Gigacycle Fatigue in Mechanical Practice Claude Bathias-Paul Paris

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Gigacycle Fatigue in

Mechanical PracticE

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Gigacycle Fatigue in

Mechanical PracticE

Claude bathias

paul C. Paris

Professor of Mechanics Institute for Technology and Advanced Materials (ITMA) Conservatoire National des Arts et Métiers Paris, France

Senior Professor of Mechanics Department of Mechanical and Aeronautical Engineering Washington University in St. Louis St. Louis, Missouri, U.S.A.

MARCEL DEKKER

NEW YORK

Cover:

Upper photo: Modern TGV high-speed train. Courtesy of Israel Marines (CNAM/ITMA, Paris, France). Lower photo: Stephenson locomotive ca. 1833, © Musée des arts et métiers/S. Pelly, Paris, France.

Although great care has been taken to provide accurate and current information, neither the author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The material contained herein is not intended to provide specific advice or recommendations for any specific situation.

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-2313-9 Marcel Dekker, 270 Madison Avenue, New York, NY 10016, USA http://www.dekker.com Distribution center: Marcel Dekker, Cimarron Road, Monticello, NY 12701 USA Copyright © 2005 by Marcel Dekker. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10

9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES OF AMERICA Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Dedication

We dedicate this book to the patient encouragement of our wives—Marie-Claude Bathias and Barbara L. Paris. We also include in our dedication our children Anne Potter, Claire Besset, Gail Paris, and Dr. Anthony J. Paris, who have also greatly inspired our effort.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Contents

Preface Acknowledgments Table of Notation 1

Introduction

2

Ultrasonic Fatigue Concepts 2.1 Introduction 2.2 Longitudinal elastic waves and resonance frequency 2.3 Analytical solution for the variable section specimen 2.4 Stress magnification factor 2.5 Analytical solution of resonance length 2.6 Methods for calculating crack tip stress intensity factor

3

Testing Machines and Their Performance 3.1 Introduction 3.2 Basic structure 3.3 Nonsymmetrical and variable amplitude test equipment 3.4 Computer control system 3.5 High temperature test equipment 3.6 Low temperature test equipment

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

iv

Contents

3.7 3.8 3.9

Thin sheet test equipment High pressure piezo-electric fatigue machine Non-axial test equipment

4

S-N 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Curve and Fatigue Strength Introduction Ferrous materials Aluminium matrix composite Non-ferrous alloys Alloys at cryogenic temperature N18 alloy at high temperature Rotating-bending internal crack stress correction Ti-Al intermetallic alloys

5

Crack Growth and Threshold 5.1 Titanium alloys 5.2 Nickel-based alloys 5.3 Aluminium alloys 5.4 Materials of b.c.c. and f.c.c crystalline structure 5.5 Low carbon steel sheet 5.6 Austenitic stainless steel 5.7 Spheroidal graphite cast iron (SGI) 5.8 Database of threshold SIF DKth 5.9 Other applications: Fretting fatigue

6

Frequency and Environmental Effects 6.1 Frequency effect 6.2 Heat effect 6.3 Cryogenic temperature 6.4 Environmental effects 6.5 S-N curve at room temperature and high pressure hydrogen for Ti-6A4V

7

Microstructural Aspects and Damage to Materials in the Gigacycle Regime 7.1 Gigacycle S-N curve shape 7.2 Mechanical aspects of initiation between 106 and 109 cycles 7.3 Initiation zone for low cycle to gigacycle failures 7.4 Initiation mechanisms at 109 cycles 7.5 Role of inclusions

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Contents

7.6 7.7

v

Gigacycle fatigue of alloys without inclusions General discussion of the gigacycle fatigue mechanisms

Appendix A1.1 A1.2 A1.3 A1.4

1 Stress Calibration Amplifying horn First calibration Second calibration Third calibration

Appendix 2 Remarks on the Statistical Prediction A2.1 Remarks on the statistical analysis in the megacycle regime References

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Preface

The photos of locomotives on the cover of this book illustrate the time period of interest in significant studies of metal fatigue. The older Stephenson Locomotive 020 of 1833, displayed at the CNAM museum in Paris, is an example from the era of first recognition of fatigue. The French TGV high speed trains of the late 20th century evolved this interest into the ‘‘gigacycle regime’’. Therefore, over 150 years ago, A. Wohler began his studies of metal fatigue for application to rail car axles. Others, such as Bauschinger, also examined fatigue phenomena later in the 19th century but were limited by the test equipment and instrumentation available. At about the beginning of the 20th century, it was found that initiation of fatigue from a smooth surface was preceded by plastic slip and later by reversals of this slip at the surface to form an intrusion leading to a crack growing failure. It was thereafter frequently concluded that below a certain stress level—the so called ‘‘endurance limit’’—this reversing slip and=or crack initiation would not occur and fatigue failure could be avoided. This concept assumed that fatigue crack initiation from imperfec-

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viii

Preface

tions in the material or due to manufacturing could be avoided and was accepted well beyond the middle of the 20th century. However, improvements in test equipment and methods, as well the motivation for improved metal structures such as aircraft, commercial power generators, high speed trains, etc., led to more intensive analyses of fatigue. In the 1960s and early 1970s the ‘‘damage tolerance’’ approach to fatigue was developed, which assumed crack-like flaws initially in a structure and calculated a safe crack growth life. Some structural situations required showing that present cracks would not grow at all or that they would be below the ‘‘crack growth threshold’’ in size and imposed stress. Both of these methods employed so called ‘‘fracture mechanics’’ methods in their approach. For components sustaining extremely high numbers of cycles of loads, manufacture without significant flaws and holding the stress levels low enough to avoid initiation remains the dominant method of approach. This motivates studies of ‘‘gigacycle fatigue’’. These requirements have also motivated this book and its presentation of results of fatigue under conditions up to 1010 cycles of loading. The development of piezo-electrically loaded fatigue machines capable of testing at the ultrasonic frequency of 20 kHz or more in the 1980s made it practical to test to such high number of cycles of load for fatigue initiation as well as for very slowly growing cracks to establish thresholds efficiently. Consequently, this is a book that is an exposition of the new concepts and data established by these new testing techniques of the last 20 years. This book not only presents results but also discusses in detail the design of these machines and the methods of using them to explore high cycle fatigue phenomena. Environmental testing and results including vacuum and temperature effects are presented and discussed. Load ratio effects and variable amplitude loading are also included. Paul C. Paris and Claude Bathias August 2004

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Acknowledgments

The authors wish to acknowledge the special efforts of Dr. Hiroshi Tada in editorial work and checking the mathematical accuracy of this book. We also wish to express special thanks to Delphine Martin, Fabrice Montembault and Emin Bayraktar for their patient assistance in preparing several drafts of the manuscript as it evolved. The research efforts of graduate students at CNAM who prepared doctoral dissertations on gigacycle fatigue providing data and experience reflected in the book include K. Saanouni from Tunisia (1981): X. Kong (1986), J. Ni (1992), T. Wu (1994), H. Tao (1996), Q. Wang (1998), Z. Sun (2000), and H. Xue (2004) from China: G. Thanigaiyarasu from Pakistan (1987): K. El Alami from Maroco (1995), G. Jago (1996), and J. Bonis from France (1997); and I. Marines from Mexico (2004) are due much thanks. We are also thankful for the suggestions of numerous colleagues. We wish to also acknowledge the assistance of our longterm friend, John Corrigan, for his help in publishing this book. Our special thanks to Joanne Jay of Dekker for extraordinary editorial effort in expediting publication.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table of Notation

a b a and b da Da dN or DN b,h c e, e ef, ef e_ ðx; tÞ o f ¼ 2p k ¼ oc l m pðxÞ r s t v u,v,w x,y,z Ai, Bi

crack size or radius Burger’s vector constants in S-N curve formulae crack growth rate rectangular cross section dimensions wave velocity engineering or true strain strain at fracture strain rate at a specific location and time frequency (cycles per second) wave vector wave length exponent in Paris crack growth law or meters S0 ðxÞ SðxÞ

parameter in specimen shape or distance from crack tip standard deviation or seconds time charge or displacement of crack surface rectangular components of displacements Cartesian coordinates constants in displacement expressions

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xii

A0 A=D, D=A Bi C1 ; C2 ; C3 ; C4 Cþþ 0 C; K E Ed F Famp ¼ VV12 G Hv J2 Hz K1 DK, DKth, DKeff L1 ; L2 ; L N; Nf ; Ni ; Np Pa R RA R1, R2 S(x) S-N U(x) UTS V ½K
2 ; ½M
c ; ½K
g a; b a; b; C l; Dli r o x; Z s sx ; sy ; txy sa syp sw ; sd t EðxÞ; SðxÞ

Notation

displacement at the end of a bar analog to digital or vice versa thickness of specimen Elastic constants computer software temperature Celsius or Kelvin modulus of elasticity dynamic modulus of elasticity force amplification factor of a horn elastic energy release rate at a crack tip Vickers hardness a twelve prong connector plug Hertz frequency crack tip stress intensity factor stress intensity range (threshold or effective) length (resonance, exponential, or specimen) number of cycles (to failure, to initiation, or in crack propagation) Pascal cyclic load ratio reduction in area radius of the specimen at the center and end cross sectional area at location x fatigue stress vs. number of cycles curves displacement at location x ultimate tensile strength voltage input Matrix (elementary rigidity, elementary mass, or geometrical) microstructure in titanium alloys or parameters in vibration equation solutions constants in Murakami’s equation eigenvalues mass density frequency non-dimensional coordinates applied normal stress rectangular components of plane stress alternating stress yield point stress fatigue failure stress shear stress strain or stress on the reduced section of a specimen

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

1 Introduction

Initially, it is of interest to note that many structural components sustain far beyond 107 cycles of loading, but materials characterization and fatigue predictions are normally based upon data limited to between 106 and 107 cycles. This is because standard fatigue testing equipment prior to the past decades was limited in speed to less than 200 cycles per second. Therefore, testing beyond 107 cycles was very time consuming. However, the fatigue life of current automobile engines ranges around 108 cycles; big diesel engines for ships or high speed trains have ranges to 109 cycles. It is further noted that at this time interest in fatigue life extends to about 1010 cycles, for example, in turbine engine components (Figure 1.1). From a historical perspective, it was established for the first time in the mid-1980s by several Japanese researchers (Ebara, 1987; Kikukawa, 1965; Murakami, 1994) that structural metal alloys can fail after 107 cycles. More recently, the phenomena of gigacycle fatigue failures in many alloys up to 1010 cycles has been extensively established by

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

2

Figure 1.1

Chapter 1

Fatigue life of machines and components.

C. Bathias and co-workers (Kong, 1987; Ni, 1991; Thanigaiyarasu, 1988; Wu, 1991). The S-N (stress=cycles) curve is often still assumed to be a rectangular hyperbolic relationship, but in reality there is not a horizontal asymptote. This means that fatigue initiation mechanisms from 106 to beyond 109 cycles are a topic of great interest for advanced structural technologies. Consequently the S-N curve, since it is not asymptotic, must be determined in order to guarantee the real fatigue strength in the very high cycle regime. The preceding view was based on assuming that fatigue initiation mechanisms leading to growing cracks must be avoided. However, if an initiated crack or pre-existing crack-like flaw grows at a very small rate, which will allow a sufficient life, then failure may also be avoided. For this reason the very slow growth of cracks in the threshold regime is also of interest herein. In the 1960s Paris and co-workers (Lindner, 1965) observed threshold region crack growth rates as low as 0.6  10 11 meters per cycle. At such rates it would usually take well over 108 cycles to grow to failure. Consequently, the subject of threshold level crack growth rates is discussed and data developed by high speed equipment are presented in Chapter 5. Both initiation and growth of fatigue cracks are important to develop a full understanding of very high cycle fatigue. If stresses are low enough to prevent initiation from usual material initiation mechanisms, then assuring below-threshold Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Introduction

3

Figure 1.2 dization.

Typical S-N curve as defined by international standar-

conditions for any possible defects will guarantee sufficient life. Design and production conditions will dictate that considering one or the other alone will be sufficient to avoid failure in practice. Emphasis will first be placed on S-N testing curve techniques and results. (Thresholds and crack growth behavior will be mainly deferred to Chapter 5.) When the fatigue curve or S-N curve is defined, it is usually done in reference to carbon steels. The S-N curve data are generally limited to 107 cycles and it is presumed, according to the standard, that a horizontal asymptote allows determination of a fatigue limit value for an alternating stress between 106 and 107 cycles. Beyond 107 cycles (Figure 1.2), it is normally considered that the fatigue life is infinite. However for other metal alloys, it is assumed that the asymptote of the S-N curve is not horizontal. For fatigue limits to 109 cycles a few results can be observed in the references (Bathias, 1993, 1998, 2004; Stanzl, 1996). Until recently, the shape of the S-N curve beyond 107 cycles was predicted by using probabilistic methods, which Copyright © 2005 by Marcel Dekker. All Rights Reserved.

