mechanical metallurgy dieter

MEC NICALMETALLURGY

SI Metric Edition

George E. DieterUniversity of Maryland

Adapted byDavid Bacon

Professor of Materials ScienceUniversity of Liverpool

McGraw-Hili Book CompanyLondon' New Yark . St Louis' San Francisco' Auckland

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McGraw-Hill Series in Materials Science and Engineering

Editorial Board

Michael B. BeverStephen M. CopleyM. E. ShankCharles A. WertGarth L. Wilkes

Brick, Pense, and Gordon: Structure and Properties of Engineering MaterialsDieter: Engineering Design: A Materials and Processing ApproachDieter: Mechanical Metallurgy

Drauglis, Gretz, and Jaffe: Molecular Processes on Solid SurfacesFlemings: Solidification ProcessingFontana: Corrosion EngineeringGaskell: Introduction to Metallurgical ThermodynamicsGuy: Introduction to Materials ScienceKehl: The Principles of Metallographic Laboratory PracticeLeslie: The Physical Metallurgy of SteelsRhines: Phase Diagrams in Metallurgy: Their Development and ApplicationRozenfeld: Corrosion InhibitorsShewmon: Transformations in MetalsSmith: Principles of Materials Science and EngineeringSmith: Structure and Properties of Engineering AlloysVander Voort: Metallography: Principles and PracticeWert and Thomson: Physics of Solids

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MECHANICAL METALLURGY SI Metric Edition

Exclusive rights by McGraw-Hili Book Co - Singapore for manufactureand export. This book cannot be re-exported from the country to whichit is consigned by McGraw-Hili.

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Copyright 1988 McGraw-Hill Book Company (UK) LimitedCopyright 1986, 1976, 1961 by McGraw-Hili Inc.All rights reserved. No part of this publication may be reproduced, storedin a retrieval system, or transmitted in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise, withoutthe 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. Strength1. Title620.1 '63

Library of Congress Cataloging-in-Publication Data

Dieter, George Ellwood.Mechanical metallurgy/George E. Dieter. - SI Metric ed./adaptedby David Bacon.(McGraw-Hill series in materials science and engineering)Bibliography:p.includes indexes.ISBN 0-07-084187-XI. Strength of materials. 2. Physical metallurgy.

1. Bacon, D. J. 11. Title. 111. SeriesTA405.D53 1988620.1'63 - dc 1988-10351

When ordering this title use ISBN 0-07-100406-8

Printed in Singapore

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II.>.~

ABOUT THE AUTHOR

George E. Dieter is currently Dean of Engineering and Professor of MechanicalEngineering 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 ResearchLaboratory, he became Head of the Metallurgical Engineering Department atDrexel University, where he later became Dean of Engineering. Professor Dieterlater joined the faculty of Carnegie-Mellon University, as Professor of Engineer-ing and Director of the Processing Research Institute. He moved to the Universityof Maryland four years later.

A former member of the National Materials Advisory Board, ProfessorDieter is a fellow of the American Society for Metals, and a member of AAAS,AIME, ASEE, NSPE, and SME.

v

Preface to the Third EditionPreface to the Second EditionPreface to the First EditionList of Symbols

Part 1 Mechanical Fundamentals

CONTENTS

Xlll

xv

XVll,

XXI

1 Introduction 3't ...._. _,,_._,. A~~'-"~-'

1)' Scope of This Book l1.:ljStrength of Mateial,s-BasicAssumptions :vr; Elastic and Plastic Behavior LJ::4JAverageStress and Strain rrS Tensile Deformation of Ductile Metal

, .,,--- .-- ~".,.- --. ",,' -- " .-' ." ," - -, -,,- ,-'

/1-6 Ductile vs. Brittle Behavior "1-7 What Constitutes Failure?; . ._,----~-c- ..0 I i)-8 Concept of Stress and the Types of Stresses Lk9j Concept ofStrain and the Types of Strain 'Ji~i61Units of Stress and Other

L.v.' ,. ,.,-Quantities

2 Stress and Strain Relationships for Elastic Behavior 17--'

Vlll CONTENTS

3 Elements of the Theory of Plasticity 69----"1 ~.------~ : - ..- --~{[ill Intr,()du~tion LUJThe Flow Curve J.--:1JTrue Strfss and True

Strain 3-4(Yielding Criteria for Ductile Metals .J.~5Combined-_.-_.. -... -" ~'---'"'-'-'.---'~'. . ..-"'-'. "'-, . '--. . .

Stress Tests i 3-6tThe YIeld Locus' 3-7'Anisotropy m YIeldmgC3:8Yield Su'a~~eand.Normality'3~Octahedral Shear Stressand Shear Strain~ Invariants ofS!!::ss and Strain{3:rD Plastic Stress-Strain Relations r3-1~ Two-Dimensional Plastic

,-=-.~.~ :.--- , '.Flow-Slip-Line Field Theory'

Part 2 Metallurgical Fundamentals4 Plastic Deformation of Single Crystals

4-1 Introduction 4-2 Concepts of Crystal Geometry 4-3 LatticeDefects 4-4 Deformation by Slip 4-5 Slip in a Perfect Lattice4-6 Slip by Dislocation Movement 4-7 Critical ResolvedShear Stress for Slip 4-8 Deformation of Single Crystals4-9 Deformation of Face-Centered Cubic Crystals4-10 Deformation by Twinning 4-11 Stacking Faults4-12 Deformation Bands and Kink Bands 4-13 MicrostrainBehavior 4-14 Strain Hardening of Single Crystals

5 Dislocation Theory5-1 Introduction 5-2 Observation of Dislocations 5-3 BurgersVector and the Dislocation Loop 5-4 Dislocations in theFace-Centered Cubic Lattice 5-5 Dislocations in the HexagonalClose-Packed Lattice 5-6 Dislocations in the Body-CenteredCubic Lattice 5-7 Stress Fields and Energies of Dislocations5-8 Forces on Dislocations 5-9 Forces between Dislocations5-10 Dislocation Climb 5-11 Intersection of Dislocations5-12 Jogs 5-13 Dislocation Sources 5-14 Multiplication ofDislocations 5-15 Dislocation-Point Defect Interactions5-16 Dislocation Pile-Ups

6 Strengthening Mechanisms6-1 Introduction 6-2 Grain Boundaries and Deformation6-3 Strengthening from Grain Boundaries 6-4 Low-Angle GrainBoundaries 6-5 Yield-Point Phenomenon 6-6 Strain Aging6-7 Solid-Solution Strengthening 6-8 Deformation of Two-PhaseAggregates 6-9 Strengthening from Fine Particles 6-10 FiberStrengthening 6-11 Strengthening Due to Point Defects6-12 Martensite Strengthening 6-13 Cold-Worked Structure6-14 Strain Hardening 6-15 Annealing of Cold-Worked Metal6-16 Bauschinger Effect 6-17 Preferred Orientation (Texture)

103

145

184

7 Fracture

CONTENTS IX

241

7-1 Introduction 7-2 Types of Fracture in Metals7-3 Theoretical Cohesive Strength of Metals 7-4 Griffith Theoryof Brittle Fracture 7-5 Fracture of Single Crystals7-6 Metallographic Aspects of Fracture 7-7 Fractography7-8 Dislocation Theories of Brittle Fracture 7-9 Ductile Fracture7-10 Notch Effects 7-11 Concept of the Fracture Curve7-12 Fracture under Combined Stresses 7-13 Effect of HighHydrostatic Pressure on Fracture

Part 3 Applications to Materials Testing8 The Tension Test

8-1 Engineering Stress-Strain Curve 8-2 True-Stress-True-StrainCurve 8-3 Instability in Tension 8-4 Stress Distribution at theNeck 8-5 Ductility Measurement in Tension Test 8-6 Effectof Strain Rate on Flow Properties 8-7 Effect of Temperature onFlow Properties 8-8 Influence of Testing Machine on FlowProperties 8-9 Constitutive Equations 8-10 FurtherConsideration of Instability 8-11 Stress Relaxation Testing8-12 Thermally Activated Deformation 8-13 Notch Tensile Test8-14 Tensile Properties of Steel 8-15 Anisotropy of TensileProperties

9 The Hardness Test9-1 Introduction 9-2 Brinell Hardness 9-3 Meyer Hardness9-4 Analysis of Indentation by an Indenter 9-5 Relationshipbetween Hardness and the Flow Curve 9-6 Vickers Hardness9-7 Rockwell Hardness Test 9-8 Microhardness Tests9-9 Hardness-Conversion Relationships 9-10 Hardness atElevated Temperatures

10 The Torsion Test10-1 Introduction 10-2 Mechanical Properties in Torsion10-3 Torsional Stresses for Large Plastic Strains 10-4 Types ofTorsion Failures 10-5 Torsion Test vs. Tension Test 10-6 HotTorsion Testing

11 Fracture Mechanics11-1 Introduction 11-2 Strain-Energy Release Rate 11-3 StressIntensity Factor 11-4 Fracture Toughness and Design11-5 K!c Plane-Strain Toughness Testing 11-6 PlasticityCorrections 11-7 Crack Opening Displacement 11-8 J Integral11-9 R Curve 11-10 Probabilistic Aspects of Fracture Mechanics11-11 Toughness of Materials

