:2:9)where u o and v o are arbitrary constant translations in the x-and y-directions.This result then specifies the general form of the displacementfield for two-dimensional rigid-body motion.We can easily verify that the displacementfield given by(2.2.9)yields zero strain.xFIGURE2-5Two-dimensional rigid-body rotation.Deformation:Displacements and Strains33For the three-dimensional case,the most general form of rigid-body displacement can beexpressed asu*¼u oÀ!z yþ!y zv*¼v oÀ!x zþ!z xw*¼w oÀ!y xþ!x y(2:2:10)As shown later,integrating the strain-displacement relations to determine the displacementfield produces arbitrary constants and functions of integration,which are equivalent to rigid-body motion terms of the form given by(2.2.9)or(2.2.10).Thus,it is important to recognizesuch terms because we normally want to drop them from the analysis since they do notcontribute to the strain or stressfields.2.3Strain TransformationBecause the strains are components of a second-order tensor,the transformation theorydiscussed in Section1.5can be applied.Transformation relation(1:5:1)3is applicable forsecond-order tensors,and applying this to the strain givese0ij¼Q ip Q jq e pq(2:3:1)where the rotation matrix Q ij¼cos(x0i,x j).Thus,given the strain in one coordinate system,we can determine the new components in any other rotated system.For the general three-dimensional case,define the rotation matrix asQ ij¼l1m1n1l2m2n2l3m3n32435(2:3:2)Using this notational scheme,the specific transformation relations from equation(2.3.1)becomee0x¼e x l21þe y m21þe z n21þ2(e xy l1m1þe yz m1n1þe zx n1l1)e0y¼e x l22þe y m22þe z n22þ2(e xy l2m2þe yz m2n2þe zx n2l2)e0z¼e x l23þe y m23þe z n23þ2(e xy l3m3þe yz m3n3þe zx n3l3)e0xy¼e x l1l2þe y m1m2þe z n1n2þe xy(l1m2þm1l2)þe yz(m1n2þn1m2)þe zx(n1l2þl1n2)e0yz¼e x l2l3þe y m2m3þe z n2n3þe xy(l2m3þm2l3)þe yz(m2n3þn2m3)þe zx(n2l3þl2n3)e0zx¼e x l3l1þe y m3m1þe z n3n1þe xy(l3m1þm3l1)þe yz(m3n1þn3m1)þe zx(n3l1þl3n1)(2:3:3)For the two-dimensional case shown in Figure2-6,the transformation matrix can be ex-pressed asQ ij¼cos y sin y0Àsin y cos y00012435(2:3:4)34FOUNDATIONS AND ELEMENTARY APPLICATIONSUnder this transformation,the in-plane strain components transform according toe 0x ¼e x cos 2y þe y sin 2y þ2e xy sin y cos ye 0y ¼e x sin 2y þe y cos 2y À2e xy sin y cos ye 0xy ¼Àe x sin y cos y þe y sin y cos y þe xy (cos 2y Àsin 2y )(2:3:5)which is commonly rewritten in terms of the double angle:e 0x ¼e x þe y 2þe x Àe y 2cos 2y þe xy sin 2y e 0y ¼e x þe y Àe x Àe y cos 2y Àe xy sin 2y e 0xy ¼e y Àe x 2sin 2y þe xy cos 2y (2:3:6)Transformation relations (2.3.6)can be directly applied to establish transformations between Cartesian and polar coordinate systems (see Exercise 2-6).Additional applications of these results can be found when dealing with experimental strain gage measurement systems.For example,standard experimental methods using a rosette strain gage allow the determination of extensional strains in three different directions on the surface of a ing this type of data,relation (2:3:6)1can be repeatedly used to establish three independent equations that can be solved for the state of strain (e x ,e y ,e xy )at the surface point under study (see Exercise 2-7).Both two-and three-dimensional transformation equations can be easily incorporated in MATLAB to provide numerical solutions to problems of interest.Such examples are given in Exercises 2-8and 2-9.2.4Principal StrainsFrom the previous discussion in Section 1.6,it follows that because the strain is a symmetric second-order tensor,we can identify and determine its principal axes and values.According to this theory,for any given strain tensor we can establish the principal value problem and solvey ′FIGURE 2-6Two-dimensional rotational