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# Creep and Relaxation of Nonlinear Viscoelastic Materials

400
by William N. Findley, Francis A. DavisWilliam N. Findley

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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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## Product Details

ISBN-13: | 9780486660165 |
---|---|

Publisher: | Dover Publications |

Publication date: | 11/30/2011 |

Series: | Dover Civil and Mechanical Engineering |

Edition description: | Revised ed. |

Pages: | 400 |

Product dimensions: | 5.50(w) x 8.50(h) x 0.82(d) |

## Table of Contents

**PREFACE**

CHAPTER 1. INTRODUCTION

1.1 Elastic Behavior

CHAPTER 1. INTRODUCTION

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 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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