275

325

338

348

X CONTENTS

12 Fatigue of Metals12-1 Introduction 12-2 Stress Cycles 12-3 The S-N Curve12-4 Statistical Nature of Fatigue 12-5 Effect of Mean Stress onFatigue 12-6 Cyclic Stress-Strain Curve 12-7 Low-Cycle Fatigue12-8 Strain-Life Equation 12-9 Structural Features of Fatigue12-10 Fatigue Crack Propagation 12-11 Effect of StressConcentration on Fatigue 12-12 Size Effect 12-13 SurfaceEffects and Fatigue 12-14 Fatigue under Combined Stresses12-15 Cumulative Fatigue Damage and Sequence Effects12-16 Effect of Metallurgical Variables and Fatigue 12-17 Designfor Fatigue 12-18 Machine Design Approach-Infinite-LifeDesign 12-19 Local Strain Approach 12-20 Corrosion Fatigue12-21 Effect of Temperature on Fatigue

13 Creep and Stress Rupture13-1 The High-Temperature Materials Problem 13-2 Time-Dependent Mechanical Behavior 13-3 The Creep Curve13-4 The Stress-Rupture Test 13-5 Structural Changes DuringCreep 13-6 Mechanisms of Creep Deformation 13-7Deformation Mechanism Maps 13-8 Activation Energy forSteady-State Creep 13-9 Superplasticity 13-10 Fracture atElevated Temperature 13-11 High-Temperature Alloys13-12 Presentation of Engineering Creep Data 13-13 Predictionof Long-Time Properties 13-14 Creep Under Combined Stresses13-15 Creep-Fatigue Interaction

14 Brittle Fracture and Impact Testing14-1 The Brittle-Fracture Problem 14-2 Notched-Bar ImpactTests 14-3 Instrumented Charpy Test 14-4 Significance ofTransition-Temperature Curve 14-5 Metallurgical FactorsAffecting Transition Temperature 14-6 Drop-Weight Test andOther Large-Scale Tests 14-7 Fracture Analysis Diagram14-8 Temper Embrittlement 14-9 Environment Sensitive Fracture14-10 Flow and Fracture under Very Rapid Rates of Loading

Part 4 Plastic Forming of Metals15 Fundamentals of Metalworking

15-1 Classification of Forming Processes 15-2 Mechanics ofMetalworking 15-3 Flow-Stress Determination15-4 Temperature in Metalworking 15-5 Strain-Rate Effects15-6 Metallurgical Structure 15-7 Friction and Lubrication15-8 Deformation-Zone Geometry 15-9 Hydrostatic Pressure15-10 Workability 15-11 Residual Stresses 15-12 ExperimentalTechniques for Metalworking Processes 15-13 Computer-AidedManufacturing

375

432

471

503

CONTENTS Xl

16 Forging 56416-1 Classification of Forging Processes 16-2 Forging Equipment16-3 Forging in Plane Strain 16-4 Open-Die Forging16-5 Closed-Die Forging 16-6 Calculation of ForgingLoads in Closed-Die Forging 16-7 Forging Defects16-8 Powder Metallurgy Forging 16-9 Residual Stresses inForgings

17 Rolling of Metals 58617-1 Classification of Rolling Processes 17-2 Rolling Mills17-3 Hot-Rolling 17-4 Cold-Rolling 17-5 Rolling of Bars andShapes 17-6 Forces and Geometrical Relationships in Rolling17-7 Simplified Analysis of Rolling Load: Rolling Variables17-8 Problems and Defects in Rolled Products 17-9 Rolling-MillControl 17-10 Theories of Cold-Rolling 17-11 Theories ofHot-Rolling 17-12 Torque and Power

18 Extrusion18-1 Classification of Extrusion Processes 18-2 ExtrusionEquipment 18-3 Hot Extrusion 18-4 Deformation, Lubrication,and Defects in Extrusion 18-5 Analysis of the Extrusion Process8-6 Cold Extrusion and Cold-Forming 18-7 Hydrostatic Extrusion18-8 Extrusion of Tubing 18-9 Production of Seamless Pipe andTubing

19 Drawing of Rods, Wires, and Tubes19-1 Introduction 19-2 Rod and Wiredrawing 19-3 Analysisof Wiredrawing 19-4 Tube-Drawing Processes 19-5 Analysisof Tube Drawing 19-6 Residual Stresses in Rod, Wire, and Tubes

20 Sheet-Metal Forming20-1 Introduction 20-2 Forming Methods 20-3 Shearing andBlanking 20-4 Bending 20-5 Stretch Forming 20-6 DeepDrawing 20-7 Forming Limit Criteria 20-8 Defects in FormedParts

21 Machining of Metals21-1 Types of Machining Operations 21-2 Mechanics ofMachining 21-3 Three-Dimensional Machining21-4 Temperature in Metal Cutting 21-5 Cutting Fluids21-6 Tool Materials and Tool Life 21-7 Grinding Processes21-8 Nontraditional Machining Processes 21-9 Economics ofMachining

616

635

651

679

Xli CONTENTS

AppendixesA The International System of Units (SI)B Problems

Answers to Selected ProblemsIndexes

Name IndexSubject Index

709712

734739

PREFACETO THE THIRD EDITION

The objective of Mechanical Metallurgy continues to be the presentation of theentire scope of mechanical metallurgy in a single comprehensive volume. Thus,the book starts with a continuum description of stress and strain, extends thisbackground to the defect mechanisms of flow and fracture of metals, and thenconsiders the major mechanical property tests and the basic metalworkingprocesses. As before, the book is intended for the senior or first-year graduate-stu-dent level. Emphasis is on basic phenomena and relationships in an engineeringcontext. Extensive references to the literature have been included to assiststudents in digging deeper into most topics.

Since the second edition in 1976 extensive progress has been made in allresearch areas of the mechanical metallurgy spectrum. Indeed, mechanical behav-ior is the category of research under which the greatest number of papers arepublished in Metallurgical Transactions. Since 1976 the field of fracture mechan-ics has grown greatly in general acceptance. In recognition of this a separatechapter on fracture mechanics has been added to the present edition, replacing achapter on mechanical behavior of polymeric materials. Other topics added forthe first time or greatly expanded in coverage are deformation maps, finiteelement methods, environmentally assisted fracture, and creep-fatigue interaction.

As an aid to the student, numerous illustrative examples have been includedthroughout the book. Answers have been provided to selected problems for thestudent, and a solutions manual is available for instructors. In this third edition,major emphasis is given to the use of SI units, as is common with mostengineering texts today.

I would like to express my thanks for the many useful comments andsuggestions provided by Ronald Scattergood, North Carolina State University,and Oleg Sherby, Stanford University, who reviewed this text during the course ofits development.

Xlll

XIV PREFACE TO THE THIRD EDITION

Acknowledgment is given to Professor Ronald Armstrong, University ofMaryland, for providing many stimulating problems, and Dr. A. Pattniak, NavalResearch Laboratory, for assistance in obtaining the fractographs. Special thanksgoes to Jean Beckmann for her painstaking efforts to create a perfect manuscript.

George E. Dieter

PREFACETO THE SECOND EDITION

In the 12 years since the first edition of Mechanical Metallurgy at least 25textbooks dealing with major segments of the book have appeared in print. Forexample, at least 10 books dealing with the mechanics of metalworking have beenpublished during this period. However, none of these books has dealt with theentire scope of mechanical metallurgy, from an understanding of the continuumdescription of stress and strain through crystalline and defect mechanisms of flowand fracture, and on to a consideration of the major mechanical property testsand the basic metalworking processes.

Important advances have been made in understanding the mechanical behav-ior of solids in the period since the first edition. The dislocation theory of plasticdeformation has become well established, with excellent experimental verificationfor most of the theory. These advances have led to a better understanding of thestrengthening mechanisms in solids. Developments such as fracture mechanicshave matured to a high level of technical sophistication and engineering useful-ness. An important development during this period has been the "materialsscience movement" in which crystalline solids, metals, ceramics, and polymers areconsidered as a group whose properties are controlled by basic structural defectscommon to all classes of crystalline solids.

In this revision of the book the emphasis is as before. The book is intendedfor the senior or first-year graduate-student level. Extensive revisions have beenmade to up-date material, to introduce new topics which have emerged asimportant areas, and to clarify sections which have proven difficult for students tounderstand. In some sections advanced material intended primarily for graduatestudents has been set in smaller type. The problems have been extensively revisedand expanded, and a solutions manual has been prepared. Two new chapters, onedealing with the mechanical properties of polymers and the other with the

xv

XVI PREFACE TO THE SECOND EDITION

machining of metals, have been added, while the chapters of statistical methodsand residual stresses have been deleted. In total, more than one-half of the bookhas been rewritten completely.

George E. Dieter

PREFACETO THE FIRST EDITION

Mechanical metallurgy is the area of knowledge which deals with the behaviorand response of metals to applied forces. Since it is not a precisely defined area, itwill mean different things to different persons. To some it will mean mechanicalproperties of metals or mechanical testing, others may consider the field restrictedto the plastic working and shaping of metals, while still others confine theirinterests to the more theoretical aspects of the field, which merge with metalphysics and physical metallurgy. Still another group may consider that mechani-cal metallurgy is closely allied with applied mathematics and applied mechanics.In writing this book an attempt has been made to cover, in some measure, thisgreat diversity of interests. The objective has been to include the entire scope ofmechanical metallurgy in one fairly comprehensive volume.