transformation.Deformation:Displacements and Strains 35the characteristic equation to explicitly determine the principal values and directions.The general characteristic equation for the strain tensor can be written asdet[e ijÀe d ij]¼Àe3þW1e2ÀW2eþW3¼0(2:4:1) where e is the principal strain and the fundamental invariants of the strain tensor can be expressed in terms of the three principal strains e1,e2,e3asW1¼e1þe2þe3W2¼e1e2þe2e3þe3e1W3¼e1e2e3(2:4:2)Thefirst invariant W1¼W is normally called the cubical dilatation,because it is related to the change in volume of material elements(see Exercise2-11).The strain matrix in the principal coordinate system takes the special diagonal forme ij¼e1000e2000e32435(2:4:3)Notice that for this principal coordinate system,the deformation does not produce anyshearing and thus is only extensional.Therefore,a rectangular element oriented alongprincipal axes of strain will retain its orthogonal shape and undergo only extensional deform-ation of its sides.2.5Spherical and Deviatoric StrainsIn particular applications it is convenient to decompose the strain tensor into two parts calledspherical and deviatoric strain tensors.The spherical strain is defined by~e ij¼13e kk d ij¼13Wd ij(2:5:1)while the deviatoric strain is specified as^e ij¼e ijÀ13e kk d ij(2:5:2)Note that the total strain is then simply the sume ij¼~e ijþ^e ij(2:5:3)The spherical strain represents only volumetric deformation and is an isotropic tensor,being the same in all coordinate systems(as per the discussion in Section1.5).The deviatoricstrain tensor then accounts for changes in shape of material elements.It can be shownthat the principal directions of the deviatoric strain are the same as those of the straintensor.36FOUNDATIONS AND ELEMENTARY APPLICATIONS2.6Strain CompatibilityWe now investigate in more detail the nature of the strain-displacement relations (2.2.5),and this will lead to the development of some additional relations necessary to ensure continuous,single-valued displacement field solutions.Relations (2.2.5),or the index notation form (2.2.6),represent six equations for the six strain components in terms of three displacements.If we specify continuous,single-valued displacements u,v,w,then through differentiation the resulting strain field will be equally well behaved.However,the converse is not necessarily true;that is,given the six strain components,integration of the strain-displacement relations (2.2.5)does not necessarily produce continuous,single-valued displacements.This should not be totally surprising since we are trying to solve six equations for only three unknown displacement components.In order to ensure continuous,single-valued displacements,the strains must satisfy additional relations called integrability or compatibility equations .Before we proceed with the mathematics to develop these equations,it is instructive to consider a geometric interpretation of this concept.A two-dimensional example is shown in Figure 2-7whereby an elastic solid is first divided into a series of elements in case (a).For simple visualization,consider only four such elements.In the undeformed configuration shown in case (b),these elements of course fit together perfectly.Next,let us arbitrarily specify the strain of each of the four elements and attempt to reconstruct the solid.For case (c),the elements have been carefully strained,taking into consideration neighboring elements so that the system fits together thus yielding continuous,single-valued displacements.However,for(b) Undeformed Configuration(c) Deformed ConfigurationContinuous Displacements (a) Discretized Elastic Solid (d) Deformed Configuration Discontinuous DisplacementsFIGURE 2-7Physical interpretation of strain compatibility.case(d),the elements have been