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MEC NICAL METALLURGY SI Metric Edition George E. Dieter University of Maryland Adapted by David Bacon...

MEC NICAL METALLURGY SI Metric Edition George E. Dieter University of Maryland Adapted by David Bacon Professor of Materials Science University of Liverpool McGraw-Hili Book Company London' New Yark. St Louis' San Francisco' Auckland Bogota' Guatemala' Hamburg· Lisbon' Madrid· Mexico Montreal· New Delhi· Panama' Paris' San Juan' Sao Paulo Singapore' Sydney' Tokyo' Toronto McGraw-Hill Series in Materials Science and Engineering Editorial Board Michael B. Bever Stephen M. Copley M. E. Shank Charles A. Wert Garth L. Wilkes Brick, Pense, and Gordon: Structure and Properties of Engineering Materials Dieter: Engineering Design: A Materials and Processing Approach Dieter: Mechanical Metallurgy Drauglis, Gretz, and Jaffe: Molecular Processes on Solid Surfaces Flemings: Solidification Processing Fontana: Corrosion Engineering Gaskell: Introduction to Metallurgical Thermodynamics Guy: Introduction to Materials Science I, I Kehl: The Principles of Metallographic Laboratory Practice Leslie: The Physical Metallurgy of Steels Rhines: Phase Diagrams in Metallurgy: Their Development and Application Rozenfeld: Corrosion Inhibitors Shewmon: Transformations in Metals Smith: Principles of Materials Science and Engineering Smith: Structure and Properties of Engineering Alloys Vander Voort: Metallography: Principles and Practice Wert and Thomson: Physics of Solids ,1. , MECHANICAL METALLURGY SI Metric Edition Exclusive rights by McGraw-Hili Book Co - Singapore for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hili. 789KHL987654 Copyright © 1988 McGraw-Hill Book Company (UK) Limited Copyright © 1986, 1976, 1961 by McGraw-Hili Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of McGraw-Hili Book Company (UK) Limited. British Library Cataloguing in Publication Data Dieter, George E.. (George Ellwood), 1928- Mechanical metallurgy. - SI Metric ed. 1. Metals & alloys. Strength 1. Title 620.1 '63 Library of Congress Cataloging-in-Publication Data Dieter, George Ellwood. Mechanical metallurgy/George E. Dieter. - SI Metric ed./adapted by David Bacon. (McGraw-Hill series in materials science and engineering) Bibliography:p. includes indexes. ISBN 0-07-084187-X I. Strength of materials. 2. Physical metallurgy. 1. Bacon, D. J. 11. Title. 111. Series TA405.D53 1988 620.1'63 - dc 1988-10351 When ordering this title use ISBN 0-07-100406-8 Printed in Singapore , I.>. ~ I  ABOUT THE AUTHOR George E. Dieter is currently Dean of Engineering and Professor of Mechanical Engineering at the University of Maryland. The author received his B.S. Met.E. degree from Drexel University, and his D.Sc. degree from Carnegie- Mellon University. After a career in industy with DuPont Engineering Research Laboratory, he became Head of the Metallurgical Engineering Department at Drexel University, where he later became Dean of Engineering. Professor Dieter later joined the faculty of Carnegie-Mellon University, as Professor of Engineer- ing and Director of the Processing Research Institute. He moved to the University of Maryland four years later. A former member of the National Materials Advisory Board, Professor Dieter is a fellow of the American Society for Metals, and a member of AAAS, AIME, ASEE, NSPE, and SME. v  CONTENTS Preface to the Third Edition Xlll Preface to the Second Edition xv Preface to the First Edition XVll , List of Symbols XXI Part 1 Mechanical Fundamentals 1 Introduction 3 't...._. _,,_._,. A~~'-"~-' 1)' Scope of This Book l1.:ljStrength of Mate£ial,s-Basic Assumptions : vr; Elastic and Plastic Behavior LJ::4JAverage Stress and Strain rrS.-'Tensile Deformation , ,,---.--. ~".,.- --.." ",,' -- " of Ductile Metal ," - -, - ,,- ,-' /1-6 Ductile vs. Brittle Behavior "1-7 What Constitutes Failure? c··-· ;... I i 0._,----~- )-8 Concept of Stress and the Types of Stresses Lk9j Concept of Strain and the Types of Strain 'Ji~i61Units L.v.' ,. ,.,- of Stress and Other Quantities 2 Stress and Strain Relationships for Elastic Behavior 17 ,. _. -'... -. - , --' generalized angles r line tension of a dislocation y shear strain ~ volume strain or cubical dilatation; finite change 8 deformation or elongation; deflection; logarithmic decrement;),. Kronecker delta ;., f general symbol for strain; natural or true strain f significant, or effective, true strain i; true-strain rate  LIST OF SYMBOLS XXIII es minimum creep rate 1/ efficiency; coefficient of viscosity () Dorn time-temperature parameter K bulk modulus or volumetric modulus of elasticity A Lame's constant; interparticle spacing }L coefficient of friction v Poisson's ratio p density a normal stress; true stress ao yield stress or yield strength a6 yield stress in plane strain a significant, or effective, true stress aI' a 2, a 3 principal stresses a' stress deviator a" hydrostatic component of stress aa alternating, or variable, stress am average principal stress; mean stress ar range of stress au ultimate tensile strength aw working stress T shearing stress; relaxation time  PART MECHANICAL FUNDAMENTALS  CHAPTER ONE INTRODUCTION 1-1 SCOPE OF THIS BOOK Mechanical metallurgy is the area of metallurgy which is concerned primarily with the response of metals to forces or loads. The forces may arise from the use of the metal as a member or part in a structure or machine, in which case it is necessary to know something about the limiting values which can be withstood without failure. On the other hand, the objective may be to convert a cast ingot into a more useful shape, such as a flat plate, and here it is necessary to know the conditions of temperature and rate of loading which minimize the forces that are needed to do the job. Mechanical metallurgy is not a subject which can be neatly isolated and studied by itself. It is a combination of many disciplines and many approaches to the problem of understanding the response of materials to forces. On the one hand is the approach used in strength of materials and in the theories of elasticity and plasticity, where a metal is considered to be a homogeneous material whose mechanical behavior can be rather precisely described on the basis of only a very few material constants. This approach is the basis for the rational design of structural members and machine parts. The topics of strength of materials, elasticity, and plasticity are treated in Part One of this book from a more generalized point of view than is usually considered in a first course in strength of materials. The material in Chaps. 1 to 3 can be considered the mathematical framework on which much of the remainder of the book rests. For students of engineering who have had an advanced course in strength of materials or machine design, it probably will be possible to skim rapidly over these chapters. However, for most students of. metallurgy and for practicing engineers in industry, it is 3 4 MECHANICAL FUNDAMENTALS worth spending the time to become familiar with the mathematics presented in Part One. The theories of strength of materials, elasticity, and plasticity lose much of their power when the structure of the metal becomes an important consideration and it can no longer be considered a homogeneous medium. Examples of this are in the high-temperature behavior of metals, where the metallurgical structure may continuously change with time, or in the ductile-to-brittle transition, which occurs in carbon steel. The determination of the relationship between mechanical behav- ior and structure (as detected chiefly with microscopic and x-ray techniques) is the main responsibility of the mechanical metallurgist. When mechanical behavior is understood in terms of metallurgical structure, it is generally possible to improve the mechanical properties or at least to control them. Part Two of this book is concerned with the metallurgical fundamentals of the mechanical behavior of metals. Metallurgical students will find that some of the material in Part Two has been covered in a previous course in physical metallurgy, since mechanical metallurgy is part of the broader field of physical metallurgy. However, these subjects are considered in greater detail than is usually the case in a first course in physical metallurgy. In addition, certain topics which pertain more to physical