Fatigue of Materials and Structures E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 628

Veröffentlichungsjahr: 2013

Beschreibung

The design of mechanical structures with improved and predictable durability cannot be achieved without a thorough understanding of the mechanisms of fatigue damage and more specifically the relationships between the microstructure of materials and their fatigue properties. Written by leading experts in the field, this book (which is complementary to Fatigue of Materials and Structures: Application to Damage and Design, also edited by Claude Bathias and André Pineau), provides an authoritative, comprehensive and unified treatment of the mechanics and micromechanisms of fatigue in metals, polymers and composites. Each chapter is devoted to one of the major classes of materials or to different types of fatigue damage, thereby providing overall coverage of the field. The book deals with crack initiation, crack growth, low-cycle fatigue, gigacycle fatigue, shorts cracks, fatigue micromechanisms and the local approach to fatigue damage, corrosion fatigue, environmental effects and variable amplitude loadings, and will be an important and much used reference for students, practicing engineers and researchers studying fracture and fatigue in numerous areas of mechanical, structural, civil, design, nuclear, and aerospace engineering as well as materials science.

Leseprobe

Table of Contents

Foreword

Chapter 1. Introduction to Fatigue: Fundamentals and Methodology

1.1. Introduction to the fatigue of materials

1.2. Mechanisms of fatigue damage

1.3. Test systems

1.4. Structural design and fatigue

1.5. Fatigue of polymers, elastomers and composite materials

1.6. Conclusion

1.7. Bibliography

Chapter 2. Modeling of Fatigue Strength and Endurance Curve

2.1. Introduction

2.2. Nature and aspect of the scatter of fatigue test results

2.3. Determination of the endurance limit

2.4. Estimation methods of fatigue resistance and standard deviation with N cycles

2.5. Mathematical representations and plotting methods of the Wöhler curve

2.6. Estimation of the cycle number N for a given level of stress amplitude

2.7. Influence of mechanical parameters on endurance

2.8. Relationship between endurance and mechanical characteristics (of steels)

2.9. Bibliography

Chapter 3. Fatigue Crack Initiation

3.1. Introduction

3.2. Physical mechanisms of crack initiation

3.3. Methods of evaluating crack initiation

3.4. Practical method of structure calculation

3.5. Bibliography

Chapter 4. Low-cycle Fatigue

4.1. Introduction

4.2. Phenomenological description of low-cycle fatigue

4.3. Adaptation mechanism and cracking during low-cycle fatigue

4.4. Conclusion

4.5. Acknowledgements

4.6. Bibliography

Chapter 5. Gigacycle Fatigue

5.1. Introducing the real-life fatigue life of machines

5.2. Testing process

5.3. Systems of piezoelectric fatigue machines

5.4. SN curves above 107 cycles

5.5. Initiation mechanism under gigacycle fatigue

5.6. Assessing fatigue strength

5.7. Conclusion

5.8. Bibliography

Chapter 6. Fatigue Crack Growth Laws

6.1. Introduction

6.2. Models describing crack propagation

6.3. Critical evaluation of the models

6.4. Future plans

6.5. Conclusion

6.6. Bibliography

Chapter 7. Short Crack Propagation

7.1. Introduction

7.2. Theoretical considerations showing the limits of LEFM

7.3. Experimental observations

7.4. Role of closure in the behavior of short cracks

7.5. Modeling of the behavior of short cracks

7.6. Conclusion

7.7. Acknowledgements

7.8. Bibliography

Chapter 8. Plastic Deformation Mechanisms at the Crack Tip

8.1. Introduction

8.2. Fatigue plastic deformation at the crack tip

8.3. Microfractographic aspects of the fatigue crack

8.4. Model based on displacement on crack tip opening

8.5. Cyclic stress hardening at the crack tip

8.6. Model based on the effective stress intensity factor

8.7. Conclusion

8.8. Bibliography

Chapter 9. Local Approach to Fatigue Crack Growth

9.1. Introduction

9.2. Plasticity at the crack tip

9.3. Cyclic plasticity at the crack tip

9.4. Local approach to fatigue crack growth

9.5. Conclusion

9.6. Bibliography

Chapter 10. Corrosion Fatigue

10.1. Introduction

10.2. Crack initiation

10.3. Short cracks

10.4. Long crack propagation

10.5. Conclusions

10.6. Bibliography

Chapter 11. Effect of Environment

11.1. Introduction

11.2. Effect of environment on lifetime under high-cycle fatigue conditions

11.3. Influence of the environment on fatigue crack propagation

11.4. Conclusion

11.5. Bibliography

Chapter 12. Fatigue under Variable Amplitude Loadings

12.1. Introduction

12.2. Variable amplitude loadings

12.3. Fatigue tests under variable amplitude loadings

12.4. Factors influencing the test results under variable amplitude loading .

12.5. Fatigue lifetime assessment under variable amplitude loading

12.6. Conclusion

12.7. Bibliography

List of Authors

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Library of Congress Cataloging-in-Publication Data

Fatigue of materials and structures / edited by Claude Bathias, André Pineau.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-051-6

1. Materials--Fatigue. 2. Materials--Mechanical properties. 3. Microstructure. I. Bathias, Claude. II. Pineau, A. (André)

TA418.38.F375 2010

620.1’12--dc22

2010002223

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-051-6

Foreword

This book, along with the forthcoming publication Fatigue of Materials and Structures: Application to Damage and Design (edited by C. Bathias), is the most comprehensive compilation of current approaches in the field of fatigue of materials and structures under repeated loads/loadings of various types. Historic methods, as well as the most recent approaches and current research, are included. Professors Claude Bathias and André Pineau have selected a group of outstanding experts on the various topics in the field for each chapter and have themselves contributed to some chapters dealing with their specialties. These books are great references for anyone wishing to be up-to-date on any topic in this field.

Although the fatigue of materials has been studied for over 150 years, many significant approaches have been developed in the past 100 years. The Coffin-Manson “plastic strain cycling approach” for low cycle failures and later the beginning of the “damage tolerance approach” through the fracture mechanics correlation of crack growth rates were suggested in the 1950s. Indeed these methods were shown to be applicable in the late 1950s but were frequently ignored until the failure of an F-111 aircraft in December 1969. This crash convinced the US Air Force to develop and use damage tolerance methods on every aircraft. The US Federal Aviation Agency was soon applying similar methods to ensure a sufficient crack-growth life in order to set up adequate inspection intervals for critical structural parts. Earlier in the 1960s Westinghouse and others used/applied fatigue crack-growth testing to ensure sufficient life in the case of various power generating systems. Since then, many novel applications of these newer methods have been developed.

