DEFORMATION THEORY OF PLASTICITY

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1 DEFORMATION THEORY OF PLASTICITY ROBERT M. JONES Professor Emeritus of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, Virginia Bull Ridge Publishing Blacksburg, Virginia United States of America copyright 2009 by Bull Ridge Publishing all rights reserved

2 Perfectly Perfectly CONTENTS PREFACE xv 1 INTRODUCTION MATERIAL MODELING EXAMPLES OF NONLINEAR MATERIAL BEHAVIOR MetalForming Operations 5 Nonlinear Materials Inherently Composite Materials 7 PROBLEMS IN PLASTICITY Elastic Plastic Problems Rigid Plastic Problems Other Material Models for Plasticity Problems Contained Plasticity Problems General Comments on Plasticity Problems A HISTORY OF THE DEFORMATION THEORY OF PLASTICITY The Origin and Meaning of Hooke's Law The Early Days of Deformation Theory Deformation Theory versus Incremental Theory Early Developments in Plasticity Comments on Naming of Theories, Laws, and Phenomena The Linear StressStrain Relation and its Limits Summary IMPORTANCE OF EXPERIMENTAL RESULTS IN DEVELOPMENT OF THEORIES Introduction Comparison of Theoretical and Experimental Results Experimental Results are Used More than 'Just' to Validate Theory Experimental Results are the Very Basis of ail Theoretical Models Summary SOME IMPORTANT DISTINCTIONS IN ENGINEERING ANALYSIS Introduction Presumption versus Assumption Presumption versus Restriction Misinterpretations and their Consequences Summary BOOK OUTLINE 49 CHAPTER 1 REFERENCES 50 vii

3 Perfectly Perfectly viii 2 CHARACTERISTICS OF NONLINEAR MATERIAL BEHAVIOR INTRODUCTION 2.2 ELEMENTS OF STRESSSTRAIN BEHAVIOR Definitions of Stress and Strain Yielding and the Yield Point Necking Poisson's Ratio Offset Yield Stress Yielding versus Collapse TYPES OF STRESSSTRAIN BEHAVIOR Ductile versus Brittle Behavior High Strength versus High Toughness Linear versus Nonlinear Elastic Behavior Proportional Limit versus Elastic Limit Effect of Temperature Effect of Moisture Anisotropy Bimodulus StressStrain Behavior Combined Response IDEALIZATIONS OF STRESSSTRAIN BEHAVIOR Elastic Plastic Linear Hardening Rigid Plastic PowerLaw Plasticity Viscoelasticity Summary of Idealizations Common Definitions LOADING AND UNLOADING BEHAVIOR OF INELASTIC MATERIALS Actual Unloading and Reloading Behavior Idealized Loading and Unloading Behavior Strain Hardening Bauschinger Effect Hysteresis Effect Anelastic Behavior SUMMARY 93 CHAPTER 2 REFERENCES 94 3 FUNDAMENTALS OF ELASTICITY INTRODUCTION THEORY OF STRESS Stress at a Point Stress Tensor Principal Stresses Stress Invariants Calculation of Principal Stresses Spherical and Deviator Stress Tensors Normal and Shear Components of Stress Octahedral Stresses Mohr's Circle 112

4 ix 3.3 THEORY OF STRAIN, Infinitesimal Strain at a Point Principal Strains and Strain Invariants Spherical and Deviator Strain Tensors Octahedral Strains ELASTIC STRESSSTRAIN RELATIONS ELASTIC STRAIN ENERGY 121 PROBLEM SET CHAPTER 3 REFERENCE YIELDING AND YIELD CRITERIA INTRODUCTION YIELDING Causes of Yielding in Metals Nature and Characteristics of Yielding YIELD CURVES AND YIELD SURFACES Biaxial Stress States (Yield Curves) Triaxial Stress States (Yield Surfaces) Projection of Triaxial Behavior on the % Plane SUBSEQUENT YIELD OR LOADING SURFACES HARDENING Loading Beyond Initial Yielding Isotropic Hardening Kinematic Hardening BatdorfBudiansky Slip Theory with Modifications for the Bauschinger Effect YIELD CRITERIA Introduction Maximum Principal Stress Yield Criterion Maximum Principal Strain Yield Criterion Maximum Shear Stress or Tresca Yield Criterion Maximum Strain Energy Yield Criterion Maximum Distortional Energy or Mises Yield Criterion Comparison of the Tresca and Mises Yield Criteria Other Yield Criteria Extension of Present Plasticity Concepts for Isotropic Materials to Nonlsotropic Materials SUMMARY 162 PROBLEM SET CHAPTER 4 REFERENCES DEFORMATION THEORY OF PLASTICITY INTRODUCTION PLASTIC INCOMPRESSIBILITY AND POISSON'S RATIO STRESSSTRAIN RELATIONS 168