metallurgy than mechanical metallurgy have been included in order to provide continuity and to assist nonmetallurgical students who may not have had a course in physical metallurgy. The last three chapters of Part Two are concerned primarily with atomistic concepts of the flow and fracture of metals. Many of the developments in these areas have been the result of the alliance of the solid-state physicist with the metallurgist. This has been an area of great progress. The introduction of transmission electron microscopy has provided an important experimental tool for verifying theory and guiding analysis. A body of basic dislocation theory is presented which is useful for understanding the mechanical behavior of crystalline solids. Basic data concerning the strength of metals and measurements for the routine control of mechanical properties are obtained from a relatively small number of standardized mechanical tests. Part Three, Applications to Materials Testing, considers each of the common mechanical tests,. not from the usual standpoint of testing techniques, but instead from the consideration of what these tests tell about the service performance of metals and how metallurgical variables affect the results of these tests. Much of the material in Parts One and Two has been utilized in Part Three. It is assumed that the reader either has completed a conventional course in materials testing or will be concurrently taking a labora- tory course in which familiarization with the testing techniques will be acquired. Part Four considers the metallurgical and mechanical factors involved in forming metals into useful shapes. Attempts have been made to present mathe- matical analyses of the principal metalworking processes, although in certF cases this has not been possible, either because of the considerable detail required or because the analysis is beyond the scope of this book. No attempt has been made to include the extensive specialized technology associated with each metal-  INTRODUCTION 5 working process, such as rolling or extrusion, although some effort has been made to give a general impression of the mechanical equipment required and to familiarize the reader with the specialized vocabulary of the metalworking field: Major emphasis has been placed on presenting a fairly simplified picture of the forces involved in each process and of how geometrical and metallurgical factors affect the forming loads and the success of the metalworking process. 1-2 STRENGTH OF MATERIALS-BASIC ASSUMPTIONS Strength of materials is the body of knowledge which deals with the relation between internal forces, deformation, and external loads. In the general method of analysis used in strength of materials the first step is to assume that the member is in equilibrium. The equations of static equilibrium are applied to the forces acting on some part of the body in order to obtain a relationship between the external forces acting on the member and the internal forces resisting the action of the external loads. Since the equations of equilibrium must be expressed in terms of forces acting external to the body, it is necessary to make the internal resisting forces into external forces. This is done by passing a plane through the body at the point of interest. The part of the body lying on one side of the cutting plane is removed and replaced by the forces it exerted on the cut section of the part of the body that remains. Since the forces acting on the "free body" hold it in equilibrium, the equations of equilibrium may be applied to the problem. The internal resisting forces are usually expressed by the stress! acting over a certain area, so that the internal force is the integral of the stress times the differential area over which it acts. In order to evaluate this integral, it is necessary to know the distribution of the stress over the area of the cutting plane. The stress distribution is arrived at by observing and measuring the strain distribution in the member, since stress cannot be physically measured. However, since stress is proportional to strain for the small deformations involved in most work, the determination of the strain distribution provides the stress distribution. The expression for the stress is then substituted into the equations of equilibrium, and they are solved for stress in terms of the loads and dimensions of the member. Important assumptions in strength of materials are that the body which is being analyzed is continuous, homogeneous, and isotropic. A continuous body is one which does not contain voids or empty spaces of any kind. A body is homogeneous if it has identical properties at all points. A body is considered to be isotropic with respect to some property when that property does not vary with direction or orientation. A property which varies with orientation with respect to some system of axes is said to be anisotropic. 1 For present purposes stress is defined as force per unit area. The companion term strain is defined as the change in length per unit length. More complete definitions will be given later.  j 1 j , 1 , 6 MECHANICAL FUNDAMENTALS While engineering materials such as steel, cast iron, and aluminum may appear to meet these conditions when viewed on a gross scale, it is readily apparent when they are viewed through a microscope that they are anything but homogeneous and isotropic. Most engineering metals are made up of more than one phase, with different mechanical properties, such that on a micro scale they are heterogeneous. Further, even a single-phase metal will usually exhibit chem- ical segregation, and therefore the properties will not be identical from point to point. Metals are made up of an aggregate of crystal grains having different properties in different crystallographic directions. The reason why the equations of strength of materials describe the behavior of real metals is that, in general, the crystal grains are so small that, for a specimen of any macroscopic volume, the materials are statistically homogeneous and isotropic. However, when metals are severely deformed in a particular direction, as in rolling or forging, the mechani- cal properties may be anisotropic on a macro scale. Other examples of anisotropic properties are fiber-reinforced composite materials and single crystals. Lack of continuity may be present in porous castings or powder metallurgy parts and, on an atomic level, at defects such as vacancies and dislocations. 1-3 ELASTIC AND PLASTIC BEHAVIOR Experience shows that all solid materials can be deformed when subjected to external load. It is further found that up to certain limiting loads a solid will recover its original dimensions when the load is removed. The recovery of the original dimensions of a deformed body when the load is removed is known as elastic behavior. The limiting load beyond which the material no longer behaves elastically is the elastic limit. If the elastic limit is exceeded, the body will experience a permanent set or deformation when the load is removed. A body which is permanently deformed is said to have undergone plastic deformation. For most materials, as long as the load does not exceed the elastic limit, the deformation is proportional to the load. This relationship is known as Hooke's law; it is more frequently stated as stress is proportional to strain. Hooke's law requires that the load-deformation relationship should be linear. However, it does not necessarily follow that all materials which behave elastically will have a linear stress-strain relationship. Rubber is an example of a material with a nonlinear stress-strain relationship that still satisfies the definition of an elastic material. Elastic deformations in metals are quite small and require very sensitive instruments for their measurement. Ultrasensitive instruments have shown that the elastic limits of metals are much lower than the values usually measured in engineering tests of materials. As the measuring devices become more sensitive, the elastic limit is decreased, so that for most metals there is only a rather narrow range of loads over which Hooke's law strictly applies. This is, however, primarilt of academic importance. Hooke's law remains a quite valid relationship for engineering design. i  INTRODUCTION 7 I - - - - L o + 8 --~ I----Lo--~ I----~p f--~P Figure I-I Cylindrical bar subjected to axial load. Figure 1-2 Free-body diagram for Fig. 1-1. 