More recently, Bathias and others have shown that the “traditional fatigue limit stress”, below which failures were regarded as not occurring, are unsafe for “very high cycle fatigue” of the order ranging from 108 to 1010 loading cycles. This evidence was determined using ultrasonic testing at 20 to 30 kHz. This field is still rapidly developing but is thoroughly covered in this book. Further discussion dealing with the historical aspects of fatigue are detailed in the introduction in Chapter of this book.

Each chapter is self-contained on the topic of interest. Each chapter is well referenced in order to provide the reader with a thorough background and to act as a source for deeper study on the topics covered in this book. As a consequence, readers can use this book as a guide to further information on the topics they are interested in. In some cases, the chapters focus on similar topics as they belong to the same general category but are written from different points of view and with different emphasis.

Chapters 2, 3, 4, 5 and 9 present the approach of failure cycles from low cycle plastic fatigue to very high cycle behavior, including the effects of notches, hardening mechanisms, etc., with many other variables involved. Within Chapter 9 fatigue crack growth mechanisms are also discussed.

Chapters 6, 7, 8 and 9 (again) deal with fatigue crack growth from small to long cracks with various models. They cover growth laws and their mechanisms. Chapters 10 and 11 provide a thorough overview of environmental factors, from aggressive to vacuum effects, leading to the initiation and growth of cracks. While Chapter 12 discusses loading interaction effects for a wide variety of structural applications and the counting methods for these various loading programs and types.

As a veteran of this field, allow me to point out the excellence of the work of some of the outstanding young stars of this field, such as Sylvie Pommier and Thierry Palin-Luc, who contributed to the writing of Chapters 9 and 12. Although these volumes present the current state of understanding, this field has many other outstanding young researchers who will develop new approaches as time goes by. However, these volumes stand as a full picture of the current “state of the art” in understanding fatigue phenomena.

Paul C. PARIS

May 2010

Chapter 1

Introduction to Fatigue: Fundamentals and Methodology

1.1. Introduction to the fatigue of materials1

1.1.1. Brief history of fatigue: its technical and scientific importance

Experience shows that fracture of structures or machine parts during regular operating conditions are most often due to fatigue. Structural integrity has always been an obstacle to industrial development. Its consequences could be seen during the development of mechanical industry in the 19th century. The industrial revolution, particularly the development of rail transportation, was affected from the start by a certain number of serious accidents, such as the one in Versailles, 1842, where the rupture of an axle caused the death of 60 people [SMI 90]. This death toll is close to that of the two Comet plane crashes that occurred in 1954.

It is known that fatigue damage costs several percent of the gross domestic product of the engineering industry. For this reason, we can understand the fact that articles and papers about this type of damage are ever increasing. Toth [TOT 01], who recently checked the COMPENDEX data base, found about 10,000 articles on this topic between 1988 and 1993, which comes to 2,000 articles a year.

According to Schütz [SCH 96], Braithwaite [BRA 1854] introduced the term “metal fatigue” in 1854. Despite this, Lemaitre [LEM 01] reckons that Poncelet mentioned this term during an engineering lecture in Metz as early as 1839, and that Rankine used it in 1843. To gain a better understanding of the work carried by Poncelet and Rankine in this field, we can refer to Timoshenko’s work dealing with the history of the strength of materials [TIM 53]. As a matter of fact, this term has probably been in use for a long time. For instance, Stendhal used it in one of his pieces “Memoirs of a tourist” published in 1838 [STE 1838]. On his way to Civitavecchia, in Italy (where he had been appointed Consul), while crossing the Loire river in La Charité one of the axles of his carriage broke. What he wrote is as follows:

“La Charité — April 13. I was riding through the small town of La Charité, when, as a reminder of the long thoughts I had in the morning about iron diseases, the axle of my carriage suddenly broke down. I have to be blamed: I swore that if I ever had my own carriage, I would get a nice Fourvoirie axle, with six mild steel rods, forged under my own eyes… I checked the iron grain of my axle; it was larger as it has apparently been used for a long time… .”

We should remember that in those times, and for many years during the 19th century, people thought that iron “crystallized” due to mechanical vibrations. The fact that Stendhal, who lived at the same time as Poncelet, already knew what fatigue was, at least in this form, is not surprising. They both campaigned for Napoleon in Russia in 1812 and we can assume that they would have discussed this subject.

Excellent reviews on the history of fatigue have been written, some of them very recently. We can for instance refer to the work of Schutz [SCH 96] which lists more than 550 references, such as Toth [TOT 01], or Schijve [SCH 03].

It is worth noting that some works on this subject have recently been published:

– Bathias and Baïlon [BAT 97];

– Bathias and Paris [BAT 05];

– Henaff and Morel [HEN 05];

– Murakami [MUR 02, MUR 03];

– Polak [POL 91];

– Reifsnider [REI 91];

– Schijve [SCH 01];

– Shaniavski [SHA 07]; and

– Suresh [SUR 98].

Here we should mention two regularly published journals that explicitly refer to the fatigue phenomenon: Fatigue and Fracture of Engineering Materials and Structures and the International Journal of Fatigue. In addition to this, in other countries scientific societies organize lectures and conferences on this subject, such as the ASTM (American Society for Testing and Materials) in the USA and the SF2M (French Society of Metallurgy and Materials) in France.

Table 1.1.A few stages and main events regarding the history of the fatigue phenomenon

Year

Event

1842

Meudon railway accident

1858

First publication by Wohler

1860-70

Wohler experiments on smooth and notched axles. Bending and torsion tests — Investigation on the effect mean stress

-1881

Study by Bauschinger which initiated low-cycle fatigue

1910

Basquin law

1913

Stress distribution within notches (Inglis)

1920

Energy balance regarding the propagation of a crack (Griffith)

1930

Stress concentration factor and endurance limit (Peterson)

1937

Neuber concept applied to notches

1939

Statistical approach Weibull law

1945

Miner concept for fatigue damage accumulation

1953-54

Low cycle fatigue. Manson — Coffin law

1954

Comet aircrafts accidents

1956

Introduction of strain energy released rate (Irwin)

1960

Servohydraulic machines

1961

Paris law

1968

Introdcution of effective stress intensity factor (Elber)

1988

Aloha B737 accident

1989

DC 10 Sioux City accident

1996

Pensacola accident

1998

ICE. Eschede railway accident

2006

Los Angeles B767 accident

Some memorable stages and events that have marked the history of fatigue are highlighted in Table 1.1. As mentioned earlier, this type of damage has clearly been of great importance during the development of rail transportation. The various ruptures that Wöhler observed in Germany led him to undertake a systematic study of this type of damage.