4

Figure 1.3

Chapter 1

Isoprobability of failure.

is also true for the fatigue limit. In principle, the fatigue limit is given for a specific number of cycles to failure. Using, for example, the staircase method, the fatigue limit is given by the average alternating stress sD and the probability of fracture is given by the standard deviation (s) of the scatter. A classical way to determine the infinite fatigue life is to use a Gaussian function. Roughly speaking, it is said that the mean endurance limit stress sD, minus 3s gives a probability of fracture close to zero (Figure 1.3). Assuming s is equal to 10 MPa, the true infinite fatigue limit should be sD 30 MPa. However, experiments data herein will show that for many alloys between sD for 106 and sD for 109 the difference is greater than 30 MPa. This so-called SD approach to the average fatigue limit is certainly not the best way to reduce the risk of rupture in fatigue (Figure 1.4) and meant as a last resort. Only direct experience can remove this ambiguity by providing some accelerated tests of fatigue. From a basic point of view, it seems that it is better to determine the real fatigue strength and not an estimated fatigue limit for a given number of cycles, especially in the gigacycle regime (Figure 1.5). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Introduction

Figure 1.4

5

Safe fatigue curve.

Today, this is possible since piezoelectric fatigue machines are very reliable and capable of producing 1010 cycles in less than 1 week (at 20 kHz), whereas the conventional systems require more than 3 years of testing for only one sample (at <200 Hz).

Figure 1.5

The concept of gigacycle S-N curve.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

6

To reemphasize the present situation, it is evident that the historical concept of a fatigue limit is bound to the hypothesis of the existence of a horizontal asymptote on the S-N curve between 106 to 107 cycles (for example, see Figure 1.1). Therefore, if a sample reaches 107 cycles and is not broken, it is considered to have an infinite life. That is a convenient and economical assumption, but it is not a rigorous approach. It is important to understand that the staircase method is popular today to determine an assumed fatigue limit only because of the convenience of this approximation. A fatigue limit determined by this method to 107 cycles requires 30 hours of test with a machine working at 100 Hz for only one sample. To reach 109 cycles, 3000 hours of testing would be necessary, which is very time consuming and expensive. It is of great importance to understand and predict a fatigue life in terms of crack initiation and small crack propagation. It has been generally accepted that at high stress levels, fatigue life is determined primarily by crack growth, while at low stress levels, the life span is mainly consumed by the process of crack initiation. Several authors have demonstrated that the portion of life attributed to crack nucleation is above 90% in the high cycle regime (106 to 107 cycles) for steel, aluminium, titanium, and nickel alloys. In cases where the crack nucleates from a defect, such as an inclusion or pore, it is said that a relation should exist between the fatigue limit and the crack growth threshold. However, the relation between crack growth and initiation is not obvious for many reasons. First, it is not certain that a sharp defect implies immediate fatigue crack growth from the very first cycle. Second, when a defect is small, a short crack does not grow at the same rate as a long crack. In particular the effects of load ratio, (R), and closure depend on the crack length. Thus, the relationship between fatigue crack growth threshold (DKth) and sD remains to be developed. The relationship between sD and DKth must be established in the gigacycle regime if a relationship does in fact exist. Indeed, the experiments show that there are several crack initiation mechanisms dependent upon the alloys and particuCopyright © 2005 by Marcel Dekker. All Rights Reserved.

lar types of defects. Consequently, it seems that there is no general relation between DKth and sD even at 109 cycles. But in the case when initiation depends on inclusions, a Murakami type model (see Section 4.2 and Section 7.52) appears to be sufficient. For all the reasons stated above, it is necessary to apply an accelerated fatigue testing method by ultrasonic fatigue test techniques to investigate the behavior of the S-N curve into the gigacycle regime. In the earlier chapters of this book, ultrasonic testing methods will be presented.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

2 Ultrasonic Fatigue Concepts

2.1. INTRODUCTION The ultrasonic fatigue test method differs from the conventional fatigue test method that has frequency limited to 100 Hz of cyclic stressing of material. The frequency of ultrasonic fatigue testing ranges from 15 kHz to 30 kHz, with a typical frequency being 20 kHz. With this high frequency, the time and cost to obtain a fatigue limit (if one does exist) or crack growth rate threshold data can be dramatically reduced. For instance, the test time for 107 cycles is within 9 minutes by ultrasonic method, while conventional fatigue testing at 100 Hz will take about 12 days. For even higher cycles—for instance, 109—the ultrasonic method requires only 14 hours, whereas it would take more than 3 years at 100 Hz for a single specimen. The ultrasonic method also provides a reliable way of testing at the extremely small rates of crack growth in the threshold regime, for example in the range of 10
9 mm=s to 10
11 mm=s. The application of ultrasonic fatigue testing started near the beginning of the 20th century (Hopkinson, 1911). Up until

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

then, the highest frequency of fatigue testing with a mechanically driven system did not exceed 33 Hz. That year Hopkinson developed a first electromagnetic resonance system of 116 Hz. (Jenkin, 1925) used similar techniques to test copper, iron, and steel wires at the frequency of 2.5 kHz. In 1929, the test machine of Jenkin and Lehmann (Jenkin, 1929) reached the frequency of 10 kHz with a pulsating air resonance system. In (Mason, 1950) the test machine marked an important point in the development of ultrasonic fatigue testing techniques. He introduced the piezo-electric and magnetostrictive types of transducers capable of translating 20 kHz electrical voltage signals into displacement controlled 20 kHz mechanical vibration, and used high power 20 kHz ultrasonic waves to induce fracture of materials in fatigue. Shortly afterward, even higher frequencies of fatigue testing were reached, for example 92 kHz (Girard, 1959) and 199 kHz (Kikukawa, 1965). However, the design of Mason’s 20 kHz machine has been used as the basis for most modern ultrasonic fatigue testing machines. In (Neppiras, 1959) a proposition to apply ultrasound to the determination of S-N curves initiated a series of research work, most of which aimed at developing methods of measuring fatigue life and fatigue limits under constant amplitude loading conditions at R ¼
1. In (Mitsche, 1973) was the first to use ultrasound for the purpose of fatigue crack propagation testing. They gave the first ultrasonic data on the curves of Da=DN versus DK, using standard procedures for low frequency loading (conventional) as a first approximation to calculate DK. More accurate methods of calculating DK in high frequency appeared rather slowly. Important work, such as that from (Saanouni, 1982), (Shoeck, 1982), (Kong, 1991), (Mayer, 1993), (Wu, 1994), and (Ni, 1996), was published between 1982 and 1996. Fatigue properties of materials under variable amplitude loading are also of great interest in practice. This is another domain to which the ultrasonic fatigue concept can contribute. Conventional tests often involve the modification of the load time sequence; however, ultrasonic tests can follow exactly the load sequences desired. Bathias with (Wu, 1994) Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Ultrasonic Fatigue Concepts

11

and (Ni, 1994) developed computer control systems for ultrasonic fatigue testing. For this purpose PC 486 (or higher) computers were used, since the speed of these computers is adequate to drive a piezo-electrical fatigue machine. Industrial applications of the ultrasonic fatigue data (constant amplitude and random amplitude) now have been extended to aircraft, automobile, railway, offshore, and other structures. Among them, some in-service loading conditions of aircraft do fall into the ultrasonic frequency region. In these domains, the importance of ultrasonic fatigue technologies is, of course, more direct. There are several articles reviewing the development of ultrasonic fatigue techniques; we refer the reader to (Bathias, 2002); (Mayer, 1999); and (Stanzl, 1996). 2.2. LONGITUDINAL ELASTIC WAVES AND RESONANCE FREQUENCY In conventional fatigue tests, the frequency is that of the external load system of the test machine, which is different from the natural frequencies of the specimen. In other words, the specimen is in forced vibration. An ultrasonic fatigue test differs from this in that the external frequency supplied by the test machine must be one of the natural frequencies of the specimen. This is the definition of free vibration. To better understand this phenomenon, it is worth briefly recalling elastic wave theory. The differential equations for a general three-dimensional isotropic elastic body in a Cartesian co-ordinate system are
 @2u E 1 @e 2 ð2:1aÞ þr u r 2 ¼ @t ð1 þ nÞ 1
2n @x
 @2v E 1 @e 2 r 2 ¼ þr v @t ð1 þ nÞ 1
2n @y r


 @2w E 1 @e 2 þ ¼ r w @t2 ð1 þ nÞ 1
2n @z

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:1bÞ

ð2:1cÞ

12

Chapter 2

where u, v, and w are displacements along x, y, and z respectively, E and n the Young’s modulus and Poisson’s ratio, r the mass density, H2 the Laplacian, and e¼

@u @v @w þ þ @x @y @z

ð2:2Þ

is the volume dilatation. Elastic wave theory indicates that the following two types of wave may exist in an infinite isotropic elastic body. 2.2.1. Longitudinal Wave For a longitudinal wave, the curl of the displacement field is zero. The velocity of wave propagation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eð1
nÞ c¼ ð1 þ nÞð1
2nÞr

ð2:3Þ

2.2.2. Transverse Wave For a transverse wave, the volume dilatation e vanishes. The velocity of wave propagation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E c¼ 2ð1 þ nÞr

ð2:4Þ

If there is a boundary, a surface wave may also be produced. This wave is similar to the gravity surface wave in a fluid. The amplitude of vibration decreases rapidly with the distance from the surface, and the wave velocity is less than the velocity inside the body. To simplify the discussion, let us start with a onedimensional specimen of straight cylinder. As shown in Figure 2.1, a longitudinal elastic wave comes from one end of the bar and travels through the length l, then it is reflected from the other end and returns to the initial place of entrance. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Ultrasonic Fatigue Concepts

Figure 2.1

13

Displacement and strain variation along an elastic bar.

The wave velocity c will be determined directly from Eq. 2.3 with n ¼ 0 for consideration of this one-dimensional example. sffiffiffiffi E c¼ ð2:5Þ r Also the differential Eqs. 2.1a to 2.1c reduce to a single equation @2u E @2u ¼ @t2 r @x2

ð2:6Þ

The solution of Eq. 2.6 is given by u¼

1 X

un ðx; tÞ

ð2:7Þ

n¼1

where un ðx; tÞ ¼


An
1 cos

 npct npct npx þ Bn
1 sin cos l l l

ð2:8Þ

The boundary conditions of ultrasonic fatigue testing require the displacement to be maximum at both ends whereas the strain vanishes at the same places. That is
 @u ¼0 ð2:9Þ @x x¼0;l Copyright © 2005 by Marcel Dekker. All Rights Reserved.

14

Chapter 2

Thus Eq. 2.8 for the first mode of vibration becomes uðx; tÞ ¼ A0 cosðkxÞ sinðotÞ

ð2:10Þ

where p k¼ ; l

o¼

pc l

ð2:11Þ

The amplitude of vibration at each point along the bar is UðxÞ ¼ A0 cosðkxÞ

ð2:12Þ

where A0 is the displacement amplitude at the end of the bar. The strain e of each point is given by eðx; tÞ ¼
kA0 sinðkxÞ sinðotÞ

ð2:13Þ

with its maximum eðxÞ ¼
kA0 sinðkxÞ

ð2:14Þ

The strain rate is e_ ðx; tÞ ¼
koA0 sinðkxÞ cosðotÞ

ð2:15Þ

with its maximum e_ ðxÞ ¼
koA0 sinðkxÞ From Eqs. 2.5 and 2.11 sffiffiffiffiffiffi 1 Ed l¼ 2f r

ð2:16Þ

ð2:17Þ

for the first mode of vibration, where f ¼ o=2p is the frequency and Ed ¼ the dynamic elastic modulus for consideration of dynamic effects. In conclusion, the length of resonance of the onedimensional specimen is given by Eq. 2.17 for the first vibration mode. The displacement node (where displacement vanishes) at the center of the specimen corresponds to the maxima of the strain, stress and strain rate. At both ends, we have the maximum of displacement and the nodes of Copyright © 2005 by Marcel Dekker. All Rights Reserved.

strain, stress and strain rate. That is u ¼ 0; for x ¼

s ¼
Ed kA0 ;

e ¼
kA0 ;

e_ ¼
koA0 ð2:18aÞ

l 2

U ¼ A0 ;

e ¼ 0;

s ¼ 0;

e_ ¼ 0;

for x ¼ 0; l

ð2:18bÞ

Eq. 2.17 indicates an important fact: The resonance length is inversely proportional to the frequency. This explains why some very high frequencies—for example, 92 kHz and 199 kHz mentioned in the previous section—are not practicable. For example, for a typical steel specimen of uniform section with E ¼ 200 000 MPa, r ¼ 7800 kg=m3, we have (from Eq. 2.17) l

2:5  106 ðmmÞ f

and f (kHz) 20 92 199

l (mm) 127 27.5 13.2

The last two values of l obviously lead to difficulties in machining, displacement, or strain measurements, as well as energy dissipation. 2.3. ANALYTICAL SOLUTION FOR THE VARIABLE SECTION SPECIMEN For a fatigue specimen of variable section, amplitudes of strain and stress vary at each section. Normally, the resonance amplitude that is a function of the specimen geometry is determined numerically. In order to obtain stress concentration in the middle of the specimen to expedite a fatigue test, the specimen for ultrasonic fatigue initiation or fatigue crack growth is, in Copyright © 2005 by Marcel Dekker. All Rights Reserved.