individually deformed without any concern for neighboring deformations.It is observed for this case that the system will notfit together without voids and gaps,and this situation produces a discontinuous displacementfield.So,we again conclude that the strain components must be somehow related to yield continuous,single-valued displacements.We now pursue these particular relations.The process to develop these equations is based on eliminating the displacements from the strain-displacement relations.Working in index notation,we start by differentiating(2.2.6) twice with respect to x k and x l:e ij,kl¼12(u i,jklþu j,ikl)Through simple interchange of subscripts,we can generate the following additional relations:e kl,ij¼12(u k,lijþu l,kij)e jl,ik¼12(u j,likþu l,jik)e ik,jl¼12(u i,kjlþu k,ijl)Working under the assumption of continuous displacements,we can interchange the order of differentiation on u,and the displacements can be eliminated from the preceding set to gete ij,klþe kl,ijÀe ik,jlÀe jl,ik¼0(2:6:1) These are called the Saint Venant compatibility equations.Although the system would lead to 81individual equations,most are either simple identities or repetitions,and only6are meaningful.These six relations may be determined by letting k¼l,and in scalar notation, they become@2e x @y2þ@2e y@x2¼2@2e xy@x@y@2e y @z2þ@2e z@y2¼2@2e yz@y@z@2e z @x2þ@2e x@z2¼2@2e zx@z@x@2e x @y@z ¼@@xÀ@e yz@xþ@e zx@yþ@e xy@z@2e y @z@x ¼@@yÀ@e zx@yþ@e xy@zþ@e yz@x@2e z @x@y ¼@@zÀ@e xy@zþ@e yz@xþ@e zx@y(2:6:2)It can be shown that these six equations are equivalent to three independent fourth-order relations(see Exercise2-14).However,it is usually more convenient to use the six second-order equations given by(2.6.2).In the development of the compatibility relations,we assumed that the displacements were continuous,and thus the resulting equations (2.6.2)are actually only a necessary condition.In order to show that they are also sufficient,consider two arbitrary points P and P o in an elastic solid,as shown in Figure 2-8.Without loss in generality,the origin may be placed at point P o .The displacements of points P and P o are denoted by u P i and u o i ,and the displacement ofpoint P can be expressed asu P i ¼u o i þðC du i ¼u o i þðC @u i @x j dx j (2:6:3)where C is any continuous curve connecting points P o and P .Using relation (2.1.6)for the displacement gradient,(2.6.3)becomesu P i ¼u o i þðC (e ij þ!ij )dx j (2:6:4)Integrating the last term by parts givesðC !ij dx j ¼!P ij x P j ÀðC x j !ij ,k dx k (2:6:5)where !P ij is the rotation tensor at point P .Using relation (2:1:7)2,!ij ,k ¼12(u i ,jk Àu j ,ik )¼12(u i ,jk Àu j ,ik )þ12(u k ,ji Àu k ,ji )¼12@@x j (u i ,k þu k ,i )À12@@x i(u j ,k þu k ,j )¼e ik ,j Àe jk ,i (2:6:6)Substituting results (2.6.5)and (2.6.6)into (2.6.4)yieldsu P i¼u o i þ!P ij x P j þðC U ik dx k (2:6:7)where U ik ¼e ik Àx j (e ik ,j Àe jk ,i ).P oFIGURE 2-8Continuity of displacements.Now if the displacements are to be continuous,single-valued functions,the line integral appearing in(2.6.7)must be the same for any curve C;that is,the integral must be independent of the path of integration.This implies that the integrand must be an exact differential,so that the value of the integral depends only on the end points.Invoking Stokes theorem,we can show that if the region is simply connected(definition of the term simply connected is postponed for the moment),a necessary and sufficient condition for the integral to be path independent is for U ik,l¼U il,ing this result yieldse ik,lÀd jl(e ik,jÀe jk,i)Àx j(e ik,jlÀe jk,il)¼e il,kÀd jk(e il,jÀe jl,i)Àx j(e il,jkÀe jl,ik) which reduces tox j(e ik,jlÀe jk,ilÀe il,jkþe jl,ik)¼0Because this equation must be true for all values of x j,the terms in parentheses must vanish, and