The book has been divided into four parts. Patt One, Mechanical Fundamen-tals, presents the mathematical framework for many of the chapters which follow.The concepts of combined stress and strain are reviewed and extended into threedimensions. Detailed consideration of the theories of yielding and an introductionto the concepts of plasticity are given. No attempt is made to carry the topics inPart One to the degree of completion required for original problem solving.Instead, the purpose is to acquaint metallurgically trained persons with themathematical language encountered in some areas of mechanical metallurgy. PartTwo, Metallurgical Fundamentals, deals with the structural aspects of plasticdeformation and fracture. Emphasis is on the atornistics of flow and fracture andthe way in which metallurgical structure affects these processes. The concept ofthe dislocation is introduced early in Part Two and is used throughout to providequalitative explanations for such phenomena as strain hardening, the yield point,dispersed phase hardening, and fracture. A more mathematical treatment of theproperties of dislocations is given in a separate chapter. The topics covered inPart Two stem from physical metallurgy. However, most topics are discussed in

XVIl

xviii PREFACE TO THE FIRST EDITION

greater detail and with a different emphasis than when they are first covered inthe usual undergraduate course in physical metallurgy. Certain topics that aremore physical metallurgy than mechanical metallurgy are included to providecontinuity and the necessary background for readers who have not studiedmodern physical metallurgy.

Part Three, Applications to Materials Testing, deals with the engineeringaspects of the common testing techniques of mechanical failure of metals.Chapters are devoted to the tension, torsion, hardness, fatigue, creep, and impacttests. Others take up the important subjects of residual stresses and the statisticalanalysis of mechanical-property data. In Part Three emphasis is placed on theinterpretation of the tests and on the effect of metallurgical variables on mechani-cal behavior rather than on the procedures for conducting the tests. It is assumedthat the actual performance of these tests will be covered in a concurrentlaboratory course or in a separate course. Part Four, Plastic Forming of Metals,deals with the common mechanical processes for producing useful metal shapes.Little emphasis is given to the descriptive aspects of this subject, since this canbest be covered by plant trips and illustrated lectures. Instead, the main attentionis given to the mechanical and metallurgical factors which control each processsuch as forging, rolling, extrusion, drawing, and sheet-metal forming.

This book is written for the senior or first-year graduate student in metal-lurgical or mechanical engineering, as well as for practicing engineers in industry.While most universities have instituted courses in mechanical metallurgy ormechanical properties, there is a great diversity in the material covered and in thebackground of the students taking these courses. Thus, for the present there canbe nothing like a standardized textbook on mechanical metallurgy. It is hopedthat the breadth and scope of this book will provide material for these somewhatdiverse requirements. It is further hoped that the existence of a comprehensivetreatment of the field of mechanical metallurgy will stimulate the development ofcourses which cover the total subject.

Since this book is intended for college seniors, graduate students, andpracticing engineers, it is expected to become a part of their professional library.Although there has been no attempt to make this book a handbook, some thoughthas been given to providing abundant references to the literature on mechanicalmetallurgy. Therefore, more references are included than is usual in the ordinarytextbook. References have been given to point out derivations or analyses beyondthe scope of the book, to provide the key to further information on controversialor detailed points, and to emphasize important papers which are worthy offurther study. In addition, a bibliography of general references will be found atthe end of each chapter. A collection of problems is included at the end of thevolume. This is primarily for the use of the reader who is engaged in industry andwho desires some check on his comprehension of the material.

The task of writing this book has been mainly one of sifting and sorting factsand information from the literature and the many excellent texts on specializedaspects of this subject. To cover the breadth of material found in this book wouldrequire parts of over 15 standard texts and countless review articles and individ-

PREFACE TO THE FIRST EDITION XIX

ual contributions. A conscientious effort has been made throughout to give creditto original sources. For the occasional oversights that may have developed duringthe "boiling-down process" the author offers his apologies. He is indebted tomany authors and publishers who consented to the reproduction of illustrations.Credit is given in the captions of the illustrations.

Finally, the author wishes to acknowledge the many friends who advised himin his work. Special mention should be given to Professor A. W. Grosvenor,Drexel Institute of Technology, Dr.G. T. Horne, Carnegie Institute of Technol-ogy, Drs. T. C. Chilton, J. H. Faupel, W. L. Phillips, W. I. Pollock, and J. T.Ransom of the du Pont Company, and Dr. A. S. Nemy of the Thompson-Ramo-Wooldridge Corp.

George E. Dieter

LIST OF SYMBOLS

A area; amplitudeG linear distance; crack length

Go interatomic spacingB constant; specimen thicknessb width or breadthb Burgers vector of a dislocationC generalized constant; specific heat

Cij elastic coefficientsc length of Griffith crack

D diameter, grain diameterE modulus of elasticity for axial loading (Young's modulus)e conventional, or engineering, linear straic.

exp base of natural logarithms (= 2.718)F force per unit length on a dislocation lineG modulus of elasticity in shear (modulus of rigidity)" crack-extension forceH activation energyh distance, usually in thickness direction

(h, k, I) Miller indices of a crystallographic planeI moment of inertiaJ invariant of the stress deviator; polar moment of inertiaK strength coefficient

Kf fatigue-notch factorK t theoretical stress-concentration factor

K Ie fracture toughnessk yield stress in pure shearL length

XXI

XXII LIST OF SYMBOLS

I, m, n direction cosines of normal to a planeIn natural logarithm

log logarithm to base 10M B bending momentM T torsional moment, torque

m strain-rate sensitivityN number of cycles of stress or vibrationn strain-hardening exponent

n' generalized constant in exponential termP load or external forceQ activation energyp pressureq reduction in area; plastic-constraint factor; notch sensitivity index in

fatigueR radius of curvature; stress ratio in fatigue; gas constantr radial distanceS total stress on a plane before resolution into normal and shear compo-

nentsSij elastic compliance

s engineering stressT temperature

Tm melting pointt time; thickness

t r time for ruptureU elastic strain energy

Uo elastic strain energy per unit volumeu, v, w components of displacement in x, y, and z directions[uvw1Miller indices for a crystallographic direction

V volumev velocity

W workZ Zener-Hollomon parametera linear coefficient of thermal expansion; phase angle

a, {3, 8, 1> generalized anglesr line tension of a dislocationy shear strain~ volume strain or cubical dilatation; finite change8 deformation or elongation; deflection; logarithmic decrement;),.

Kronecker deltaf general symbol for strain; natural or true strainf significant, or effective, true straini; true-strain rate

,~1

;.,

es minimum creep rate1/ efficiency; coefficient of viscosity() Dorn time-temperature parameterK bulk modulus or volumetric modulus of elasticityA Lame's constant; interparticle spacing}L coefficient of frictionv Poisson's ratiop densitya normal stress; true stress

ao yield stress or yield strengtha6 yield stress in plane straina significant, or effective, true stress

aI' a2, a3 principal stressesa' stress deviatora" hydrostatic component of stressaa alternating, or variable, stressam average principal stress; mean stressar range of stressau ultimate tensile strengthaw working stress

T shearing stress; relaxation time

LIST OF SYMBOLS XXIII

PART

MECHANICALFUNDAMENTALS

CHAPTER

ONEINTRODUCTION

1-1 SCOPE OF THIS BOOK

Mechanical metallurgy is the area of metallurgy which is concerned primarily withthe response of metals to forces or loads. The forces may arise from the use of themetal as a member or part in a structure or machine, in which case it is necessaryto know something about the limiting values which can be withstood withoutfailure. On the other hand, the objective may be to convert a cast ingot into amore useful shape, such as a flat plate, and here it is necessary to know theconditions of temperature and rate of loading which minimize the forces that areneeded to do the job.

Mechanical metallurgy is not a subject which can be neatly isolated andstudied by itself. It is a combination of many disciplines and many approaches tothe problem of understanding the response of materials to forces. On the onehand is the approach used in strength of materials and in the theories of elasticityand plasticity, where a metal is considered to be a homogeneous material whosemechanical behavior can be rather precisely described on the basis of only a veryfew material constants. This approach is the basis for the rational design ofstructural members and machine parts. The topics of strength of materials,elasticity, and plasticity are treated in Part One of this book from a moregeneralized point of view than is usually considered in a first course in strength ofmaterials. The material in Chaps. 1 to 3 can be considered the mathematicalframework on which much of the remainder of the book rests. For students ofengineering who have had an advanced course in strength of materials or machinedesign, 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 inPart One.

The theories of strength of materials, elasticity, and plasticity lose much oftheir power when the structure of the metal becomes an important considerationand it can no longer be considered a homogeneous medium. Examples of this arein the high-temperature behavior of metals, where the metallurgical structure maycontinuously change with time, or in the ductile-to-brittle transition, which occursin carbon steel. The determination of the relationship between mechanical behav-ior and structure (as detected chiefly with microscopic and x-ray techniques) is themain responsibility of the mechanical metallurgist. When mechanical behavior isunderstood in terms of metallurgical structure, it is generally possible to improvethe mechanical properties or at least to control them. Part Two of this book isconcerned with the metallurgical fundamentals of the mechanical behavior ofmetals. Metallurgical students will find that some of the material in Part Two hasbeen covered in a previous course in physical metallurgy, since mechanicalmetallurgy is part of the broader field of physical metallurgy. However, thesesubjects are considered in greater detail than is usually the case in a first course inphysical metallurgy. In addition, certain topics which pertain more to physicalmetallurgy than mechanical metallurgy have been included in order to providecontinuity and to assist nonmetallurgical students who may not have had a coursein physical metallurgy.