16

Chapter 2

most cases, designed with a reduced section in the center as shown in Figure 2.2. We call the length L1 the resonance length, the determination of which involves a numerical approach, such as the finite element method (FEM). But if the center part is in an exponential form, an analytical solution of the resonance length can be obtained (Kong, 1987). The longitudinal wave equation for a specimen with a varying cross section can be written as rSðxÞ

@ 2 u @f ¼ @t2 @x

ð2:19Þ

where S(x) is the area of cross section at position x, and f ¼ Ed SðxÞ

@u @x

ð2:20Þ

is the force acting on the section. Thus we have   @ 2 uðx; tÞ @uðx; tÞ @ 2 uðx; tÞ 2 ¼ c pðxÞ ¼0 ð2:21Þ þ @t2 rffiffiffiffiffiffi @x @x2 S0 ðxÞ where c ¼ Erd , as before and pðxÞ ¼ SðxÞ Under boundary conditions that ultrasonic fatigue specimens must satisfy, the solution to Eq. 2.21 takes the form u(x,t) ¼ U(x) sin(ot), and the equation for the amplitude of vibration U(x) (see Eq. 2.12) at each point along the specimen can be easily obtained U 00 ðxÞ þ pðxÞU 0 ðxÞ þ k2 UðxÞ ¼ 0

ð2:22Þ

Figure 2.2 Ultrasonic specimen geometry: A. Endurance specimen. B. Crack growth specimen. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

where k¼

o c

ð2:23Þ

To obtain the solution of Eq. 2.22, we must define the curve in the central part of the specimens given in Figure 2.2. For the axisymmetric specimen in Figure 2.2A, if the curve is of a profile of hyperbolic cosine, i.e., L2 < jxj  L

yðxÞ ¼ R2 ;

yðxÞ ¼ R1 coshðaxÞ;

ð2:24aÞ

jxj  L2

ð2:24bÞ


 1 R2 a¼ arccosh L2 R1

ð2:25Þ

where L ¼ L1 þ L2 ;

Then from the boundary conditions of Eq. 2.22, we can find the specimen’s resonance length.   1 1 ð2:26Þ L1 ¼ arctan ½b cothðbL2 Þ
a tanhðaL2 Þ k k where b¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2
k2

ð2:27Þ

The solution of Eq. 2.22 is UðxÞ ¼ A0 jðL1 ; L2 Þ

sinhðbxÞ ; coshðaxÞ

UðxÞ ¼ A0 cosðkðL
xÞÞ;

jxj  L2

ð2:28aÞ

L2 < jxj  L

ð2:28bÞ

where jðL1 ; L2 Þ ¼

cosðkL1 Þ coshðaL2 Þ sinhðbL2 Þ

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ð2:29Þ

With this solution, it is easy to obtain the strain and stress for the reduced section part and for the cylindrical part. The results are as follows. For the reduced section part (jxj  L2): eðxÞ ¼ A0 jðL1 ; L2 Þ

½b coshðbxÞ coshðaxÞ
a sinhðbxÞ sinhðaxÞ cosh2 ðaxÞ ð2:30aÞ

sðxÞ¼ Ed A0 jðL1 ;L2 Þ

½bcoshðbxÞcoshðaxÞ
asinhðbxÞsinhðaxÞ cosh2 ðaxÞ ð2:30bÞ

For the cylindrical part (L2 < jxj  L): eðxÞ ¼ kA0 sinðkðL
xÞÞ

ð2:31aÞ

sðxÞ ¼ Ed kA0 sinðkðL
xÞÞ

ð2:31bÞ

The difference between the revolution surfaces with the hyperbolic cosine profile and a circular profile is very small for the axisymmetric ultrasonic fatigue specimen. Therefore, we can use the analytical solution given above to avoid the numerical calculation of the resonance length and the stress field of the specimen. For the plane stress specimen of Figure 2.2B, if the central part has an exponential profile, i.e., yðxÞ ¼ R2 ;

L2 < jxj  L

yðxÞ ¼ R1 expð2a1 xÞ; with a1 ¼

jxj  L2


 1 R2 ln 2L2 R1

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ð2:32aÞ ð2:32bÞ

ð2:33Þ

then we can find the specimen’s resonance length as follows.   1 1 L1 ¼ arctan ½b1 cothðb1 L2 Þ
a1  ð2:34Þ k k where b1 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21
k2

ð2:35Þ

The solution of Eq. 2.22 is UðxÞ ¼ A0 j1 ðL1 ; L2 Þ sinhðb1 xÞ expð
a1 xÞ;

jxj  L2 ð2:36aÞ

UðxÞ ¼ A0 cosðkðL
xÞÞ;

L2 < jxj  L

ð2:36bÞ

where j1 ðL1 ; L2 Þ ¼

cosðkL1 Þ expða1 L2 Þ sinhðb1 L2 Þ

ð2:37Þ

The strain and stress are as follows. For the reduced section part (jxj  L2): eðxÞ ¼ A0 j1 ðL1 ; L2 Þ½b1 coshðb1 xÞ
a1 sinhðb1 xÞ expð
a1 xÞ ð2:38aÞ sðxÞ¼Ed A0 j1 ðL1 ;L2 Þ½b1 coshðb1 xÞ
a1 sinhðb1 xÞexpð
a1 xÞ ð2:38bÞ The strain and stress in the rectangular part (L2 <jxj  L) are again given by Eq. 2.31, and the detailed solution procedure is given in Section 2.5. Because of the surface finishing difficulties of specimens with a hyperbolic cosine (catenary) profile in the center, we examine the difference between this geometry and a circular profile specimen that is commonly and economically used in practice. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In Figure 2.3, we draw two curves, a circle x2 þ ðb
yÞ2 ¼ r2

ð2:39Þ

and a catenary y ¼ ðb
rÞ coshðaxÞ

ð2:40Þ

with b ¼ 11.5, r ¼ 10 and a ¼ 0.249843 (in mm). The values of these two equations and the difference in y are summed up in Table 2.1 (Wu, 1992), which shows that the largest difference between the two curves is approximately 1.8% and situated in 4 mm from the specimen center. We consider therefore that, for a length of 2 mm from the center (a total of 4 mm), the catenoidal profile agrees well with that of the circle. In other words, the analytical solution of the specimen geometry described in this section can be used as a good approximation for the actual specimen without performing costly numerical simulations. Besides catenoidal approximation, other forms of reduced section profile may be used; for example, the forms of simple dumbbell, hollow dumbbell, exponential dumbbell, half dumbbell, and half straight as summarized in (Wu, 1994). Analytical solutions of resonance length exist for some of them. For example, (Koslov, 1988) gives an analytical solution for exponential profile.

Figure 2.3

Catenoidal approximation of a circle.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 2.1 Difference Between a Circle and Its Catenoidal Approximation x

y circle

y catenary

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

1.49999 1.51277 1.55122 1.61563 1.70655 1.82472 1.97118 2.14731 2.35484 2.59604 2.87379 3.19186 3.55523 3.97063 4.44756 4.99999

1.5 1.51172 1.54706 1.60658 1.69119 1.80224 1.94144 2.11099 2.31351 2.55219 2.83075 3.15354 3.52561 3.95277 4.44169 5.00001

Dy

Dy (%)


1.43051e-5 1.04845e-3 4.15492e-3 9.05788e-3 0.0153511 0.0224781 0.0297375 0.0363224 0.0413277 0.0438468 0.0430427 0.0383225 0.0296195 0.0178688 5.87225e-3
2.3365e-5


9.53674e-4 0.0693545 0.268569 0.563801 0.907705 1.24723 1.53172 1.72063 1.78636 1.71801 1.52054 1.21522 0.840123 0.452057 0.132208
4.67299e-4

2.4. STRESS MAGNIFICATION FACTOR The stress magnification factor is defined as the ratio of the maximum stress in the reduced section specimen to that in the constant section specimen having the same length, boundary, and excitation conditions. For the specimen with surface of revolution of a hyperbolic cosine profile in the center, the magnification factor of stress, from Eqs. 2.30b and 2.14, is F¼

b cosðkL1 Þ coshðaL2 Þ k sinhðbL2 Þ

ð2:41Þ

For a plane stress specimen, this factor is F¼

b1 cosðkL1 Þ expða1 L2 Þ k sinhðb1 L2 Þ

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:42Þ

2.5. ANALYTICAL SOLUTION OF RESONANCE LENGTH In the following section, we give the complete procedures for obtaining the solution of a longitudinal wave Eq. 2.22 and the resonance length L1 for both endurance and crack growth specimens. 2.5.1. Endurance Specimen The endurance specimen is axi-symmetric (Figure 2.2A). From Eq. 2.24 that defines the profile of a longitudinal section, we have SðxÞ ¼ pR22 ;

L2 < jxj  L

SðxÞ ¼ pR21 cosh2 ðaxÞ;

jxj  L2

ð2:43Þ ð2:44Þ

For the cylindrical part, the section area S(x) keeps constant, so that p(x) in Eq. 2.22 vanishes pðxÞ ¼

S0 ðxÞ ¼0 SðxÞ

ð2:45Þ

and Eq. 2.22 becomes U 00 ðxÞ þ k2 UðxÞ ¼ 0

ð2:46Þ

This ordinary differential equation of second order has the solution in a general form UðxÞ ¼ C1 cosðkxÞ þ C2 sinðkxÞ

ð2:47Þ

For the reduced section part that has a hyperbolic cosine profile, we have from Eq. 2.44 pðxÞ ¼

S0 ðxÞ ¼ 2a tanhðaxÞ SðxÞ

ð2:48Þ

Combining Eq. 2.48 with Eq. 2.22 results in U 00 ðxÞ þ 2a tanhðaxÞU 0 ðxÞ þ k2 UðxÞ ¼ 0 Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:49Þ

Introducing a function wðxÞ ¼ coshðaxÞUðxÞ

ð2:50Þ

leads to w0 ðxÞ ¼ U 0 ðxÞ coshðaxÞ þ a sinhðaxÞUðxÞ

ð2:51Þ

w00 ðxÞ ¼ ½U 00 ðxÞ þ 2a tanhðaxÞU 0 ðxÞ þ a2 UðxÞ coshðaxÞ ð2:52Þ Comparing Eq. 2.49 to Eq. 2.52, we get w00 ðxÞ ¼ ða2
k2 Þ coshðaxÞUðxÞ or w00 ðxÞ
ða2
k2 ÞwðxÞ ¼ 0

ð2:53Þ

This equation has the general solution wðxÞ ¼ C3 expðbxÞ þ C4 expð
bxÞ

ð2:54Þ

where b is defined in Eq. 2.27. The definition (Eq. 2.50) gives the general solution of Eq. 2.49 UðxÞ ¼

C3 expðbxÞ þ C4 expð
bxÞ coshðaxÞ

ð2:55Þ

This general solution was given in Wu (1992). Now we have the solutions for the cylindrical part (Eq. 2.47) and for the reduced section part (Eq. 2.55) with four constants C1, C2, C3 and C4. These constants can be determined by appropriate boundary and continuity conditions. The boundary conditions (Eqs. 2.9 and 2.18b) in the end of specimen (note that the difference between the co-ordinate systems of Figures 2.1 and 2.2) require, from the solution to Eq. 2.47, that C1 cosðkLÞ þ C2 sinðkLÞ ¼ A0

ð2:56Þ

C2 ¼ C1 tanðkLÞ

ð2:57Þ

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

From these two relations, we have C1 ½cosðkLÞ þ sinðkLÞ tanðkLÞ ¼ A0

ð2:58Þ

so that C1 ¼ A0 cosðkLÞ;

C2 ¼ A0 sinðkLÞ

ð2:59Þ

Consequently, the solution of displacement amplitude for the cylindrical part is UðxÞ ¼ A0 cos½kðL
xÞ

ð2:60Þ

This is Eq. 2.28b. The center of the specimen is a node of displacement [U(0) ¼ 0]; therefore, from the general solution for this part (Eq. 2.55), we obtain C3 þ C4 ¼ 0

ð2:61Þ

and Eq. 2.55 is now rewritten as UðxÞ ¼

2C3 sinhðbxÞ coshðaxÞ

ð2:62Þ

Now the continuity conditions at x ¼ L2 require that the displacement amplitude U(L2) and strain amplitude U0 (L2) calculated from solutions (Eqs. 2.60 and 2.62) must be equal. This gives A0 cos½kðL
L2 Þ ¼

2C3 sinhðbL2 Þ coshðaL2 Þ

ð2:63Þ

kA0 sin½kðL
L2 Þ 2C3 ½b coshðbL2 Þ coshðaL2 Þ
a sinhðbL2 Þ sinhðaL2 Þ ¼ cosh2 ðaL2 Þ ð2:64Þ Remember that L ¼ L1 þ L2 and, if we compare the two expressions of Eqs. 2.63 and 2.64, we have k tanðkL1 Þ ¼ b cothðbL2 Þ
a tanhðaL2 Þ Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:65Þ

or, again, Eq. 2.26 for the resonance length   1 1 L1 ¼ arctan ½b cothðbL2 Þ
a tanhðaL2 Þ k k Finally, Eqs. 2.61 and 2.63 give constants C3 and C4 C3 ¼
C4 ¼