after some index renaming this gives the identical result previously stated by the compati-bility relations(2.6.1):e ij,klþe kl,ijÀe ik,jlÀe jl,ik¼0Thus,relations(2.6.1)or(2.6.2)are the necessary and sufficient conditions for continuous, single-valued displacements in simply connected regions.Now let us get back to the term simply connected.This concept is related to the topology or geometry of the region under study.There are several places in elasticity theory where the connectivity of the region fundamentally affects the formulation and solution method. The term simply connected refers to regions of space for which all simple closed curves drawn in the region can be continuously shrunk to a point without going outside the region. Domains not having this property are called multiply connected.Several examples of such regions are illustrated in Figure2-9.A general simply connected two-dimensional region is shown in case(a),and clearly this case allows any contour within the region to be shrunk to a point without going out of the domain.However,if we create a hole in the region as shown in case(b),a closed contour surrounding the hole cannot be shrunk to a point without going into the hole and thus outside of the region.Thus,for two-dimensional regions,the presence of one or more holes makes the region multiply connected.Note that by introducing a cut between the outer and inner boundaries in case(b),a new region is created that is now simply connected. Thus,multiply connected regions can be made simply connected by introducing one or more cuts between appropriate boundaries.Case(c)illustrates a simply connected three-dimensional example of a solid circular cylinder.If a spherical cavity is placed inside this cylinder as shown in case(d),the region is still simply connected because any closed contour can still be shrunk to a point by sliding around the interior cavity.However,if the cylinder has a through hole as shown in case(e),then an interior contour encircling the axial through hole cannot be reduced to a point without going into the hole and outside the body.Thus,case(e)is an example of the multiply connected three-dimensional region.It was found that the compatibility equations are necessary and sufficient conditions for continuous,single-valued displacements only for simply connected regions.However, for multiply connected domains,relations(2.6.1)or(2.6.2)provide only necessary but。
材料力学双语教学学习资料(英汉对照)
材料力学双语教学学习资料第一章绪论Chapter 1 Introduction§1-1 材料力学的任务The Tasks of Mechanics of Materials1＊. 材料力学: Mechanics of Materials2. 构件: Structural Members3. 变形: Deformation4＊. 强度: Strength5＊. 刚度: Rigidity6＊. 稳定性: Stability§1-2 变形固体的基本假设Fundamental Assumptions of SolidDeformation Bodies1. 连续性假设: Continuity2. 均匀性假设: Homogeneity3. 各向同性假设: Isotropy§1.3 外力及其分类External Forces and Classification1. 分布力: Distributed Force2. 集中力: Point Force3. 静载荷: Static Load4. 动载荷: Dynamic Load§1.4 内力、截面法和应力的概念Concepts of Internal Forces，Method ofSection and Stress1＊. 内力: Internal Force2＊. 截面法: Method of Section3. 截面法的三个步骤：截开，代替，平衡Three steps of method of section: cut off, substitute , and equilibrium.4＊. 应力: Stress5. 平均应力：Average stress6. 应力（全应力）：Whole stress（sum stress）7＊. 正应力: Normal Stress8＊. 剪应力(切应力)：Shearing Stress§1.5 变形与应变Deformation and Strain1．线应变: Strain2．剪应变: Shearing Strain§1.6 杆件变形的基本形式Basic Types of Deformations of Rods1＊. 拉伸或压缩: Tension or Compression2＊. 剪切: Shear3＊. 扭转: Torsion4＊. 弯曲: Bending第二章拉伸、压缩与剪切Chapter 2 Tension，Compression andShear§2.1 轴向拉伸与压缩的概念和实例The Concept and Examples of AxialTension and Compression1. 拉杆: Tensile Rod2. 压杆: Compressive Rod3. 受力特点：外力合力的作用线与杆轴线重合Characteristic of the External Forces: The acting line of the resultant of external forces is coincided with the axis of the rod.4. 