1-4 AVERAGE STRESS AND STRAIN As a starting point in the discussion of stress and strain, consider a uniform cylindrical bar which is subjected to an axial tensile load (Fig. 1-1). Assume that.two gage marks are put on the surface of the bar in its unstrained state and that La is the gage length between these marks. A load P is applied to one end of the bar, and the gage length undergoes a slight increase in length and decrease in diameter. The distance between the gage marks has increased by an amount 8, called the deformation. The average linear strain e is the ratio of the change in length to the original length. L-L a e= -- (1-1) Strain is a dimensionless quantity since both 8 and La are expressed in units of length. Figure 1-2 shows the free-body diagram for the cylindrical bar shown in Fig. 1-1. The external load P is balanced by the internal resisting force fa dA, where a is the stress normal to the cutting plane and A is the cross-sectional area of the bar. The equilibrium equation is P = jadA (1-2) If the stress is distributed uniformly over the area A, that is, if a is constant, Eq. (1-2) ~ecomes P = a j dA = aA P a =- (1-3) A In general, the stress will not be uniform over the area A, and therefore Eg. (1-3) represents an average stress. For the stress to be absolutely uniform, every longitudinal element in the bar would have to expe!'ience exactly the same strain, and the proportionality between stress and strain would have to be identical for each element. The inherent anisotropy between grains in a polycrystalline metal rules out the possibility of complete uniformity of stress over a body of macro- 8 MECHANICAL FUNDAMENTALS scopic size. The presence of more than one phase also gives rise to nonuniformity of stress on a microscopic scale. If the bar is not straight or not centrally loaded, the strains will be different for certain longitudinal elements and the stress will not be uniform. An extreme disruption in the uniformity of the stress pattern occurs when there is an abrupt change in cross section. This results in a stress raiser or stress concentration (see Sec. 2-15). Below the elastic limit Hooke's law can be considered valid, so that the average stress is proportional to the average strain, (J - = E = constant (1-4) e The constant E is the modulus of elasticity, or Young's modulus. 1-5 TENSILE DEFORMATION OF DUCTILE METAL The basic data on the mechanical properties of a ductile metal are obtained from a tension test, in which a suitably designed specimen is subjected to increasing axial load until it fractures. The load and elongation are measured at frequent intervals during the test and are expressed as average stress and strain according to the equations in the previous section. (More complete details on the tension test are given in Chap. 8.) The data obtained from the tension test are generally plotted as a stress-strain diagram. Figure 1-3 shows a typical stress-strain curve for a metal such as aluminum or copper. The initial linear portion of the curve OA is the elastic region within which Hooke's law is obeyed. Point A is the elastic limit, defined as the greatest stress that the metal can withstand without experiencing a permanent strain when the load is removed. The determination of the elastic limit is quite tedious, not at all routine, and dependent on the sensitivity of the strain-measur- ing instrument. For these reasons it is often replaced by the proportional limit, point A'. The proportional limit is the stress at which the stress-strain curve deviates from linearity. The slope of the stress-strain curve in this region is the modulus of elasticity. O"max Fracture oc Strain e Figure 1-3 Typical tension stress-strain curve. "-  INTRODUCTION 9 For engineering purposes the limit of usable elastic behavior is described by the yield strength, point B. The yield strength is defined as the stress which will produce a small amount of permanent deformation, generally equal to a strain of. 0.002. In Fig. 1-3 this permanent strain, or offset, is OC. Plastic deformation begins when the elastic limit is exceeded. As the plastic deformation of the specimen increases, the metal becomes stronger (strain hardening) so that the load required to extend the specimen increases with further straining. Eventually the load reaches a maximum value. The maximum load divided by the original area of the specimen is the ultimate tensile strength. For a ductile metal the diameter of the specimen begins to decrease rapidly beyond maximum load, so that the load required to continue deformation drops off until the specimen fractures. Since the average stress is based on the original area of the specimen, it also decreases from maximum load to fracture. 1-6 DUCTILE VS. BRITTLE BEHAVIOR The general behavior of materials under load can be classified as ductile or brittle depending upon whether or not the material exhibits the ability to undergo plastic deformation. Figure 1-3 illustrates the tension stress-strain curve of a ductile material. A completely brittle material would fracture almost at the elastic limit (Fig. 1-4a), while a brittle metal, such as white cast iron, shows some slight measure of plasticity before fracture (Fig. 1-4b). Adequate ductility is an im- portant engineering consideration, because it allows the material to redistribute localized stresses. When localized stresses at notches and other accidental stress concentrations do not have to be considered, it is possible to design for static situations on the basis of average stresses. However, with brittle materials, localized stresses continue to build up when there is no local yielding. Finally, a crack forms at one or more points of stress concentration, and it spreads rapidly over the section. Even if no stress concentrations are present in a brittle material, fracture will still occur suddenly because the yield stress and tensile strength are practically identical. It is important to note that brittleness is not an absolute property of a metal. A metal such as tungsten, which is brittle at room temperature, is ductile at an elevated temperature. A metal which is brittle in tension may be ductile under hydrostatic compression. Furthermore, a metal which is ductile in tension at room '"'"cu -L ( /) Figure 1-4 (a) Stress-strain curve for completely Strai n Strain brittle material (ideal behavior); (b) stress-strain (a) (b) curve for brittle metal with slight amount of ductility. 10 MECHANICAL FUNDAMENTALS temperature can become brittle in the presence of notches, low temperature, high rates of loading, or embrittling agents such as hydrogen. 1-7 WHAT CONSTITUTES FAILURE? Structural members and machine elements can fail to perform their intended functions in three general ways: 1. Excessive elastic deformation 2. Yielding, or excessive plastic deformation 3. Fracture ' An understanding of the common types of failure is important in good design because it is always necessary to relate the loads and dimensions of the member to some significant material parameter which limits the load-carrying capacity of the member. For different types of failure, different significant parameters will be important. Two general types of excessive elastic deformation may occur: (1) excessive deflection under condition of stable equilibrium, such as the deflection of beam under gradually applied loads; (2) sudden deflection, or buckling, under condi- tions of unstable equilibrium. Excessive elastic deformation of a machine part can mean failure of the machine just as much as if the part completely fractured. For example, a shaft which is too flexible can cause rapid wear of the bearing, or the excessive deflection of closely mating parts can result in interference and damage to the parts. The sudden buckling type of failure may occur in a slender column when the axial load exceeds the Euler critical load or when the external pressure acting against a thin-walled shell exceeds a critical value. Failures due to excessive elastic deformation are controlled by the modulus of elasticity, not by the strength of the material. Generally, little metallurgical control can be exercised over the elastic i modulus. The most effective way to increase the stiffness of a member is usually by changing its shape and increasing the dimensions