A0 cosðkL1 Þ coshðaL2 Þ 2 sinhðbL2 Þ

ð2:66Þ

The solution (Eq. 2.62) now reads UðxÞ ¼ A0

cosðkL1 Þ coshðaL2 Þ sinhðbxÞ sinhðbL2 Þ coshðaxÞ

ð2:67Þ

which is Eq. 2.28a. 2.5.2. Crack Growth Specimen The solution steps for the crack growth specimen of Figure 2.2B are similar to those for the endurance specimen discussed above. The areas of cross sections are given by SðxÞ ¼ 2R2 L3 ;

L2 < j xj  L

SðxÞ ¼ 2R1 L3 expð2a1 xÞ;

j xj  L2

ð2:68Þ ð2:69Þ

and correspondingly pðxÞ ¼ 0; pðxÞ ¼ 2a1 ;

L2 < j xj  L

ð2:70Þ

j xj  L2

ð2:71Þ

For the constant section part (L2 < jxj  L), the differential equation is again Eq. 2.46. Moreover, because the boundary conditions at the end of the specimen are the same as those for the endurance specimen, so the solution of displacement amplitude is also expressed by Eq. 2.60. For the central part with exponential profile, the differential equation is U 00 ðxÞ þ 2a1 U 0 ðxÞ þ k2 UðxÞ ¼ 0 Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:72Þ

Now introduce a function w1 ðxÞ ¼ expða1 xÞUðxÞ

ð2:73Þ

Then w01 ðxÞ ¼ ½U 0 ðxÞ þ a1 UðxÞ expða1 xÞ

ð2:74Þ

w001 ðxÞ ¼ ½U 00 ðxÞ þ 2a1 U 0 ðxÞ þ a21 UðxÞ expða1 xÞ

ð2:75Þ

Comparing Eq. 2.75 to Eq. 2.72, we have the equation for w1(x)   ð2:76Þ w001 ð xÞ
a21
k2 w1 ð xÞ ¼ 0 with the general solution w1 ðxÞ ¼ C5 expðb1 xÞ þ C6 expð
b1 xÞ

ð2:77Þ

where b1 is defined by Eq. 2.35. Substituting Eq. 2.77 into Eq. 2.73 gives the general solution of Eq. 2.72 U ð xÞ ¼ ½C5 expðb1 xÞ þ C6 expð
b1 xÞ expð
a1 xÞ

ð2:78Þ

The condition that U(0) ¼ 0 results in C5 þ C6 ¼ 0

ð2:79Þ

From Eqs. 2.78 and 2.79, we have UðxÞ ¼ 2C5 sinhðb1 xÞ expð
a1 xÞ

ð2:80Þ

Now our task is to determine the constant C5 from the continuity conditions that U(x) and U0 (x) must satisfy at x ¼ L2. This gives, from Eqs. 2.60 and 2.80 A0 cosðkL1 Þ ¼ 2C5 sinhðb1 L2 Þ expð
a1 L2 Þ A0 k sinðkL1 Þ ¼ 2C5 ½b1 coshðb1 L2 Þ
a1 sinhðb1 L2 Þ expð
a1 L2 Þ Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:81Þ ð2:82Þ

From these two relations, we find the resonance length of the crack growth specimen and the constant C5   1 1 L1 ¼ arctg ½b1 cothðb1 L2 Þ
a1  k k

C5 ¼
C6 ¼

ð2:83Þ

A0 cosðkL1 Þ expða1 L2 Þ 2 sinhðb1 L2 Þ

ð2:84Þ

Substituting Eq. 2.84 into Eq. 2.80 gives the solution of Eq. 2.72 UðxÞ ¼ A0 j1 ðL1 ; L2 Þ sinhðb1 xÞ expð
a1 xÞ;

j xj  L2 ð2:85Þ

This is the solution given in Eq. 2.36a. 2.5.3. Transition Section In a vibration system, we must reduce the number of displacement nodes because it is easier to excite the system when the order of vibration mode is low. For a fatigue test, the specimen center must be a displacement node because of the mechanical symmetry of the specimen. The vibration at the two (or one) positions where static load is applied has to be avoided. Thus, there will be three displacement nodes in the test domain. Between the two neighboring nodes, there is a stress node at each end of the specimen (Figure 2.4). To conclude, there are generally four nodes of displacement (DN) and four nodes of stress (SN), as shown in Figure 2.5. Thus, the test domain is composed of a specimen in the middle and two transition sections in two end parts. The intrinsic frequency for all parts must be the same (20 kHz). The geometry of the transition section can be determined in the same way as for the specimen itself. These discussions are also valid when there is only one transition section, with the load ratio R ¼
1. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.4

Distributions of displacement and stress in test domain.

2.5.4. About the Effect of Transverse Motion In the one-dimensional formulas of elastic wave discussed so far, we have neglected the transverse motion of the media. This implies that we consider a thin bar which is much longer than it is wide. In other words, physically, we omit the effect of the Poisson ratio. Ni (1996) demonstrates that this treatment is reasonable and the difference in eigenvalues between a one-dimensional model and a fully three-dimensional model is small. For a cylinder of length l and radius r, Ni   (1996) finds that, if the terms equal to or higher than O2 rl are omitted in the development of Bessel functions of the solution, the eigenvalue problems for the two cases are identical, and that,  2 r if the term O l is considered, we have the following relation (Ni, 1996): r 2 fn 1 ¼ 1
n 2 p2 f 4 l

ð2:86Þ

where fn is the eigen frequency of the first mode of longitudinal vibration taking into account the transverse dimensions of the specimen, and f is that of a very thin cylinder with a zero Poisson ratio calculated by the formula in Eq. 2.17. From the formula in Eq. 2.86, data from Table 2.2, and a Poisson ratio 0.3, we find that for specimens of T6A4V and Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.5

Displacement and stress nodes.

U500, the fn =f values are, respectively, 0.998686 and 0.998547. These results justify the use of one-dimensional approximation in the determination of resonance length of an ultrasonic fatigue specimen. Tables 2.2 and 2.3 list dimensions of some specimens tested in the CNAM=ITMA laboratory (see Figure 2.2 for the meaning of symbols). The circular part of the specimen center is approximated by a catenoidal profile, and the resonance length L1 is determined analytically. The values of the stress magnification factor range from 5 to 10 for these Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 2.2

Dimensions of Fatigue Life Specimens (length in mm)

Material T6A4V U500 17-4PH T6A4V IN718 U500 T6A4V 42CrMo4U-Rep B 42CrMo4U-Rep C

L1

L2

B0

B1

B2

r (kg=m3)

Ed (GPa)

18.19 16.60 16.14 15.03 18.15 30.704 29.01 16.94 17.14

14.31 14.31 14.31 14.31 15.00 7.5 7.5 14.31 14.31

31 31 31 31 25 10 10 31 31

1.5 1.5 1.5 1.5 2.5 1.5 1.5 1.5 1.5

5 5 5 5 7.5 5 5 5 5

4420 8020 7830 4420 8200 8020 4420 7820 7870

110 214 203 108 215 214 108 211 216

specimens. The dimensions of the ultrasonic fatigue threepoint bending specimen of aluminium alloy-based metal– matrix composites are given in Chapter 3 and the test results discussed subsequently. 2.6. METHODS FOR CALCULATING CRACK TIP STRESS INTENSITY FACTOR In the field of fracture mechanics, formulas have been available to calculate stress intensity factor K for various specimens (Paris, 1965; Tada, 2000). However, it is generally believed that all formulas were established for static- or low-frequency cyclic loading rather than vibratory Table 2.3 Dimensions of Fatigue Crack Growth Specimens (length in mm) Material

W

L1

L2

B0

B1

B2

r (kg=m3)

Ed (GPa)

T6A4V U500 U500 17-4PH T6A4V Al2017A Astroloy Al-Li8090 Cr-Si

14 14 14 14 14 14 14 14 11

41.81 44.53 44.53 43.75 43.483 64.1 44.61 39.71 22.31

12.20 12.20 5 12.20 5 33.9 12.20 21.07 12.20

31 31 6 31 6 90 31 90 31

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

4 4 4 4 4 4 4 4 4

4420 8020 8020 7830 4420

108 214 214 203 108

8000 2350 7850

214 86 210

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 2.4

Specimen Dimensions (mm)

U500 Astroloy (20 C) Astroloy (400 C) 17-4PH T6A4V Al-Li8090

B

B1

B2

L1

L2

31 31 31 31 31 31

1.5 1.5 1.5 1.5 1.5 1.5

4 4 4 4 4 4

12.2 12.2 12.2 12.2 12.2 12.2

45.05 42.50 42.50 44.40 42.38 52.00

wave excitations. In a wave vibration system, the situations are more complex in the presence of an inertia force. Since the use of the 20 kHz-resonance method was suggested by the Fatigue Crack Growth (FCG) study in the 1970s, the determination of K1 in a vibration regime has always been a key problem, that many researchers must solve in the case of ultrasonic fatigue. It is well known that SIF is a mechanical parameter characterizing the intensity of a stress field around a crack tip. In the handbook of Tada, Paris, and Irwin (Tada, 2000), the geometric shape and the state of applied force are considered. Formulas are obtained using numerical methods, such as the boundary collocation, Green’s function, asymptotic approximation and FEM (finite element method). However, to our knowledge, literature and handbook data do not adequately answer problems related to the determination of SIF in the regime of ultrasonic fatigue. We will present methods, developed mainly in the CNMA= ITMA laboratory by Kong, Wu, and Ni, for calculating SIF in an ultrasonic frequency regime. These approaches will provide not only numerical tools such as efficient algorithms and interfaces to commercial FEM software but also practical formulas (Kong, 1987; Ni, 1995, 1996a, 1996b; Bathias, 1997; Wu, 1994). 2.6.1. Two-Dimensional Approach Based on Fracture Mechanics Concepts Several researchers (Mayer, 1992; Mayer et al., 1993) have used an approximation to compute K1: pffiffiffiffiffiffi K1 ¼ s paYða=wÞ Copyright © 2005 by Marcel Dekker. All Rights Reserved.

where Y(a=w) is established for a static state. However, in a vibration system, s is not a measurable quantity and is not the same in two cross-sections along the specimen even if the area of the section is constant. Because the specimens used in ultrasonic fatigue testing often have special geometries, certain numerical methods are required to obtain the stress solution; the FEM is a suitable method (Wu, 1994; Mayer, 1992). 2.6.1.1. Difficulties of Calculation Figure 2.6 gives a special type of specimen for FCG test at 20 kHz, designed with the mechanical symmetry about the crack line and with a variable thickness to reduce the resonance length of the specimen.

Figure 2.6

Specimen geometry.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

To calculate K1 in a two-dimensional vibrating system, the computation must incorporate the effects of the following five features: 1. 2. 3. 4. 5.

Presence of crack Non-uniform thickness Vibration Higher order vibration mode Influence of temperature on the experiments.

2.6.1.2. Development of Computation A specimen with a crack is not a linear vibration system when both opening and closing of the crack occur. In the case of a small crack, we can consider the vibrating system as two different linear systems corresponding respectively to a system with an opened and with a closed crack. For the fracture tests, the opened state is relevant to the SIF. The computation is based on the FEM using the linear elasticity theory for the plane strain problem. Eight-node isoparametric quadratic elements are employed. For vibration fatigue testing at ultrasonic frequency and at R ¼
1, K1max is considered to be DK1max and, thus, FEM can be applied to obtain it. It is well known that there is a singular stress field around the pffiffiffi crack tip. The elastic stresses are inversely proportional to r where r is the distance from the crack pffiffiffi tip, whereas the displacements are directly proportional to r  K1 can be found by stress s or crack-opening displacement v, that is in the polar coordinates (r, y) at the crack tip (Figure 2.7). pffiffiffiffiffiffiffiffi K1 ¼ lim 2prsy ; r!0

E K1 ¼ lim r!0 1
n 2

at y ¼ 0

rffiffiffiffiffi p v ; 2r 2

at y ¼ p

ð2:87Þ

ð2:88Þ

To simulate the singular field of elastic stress, two singular elements are disposed at the crack tip (Figure 2.7), and the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.7

Singular elements.

formula calculating K1 is E K1 ¼ 1
v2

rffiffiffiffiffiffi p vd 2d 2

ð2:89Þ

where d is the width of the element, and vd is the crack opening displacement at point D, which can be computed by means of FEM. For a better simulation of the r
1=2 singularity of elastic stress field around the crack tip, we can also use the so-called transition elements that are adjacent to the singular element. This has been done by Wu (1992) in a study of the stress field for an ultrasonic fatigue crack. For the specimen in Figure 2.6, the equilibrium equations become @sx @txy txy dt þ fx ¼ 0 þ þ @x @y t dy

