Creep and Relaxation of Nonlinear Viscoelastic Materials

PREFACE CHAPTER 1. INTRODUCTION 1.1 Elastic Behavior 1.2 Plastic Behavior 1.3 Viscoelastic Behavior 1.4 Creep 1.5 Recovery 1.6 Relaxation I.7 Linearity CHAPTER 2. HISTORICAL SURVEY OF CREEP 2.1 Creep of Metals 2.2 Creep under Uniaxial Stress 2.3 Creep under Combined Stresses 2.4 Creep under Variable Stress 2.5 Creep of Plastics 2.6 Mathematical Representation of Creep of Materials 2.7 Differential Form 2.8 Integral Form 2.9 Development of Nonlinear Constitutive Relations CHAPTER 3. STATE OF STRESS AND STRAIN 3.1 State of Stress 3.2 Stress Tensor 3.3 Unit Tensor 3.4 Principal Stresses 3.5 Mean Normal Stress Tensor and Deviatoric Stress Tensor 3.6 Invariants of Stress 3.7 Traces of Tensors and Products of Tensors 3.8 Invariants in Terms of Traces 3.9 Hamilton-Cayley Equation 3.10 State of Strain 3.11 Strain-Displacement Relation 3.12 Strain Tensor CHAPTER 4. MECHANICS OF STRESS AND DEFORMATION ANALYSES 4.1 Introduction 4.2 Law of Motion CONTENTS ix 4.3 Equations of Equilibrium 4.4 Equilibrium of Moments 4.5 Kinematics 4.6 Compatibility Equations 4.7 Constitutive Equations 4.8 Linear Elastic Solid 4.9 Boundary Conditions 4.10 The Stress Analysis Problem in a Linear Isotropic Elastic Solid CHAPTER 5. LINEAR VISCOELASTIC CONSTITUTIVE EQUATIONS 5.1 Introduction 5.2 Viscoelastic Models 5.3 The Basic Elements: Spring and Dashpot 5.4 Maxwell Model 5.5 Kelvin Model 5.6 Burgers or Four-element Model 5.7 Generalized Maxwell and Kelvin Models 5.8 Retardation Spectrum for tn 5.9 Differential Form of Constitutive Equations for Simple Stress States 5.10 Differential Form of Constitutive Equations for Multiaxial Stress States 5.11 Integral Representation of Viscoelastic Constitutive Equations 5.12 Creep Compliance 5.13 Relaxation Modulus 5.14 Boltzmann's Superposition Principle and Integral Representation 5.15 Relation Between Creep Compliance and Relaxation Modulus 5.16 Generalization of the Integral Representation to Three-Dimensions 5.17 Behavior of Linear Viscoelastic Material under Oscillating Loading 5.18 Complex Modulus and Compliance 5.19 Dissipation 5.20 Complex Compliance and Complex Modulus ofSome Viscoelastic Models 5.21 Maxwell Model 5.22 Kelvin Model 5.23 Burgers Model 5.24 Relation Between the Relaxation Modulus and the Complex Relaxa­tion Modulus 5.25 Relation Between Creep Compliance and Ccmplex Compliance 5.26 Complex Compliance for In 5.27 Temperature Effect and Time-Temperature Superposition Principle CHAPTER 6. LINEAR VISCOELASTIC STRESS ANALYSIS 108 6.1 Introduction 108 6.2 Beam Problems 109 6.3 Stress Analysis of Quasi-static Viscoelastic Problems Using the Elastic-Viscoelastic Correspondence Principle 119 6.4 Thick-walled Viscoelastic Tube 122 6.5 Point Force Acting on the Surface of a Semi-infinite Viscoelastic Solid 128 6.6 Conduding Remarks 130 x CONTENTS CHAPTER 7. MULTIPLE INTEGRAL REPRESENTATION 7.1 Introduction 7.2 Nonlinear Viscoelastic Behavior under Uniaxial Loading 7.3 Nonlinear Viscoelastic Behavior under Multiaxial Stress State 7.4 A Linearly Compressible Material 7.S Incompressible Material Assumption 7.6 Linearly Compressible 7.7 Constant Volume 7.8 Incompressible and Linearly Compressible Creep 7.9 Incompressible and Linearly Compressible Relaxation 7.10 Constitutive Relations under Biaxial Stress and Strain 7.11 Constitutive Relations under Uniaxial Stress and Strain 7.12 Strain Components for Biaxial and Uniaxial Stress States, Compres­sible Material 7.13 Strain Components for Biaxial and Uniaxial Stress States, Linearly Compressible Material 7.14 Stress Components for Biaxial and Uniaxial Strain States 7.15 Approximating Nonlinear Constitutive Equations under Short Time Loading 7.16 Superposed Small Loading on a Large Constant Loading 7.17 Other Representations 7.18 Finite Linear Viscoelasticity 7.19 Elastic Fluid Theory 7.20 Thermodynamic Constitutive Theory CHAPTER 8. NONLINEAR CREEP AT CONSTANT STRESS AND RELAXATION AT CONSTANT STRAIN 8.1 Introduction 8.2 Constitutive Equations for 3 X 3 Matrix 8.3 Components of Strain for Creep at Constailt Stress 8.4 Components of Stress for Relaxation at Constant Strain 8.5 Biaxial Constitutive Equations for 2 x 2 Matrix 8.6 Components of Strain (or Stress) for Biaxial States for 2x2 Matrix 8.7 Constitutive Equations for Linearly Compressible Material 8.8 Components of Strain for Creep of Linearly Compressible Material 8.9 Components of Stress for Relaxation of Linearly Compressible Material 8.10 Poisson's Ratio 8.11 Time Functions 8.12 Determination of Kc,lrnel Functions for Constant Stress Creep 8.13 Determination of Kernel Functions for Constant-Strain Stress-Relaxation 8.14 Experimental Results of Creep CHAPTER 9. NONLINEAR CREEP (OR RELAXATION) UNDER VARIABLE STRESS (OR STRAIN) 9.1 Introduction 9.2 Direct Determination of Kernel Functions 9.3 Product-Form Approximation of Kernel Functions 9.4 Additive Forms of Approximation of Kernel Functions 9.5 Modified Superposition Method 9.6 Physical Linearity Approximation of Kernel Functions 233 9.7 Comparison 235 CHAPTER 10. CONVERSION AND MIXING OF NONLINEAR CREEP AND RELAXATION 10.1 Introduction 10.2 Relation Between Creep and Stress Relaxation for Uniaxial Nonlinear Viscoelasticity 10.3 Example: Prediction of Uniaxial Stress Relaxation from Creep of Nonlinear Viscoelastic Material 10.4 Relation Between Creep and Relaxation for Biaxial Nonlinear Visco­elasticity 244 10.5 Behavior of Nonlinear Viscoelastic Material under Simultaneous Stress Relaxation in Tension and Creep in Torsion 10.6 Prediction of Creep and Relaxation under Arbitrary Input CHAPTER 11. EFFECT OF TEMPERATURE ON NONLINEAR VISCOELASTIC MATERIALS 11.1 Introduction 11.2 Nonlinear Creep Behavior at Elevated Temperatures 11.3 Determination of Temperature Dependent Kernel Functions 11.4 Creep Behavior under Continuously Varying Temperature-Uniaxial Case 11.5 Creep Behavior under Continuously Varying Temperature for Com­bined Tension and Torsion 11.6 Thermal Expansion Instability CHAPTER 12. NONLINEAR VISCOELASTIC STRESS ANALYSIS 12.1 Introduction 12.2 Solid Circular Cross-section Shaft under Twisting 12.3 Beam under Pure Bending 12.4 Thick-walled Cylinder under Axially Symmetric CHAPTER 13. EXPERIMENTAL METHODS 13.1 Introduction 13.2 Loading Apparatus for Creep 13.3 Load Application 13.4 Test Specimen 13.5 Uniform Stressing or Straining 13.6 Strain Measurement 13.7 Temperature Control 13.8 Humidity and Temperature Controlled Room 13.9 Internal Pressure 13.10 Strain Control and Stress Measurement for Relaxation 13.11 A Machine for Combined Tension and Torsion APPENDIX AI. LIST OF SYMBOLS APPENDIX A2. MATHEMATICAL DESCRIPTION OF NONLINEAR VISCOELASTIC CONSTITUTIVE RELATION APPENDIX A3. UNIT STEP FUNCTION AND UNIT IMPULSE FUNCTION APPENDIX A4. LAPLACE TRANSFORMATION APPENDIX A5. DERIVATION OF THE MODIFIED SUPERPOSITION PRINCIPLE FROM THE MULTIPLE INTEGRAL REPRESENTATION APPENDIX A6. CONVERSION TABLES BIBLIOGRAPHY SUBJECT INDEX AUTHOR INDEX