DEFORMATION THEORY OF PLASTICITY ROBERT M. JONES Professor Emeritus of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia 240610219 Bull Ridge Publishing Blacksburg, Virginia United States of America copyright 2009 by Bull Ridge Publishing all rights reserved
Perfectly Perfectly CONTENTS PREFACE xv 1 INTRODUCTION 1 1.1 1.2 1.3 MATERIAL MODELING EXAMPLES OF NONLINEAR MATERIAL BEHAVIOR 1.2.1 MetalForming Operations 5 Nonlinear Materials 7 1.2.2 Inherently 1.2.3 Composite Materials 7 PROBLEMS IN PLASTICITY 1.3.1 Elastic Plastic Problems 9 1.3.2 Rigid Plastic Problems 11 1.3.3 Other Material Models for Plasticity Problems 12 1.3.4 Contained Plasticity Problems 12 1.3.5 General Comments on Plasticity Problems 14 3 4 9 1.4 A HISTORY OF THE DEFORMATION THEORY OF PLASTICITY... 15 1.4.1 The Origin and Meaning of Hooke's Law 15 1.4.2 The Early Days of Deformation Theory 16 1.4.3 Deformation Theory versus Incremental Theory 17 1.4.4 Early Developments in Plasticity 18 1.4.5 Comments on Naming of Theories, Laws, and Phenomena 23 1.4.6 The Linear StressStrain Relation and its Limits 25 1.4.7 Summary 28 1.5 IMPORTANCE OF EXPERIMENTAL RESULTS IN DEVELOPMENT OF THEORIES 29 1.5.1 Introduction 29 1.5.2 Comparison of Theoretical and Experimental Results 30 1.5.3 Experimental Results are Used More than 'Just' to Validate Theory 31 1.5.4 Experimental Results are the Very Basis of ail Theoretical Models 33 1.5.5 Summary 35 1.6 SOME IMPORTANT DISTINCTIONS IN ENGINEERING ANALYSIS 36 1.6.1 Introduction 36 1.6.2 Presumption versus Assumption 36 1.6.3 Presumption versus Restriction 43 1.6.4 Misinterpretations and their Consequences 45 1.6.5 Summary 47 1.7 BOOK OUTLINE 49 CHAPTER 1 REFERENCES 50 vii
Perfectly Perfectly viii 2 CHARACTERISTICS OF NONLINEAR MATERIAL BEHAVIOR 53 2.1 INTRODUCTION 2.2 ELEMENTS OF STRESSSTRAIN BEHAVIOR 2.2.1 Definitions of Stress and Strain 54 2.2.2 Yielding and the Yield Point 57 2.2.3 Necking 57 2.2.4 Poisson's Ratio 58 2.2.5 Offset Yield Stress 58 2.2.6 Yielding versus Collapse 60 53 54 2.3 TYPES OF STRESSSTRAIN BEHAVIOR 61 2.3.1 Ductile versus Brittle Behavior 62 2.3.2 High Strength versus High Toughness 63 2.3.3 Linear versus Nonlinear Elastic Behavior 64 2.3.4 Proportional Limit versus Elastic Limit 65 2.3.5 Effect of Temperature 65 2.3.6 Effect of Moisture 67 2.3.7 Anisotropy 68 2.3.8 Bimodulus StressStrain Behavior 71 2.3.9 Combined Response 73 2.4 IDEALIZATIONS OF STRESSSTRAIN BEHAVIOR 77 2.4.1 Elastic Plastic 77 2.4.2 Linear Hardening 77 2.4.3 Rigid Plastic 78 2.4.4 PowerLaw Plasticity 78 2.4.5 Viscoelasticity 78 2.4.6 Summary of Idealizations 79 2.4.7 Common Definitions 80 2.5 LOADING AND UNLOADING BEHAVIOR OF INELASTIC MATERIALS 82 2.5.1 Actual Unloading and Reloading Behavior 82 2.5.2 Idealized Loading and Unloading Behavior 85 2.5.3 Strain Hardening 87 2.5.4 Bauschinger Effect 89 2.5.5 Hysteresis Effect 91 2.5.6 Anelastic Behavior 92 2.6 SUMMARY 93 CHAPTER 2 REFERENCES 94 3 FUNDAMENTALS OF ELASTICITY 95 3.1 INTRODUCTION 95 3.2 THEORY OF STRESS 95 3.2.1 Stress at a Point 95 3.2.2 Stress Tensor 97 3.2.3 Principal Stresses 98 3.2.4 Stress Invariants 100 3.2.5 Calculation of Principal Stresses 101 3.2.6 Spherical and Deviator Stress Tensors 106 3.2.7 Normal and Shear Components of Stress 108 3.2.8 Octahedral Stresses 109 3.2.9 Mohr's Circle 112