ð2:90Þ

@txy @sy sy dt þ fy ¼ 0 þ þ @x @y t dy

ð2:91Þ

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

where fx and fy are the body forces, and t ¼ t(y) is the specimen’s thickness. The additional terms txy dt t dy

sy dt t dy

and

appear for a specimen with non-uniform thickness. In two-dimensional FEM, the elementary stiffness matrix and mass matrix are respectively presented by e

½K  ¼

Z

Z

1
1

e

½M ¼ r

Z

1

½BT ½ D½BtjJ jdxdZ

ð2:92Þ


1 1


1

Z

1

½ N T ½ D½ N tjJ jdxdZ

ð2:93Þ


1

with x¼

2x
1; w

Z¼

y ; L

L ¼ L1 þ L2

where r is the mass density; [B], [D], [N] are geometrical, elasticity, interpolation matrices, and jJj is the Jacobian. For our specimen, these matrices must be integrated numerically. In a specimen with a crack, the vibration problem is complicated. The specimen can be considered to be two linear mechanical systems—according to whether the crack is opened or closed—with different intrinsic frequencies and modes. Because the crack growth takes place when the crack is opened, we are only interested in the mode for the opened crack. By means of the FEM, the following numerical equations of vibration without damping can be used ½ K fug ¼ o2 ½M fug

ð2:94Þ

or, with 1=o2 ¼ l, l½ K fug ¼ ½M fug Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð2:95Þ

where [K] and [M] are global stiffness and mass matrices. We now have the problem of finding the eigenmatrix and eigenvector. Solving this consists of finding the values l and fug. The inverse iteration method can give the largest eigenvalue l1 and the corresponding eigenvector fu1g (Eq. 2.95); the largest eigenvalue corresponds to the fundamental frequency of the mechanical system. Since the intrinsic frequency of about 20 kHz is not the fundamental frequency, the inverse iteration method cannot be directly used and needs a transformation. Suppose the frequency of about 20 kHz corresponds to 2 oe . We have the following inequalities 0  o21  o22      o2e      o2n

ð2:96Þ

If we introduce o2i ¼ o2 þ

1 Dli

ð2:97Þ

then o2 can be chosen so that jDle j ¼ max jDli j i¼1;...;n

ð2:98Þ

Eq. 2.95 becomes Dl½K  fug ¼ ½M fug

ð2:99Þ

with ½K   ¼ ½ K 
o2 ½M 

ð2:100Þ

For Eq. 2.99, the largest eigenvalue Dle corresponds to the mode with an intrinsic frequency of about 20 kHz. The convergence of the inverse iteration method is rapid and the cumulative error is minimized. In the case of a turbine disk, it is customary to study the FCG rate at elevated temperatures. The specimen is heated by an inductor in its center with a temperature gradient between the middle and the end of the specimen. Since the end is free, and plasticity and thermal-mechanical coupling at high strain rate is not taken into account in the present Copyright © 2005 by Marcel Dekker. All Rights Reserved.

elastic analysis, there is no extra thermal stress in the specimen. This does not mean that heat effect is totally negligible, however; for vibration fatigue, the material rigidity, such as Young’s modulus, is a function of temperature. That is to say, the vibration mode may change. So we have to consider the temperature gradient to obtain a correct vibration mode when the elementary stiffness matrix [K]e is determined. 2.6.1.3. Numerical Results A typical finite element mesh is presented in Figure 2.8. To check this mesh, an example is computed in which a constant thickness specimen is loaded with static tensile stress at the end. A formula to calculate K for this case is given Wu (1992). The error of the FEM result is small, less than 5.7% for a=w ranging from 0.15 to 0.5. The singular element size

Figure 2.8

A finite element mesh.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

is small. Figure 2.9 gives vibration modes. Note the boundary condition that there is no horizontal constraint, and the inertia force balances automatically in the horizontal direction. For displacement amplitude at the specimen end U0 ¼ 1 mm, the relations of DK versus a=w are presented in Figure 2.10. From these curves, the experimental results, da=dN vs DK, were plotted in Figure 2.11 for Astroloy, at both room and elevated temperatures. The threshold is usually lower at 400 C than at 20 C. 2.6.1.4. Formula for K In order to simplify the calculation of K for ultrasonic fatigue, we propose a formula. For FCG rate tests at 20 kHz, it is possible to obtain a standard formula for the specimen shown in Figure 2.4 at room temperature, where Ed and r are constant

Figure 2.9

Vibration modes.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.10 Curves of DK versus a=w.

and Poisson’s ratio is about 0.3. For this purpose, Eq. 2.94 can be written as
rffiffiffiffiffiffi2 r 0 ½K fug ¼ o ½M 0 fug ð2:101Þ Ed Matrices [K0 ] and [M0 ] depend only on geometrical dimensions. For this specimen, the resonance length L1 is uniquely

Figure 2.11 Test data at 20 kHz for Astroloy. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

determined so that it p has a 20 kHz intrinsic frequency. It ffiffiffiffiffiffiffiffiffiffiffi depends directly upon Ed =r, sound velocity in a material. Eq. 2.102 gives the relation, calculated by the FEM with the aid of dichotomizing search. sffiffiffiffiffiffi Ed ð2:102Þ L1 ¼
20:327 þ 12:765 r From dimensional analysis considerations, the following formula is introduced rffiffiffi Ed p A0 f ða=wÞ ð2:103Þ DK1 ¼ 2 a 1
n where A0 is a measurable quantity and f(a=w) is a dimensionless correction factor. This formula describes only the positive value of DK1. Generally, the function f(a=w) depends on the resonance length. However, as observed in Figure 2.12, its dependency pffiffiffiffiffiffiffiffiffiffiffi Ed =r ¼ on L1 does not seem to be significant for 4:8 5:7 km=sec and for a=w ¼ 0 0.5. Therefore, the follow-

Figure 2.12

Correction factors (Kong, 1991).

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ing polynomial expression (Eq. 2.104) is established as correction factor. When the temperature field is not uniform, it is difficult to obtain a standard formula. But, if the temperature field is uniform, the formula (Eq. 2.103) is still suitable even at a higher temperature. f ða=wÞ ¼ 0:64ða=wÞ þ 1:73ða=wÞ2
3:98ða=wÞ3 þ 1:96ða=wÞ4

ð2:104Þ

2.6.1.5. Calculation by ANSYS The previous calculation was for cyclic loading with R ¼
1. To determine the SIF in the presence of a static load (Sun, 2001), instead of developing our own program, the ANSYS program is used. In this case, the SIF is not a simple superposition of static and dynamic components, because a geometrical matrix depending on the static force influences is added to the vibration equation. Since the experimental system is mechanically symmetrical about the crack, only half of the specimen will be taken as the calculation model, as was done in the above for R ¼
1. But, if half the specimen alone is taken, a more complex vibration equation will result with a non-zero matrix on the right side and the calculation will be more involved. If the specimen and the cone are calculated together, the equation becomes simpler  ð2:105Þ ½ K  þ ½ K g fug
o2 ½ M fug ¼ f0g where [K]g is a geometrical matrix. The right side is zero, because the dynamical displacements are zero at the ends of this model and the force does no work (Figure 2.13). In this case, computation is easier. When using ANSYS, isoparametric elements of eight nodes (STIF82) simulate the specimen, and beam elements (STIF3) simulate the cone. An elastic tee is introduced to connect the two types of the elements. To do so, it is necessary to introduce two degenerated STIF82 elements in the specimen. When conducting experiments, a screw connects the specimen Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.13

Calculation model using ANSYS.

and the cone. Since the dynamical stress of the connection zone is theoretically very small, this tee connection would not cause significiant errors. The constraint equations representing the relation of displacement and rotation in the connection are as follows Ux4 ¼ Ux2

ð2:106Þ

1 Uy4 ¼ ðUy1 þ Uy2 þ Uy3 Þ 3

ð2:107Þ

ROTz4 ¼

Uy3
Uy1 x3
x1

ð2:108Þ

where the numerical subscripts 1 through 4 correspond to those in Figure 2.13. In vibratory fatigue, the stress ratio R can be defined only by the ratio of stress intensity factors as R¼

Kmin Kmax

with Kmin ¼ Ks
Ka and Kmax ¼ Ks þ Ka where Ks and Ka are computed as follows. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.14 SIF amplitude and static SIF for T6A4V.

First, the static SIF—called mean factor, Ks, which is proportional to tension force—is determined by Eq. 2.109 ½Kfus g ¼ ff g

ð2:109Þ

where ffg and fusg are vectors of static force and displacement, respectively. Secondly, the SIF amplitude, Ka, is determined by Eq. 2.105. Figure 2.14 gives curves determined for titanium alloy T6A4V. Ka is proportional to the displacement amplitude at the specimen’s end. An optical sensor measures this amplitude. We observe that the greater the static force, the smaller the Ka for a given crack length. 2.6.2. Three-Dimensional Computation of Stress Intensity Factor In the previous section, half a specimen is taken into account and, by a self-compiling two-dimensional finite element code, K1 values are evaluated using the opening displacement of a node close to the crack tip. Although the basic idea of this approach is acceptable with the newly defined function f(a=w) for use in ultrasonic fatigue, there is also room for improvement. First, only one Copyright © 2005 by Marcel Dekker. All Rights Reserved.

half of the specimen is computed on the assumption that the displacement and deformation distribution along the specimen is symmetric about its center line. However, the lower side of the specimen is linked elastically with an amplifying horn while the upper side is free, and so the displacement and strain will not be symmetric in the upper and the lower parts of the specimen. Secondly, the analysis is carried out in two-dimensions, neglecting the Poisson’s contraction effect in thickness. And thirdly, the finite element mesh close to the crack tip is not fine enough. In this section, methods for evaluating K1 by means of displacement and energy approaches are described (Ni, 1996; Wu, 1992). 2.6.2.1. Three-Dimensional Finite Element Models The ultrasonic FCG specimen, shown in Figure 2.15, is designed to have exactly two quarter-wavelengths, with symmetry about the middle section. A lateral notch is made in the central section to facilitate and localize the crack initiation. The specimen is linked elastically with the amplifying horn(s)—either at one end for load ratio R ¼
1, or at both ends for R >
1. The dimensions in Figure 2.15 are

Figure 2.15 Ultrasonic FCG specimen coupled with amplifying horns. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

R0 ¼ 31 mm, B1 ¼ 3.0 mm, B2 ¼ 8.0 mm, and L1 ¼ 12.2 mm, and W ¼ 14 mm. And, for the specimen of T6A4V and the amplifying horn studied here, D1 ¼ 18.0 mm, D2 ¼30.38 mm, and L3 ¼ 75.51 mm, with the resonant length L2 determined to be L2 ¼ 41.88 mm. Software ALGOR is adopted in computation. The finite element mesh for the model has the following two features: 1. The whole specimen and the amplifying horn(s) are considered together. The finite element mesh is symmetric about the central line of the specimen. 2. A group of finite element meshes with minute dimensions is installed all around the crack tip, and it is movable with the crack tip. The smallest dimension of the element meshes close to the crack tip is 1=224 of the specimen width, W. 2.6.2.2. SIF Determined by Displacement and Energy Approaches Displacement approach involves a singular element modeling of r
1=2 singularity in the elastic stress field around the crack tip, the approach used in the previous section. The relations between the SIF and the crack length obtained by the displacement method are given in Figure 2.16 The energy method is a more global approach than the displacement method. The basic idea of using strain energy is particularly well suited for in finite element simulation. The elastic energy, G, made available for crack extension is given by
 @V ð2:110Þ G¼þ @A when the load is kept constant, and where V is the elastic strain energy of the body and A is crack surface area. In our three-dimensional finite element calculation, the energy rate, G, is determined as follows. First, the strain Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.16 K1 versus a by displacement approach: (A) R ¼
1; pffiffiffiffiffi (B) R >
1 (U0 ¼ 1 mm, K1 in MPa m).

energy V is computed when the crack area is A by the formula 1 V ¼ fugT ½Kfug 2

ð2:111Þ

Note that the stiffness matrix [K] is a function of A. Then under the same displacement fug, the strain energy V þ DV corresponding to a crack area A þ DA is calculated. The elastic energy rate, G, is determined by
 DV ð2:112Þ G¼
DA In plane stress conditions, the relation between G and K1 is given by pffiffiffiffiffiffiffiffiffiffi K1 ¼ E d G ð2:113Þ An interface routine interconnected with ALGOR is compiled for calculating G as well as K1 values in this ultrasonic FCG study. The relationship between K1 and crack length a computed by the energy method is given in Figure 2.17. The determination of K1 by means of the energy method may achieve a better precision than the displacement method, since, in the energy approach, the evaluation of the displacement distribution is taken into consideration over the entire vibration body (specimen plus horns), while in the displaceCopyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.17 K1 versus a by p energy approach. (A) R ¼
1; ffiffiffiffiffi (B) R >
1 (U0 ¼ 1 mm, K1 in MPa m).

ment method, only displacements of nodes near the crack tip are used. In the case of R ¼
1, K1 determined by energy approach increases progressively with the crack length, and then develops a decreasing tendency for crack lengths greater than 6.5 mm (Figure 2.17). This decreasing characteristic is not observed in the displacement approach (Figure 2.15). In fact, as a fatigue crack initiates and then propagates in ultrasonic resonance vibration, the vibration energy transformed through the continuous medium is reflected at crack lips. As a result, the system in resonant vibration becomes more and more detuned and deviates from its proper frequency, resulting in a hybrid effect of vibration modes. Figure 2.18 follows the evolution of the resonance frequency as the crack length increases. Once the crack length reaches a certain value, the resonant system in ultrasonic vibration will be destroyed and the vibration energy will decrease. This effect has been observed in experiments and is easily simulated numerically (Figure 2.19). It is mandatory, therefore, to control carefully the frequency of the machine when the crack is propagating. An error in the frequency induces an error in the stress intensity factor. In order to avoid such errors, the piezoelectric machine must be controlled with a computer and the test has to be stopped when the crack length reaches one half of the specimen width. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 2.18 Resonant frequency versus crack length a without frequency control.