Along with trains and many other mechanical structures, aircraft were also readily affected by the fatigue phenomenon. The first serious accidents that occurred are those involving two Comet aircraft in 1954. A more recent example was the Aloha accident in 1988, which involved a Boeing 737. The damage was really serious, as we can see in Figure 1.1. This accident was caused by the formation of cracks due to fatigue and corrosion in the assembly rivets area within the fuselage. As a result, numerous studies have been carried out regarding the issue of multiple site damage.

Figure 1.1.The Aloha Airlines Boeing 737at Honolulu international airport, Hawaii, following the accident on April 28, 1988

Another example concerns the MacDonald Douglas DC 10 crash, which occurred in Sioux City in Iowa in 1989 (see Figure 1.2). The explosion of one of the engines led to this tragic accident. Even more recent was the Pensacola crash, when one of the engines broke apart due to cracking initiation caused by a drilling defect within a fan disk (see Figure 1.3).

These three examples from the aeronautical industry should not lead people to think that aircrafts as a means of transportation are dangerous and the only means affected by fatigue phenomenon. If we calculate the distance to passenger ratio, flying remains the safest means of transport. Nevertheless, due to its rapid development and despite the work being done on its design, manufacturing and maintenance, we can predict that in about 10 years’ time a major aircraft accident is likely to occur every week (see Figure 1.4). Let us keep in mind that human error is the main cause of accidents involving aircraft. Accidents caused by defects in the materials are still occurring in spite of improved manufacturing processes.

Figure 1.2.DC 10 aircraft crash. Part of a detached engine. Sioux City Airport, July 19, 1989

Figure 1.3.Pensacola Crash (Florida, USA), July 6, 1996, was due to a failure during thetake off of a Delta Airlines MD-88 aircraft

Fatigue also affects many other fields of transport, as shown in Figure 1.5 where cylinder heads of diesel engines subjected to increasing thermo-mechanical loading can break due to thermal and mechanical fatigue cracking if their design is wrong [SAL 07].

Figure 1.4.Statistical study of the evolution of air traffic and of the number of crashes (MANHIRP, 2001, see also 1001crash.com)

Figure 1.5.Cracking within the cylinder head of a diesel engine [SAL 07]

1.1.2. Definitions

Fatigue or fatigue damage refers to the modification of the properties of materials due to the application of stress cycles whose repetition can lead to fracture.

– asymmetrically reversed: 0 < σm < σa, −1 < R < 0;

– alternating tension: σm > σa, 0 < R < 1.

Figure 1.6.Different cases of fatigue stresses: load-time; force-strain

Plastic deformations occur with low-cycle fatigue (see Chapter 4). Usually, the fatigue phenomenon occurs without any general plastic deformation, which makes it less likely to be noticed. Nevertheless this phenomenon occurs with a localized plastic deformation around pre-existing defects within the materials, at the notches of the structures, or at the tips of a crack when it has already been initiated.

For multi-axial loading, which will be presented in [BAT 10], the definition of a strain amplitude is much more subtle, especially when loading is not proportional.

Fatigue is rarely perfectly cyclic (of constant amplitude and frequency), as shown in Figure 1.6. In many cases (thermal engines, bridges, etc.), loads have variable amplitudes and frequencies. These kinds of loads are examined in detail in Chapter 12.

Theoretically, fatigue damage only depends on the number of cycles and not on their frequency. As a matter of fact in most cases frequency does have a consequence. This is the case when environmental and visco-plasticity effects at high temperatures are involved (see Chapters 10 and 11).

In general, lifetime is measured using the number of cycles to failure, NF. When N cycles have occurred (N < NF) a given damage is accumulated and has to be evaluated. This allows us to determine the residual lifetime of the structure and the management of its operation, such as the timing between aircraft inspections.

We define endurance as the strength capability of components and entire structures before fatigue develops.

Thus, in general, fatigue occurs as soon as time dependent forces are involved. As a consequence, fatigue damage is characterized by its danger, which is basically that fracture can occur at low cycle stresses that in most cases are lower than the tensile strength and even lower than the elastic limit of the material.

1.1.3. Endurance diagrams

The easiest fatigue test consists of subjecting each specimen to periodic loading cycles, most frequently sinusoidal, with a maximum amplitude Ja, along with a constant frequency. The number of cycles is measured once rupture starts to occur (NF). We then obtain a curve which looks like the one plotted in Figure 1.7.

Figure 1.7.Wöhler curve and definition of the various endurance areas

With this curve, known as the Wöhler curve, SN curve (stress-number of cycles) or endurance curve, we can differentiate four different regions:

– with a high stress, we get low cycle plastic fatigue. Within this region, studied later on in Chapter 4, fracture occurs after a relatively low number of cycles (102 to 104) along with a significant plastic deformation. This type of damage has been studied since the 1950s, following Manson [MAN 52] and Coffin [COF 54] who introduced the Coffin-Manson law;

– with a lower stress there is a fatigue region where endurance is limited. Within this region, fracture occurs for a given number of cycles. The lower the stress amplitude, the higher the number of cycles. The region of limited endurance is presented in Chapter 2;

– an endurance region, which has been considered as an infinite lifetime, or safety region, corresponding to what is called the endurance limit. For steels, this region is reached after 106 to 107 cycles. In reality metal alloys have no real endurance limit. This has led us to consider the fourth (gigacycle) region in the past 10 years;

– a region corresponding to the gigacycle fatigue, which is significant for a given number of applications. Within this region, studied in Chapter 5, we often see that the “endurance limit” still decreases when the number of fracture cycles increases.

1.2. Mechanisms of fatigue damage

1.2.1. Introduction/background

On the Wöhler curve (see Chapter 2) we can see four stages, as shown in Figure 1.8, where, in contradiction with what is described in the previous figure, an endurance limit σd is highlighted for demonstration purposes. The upper region I corresponds to conditions in which specimens are broken. The lower region IV corresponds to the cases of unbroken specimens, where curve A separates both regions. Within the area directly below curve A, we can see two new regions located above the endurance limit: region III corresponding to the initiation of a crack, and region II associated with the propagation of this crack, the number of corresponding cycles being Np. We can also see that initiation Ni represents the main part of the lifetime when the number of cycles to failure NF, is high.