ix 3.3 THEORY OF STRAIN, 113 3.3.1 Infinitesimal Strain at a Point 113 3.3.2 Principal Strains and Strain Invariants 116 3.3.3 Spherical and Deviator Strain Tensors 117 3.3.4 Octahedral Strains 118 3.4 ELASTIC STRESSSTRAIN RELATIONS 119 3.5 ELASTIC STRAIN ENERGY 121 PROBLEM SET 3 123 CHAPTER 3 REFERENCE 124 4 YIELDING AND YIELD CRITERIA 125 4.1 INTRODUCTION 125 4.2 YIELDING 126 4.2.1 Causes of Yielding in Metals 126 4.2.2 Nature and Characteristics of Yielding 127 4.3 YIELD CURVES AND YIELD SURFACES 129 4.3.1 Biaxial Stress States (Yield Curves) 129 4.3.2 Triaxial Stress States (Yield Surfaces) 132 4.3.3 Projection of Triaxial Behavior on the % Plane 136 4.4 SUBSEQUENT YIELD OR LOADING SURFACES HARDENING... 140 4.4.1 Loading Beyond Initial Yielding 140 4.4.2 Isotropic Hardening 141 4.4.3 Kinematic Hardening 141 4.4.4 BatdorfBudiansky Slip Theory with Modifications for the Bauschinger Effect 144 4.5 YIELD CRITERIA 146 4.5.1 Introduction 146 4.5.2 Maximum Principal Stress Yield Criterion 147 4.5.3 Maximum Principal Strain Yield Criterion 148 4.5.4 Maximum Shear Stress or Tresca Yield Criterion 148 4.5.5 Maximum Strain Energy Yield Criterion 150 4.5.6 Maximum Distortional Energy or Mises Yield Criterion 151 4.5.7 Comparison of the Tresca and Mises Yield Criteria 154 4.5.8 Other Yield Criteria 158 4.5.9 Extension of Present Plasticity Concepts for Isotropic Materials to Nonlsotropic Materials 160 4.6 SUMMARY 162 PROBLEM SET 4 163 CHAPTER 4 REFERENCES 163 5 DEFORMATION THEORY OF PLASTICITY 165 5.1 INTRODUCTION 165 5.2 PLASTIC INCOMPRESSIBILITY AND POISSON'S RATIO 166 5.3 STRESSSTRAIN RELATIONS 168
X 5.4 THE STRESS AND STRAIN INTENSITIES AND THE UNIVERSAL STRESSSTRAIN CURVE 171 5.4.1 Elasticity Preliminaries 171 5.4.2 Transition to the Deformation Theory of Plasticity 172 5.4.3 The Universal StressStrain Curve 175 5.4.4 Historical Review and Comments 178 5.5 NONLINEAR STRESSSTRAIN BEHAVIOR CURVE MODELS 179 5.5.1 Linear StrainHardening StressStrain Curve Model 179 5.5.2 PowerLaw StressStrain Curve Model 180 StressStrain Curve Model 182 5.5.4 Nadai StressStrain Curve Model 184 5.5.3 RambergOsgood 5.5.5 Generalized Nadai StressStrain Curve Model 186 5.5.6 NadaiJones StressStrain Curve Model 191 5.5.7 Summary Remarks on StressStrain Curve Modeling 194 5.6 COMMENTS ON HENCKY'S AND NADAI'S CONTRIBUTIONS 194 5.7 SUMMARY REMARKS ON DEFORMATION THEORY 195 PROBLEM SET 5 196 CHAPTER 5 REFERENCES 197 6 INCREMENTAL THEORY OF PLASTICITY 199 