Figure 2.19 Detuning effect of the resonant vibration due to a crack (a ¼ 5.5 mm). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In the case of R >
1, K1 increases continuously as the fatigue crack propagates from 0.5 mm to 7.5 mm. When a static tensile load is applied to the ultrasonic vibration system, the fatigue crack opening displacement at the crack tip may be greater than that obtained for R ¼
1, even if the vibration energy sent through the cross-section of the specimen decreases. Experimental data indicate that the FCG behaviors at ultrasonic fatigue frequency are comparable to those observed in conventional fatigue tests regardless of the R ratio. In ultrasonic resonance vibration, as a fatigue crack initiates and then propagates, the vibration energy is transmitted through the continuous medium and is reflected at crack facets. It is not clear why the FCG at ultrasonic frequency is not affected by the high frequency. The effects of the following several factors are discussed later.

The effect of the deformation rate de=dt on the plastic zone size at the crack tip;

The effect of vibration on the residual stresses at the crack tip;

The effect of the plastic zone size and the residual stresses on the crack opening and the crack closure;

The effect of the environment at high frequency.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

3 Testing Machines and Their Performance

3.1. INTRODUCTION Up to now, there have been no standards for testing procedures and testing machines of ultrasonic fatigue, although efforts are in progress within ASTM to provide a recommended practice and ultimately a testing standard (Bathias, 1998). Because of this, laboratories must develop their own machines and design practical test procedures. The laboratories of Willertz in the United States, Stanzl in Austria, Bathias in France, Ni in China, Ishii in Japan, and Puskar in Slovakia are among the leading laboratories in this field. Although ultrasonic fatigue test machines in these laboratories are not the same, some components are common to all machines. The three most important are: (1) a high frequency generator that generates 20 kHz sinusoidal electrical signal, (2) a transducer that transforms the electrical signal into mechanical vibration, and (3) a control unit. Early ultrasonic fatigue machines performed only uni-axial

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

(one-dimensional) and constant amplitude tests, so the control unit and other parts were not very complicated. In the last two decades, progress has been made to extend the ultrasonic fatigue technique to variable amplitude loading conditions, low or high temperature environments, torsional or multi axial tests, and so on. Thus, designing a modern ultrasonic fatigue test machine may involve mechanical, electrical, optical, magnetic, and thermal considerations. In France, Bathias used a first ultrasonic fatigue test machine in 1967 on the principle used by Mason (Bathias, 1998). As indicated in an early review paper (Stanzl, 1996), the rather restrictive uses of the ultrasonic fatigue test method appeared to be partly due to the lack of commercially available test equipment, forcing the individual investigators to work with improvized facilities not readily amenable to standardized experimental conditions. 3.2. BASIC STRUCTURE As stated above, an ultrasonic fatigue test machine must include the following three common components: 1.

2.

3.

A power generator that transforms 50 or 60 Hz voltage signal into ultrasonic 20 kHz electrical sinusoidal signal. A piezoelectric (or magnetostrictive) transducer excited by the power generator, which transforms the electrical signal into longitudinal ultrasonic waves and mechanical vibration of the same frequency. An ultrasonic horn that amplifies the vibration coming from the transducer in order to obtain the required strain amplitude in the middle section of the specimen.

These three parts are special devices required for the production of ultrasonic fatigue load. Other components of an ultrasonic fatigue test machine may include recording systems (amplitude control unit voltmeter, frequency control unit, cycle counter and oscilloscope) and measuring systems (displacement sensor and video camera observation unit). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

The function of the system shown in Figure 3.1 is to make the specimen vibrate in ultrasonic resonance at one of its longitudinal modes. The displacement amplitude reaches its maximum U0 at the end of the specimen, which can be measured by means of a dynamic sensor, while the strain excitation in push–pull cycles (load ratio R ¼
1) attains the maximum in the middle section of the specimen that produces the required high frequency fatigue stress. The video camera supervision system in Figure 3.2 is used in the fatigue crack growth test for observing and recording crack initiation and propagation processes. The information recorded may include

Figure 3.1 Full resonance system in Bathias’s laboratory with schematic view of apparatus. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.2

Diagram of equipment with computer control.

ultrasonic cyclic displacement amplitude U0, the evolution of the fatigue crack growth, which enables us to determine the fatigue crack growth rate da=dN, and the stress intensity factor Kmax, calculated by analytical or numerical methods (Wu, 1991). During ultrasonic fatigue tests, the maximum strain values can be measured directly using miniature strain gauges, suitably positioned on the sample surface. For example, a measuring system consists of a Wheatstone bridge amplifier, dynamic strain gauges (0.79 mm by 0.81 mm), and a digital oscilloscope (two channels, 40 K memory for each) has been built for direct strain measurements in Bathias’s laboratory. In the same laboratory, the dynamic displacement amplitude at the specimen extremity, U0, is measured by an optic fiber sensor, which permits measurements of the displacement from 1 mm to 199.9 mm, with a resolution of 0.1 mm. The magnification factor of stress can then be calculated according to these measurements. For a virgin specimen (i.e., without a crack), the vibratory stress and strain can also be determined at the midsection. The maximum strain value thus determined is then confirmed to be accurate by use of the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

above-mentioned strain gauge. In addition, a system of video-camera–television has been used for the control of crack initiation and propagation. This system refines events to 1=25th of a second and magnifies specimen surface 140  200 times. 3.3. NONSYMMETRICAL AND VARIABLE AMPLITUDE TEST EQUIPMENT Because specimens of ultrasonic fatigue vibrate in resonance, a free end is sufficient for symmetric loading conditions (R ¼
1). This avoids the large and cumbersome arrangements for gripping the specimen that is often encountered in conventional fatigue testing (Figure 3.3). If there is a static load, the situation is different.

Figure 3.3 Vibratory stress and displacement field and computer control system. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.4 Vibration system for ultrasonic fatigue superposed on static loading.

Superposing a mean stress or displacement upon the symmetric tension–compression cycle can create a complex cyclic loading with R >
1. Therefore, an additional horn will be added at the other end of the specimen, as shown in Figure 3.4 Such a test machine is particularly useful to efficiently study fatigue endurance and fatigue crack growth behavior of materials subjected to elasto-plastic low cycle fatigue loading superposed on vibration stress cycles with high frequency (Figure 3.4). 3.4. COMPUTER CONTROL SYSTEM A computer control system is of great importance in programming and controlling the load as well as in data acquisition. Before 1981, there was an ultrasonic fatigue machine with a computer control unit (Kong, 1987). Here, however, we discuss the machine built in Bathias’ laboratory (Wu, 1992). This computer control system uses an IBM PC computer and a data acquisition system composed mainly of a 12 bit A=D converter, a 12 bit digital to analogue (D=A) converter, a strain gauge board, and a thermocouple board. The A=D board converts the data from analogue to digital in 20 ms. When using the program with necessary commands, the acquisition Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.5

Flow diagram for test control program.

time for TURBOBASIC language is 280 ms. This acquisition time does not agree with the sampling theory, and with the additional control operations, the time becomes much longer. If sampling were directly executed for the 20 kHz sinusoidal wave, auto-control would be impossible. For fatigue tests, the most important factor is load; that is, the amplitude of the sinusoidal wave. Therefore, a rectifier with a filter is installed between the transducer and A=D converter, from which the d–c output-voltage proportional to the vibrating amplitude is obtained. At any time, the A=D converter can detect the test load, and with this information, the computer gives a control signal to the power amplifier at the D=A board to maintain constant amplitude or to change the amplitude. Figure 3.5 is the flow chart of the control program. When the operator places the computer on line, by interrupting the power amplifier potentiometer, the program demands an input load with the expected amplitude to begin the control test and the specimen vibrates with the amplitude of the load. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In order to avoid the influence of a parasite signal for the system input read by A=D, a hardware filter has been installed before the A=D. In other words, in this control program there is another software numerical filter that selects the medium value among three values of every sampling to obtain a true load signal. The adjustment of the D=A output voltage is made to reduce the observed difference between the desired load and actual feedback reading of the A=D load channel. A software gain is introduced in the program so that the D=A command to the amplifier is the product of a software loop gain and the expected vibration amplitude of the specimen. The computer is an integration link in this control loop. Corrections to the D=A voltage are made automatically ten times per second. The testing load can be modified while the test is in progress. During this mode of operation, the operator has the option of either changing vibrating amplitude or stopping the test at the current mean load level. A control loop is composed of the specimen, transducer, voltmeter and rectifier, computer, power amplifier, and vibrator. The regulation of some parameters in the program is very important for this computer control system. The system presented in Figure 3.5 permits entry at expected vibrating amplitude by a computer keyboard, so that the input of the system is a jump function. The output of the system (that is, the vibrating amplitude measured by a capacitive sensor) responds differently to the different control parameters in the program. Figures 3.6 and 3.7, respectively, present the responses to the first two groups of parameters where the output voltage (ordinate), which is the input of the A=D converter, is directly proportional to the specimen vibrating amplitude. Interval 40–60 represents the jump function from 40  60 mV and so on, and the curves represent the autocontrol system response to the jump function. In Figure 3.6, the amplitude response converges with oscillation and the convergence is slow (20 s). In Figure 3.7, there is no oscillation, but the convergence is still slow (13 s). For fatigue tests, the convergence must be rapid and the curve smooth to avoid the overload. When the parameters are chosen properly, the satisfactory system response will be obtained. Figure 3.8 gives Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.6

Response curve under the first parameters.

the responses under the final parameters chosen; that is, the signal gain to the D=A converter depends on the difference between the desired load and actual load. Clearly, the convergence time is significantly shorter—it is 3 seconds, which is much shorter than the time of manual operation. Another important aspect is data acquisition. For ultrasonic fatigue crack tests, the specimen is relatively small because it must have a high intrinsic frequency. A typical

Figure 3.7

Response curve under the second parameters.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.8

Response curve under the final parameters chosen.

specimen width is approximately 14 mm. The effective crack propagation range is only about 4 mm. When a crack propagates rapidly under a high load, it may cause a rapid change in the vibratory amplitude because of the decrease of the specimen intrinsic frequency. This amplitude change may go beyond the auto-control range of the computer and an unsatisfactory experiment will result. Therefore, it is important to distinguish the satisfactory period from the unsatisfactory period, and to record the results for out-of-line analyses after the tests. Because the entire process takes only 10 seconds or so, the data acquisition using this control program is very helpful. Figure 3.9 shows the testing charge recorded during 60 s and its relation with the crack length recorded by the video system. These records provide the information needed in calculation of the stress intensity factor and in determination of the crack growth rate. The decrease in testing load after 40 s (when the machine stops automatically in case of crack) occurs because the intrinsic frequency of the specimen begins to depart significantly from the designed frequency (error more that 5%). These results are then unusable. The input data for the control system are elastic modulus, mass density of the material to be tested, and the desired Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.9 Charge recorded by computer and corresponding crack length (without frequency control).

test stress. Several forms of amplifying horn can be presented over a visual interface on the computer screen. This permits one to calculate the amplification of displacement and stress range in comparison with the tension measured by the interface J2 (described later in this chapter). The computer starts the control process by activating the weakest vibration with the help of a card piloted by a numerical exit. After 50 ms, the stress of vibration attains the recorded level picked up from an input curve of stress signal. Figure 3.10 shows a response signal to the start-up (Wu, 1992). The response signal comes from plug 9 of interface J2 (see Figure 3.13). We observe that the response curve has a plateau at 50% power before attaining 100% power. The time for reaching 100% is 85 ms, and there is no overload. Then, the software maintains constant amplitude of vibration and numerical filter eliminates false acquisitions. With plugs in J2, we know the actual frequency and power. As soon as a macro crack appears, the ultrasonic generator automaticaly cuts off the current and the computer gives the fatigue life of the specimen. This function also ensures that the test conditions are maintained. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.10

Response signal to the start-up.