Figure 1.8.Wöhler SN curve (A) and number of crack initiation cycles (B)

Numerous fatigue damage indicators, in addition to crack initiation and propagation, have been studied, such as electrical resistivity. For a decade, infrared thermography has been used, providing researchers with encouraging results [DOU 04, LUO 95, LUO 98]. However, it is still too early to know whether this method can provide reliable results, and especially whether it can be used to speed up the determination of endurance curves.

1.2.2. Initiation of fatigue cracks

Since the first observations were made using optical microscopy in 1903 by Ewing and Humphrey [EWI 03], the initiation of fatigue cracks has been widely studied. In the mid-1970s the articles published by Thompson and Wadworth [THO 58] and by Laird and Duquette [LAI 71] enabled a review to be written on this subject. Since then, significant efforts have been devoted to this stage of fatigue damage.

Work carried out by Forsyth [FOR 51, FOR 53] showed that fatigue damage is mainly surface related. On the polished surface of the specimens, we can observe steps due to the formation of localized deformation bands, known as persistent slip bands. These bands are formed on the sliding planes with a maximum resolved shear stress. The mechanisms by which these bands form are presented in Chapter 4. Topography of the surface reveals the formation of intrusions and extrusions, as shown in Figure 1.9.

Figure 1.9.a) Initiation of micro-cracks due to the sliding of alternate planes and to the formation of intrusions and extrusions at the free surface (cross-section); b) formation of a main crack from micro-cracks; c) characteristic formation of stage I intrusions and extrusions at the surface of a fatigue specimen made of copper

During a uniaxial test on polycrystalline specimens, these bands, which will lead to the formation of stage I micro-cracks, appear at a 45 degree angle to the tensile axis. Only a few grains are involved in the formation of these bands. Orientation of the persistent bands and of the stage I cracks is significant not only in the case of a uniaxial loading (tension or torsion), but also in the case of a multi-axial loading where the directional characteristic of fatigue damage is essential. Brown and Miller [BRO 79, MIL 91] have introduced the really useful notion of type A and B facets regarding multi-axial loading, as shown in Figure 1.10. As expected, type B facets, whose slip vector goes into the material, are usually more dangerous than type A facets, whose slip direction is tangent to the free surface of the specimen. Unfortunately, few studies have been carried out regarding this directional characteristic that occurs at the start of fatigue damage. We can nevertheless list some studies, such as the one carried out by Jacquelin [JAC 85].

Figure 1.10.Directional characteristic of fatigue damage. The significance of the orientation of the strain field compared to the surface plane and free surface plane (cross hatched); a) type A facet; b) type B facet

Intrusions and extrusions, associated with persistent slip bands and the micro-propagation of stage I cracks, extend over a distance of the order of the grain size. Indeed, as soon as this micro-crack of significant crystallographic nature hits the first grain boundary it branches off, following a stage II course and then propagates perpendicularly to the direction of the maximum principal stress.

The definition of initiation remains ambiguous as it depends on the chosen scale. We usually define this damage stage as corresponding to the number of cycles, Ni, that have to be applied before the crack branches to stage II. Here, the corresponding distance is similar to the size of a grain. To the best of our knowledge, this is the most plausible definition. The most commonly accepted definition, which corresponds to the failure of a specimen or to the reduction of the maximum tensile stress by a certain amount (for example 5%), is not accurate enough.

1.2.3. Propagation of fatigue cracks

Stage II crack propagation according, that is to say in mode I, has been studied frequently since the early work by Paris and Erdogan in 1963 [PAR 63]. When these cracks are long enough, the rate of crack propagation, as presented in Chapter 4, can be described using Paris’ law:

[1.1]

where a stands for the length of the crack and ΔK for the variation of the stress intensity factor K, whereas C and m are constants, depending on the material. The loading ratio R is also important. The various laws proposed are listed in Chapter 4.

A region between stage I and the long crack stage is known as the short crack stage. It has been extensively studied since Pearson published his work on the topic in 1975 [PEA 75]. He proved that short cracks propagate faster than long cracks at the same apparent value, ΔK. This short-crack phenomenon is significant and is presented in Chapter 7. It increasingly appears that the particular behavior of short cracks, or at least those termed “physically short”, can mainly be explained by their tri-dimensional aspect and by the crack closure phenomenon [LIN 95, PIN 86] concept introduced by Elber in 1970 [ELB 70].

Paris’ law is purely phenomenological. Since then, some authors have tried to develop and improve this law using the properties of materials. These authors have developed what we call a local approach to fatigue crack. The principle of this approach is to start from the crack tip stress-strain field ahead of the crack-tip (see Chapters 8 and 9) and then introduce a fatigue failure criterion. The first model of this type was proposed by McClintock in 1963 [CLI 63], who assumed that the crack propagates in successive stages under the effect of low-cycle fatigue-type damage. The law thus formulated by McClintock considers that the exponent m in Paris’ law [1.1] is equal to four and a non-propagation crack threshold ΔKth is involved. A second model was also suggested by McClintock in 1967 [CLI 67], due to the observations of Pelloux [PEL 64] and McMillan [MIL 67]. These authors showed that fatigue failure surfaces were covered with striations — one striation corresponding to one cycle — at least in a certain region of crack propagation rates (0.01 to 1 µm/cycle). McClintock then related crack growth rate per cycle, being the distance between the striations, to the blunting at the crack tip (see Chapter 6). This model then considers that the slope m of Paris’ law is equal to two. In practice, we can observe that the value of this exponent goes from two to six in the case of most materials. Since McClintock developed his models, others have been proposed (see Chapters 6, 8 and 9 for more information).

1.3. Test systems

The most commonly used method to obtain endurance curves is the rotating bending or plane bending test (see Figure 1.11). The machines used for these tests allow for frequencies close to 20 Hz.

Figure 1.11.Principle of rotary bending tests

These machines are simple to use and relatively inexpensive. We can also use these machines to perform some traction/compression tests. The advantage of this is that the effect of a constant stress through the section of the specimen can be observed. The choice of specimen and loading mode is significant because a size effect is involved. Thus, with similar conditions, and especially with a similar cyclic stress, the tension-compression endurance is lower than that calculated in the case of rotary bending, itself being lower than that measured using plane bending.

Resonance machines have been developed in order to perform higher frequency tests on metallic materials. Some specific machines can enable much higher frequency tests (close to 20 KHz) and thus allow us to study the region of gigacycle fatigue (see Figure 1.7). Chapter 5 deals with this particular topic.