6.1 INTRODUCTION 199 6.2 STRESS HISTORY IMPORTANCE 200 6.3 STRESS AND STRAIN INCREMENTS UNDER UNIAXIAL STRESS 202 6.4 STRAIN INCREMENTS UNDER MULTIAXIAL STRESSES 205 6.5 VARIATIONAL APPROACH TO PLASTIC STRAIN INCREMENTS FOR MULTIAXIAL STRESS INCREMENTS 209 6.6 LOADING, UNLOADING, AND NEUTRAL LOADING 213 6.7 PLASTIC WORK AND HARDENING 214 6.8 SUMMARY REMARKS 219 PROBLEM SET 6 220 CHAPTER 6 REFERENCES 220 7 SOLUTION OF PLASTICITY PROBLEMS 221 7.1 INTRODUCTION 221 7.2 DEGREE OF MATERIAL NONLINEARITY 221 7.3 PLASTIC STRAIN VERSUS TOTAL STRAIN RELATIONS 222 7.4 METHOD OF SUCCESSIVE ELASTIC SOLUTIONS 226 7.5 COMMENTS ON LOADING PATH AND HISTORY 228 7.5.1 Introduction 228 7.5.2 Generalization of the Deformation Theory Restriction to Proportional Loading Paths 228 7.5.3 Generalization of the Loading and Unloading Paths 228 7.5.4 General Comments 230 7.6 MATERIAL STRESSSTRAIN BEHAVIOR VERSUS STRUCTURAL RESPONSE 232 7.7 THE ELASTIC STRAIN RESPONSE APPROXIMATION 234 7.8 SUMMARY 234 CHAPTER 7 REFERENCES 235
PERFECTLY PERFECTLY xi 8 THICKWALLED SPHERICAL SHELLS UNDER INTERNAL PRESSURE AND/OR INNERSURFACE HEATING 237 8.1 INTRODUCTION 237 8.2 FUNDAMENTAL RELATIONS AND CHARACTERISTICS 239 8.3 BEHAVIOR OF ELASTIC PLASTIC SHELLS UNDER INTERNAL PRESSURE 241 8.3.1 Linear Elastic Behavior 241 8.3.2 Limit of Linear Elastic Behavior (Yielding) 243 8.3.3 Collapse Pressure 244 8.3.4 Behavior between Yielding and Collapse 246 8.3.5 Residual Stresses 253 8.4 BEHAVIOR OF STRAINHARDENING SHELLS UNDER INTERNAL PRESSURE 261 8.4.1 General StrainHardening Behavior 261 8.4.2 Linear StrainHardening Behavior 263 8.4.3 Summary of StrainHardening Effects 279 8.5 BEHAVIOR OF ELASTIC PLASTIC SHELLS UNDER STEADYSTATE INNERSURFACE HEATING 280 8.5.1 Linear Elastic Behavior 280 8.5.2 Limit of Linear Elastic Behavior (Yielding) 284 8.5.3 Comments on Initial Yielding of Elastic ThickWalled Spherical Shells under Combined Heating and Pressurization 287 8.5.4 PostYielding Behavior under InnerSurface Heating 293 8.5.5 Example with b/a = 2 301 8.5.6 Example with b/a = 3 314 8.5.7 Summary of Heating Studies 316 8.6 SUMMARY 317 8.6.1 InnerSurface Pressurization Results 317 8.6.2 InnerSurface Heating Results 318 8.6.3 Relation of Present Work to Past Work 319 8.6.4 Possible Future Research 319 PROBLEM SET 8 321 CHAPTER 8 REFERENCES 322 9 PLASTIC BUCKLING OF BARS, PLATES, AND SHELLS 323 9.1 INTRODUCTION 323 9.2 PLASTIC BUCKLING OF BARS 325 9.2.1 Introduction 325 9.2.2 Governing Equation and Solution for Elastic Buckling 325 9.2.3 Governing Equation for Plastic Buckling 330 9.2.4 ReducedModulus Theory 331 9.2.5 TangentModulus Theory 333 9.2.6 Transcendental Plastic Buckling Equation 333 9.2.7 Solution Strategy for the Buckling Criterion 335 9.2.8 Numerical Results 338 9.2.9 Comparison of ReducedModulus Theory with TangentModulus Theory 339 9.2.10 Summary 340 Problem Set 9.2 340