A generator with a converter composed of six piezo-ceramics is chosen to provide vibration energy. This ultrasonic generator 900BA is made by Branson Ultrasonic Corporation. It has a maximum power of 2 kW and provides a sinusoidal signal for the converter that is the source of mechanical vibration. This amplifier automatically maintains the intrinsic frequency of the mechanical system in the range of 19.5 kHz  20.5 kHz. The converter, horn, and specimen form a mechanical vibration system with four stress nodes (null stress) and three displacement nodes (null displacement) for an intrinsic frequency of 20 kHz. Here, the stress and displacement are considered to be longitudinal. In Figure 3.11, points B and C (connected points), and point A and converter top are stress nodes. The specimen center is a displacement node; there the stress is at the maximum. The horn must vibrate at a frequency of 20 kHz. Depending on the specimen loading, the horn is designed so that the displacement is amplified between B and C, usually 3 to 9 times, meaning that the geometry between B and C must determined accordingly. The finite element method may be required when the geometrical shape is complex. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.11 Vibratory stress and displacement field.

The mechanical system composed of a converter, a horn, and a specimen is linear, and all stress and displacement fields are linear. It is necessary only to measure the amplitude of one of them. To determine an S-N curve, one needs to know the stress amplitude with good accuracy. However, at high frequency and low temperature, it is difficult to measure the stress amplitude. Therefore, the stress in the midsection of the specimen is computed from the displacement of the piezo-ceramics system. Piezo-ceramics expand or contract when an electrically induced tension is applied. The tension is proportional to expansion or contraction; that is, the tension is proportional to the displacement in the mechanical system. It is strictly proportional to the expansion or contraction of the converter and to the displacement of point C. In other words, electrical current depends on the damping of the horn and specimen. The damper is installed on the converter. In the generator, an interface called J2 has been set up, which has a plug with 0  10 volt (DC tension) corresponding to 0  100% of vibration amplitude of the converter. This output is calibrated with the displacement of the horn end (point B), to determine the stress in the specimen using a computer that acquires this Copyright © 2005 by Marcel Dekker. All Rights Reserved.

DC tension. The stress can be calculated by the following equation s ¼ Eks kh UC100%

V 10

where ks, is a factor of the specimen depending on the geometrical form, kh is the ratio of amplitude amplification, UC100% is the maximum amplitude at point C, which is constant, and V is DC tension acquired by the computer. According to this formula, the test stress for a certain specimen can be altered not only by changing output power but also by replacing the horn. For calibration, a simple cylindrical specimen was used with a gauge mounted in the middle. The strain measured by this gauge and displacement of horn end (point B) UB is given by the relation below. rffiffiffiffi r e ¼ 2p f UB E where r is mass density. When the DC output is calibrated according to this measure, a comparison between measured strain in liquid nitrogen and strain calculated by control computer for different power is presented in Figure 3.12. It shows a good linear relationship between measured and calculated strains. Another group of calibration tests was made with an optical sensor that measures displacement of the specimen end at room temperatures. It is possible to apply a correction from room temperature to low temperature, since the amplification ratio is known for different temperatures. The results were also satisfactory. Furthermore, interface J2 installed in the command box (Figure 3.13) makes the computer control possible. In this connector, plug 2 supplies a DC tension of 10 volts with which we can use a potentiometer to control the equipment manually. Plug 8 can be loaded with a DC tension of 0  10 volts corresponding to 50% and 100% of the vibration amplitude. Plug 9 also gives a DC power of 0  10 volts proportional to Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.12 Comparison of measured and calculated values of strain at 77K .

the vibration amplitude regardless of the magnitude of the mechanical excitation. Plugs 3 and 10 indicate the power and frequency, respectively. In general, direct control for 20 kHz is very difficult. Thus, it relies on the use of d=c signal proportional to the amplitude of the alternation current

Figure 3.13 Command box and interface J2. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

signal (Wu, 1992). A normal A=D and D=A converter card connecting connector J2 and a PC (Figure 3.1) can achieve a computer-controlled test at 20 kHz. A control program has been written with Turbo Cþþ language, which calculates the vibration stress in the specimen. The test starts by giving a test stress and the real stress rises within 85 ms to the expected level without overloading. Then, the stress is held constant with a control accuracy  3 MPa. When a crack appears the testing system stops automatically because its intrinsic frequency decreases and it gives the fatigue life. With this software, a fatigue test between 105 to 1010 cycles can be performed. A generator with a converter composed of six piezo-ceramics is chosen to provide vibration energy. It has a maximum power of 2 kW. The computer testing system described above has the following advantages: 1.

2.

3.

4.

5.

The output signal to the power amplifier from the D=A converter does not correspond directly to the input of the A=D converter; that is, the adjustment is done according to the difference between the input signal and expected amplitude. If the electronic drift and main system error are caused by the temperature in the vibrator or in the power amplifier, the computer control system compensates for those errors so that the experiments are accurate. The program can easily compensate for the nonlinearity of the transducer-rectifier–filter after the input and output relationship is calibrated. This system possesses a better convergence tendency, a rapid convergence velocity, and a steady amplitude error within 5%. The system applies a block spectrum with gradual change for ultrasonic fatigue endurance tests, thus making programming easy. When normal frequency drops off and the mechanical system stops vibrating, the computer can automatically stop the power amplifier to protect the equipment.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

6. The design of the computer control system respects the integrity of the original machine. The test machine can run without the computer. 7. The change of test parameters is continuous rather than stepwise in some multi-stage control systems, such as that of (Stanzl, 1981). 3.5. HIGH TEMPERATURE TEST EQUIPMENT Figure 3.14 is a diagram of a system for high temperature tests, where the temperature in the specimen is constant along 5 to 6 mm. Figure 3.15 is a photograph of this same system. The test equipment consists of a heating device in the middle, a capacity transducer above, and a video camera with an enlargement factor of 200 on the right. The television images can be recorded on videocassette during the tests. The fatigue crack growth rate can be determined easily up

Figure 3.14 alloy.

Evolution of the specimen temperature for a Ni base

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.15 Specimen installed in the machine including measuring and heating devices.

to the order of 10
9 mm=cycle. For the experiments at elevated temperatures, a high-frequency inductor made by CELES is used; test temperatures can then reach 1000 C without problems. The computer system has a thermocouple board that can send the temperature analogue signal to an A=D converter, so that the computer is able to control heating of the specimen by using a disjunctor. Because the Young’s modulus decreases at high temperatures, the resonance length of ultrasonic fatigue specimen will be shorter than that at ambient temperature. For temperature sensitive materials, this change must be taken into account. For example, in a crack growth experiment, we Copyright © 2005 by Marcel Dekker. All Rights Reserved.

usually start crack initiation at ambient temperature with adequate resonance length. Then, by cutting off both ends, we can obtain the resonance length at high temperature. An early design of ultrasonic fatigue equipment at high temperature can be found in reports by Ebara (1994). The system described there is capable of studying fatigue crack growth rate and fatigue thresholds at 22 kHz, at elevated temperatures of 200  500 C in an argon environment in a heat chamber, and at 20 C using water as a coolant. 3.6. LOW TEMPERATURE TEST EQUIPMENT Let us now discuss the possibility of testing materials at low temperatures. A system for ultrasonic fatigue tests at cryogenic temperatures has also been developed in Bathias’s laboratory (Tao, 1996). In the laboratory, liquid nitrogen, liquid hydrogen, and liquid helium are used to create a cryogenic temperature atmosphere. Liquefied gasses are costly, especially liquid helium. If conventional fatigue testing were employed, the fatigue tests at very low temperatures for titanium alloys used in space rockets would require a large amount of liquefied gas because the tests would take a very long time. This is another advantage of the ultrasonic fatigue method, which substantially reduces testing time. The machine with a computer control system works at 20 kHz and at cryogenic temperatures (77 K and 20 K) for studying fatigue behaviors of the titanium alloys used in rocket engines. The device consists of three parts: a cryostat, a mechanical vibrator, and a controlled power generator. Figure 3.16 shows the principal aspects of this machine, which is simpler than a conventional hydraulic machine. The function of the converter and the horn are the same as in other ultrasonic fatigue apparatus: The converter changes an electronic signal into a mechanical vibration and the horn plays the role of displacement amplifier. A dewar cryostat contains liquefied gasses to keep the testing temperature constant. Ultrasonic fatigue tests at cryogenic temperatures for load ratio R >
1 are also possible by adding a second horn to the other end of the specimen (Figure 3.16). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.16 Low temperature and high frequency fatigue testing machine.

Another example of low temperature test system is that of Stanzl’s laboratory (Buchinger, 1984), where a temperature environment of 77 K is guaranteed by liquid nitrogen. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

3.7. THIN SHEET TEST EQUIPMENT The geometry of ultrasonic fatigue specimens is usually either a cylindrical or plane form with a reduced section in the central part to form a higher stress area to accelerate the test process. Theoretically, as the excitation frequency of the machine coincides with one of the resonance frequencies of the specimen, the specimen will vibrate in that frequency. To put this theory into practice, however, is not an easy task. On the one hand, the finishing of specimens precisely to the design requires delicate work. On the other hand, the horn where the specimen is installed may change the real frequency of vibration of the system if the connection is not well designed. This is the case especially for load ratio R >
1 when another horn is necessary or when plane specimens are very thin. It was indicated long ago (Ebara, 1994) that the use of a two horn system and a positive constant mean stress is favored for avoiding transverse vibration. For most plane specimens of ultrasonic fatigue, the ratio of the thickness to the largest dimension i.e, w=2ðL1 þ L2 Þ, is about 6%  8%. Our experience shows that a thickness–length ratio in this range does not pose severe problems to the machine and control system in maintaining the desired frequency of about 20 kHz. But, as the thickness–length ratio of a plane specimen decreases by about one order of magnitude, say to 0.7%, the perturbation of vibration frequency of the system becomes so great that the test could not be performed at all if special measures were not taken. A series of thin sheet tests of ferrous materials (Wang, 1996) has been conducted. The purpose is to determine the fatigue strength (or S-N curve) at 109 cycles and the threshold of cracking at a small propagation speed of 10
12 m=cycle, both with R ¼ 0.1. Figures 3.17 and 3.18 present dimensions of two types of specimen. The ratio of thickness to length is 0.7%, an order of magnitude smaller than that of ordinary specimens. The key to the execution of the test is how to fix the thin specimen to the ends of amplifying horns. We cannot use the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.17

Fatigue life specimen of thin sheet.

same kind of set screw as that used in the attachment of ordinary specimens (i.e., specimens with the thickness–length ratio of 6%  8%). Among the methods for linking two steel components are riveting, bolt jointing, welding, and gluing. We have tried different methods and secured a special type of screw and structural glue for the thin sheet specimens. The glue is soluble in acetone, which reduces the number of screws required. In this way, we minimize the influence on the frequency of the test machine used for thin sheet specimens that have small transverse rigidity, and tests are therefore successfully performed. Figure 3.19 presents the geometry of the special screw and the connection with the horn. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.18 Crack growth specimen of thin sheet.

Figure 3.20 shows the test system. The experimental results will be discussed in the next chapter. 3.8. HIGH PRESSURE PIEZO-ELECTRIC FATIGUE MACHINE It is well known that it is difficult to conduct a fatigue test under high pressure with a conventional machine. The problem stems from the displacement of an actuator through the wall of an autoclave. The use of a piezo-electric fatigue system eliminates this problem because it is easy to get zero displacement at the location where the sonotrode under pressure crosses the wall of the autoclave. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.19

Special screw and the connection.

Figure 3.20

Test system of thin sheet.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.21 Autoclave description.

Figure 3.22 pressure.

Wo¨hler Curve–INCONEL 718:Effect of hydrogen

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

A high pressure piezo-electric fatigue machine that works under a pressure up to 300 bar has been built in Bathias’s laboratory. The design is shown in Figure 3.21. With this device, it has been shown that hydrogen under a pressure of 100 bars has an effect on the S-N curve of IN 718 at room temperature. In Figure 3.22, S-N data in hydrogen and in helium are compared to show the effect of hydrogen between 106 and 109 cycles. 3.9. NON-AXIAL TEST EQUIPMENT 3.9.1. Ultrasonic Fretting Fatigue Testing Fretting fatigue is generally promoted by high frequency low amplitude vibratory motions and commonly occurs in clamped joints and shrunk-on components (Lindley, 1997). The surface damage produced by fretting can take the form of fretting wear or fretting fatigue where the material’s fatigue properties can be seriously degraded. Some practical examples of fretting fatigue failures are observed in wheel shaft, steam and gas turbines, bolted plates, wire ropes, and springs. Fretting fatigue is a combination of fretting friction and the fatigue process and involves a number of factors including the magnitude and distribution of contact pressure, the amplitude of relative slip, friction forces, surface conditions, contact materials, cyclic frequency, and environment. Great efforts have been made to quantify fretting fatigue in terms of these factors, but limited success has been achieved. More often, fretting fatigue characteristics are studied in the laboratory experimentally by using a contact pad clamped to a fatigue specimen in order to determine S-N curves with fretting and thereby to establish the fatigue strength reduction factor for a particular material. But these studies, generally performed on the conventional tension–compression fatigue machine with a low frequency, have some drawbacks: 1.

