Unit 2 Stress and Strain应力和应变
1. Introduction to mechanics of materials材料力学的简介
Mechanics of material is a branch of applied mechanics that deals with the behaviour of solid bodies subjected to various types of loading. It is a field of study that is known by a variety of names,including “strength of materials”and “mechanics of deformable bodies .” The solid bodies considered in this book include axially-loaded bars , shafts ,beams ,and columns ,as well as structures that are asmblies of the components. Usually the objective of our analysis will be the determination of the stress,strains,anddeformations produced by the loads ;if the quantities can be found for all values of load up to the failure load ,then we will have obtained a complete picture of the mechanical behaviour of body. 材料力学是研究固体在各种载荷作用下的力学性能，是应用力学的一个分支。它是常被冠以像材料强度和变形体的力学等各种名称的一个研究领域。本文所讨论的固体包括受轴向载荷的杆件，轴，梁，柱以及这些结构的组成体。通常我们的分析目的是确定载荷失效之前在各种载荷值作用下构件所产生的应力，应变，和变形，如果这些参数值都能够成功获得
的话，那么我们就可以得到一个完整的力学性能图
Theoretical analys and experimental results have equally important roles in the study of mechanics of materials. 。On many occasions we will make logical derivations to obtain formulas and equations for predicting mechanical behaviour ,but at the same time we must recognize that the formulas cannot be ud in a realistic way unless certain properties of the material are known. The properties are available to us only after suitable experiments have been made in the laboratory. Also, many problems of importance in engineering cannot be handled efficiently by theoretical means , and experimental measurements become a practical necessity . The historical development of mechanics of materials is a fascinating blend of both theory and experiment , with experiments pointing the way to uful results in some instances and with theory doing so in others . Such famous men as Leonardo da Vinci (1452-1519)and Galileo Galilei (1564-1642) made experiments to determine the strength of wires ,bars ,and beams ,although they did not develop any adequate theories (by today’s standards)to explain their test results . By contrast ,the famous mathematician Leonhard Euler (1707-1783)de
veloped the mathematical theory of columns and calculated the critical load of a column in 1744 ,long before any exper imental evidence existed to show the signifi cance of his results. Thus ,Euler’s theoretical results remained unud for many years ,although today they form the basis of column theory . 理论分析和实验结论在材料力学的研究中出于同样重要的地位。在许多情况下为了预测机械性能我们使用论及推理来获得相关的方程式和公式，但是同时必须意识到除非材料 的某些性能已知，那么这些公式才能在实际情况下使用。这些性能只能在实验中通过合适的实验验证之后，对我们来讲才是可用的。而且，工程上许多重要的问题使用理论手段也是不能有效地解决，因此，实验验证就变得很重要。材料力学的发展历史是理论与实验极有趣的结合，在这些情况下市实验指明了得到有用结果的道路，在另一些情况下则是理论来做这些事。想家里略和达芬奇这样的名人都曾经通过做实验来测定金属丝，杆件，梁的强度， 尽管用今天的标注来看，他们没有发现足够的理论来解释他们的实验结论。相反，著名的数学家里昂哈德.欧拉在1744 年就提出了柱体的数学理论并计算出其极限载荷，而过了很久才有实验证明其结论的重要性。虽然，欧拉的结论构成了现在柱体理论的基础，在许多年里他的结论没有什么实用价值。
The importance of combining theoretical derivations with experimentally determined prop
erties of materials will be evident as we proceed with our study of the subject. In this ction we will begin by discussing some fundamental concepts ,such as stress and strain ,and then we will investigate the behaviour of simple sturctural elements subjected to tension ,compression ,and shear. 随着研究的不断深入，把理论推导和实验上的已确定的材料性质结合起来研究的重要性是显然的。.在这一章我们从讨论像应力，应变这样的基本概念开始着手，然后研究承受拉伸压缩和剪切的简单的结构的元件的特性 。
2. Stress
The concepts of stress and strain can be illustrated in an elementary way by considering the extension of a prismatic bar [e Fig .1.4(a)] . A prismatic bar is one that has constant cross ction throughout its length and a straight axis . In this illustration the bar is assumed to be loaded at its ends by axial forces P that produce a uniform stretching .or tension ,of the bar . By making an artificial cut (ction m m) through the bar at right angles to its axis ,we can isolate part of the bar as a free body [Fig. 1.4 (b)]. At the right-hand end the tensile fore P is applied ,and at the other end there are forces reprenting