The last three chapters of Part Two are concerned primarily with atomisticconcepts of the flow and fracture of metals. Many of the developments in theseareas have been the result of the alliance of the solid-state physicist with themetallurgist. This has been an area of great progress. The introduction oftransmission electron microscopy has provided an important experimental toolfor verifying theory and guiding analysis. A body of basic dislocation theory ispresented which is useful for understanding the mechanical behavior of crystallinesolids.

Basic data concerning the strength of metals and measurements for theroutine control of mechanical properties are obtained from a relatively smallnumber of standardized mechanical tests. Part Three, Applications to MaterialsTesting, considers each of the common mechanical tests,. not from the usualstandpoint of testing techniques, but instead from the consideration of what thesetests tell about the service performance of metals and how metallurgical variablesaffect the results of these tests. Much of the material in Parts One and Two hasbeen utilized in Part Three. It is assumed that the reader either has completed aconventional 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 informing metals into useful shapes. Attempts have been made to present mathe-matical analyses of the principal metalworking processes, although in certFcases this has not been possible, either because of the considerable detail requiredor because the analysis is beyond the scope of this book. No attempt has beenmade to include the extensive specialized technology associated with each metal-

INTRODUCTION 5

working process, such as rolling or extrusion, although some effort has been madeto give a general impression of the mechanical equipment required and tofamiliarize the reader with the specialized vocabulary of the metalworking field:Major emphasis has been placed on presenting a fairly simplified picture of theforces involved in each process and of how geometrical and metallurgical factorsaffect 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 relationbetween internal forces, deformation, and external loads. In the general methodof analysis used in strength of materials the first step is to assume that themember is in equilibrium. The equations of static equilibrium are applied to theforces acting on some part of the body in order to obtain a relationship betweenthe external forces acting on the member and the internal forces resisting theaction of the external loads. Since the equations of equilibrium must be expressedin terms of forces acting external to the body, it is necessary to make the internalresisting forces into external forces. This is done by passing a plane through thebody at the point of interest. The part of the body lying on one side of the cuttingplane is removed and replaced by the forces it exerted on the cut section of thepart of the body that remains. Since the forces acting on the "free body" hold itin equilibrium, the equations of equilibrium may be applied to the problem.

The internal resisting forces are usually expressed by the stress! acting over acertain area, so that the internal force is the integral of the stress times thedifferential area over which it acts. In order to evaluate this integral, it isnecessary 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 straindistribution in the member, since stress cannot be physically measured. However,since stress is proportional to strain for the small deformations involved in mostwork, 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 themember.

Important assumptions in strength of materials are that the body which isbeing analyzed is continuous, homogeneous, and isotropic. A continuous bodyis one which does not contain voids or empty spaces of any kind. A body ishomogeneous if it has identical properties at all points. A body is considered to beisotropic with respect to some property when that property does not vary withdirection or orientation. A property which varies with orientation with respect tosome system of axes is said to be anisotropic.

1 For present purposes stress is defined as force per unit area. The companion term strain isdefined as the change in length per unit length. More complete definitions will be given later.


While engineering materials such as steel, cast iron, and aluminum mayappear to meet these conditions when viewed on a gross scale, it is readilyapparent when they are viewed through a microscope that they are anything buthomogeneous and isotropic. Most engineering metals are made up of more thanone phase, with different mechanical properties, such that on a micro scale theyare heterogeneous. Further, even a single-phase metal will usually exhibit chem-ical segregation, and therefore the properties will not be identical from point topoint. Metals are made up of an aggregate of crystal grains having differentproperties in different crystallographic directions. The reason why the equationsof strength of materials describe the behavior of real metals is that, in general, thecrystal grains are so small that, for a specimen of any macroscopic volume, thematerials are statistically homogeneous and isotropic. However, when metals areseverely deformed in a particular direction, as in rolling or forging, the mechani-cal properties may be anisotropic on a macro scale. Other examples of anisotropicproperties are fiber-reinforced composite materials and single crystals. Lack ofcontinuity may be present in porous castings or powder metallurgy parts and, onan 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 toexternal load. It is further found that up to certain limiting loads a solid willrecover its original dimensions when the load is removed. The recovery of theoriginal dimensions of a deformed body when the load is removed is known aselastic behavior. The limiting load beyond which the material no longer behaveselastically is the elastic limit. If the elastic limit is exceeded, the body willexperience a permanent set or deformation when the load is removed. A bodywhich is permanently deformed is said to have undergone plastic deformation.

For most materials, as long as the load does not exceed the elastic limit, thedeformation is proportional to the load. This relationship is known as Hooke'slaw; it is more frequently stated as stress is proportional to strain. Hooke's lawrequires that the load-deformation relationship should be linear. However, it doesnot necessarily follow that all materials which behave elastically will have a linearstress-strain relationship. Rubber is an example of a material with a nonlinearstress-strain relationship that still satisfies the definition of an elastic material.

Elastic deformations in metals are quite small and require very sensitiveinstruments for their measurement. Ultrasensitive instruments have shown thatthe elastic limits of metals are much lower than the values usually measured inengineering 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 narrowrange of loads over which Hooke's law strictly applies. This is, however, primariltof academic importance. Hooke's law remains a quite valid relationship forengineering design.

j1j,

1,

i

INTRODUCTION 7

I----Lo +8--~I----Lo--~

f--~P

I----~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 uniformcylindrical 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 thatLa is the gage length between these marks. A load P is applied to one end of thebar, and the gage length undergoes a slight increase in length and decrease indiameter. 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 inlength to the original length.

e= --L-La (1-1)

Strain is a dimensionless quantity since both 8 and La are expressed in units oflength.

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 ais the stress normal to the cutting plane and A is the cross-sectional area of thebar. 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 = aAP

a =-A

(1-3)

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, everylongitudinal 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 foreach element. The inherent anisotropy between grains in a polycrystalline metalrules out the possibility of complete uniformity of stress over a body of macro-


scopic size. The presence of more than one phase also gives rise to nonuniformityof 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 willnot be uniform. An extreme disruption in the uniformity of the stress patternoccurs when there is an abrupt change in cross section. This results in a stressraiser or stress concentration (see Sec. 2-15).

Below the elastic limit Hooke's law can be considered valid, so that theaverage stress is proportional to the average strain,

(J- = E = constante

The constant E is the modulus of elasticity, or Young's modulus.

1-5 TENSILE DEFORMATION OF DUCTILE METAL

(1-4)

The basic data on the mechanical properties of a ductile metal are obtained froma tension test, in which a suitably designed specimen is subjected to increasingaxial load until it fractures. The load and elongation are measured at frequentintervals during the test and are expressed as average stress and strain accordingto the equations in the previous section. (More complete details on the tensiontest are given in Chap. 8.)

The data obtained from the tension test are generally plotted as a stress-straindiagram. Figure 1-3 shows a typical stress-strain curve for a metal such asaluminum or copper. The initial linear portion of the curve OA is the elasticregion within which Hooke's law is obeyed. Point A is the elastic limit, defined asthe greatest stress that the metal can withstand without experiencing a permanentstrain when the load is removed. The determination of the elastic limit is quitetedious, 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 curvedeviates from linearity. The slope of the stress-strain curve in this region is themodulus 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 bythe yield strength, point B. The yield strength is defined as the stress which willproduce 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 deformationbegins when the elastic limit is exceeded. As the plastic deformation of thespecimen increases, the metal becomes stronger (strain hardening) so that the loadrequired to extend the specimen increases with further straining. Eventually theload reaches a maximum value. The maximum load divided by the original areaof the specimen is the ultimate tensile strength. For a ductile metal the diameter ofthe specimen begins to decrease rapidly beyond maximum load, so that the loadrequired to continue deformation drops off until the specimen fractures. Since theaverage stress is based on the original area of the specimen, it also decreases frommaximum load to fracture.

1-6 DUCTILE VS. BRITTLE BEHAVIOR

The general behavior of materials under load can be classified as ductile or brittledepending upon whether or not the material exhibits the ability to undergo plasticdeformation. Figure 1-3 illustrates the tension stress-strain curve of a ductilematerial. 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 slightmeasure of plasticity before fracture (Fig. 1-4b). Adequate ductility is an im-portant engineering consideration, because it allows the material to redistributelocalized stresses. When localized stresses at notches and other accidental stressconcentrations do not have to be considered, it is possible to design for staticsituations on the basis of average stresses. However, with brittle materials,localized stresses continue to build up when there is no local yielding. Finally, acrack forms at one or more points of stress concentration, and it spreads rapidlyover 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 arepractically 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 anelevated temperature. A metal which is brittle in tension may be ductile underhydrostatic compression. Furthermore, a metal which is ductile in tension at room

Strai n(a)

'"'"cuL

-(/)

Strain(b)

Figure 1-4 (a) Stress-strain curve for completelybrittle material (ideal behavior); (b) stress-straincurve for brittle metal with slight amount of ductility.


temperature can become brittle in the presence of notches, low temperature, highrates of loading, or embrittling agents such as hydrogen.

1-7 WHAT CONSTITUTES FAILURE?

Structural members and machine elements can fail to perform their intendedfunctions in three general ways:

1. Excessive elastic deformation2. Yielding, or excessive plastic deformation3. Fracture '

An understanding of the common types of failure is important in good designbecause it is always necessary to relate the loads and dimensions of the memberto some significant material parameter which limits the load-carrying capacity ofthe member. For different types of failure, different significant parameters will beimportant.