All of these tests have been carried out with a load or imposed strain within the elastic region. Low-cycle fatigue tests, however, are performed under strain control, meaning that an extensometer system has to be used. The servo-hydraulic machines were developed in the mid-1960s for that purpose. A description of these tests, specimens and of extensometers can be found in Chapter 4.

More specific and less common machines have to be used in order to carry out multi-axial fatigue tests. Tension-torsion fatigue tests on thin tubes are well developed. The introduction of an internal/external pressure within the traction-torsion stressed tubes allows the degree of multi-axiality to be increased. These machines are less commonly found in practice, especially the ones allowing a range of temperatures to be studied. The downside (or the advantage, depending on the context) of these tests, is that the reference of the main stresses is not fixed. This issue can be overcome if a machine that can exert independent forces along three orthogonal directions (XYZ) is used. Only one machine of this type can be found in France. Biaxial machines (XY), however, are easier to use in order to test relatively thin materials.

1.4. Structural design and fatigue

Since Wöhler, engineers have developed new mechanical parts that are capable of resisting fatigue. The improvement of fatigue tests came hand-in-hand with the improvement in design methods.

The initial idea of an infinite lifetime or a fatigue limit under which a propagating fatigue crack could not form within a metal, has been questioned. At the same time, experiments based on Gauss statistics around the notion of a fatigue limit in order to predict an infinite lifetime related to a quasi-asymptotic SN curve were questioned. Statistics therefore had to be introduced to fatigue test results and the design methods were no longer purely deterministic but probabilistic.

As an example, we can consider a research engineer working in the aeronautical industry. He designed the “Caravelle” aircraft, in the 1950s, as beams put together under bending conditions, using Timoshenko’s strength of materials approach. “Concorde” was the first aircraft in which the fracture mechanics was partially applied in 1970. The design of the European supersonic aircraft was first based on the conventional concept of strength of materials but was then improved using the notion of fail safe, which was developed from the application of fracture mechanics. Following this, the “Airbus” was the first aircraft to be built using the damage tolerance design in 1980.

In scientific terms, the main stages in the development of mechanical design are listed below chronologically:

– safe life: the structure is designed so that the stresses remain lower than the endurance limit. No defect is considered or accepted;

– fail safe: the structure is designed in relation to the residual strength of the metal when a planar defect, representing a crack, is involved. Some defects are accepted;

– safe crack growth: the failure mechanism allows the crack growth instability to be deterministically predicted.

A fourth approach — the local approach to failure — has recently been developed in France to overcome problems in the analysis of the fatigue of metals. In addition to this, the discovery of gigacycle fatigue in the 1990s highlighted the critical role of microscopic defects, such as inclusions, porosities, large grains due to forging, etc.

Micro-cracks remain the most harmful lifetime defects in the region of a few million cycles and micro-defects control billion cycle lifetime, the latter being the lifetime of engines, turbines and bearings. The initiation of distortion bands around the defects is then essential. Control of resistance to fatigue caused by micro-defects has been extensively studied. The notion of infinite lifetime obviously has to be discarded and the failure mechanism along with its local approach have to be added to the study of the resistance of materials to fatigue failure.

1.5. Fatigue of polymers, elastomers and composite materials

As modern industry commonly replaces one material with another, the fatigue of non-metallic materials is worth mentioning. Moreover, a global representation of material fatigue is fundamentally worth studying. This topic will be covered in [BAT 10].

All materials are prone to fatigue damage in their own way.

We can basically say that the fatigue of metals arises from:

– plasticity;

– general plasticity during low-cycle fatigue;

– plasticity during mega-cycle fatigue under plane stress; and

– micro-plasticity on a grain-scale during gigacycle fatigue.

In any case, dislocation debris accumulates in permanent slip bands before the first cracks develop (see Figure 1.12).

It would be a mistake to consider the fatigue of non-metallic materials as related to plasticity because there is no dislocation within amorphous or semi-crystalline polymers. Damage mechanisms in polymers are related to the formation of cavities or cavitations, two terms used to differentiate the effects of stress tri-axiality and the hydrostatic tensor. When plane stress prevails, fatigue quasi-cleavages can appear within some polymers. All these mechanisms are related to the progressive degradation of macromolecules and have no physical relationship to the plasticity of metals (see Figures 1.13 and 1.14).

Figure 1.12.Slip bands within low carbon steel

Figure 1.13.Cavitations due to hydrostatic tension in an NR rubber

Figure 1.14.Shear damage in epoxy matrix glass fiber composites

Basically, fatigue of metals is modeled from von Mises plasticity criteria, based on the stress tensor deviator. This approach is totally justified by solid physics, which confirms that dislocation slip is governed by the shear components of the stress tensor. Experiments have shown that the formation of cavities, cavitations and quasi-cleavages within polymer materials do not just depend on shearing, but also on the principal main strains and the hydrostatic part of the stress tensor. As a consequence, von Mises’ criterion and its variations do not apply to the modeling of plastics, rubbers and fibrous composites. This has many important consequences. For instance, an elastomer can crack due to compressive fatigue or under hydrostatic loading, which is not the case with metals. Another example is the substitution of a metallic part with a glass-fiber epoxy matrix in order to overcome the issue of fatigue as fatigue will be increased due to tension but decreased due to compression.

We therefore have to keep in mind that the stress deviator is a main parameter in metal fatigue.

1.6. Conclusion

Fatigue is of major technical and scientific importance. The design of many components is directly related to this type of damage. Fatigue is still the subject of numerous studies at various scales, from dislocations and point defects to macroscopic cracks.

In this book, we have gathered the reviews of experts in this type of loading and damage. The first volume mainly deals with metallic materials whose various damage sequences (initiation of stage I cracks and stage II propagation) are well known. For greater elucidation on the topics covered in this book, we have quoted experts in physical and mechanical metallurgy, fracture mechanisms and the local approach to failure. Environment and corrosion effects, which are of particular significance, are also discussed.

Finally, a whole chapter has been dedicated to the case of fatigue due to loading of variable amplitudes, which is commonly found in practice.

1.7. Bibliography

[BAT 10] C. Bathias, Fatigue of Materials and Structures: Application to Damage and Design, ISTE, London, John Wiley, New York, 2010.

[BAT 05] C. Bathias, P.C. Paris, Gigacycle Fatigue in Mechanical Practice, Marcel Dekker, New York, 2005.