5 X 5.4 THE STRESS AND STRAIN INTENSITIES AND THE UNIVERSAL STRESSSTRAIN CURVE Elasticity Preliminaries Transition to the Deformation Theory of Plasticity The Universal StressStrain Curve Historical Review and Comments NONLINEAR STRESSSTRAIN BEHAVIOR CURVE MODELS Linear StrainHardening StressStrain Curve Model PowerLaw StressStrain Curve Model 180 StressStrain Curve Model Nadai StressStrain Curve Model RambergOsgood Generalized Nadai StressStrain Curve Model NadaiJones StressStrain Curve Model Summary Remarks on StressStrain Curve Modeling COMMENTS ON HENCKY'S AND NADAI'S CONTRIBUTIONS SUMMARY REMARKS ON DEFORMATION THEORY 195 PROBLEM SET CHAPTER 5 REFERENCES INCREMENTAL THEORY OF PLASTICITY INTRODUCTION STRESS HISTORY IMPORTANCE STRESS AND STRAIN INCREMENTS UNDER UNIAXIAL STRESS STRAIN INCREMENTS UNDER MULTIAXIAL STRESSES VARIATIONAL APPROACH TO PLASTIC STRAIN INCREMENTS FOR MULTIAXIAL STRESS INCREMENTS LOADING, UNLOADING, AND NEUTRAL LOADING PLASTIC WORK AND HARDENING SUMMARY REMARKS 219 PROBLEM SET CHAPTER 6 REFERENCES SOLUTION OF PLASTICITY PROBLEMS INTRODUCTION DEGREE OF MATERIAL NONLINEARITY PLASTIC STRAIN VERSUS TOTAL STRAIN RELATIONS METHOD OF SUCCESSIVE ELASTIC SOLUTIONS COMMENTS ON LOADING PATH AND HISTORY Introduction Generalization of the Deformation Theory Restriction to Proportional Loading Paths Generalization of the Loading and Unloading Paths General Comments MATERIAL STRESSSTRAIN BEHAVIOR VERSUS STRUCTURAL RESPONSE THE ELASTIC STRAIN RESPONSE APPROXIMATION SUMMARY 234 CHAPTER 7 REFERENCES 235

6 PERFECTLY PERFECTLY xi 8 THICKWALLED SPHERICAL SHELLS UNDER INTERNAL PRESSURE AND/OR INNERSURFACE HEATING INTRODUCTION FUNDAMENTAL RELATIONS AND CHARACTERISTICS BEHAVIOR OF ELASTIC PLASTIC SHELLS UNDER INTERNAL PRESSURE Linear Elastic Behavior Limit of Linear Elastic Behavior (Yielding) Collapse Pressure Behavior between Yielding and Collapse Residual Stresses BEHAVIOR OF STRAINHARDENING SHELLS UNDER INTERNAL PRESSURE General StrainHardening Behavior Linear StrainHardening Behavior Summary of StrainHardening Effects BEHAVIOR OF ELASTIC PLASTIC SHELLS UNDER STEADYSTATE INNERSURFACE HEATING Linear Elastic Behavior Limit of Linear Elastic Behavior (Yielding) Comments on Initial Yielding of Elastic ThickWalled Spherical Shells under Combined Heating and Pressurization PostYielding Behavior under InnerSurface Heating Example with b/a = Example with b/a = Summary of Heating Studies SUMMARY InnerSurface Pressurization Results InnerSurface Heating Results Relation of Present Work to Past Work Possible Future Research 319 PROBLEM SET CHAPTER 8 REFERENCES PLASTIC BUCKLING OF BARS, PLATES, AND SHELLS INTRODUCTION PLASTIC BUCKLING OF BARS Introduction Governing Equation and Solution for Elastic Buckling Governing Equation for Plastic Buckling ReducedModulus Theory TangentModulus Theory Transcendental Plastic Buckling Equation Solution Strategy for the Buckling Criterion Numerical Results Comparison of ReducedModulus Theory with TangentModulus Theory Summary 340 Problem Set