Two general types of excessive elastic deformation may occur: (1) excessivedeflection under condition of stable equilibrium, such as the deflection of beamunder 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 themachine just as much as if the part completely fractured. For example, a shaftwhich is too flexible can cause rapid wear of the bearing, or the excessivedeflection of closely mating parts can result in interference and damage to theparts. The sudden buckling type of failure may occur in a slender column whenthe axial load exceeds the Euler critical load or when the external pressure actingagainst a thin-walled shell exceeds a critical value. Failures due to excessive elasticdeformation are controlled by the modulus of elasticity, not by the strength of thematerial. Generally, little metallurgical control can be exercised over the elasticmodulus. The most effective way to increase the stiffness of a member is usuallyby changing its shape and increasing the dimensions of its cross section.

Yielding, or excessive plastic deformation, occurs when the elastic limit of themetal has been exceeded. Yielding produces permanent change of shape, whichmay prevent the part from functioning properly any longer. In a ductile metalunder conditions of static loading at room temperature yielding rarely results infracture, because the metal strain hardens as it deforms, and an increased stress isrequired to produce further deformation. Failure by excessive plastic deformationis 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 significantparameter, but it must be used with a suitable failure criterion (Sec. 3-4). Attemperatures significantly greater than room temperature metals no longer exhibitstrain hardening. Instead, metals can continuously deform at constant stress in ftime-dependent yielding known as creep. The failure criterion under creep condi-

i

INTRODUCTION 11

tions is complicated by the fact that stress is not proportional to strain and thefurther fact that the mechanical properties of the material may change apprecia-bly during service. This complex phenomenon will be considered in greater detailin 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 isloaded statically rarely fractures like a tensile specimen, because it will first fail byexcessive plastic deformation. However, metals fail by fracture in three generalways: (1) sudden brittle fracture; (2) fatigue, or progressive fracture; (3) delayedfracture. In the previous section it was shown that a brittle material fracturesunder static loads with little outward evidence of yielding. A sudden brittle typeof fracture can also occur in ordinarily ductile metals under certain conditions.Plain carbon structural steel is the most common example of a material with aductile-to-brittle transition. A change from the ductile to the brittle type offracture 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. Thisproblem is considered in Chap. 14. A powerful and quite general method ofanalysis 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 inparts which are subjected to alternating, or fluctuating, stresses. A minute crackstarts at a localized spot, generally at a notch or stress concentration, andgradually spreads over the cross section until the member breaks. Fatigue failureoccurs without any visible sign of yielding at nominal or average stresses that arewell below the tensile strength of the metal. Fatigue failure is caused by a criticallocalized tensile stress which is very difficult to evaluate, and therefore design fOffatigue 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 occurswhen a metal has been statically loaded at an elevated temperature for a longperiod of time. Depending upon the stress and the temperature there may be noyielding prior to fracture. A similar type of delayed fracture, in which there is nowarning by yielding prior to failure, occurs at room temperature when steel isstatically 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 themost simple member. Allowance must be made for the possibility of accidentalloads of high magnitude. ~hus, in order to provide a margin of safety and toprotect against failure from unpredictable causes, it is necessary that the allow-able stresses be smaller than the stresses which produce failure. The value of stressfor a particular material used in a particular way which is considered to be a safestress is usually called the working stress aw For static applications the workingstress of ductile metals is usually based on the yield strength ao and for brittle


metals on the ultimate tensile strength au. Values of working stress are establishedby local and federal agencies and by technical organizations such as the AmericanSociety of Mechanical Engineers (ASME). The working stress may be consideredas either the yield strength or the tensile strength divided by a number called thefactor of safety.

aoa =

w N.o

auor a =

W Nu

(1-5)

where aw = working stressao= yield strengthau = tensile strength

No = factor of safety based on yield strengthNu = factor of safety based on tensile strength

The value assigned to the factor of safety depends on an estimate of all thefactors discussed above. In addition, careful consideration should be given to theconsequences, which would result from failure. If failure would result in loss oflife, the factor of safety should be increased. The type of equipment will alsoinfluence the factor of safety. In Inilitary equipment, where light weight may be aprime consideration, the factor of safety may be lower than in commercialequipment. 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 amachine, 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 beuniformly distributed over the cross-sectional area of the member. However, thisis not the general case. Figure I-Sa represents a body in equilibrium under theaction of external forces PI' P2 , , Ps. There are two kinds of external forceswhich may act on a body: surface forces and body forces. Forces distributed overthe surface of the body, such as hydrostatic pressure or the pressure exerted byone body on another, are called surface forces. Forces distributed over the volumeof a body, such as gravitational forces, magnetic forces, or inertia forces (for abody in motion), are called body forces. The two most common types of bodyforces encountered in engineering practice are centrifugal forces due to high-speedrotation and forces due to temperature differential over the body (thermal stress).

In general the force will not be uniformly distributed over any cross sectionof the body illustrated in Fig. I-Sa. To obtain the stress at some point 0 in aplane such as mm, part 1 of the body is removed and replaced by the system ofexternal forces on mm which will retain each point in part 2 of the body in thesame 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

m

CDm

CDn

P,

y

(a) (b)

Figure 1-5 (a) Body in equilibrium under action of external forces PI"'" Ps; (b) forces acting onparts.

on this area. If the area ~A is continuously reduced to zero, the limiting value ofthe ratio ~P / ~A is the stress at the point 0 on plane mm of body 2.

~plim = (J~A->O ~A (1-6)

The stress will be in the direction of the resultant force P and will generally beinclined at an angle to ~A. The same stress at point 0 in plane mm would beobtained 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 tothe 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. Theforce P makes an angle 0 with the normal z to the plane of the area A. Also, theplane 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.


makes an angle

INTRODUCTION 15

IIIII

II

-'> Bk--h II

I II I

A ~...~-- Figure 1-7 Shear strain.

between any two lines. The angular change in a right angle is known as shearstrain. Figure 1-7 illustrates the strain produced by the pure shear of one face of acube. The angle at A, which was originally 90 0 , is decreased by the application ofa shear stress by a small amount f). The shear strain y is equal to the displace-ment a divided by the distance between the planes, h. The ratio a/h is also thetangent of the angle through which the element has been rotated. For the smallangles usually involved, the tangent of the angle and the angle (in radians) areequal. Therefore, shear strains are often expressed as angles of rotation.

ay=-=tanf)=f)

h(1-12)

1-10 UNITS OF STRESS AND OTHER QUANTITIES

The International System of Units, usually called SI (for Systeme International),is followed in this book and is summarized in Appendix A. There are seven baseSI units for measuring quantities, namely meter (m) for length, kilogram (kg) formass, second (s) for time, ampere (A) for electric current, kelvin (K) for thermo-dynamic temperature, mole (mol) for amount of a substance, and candela (cd) forluminous intensity. All other units are derived from these.

Some of the derived units have special names. For example, frequency ismeasured in units of s -1 (i.e. per second), and this unit is known as the hertz(Hz); force is measured in units of m kg S-2 (i.e. the force required to accelerateone kg to an acceleration of one meter per second squared), and this derived unitis called the newton (N); stress has dimensions of force per unit area and istherefore measured in units of N m - 2, which is known as the pascal (Pa). Notethat since the acceleration due to gravity is 9.807 m s- 2, a load of 1 kg exerts aforce of 9.807 N. It can be seen from Appendix A that although different quan-tities may be expressed in different derived units, the units may be identical whenexpressed in base units (e.g. stress and energy density).

In order to avoid describing quantities by very large or small numbers explic-itly, a system of prefixes is used to indicate multiples of a unit (see Appendix A).For example, a stress of 1 Pa is very small, and most stresses of practical interestare in the range 106 Pa to 1011 Pa. This range is more conveniently expressedusing multiples as 1 MPa to 100 GPa. Units in a product are separated in this


book by a space, e.g. MJ m - 3 not MJm - 3: this is not strictly necessary but is anaid to clarity.

Finally it should be reemphasized that strain is simply a number. It is adimensionless quantity and is not expressed in physical units.

Example The shear stress required to nucleate a grain boundary crack inhigh-temperature deformation has been estimated to be

1/27"=

where 'Yb is the grain boundary surface energy, let us say 2 J m- 2 ; G is theshear modulus, 75 GPa; L is the grain boundary sliding distance, assumedequal to the grain diameter 0.01 mm, and v is Poisson's ratio, v = 0.3. To cal-culate 7" we need to be sure the units are consistent and that the prefixes havebeen properly evaluated.

To check the equation express all units in newtons and meters.

Nm N 1/2

m2x

m2 N 2 1/2 N7"= -

-

m4 m2m

Note that a joule (J) is a unit of energy; J = N m (see Appendix A)3n x 2 x 75 X 109 1/2

7" = = (252.4 X 1014)1/28(1 - 0.3) x 10- 2 x 10- 3

= 15.89 X 107 N m- 2

= 158.9 MN m- 2 = 158.9 MPa

BIBLIOGRAPHY

Caddell, R. M.: "Deformation and Fracture of Solids," Prentice-Hall Inc., Englewood Cliffs, N.J.,1980.

Felbeck, D. K., and A. G. Atkins: "Strength and Fracture of Engineering Solids," Prentice-Hall Inc.,Englewood Cliffs, N.J., 1984.