of its cross section. Yielding, or excessive plastic deformation, occurs when the elastic limit of the metal has been exceeded. Yielding produces permanent change of shape, which may prevent the part from functioning properly any longer. In a ductile metal under conditions of static loading at room temperature yielding rarely results in fracture, because the metal strain hardens as it deforms, and an increased stress is required to produce further deformation. Failure by excessive plastic deformation is controlled by the yield strength of the metal for a uniaxial condition of loading. For more complex loading conditions the yield strength is still the significant parameter, but it must be used with a suitable failure criterion (Sec. 3-4). At temperatures significantly greater than room temperature metals no longer exhibit strain hardening. Instead, metals can continuously deform at constant stress in f time-dependent yielding known as creep. The failure criterion under creep condi-  INTRODUCTION 11 tions is complicated by the fact that stress is not proportional to strain and the further fact that the mechanical properties of the material may change apprecia- bly during service. This complex phenomenon will be considered in greater detail in Chap. 13. The formation of a crack which can result in complete disruption of continu- ity of the member constitutes fracture. A part made from a ductile metal which is loaded statically rarely fractures like a tensile specimen, because it will first fail by excessive plastic deformation. However, metals fail by fracture in three general ways: (1) sudden brittle fracture; (2) fatigue, or progressive fracture; (3) delayed fracture. In the previous section it was shown that a brittle material fractures under static loads with little outward evidence of yielding. A sudden brittle type of fracture can also occur in ordinarily ductile metals under certain conditions. Plain carbon structural steel is the most common example of a material with a ductile-to-brittle transition. A change from the ductile to the brittle type of fracture is promoted by a decrease in temperature, an increase in the rate of '. loading, and the presence of a complex state of stress due to a notch. This problem is considered in Chap. 14. A powerful and quite general method of analysis for brittle fracture problems is the technique called fracture mechanics. This is treated in detail in Chap. 11. Most fractures in machine parts are due to fatigue. Fatigue failures occur in parts which are subjected to alternating, or fluctuating, stresses. A minute crack starts at a localized spot, generally at a notch or stress concentration, and gradually spreads over the cross section until the member breaks. Fatigue failure occurs without any visible sign of yielding at nominal or average stresses that are well below the tensile strength of the metal. Fatigue failure is caused by a critical localized tensile stress which is very difficult to evaluate, and therefore design fOf fatigue failure is based primarily on empirical relationships using nominal stresses. Fatigue of metals is discussed in greater detail in Chap. 12. One common type of delayed fracture is stress-rupture failure, which occurs when a metal has been statically loaded at an elevated temperature for a long period of time. Depending upon the stress and the temperature there may be no yielding prior to fracture. A similar type of delayed fracture, in which there is no warning by yielding prior to failure, occurs at room temperature when steel is statically loaded in the presence of hydrogen. All engineering materials show a certain variability in mechanical properties, which in turn can be influenced by changes in heat treatment or fabrication. Further, uncertainties usually exist regarding the magnitude of the applied loads, and approximations are usually necessary in calculating stresses for all but the most simple member. Allowance must be made for the possibility of accidental loads of high magnitude. ~hus, in order to provide a margin of safety and to protect against failure from unpredictable causes, it is necessary that the allow- able stresses be smaller than the stresses which produce failure. The value of stress for a particular material used in a particular way which is considered to be a safe stress is usually called the working stress aw For static applications the working stress of ductile metals is usually based on the yield strength ao and for brittle 12 MECHANICAL FUNDAMENTALS metals on the ultimate tensile strength au. Values of working stress are established by local and federal agencies and by technical organizations such as the American Society of Mechanical Engineers (ASME). The working stress may be considered as either the yield strength or the tensile strength divided by a number called the factor of safety. ao au a w = N.o or aW = N (1-5) u where aw = working stress ao= yield strength au = tensile strength No = factor of safety based on yield strength Nu = factor of safety based on tensile strength The value assigned to the factor of safety depends on an estimate of all the factors discussed above. In addition, careful consideration should be given to the consequences, which would result from failure. If failure would result in loss of life, the factor of safety should be increased. The type of equipment will also influence the factor of safety. In Inilitary equipment, where light weight may be a prime consideration, the factor of safety may be lower than in commercial equipment. The factor of safety will also depend on the expected type of loading. For static loading, as in a building, the factor of safety would be lower than in a machine, which is subjected to vibration and fluctuating stresses. 1-8 CONCEPT OF STRESS AND THE TYPES OF STRESSES Stress is defined as force per unit area. In Sec. 1-4 the stress was considered to be uniformly distributed over the cross-sectional area of the member. However, this is not the general case. Figure I-Sa represents a body in equilibrium under the action of external forces PI' P2 , , Ps. There are two kinds of external forces which may act on a body: surface forces and body forces. Forces distributed over the surface of the body, such as hydrostatic pressure or the pressure exerted by one body on another, are called surface forces. Forces distributed over the volume of a body, such as gravitational forces, magnetic forces, or inertia forces (for a body in motion), are called body forces. The two most common types of body forces encountered in engineering practice are centrifugal forces due to high-speed rotation and forces due to temperature differential over the body (thermal stress). In general the force will not be uniformly distributed over any cross section of the body illustrated in Fig. I-Sa. To obtain the stress at some point 0 in a plane such as mm, part 1 of the body is removed and replaced by the system of external forces on mm which will retain each point in part 2 of the body in the same position as before the removal of part 1. This is the situation in Fig. 1-~b. We then take an area ~A surrounding the point 0 and note that a force ~P a~ts  INTRODUCTION 13 y CD n m CD P, m (a) (b) Figure 1-5 (a) Body in equilibrium under action of external forces PI"'" Ps; (b) forces acting on parts. on this area. If the area ~A is continuously reduced to zero, the limiting value of the ratio ~ P/ ~A is the stress at the point 0 on plane mm of body 2. ~p lim = (J (1-6) ~A->O ~A The stress will be in the direction of the resultant force P and will generally be inclined at an angle to ~A. The same stress at point 0 in plane mm would be obtained if the free body were constructed by removing part 2 of the solid body. However, the stress will be different on any other plane passing through point 0, such as the plane nn. It is inconvenient to use a stress which is inclined at some arbitrary angle to the area over which it acts. The total stress can be resolved into two components, a normal stress (J perpendicular to ~A, and a shearing stress (or shear stress) 7" lying in the plane mm of the area. To illustrate this point, consider Fig. 1-6. The force P makes an angle 0 with the normal z to the plane of the area A. Also, the plane containing the normal and P intersects the plane A along a dashed line that...... ~ ,... ,..."" \ P \ " z \ -- ~8 Figure 1-6 Resolution of total stress into its compo- nents. 