xii 9.3 PLASTIC BUCKLING OF PLATES 341 9.3.1 Introduction 341 9.3.2 Derivation of the Buckling Criterion 342 9.3.3 Solution Strategy for the Buckling Criterion 348 9.3.4 Numerical Results 350 9.3.5 Comparison of Theoretical and Experimental Results 352 9.3.6 Summary 356 Problem Set 9.3 356 9.4 PLASTIC BUCKLING OF SHELLS 357 9.4.1 Introduction 357 9.4.2 Derivation of the Buckling Criterion 358 9.4.3 Solution Strategy for the Buckling Criterion 363 9.4.4 Numerical Results 365 9.4.5 Summary 368 Problem Set 9.4 368 9.5 PLASTIC THERMAL BUCKLING OF BARS WITH TEMPERATUREDEPENDENT MATERIAL PROPERTIES... 369 9.5.1 Introduction 369 9.5.2 Plastic Thermal Buckling Analysis 371 9.5.3 Solution Strategy 378 9.5.4 Modeling Nonlinear StressStrain Behavior and Thermal Expansion Behavior 381 9.5.5 Numerical Results and Observations 389 9.5.6 Summary 395 9.6 PLASTIC THERMAL BUCKLING OF PLATES WITH TEMPERATUREDEPENDENT MATERIAL PROPERTIES... 397 9.6.1 Introduction 397 9.6.2 Plastic Thermal Buckling Analysis 401 9.6.3 Solution Strategy 414 9.6.4 Nonlinear TemperatureDependent Material Model 418 9.6.5 Numerical Results and Observations 425 9.6.6 Summary and Perspective Remarks 439 9.7 SUMMARY REMARKS ON MECHANICAL AND THERMAL PLASTIC BUCKLING OF BARS, PLATES, AND SHELLS 447 9.7.1 General Summary 447 9.7.2 Other Significant Contributions 447 9.7.3 Perspective or Design Significance 448 9.7.4 Design Considerations and Factor of Safety 450 9.7.5 Comparison with Experimental Results 452 CHAPTER 9 REFERENCES 453 10 NONCLASSICAL PLASTICITY 457 10.1 INTRODUCTION 457 10.2 THE NONLINEAR BIMODULAR MATERIAL MODEL 459 10.2.1 Linear Bimodular Behavior Models 459 10.2.1.1 Introduction 459 10.2.1.2 Criteria for a Consistent Model 460 10.2.1.3 Ambartsumyan Material Model 463 10.2.1.4 WeightedComplianceMatrix Material Model 467 10.2.1.5 Summary of the Linear Bimodular Material Models 471
xiit 10.2.2 Material Models for StressStrain Curve Nonlinearities 472 10.2.3 Extended StressStrain Curve Approach 476 10.2.4 Iteration Procedure for Nonlinear Bimodular Material Models 479 10.2.5 Convergence of the Iteration Procedure 481 10.2.6 Summary and Validation Procedure for Material Models 482 10.3 APPLICATION TO LAMINATED STRUCTURES 483 10.3.1 Nonlinear Lamina Model 483 10.3.1.1 Introduction 483 10.3.1.2 Theory of Uniaxial OffAxis Loading 484 10.3.1.3 BoronEpoxy Lamina Results 485 10.3.1.4 GraphiteEpoxy Lamina Results 489 10.3.1.5 Summary of the Nonlinear Lamina Model 491 10.3.2 Nonlinear Laminate Model 492 10.3.2.1 Introduction 492 10.3.2.2 Derivation of Theory 505 10.3.2.3 Nonlinear Response of Symmetric BoronEpoxy Laminates to Unixial Loading 501 10.3.2.4 Summary of the Nonlinear Laminate Model 504 10.3.3 Buckling of Laminated Plates 505 10.3.3.1 Introduction 505 10.3.3.2 Derivation and Solution of the Buckling Criterion 507 10.3.3.3 Numerical Results 514 10.3.3.4 Summary of Buckling of Laminated Plates 