Preface
This book is an attempt to provide a unified methodology to derive models for fatigue life. This includes S-N, ε-N and crack propagation models. This is not a conventional book aimed at describing the fatigue fundamentals, but rather a book in which the basic models of the three main fatigue approaches, the stress-based, the strain-based and the fracture mechanics approaches, are contemplated from a novel and integrated point of view.
On the other hand, as an alternative to the preferential attention paid to deterministic models based on the physical, phenomenological and empirical description of fatigue, their probabilistic nature is emphasized in this book, in which stochastic fatigue and crack growth models are presented.
This book is the result of a long period of close collaboration between its two authors who, although of different backgrounds, mathematical and mechanical, both have a strong sense of engineering with respect to the fatigue problem. When the authors of this book first approached the fatigue field in 1982 (twenty six years ago), they found the following scenario:
- Linear, bilinear or trilinear models were frequently proposed by relevant laboratories and academic centers to reproduce the W ̈ohler field. This was the case of well known institutions, which justified these models based on client requirements or preferences. This led to the inclusion of such models and methods as, for example, the up-and-down, in standards and official practical directives (ASTM, Euronorm, etc.), which have proved to be unfortunate.
- The evaluation of the S-N field lacked models not arising from arbitrary hypotheses. At that time the ASTM (1963) suggested a non explicit cer- tain relation between the statistical distributions of Δσ for fixed N and N for fixed Δσ.
- The up-and-down method, clearly inefficient from the cost, reliability and extrapolation to other conditions point of view, was commonly used.
- The existence of proposals for taking into account the length effect was based on families of distributions, such as the normal, log-normal, etc., that are not stable with respect to minimum operations.
- The existence of models, which did not contemplate the compatibility con- dition, led to contradictions and inconsistencies in the cumulated damage evaluations.
- Models based on micro-mechanical considerations combined with specu- lative assumptions, but such that they satisfied the compatibility require- ments though without an explicit formulation of this very important con- dition, were unfortunately considered as excessively theoretical and useless for practical application.
- The ε-N field was treated based on the Morrow linear elastic-plastic model, which apart from depending on a relatively large number of parameters (four) for the reduced information it supplies, only provides the mean curve, thus requiring additional methods to deal with percentile curves.
- Crack growth appeared as a completely different and unrelated problem to the S-N and the ε-N approaches. While the first was considered as a fracture mechanics based problem, the last two were treated as phe- nomenological approaches to fatigue of a second order scientific level, and this occurred in spite of the fact that the three problems are different ways of contemplating the same fatigue phenomena.
Book Contents:
I Introduction and Motivation of the Fatigue Problem
1 An Overview of Fatigue Problems
1.1 Introduction
1.2 Models with dimensionless variables
1.3 S-N or W ̈ohler curves
1.3.1 Compatibility condition of N∗|Δσ and Δσ|N∗
1.3.2 Statistical considerations
1.4 ε-N curves
1.5 Stress level effect
1.5.1 Compatibility condition of S-N curves for constant σ∗m and S-N curves for constant σ∗M
1.6 Crack growth curves
1.6.1 Crack growth curves for a constant stress pair T
1.6.2 Crack growth curves for a varying stress pair T
1.6.3 Compatibility of crack growth and S-N models
1.7 Crack growth rate curves
1.8 Size effect
1.9 Normalization
1.9.1 Percentile based normalizations
1.9.2 Stress range and lifetime based normalizations
1.9.3 Extended percentile normalization
1.10 Damage measures and damage accumulation
II Models Used in the Stress Based Approach
2 S-N or W ̈ohler Field Models
2.1 Introduction
2.2 Dimensional analysis
2.3 Extreme models in fatigue
2.3.1 The Weibull model
2.3.2 The minimal Gumbel model
2.4 Model for constant stress range and level
2.4.1 Derivation of the model
2.4.2 Parameter estimation
2.4.3 Alternative methods for dealing with run-outs
2.5 Model for varying range and given stress level
2.5.1 Derivation of the model
2.5.2 Some weaknesses of the proposed model
2.5.3 Parameter estimation
2.5.4 Use of the model in practice
2.5.5 Example of application
2.6 Model for varying stress range and level
2.7 Dimensional Weibull and Gumbel models
2.8 Properties of the model
2.8.1 Parameter estimation
2.8.2 Use of the model in practice
2.8.3 Example of applications
2.9 Concluding remarks
2.10 Appendix A: Derivation of the general model
2.11 Appendix B: S-N curves for the general model
3 Length Effect
3.1 Introduction
3.2 Modeling the S-N field for different lengths
3.2.1 A previous example
3.2.2 General model for different lengths
3.2.3 Parameter estimation
3.3 Examples of Application
3.3.1 Prestressing wires
3.3.2 Prestressing strands
III Models Used in the Strain Based Approach 111
4 Log-Weibull ε-N Model
4.1 Introduction
4.2 Model for constant strain range and level
4.2.1 Practical example
4.3 Model for varying strain range and level
4.4 Converting strain- into stress-life curves
4.4.1 Practical example
4.5 Concluding remarks
IV Models Used in the Fracture Mechanics Approach127
5 Crack Growth Models
5.1 Introduction and motivation
5.2 Building crack growth models
5.3 Crack growth curves approach I
5.3.1 Crack growth curves for constant Δσ∗ and σ∗
5.3.2 Crack growth curves for varying Δσ∗ and σ∗
5.3.3 Compatibility of crack growth and S-N models
5.4 Crack growth curves approach II
5.4.1 Crack growth curves for constant Δσ∗ and σ∗
5.4.2 Crack growth curves for varying Δσ∗ and σ∗
5.4.3 Statistical distributions of a∗|N∗ and N∗|a∗
5.4.4 Learning and estimating the model
5.4.5 Compatibility of approaches I and II
5.5 Example of application
5.6 Summary and future work
V Damage and Damage Accumulation Models 159
6 Damage Measures
6.1 Introduction
6.2 Normalization
6.3 Damage measures
6.3.1 Some requirements for a damage measure
6.3.2 Some damage measures
6.4 Concluding remarks
7 Damage Accumulation
7.1 Damage accumulation
7.1.1 Accumulated damage after a constant stress range load step
7.1.2 Accumulated damage after block loading
7.1.3 Fatigue under a general loading history
7.1.4 Random loading
7.2 Crack growth damage for any load history
VI Appendices 191
A Models Used in Fatigue
A.1 Introduction
A.2 S-N curve models
A.2.1 The W ̈ohler model
A.2.2 The Basquin model
A.2.3 The Strohmeyer model
A.2.4 The Palmgren model
A.2.5 The St ̈ussi model
A.2.6 The Weibull model
A.2.7 The Spindel and Haibach model
A.2.8 The Kohout and Vechet model
A.3 Stress field models
A.3.1 The Pascual and Meeker model
A.3.2 The Bastenaire model
A.3.3 The Castillo et al. (1985) model
A.4 Fatigue limit models
A.4.1 The up-and-down method
B Notation Used in This Book
Bibliography
Index
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A Unified Statistical Methodology for Modeling Fatigue Damage