14 MECHANICAL FUNDAMENTALS makes an angle 3h g ~ 1.8 U U ",0 "'~ \. ~ 1.4 +- / Vl 1.0 I I a 0.1 0.2 0.3 0.4 0.5 rlh (d) :.c3.0 l..- E 2.8.. U.E c: 2.6 ',1 o \. ~2.4 l..- +- r~ ~ 2.2 u c: -\--'d---$=~ -+- 82.0 :'" I..- 1. 8 Ui 1.6 a 0.10 0.20 0.30 aid. (e) 3.4 ~ ~ 0 3.0 = 2.00 o +- I..- U o de d ~ c: 26. r.,g o \#v ~ = '.20 ;: 2.2 -- ~ c - d -0 V '"c: U 0 Mr MT 8 1.8..... ~ d,=!.09 "" '"'" ~ 1.4 't-"..... ""1>"- I Vl.l.. 1.0 I o 0.04 0.08 0.12 rid (f) Figure 2-21 (Continued) ,, I I I  STRESS AND STRAIN RELATIONSHIPS FOR ELASTIC BEHAVIOR 65 Node 2 ,..--.,-_ _--l~-_ , Node 1 ? + "---....----~--..J _ _ x -I I- u, -I (a) u2 I- Node 3 Node 1'....... Element 1 Element 2... 1\\, Node 2 (b) Figure 2-22 (a) Simple rectangular element to illustrate finite element analysis; (b) two elements joined to model a structure. will not develop the full theoretical stress-concentration factor. However, redistri- bution of stress will not occur to any extent in a brittle material, and therefore a stress concentration of close to the theoretical value will result. Although stress raisers are not usually dangerous in ductile materials subjected to static loads, appreciable stress-concentration effects will occur in ductile materials under fatigue conditions of alternating stresses. Stress raisers are very important in the fatigue failure of metals and will be discussed further in Chap. 12. 2-16 FINITE ELEMENT METHOD The finite element method (FEM) is a very powerful technique for determining stresses and deflections in structures too complex to analyze by strictly analytical methods. With this method the structure is divided into a network of small elements connected to each other at node points. Finite element analysis grew out of matrix methods for the analysis of structures when the widespread availability of the digital computer made it possible to solve systems of hundreds of simultaneous equations. More recent advances in computer graphics and availa- bility of powerful computer work stations have given even greater emphasis to the spread of finite element methods throughout engineering practice. 1 A very simplified concept of the finite element method is given in Fig. 2-22. A simple one-dimensional two-node element is shown in Fig. 2-22a. For this element, each node has one degree of freedom as it is displaced by U 1 and u 2 The 1 O. Zienkiewicz, "The Finite Element Method," 3d ed., McGraw-Hill, New York, 1977; L J. Segerlind, "Applied Finite Element Anal:, sis," John Wiley & Sons, New York, 1976; K. H. Heubner, and E. A. Thornton, "The Finite Element Method for Engineers," John Wiley & Sons, New York, 1982. 66 MECHANICAL FUNDAMENTALS equations relating the forces applied to the nodes to their displacements are PI = kUu 1 + k 12 u 2 (2-106) The stiffness coefficients k ij are calculated by the computer program based on the elastic properties of the material and the geometry of the finite element. The stiffness equations above are manipulated in the computer in matrix form. (2-107) When a second element is added to the first, Fig. 2-22b, a new set of matrix equations is generated. (2-108) When these two elements are combined into a structure we can use the principle of superposition to arrive at the stiffness for the two-element structure. (2-109) Finite element analysis was originally developed for two-dimensional (plane- stress) situations. A three-dimensional structure causes orders of magnitude increase in the number of simultaneous equations; but by using higher order elements and faster computers, these problems are being handled by the FEM. Figure 2-23 shows a few of the elements available for FEM analysis. Figure 2-23a is the basic triangular element. It is the simplest two-dimensional element, and it is also the element most often used. An assemblage of triangles can always represent a two-dimensional domain of any shape. The six-node triangle (b) increases the degrees of freedom available in modeling. The quadrilateral element (c) is a combination of two basic triangles. Its use reduces the number of elements necessary to model some situations. Elements (d) and (e) are three- dimensional but require only two independent variables for their description. These elements are used for problems that possess axial symmetry in cylindrical coordinates Figure 2-23d is a one-dimensional ring element and (e) is a two- dimensional triangular element. Three-dimensional FEM models are best con- structed from isoparametric elements with curved sides. Figure 2-23/ is an isoparametric triangle; (g) is a tetrahedron; and (h) is a hexahedron. These elements are most useful when it is desirable to approximate curved boundaries with a minimum number of elements. A finite element solution involves calculating the stiffness matrices for every element in the structure. These elements are then assembled into an overall.~ , Ii.~ ,~.1  STRESS AND STRAIN RELATIONSHIPS FOR ELASTIC BEHAVIOR 67 (a) (d) (I) (b) (9) ------ ,.......3. (e) (c) (h) Figure 2-23 Some common elements used in FEM analysis. stiffness matrix [K] for the complete structure. {P} = [K]{u} (2-110) The force matrix is known because it consists of numerical values of loads and reactions computed prior to the start of the finite element analysis. The displace- ments {u} are the unknowns and they are solved for in Eq. (2-110) to give the displacements of all the nodes. When this is multiplied by the matrix of coordi- nate positions of the nodes [B] and the matrix of elastic constants [D] the stress is known at every nodal point. {a} = [D][B]{u} (2-111 ) A cumbersome part of the finite element solution is the preparation of the input data. The topology of the element mesh must be described in the computer program with the node numbers and the coordinates of the node points, along with the element numbers and the node numbers associated with each element. Tabulating all of this information is an extremely tedious bookkeeping task which IS very error prone for a structure containing hundreds of nodes. Fortunately, modern technology has eliminated these problems and in the process has greatly expanded the utilization of finite element methods. Preprocessors allow the finite element mesh to be positioned on a drawing of the structure and the nodal Coordinates and element connectivity to be automatically input. Postprocessing routines display the output of the finite element analysis in graphic form, allowing the user to quickly evaluate the information instead of wading through reams of numerical printouts.  I I" j 68 MECHANICAL FUNDAMENTALS , ~, BIBLIOGRAPHY Jaeger, J. c.: "Elasticity, Fracture, and Flow," 3d ed., Methuen & Co., Ltd., London, 1971. Love, A. E. H.: "A Treatise on the Mathematical Theory of Elasticity," 4th ed., Dover Publications, Inc., New York, 1949. Reid, C. N.: "Deformation Geometry for Materials Scientists," Pergamon Press, New York, 1973. Timoshenko, S. P., and J. N. Goodier: "Theory of Elasticity," 3d ed., McGraw-Hill Book Company, New York, 1961. Wang, C. T.: "Applied Elasticity," McGraw-Hill Book Company, New York, 1953. Urgarl, A. c., and S. K. Fenster: "Advanced Strength of Materials and Applied Elasticity," Elsevier, 1975. , I I I , ' i, :~ ,I , ! I -  CHAPTER THREE ELEMENTS OF THE THEORY OF PLASTICITY 3-1 INTRODUCTION The theory of plasticity deals with the behavior of materials at strains where Hooke's law is no longer valid. A number of aspects of plastic deformation make the mathematical formulation of a theory of plasticity more difficult than the description of the behavior of an elastic solid. For example, plastic def0rmation is not a reversible process like elastic deformation. Elastic deformation depends only on the initial and final states of stress and strain, while the plastic strain depends on the loading path by which the final state is achieved. Moreover, in plastic deformation there is no easily measured constant relating stress to strain as with Young's modulus for elastic deformation. The phenomenon of strain harden- ing is difficult to accommodate within the theory of plasticity without introducing.considerable mathematical complexity. Also, several aspects of real material.behavior, such as plastic anisotropy, elastic hysteresis, and the Bauschinger effect (see Sec. 3-2) cannot be treated easily by