This page intentionally left blank MECHANICAL BEHAVIOR OF MATERIALS The term mechanical behavior encompasses the response of materials to external forces. This text considers a wide range of topics, including mechanical testing to determine material properties, plasticity needed for FEM analyses of automobile crashes, means of altering mechanical properties, and treatment of several modes of failure. This text fits courses on mechanical behavior of materials taught in departments of mechanical engineering and materials science. It includes numerous examples and problems for student practice and emphasizes quantitative problem solving. End of chapter notes are included to increase students’ interest. W. F. Hosford is Professor Emeritus of Materials Science and Engineering at the Uni- versity of Michigan. He is the author of a number of books, including Metal Forming: Mechanics and Metallurgy, 2nd ed. (with R. M. Caddell), Mechanics of Crystals and Textured Polycrystals, and Mechanical Metallurgy. Mechanical Behavior of Materials WILLIAM F. HOSFORD University of Michigan Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge ,UK First published in print format - ---- - ---- © Cambridge University Press 2005 2005 Information on this title: www.cambrid g e.or g /9780521846707 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. - --- - --- Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (NetLibrary) eBook (NetLibrary) hardback Contents Preface pagexv NotetotheReader xix 1StressandStrain 1 Introduction 1 Stress 2 SignConvention3 Transformation of Axes 4 Principal Stresses 6 Mohr’s Stress Circles 6 Strains 9 Small Strains 11 Transformation of Axes 12 Mohr’s Strain Circles 14 Force and Moment Balances 15 Boundary Conditions 16 Note 17 Problems 18 2 Elasticity . 21 Introduction 21 Isotropic Elasticity 21 Variation of Young’s Modulus 24 Isotropic Thermal Expansion 26 Anisotropic Elasticity 27 Orientation Dependence of Elastic Response 29 Orientation Dependence in Cubic Crystals 31 Orientation Dependence in Noncubic Crystals 32 v viCONTENTS Orientation Dependence in Materials Other Than Single Crystals 34 Anisotropic Thermal Expansion 34 Notes 35 References 36 Problems 36 3TensileTesting 39 Introduction 39 Tensile Specimens 39 Stress–Strain Curves 40 Ductility 43 True Stress and Strain 44 The Bridgman Correction 45 Temperature Rise 47 Sheet Anisotropy 47 Measurement of Force and Strain 48 Axial Alignment 49 Special Problems 49 Notes 50 References 51 Problems 51 4 Other Tests of Plastic Behavior 53 Introduction 53 Compression Test 53 Plane-Strain Compression 56 Plane-Strain Tension 57 Biaxial Tension (Hydraulic Bulge Test) 57 Torsion Test 59 Bend Tests 61 Hardness Tests 62 Mutual Indentation Hardness 66 Note 67 References 67 Problems 67 5 Strain-Hardening of Metals 70 Introduction 70 Mathematical Approximations 70 Power-Law Approximation 72 Necking 73 Work per Volume 75 Localization of Strain at Defects 75 Notes 76 Problems 77 P1: GDZ 0521846706agg.xml CB818-Hosford 0 521 84670 6 March 14, 2005 12:1 CONTENTS vii 6 Plasticity Theory 80 Introduction 80 Yield Criteria 80 Tresca Maximum Shear Stress Criterion 81 Von Mises Criterion 82 Flow Rules 84 Principle of Normality 85 Effective Stress and Effective Strain 86 Other Isotropic Yield Criteria 89 Anisotropic Plasticity 90 Effect of Strain-Hardening on the Yield Locus 93 Notes 93 References 94 Problems 94 7 Strain-Rate and Temperature Dependence of Flow Stress 99 Introduction 99 Strain-Rate Dependence of Flow Stress 99 Superplasticity 102 Combined Strain and Strain-Rate Effects 106 Strain-Rate Sensitivity of bcc Metals 107 Temperature Dependence 110 Combined Temperature and Strain-Rate Effects 111 Hot Working 115 Notes 116 References 116 Problems 116 8 Slip 120 Introduction 120 Slip Systems 120 Schmid’s Law 121 Strains Produced by Slip 123 Strain-Hardening of fcc Single Crystals 125 Tensile Deformation of fcc Crystals 126 Slip in bcc Crystals 128 Slip in hcp Crystals 128 Lattice Rotation in Tension 129 viiiCONTENTS Lattice Rotation in Compression 131 Texture Formation in Polycrystals 132 Approximate Calculation of R-Values 133 Notes 134 References 135 Problems 135 9 Dislocation Geometry and Energy 139 Introduction 139 Theoretical Strength of Crystals 139 The Nature of Dislocations 141 Burgers Vectors 142 Energy of a Screw Dislocation 144 Reactions between Parallel Dislocations and Frank’s Rule 146 Stress Fields around Dislocations 147 Forces on Dislocations 149 Partial Dislocations in fcc Crystals 150 Stacking Faults 151 Notes 154 References 154 Problems 156 10 Dislocation Mechanics 158 Introduction 158 Frank–Read Sources 158 Dislocation Pile-Ups 161 Cross-Slip 161 Dislocation Intersections 163 Climb 166 Notes 166 References 167 Problems 167 11MechanicalTwinningandMartensiticShear 170 Introduction 171 Formal Notation 171 Twinning Shear 172 Twinning in fcc Metals 173 Twinning in bcc Metals 173 Twinning in hcp Metals 175 Shapes of Twins 178 Mechanism of Twinning 179 Martensite Transformation 182 Shape Memory and Superelasticity 183 [...]