Gordon, J. E.: "Structures-or Why Things Don't Fall Through the Floor," Penguin Books, London,1978.

Hertzberg, R. W.: "Deformation and Fracture Mechanics of Engineering Materials," 2d ed., JohnWiley & Sons, New York, 1983.

LeMay, I.: "Principles of Mechanical Metallurgy," Elsevier North-Holland Inc., New York, 1981.Meyers, M. A., and K. K. Chawla: "Mechanical Metallurgy: Principles and Applications," Prentice-

Hall Inc., Englewood Cliffs, N.J., 1984.Polakowski, N. H., and E. J. Ripling: "Strength and Structure of Engineering Materials," Prentice-Hall

Inc., Englewood Cliffs, N.J., 1964.

, .

-,j.1,'!'I

I

CHAPTER.

TWOSTRESS AND STRAIN RELATIONSHIPS FOR

ELASTIC BEHAVIOR

2-1 INTRODUCTION

The purpose of this chapter is to present the mathematical relationships forexpressing the stress and strain at a point and the relationships between stress andstrain in a solid which obeys Hooke's law. While part of the material covered inthis chapter is a review of information generally covered in strength of materials,the subject is extended beyond this point to a consideration of stress and strain inthree dimensions. The material included in this chapter is important for anunderstanding of most of the phenomenological aspects of mechanical metallurgy,and for this reason it should be given careful attention by those readers to whomit is unfamiliar. In the space available for this subject it has not been possible tocarry it to the point where extensive problem solving is possible. The materialcovered here should, however, provide a background for intelligent reading of themore mathematical literature in mechanical metallurgy.

It should be recognized that the equations describing the state of stress orstrain in a body are applicable to any solid continuum, whether it be an elastic orplastic solid or a viscous fluid. Indeed, this body of knowledge is often calledcontinuum mechanics. The equations relating stress and strain are called constitu-tive equations because they depend on the material behavior. In this chapter weshall only consider the constitutive equations for an elastic solid.

2-2 DESCRIPTION OF STRESS AT A POINT

As described in Sec. 1-8, it is often convenient to resolve the stresses at a pointinto normal and shear components. In the general case the shear components areat arbitrary angles to the coordinate axes, so that it is convenient to resolve each

17


z

!

x

--~y

Figure 2-1 Stresses acting on an ele-mental cube.

shear stress further into two components. The general case is shown in Fig. 2-l.Stresses acting normal to the faces of the elemental cube are identified by thesubscript which also identifies the direction in which the stress acts; that is ax isthe normal stress acting in the x direction. Since it is a normal stress, it must acton the plane perpendicular to the x direction. By convention, values of normalstresses greater than zero denote tension; values less than zero indicate compres-sion. All the normal stresses shown in Fig. 2-1 are tensile.

Two subscripts are needed for describing shearing stresses. The first subscriptindicates the plane in which the stress acts and the second the direction in whichthe stress acts. Since a plane is most easily defined by its normal, the firstsubscript refers to this normal. For example, 1"yz is the shear stress on the planeperpendicular to the y axis in the direction of the z axis. 1"yx is the shear stress ona plane normal to the y axis in the direction of the x axis.

A shear stress is positive if it points in the positive direction on the positiveface of a unit cube. (It is also positive if it points in the negative direction 01). thenegative face of a unit cube.) All of the shear stresses in Fig. 2-2a are positiveshear stresses regardless of the type of normal stresses that are present. A shearstress is negative if it points in the negative direction of a positive face of a unitcube and vice versa. The shearing stresses shown in Fig. 2-2b are all negativestresses.

+y

-x -++--+--++- +x

-y( a)

+y

- x-----1+--+---++- +x

-y(b)

Figure 2-2 Sign convention for shearstress. (a) Positive;, (b) negative. j

1,,,,

i

j

STRESS AND STRAIN RELATIONSHIPS FOR ELASTIC BEHAVIOR 19

The notation for stress given above is the one used by Timoshenkd and mostAmerican workers in the field of elasticity. However, many other notations havebeen used, some of which are given below.

-

-

-

-

-

(2-1)

It can be seen from Fig. 2-1 that nine quantities must be defined in order toestablish the state of stress at a point. They are ox' 0y' 0z' 'Txy ' 'Txz ' 'Tyx ' 'Tyz ' 'Tzx 'and 'Tzy However, some simplification is possible. If we assume that the areas ofthe faces of the unit cube are small enough so that the change in stress over theface is negligible, by taking the summation of the moments of the forces about thez axis it can be shown that 'Txy = 'Tyx '

('TXY Ilyllz) Ilx = ('TyX Ilx Ilz) Ily

and in like manner

Thus, the state of stress at a point is completely described by six components:three normal stresses and three shear stresses, ox' 0y' (Jz, 'Txy ' 'Txz ' 'Tyz .

2-3 STATE OF STRESS IN TWO DIMENSIONS (PLANE STRESS)

,

Many problems can be simplified by considering a two-dimensional state ofstress. This condition is frequently approached in practice when one of thedimensions of the body is small relative to the others. For example, in a thin plateloaded in the plane of the plate there will be no stress acting perpendicular to thesurface of the plate. The stress system will consist of two normal stresses Ox and0y and a shear stress 'Txy" A stress condition in which the stresses are zero in oneof the primary directions is called plane stress.

Figure 2-3 illustrates a thin plate with its thickness normal to the plane of thepaper. In order to know the state of stress at point 0 in the plate, we need to beable to describe the stress components at 0 for any orientation of the axesthrough the point. To do this, consider an oblique plane normal to the plane ofthe paper at an angle () between the x axis and the outward normal to the obliqueplane. Let the normal to this plane be the x' direction and the direction lying in

1 S. P. Timoshenko, and 1. N. Goodier, "Theory of Elasticity," 2d ed., McGraw-Hill BookCompany, New York, 1951.

,


yx'

TxyUx -.-+

o"---'-:::::::==~~ --)0)0- x< TyX

tUy Figure 2-3 Stress on oblique

plane (two dimensions).

the oblique plane the y' direction. It is assumed that the plane shown in Fig. 2-3is an infinitesimal distance from 0 and that the element is so small that variationsin stress over the sides of the element can be neglected. The stresses acting on theoblique plane are the normal stress (J and the shear stress 'T. The direction cosinesbetween x' and the x and y axes are I and m, respectively. From the geometry ofFig. 2-3, I = cos () and m = sin (). If A is the area of the oblique plane, the areasof the sides of the element perpendicular to the x and y axes are Al and Am.

Let Sx and Sy denote the x and y components of the total stress acting onthe inclined face. By taking the summation of the forces in the x direction andthe y direction, we obtain

SxA = (JxAI + 'TxyAmSyA = (JyAm + 'TxyAI

or Sx = (Jx cos () + 'Txy sin ()Sy = (Jy sin () + 'Txy cos ()

The components of Sx and Sy in the direction of the normal stress (J areSxN = Sx cos () and SyN = Sy sin ()

so that the normal stress acting on the oblique plane is given by(Jx' = Sx cos () + Sy sin ()(Jx' = (Jx cos 2 () + (Jy sin2 () + 2'Txy sin () cos ()

(2-2)

(2-3)

The shearing stress on the oblique plane is given by'Tx'y' = Sy cos () - Sx sin ()'Tx'y' = 'Txy(cos 2 () - sin2 ()) + ((Jy - (Jx)sin()cos()

The stress (Jy' may be found by substituting () + 7T/2 for () in Eq. (2-2), since (Jy'is orthogonal to (Jx"(Jy' = (Jx cos 2 (() + 7T/2) + (Jy sin2 (() + 7T/2) + 2'Txy sin (() + 7T/2) cos (() + 7T/2) .and since sin () + 7T/2) = cos () and cos () + 7T/2) = - sin (), we obtain

(2-4),

,

'I

i


Equations (2-2) to (2-4) are the transformation of stress equations which give thestresses in an x'y' coordinate system if the stresses in an xy coordinate systemand the angle 0 are known. .

To aid in computation, it is often convenient to express Eqs. (2-2) to (2-4) interms of the double angle 20. This can be done with the following identities:

cos20 + 1cos 2 0= ----

21 - cos 20

sin2 0=----2

2 sin 0 cos 0 = sin 20cos 2 0 - sin2 0 = cos 20

The transformation of stress equations now become(Jx + (Jy (Jx - 0y

(Jx, = 2 + 2 cos 20 + 'Txy sin 20

(Jx + (Jy (Jx - 0y(Jy' = 2 - 2 cos 20 - 'Txy sin 20

(2-5)

(2-6)

(2-7)>

It is important to note that (Jx' + 0y' = (Jx + 0y- Thus the sum of the normalstresses on two perpendicular planes is an invariant quantity, that is, it isindependent of orientation or angle O.

Equations (2-2) and (2-3) and their equivalents, Eqs. (2-5) and (2-7), describethe normal stress and shear stress on any plane through a point in a bodysubjected to a plane-stress situation. Figure 2-4 shows the variation of normalstress and shear stress with 0 for the biaxial-plane-stress situation given at the topof the figure. Note the following important facts about this figure:

1. The maximum and minimum values of normal stress on the oblique planethrough point 0 occur when the shear stress is zero.

2. The maximum and minimum values of both normal stress and shear stressoccur at angles which are 90 apart.

3. The maximum shear stress occurs at an angle halfway between the maximumand minimum normal stresses.

4. The variation of normal stress and shear stress occurs in the form of a sinewave, with a period of 0 = 180. These relationships are valid for any state ofstress.

