Fatigue, Stress, and Strain of Rubber Components

Inhaltsverzeichnis

1 Introduction

1.1 Objective

1.2 Discovery

1.2.1 First Vulcanization

1.2.2 Early Manufacture of Rubber Products

1.2.3 Discovery of Reinforcement

1.2.4 Production of Rubber

1.3 The Rubber Molecule

1.4 Synthetics

1.4.1 Curing and Crosslinking

1.4.2 Fillers and Reinforcement

1.4.3 Curing Ingredients

1.4.4 Other Additives

1.5 Principal Uses of Several Elastomers

Bibliography

2 Rubber Stress-Strain Behavior

2.1 Challenges of Rubber Behavior

2.2 Characteristics of Stress-Strain Behavior

2.2.1 Low Elastic Modulus, High Elongation at Break, and Non-Linearity

2.2.2 Hysteresis

2.2.3 Stress Relaxation

2.2.4 Creep

2.2.5 Mullins Effect

2.2.6 Reinforcement

2.2.7 Cyclic Frequency and Strain Rate

2.2.8 Temperature

2.2.9 Immersion Effects

2.2.10 Strain Crystallization

2.2.11 Permanent Set

2.2.12 Recovery

Bibliography

3 ATheory of the Elastomer Stress-Strain Curve

3.1 Introduction

3.2 The Internal Structure of the Vulcanized Elastomer

3.3 Assumptions and Hypotheses

3.3.1 The Coil Spring Analogy

3.3.2 Chain Segments and Terminations

3.3.3 Statistical Distribution of Chains in Length and End Point Separation

3.3.4 The Presence of van der Waals Bonds

3.3.5 Reinforcement by Particle Rotation

3.3.6 Migration of Entanglements

3.3.7 Temperature-Induced Chain Vibration

3.3.8 Bond Breaking and Remaking in Deformation

3.3.9 Parallelism-Induced Crystallization

3.4 Elastomer Behaviors

3.4.1 The Non-Linear Stress-Strain Curve

3.4.2 The Mullins Effect

3.4.3 Low Elastic Modulus and High Elongation at Break

3.3.4 Hysteresis

3.4.5 Stiffening by Reinforcing Fillers

3.4.6 Strain Rate Stiffening

3.4.7 Temperature Response

3.4.8 Stress Relaxation and Cyclic Stress Relaxation

3.4.9 Creep and Creep under Cyclic Conditions

3.4.10 Permanent Set

3.4.11 Recovery

3.4.12 Strain Crystallization

Acknowledgements

References

4 Stress-Strain Testing

4.1 Introduction

4.2 Tensile Testing

4.2.1 Specimens

4.2.2 Testing with the Dumbbell Specimen

4.2.3 Testing with the Planar Stress Specimen

4.2.4 Testing with the Loop Specimen

4.3 Shear Testing

4.3.1 Stress-Strain State

4.3.2 Specimens

4.4 Biaxial Strain Testing

4.4.1 The Bubble Test

4.4.2 The Cross Specimen

4.5 Compression Testing

4.6 Summary

References

5 Design Equations

5.1 Introduction

5.1.1 Use of Design Equations

5.1.2 Elastic Constants

5.2 Design Equations for Various Geometries

5.2.1 Pads in Shear

5.2.2 Pads in Torsion

5.2.3 Bushings

5.2.4 Pads in Compression

5.2.5 Compression of a Long Strip

5.2.6 Solid Rubber Rollers

5.2.7 Rubber-Covered Rollers

5.2.8 Compression of a Rubber Sphere

5.2.9 Compression of Solid Rubber Tire

5.2.10 Compression of Solid Rubber Ring of Circular Cross-Section

5.2.11 Solid Rubber Ring with Rectangular Cross-Section

5.2.12 Indenter, Flat Ended Cylinder

5.2.13 Indenter, Spherical Head

5.2.14 Indenter, Conical

5.2.15 Indenter, Long Narrow Flat End

5.2.16 Protrusion Through a Round Hole

5.2.17 Protrusion Through Long Narrow Gap

5.3 Summary

References

6 CalculationMethods for Spherical Elastomer Bearings

6.1 Introduction

6.2 History of the Spherical Bearing

6.3 Mathematical Description of the Bearing

6.3.1 Overall Bearing Parameters

6.3.2 Parameters of Particular Pads

6.3.3 Angular Moment

6.4 Shear Strain of Pads under Angular Deflection

6.5 Axial Loads

6.5.1 Compression of Pads under Axial Force

6.5.2 Bulge Shear Strain

6.5.3 Summary of Calculations

6.6 Torsional Loads

6.6.1 Shear Strain of Pads under Torsional Rotation

6.6.2 Computational Procedure

6.6.3 Limitations

References

7 Finite Element Analysis

7.1 Introduction

7.2 Procedure

7.2.1 Symmetry

7.2.2 Loads and Boundary Conditions

7.2.3 Element Selection and Meshing

7.3 Material Model or Constitutive Equations

7.3.1 Simpler Constitutive Equations

7.3.2 Higher Order Constitutive Equations

7.4 Fitting Equations to Test Data

7.5 O-Ring Seal with Pressure

7.6 Rubber Boot

7.7 Summary

Acknowledgements

References

8 Fatigue Testing

8.1 Introduction

8.2 Parameters Affecting the Strain-Life Curve

8.2.1 Parameters to Be Specified

8.2.2 Selecting Strain Amplitude

8.3 Failure Criteria

8.4 R-Ratio

8.5 Combined Strain State

8.6 Wave Form

8.7 Creep and Stress Relaxation

8.8 Frequency and Strain Rate

8.9 Effect of Temperature

8.10 Liquid Immersion

8.11 Recovery

8.12 Scragging

8.13 Batch Variation

8.14 Storage

Acknowledgements

References

9 Fitting the Strain-Life Curve

9.1 Introduction

9.2 Development of an Equation for N in ?a , R and T

9.3 The Strain-Life Curve Equation with Nagel’s Equation for Temperature

9.4 Employing the Simple Empirical Formula for Temperature

Acknowledgements

References

10 Fatigue Life Estimation

10.1 Introduction

10.2 Single Wave Form, the ?-N Method

10.3 The Miner’s Number

10.4 The Deterministic Fatigue Spectrum

10.5 Sample Calculation of the Miner’s Number

10.6 White Noise

10.6.1 Rainflow Counting

11 Fatigue Crack Growth and Tearing Energy

11.1 Introduction

11.2 Griffith Strain Energy Release Rate

11.2.1 Griffith Criterion

11.2.2 Derivation

11.2.3 Griffith Condition for Fracture

11.2.4 Critical Assumptions

11.3 Rivlin and Thomas and Tearing Energy

11.3.1 Modification of Griffith’s Criterion for Fracture ofMetals

11.3.2 Application to Rubber

11.3.3 State of Critical Assumptions

11.4 Shortcut Formulas for T

11.5 Tearing Energy Applied to Fatigue Crack Growth

11.5.1 Pioneering Developments in Fatigue

11.5.2 The Change in Definition of Tearing Energy

11.6 Limitations

11.6.1 Fatigue Crack Growth Parameter

11.6.2 Cycles to Failure by T or ?a ?

11.7 Summary and Conclusions

Acknowledgements

References

Appendix I. Rubber Nomenclature

Appendix 2. Fatigue Terminology

Appendix 3. English to Metric Conversion

Appendix 4. Fitting the Strain-Life Curve

Appendix 5. Derivation of Tearing Energy Equations

Appendix 6. Derivation of Equations for Spherical Elastomer Bearings

Index