... so that it recrystallizes All of these statements are incorrect They are examples of incomplete understanding of the mechanical behavior of materials It is to be hoped that this book will clear up these misconceptions 1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials The term mechanical behavior refers to the response of materials to forces Under load,... 421 xiii Preface The term mechanical behavior encompasses the response of materials to external forces This text considers a wide range of topics These include mechanical testing to determine material properties, plasticity for FEM analyses of automobile crashes, means of altering mechanical properties, and treatment of several modes of failure The two principal responses of materials to external forces... products should understand how materials respond to applied stresses and strains to avoid unexpected deflection, deformation, and failure Understanding material behavior is also essential for shaping material, improving the mechanical behavior of materials for specific applications, and failure analysis Some engineers have many misconceptions about the mechanical behavior of materials The following statements... Weibul analysis is presented here Chapter 19 also covers methods of improving toughness of ceramics and the role of thermally induced stresses The most important aspect of the mechanical behavior of polymers is their great time dependence and the associated temperature dependence The effects of pressure on yielding and the phenomenon of crazing are also unique Rubber elasticity is very different from... distortion of a two-dimensional body dy u D’ A’ V A u +(∂u/∂x)dx D dx x 12 MECHANICAL BEHAVIOR OF MATERIALS a b c Figure 1.12 Illustration of shear and rotation With small deformations, (a) differs from (c) only by a rotation, (b) The total engineering shear strain, γ yx , is the sum of these angles, γ yx = ∂v/∂ x + ∂u/∂ y = γx y (1.23) Figure 1.12 shows that this definition excludes the effects of rotation... orientation and length of the fibers control properties The volume fraction of the stronger, stiffer phase controls the overall properties of all composites The final chapter on metal forming analyzes bulk-forming and sheet-forming operations This text differs from other books on mechanical behavior in several aspects The treatment of plasticity has a greater emphasis on the interrelationship of the flow, effective... competing books There is less coverage of fatigue failure and fracture mechanics This book may contain more material than can be covered in a single course Depending on the focus of the course, various chapters or portions of chapters may be omitted It is hoped that this book will be of value to mechanical engineers as well as materials engineers If the book is used in a mechanical engineering course, the... applications of a somewhat lower stress may cause fracture This is called fatigue The amount of deformation that a material undergoes is described by strain The forces acting on a body are described by stress Although the reader should already be familiar with these terms, they will be reviewed in this chapter 1 2 MECHANICAL BEHAVIOR OF MATERIALS Stress Stress, σ , is defined as the intensity of force at... are the values of σ x and σ y for this value of φ, σ1,2 = (σx + σ y )/2 ± [(σx − σ y )/2] cos 2θ + τx y sin 2θ or σ1,2 = (σx + σ y )/2 ± (1/2)[(σx − σ y )2 + 4τx y 2 ]1/2 (1.17) A Mohr’s circle diagram is a graphical representation of Equations (1.16) and (1.17) It plots as a circle with radius (σ 1 − σ 2 )/2 centered at (σ1 + σ2 )/2 = (σx + σ y )/2 (1.17a) 8 MECHANICAL BEHAVIOR OF MATERIALS τ σx... book have had both an introductory materials science course and a “strength of materials course From the strength of materials course, they can be expected to know basic concepts of stress and strain, how to resolve stresses from one axis system to another, and how to use Hooke’s laws in three dimensions They should be familiar with force and moment balances From the materials science course, they should . of automobile crashes, means of altering mechanical properties, and treatment of several modes of failure. This text fits courses on mechanical behavior of materials taught in departments of mechanical engineering. left blank MECHANICAL BEHAVIOR OF MATERIALS The term mechanical behavior encompasses the response of materials to external forces. This text considers a wide range of topics, including mechanical. (with R. M. Caddell), Mechanics of Crystals and Textured Polycrystals, and Mechanical Metallurgy. Mechanical Behavior of Materials WILLIAM F. HOSFORD University of Michigan