For any state of stress it is always possible to define a new coordinate systemwhich has axes perpendicular to the planes on which the maximum normalstresses act and on which no shearing stresses act. These planes are called theprincipal planes, and the stresses normal to these planes are the principal stresses.For two-dimensional plane stress there will be two principal stresses (Jl and (J2which occur at angles that are 90 apart (Fig. 2-4). For the general case of stress


-40

Figure 2-4 Variation of normal stress and shear stress on oblique plane with angle B.

(2-8)tan20 =

in three dimensions there will be three principal stresses aI' 0'2' and 0'3' Accordingto convention, 0'1 is the algebraically greatest principal stress, while 0'3 is thealgebraically smallest stress. The directions of the principal stresses are theprincipal axes 1, 2, and 3. Although in general the principal axes 1, 2, and 3 donot coincide with the cartesian-coordinate axes x, y, Z, for many situations thatare encountered in practice the two systems of axes coincide because of symmetryof loading and deformation. The specification of the principal stresses and theirdirection provides a convenient way of describing the state of stress at a point.

Since by definition a principal plane contains no shear stress, its angularrelationship with respect to the xy coordinate axes can be determined by findingthe values of 0 in Eq. (2-3) for which 'Tx'y' = O.

'TXy(cos 2 0 - sin2 0) + (ay - ax)sinOcosO = 0'Txy sin ocos 0 Hsin20) 1

O=-tan20

ax - ay cos 2 0 - sin2 0 cos2 2

2'Txy

J


Since tan 20 = tan (71 + 20), Eq. (2-8) has two roots, 01 and O2 = 01 + n7T/2.These roots define two mutually perpendicular planes which are free from shear.

Equation (2-5) will give the principal stresses when values of cos 20 andsin 20 are substituted into it from Eq. (2-8). The values of cos 20 and sin 20 arefound from Eq. (2-8) by means of the pythagorean relationships.

7"xysin 20 = + -=-[-;.---)-2--'----2-=-]:-1/=2

~Ox - 0y /4 + 7"xy(Ox - 0y)/2

cos 20 = + ---------''--------,..--..".-[( ) 2 2 ] 1/2Ox - 0y /4 + 7"xy

Substituting these values into Eq. (2-5) results in the expression for the maximumand minimum principal stresses for a two-dimensional (biaxial) state of stress.

0max = 1min = 2 2

1/2(2-9)

The direction of the principal planes is found by solving for 0 in Eq. (2-8),taking special care to establish whether 20 is between 0 and 71/2, 71, and 371/2,etc. Figure 2-5 shows a simple way to establish the direction of the largestprincipal stress 1 ~ 1 will lie between the algebraically largest normal stress andthe shear diagonal. To see this intuitively, consider that if there were no shearstresses, then Ox = 1, If only shear stresses act, then a normal stress (the principalstress) would exist along the shear diagonal. If both normal and shear stresses acton the element, then 1 lies between the influences of these two effects.

To find the maximum shear stress we return to Eq. (2-7). We differentiate theexpression for 7"x'Y' and set this equal to zero.

d~.y. .dO = (Oy - oJcos20 - 2'Txy sm20 = 0

(2-10)

Comparing this with the angle at which the principal planes occur, Eq. (2-8),tan 20n = 27"xy/( Ox - Oy), we see that tan 20s is the negative reciprocal of tan 20n.

I-

Ux > "y

Sheardiagonal

Figure 2-5 Method of establishing direction of Cl j .


This means that 20s and 20n are orthogonal, and that Os and On are separated inspace by 45 0. The magnitude of the maximum shear stress is found by substitut-ing Eq. (2-10) into Eq. (2-7).

2

2 1/22+ Txy (2-11)

Example The state of stress is given by Ox = 25P and 0y = 5p plus shearingstresses Txy On a plane at 45 counterclockwise to the plane on which Ox actsthe state of stress is 50 MPa tension and 5 MPa shear. Determine the valuesof ox, Oy, Txy '

From Eqs. (2-5) and (2-7)Ox + 0y Ox - Oy .

Ox' = 2 + 2 cos 20 + Txy sm20 Eq.(2-5)

25p + 5P 25P - 5p50 X 10 6 = 2 + 2 cos 90 + Txy sin 90

15p + r xy = 50 x 106 Pa

Eq. (2-7),

5 X 10 6 =5p - 25p

sin 90 + Txy cos 902

-lOp = 5 x 106 p = -5 X 105 Pa

Ox = 25( -5 X 10 5 ) = -12.5 MPa0y = 5( p) = - 2.5 MPa

Txy = 50 X 10 6 - 15( -5 X 10 5 )

= 50 X 10 6 + 7.5 X 10 6 = 57.5 MPa

We also can find Oy" orthogonal to Ox' = 50 MPa, since Ox + 0y = Ox' + 0y'

-12.5 - 2.5 = 50 + Oy'

0y' = -65 MPa

2-4 MOHR'S CIRCLE OF STRESS-TWO DIMENSIONS

A very useful graphical method for representing the state of stress at a point onan oblique plane through the point was suggested by O. Mohr. The transforma-

STRESS AND STRAIN RELATIONSillPS FOR ELASTIC BEHAVIOR 25

tion of stress equations, Eqs. (2-5) and (2-7), can be rearranged to give

ay - ax .'r:, , = SIn 28 + Txy cos 28yx 2

We can solve for ax, in terms of Tx'y' by squaring each of these equations andadding

2(2-12)

Equation (2-12) is the equation of a circle of the form (x - h)2 + y2 = ,2. Thus,Mohr's circle is a circle in ax" Tx'y' coordinates with a radius equal to Tmax and thecenter displaced (ax + ay )/2 to the right of the origin.

In working with Mohr's circle there are only a few basic rules to remember.An angle of 8 on the physical element is represented by 28 on Mohr's circle. Thesame sense of rotation (clockwise or counterclockwise) should be used in eachcase. A different. convention to express shear stress is used in drawing andinterpreting Mohr's circle. This convention is that a shear stress causing aclockwise rotation about any point in the physical element is plotted above thehorizontal axis of the Mohr's circle. A point on Mohr's circle gives the magnitudeand direction of the normal and shear stresses on any plane in the physicalelement.

Figure 2-6 illustrates the plotting and use of Mohr's circle for a particularstress state shown at the upper left. Normal stresses are plotted along the x axis,shear stresses along the y axis. The stresses on the planes normal to the x and y

cr2B f\

\ Tmax+Txy\\ D\ cr+

t-

~ 0 rrl:---:::::::::::=l a --'" - II

: 1>0I'V!-~~-l Figure 2-19 Interaction energy vs. separation between

atoms

where ( u) is the interaction bond energy at a displacement u. Thus, the force ona bond is a function of displacement u. For each displacement there is acharacteristic value of force P( u). Moreover, the deformation of the bondsbetween atoms is reversible. When the displacement returns to some initial valueu1 after being extended to U 2 the force returns to its previous value P( ud.

In an elastic solid the bond energy is a continuous function of displacement. lThus, we can express (u) as a Taylor series

-, '

( u) = 0 + d 1u+-du 0 2 (2-88)

where 0 is the energy at u = 0 and the differential coefficients are measured atu = O. Since the force is zero when a = ao, d/du = 0

--~

1 ( u) = 0 + -2

d(u)P=----

du

(2-89),i

I

The coefficient (d 2/du 2 )0 is the curvature of the energy-distance curve atu = ao' Since it is independent of u, the coefficient is a constant, and Eq. (2-89) isequivalent to P = ku, which is Hooke's law in its original form. When Eq. (2-89)is expressed in terms of stress and strain, the coefficient is directly proportional tothe elastic constant of the material. It has the same value for both tension andcompression since it is independent of the sign of u. Thus, we have shown thatthe elastic constant is determined by the sharpness of curvature of the minimumin the energy-distance curve. It is therefore a basic property of the material, notreadily changed by heat treatment or defect structure, although it would beexpected to decrease with increasing temperature. Moreover, since the bindingforces will be strongly affected by distance between atoms, the elastic constantswill vary with direction in the crystal lattice.

1 This development follows that given by A. H. Cottre)l, "The Mechanical Properties of Matter,"pp. 84-85, John Wiley & Sons, Inc., New York, 1964. '


In the generalized easel Hooke's law may be expressed as

(2-90)and

(2-91)where S,jk/ is the compliance tensor and Cijk / is the elastic stiffness (often calledjust the elastic constants). Both Sijk/ and Cijk / are fourth-rank tensor quantities.If we expanded Eq. (2-90) or (2-91), we would get nine equations, each with nineterms, 81 constants in all. However, we know that both ij and (Jij are symmetrictensors, that is, (Jij = (Jji' which immediately leads to appreciable simplification.Thus, we can write

and since

Also, we could write

Eij = Sijk/(Jk/ or ij = Sij/k(J/k

Sijk/(Jk/ = Sij/k(J/k

(J k/ = (J/k and Sijk/ = Sijlk

ij = Sijk/(Jk/ = ji = Sjik/(Jk/

Sijk/ = Sjik/

Therefore, because of the symmetry of the stress and strain tensors, only 36 of thecomponents of the compliance tensor are independent and distinct terms. Thesame is true of the elastic stiffness tensor.