[BAT 97] C. Bathias, J.P. Baïlon, La Fatigue des Matériaux et des Structures, Hermes, Paris, 1997.

[BRA 1854] F. Braithwaite, “On the fatigue and consequent fracture of metals”, Institution ofCivil Engineers, Minutes of Proceedings, vol. XIII, p. 463–474, London, 1854.

[BRO 73] M.W. Brown, K.J. Miller, “A theory for fatigue failure under multiaxial stress-strain conditions”, Proceedings of the Institution of Mechanical Engineers, vol. 187, p. 745–755, 1973.

[CLI 63] F.A. McClintock, “On the plasticity of the growth of fatigue cracks”, Fracture of Solids, John Wiley, New York, 1963.

[CLI 67] F.A. McClintock, “Fatigue crack propagation”, ASTM STP 415, written discussion, p. 170, 1967.

[COF 54] L.F. Coffin, “A study of the effect of cyclic thermal stresses on a ductile metal”, Trans. ASME, vol. 76, p. 931–950, 1954.

[DOU 04] C. Doudard, S. Calloch, F. Hild, P. Cugy, A. Galtier, “Identification of the scatter in high cycle fatigue from temperature measurements”, C.R. Mécanique, vol. 332, p. 795–801, 2004.

[ELB 70] W. Elber, “Fatigue crack closure under cyclic tension”, Eng. Fract. Mech., vol. 2, p. 37–45, 1970.

[EWI 03] J.A. Ewing, J.C.W. Humphrey, “The fracture of metals under repeated alternations of stress”, Phil. Trans. Royal Soc. London, vol A.200, p. 241–250, 1903.

[FOR 51] P.J.E. Forsyth, “Some metallographic observations on the fatigue of metals”, J. Inst. Met., vol. 80, p. 181, 1951.

[FOR 53] P.J.E. Forsyth, “Exudation of material form slip bands at the surface of fatigued crystals of an aluminum copper alloy”, Nature, vol. 171, p. 172–173, 1953.

[HEN 05] G. Henaff, F. Morel, Fatigue des Structures, Ellipses, Paris, 2005.

[JAC 85] B. Jacquelin, F. Hourlier, A. Pineau, “Crack initiation under low cycle multiaxial fatigue”, ASTM-STP 853, p. 285–313, 1985.

[LAI 71] C. Laird, D.J. Duquette, Proc. 2nd Int. Conf. Corrosion Fatigue, NACE, Houston, pp. 88–117,1971.

[LEM 01] J. Lemaitre, “Petite histoire de l’expérimentation en mécanique des solides”, Meccanica, vol. 36, p. 13–51, 2001.

[LIN 95] T. Lindley, A. Pineau, “Short crack effects in fracture and fatigue”, Rev. de Mét., vol. 92, p. 187–201, 1995.

[LUO 95] M.P. Luong, “Infrared thermographic scanning of fatigue in metals”, Nucl. Eng. Design, vol. 158, p. 363–376, 1995.

[LUO 98] M.P. Luong, “Fatigue limit evolution of metals using an infrared thermographic technique”, Mech. Mat., vol. 28, p. 155–163, 1998.

[MAN 52] S.S. Manson, “Behavior of materials under conditions of thermal stress”, Heat Transfer Symposium, University of Michigan Engineering Research Institute, p. 9–71, 1953.

[MIL 63] J.C. Mc Millan, R.M. Pelloux, “Fatigue crack propagation under program and random loads”, ASTM STP 415, p. 505–535, 1963.

[MIL 91] K.L. Miller, “Metal fatigue-past, current and future”, Proc. Instn. Mech. Engrs., vol. 205, p. 1–14, 1991.

[MUR 02] Y. Murakami, Metal fatigue: Effect of Small Defects and Nonmetallic Inclusions, Elsevier, Amsterdam, 2002.

[MUR 03] Y. Murakami, “Cyclic loading and fatigue”, in Ritchie R.O. (ed.), Comprehensive Structural Integrity, Fracture of Materials from Nano to Macro, vol. 4, Elsevier, Amsterdam, 2003.

[PAR 63] P. Paris, F. Erdogan, “A critical analysis of crack growth propagation laws”, J. Basic Eng, vol. 85, p. 528–534, 1963.

[PEA 75] S. Pearson, “Initiation of fatigue cracks in commercial aluminum alloy and the subsequent propagation of very short cracks”, Eng. Fract. Mech., vol. 7, p. 235–247, 1975.

[PEL 64] R.M. Pelloux, “Fractographic analysis of the influence of constituent particles on fatigue crack propagation in aluminum alloys”, ASM Trans. Quart., vol. 57, p. 511, 1964.

[PIN 86] A. Pineau, “Short fatigue crack behaviour in relation to three-dimensional aspects and crack closure effects”, in Ritchie R.O, Lankford J. (ed.), Small Fatigue Cracks, Metallurgical Society of AIME, Warrendale, PA, p. 191–211, 1986.

[POL 91] J. Polak, Cyclic Plasticity and Low Cycle Fatigue Life of Metals, 2nd edition, Academia, Prague, 1991.

[REI 91] K.L. Reifsnider, Fatigue of Composite Materials, Elsevier, Amsterdam, 1991.

[SAL 07] R. Salapete, PhD thesis, École des Mines, Paris, December 17, 2007.

[SCH 01] J. Schijve, Fatigue of Structures and Materials, Kluwer Academic Publishers, Dordrecht, 2001.

[SCH 03] J. Schijve, “Fatigue of structures and materials in the 20th century and the state of the art”, Int. J. Fatigue, vol. 25, p. 679–702, 2003.

[SCH 96] W. Schutz, “A history of fatigue”, Eng. Fract. Mech., vol. 54, p. 263–300, 1996.

[SHA 07] S.A.A. Shaniavski, Modeling of Fatigue Cracking of Metals, Synergetics for Aviation, Ufa, Publishing House of Scientific and Technical Literature “Monography”, 2007.

[SMI 90] R.A. Smith, “The Versailles railway accident of 1842 and the first research into metal fatigue”, Fatigue’90, vol. IV, EMAS, Birmingham, p. 2033–2041, 1990.

[STE 1838] Stendhal, Mémoires d’un Touriste, Charité-sur-Loire, 13 Avril 1837, Ed Ambroise Dupont, p. 16, 1838.

[SUR 98] S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, 1998.

[THO 58] N. Thompson, N.J. Wadworth, “Metal fatigue”, Adv. Phys., vol. 7, p. 72–169, 1958.

