Creep And Relaxation Of Nonlinear Viscoelastic Materials With An Introduction To Linear Viscoelasticity, Volume 18

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

1.7 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

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 of Some 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 Complex Compliance

5.26 Complex Compliance for tn

5.27 Temperature Effect and Time-Temperature Superposition Principle

Chapter 6. Linear Viscoelastic Stress Analysis

6.1 Introduction

6.2 Beam Problems

6.3 Stress Analysis of Quasi-static Viscoelastic Problems Using the Elastic-Viscoelastic Correspondence Principle

6.4 Thick-walled Viscoelastic Tube

6.5 Point Force Acting on the Surface of a Semi-infinite Viscoelastic Solid

6.6 Concluding Remarks

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

7.5 Incompressible Material Assumption

7.6 Linearly Compressible, II

7.7 Constant Volume

7.8 Incompressible and Linearly Compressible Creep in Terms of σ

7.9 Incompressible and Linearly Compressible Relaxation in Terms of ε

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 ×3 Matrix

8.3 Components of Strain for Creep at Constant Stress

8.4 Components of Stress for Relaxation at Constant Strain

8.5 Biaxial Constitutive Equations for 2 ×2 Matrix

8.6 Components of Strain (or Stress) for Biaxial States for 2 ×2 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 Kernel 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

9.7 Comparison

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

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 Loading

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 A1. List of Symbols

Appendix A2. Mathematical Description of Nonlinear Viscoelastic Constitutive Relation

A2.1 Introduction

A2.2 Material Properties

A2.3 Multiple Integral Representation of Initially Isotropic Materials (Relaxation Form)

A2.4 The Inverse Relation (Creep Form)

Appendix A3. Unit Step Function and Unit Impulse Function

A3.1 Unit Step Function or Heaviside Unit Function

A3.2 Signum Function

A3.3 Unit Impulse or Dirac Delta Function

A3.4 Relation Between Unit Step Function and Unit Impulse Function

A3.5 Dirac Delta Function or Heaviside Function in Evaluation of Integrals

Appendix A4. Laplace Transformation

A4.1 Definition of the Laplace Transformation

A4.2 Sufficient Conditions for Existence of Laplace Transforms

A4.3 Some Important Properties of Laplace Transforms

A4.4 The Inverse Laplace Transform

A4.5 Partial Fraction Expansion

A4.6 Some Uses of the Laplace Transform

Appendix A5. Derivation of the Modified Superposition Principles from the Multiple Integral Representation

A5.1 Second Order Term

A5.2 Third Order Term

A5.3 Application to Third Order Multiple Integrals for Creep

Appendix A6. Conversion Tables

Bibliography

Subject Index

Author Index

Creep and Relaxation of Nonlinear Viscoelastic Materials with an Introduction to Linear Viscoelasticity deals with nonlinear viscoelasticity, with emphasis on creep and stress relaxation. It explains the concepts of elastic, plastic, and viscoelastic behavior, along with creep, recovery, relaxation, and linearity. It also describes creep in a variety of viscoelastic materials, such as metals and plastics. Organized into 13 chapters, this volume begins with a historical background on creep, followed by discussions about strain and stress analysis, linear viscoelasticity, linear viscoelastic stress analysis, and oscillatory stress and strain. It methodically walks the reader through topics such as the multiple integral theory with simplifications to single integrals, incompressibility and linear compressibility, and the responses of viscoelastic materials to stress boundary conditions (creep), strain boundary conditions (relaxation), and mixed stress and strain boundary conditions (simultaneous creep and relaxation). The book also looks at the problem of the effect of temperature, especially variable temperature, on nonlinear creep, and describes methods for the characterization of kernel functions, stress analysis of nonlinear viscoelastic materials, and experimental techniques for creep and stress relaxation under combined stress. This book is a useful text for designers, students, and researchers.