Expanding Eq. (2-91) and taking into account the above relationships givesequations like

(Jll = Cllllll + CI122 22 + C1l3333 + C1l23(223) + Cll13(213) + Cll12 (212) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .. . . .. .. .. .. . .. . . .. .. .. .. .. . .. .. .. .. .. . . .. .. .. .. .. .. .. .. .. .. . . . .. .. . .. .. .. . . . .. .. .. .. .. .

.. .. .. .. . .. . .. .. .. .. .. . .. . . .. . .. . .. .. .. . . . .. . . . .. . .. . .. .. . . .. .. .. .. .. . .. . .. . .. . .. . .. .. .. . .. ..

(2-92)These equations show that, in contrast to the situation for an isotropic elasticsolid, Eq. (2-72), for an anisotropic elastic solid both normal strains and shearstrains are capable of contributing to a normal stress.

I An excellent text that deals with the anisotropic properties of crystals in tensor notation is J. F.Nye, "Physical Properties of Crystals," Oxford University Press, London, 1957. For a treatment ofanisotropic elasticity see R. F. S. Hearmon, "An Introduction to Applied Anisotropic Elasticity,"Oxford University Press, London, 1961. A fairly concise but complete discussion of crystal elasticity isgiven by S. M. Edelglass, "Engineering Materials Science," pp. 277-301. The Ronald Press Company,New York, 1966.


In expanding Eq. (2-90), we express the shearing strains by the moreconventional engineering shear strain y = 2E.

En = Snnan + Sll22a22 + S1l33a 33 + 2S1123a23 + 2Slll3a 13 + 2S11l2a l2

Y23 = 2E 23 = 2S23lla ll + 2S2322a 22 + 2S233 P33 +4S2323a23+4S23l3a l3 + 4S231Pl2

(2-93)The usual convention for designating components of elastic compliance and

elastic stiffness uses only two subscripts instead of four. This is called thecontracted notation. The subscripts simply denote the row and column in thematrix of components in which they fall.

an = CllEll + Cl2 E22 + C l3 E33 + C14Y23 + C1S Y13 + C 16 Yl2

(2-94)

and

(2-95)

By comparing coefficients in Eqs. (2-92) and (2-94) and Eqs. (2-93) and (2-95) wenote, for example, that

C2322 = C42Sll22 = Cl2

C1l22 = Cl22S2311 = C41

The elastic stiffness constants are defined by equations like6.all

Cll = A all Eij constant except EllUEll

Unfortunately, a measurement such as this is difficult to do experimentally since

all O'ii constant except all


the specimen must be constrained mechanically to prevent strains such as E23 . It ismuch easier to experimentally determine the coefficients of the elastic compliancefrom equations of the type

LlEUSu = A

L.1O'u

If the components of Sii have been determined experimentally, then the compo-nents of Cii can be determined by matrix inversion.

At this stage we have 36 independent constants, but further reduction in thenumber of independent constants is possible. By using the relationship given inEq. (2-86), we can show that the constants are symmetrical, that is, Cii = ~i' Forexample,

au

au

In general, Cii = ~i and Sij = Sii. Now, we start with 36 constantsCii , but ofthese there are six consants where i = j. This leaves 30 constants where i =1= j, butonly ope-half of these are independent constants since Cii = ~i. Therefore, forthe general anisotropic linear elastic solid there are 30/2 + 6 = 21 independentelastic constants.

As a result of symmetry conditions found in different crystal structures thenumber of independent elastic constants can be reduced still further.

Crystal structure

TriclinicMonoclinicOrthorhombicTetragonalHexagonalCubicIsotropic'

Rotationalsymmetry

None1 twofold rotation2 perpendicular twofold rotations1 fourfold rotation1 sixfold rotation4 threefold rotations

Number of independentelastic constants

2113

96532


Table 2-2 Stiffness and compliance constants forcubic crystals

Metal C l1 C l2 C44 811 8' 2 844

Aluminum 108.2 61.3 28.5 15.7 -5.7 35.1Copper 168.4 121.4 75.4 14.9 -6.2 13.3Iron 237.0 141.0 116.0 8.0 -2.8 8.6Tungsten 501.0 198.0 151.4 2.6 -0.7 6.6

Stiffness constants in units of GPa.Compliance constants in units of TPa - 1.

For a cubic crystal structure

(2~~6)

/

Jl

The modulus of elasticity in any direction of a cubic crystal (described by thedirection cosines I, m, n) is given by

(2-97)

Typical values of elastic constants for cubic metals are given in Table 2-2.By comparing the generalized Hooke's law Eqs. (2-95) with the equations

using the common technical moduli Eq. (2-64) we can conclude that the elasticconstants for an isotropic material are given by

1Sll = E

P--

E1

S =-44 GSince Sll and S12 are the independent constants, their relationship to S44 can be .obtained from Eq. (2-68)

or

E 1G= =------

2(1+p) 2(1/E+p/E)1 1

G= ------S44 2(Sll - S12)

S44 = 2(Sll - S12) (2-98)Comparable equations relating the elastic stiffness constants can be developed j

1.,

'IJI

I


from Eqs. (2-95) and (2-74).Cll = A Lame's constant

Cll = 2G + A (2-99)

C44 = ~(Cll - Cll )The technical elastic moduli E, P, and G are usually measured by direct

static measurements in the tension or torsion tests. However, where more precisemeasurements are required or where measurements are required in small single-crystal specimens cut along specified directions, dynamic techniques usingmeasurement of frequency or elapsed time are frequently employed. Dynamicmeasurements involve very small atomic displacements and low stresses comparedwith static modulus measurements. The velocity of propagation of a displacementdown a cylindrical-crystal specimen is given by

\

WAu =

x 217 p(2-100)

,

(2-101)

where W is the natural frequency of vibration of a stress pulse of wavelength A ina crystal of density p. Dynamic techniques consist of measuring either the naturalfrequency of vibration or the elapsed time for an ultrasonic pulse to travel downthe specimen and return. Because the strain cycles produced in dynamic testingoccur at high rates, there is very little time for heat transfer to take place. Thus,dynamic measurements of elastic constants are obtained under adiabatic condi-tions, while static elastic measurements are obtained under essentially isothermalconditions. There is a small difference between adiabatic and isothermal elasticmoduli. l

E isoEadi = ----~2:EisoTa1----9c

where a is the volume coefficient of thermal expansion and c is the specific heat.Since the specific heat of a solid is large compared to a gas, the difference betweenadiabatic and isothermal moduli is not great and can be ignored for practicalpurposes.

Example Determine the modulus of elasticity for tungsten and iron in the(111) and (100) directions. What conclusions can be drawn about theirelastic anisotropy? From Table 2-2

Fe:W:

8.02.6

-2.8-0.7

8.66.6

I For a derivation of Eq. (2-101) see S. M. Edelgass, op. cit., pp. 294-297.


The direction cosines for the chief directions in a cubic lattice are:

Directions I m n(100) 1 0 0(110) 1/12 1/12 0(111) 1//3 1//3 1//3

For iron:1 1 1 1

- = 8.0 - 2{(8.0 + 2.8) - 8.6/2} - + - +-E ll1 9 9 9

111E = 8.0 - 2(10.8 - 4.3) 3 = 8.0 - 13.0 3

111

= 8.0 - 4.3 = 3.7 TPa- 1

1E 111 = TPa = 270 GPa3.7

E1

= 8.0 - 13.0(0) = 8.0 TPa- 1100

For tungsten:

E100 = 125 GPa

.

,

1 6.6 1- = 2.6 - 2 (2.6 + 0.7) - -E111 2 3

1 1- = 2.6 - ,2{3.3 - 3.3} - = 2.6 TPa- 1E ll1 3

1E ll1 = 0.26 TPa = 385 GPa

1 _ _ \ _ 6.6 _ _ 1E - 2.6 2 (2.6 + 0.7; 2 (0) - 2.6 TPa100

1E100 = 0.26 TPa = 385 GPa

Therefore, we see that tungsten is elastically isotropic while iron iselastically anisotropic.

2-15 STRESS CONCENTRATION

A geometrical discontinuity in a body, such as a hole or a notch, results in anonuniform stress distribution at the vicinity of the discontinuity. At some regionnear the discontinuity the stress will be higher than the average stress at distancesremoved from the discontinuity. Thus, a stress concentration occurs at the

jjll1;i

ySTRESS AND STRAIN RELATIONSHIPS FOR ELASTIC BEHAVIOR 61

y

0''\~./ -1 o-m, O"nom~7A- - - t

-.. 8;X r ere8

'\.,o-r

,

./ ~~

- -

'bI--a-> i

I

)(

(a) (b)

Figure 2-20 Stress distributions due to (a) circular hole and (b) elliptical hole.

discontinuity, or stress raiser. Figure 2-20a shows a plate containing a circularhole which is subjected to a uniaxial load. If the hole were not present, the stresswould be uniformly distributed over the cross section of the plate and it would beequal to the load divided by the cross-sectional area of the plate. With the holepresent, the distribution is such that the axial stress reaches a high value at theedges of the hole and drops off rapidly with distance away from the hole.

The stress concentration is expressed by a theoretical stress-concentrationfactor K t Generally K t is described as the ratio of the maximum stress to thenominal stress based on the net section, although some workers use a value ofnominal stress based on the entire cross section of the member in a region wherethere is no stress concentrator.

amaxKt=---

anominal(2-102)

In addition to producing a stress concentration, a notch also creates alocalized condition of b

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