变形特点：杆沿轴向伸长或缩短Characteristic of Deformation: Rod will elongate or contract along the axis of the rod.§2.2 轴向拉伸或压缩时横截面上的内力和应力Internal Force and Stress of Axial Tension or Compression on the Cross Section1＊. 横截面: Cross Section2＊. 轴力: Normal Force3＊. 轴力图: Diagram of Normal Force§2.3 直杆轴向拉伸或压缩时斜截面上的应力Stress of Axial Tension or Compressionon the Skew Section1. 斜截面: Skew Section2.ασσα2cos = αστα2s i n 2=§2.4 材料在拉伸时的力学性能Mechanical Properties of Materialswith Tensile Load1. 标准试件: Specimen2. 低碳钢(C ≤0.3%): Low Carbon Steel3. 弹性阶段:Elastic Region4. 屈服阶段:Yielding Stage5. 强化阶段:Hardening Stage6. 颈缩阶段: Necking Stage 7＊.σp ----比例极限: Proportional Limit 8＊.σe ----弹性极限: Elastic Limit 9＊.σs ----屈服极限: Yielding Stress 10＊.σb ----强度极限: Ultimate Stress 11. 延伸率: Percent Elongation12. 断面收缩率: Percent Reduction of Area 13. 塑性材料: Ductile Materials 14. 脆性材料: Brittle Materials 15. 铸铁:Cast iron§2.7 失效、安全系数和强度计算 Failure, Safety factor and Strengthcalculation1＊. 许用应力: Allowable Stress 2. 安全系数: Safety Factor 3＊. 强度条件: Strength Condition][max σσ≤=AF N4＊. 强度校核: Check strength][max σσ≤5＊. 截面设计: Section design][σNF A ≥6＊. 确定许可载荷:Determine allowable load][σA F N ≤§2.8 轴向拉伸或压缩时的变形 Deformation in Axial Tension orCompression1. 弹性变形: Elastic Deformation2. 塑性变形: Plastic Deformation3. 纵向应变: Longitudinal Strainll l l l -=∆=1ε 4. 横向应变: Lateral Straindd d d d -=∆=''ε5．线弹性变形:Linear Elastic Deformation6.泊松比:Poisson’s ratioεεμ'=7＊.弹性模量-E ：表示材料抵抗拉压变形的 能力 E - modulus of elasticity ：Indicates the capability of materials for resisting tension or compression 8＊.抗拉刚度-EA :表示构件抵抗拉压变形的能力EA -the axial rigidity: Indicates the capability of constructive members for resisting tension or compression 9＊. 胡克定律（Hooke’s Law ）:当应力不超过材料的比例极限时，应力与应变成正比.The stress is proportional to the strain within the elastic region.εσE =§2.12 应力集中的概念The Concept of Stress Concentration 1.由于截面尺寸的突然变化，使截面上的应力分布不再均匀，在某些部位出现远大于平均值的应力，称应力集中。
弹性力学课件 第1章 绪论
3. 各向同性(isotropy)假设
＊假定物体在各个不同的方向上具有相同的物理性质 物体的弹性常数将不随坐标方向的改变而变化。 ＊宏观假设，材料性能是显示各向同性 ＊木材，竹子以及纤维增强材料等，属于各向异性材料 ＊这些材料的研究属于复合材料力学研究的对象。
4.完全弹性（线弹性linear elasticity)假设
n阶张量：有n个自由指标的量，如四阶弹性系数Dijkl
3. 应变 (1) 一点应变的度量
是描述物体受力后发生变形的相对概念的力学量 正应变——棱边的伸长和缩短
x , y , z
xy , yz , zx
z C
切应变——棱边之间夹角（直角）改变 应变的正负： 线应变：伸长时为正，缩短时为负；
＊对应一定的温度，如果应力和应变之间存在一一对 应关系，而且这个关系和时间无关，也和变形历史 无关，称为完全弹性材料。 材料弹性常数不随应力或应变的变化而改变 ＊完全弹性分为线性弹性和非线性弹性 ＊弹性力学研究限于线性的应力与应变关系
5. 小变形(small deformation)假设
＊假设在外力或者其他外界因素（如温度等）的影响下， 物体的变形与物体自身几何尺寸相比属于高阶小量。 ＊在弹性体的平衡等问题讨论时，可以不考虑因变形 所引起的尺寸变化 ＊忽略位移、应变和应力等分量的高阶小量，使基本 方程成为线性的偏微分方程组。
铁木辛柯(S.P.Timoshenko)做出了贡献。
中国科学家钱伟长，钱学森，徐芝伦，胡海昌,等在弹性
力学的发展，特别是在中国的推广应用做出了重要贡献。
钱学森
钱伟长
胡海昌
徐芝伦
杨桂通
弹性力学——促进数学和自然科学基本理论的建立和发展 广泛工程应用——造船、建筑、航空和机械制造等。 发展——形成了一些专门的分学科； 现代科学技术和工程技术——仍然提出新的理论和工程问题。 对于现代工程技术和科研工作者的培养——对于专业基础， 思维方法以及独立工作能力都有不可替代的作用。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
extbook: Applied Elasticity 徐芝纶 中文教材： 中文教材： 弹性力学简明教程 徐芝纶
Chapter 1. Introduction 第一章 绪论
•A prismatical tension member with a small hole •It is assumed in mechanics of materials that the tensile stresses are uniformly distributed across the net section of the member. •The analysis in elasticity shows that the stresses are by no means uniform, but are concentrated near the hole.
Three branches of solid mechanics 固体力学的三个分枝 固体力学的三个分枝
• Mechanics of materials 材料力学, 材料力学, Structural Mechanics 结构力学 Elasticity 弹性力学
•
•
What does the Elasticity deal with? It deals with the stresses, deformations and displacements in elastic solids produced by external forces or changes in temperature. 研究弹性体由于外力和温度改变而引起的应力， 由于外力和温度改变而引起的应力 研究弹性体由于外力和温度改变而引起的应力， 形变和位移。 形变和位移。 It analyzes the stresses, deformations and displacements of structural elements within the elastic range and thereby to check the sufficiency of their strength, stiffness and stability. 分析结构的应力，形变和位移， 分析结构的应力，形变和位移，检查是否满足强 刚度和稳定性条件。 度，刚度和稳定性条件。
Body force Fig.
体力定义图。 体力定义图。
F=lim ∆Q/ ∆V
∆v 0
F--body force vector at p, The vector quantity F is the intensity of body force at P F-- P点的体力矢量 点的体力矢量 ∆V--an elementary volume of the body around point p 包含P点的小体积 ∆V--包含 点的小体积 包含 ∆Q--body force acting on ∆ V 作用在∆ 上的体力的合力 ∆Q--作用在∆ V上的体力的合力 作用在
1.2 some important concepts in theory of elasticity 1.2 弹性力学中的几个重要概念 A. External Forces 外力 B. Stress 应力 C. Deformation(Strain) 形变 应变 形变(应变 应变) D. Displacement 位移
• Mechanics of materials deals essentially with the stresses and displacements of structural element in the shape of a bar, straight or curved, which is subjected to tension, compression, shear, bending, or torsion. 材料力学研究受到拉、 材料力学研究受到拉、压、剪、弯或扭的直杆 或曲杆的应力和位移。 或曲杆的应力和位移。
• Elasticity also analyze a bar element thoroughly and precisely. • 弹性力学对杆件作更精确分析
2. methods of analysis:研究方法 研究方法
(1) Similarity--- 相同点 :
equilibrium aspects geometrical aspects physical aspects
• Elasticity deals with the stresses and displacements of the structural elements such as blocks, plates and shells, which are not in the form of a bar. • 弹性力学研究块体、板和壳体的应力和位移。 弹性力学研究块体、板和壳体的应力和位移。
Comparison among the three courses in solid mechanics 固体力学三门学科的比较
• Three branches have the same purpose and do differ from one another both in objects studied and the methods of analysis used. • 1. Objects studied 研究对象 2. Methods of analysis 研究方法
•The problem of bending of a straight beam under transverse loads. •It is assumed in mechanics of materials that a plane section of the beam remains plane after bending, which leads to the linear distribution of bending stresses. •No assumption, that a plane section of the beam remains plane after bending, is made in Elasticity.
Elasticity： ： no assumptions on the strain condition or the stress condition are made. The results obtained in elasticity are more accurate and may be used to check the approximate results obtained in Mechanics of materials.
1. objects studied:--研究对象 研究对象: 研究对象 (1) Similarity---------相同点 相同点
to deal with the elastic solids 都是研究弹性体
(2)objects studied--difference研究对象 不同点 研究对象--不同点 研究对象 Mechanics of materials : bar element 材料力学 单根杆件 Structural bar systems:-Mechanics : truss, rigid frame 杆件系统： 桁架，刚架。 结构力学 杆件系统： 桁架，刚架。 Elasticity: 1. plates and shells 板，壳 2.blocks: 块体 弹性力学 e.g. dams,foundations 坝，基础 3.analyze bar element precisely 对杆件作精确分析
A. external forces 外力
1. Body forces 2. Surface forces
体积力，体力 体积力， 表面力， 表面力，面力
1. Body forces
体力。 体力。
External forces or the loads,distributed over the volume of the body,are called body forces. 分布在物体体内的外力叫体力 E.g. gravitational forces, or inertia forces in the case of a body in motion. 例如： 重力， 例如： 重力， 惯性力
1.1 Contents of Theory of Elasticity 1.1 弹性力学的内容 • NAME Theory of elasticity is often called elasticity for short. It is a branch of solid mechanics. 弹性力学的理论简称为弹性理论或弹性力学. 弹性力学的理论简称为弹性理论或弹性力学. 它是固体力学的一个分枝
Body force components 体力分量 • F=X i+Y j+Z k=(X,Y,Z) • The projections of F on the x,y,and z axes are called the body force components at P. 体力在坐标轴上的投影叫体力分量。 体力在坐标轴上的投影叫体力分量。 • The body force components will be denoted by X,Y and Z 体力分量用X,Y,Z表示 体力分量用 表示
静力学方面 几何学方面 物理学方面
•equilibrium aspects --equilibrium of forces of an isolated body 静力学方面--脱离体力的平衡 静力学方面 脱离体力的平衡 • geometrical aspects --the relations between displacements and strains. 几何学方面--位移和应变的关系 几何学方面--位移和应变的关系 • physical aspects-- -the relations between stresses and strains 物理学方面--应力和应变的关系 物理学方面 应力和应变的关系