the action of the removed portion of the bar upon the part that that remains The force will be continuously distributed over the cross ction ,analogous to the continuous distribution of hydros tatic pre ssure over a submerged surface . The intensity of force ,that is ,the per unit area ,is called the stress and is commonly denoted by the Greek letter Q. Assuming that the stress has a uniform distribution over the cross ction [e Fig .1.4(b)], we can readily e that its resultant is equal to the intensity Q times the cross-ctional area A of the bar . Furthermore ,from the equilibrium of the body shown in Fig .1.4(b),we can also e that this resultant must be equal in magnitude and opposite in direction to the force P. Hence ,we obtain as the equation for the uniform stress in a prismatic bar . This equation shows that stress has units of force divided by area –for example ,Newtons per square millimeter( N/mm) or pounds per square inch( psi). When the bar is being stretched by the forces P,as shown in the figure ,the resulting stress is a tensile stress;if the forces are reverd in direction ,causing the bar to be compresd ,they are called compressive stress . 应力和应变的概念可以通过考虑等截面杆受拉伸这种基本方式来进行阐述(见图1.4a)。.开朗的近义词所谓等截面杆就是在其整个长度上有相同的横截面和
柠檬水的功效与作用有一根直轴.在这个图例中假定杆件在端部受到一轴向力P，该力使杆件产生一个均匀的伸长或拉伸。人为的在杆件上与轴线成直角方向上取一段截面，我们把被隔离出来的杆件的一部分作为一个自由段[图张学良简介1.4b]。.在右端有拉力P任的多音字组词作用着，在另一段也有表示被移走的杆件部分对剩余杆件作用的力存在。这些里连续的分布在整个横街面上就如流体静压力连续分布在一个沉浸面上一样。力的强度，也即单位面积上的力被称为应力，通常用希腊字母 表示。假定应力在整个横截面上均匀分布(见图1.4b),那么合力应该为应力乘杆件截面的面积。而且从图1.4（b）中物体平衡，我们同样知道合理同外力P大小相等，反方向相反。.因此得到 作为等截面杆上均匀应力的计算公式。这个公式表明应力单位为力除以面积，例如牛顿每平方毫米或者是磅每平方英寸。当杆件如图所示受P拉伸，则产生的应力称为拉应力，如果作用在相反方向，使杆件被压缩则成为压应力。
A necessary condition for Eq.( 1.3) to be valid is that the stress Q must be uniform over the cross ction of the bar. This condition will be realized if the axial force P acts through the centroid of the cross ction ,as can be demonstrated by statics . When the load P does not act at the centroid ,bending of the bar will result ,and a more complicated analysis is necessary. Throughout this book ,however ,it is assumed that all axial forces a
re applied at the centroid of the cross ction unless specifically stated to the contrary Also ,unless stated otherwi ,it is generally assumed that the weigh of the object itlf is neglected ,as was done when discussing the bar in Fig .1.4 . 公式1.3可使用的一个必要条件是压力Q 在整个杆件的横街面上的分布式均匀的。若轴向力作用在杆件横街面的行心上，这一点可以用静力学来验证。若载荷P没有作用在行心，那么杆件将会发生弯曲，那么就必须进行更复杂的分析。然而，如果没有特殊说明，书中假定所有轴向力都作用在横截面的行心。同样除非特殊说明，物体的质量都是可以忽略的，就像我们在研究图孕期可以喝茶吗>显卡温度多少是正常的1.4中的杆件一样.
3.Strain
The total elongation of a bar carrying an axial force will be denoted by the Greek letter Q [e Fig . 1.4 ],and the elongation per unit length ,or strain ,is then determined by the equation where L is the total length of the bar . Note that the strain is a nondimensional quantity . It can be obtained accurately from Eq . (1.4) as long as the strain is uniform throughout the length of the bar . If the bar is in tedsion , the strain is a te
nsile strain , reprenting an elongation or a stretching of the material ; if the bar is in compression ,the strain is a comprjacent strain ,which means that adjacent cross ctions of the bar move clor to one another . 在轴向力作用下，杆件的总伸长可用希腊字母表示(见图一个员一个力4.1 a) ，那么单位长度的伸长或者说应变则可用公式 表示。这里L是指杆件的总长.注意应变是一个无刚量的量。只要整个杆件长度上应变是均匀，那么就可以正确的用公式1.4菜饭骨头汤得到应变的大小。一般说来，若杆件是拉伸的，那么为拉应变，代表材料被拉伸或延长；若杆件受 压缩，为压应变也就是杆件的相邻横截面彼此之间被拉的更近。