526 10.3.4 Summary Remarks on Nonlinear Laminated Structures 526 10.4 APPLICATION TO SOLID BODIES 528 10.4.1 Introduction 528 10.4.2 Modeling Particulate Composite Materials 528 10.4.2.1 Introduction 528 10.4.2.2 Establishing the Material Model 529 10.4.2.3 Uniaxial OffAxis Loading 531 10.4.2.4 Biaxial Behavior Characteristics 536 10.4.3 ATJS Graphite Tube under Biaxial Loading 539 10.4.3.1 Test Specimen 539 10.4.3.2 Biaxial Strain Correlations at Room Temperature for the Nonlinear Model with only Tensile Properties 540 10.4.3.3 Biaxial Strain Correlations at Room Temperature with the WCM Material Model 543 10.4.3.4 Biaxial Strain Correlations at 2000T (1100 C) with the RCM Material Model 545 10.4.3.5 Summary of Biaxial Strain Correlations 547 10.4.4 ATJS Graphite Annular Disk under Thermal Loading 548 10.4.4.1 Introduction 548 10.4.4.2 Temperature and Deformation Measurements 549 10.4.4.3 ATJS Graphite Material Characteristics 549 10.4.4.4 Temperature Interpolation of Mechanical Properties 554 10.4.4.5 InnerDiameter Change Predictions 555 10.4.4.6 Stress and Strain Predictions 557 10.4.4.7 Summary 559 10.4.5 OffAxis Loading of CarbonCarbon Materia! 560 10.4.5.1 Introduction 560 10.4.5.2 CarbonCarbon Nonlinear Bimodular Material Model 562
xiv 10.4.5.3 Effect of ShearExtension Coupling on Deformation under Uniaxial OffAxis Loading 569 10.4.5.4 Predicted and Measured Strain Response for Uniaxial OffAxis Loading 572 10.4.5.5 Summary of CarbonCarbon Material Modeling 578 10.5 CONCLUDING REMARKS ON NONCLASSICAL PLASTICITY... 579 CHAPTER 10 REFERENCES 580 587... 11 SUMMARY REMARKS AND OTHER TOPICS 583 11.1 INTRODUCTION 583 11.2 SUMMARY OF BOOK CONTENTS 584 11.2.1 Introduction 584 11.2.2 Characteristics of Nonlinear Material Behavior 584 11.2.3 Fundamentals of Elasticity 584 11.2.4 Yielding, Yield Criteria, and Loading Surfaces 584 11.2.5 Deformation Theory of Plasticity 585 11.2.6 Incremental Theory of Plasticity 585 11.2.7 Solution of Plasticity Problems 585 11.2.8 ThickWalled Spherical Shells under Internal Pressure 585 11.2.9 Plastic Buckling of Bars, Plates, and Shells 586 11.2.10 Nonclassical Plasticity 586 11.3 ON THE TRANSITION TO COMPUTATIONAL APPROACHES 11.3.1 How to Learn Complex Plastic Behavior 587 11.3.2 Simple Analytical Approaches vs. Numerical Approaches 589 11.3.3 Computational Tools Available 590 11.3.4 Replacement of Measured Behavior by Computer Simulation 592 11.3.5 Designing Structures 594 11.4 A SURVEY OF PLASTICITY ANALYSES AND CAPABILITIES... 596 11.5 HISTORY OF PLASTICITY SOURCES 597 11.6 SUMMARY OF HISTORY OF PLASTICITY 599 11.7 CONCLUDING REMARKS ON NONCLASSICAL PLASTICITY 601... CHAPTER 11 REFERENCES 601 APPENDIX A: ABSOLUTE MINIMUM OF A FUNCTION OF TWO INTEGER VARIABLES 603 APPENDIX B: NONLINEAR REGRESSION ANALYSIS FOR A, B, AND C 610 APPENDIX C: SELECTED BIBLIOGRAPHY 612 APPENDIX D: SUBROUTINE IHALVE 614 INDEX 615