[TIM 53] S. Timoshenko, History of Strength of Materials, McGraw-Hill, New York, 1953.

[TOT 01] L. Totoh, “Fatigue crack growth laws and their material parameters”, Zeszyty Naukowe Politechniki Opolskiej. Siria: Mekanika z. 67, vol. 269, p. 349–368, 2001.

1Chapter written by André Pineau and Claude BATHIAS.

Chapter 2

Modeling of Fatigue Strength and Endurance Curve1

2.1. Introduction

Scatter of the results of fatigue tests is now accepted to be an experimental and physical fact. In the past, following a deterministic way of thinking, such scatter was considered to be due to imperfections in the test conditions and, as a consequence, it was assumed that scatter could be reduced ad libitum. The scatter is to be considered as a physical aspect of fatigue phenomenon. Moreover, from a practical point of view it is generally far too difficult and/or expensive to entirely remove some of the causes of experimental error, even if it is possible in theory. The simultaneous action of these two kinds of causes, both experimental and physical, leads to a scatter of test results that are rarely negligible with regard to the amounts being measured. On the contrary, in most cases such scatter is often important, and sometimes very significant.

Figure 2.1 gives an example of the scatter observed during some tests performed on a steel specimen.

This is why statistical methods have to be used in order to experimentally determine the characteristics of the fatigue phenomenon. These methods allow us to estimate either the fatigue strength at N cycles along with the corresponding standard deviation or to draw the curve resulting from the amplitude of cyclic stress to the number of cycles to failure.

Figure 2.1.Presentation of the test results using a Wöhler diagram (S-log N)

Various factors influence fatigue resistance; in particular, the conditions of cyclic stress applications, which can relatively modify the fatigue strength of a piece or an entire structure.

In addition, many studies show that the endurance of a material is related to its ultimate tensile strength and ductility.

This is why, in the case of steels, several authors have suggested a rough estimation of the endurance limit by relating it to the characteristics measured during a tension test.

This chapter aims to present the main statistical methods that can be used to characterize fatigue behavior under simple loading of a material from test results and to specify the influence of the application conditions of cyclic stress.

2.2. Nature and aspect of the scatter of fatigue test results

The scatter we observe results directly from the nature and physiology of a material, which we can split logically into three parts:

– internal to the material;

– due or related to the preparation of the specimens or pieces;

– external to the specimen.

The nature and mode of action relating to the preparation of an environment surrounding the specimen seem to be the most obvious causes of scatter. The setting operations, which include turning, milling and straightening, for instance, relate to preparation of specimens/pieces. The settings are known to influence the endurance of the pieces, as well as the thermal treatments, so the results cannot be rigorously reproduced on other identical specimens from the same batch.

External causes of scatter include, in particular, the uncertainty of the setting within the test machines, and of adjustment of the applied loading, cycle frequency, etc., in addition to the fact that the influence of the surroundings is not negligible.

The causes of scatter, whose mechanisms of action are less well known, are those within the material: inclusions, structure heterogenities, etc. Moreover, these causes are not independent from the influence of preparation settings, which can modify the material and its heterogenity (especially the thermal treatments). Finally, the mechanism of damage itself involves a combination of the various causes of scatter.

Experience has shown us that the result of these combined actions generally leads to the characteristic response shape of SN curves.

If a given number of stress cycles N is set (see Figure 2.2a) and the observed event being the failure or non-failure of the specimen before this cycle number, we define the response to the SN curve — the curve representing the probability of this event — as a function of the amplitude of the cycles.

Various methods have shown that these curves are normal sigmoid shapes that can be represented by the distribution function of a variable with a normal distribution. The failure probability can then be expressed as a function of the amplitude S of the stress using the following equation:

[2.1]

where, µ represents the stress amplitude where the failure probability is equal to 0.5 and σ the parameter characteristic of the scatter.

The fact that failure probability follows equation [2.1] is similar to the “all or nothing” aspect and allows the application of various statistical techniques to the study of fatigue phenomenon.

Various methods used to deal with fatigue test results aim to estimate the parameters µ and σ of a response to the stress curve for a given number of cycles.

Estimation σD of the conventional endurance limit and s of the standard deviation of the endurance region is a particular case (see Figure 2.2a).

Every response to the stress curve allows a value of the amplitude of the stress Sp, whose failure probability is P, to be defined. However, Sp is obviously a function Sp(N), of the number of cycles N and defines the curve of isoprobability of failure corresponding to P.

2.3. Determination of the endurance limit

As we just stressed, a normal sigmoid response curve is defined by two parameters: its median and its standard deviation. Figures 2.2a and 2.2b show that these parameters are obviously a function of lifetime N, set for the tests.

In the case of steels, every failure equiprobability curve is supposed to present an asymptote; in particular, the equiprobability curve 0.5, which tends towards a limit called the “endurance limit”.

In practice this limit cannot be reached and instead we use a conventional endurance limit σD, relative to a significant number of cycles N.

This value σD is related to a standard deviation called the “standard deviation of the endurance region” (standard deviation of the response curve, defined by equation [2.1] for the considered number of cycles N). Estimation of the standard deviation of the endurance region is represented by the symbol s.

2.4. Estimation methods of fatigue resistance and standard deviation with N cycles

The previous sections have shown that the estimation of fatigue resistance with N cycles and of its standard deviation or endurance limit, and the standard deviation of the endurance region, are actually the same problem.

The following estimation methods will be described and explained:

– the probit method;

– the staircase method;

– the iteration method;

– the method with K non-failed specimens.

The application of one of these methods, however, means that we need to know in advance the order of magnitude of fatigue resistance we want to study and the standard deviation of its response curve in order to choose the stress levels of the first tests and their spacing.

2.4.1. Probit method

2.4.1.1. Principle

The probit method is a method of numerical calculation used to estimate response curve parameters. We should make sure we do not confuse this with the test plane, which is supposed to provide the results that are going to be submitted. This confusion is common, however, because of a misuse of language. The test plane is itself really easy to calculate; or at least it seems to be. Its aim is simply to determine the response curve being sought.

As a stress amplitude has been chosen, n specimens are tested until a given number of cycles, N, selected for the tests. In these conditions, the number k of the specimens which, among these n, fail before N cycles, follows binomial probability distribution (the probability law of the number of specimens which, among n, lead to the birth of an event with a given probability).

The properties of the binomial law are well known. If p is the probability of the observed event, k will be, on average, equal to np1. Nevertheless, k is a random variable presenting a given scatter that we can characterize using either variance or standard deviation k.

