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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...

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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 University 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

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page xv Note to the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1 Stress 2 Sign Convention 3 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

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Orientation Dependence in Materials Other Than Single Crystals Anisotropic Thermal Expansion Notes References Problems

34 34 35 36 36

3 Tensile Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

CONTENTS

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

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Lattice Rotation in Compression Texture Formation in Polycrystals Approximate Calculation of R-Values Notes References Problems

131 132 133 134 135 135

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 11 Mechanical Twinning and Martensitic Shear . . . . . . . . . . . 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

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CONTENTS

Notes References Problems

184 185 186

12 Hardening Mechanisms in Metals . . . . . . . . . . . . . . . . . . . . . . 188 Introduction 188 Deformation of Polycrystals 188 Texture Strengthening 190 Crystal Structure 191 Grain Size 193 Strain-Hardening 194 Solid-Solution Strengthening 195 Dispersion Strengthening 196 Yield Points and Strain-Aging 199 Dynamic Strain-Aging 201 Combined Effects 202 Notes 206 References 206 Problems 206 13 Ductility and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Introduction 210 Ductile Fracture 212 Brittle Fracture 217 Impact Energy 220 Notes 224 References 224 Problems 225 14 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Introduction 227 Theoretical Fracture Strength 227 Stress Concentration 229 Griffith Theory 230 Orowan Theory 231 Fracture Modes 231 Irwin’s Fracture Analysis 231 Plastic Zone Size 233 Thin Sheets 235 Temperature and Loading Rate 236 Metallurgical Variables 237 Fracture Mechanics in Design 237 Compact Tensile Specimens 239 Strain-Energy Release 240 The J-integral 241

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Notes References Problems Appendix I Size and Shape of the Plastic Zone at a Crack Tip

242 243 243 245 245

15 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Introduction 247 Rheological Models 247 Series Combination of a Spring and Dashpot 248 Parallel Combination of a Spring and Dashpot 249 Combined Parallel–Series Model 250 More Complex Models 251 Damping 251 Natural Decay 253 Elastic Modulus – Relaxed Versus Unrelaxed 254 Thermoelastic Effect 255 Snoek Effect in bcc Metals 256 Other Damping Mechanisms 257 References 258 Notes 258 Problems 259 16 Creep and Stress Rupture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Introduction 262 Creep Mechanisms 262 Temperature Dependence of Creep 266 Deformation Mechanism Maps 267 Cavitation 268 Rupture Versus Creep 269 Extrapolation Schemes 270 Alloys for High-Temperature Use 273 Notes 275 References 275 Problems 276 17 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Introduction 279 Surface Observations 279 Nomenclature 281 S–N Curves 282

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Effect of Mean Stress The Palmgren–Miner Rule Stress Concentration Surfaces Design Estimates Metallurgical Variables Strains to Failure Crack Propagation Cyclic Stress–Strain Behavior Temperature and Cycling Rate Effects Fatigue of Polymers Fatigue Testing Design Considerations Summary Notes References Problems

283 285 286 288 290 291 291 294 296 297 300 302 302 303 303 304 304

18 Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Introduction 308 Small-Scale Stresses 308 Bauschinger Effect 311 Nonuniform Cooling 312 Nonuniform Material 313 Stresses from Welding 313 Stresses from Mechanical Working 314 Consequences of Residual Stresses 316 Measurement of Residual Stresses 317 Relief of Residual Stresses 319 Notes 321 References 321 Problems 321 19 Ceramics and Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Introduction 324 Elastic Properties 324 Slip Systems 325 Hardness 326 Weibull Analysis 326 Testing 329 Porosity 329 High-Temperature Behavior 331 Fracture Toughness 331 Toughening of Ceramics 334

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Fatigue Silicate Glasses Strength of Glasses Thermally Induced Stresses Delayed Fracture Glassy Metals Notes References Problems

335 335 338 339 340 341 342 343 343

20 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Introduction 345 Elastic Behavior 345 Rubber Elasticity 350 Damping 353 Yielding 353 Effect of Strain Rate 356 Effect of Pressure 356 Crazing 361 Yielding of Fibers in Compression 363 Fracture 364 Deformation Mechanism Maps 366 Shape-Memory Effect 367 References 368 Notes 368 Problems 370 21 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Introduction 372 Fiber-Reinforced Composites 372 Elastic Properties of Fiber-Reinforced Composites 372 Strength of Fiber-Reinforced Composites 376 Volume Fraction of Fibers 377 Orientation Dependence of Strength 378 Fiber Length 379 Failures with Discontinuous Fibers 381 Failure under Compression 382 Typical Properties 383 Particulate Composites 384 Brick-Wall Model 385 Lamellar Composites 387 Notes 388 References 389 Problems 389