plasticity theory. Nevertheless, the , theory of plasticity has been one of the most active areas of continuum mecha- nics, and considerable progress has been made in developing a theory which can solve important engineering problems. The fheory of plasticity is concerned with a number of different types of problems. From the viewpoint of design, plasticity is concerned with predicting the maximum load which can be applied to a body without causing excessive yielding. The yield criterion must be expressed in terms of stress in such a way that it is valid for all states of stress. The designer is also concerned with plastic deformation in problems where the body is purposely stressed beyond the yield stress into the plastic region. For example, plasticity must be considered in designing for processes such as autofrettage, shrink fitting, and the overspeeding 69  70 MECHANICAL FUNDAMENTALS A A ~_ I ---I + ~ I I t 0"0 I I I t 0"0 I I I ~ ~-j (a 1 (b) k) Figure 3-1 Typical true stress-strain curves for a ductile metal. of rotor disks. The consideration of small plastic strains allows economies in building construction through the use of the theory of limit design. The analysis of large plastic strains is required in the mathematical treatment of the plastic forming of metals. This aspect of plasticity will be considered in Part Four. It is very difficult to describe, in a rigorous analytical way, the behavior of a metal under these conditions. Therefore, certain simplifying as-. sumptions are usually necessary to obtain a tractable mathematical solution. Another aspect of plasticity is concerned with acquiring a better understand- ing of the mechanism of the plastic deformation of metals. Interest in this field is i ; centered on the imperfections in crystalline solids. The effect of metallurgical variables, crystal structure, and lattice imperfections on the deformation behavior are of chief concern. This aspect of plasticity is considered in Part Two. 3-2 THE FLOW CURVE The stress-strain curve obtained by uniaxial loading, as in the ordinary tension test, is of fundamental interest in plasticity when the curve is plotted in terms of true stress (J and true strain f. True stress and true strain are discussed in the next section. The purpose of this sectiorr is to describe typical stress-strain curves for real metals and to compare them with the theoretical flow curves for ideal materials. The true stress-strain curve for a typical ductile metal, such as aluminum, is illustrated in Fig. 3-1a. Hooke's law is followed up to some yield stress (Jo. (The value of (Jo will depend upon the accuracy with which strain is measured.) Beyond (Jo, the metal deforms plastically. Most metals strain-harden in this region, so that increases in strain require higher values of stress than the initial yield stress (Jo' However, unlike the situation in the elastic region, the stress and strain are not related by any simple constant of proportionality. If the metal is strained to point 1 See Chap. 8 for a more complete discussion of the mathematics of the true stress-strain curve. j ; , 1 1 ,  ELEMENTS OF THE THEORY OF PLASTICITY 71 A, when the load is released the total strain will immediately decrease from £1 to £2 by an amount a/E. The strain decrease £1 - £2 is the recoverable elastic strain. However, the strain remaining is not all permanent plastic strain. Depending upon the metal and the temperature, a small amount of the plastic strain £2 - £3 will disappear with time. This is known as anelastic behavior. Generally the anelastic strain is neglected in mathematical theories of plasticity. Usually the stress-strain curve on unloading from a plastic strain will not be exactly linear and parallel to the elastic portion of the curve (Fig. 3-1b). Moreover, on reloading the curve will generally bend over as the stress ap- proaches the original value of stress from which it was unloaded. With a little additional plastic strain the stress-strain curve becomes a continuation of what it would have been had no unloading taken place. The hysteresis behavior resulting from unloading and loading from a plastic strain is generally neglected in plasticity theories. If a specimen is deformed plastically beyond the yield stress in one direction, e.g., in tension, and then after unloading to zero stress it is reloaded in the opposite direction, e.g., in compression, it is found that the yield stress on reloading is less than the original yield stress. Referring to Fig. 3-1c, a b < 0a' This dependence of the yield stress on loading path and direction is called the Bauschinger effect. The Bauschinger effect is commonly ignored in plasticity theory, and it is usual to assume that the yield stress in tension and compression are the same. A true stress-strain curve is frequently called a flow curve because it gives the stress required to cause the metal to flow plastically to any given strain. Many attempts have been made to fit mathematical equations to this curve. The most common is a power expression of the form (3-1) where K is the stress at £ = 1.0 and n, the strain-hardening coefficient, is the slope of a log-log plot of Eq. (3-1). This equation can be valid only from the beginning of plastic flow to the maximum load at which the specimen begins to neck down. Even the simple mathematical expression for the flow curve that is given by Eq. (3-1) can result in considerable mathematical complexity when it is used with the equations of the theqry of plasticity. Therefore, in this field it is common practice to devise idealized flow curves which simplify the mathematics without deviating too far from physical reality. Figure 3-2a shows the flow curve for a rigid, perfectly plastic material. For this idealized material, a tensile specimen is completely rigid (zero elastic strain) until the axial stress equals 00' whereupon the material flows plastically at a constant flow stress (zero strain hardening). This type of behavior is approached by a ductile metal which is in a highly cold worked condition. Figure 3-2b illustrates the flow curve for a perfectly plastic material with an elastic region. This behavior is approached by a material such as plain carbon steel which has a pronounced yield-point elongation (see Sec. 6-5). A more realistic approach is to approximate the flow curve by two straight lines  72 MECHANICAL FUNDAMENTALS (To l a) ( b) (c ) Figure 3-2 Idealized flow curves. (a) Rigid ideal plastic material; (b) ideal plastic material with elastic region; (c) piecewise linear (strain-hardening) material. corresponding to the elastic and plastic regions (Fig. 3-2c). This type of curve results in somewhat more complicated mathematics. 3-3 TRUE STRESS AND TRUE STRAIN The engineering stress-strain curve does not give a true indication of the deforma- ,I tion characteristics of a material because it is based entirely on the original dimensions of the specimen, and these dimensions change continuously during the test. Also in metalworking processes, such as wiredrawing, the workpiece undergoes appreciable change in cross-sectional area. Thus, measures of stress , ,, , and strain which are based on the instantaneous dimensions are needed. Since , dimensional changes are small in elastic deformation, it was not necessary to ,, make this distinction in the previous chapter. Equation (1-1) describes the conventional concept of unit linear strain, I, namely, the change in length referred to the original unit length. e = tiL = 1 fL dL Lo Lo Lo This definition of strain is satisfactory for elastic strains where tiL is very small. However, in plastic deformation the strains are frequently large, and during the 1 extension the gage length changes considerably. Ludwik first proposed the definition of true strain, or natural strain, E, which obviates this difficulty. In this definition of strain the change in length is referred ·to the instantaneous gage length, rather than to the original gage length. "\' L 1 - Lo L2 - L1 L3 - L2 E=1..J + + +... (3-2) Lo L1 L2 i ! i [ or E = fL dL = In _L_ (3-3) i Lo L Lo I, i ,I 1 P. Ludwik, "Elemente der technologischen Mechanik," Springer-Verlag OHG, Berlin, 1909. I  ELEMENTS OF THE THEORY OF PLASTICITY 73 l The relationship between true strain and conventional linear strain follows from Eq. (1-1). L - La LDoL e=--------1 La La La L e+ 1 =- La L e=ln =In{e+l) (3-4) La Values of true strain and conventional linear strain are given for comparison: True strain E 0.01 0.10 0.20 0.50 1.0 4.0. Conventional strain e 0.01 0.105 0.22 0.65 1.72 53.6 Thus, the two measures of strain give nearly identical results up to a strain of 0.1. The advantage of using true strain should be apparent from the following example: Consider a uniform cylinder which is extended to twice its original length. The linear strain is then e = (2L a - La)!La = 1.0, or a strain of 100 percent. To achieve the same amount of negative linear strain in compression, the cylinder would have to be squeezed to zero thickness. However, intuitively we should expect that the strain produced in compressing a cylinder to half its original length would be the same as, although opposite in sign to, the strain produced by extending the cylinder to twice its length. If true strain is used, equivalence is obtained for the two cases. For extension to twice the original length, e = In (2L a/ La) = In 2. For compression to half the original length, e = In[(L a/2)/L al = In ~ = -ln2. Another advantage of working with true strain is that the total true strain is equal to the sum of the incremental true strains. This can be seen from the following example. Consider a rod initially 50 mm long that is elongated in three increments, each increment being a conventional strain of e = 0.1. Increment Length of rod o 50 1 55 e O_ 1 = 5/50 = 0.1 2 60.5 el _ 2 = 5.5/55 = 0.1 3 66.5 e 2_ 3 = 6.05/60.5 = 0.1 1The reader is warned against confusion in notation for various measures of strain. We in general shall use E for strain. The true strain will be implied unless otherwise specified. For elastic strains (E < 0.01) the numerical values of E and e are identical and on occasion we may use e for linear strain, especially where we want to designate a small elastic strain. In other texts true strain is SOmetimes denoted by 8 or f.  11 ,, 74 MECHANICAL FUNDAMENTALS We note that the total conventional strain eO~3 = 16.55/50 = 0.331 is not equal to eO_ 1 + e 1 _ 2 + e 2 _ 3 However, if we use true strain, the sum of the increments equals the total strain. 55 60.5 66.55 66.55 £0-1 + £1_2 + £2_3 = In 50 + In 55 + In 60.5 = In 50 = £0~3 = 0.286 One of the basic characteristics of plastic deformation is that a metal is essentially incompressible. The density changes measured on metals after large plastic strain are less than 0.1 percent. Therefore, as a good engineering ap- proximation we can consider that the volume of a solid remains constant during plastic deformation. In Sec. 2-8 we determined the volume strain by considering a cube of initial volume dx dy dz which when deformed had a volume dx (1 + ex) dy (1 + ey) dz (1 + e z). The volume strain ~ is given by ~V (1 + eJ(1 + ey )(1 + eJ dxdydz - dxdydz ~= - V dxdydz or ~ = (1 + eJ (1 + ey) (1 + ez) - 1 When we previously determined ~ for small elastic strains, it was permissible to neglect products of strains compared to the strain itself, but this is no longer possible when the larger plastic strains are being considered. Since the volume , , change is zero for plastic deformation, II, "'- " ~ + 1 = 0 + 1 = (1 + ex )(1 + ey )(1 + ez ) or Inl = 0 = In(1 + eJ + In (1 + ey ) + In(1 + ez ) (3-5) Equation (3-5) represents the first invariant of the strain tensor when strain is expressed as true strain. It is a very useful relationship in plasticity problems. Note particularly that Eq. (3-5) is not valid for elastic strains since there is an appreciable volume change relative to the magnitude of the elastic strains. Thus, if we add up the three equations for Hooke's law (2-64) 1 - 2p ~ = ex + ey + ez = E ( lJx + lJy + lJz) we see that ~ can be zero only if p = i. This result can be interpreted that Poisson's ratio is equal to i for a plastic material for which ~ = O. Because of constancy of volume AoL o = AL, and Eq. (3-3) can be written in terms of either length or area. L Ao £=In-=ln- , (3-6) Lo A  ELEMENTS OF THE THEORY OF PLASTICITY 75 True stress is the load at any instant divided by the cross-sectional area over which it acts. The engineering stress, or conventional stress, is the load divided by the original area. In considering elastic behavior it was not necessary.to make this. distinction, but in certain problems in plasticity, particularly when dealing with the mathematics of the tension test (Chap. 8), it is important to distinguish between these two definitions of stress. True stress will be denoted by the familiar symbol a, while engineering stress will be denoted by s. p True stress a= - (3-7) A p Engineering stress s= (3-8) Ao The true stress may be determined from the engineering stress as follows: p a - --- A But, by the constancy-of-volume relationships =e+l p a= (e+l)=s(e+l) (3-9) Ao Example A tensile specimen with a 12 mm initial diameter and 50 mm gage length reaches maximum load at 90 kN and fractures at 70 kN. The minimum diameter at fracture is 10 mm. Determine the engineering stress at maximum load (the ultimate tensile strength) and the true fracture stress. Engineering stress P max 90 X 10 3 90 X 10 3 at maximum load - A max - n(12 X 10- 3)2/4 - 113 x 10-6 = 796 MPa P 70 X 10 3 70 X 10 3 True fracture stress = Af = n ( 10 x 10- 3)2/4 - 78 X 10- 6 = 891 MPa f Determine the true strain at fracture A 12 2 8f = In A; = In 10 = 2 In 1.2 = 2(0.182) = 0.365 What is the engineering strain at fracture? 8 = In (1 + e); exp (8) = (1 + e); exp (0.365) = 1 + ef 8f = 1.44 - 1.00 = 0.44  76 MECHANICAL FUNDAMENTALS 3.4 YIELDING CRITERIA FOR DUCTILE METALS The problem of deducing mathematical relationships for predicting the conditions at which plastic yielding begins when a material is subjected to any possible combination of stresses is an important consideration in the field of plasticity. In uniaxial loading, as in a tension test, macroscopic plastic flow begins at the yield stress ao' It is expected that yielding under a situation of combined stresses can be related to some particular combination of principal stresses. There is at present no theoretical way of calculating the relationship between the stress components to correlate yielding for a three-dimensional state of stress with yielding in the uniaxial tension test. The yielding criteria are essentially empirical relationships. However, a yield criterion must be consistent with a number of experimental observations, the chief of which is that pure hydrostatic pressure does not cause yielding in a continuous 1 solid. As a result of this, the hydrostatic component of a complex state of stress does not influence the stress at which yielding occurs. Therefore, we look for the stress deviator to be involved with yielding. Moreover, for an isotropic material, the yield criterion must be independent of the choice of axes, i.e., it must be an invariant function. These considerations lead to the conclusion that the yield 'I criteria must be some function of the invariants of the stress deviator. At present there are two generally accepted criteria for predicting the onset of yielding in :I ductile metals. ,,! "'i Von Mises' or Distortion-Energy Criterion Von Mises (1913) proposed that yielding would occur when the second invariant of the stress deviator J2 exceeded some critical value. (3-10) where J 2 = i[(a 1 - ( 2 )2 + (a2 - ( 3 )2 + (a 3 - ( 1 )2]. To evaluate the constant k and relate it to yielding in the tension test, we realize that at yielding in uniaxial tension a 1 = ao, a2 = a3 = 0 al + al = 6k 2 (3-11) Substituting Eq. (3-11) in Eq. (3-10) results in the usual form of the von Mises' yield criterion (3-12) A significant influence of hydrostatic or mean stress of modest values on yielding has been 1 I observed in glassy polymers such as PMMA. S. S. Stemstein and L. Ongchin, Polyrn, Prepr. Arn. I Chern. Soc. Diu. Polyrn, Chern., September 1969. ! i, , ,,  ELEMENTS OF THE THEORY OF PLASTICITY 77 or from Eq. (2-61) 00 = ~ [( Ox - Oy)2 + (Oy - oJ 2 + (oz - oJ 2 + 6( 1';y + 1'y~ + 1';z) f/2 (3-13) Equation (3-12) or (3-13) predicts that yielding will occur when the differences of stresses on the right side of the equation exceed the yield stress in uniaxial tension °0' Example Stress analysis of a spacecraft structural member gives the state of stress shown below. If the part is made from 7075-T6 aluminum alloy with 00 = 500 MPa, will it exhibit yielding? If not, what is the safety factor? Oz = 50 MPa ~f--" 0y = 100 MPa , L __-.J~'T~XY~ 30 MPa Ox = 200 MPa , From Eq. (3-13) 1 2 00 = fi [(200 - 100)2 + (100 - ( - 50))2 + (- 50 - 200)2+ 6(30)2r/ 1 316.859 °0 = fi (100,400)1/2 = fi = 224 MPa Since the value of 00 calculated from the yield criterion is less than the yield strength of the aluminum alloy, yielding will not occur. The safety factor is 500/224 = 2.2. To identify the constant k in Eq. (3-10), consider the state of stress in pure shear, as is produced in a torsion test. 01 = - 03 = l' 02 = 0 at yielding of + of + 4of = 6k 2... 