Theory shows that the value of this standard deviation σ(k) is:

[2.2]

The real value p of the failure probability before N cycles under stress amplitude S is obviously unknown and tests are carried out in order to estimate it.

We can show that, using probability calculation, the estimation of p is given by frequency f:

[2.3]

That is to say, in our case, by the ratio of the number of failed specimens k to the number n of those tested during similar tests.

If k is random, f is also random and the standard deviation of the frequency σ(f) is given by:

[2.4]

If we consider that the distribution of f can be roughly characterized as twice the standard deviation, we can see that f is a poor estimation of p.

On the other hand, we understand that to define the studied response curve we need at least four or five points. For this reason we need to carry out about 100 tests. We will see that, despite the use of a statistical method of estimation allowing us to get the most information from the observations, it is hard to appreciably reduce this figure.

2.4.1.2. Graphic representation of a response and test result sigmoid curve

2.4.1.2.1. Galton’s anamorphosis

If we define:

[2.5]

Equation [2.1], representing a response curve with parameters µ and σ can be written as:

[2.6]

The advantage of this representation is that, with the second member of the equation, we can use a function called the reduced Laplace-Gauss integral. This integral only depends on the upper integration end-point (S — µ)/σ and has tables that are known.

To simplify this, we consider:

[2.7]

in such a way that equation [2.6] becomes:

[2.8]

Figure 2.3.Reduced Gauss integral

Table 2.1.Characteristic points of G(u)

We can observe that G is an increasing monotonic function, which enables us to reverse it; that is to say we can match a chosen value to every value p a priori given (0 < p < 1) using up such as:

[2.9]

We define up as the reduced deviation corresponding to p.

As the value of up is unique, equation [2.8] of a response curve with parameters u and σ can also be written as:

[2.10]

or as:

[2.11]

The above equation is of high practical importance as it enables us, using an appropriate choice of scale for p values, to replace the direct representation of a sigmoid curve (with p(S), S as coordinates) by a line (with up, S as coordinates).

It is worth noting that this linear representation of a sigmoid curve can either use linear scales for S and up or directly display the value of p on the scale of up (see Figure 2.4).

The method dealing with the replacement of p with up is called Galton’s anamorphosis and is widely used in order to represent response curves.

We can also observe that, when the ordinates represent up,equation [2.9] can be written as:

[2.12]

The slope of the line representing the response curve then has a value of 1/σ. The higher this slope is, the lower the scatter σ is.

2.4.1.2.2. Plotting of experimental points

We already know (section 2.4.1.1) that failure frequencies with N cycles among n tests (k/n), is an estimation of the basic failure probability p.

Frequencies fi, observed at different stages Si of the stress amplitude and plotted within the same graph will then be averaged but a given scatter, on the line will represent the response curve (see Figure 2.4).

2.4.1.3. Estimation of parameters µ and σ

We then have, according to equation [2.11]:

and:

therefore:

Figure 2.4.Proportion of failed specimens with fewer N cycles as a function of the stress amplitude. XC100 (N= 30,000 to 107cycles)

Nevertheless, the graphic estimation of µ and σ is a rough method and we recommend the use of a statistical method for calculating the estimations m and s of µ and σ.

Two of these statistical methods can be found: the “probit” method and the “maximum likelihood” method. The study of these methods is not our goal, however. Useful information about these two methods can be found in application examples in [FIN 72] and [ULM 52].

The observations that have been made above regarding the number of tests needed enable us to understand these methods are not commonly used for the study of fatigue. Nevertheless, we appreciate that when these tests have been performed there can be a genuine reason for the precise estimation of µ and σ and that, as a consequence, we should not stop at the graphic representation but should go further using numerical calculation.

2.4.1.4. Conclusions

The “probit” method (wrongly termed as we do not proceed to a numerical operation of the test results using the real “probit” method [FIN 72, ULM 52]) takes a long time to be carried out. It is also expensive as a high number of tests is required and is hard to perform. It will therefore only be used for fundamental studies, when we are seeking sufficiently precise determinations of fatigue resistance and especially the standard deviation of response to the stress curves. It can also be used if we want to test, at the same time, the normal sigmoid shape of these curves (using χ2 tests) [FIN 72, ULM 52].

In other cases, it is better to apply application methods that are simpler, like ones presented in the next section.

2.4.2. Staircase method

The drawbacks of the “probit” method led to the birth of other application methods that are simpler and cheaper.

In 1948, Dixon and Mood [DIX 48] proposed the “staircase” method. The test stress stages can be found much more easily using this method. It can be carried out automatically and gives the user a wide choice of the number of tests required (which can be much lower than the “probit” method). However, if the method can be performed with a low number of tests, we should still bear in mind that the precision (accuracy and fidelity) of the results obtained will strongly depend on it.

2.4.2.1. Rules of the staircase method

As was the case regarding the “probit” method, we set a maximum test duration N and define a spacing of the stain amplitudes through arithmetic evolution, whose pitch has a value similar to that of the standard deviation σ, of the response curve.

The first test will be carried out at the level of spacing a priori thought to be the closest to the targeted median µ. From the second test, the stress amplitude level can be chosen:

– for the new test, if the previous test did not lead to failure we should choose a higher spacing stresses stage than the one used during the last test;

– if the previous test led to failure, we should choose the stage that is lower.

In other terms, as d stands for the spacing pitch of the stresses and Si for the accepted value regarding the i-th test, we will consider that:

The tests can then be carried out following this rule, one after another, until all n available specimens have been used.

Figure 2.5 gives an example of a batch of this type of test.

2.4.2.2. Use of the results

Test results obtained using this method enables us to estimate the median resistance of fatigue with the number of given cycles and, usually, the standard deviation. The calculation to be performed is simple. We first determine the type of event that occurred less frequently during the test batch: failure or non-failure. The results of this type are enough for the calculation and are more acceptable compared to other ones. The fact that the results of a given type (failures, for instance) outnumber others is often due to the choice of a starting stage that is a long way from the value to be estimated. The sequence then begins with an uninterrupted test from a batch of the same type whose inclusion would lead to a systematic error.

Figure 2.5.Staircase method

The table in Figure 2.5 shows us how to perform these calculations in the case where the results obtained are related to the failures.

Estimation of m and µ is obtained using the following equation:

[2.13]

where:

– S0: index 0 stage, as defined above (the lowest stage used allowing us to obtain at least an accepted type of result);

– d: stage spacing: