Creep and Relaxation of Nonlinear Viscoelastic Materials

Overview

This pioneering book presents the basic theory, experimental methods, experimental results and solution of boundary value problems in a readable, useful way to designers as well as research workers and students. The mathematical background required has been kept to a minimum and supplemented by explanations where it has been necessary to introduce specialized mathematics. Also, appendices have been included to provide sufficient background in Laplace transforms and in step ...

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Creep and Relaxation of Nonlinear Viscoelastic Materials

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Overview

This pioneering book presents the basic theory, experimental methods, experimental results and solution of boundary value problems in a readable, useful way to designers as well as research workers and students. The mathematical background required has been kept to a minimum and supplemented by explanations where it has been necessary to introduce specialized mathematics. Also, appendices have been included to provide sufficient background in Laplace transforms and in step functions.
Chapters 1 and 2 contain an introduction and historic review of creep. As an aid to the reader a background on stress, strain, and stress analysis is provided in Chapters 3 and 4, an introduction to linear viscoelasticity is found in Chapter 5 and linear viscoelastic stress analysis in Chapter 6. In the next six chapters the multiple integral representation of nonlinear creep and relaxation, and simplifications to single integral forms and incompressibility, are examined at length. After a consideration of other representations, general relations are derived, then expanded to components of stress or strain for special cases. Both constant stress (or strain) and variable states are described, together with methods of determining material constants. Conversion from creep to relaxation, effects of temperature and stress analysis problems in nonlinear materials are also treated here.
Finally, Chapter 13 discusses experimental methods for creep and stress relaxation under combined stress. This chapter considers especially those experimental problems which must be solved properly when reliable experimental results of high precision are required. Six appendices present the necessary mathematical background, conversion tables, and more rigorous derivations than employed in the text. An extensive updated bibliography completes the book.

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Editorial Reviews

Booknews
A corrected reprint of the Elsevier edition of 1976 with additional literature citations added to Chapter 2. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

Table of Contents


 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 
 
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