01 = k so that k represents the yield stress in pure shear (torsion). Therefore, the von Mises' criterion predicts that the yield stress in torsion will be less than in uniaxial tension according to 1 k = 13 °0 = 0.57700 (3-14) To summarize, note that the von Mises' yield criterion implies that yielding is not dependent on any particular normal stress or shear stress, but instead,  78 MECHANICAL FUNDAMENTALS yielding depends on a function of all three values of principal shearing stress. Since the yield criterion is based on differences of normal stresses, a 1 - a2 , etc., the criterion is independent of the component of hydrostatic stress. Since the von Mises' yield criterion involves squared terms, the result is independent of the sign of the individual stresses. This is an important advantage since it is not necessary to know which are the largest and smallest principal stresses in order to use this. yield criterion. Von Mises originally proposed this criterion because of its mathematical simplicity. Subsequently, other workers have attempted to give it physical mean- ing. Hencky (1924) showed that Eq. (3-12) was equivalent to assuming that yielding occurs when the distortion energy reaches a critical value. The distortion energy is that part of the total strain energy per unit volume that is involved in change of shape as opposed to a change in volume. Example The fact that the total strain energy can be split into a term. depending on change of volume and a term depending on distortion can be seen by expressing Eq. (2-84) in terms of principal stresses. Vo = 2~ [af + af + al- 2,,(a1 a2 + a2a3 + a1a3)] (3-15) or expressing in terms of the invariants of the stress tensor 1 2 Vo = 2 [11 - 212(1 + II)] (3-16) E. ,'I' This equation is more meaningful if we express it in terms of the bulk , t modulus (volume change) and the shear modulus (distortion). From Sec. 2-11, " I, 9GK 3K - 2G T E= ,,= ---- , ,j. 3K +G 6K + 2G ,:1 I, , Substituting intoEq. (3-16) (3-17). Equation (3-17) is important because it shows that the total strain energy can be split into a term depending on change of volume and a term depending on distortion. 1 (VO)distortion = 6G (af + af + al - a 1a2 - a2a3 - a 1a3) or.. 1 [ 2 2 2] (VO)distortion = 12G (a 1 - a2 ) + (a 2 - a3) + (a 3 - a 1) (3-18) For a uniaxial state of stress, a 1 = ao, a2 = a3 = 0 j ,, j 1 j , (VO)distortion = 12G 2ai  ELEMENTS OF THE THEORY OF PLASTICITY 79 or (3-19) Another physical interpretation given to the von Mises' yield criterion is that it represents the critical value of the octahedral shear stress (see Sec. 3-9). This is the shear stress on the octahedral planes which make equal angles with the principal axes. Still another interpretation is that it represents the mean square of the shear stress averaged over all orientations in the solid. 1. Maximum-Shear-Stress or Tresca Criterion This yield criterion assumes that yielding occurs when the maximum shear stress reaches the value of the shear stress in the uniaxial-tension test. From Eq. (2-21), the maximum shear stress is given by (3-20) where °1 is the algebraically largest and 03 is the algebraically smallest principal stress. For uniaxial tension, 01 = 00' 02 = 03 = 0, and the shearing yield stress 'To is equal to °0/2. Substituting in Eq. (3-20), = 'To = 2 2 Therefore, the maximum-shear-stress criterion is given by (3-21) For a state of pure shear, 01 = - 03 = k, 02 = 0, the maximum-shear-stress criterion predicts that yielding will occur when 01 - 03 = 2k = 00 00 or k =- 2 so that the maximum-shear-stress criterion may be written o1 - 03 = 0'1 - 0'3 = 2k -= 1', l/f'; (3-22) ~; We note that the maximum-shear-stress criterion is less complicated mathe- matically than the von Mises' criterion, and for this reason it is often used in. engineering design. However, the maximum-shear criterion does not take into consideration the intermediate principal stress. It suffers from the major difficulty that it is necessary to know in advance which are the maximum and minimum principal stresses. Moreover, the general form of the maximum-shear-stress cri- terion, Eq. (3-23), is far more complicated than the von Mises' criterion, Eq. 1 See G. Sines, "Elasticity and Strength," pp. 54-56, Allyn and Bacon, Inc., Boston, 1969.  80 MECHANICAL FUNDAMENTALS (3-10), and for this reason the von Mises' criterion is preferred in most theoretical work. (3-23 ) Example Use the maximum-shear-stress criterion to establish whether yield- ing will occur for the stress state shown in the previous example. ax - az aD 2 2 200 - ( - 50) = aD aD = 250 MPa Again, the calculated value of aD is less than the yield strength of the material. 3-5 COMBINED STRESS TESTS The conditions for yielding under states of stress other than uniaxial and torsion :," loading can be studied conveniently with thin-wall tubes. Axial stress can be combined with torsion to produce various combinations of shear stress to normal stress intermediate between the values obtained separately in tension and, torsion. Alternatively, a hydrostatic pressure may be introduced to produce a circumferen- 1 tial hoop stress in the tube. ,ii For the stresses shown in Fig. 3-3, from Eq. (2-9) the principal stresses are 'I " 2 1/2 ax ax 2 a 1 -- + + Txy 2 4 I, a2 = 0 (3-24) ,I I 2 1/2 ax ax ":i, , 2 a3 = + Txy ," 2 4 Therefore, the maximum-shear-stress criterion of yielding is given by , a 2 T 2 , " x +4 xy = 1 (3-25) aD aD - and the distortion-energy theory of yielding is expressed by " " +3 = 1 (3-26) ,·i 1 See for example S. S. Hecker, Metall. Trans., vol. 2, pp. 2077-2086, 1971. A unique method for determining the yield locus of a flat sheet has been presented by D. Lee and W. A. Backofen, Trans. Metall. Soc. AIME, vol. 236, pp. 1077-1084, 1966. This method is well suited for studying the anisotropy of rolled sheet.  ELEMENTS OF THE THEORY OF PLASTICITY 81 y P ~f-- -~,,";o- p --x Mr y Figure 3-3 Combined tension and torsion in a thin-walled tube. 0.6 O' I. I I ~ Istortlon energy 0.5 ~ ~ ~ 0.4 Maximum shear stres;'> 0.2 "'~r\ 0.1 1\\ o \1 Figure 3-4 Comparison between o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 maximum-shear-stress theory and ax jeTo distortion-energy (von Mises') theory. 1 "Both equations define an ellipse. Figure 3-4 shows that the experimental results agree best with the distortion-energy theory. 3-6 THE YIELD LOCUS For a biaxial plane-stress condition «(J2 = 0) the von Mises' yield criterion can be expressed mathematically as (3-27) This is the equation of an ellipse whose major semiaxis is Ii (Jo and whose minor semiaxis is If (Jo' The plot of Eq. (3-27) is called a yield locus (Fig. 3-5). Several important points on the yield ellipse corresponding to particular stress-ratio loading paths are noted on the figure. The yield locus for the maximum-shear-stress criterion falls inside of the von Mises' yield ellipse. Note that the two yielding criteria predict the same yield stress for conditions of uniaxial stress and balanced biaxial stress (0"1 = (J3)' The greatest divergence between the two criteria occurs for pure shear «(Jl = - (J3)' The yield stress predicted by the von Mises' criterion is 15.5 percent greater than the yield stress predicted by the maximum-shear-stress criterion. 1 G. 1. Taylor and H. Quinney, Proc. R. Soc. London Ser. A., vol. 230A, pp. 323-362, 1931.  82 MECHANICAL FUNDAMENTALS , , ·, 0 3 = 01 , o-o+-~/ von Mises criterion A~:::::1-==~ " 01 = 2 -I Figure 3-12 Mohr's circle representation of stresses in Fig.~ 3-\\ plane strain are p k 0 (1.=k IJ pO o 0 P Mohr's circle representation for the state of stress given in Fig. 3-11 is shown in Fig. 3-12. If (11= -Q and (13 = -P, then (12 = (-Q - P)/2 = -po This follows because (11 + (12 + (13 1 Q P P = (1m = - -- Q+ - + - +P 3 3 2 2 Q+P p= - -(1