7 xii 9.3 PLASTIC BUCKLING OF PLATES Introduction Derivation of the Buckling Criterion Solution Strategy for the Buckling Criterion Numerical Results Comparison of Theoretical and Experimental Results Summary 356 Problem Set PLASTIC BUCKLING OF SHELLS Introduction Derivation of the Buckling Criterion Solution Strategy for the Buckling Criterion Numerical Results Summary 368 Problem Set PLASTIC THERMAL BUCKLING OF BARS WITH TEMPERATUREDEPENDENT MATERIAL PROPERTIES Introduction Plastic Thermal Buckling Analysis Solution Strategy Modeling Nonlinear StressStrain Behavior and Thermal Expansion Behavior Numerical Results and Observations Summary PLASTIC THERMAL BUCKLING OF PLATES WITH TEMPERATUREDEPENDENT MATERIAL PROPERTIES Introduction Plastic Thermal Buckling Analysis Solution Strategy Nonlinear TemperatureDependent Material Model Numerical Results and Observations Summary and Perspective Remarks SUMMARY REMARKS ON MECHANICAL AND THERMAL PLASTIC BUCKLING OF BARS, PLATES, AND SHELLS General Summary Other Significant Contributions Perspective or Design Significance Design Considerations and Factor of Safety Comparison with Experimental Results 452 CHAPTER 9 REFERENCES NONCLASSICAL PLASTICITY INTRODUCTION THE NONLINEAR BIMODULAR MATERIAL MODEL Linear Bimodular Behavior Models Introduction Criteria for a Consistent Model Ambartsumyan Material Model WeightedComplianceMatrix Material Model Summary of the Linear Bimodular Material Models 471

8 xiit Material Models for StressStrain Curve Nonlinearities Extended StressStrain Curve Approach Iteration Procedure for Nonlinear Bimodular Material Models Convergence of the Iteration Procedure Summary and Validation Procedure for Material Models APPLICATION TO LAMINATED STRUCTURES Nonlinear Lamina Model Introduction Theory of Uniaxial OffAxis Loading BoronEpoxy Lamina Results GraphiteEpoxy Lamina Results Summary of the Nonlinear Lamina Model Nonlinear Laminate Model Introduction Derivation of Theory Nonlinear Response of Symmetric BoronEpoxy Laminates to Unixial Loading Summary of the Nonlinear Laminate Model Buckling of Laminated Plates Introduction Derivation and Solution of the Buckling Criterion Numerical Results Summary of Buckling of Laminated Plates Summary Remarks on Nonlinear Laminated Structures APPLICATION TO SOLID BODIES Introduction Modeling Particulate Composite Materials Introduction Establishing the Material Model Uniaxial OffAxis Loading Biaxial Behavior Characteristics ATJS Graphite Tube under Biaxial Loading Test Specimen Biaxial Strain Correlations at Room Temperature for the Nonlinear Model with only Tensile Properties Biaxial Strain Correlations at Room Temperature with the WCM Material Model Biaxial Strain Correlations at 2000T (1100 C) with the RCM Material Model Summary of Biaxial Strain Correlations ATJS Graphite Annular Disk under Thermal Loading Introduction Temperature and Deformation Measurements ATJS Graphite Material Characteristics Temperature Interpolation of Mechanical Properties InnerDiameter Change Predictions Stress and Strain Predictions Summary OffAxis Loading of CarbonCarbon Materia! Introduction CarbonCarbon Nonlinear Bimodular Material Model 562

9 xiv Effect of ShearExtension Coupling on Deformation under Uniaxial OffAxis Loading Predicted and Measured Strain Response for Uniaxial OffAxis Loading Summary of CarbonCarbon Material Modeling CONCLUDING REMARKS ON NONCLASSICAL PLASTICITY CHAPTER 10 REFERENCES SUMMARY REMARKS AND OTHER TOPICS INTRODUCTION SUMMARY OF BOOK CONTENTS Introduction Characteristics of Nonlinear Material Behavior Fundamentals of Elasticity Yielding, Yield Criteria, and Loading Surfaces Deformation Theory of Plasticity Incremental Theory of Plasticity Solution of Plasticity Problems ThickWalled Spherical Shells under Internal Pressure Plastic Buckling of Bars, Plates, and Shells Nonclassical Plasticity ON THE TRANSITION TO COMPUTATIONAL APPROACHES How to Learn Complex Plastic Behavior Simple Analytical Approaches vs. Numerical Approaches Computational Tools Available Replacement of Measured Behavior by Computer Simulation Designing Structures A SURVEY OF PLASTICITY ANALYSES AND CAPABILITIES HISTORY OF PLASTICITY SOURCES SUMMARY OF HISTORY OF PLASTICITY CONCLUDING REMARKS ON NONCLASSICAL PLASTICITY 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

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