CONTENTS

22 Mechanical Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Introduction 392 Bulk-Forming Energy Balance 392 Deformation Zone Geometry 396 Friction in Bulk Forming 399 Formability 400 Deep Drawing 401 Stamping 403 Notes 407 References 409 Problems 409 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 are deformation and fracture. The deformation may be elastic, viscoelastic (time-dependent elastic deformation), plastic, or creep (time-dependent plastic deformation). Fracture may occur suddenly or after repeated application of loads (fatigue). For some materials, failure is timedependent. Both deformation and fracture are sensitive to defects, temperature, and rate of loading. The key to understanding these phenomena is a basic knowledge of the threedimensional nature of stress and strain and common boundary conditions, which are covered in the first chapter. Chapter 2 covers elasticity, including thermal expansion. Chapters 3 and 4 treat mechanical testing. Chapter 5 is focused on mathematical approximations to stress–strain behavior of metals and how these approximations can be used to understand the effect of defects on strain distribution in the presence of defects. Yield criteria and flow rules are covered in Chapter 6. Their interplay is emphasized in problem solving. Chapter 7 treats temperature and strain-rate effects and uses an Arrhenius approach to relate them. Defect analysis is used to understand superplasticity as well as strain distribution. Chapter 8 is devoted to the role of slip as a deformation mechanism. The tensor nature of stresses and strains is used to generalize Schmid’s law. Lattice rotations caused by slip are covered. Chapters 9 and 10 treat dislocations: their geometry, movement, and interactions. There is a treatment of stacking faults in fcc metals and how they affect strain-hardening. Hardening by intersections of dislocations is emphasized. Chapter 11 treats the various hardening mechanisms in metallic materials. Mechanical twinning is covered in Chapter 12. Chapter 13 presents a phenomenological and qualitative treatment of ductility, and Chapter 14 focuses on quantitative coverage of fracture mechanics. Viscoelasticity (time-dependent elasticity) is treated in Chapter 15. Mathematical models are presented and used to explain stress and strain relaxation as well as damping xv

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PREFACE

and rate dependence of the elastic modulus. Several mechanisms of damping are presented. Chapter 16 is devoted to creep (time-dependent plasticity) and stress rupture. The coverage includes creep mechanisms and extrapolation techniques for predicting life. Failure by fatigue is the topic of Chapter 17. The chapter starts with a phenomenological treatment of the S–N curve and the effects of mean stress, variable stress amplitude, and surface condition. The important material aspects, Coffin’s law, and the crack propagation rate are treated. Chapter 18 covers residual stresses and their origins, effects, measurement, and removal. Chapters 19, 20, and 21 cover ceramics, polymers, and composites. Separate chapters are devoted to these materials because their mechanical behavior is very different from that of metals, which were emphasized in the earlier chapters. Because ceramics and glass are brittle and their properties are variable, 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 Hookean elasticity. Composites may be divided into fiber, sheet, and particulate composites. With fiber-reinforced composites, the 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 strain, and effective stress and their use in conjunction with yield criteria to solve problems. The treatment of defects is new. Schmid’s law is generalized for complex stress states. Its use with strains allows prediction of R-values for textures. Another feature is the treatment of lattice rotations and how they lead to deformation textures. Most texts treat only strain relaxation and neglect stress relaxation. The chapter on fracture mechanics includes Gurney’s approach. Most books omit any coverage of residual stresses. Much of the analysis of particulate composites is new. Few books include anything on metal forming. Throughout this book, more emphasis is placed on quantitative problem-solving than in most other books. The notes at the ends of the chapters are included to increase reader interest in the subject. As a consequence of the increased coverage in these areas, the treatment of some other topics is not as extensive as in 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 instructor may wish to skip some chapters. In particular, Chapters 8 through 11 may be omitted. If the book is used in a materials science course, the instructor may wish to omit Chapters 10, 18, and 22. Both may wish to skip Chapter 11 on twinning and memory metals. Even

PREFACE

xvii

though it was realized that most users may wish to skip this chapter, it was included for completeness and in the hope that it may prove useful as a reference. It is assumed that the students who use this 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 have acquired the understanding that most materials are crystalline and that crystalline materials deform by slip as a result of the movement of dislocations. They should also be familiar with such concepts as substitutional and interstitial solid solutions and diffusion. Appendices A (Miller Indices) and B (Stereographic Projection) are available for students not familiar with these topics. W. F. Hosford Ann Arbor, Michigan

Note to the Reader

Engineers who design 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 are typical: This spring isn’t stiff enough. We should use a harder steel. Perhaps a stronger steel would make a tougher pipeline. If something breaks, use a stronger material. This tungsten wire isn’t ductile enough. Let’s anneal it at a higher temperature. We are having too many fatigue failures. We’d get a longer life if we used a tougher grade of steel. I know that you are having trouble with the bar splitting when you roll it. Ma...