The slip amplitude of fretting fatigue is usually coupled with the fatigue stress, and to change the

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

slip amplitude, pads with different gauge lengths are needed. 2. The frequency is low and is not appropriate to simulate the high frequency small elastic vibration cycles of mechanical, acoustic, or aerodynamic origin. On the other hand, in some industries such as the automobile and railway industries, the determination of high cyclic fretting fatigue properties up to 108 or even 109 cycles is necessary. This experiment is bound to be time-consuming and uneconomical. In Bathias’s laboratory, an ultrasonic fretting fatigue test technique at a frequency of 20 kHz has been developed, in which the fretting slip amplitude can be changed without changing the fretting pads. Experiments were performed on a high strength steel and the results were analyzed. The fretting pad has a cylindrical gauge profile. It is made of the same materials as the specimen. A pair of opposing pads are held on the sides of the specimen by springs. Figure 3.23 shows a schematic diagram of an experimental set-up, consisting of two parts. The first part is the ultrasonic fatigue test machine that has been widely used in fatigue tests for both endurance and crack propagation. Each element in the machine is designed to have a resonant frequency of about 20 kHz and an automatic unit maintains the whole system operating at the resonant frequency. The second part is a fixture to hold the two cylinder pads pressed against the specimen by two springs. The normal contact force is measured and controlled by the displacement of the springs. Moreover, the use of the springs eliminates a discernible changes in load should wear occur. The whole experimental system is controlled by a PC. The specimen of ultrasonic fretting fatigue has a cylindrical form with uniform section and is longitudinally asymmetric to amplify the fatigue stress in the gauge length (see the distribution of the vibration displacement and stress in Figure 3.24). The specific length L is determined by the requirement that the resonance frequency of the specimen in the first mode of longitudinal vibration is 20 kHz: Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.23 ting fatigue.

Schematic experimental system for ultrasonic fret-

p  L1 L2  S þ k S1 S2 pffiffiffiffiffiffiffiffiffiffiffi where k is a material constant, k ¼ 2pf r=Ed , S is the section area of the cylinder. Figure 3.25 shows the details of the fretting system. The test system has the functions of regulating test parameters and recording the relative slip amplitude, normal force, and stress. By changing the position of pads along the specimen axis, the desired value of relative slip amplitude can be obtained. The stress of fretting fatigue test is determined by the position of the pads for the load ratio R ¼
1. For R >
1, the total stress is the superposition of static L ¼ X1 þ X2 ¼

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.24 Pad and specimen.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.25

Fretting system for piezo-electric machine.

stress applied by the traction machine and the dynamic stress of vibration. The geometry of the pad and a typical specimen are illustrated in Figure 3.25. With this machine, we can choose

Figure 3.26

Fretting system.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

independently the fatigue displacement and stress in a single specimen. According to the position of pads in the specimen axis, the displacement that provokes the fretting can be chosen between 0.1 mm and tens of microns. By a gauge mounted on the specimen at point xi ¼ 0, vibration deformation of the specimen can be measured. Figure 3.26 shows another proposed fretting system. Here, the normal force is applied by a ring and two screws. Two small spheres are used to avoid the tipping of the pads with the screws and to maintain a consistent distribution of the contact pressure on the specimen. 3.9.2. Fretting Wear Testing A study of fretting wear (Mason, 1982) is an early example of the use of ultrasonic fatigue techniques. The purpose was to explore the possibilities of using ultrasonic techniques as a means of achieving accelerated fretting wear testing conditions and to study how the increased severity of the contact conditions would affect the fretting. As described in (Mason, 1982), during testing the vibrating specimen is clamped between two stationary specimens. The upper one is mounted on a traveling yoke that slides on two vertical rails. The desired normal load is applied by simply mounting dead weights on the yoke. As for the lower stationary specimen, it is fixed in position and used only as a support for avoiding high bending stress in the vibrating specimen. It is worth recalling that the reason for Mason’s pioneer work on the ultrasonic fatigue machine was to study fretting wear (Mason, 1982). 3.9.3. Torsion Fatigue Testing In a review article, Stanzl indicated that recently a new technique and equipment have been developed (Stanzl, 1986) that allow one to perform torsion fatigue testing at ultrasonic frequencies. The mechanical parts of the equipment must be designed so that the torsion resonance vibration can be generated (Figure 3.27). Because the shear modulus is smaller than Young’s modulus, all vibrating parts, including the speciCopyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.27 Ultrasonic torsion test equipment in Bathias’s laboratory.

mens, must be smaller in order to obtain the resonance. Besides this difference, the other experimental details such as amplitude measurement and control are much the same as those for the axial ultrasound fatigue loading. The superposition of axial load is possible and has been investigated in experiments on ceramic materials. Superposition of small compressive loads leads to a lifetime twice as long as that of pure cyclic 20 kHz torsion loads because of the increased friction forces (Mayer, 1994).

3.9.4. Three-Point Bending Fatigue Testing The three-point bending ultrasonic fatigue testing system developed in Bathias’s laboratory is illustrated in Figure 3.28 (Bathias, 2002). This system was developed for testing certain aluminium alloy-based metal-matrix composites used in the automobile industry. Figure 3.29 shows the variation of the displacement amplitude from the converter to the specimen, and Figure Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.28 System for ultrasonic fatigue experiments in threepoint bending.

Figure 3.29 Variation of displacement amplitude along the acoustic wave train. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 3.30 tional mode.

Specimen dimension and the first bending vibra-

3.30 shows the geometry of the specimen and the first bending vibrational mode. The solution procedure for the eigenvalue problem (the resonance length L for bending the specimen of Figure 3.30) is given below. The free flexural wave equation for a beam of uniform section is EI

@ 4 uðx; tÞ @ 2 uðx; tÞ þ rhb ¼0 @x4 @t2

ð3:1Þ

EI is the flexural rigidity of the beam, and bh3 12 Separating variables by I¼

ð3:2Þ

uðx; tÞ ¼ UðxÞ sinðotÞ

ð3:3Þ

we have the equation for U(x) @ 4 UðxÞ
k4 UðxÞ ¼ 0 @x4 where  1=4 12o2 r k¼ Eh2 The general solution of Eq. 3.4 takes the form Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð3:4Þ

ð3:5Þ

UðxÞ ¼ C1 sinðkxÞ þ C2 cosðkxÞ þ C3 sinhðkxÞ þ C4 coshðkxÞ ð3:6Þ Considering boundary conditions UðxÞ ¼ Uð
xÞðsymmetryÞ

ð3:7aÞ

ðU 00 ðxÞÞx¼L ¼ 0 ðzero moment at free endÞ

ð3:7bÞ

ðU 000 ðxÞÞx¼L ¼ 0 ðzero shear force at free endÞ

ð3:7cÞ

We can determine constants C1, C3 C1 ¼ C3 ¼ 0

ð3:8aÞ

and have the relations C4 ¼ C2

cosðkLÞ coshðkLÞ

ð3:8bÞ

tanðkLÞ þ tanhðkLÞ ¼ 0

ð3:8cÞ

So, we have from Eq. 3.6   cosðkLÞ UðxÞ ¼ C2 cosðkxÞ þ coshðkxÞ coshðkLÞ

ð3:9Þ

Other conditions the vibration mode must satisfy are Uð0Þ ¼ A0

ð3:10aÞ

UðL0 Þ ¼ 0

ð3:10bÞ

These give  C 2 ¼ A0

coshðkLÞ cosðkLÞ þ coshðkLÞ



Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ð3:11aÞ

Table 3.1 Dimensions of Ultrasonic Fatigue Specimen of Threepoint Bending h  b (mm2) SiCp=2124

SiCw=AC4CH

SiCw=AC8C

Al2O3=AS7G06

cosðkL0 Þ þ

4 4 8 4 4 8 4 4 8 4 4 8

2L0 (mm)

2L (mm)

E (GPa)

r (kg=m3)

21 21 30 20 20 28 20 20 28 19 19 27

38 38 54 36 36 50 36 36 51 35 35 49

131

2800

101

2800

110

2800

93

2750

7 10 5 7 10 5 7 10 5 7 10 5

cosðkLÞ coshðkL0 Þ ¼ 0 coshðkLÞ

ð3:11bÞ

Substituting Eq. 3.11a into 3.9 we find the solution 

 coshðkLÞ UðxÞ ¼ A0 cosðkLÞ þ coshðkLÞ   cosðkLÞ coshðkxÞ  cosðkxÞ þ coshðkLÞ

ð3:12Þ

Resolution of transcendental Eqs. 3.8c and 3.11b give the resonance length of the specimen  2 1=4 Eh 2L ¼ 0:506925 ð3:13aÞ rf 2 and  2 1=4 Eh 2L0 ¼ 0:27966 rf 2

ð3:13bÞ

Table 3.1 presents the dimensions of three-point bending ultrasonic fatigue specimen. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

4 S-N Curve and Fatigue Strength

4.1. INTRODUCTION The ultrasonic fatigue technique can be used in traditional fatigue testing, often more economically and efficiently than other techniques. However, only ultrasonic technique is practical for reaching very high number of cycles of fatigue load, for example in the gigacycle regime. Chapters 4 and 5 give some experimental and numerical results obtained by ultrasonic fatigue technique. They are mostly related to the S-N curve and fatigue limit as well as the crack growth and threshold in terms of the stress intensity factor (SIF). Some results are compiled to provide a database for practical and industrial use. The materials studied by ultrasonic fatigue (typically 20 kHz) include:
Ferrous materials 4240U, 4240R, SGI52, Cr-V, Cr-Si, steel 304, steel 17-4PH, mild steel
Titanium alloys Ti6246, T6A4V
Nickel alloys Udimet 500, Inconel 706, N18

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Aluminium alloys AC4CH, AC8C, Al2024, AlSiII, Al6061-T6, Al-Li8090
Polycrystalline copper. These high-performance materials with a variety of microstructures are widely used in many industries such as aeronautics, aerospace, automotive, and railway. They are important materials in manufacturing key equipment such as helicopter cyclic trays, turbine engines, cryogenic pumps, disks, and blades, some of which do operate in ultrasonic vibration conditions. Test environments vary greatly. The load ratio R ranges from –1 to 0.9, and the temperature from as low as 20 K to as high as 700 C. The necessity for testing in such wide ranges of environment has yielded not only the state-of-the-art of ultrasonic fatigue technique, but also a practical guide for researchers and engineers. Safe-life design based on the infinite-life criterion was initially developed in the 1800s through the early 1900s, an example being the stress-life or S-N approach related to the asymptotic behavior of steels. Many materials display an apparent fatigue limit or ‘‘endurance’’ limit at a high number of cycles (typically >106). Other materials do not exhibit such a limit, but instead display a continuously decreasing stress-life S-N curve, even at a great number of cycles (106 to 109). Therefore, each point of an S-N curve is more appropriately designated as a fatigue strength at a given number of cycles (Figure 4.1). Time and cost constraints usually rule out the use of conventional fatigue tests for more than 107 cycles to evaluate structural materials. In contrast to conventional fatigue tests that require a long duration of test to reach such high numbers of load cycles with low frequencies (typically <50 Hz), the piezoelectric fatigue technique described herein operates at much higher frequency (about 20 kHz). Therefore the time required for measurements in the high cycle fatigue range is reduced to a small fraction of that required for a conventional test.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.1

ASM handbook S-N data and fatigue limit modeling.

The form of the S-N curve between 106 and 1010 cycles is another method that can be used to help in the prediction of risk in fatigue cracking. Since Wohler’s work, the S-N curve has been conventionally represented by a hyperbola or one of its modifications as indicated below: Hyperbola: Wohler: Basquin: Stromeyer:

ln ln ln ln

Nf Nf Nf Nf

¼ ln a  ln sa ¼ a  bsa ¼ a  b ln sa ¼ a  b ln (sa  c)

The extrapolation of the life range between 106 and 109 cycles has created safer modeling. For the reasons stated above, it is necessary to apply an accelerated fatigue testing method or ultrasonic fatigue technique to investigate the full behavior of the S-N curve. We give the results of gigacycle fatigue of several typical alloys in the following discussions.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

4.2. FERROUS MATERIALS The S-N curves of steel are said to be asymptotic after 106. In fact, the decrease between megacycles (106) and gigacycles (109) continues up to 109 cycles; there is no proof to date that this will not also happen beyond 1010 cycles, even for carbon steels. To investigate the fatigue behavior in this range, several ferrous materials have been tested by the ultrasonic fatigue method. High strength steels and spring steels with tensile strengths ranging from 1000 MPa to 1800 MPa were tested between 105 and 109 cycles through ultrasonic fatigue test at 20 kHz. The S-N curves for the steels were obtained, indicating that the specimens continued to fail over 107, even 108 stress cycles. The mechanisms of fatigue cracks have been determined by microfractographic analyses with the scanning electron microscope. The fatigue strength is also estimated by the Murakami model, which has been applied to the fatigue of many high strength steels with nonmetallic inclusions or small defects. The model does not fit the data well for the gigacyclic fatigue; therefore, modified empirical formula has been proposed to predict the high cycle fatigue life of high strength steels (Bathias, 2001). Investigations on long fatigue lives (>107 cycles) are relatively rare. The reason is obvious; the time and costs are prohibitive to perform the fatigue tests over 108 cycles using a conventional testing machine. The experimental results have shown that when fatigue fracture does occur beyond 107 cycles in the steels, the origin of this fracture is not at the surface but at the interior of the specimen. Indeed, in the high and low cycle regimes, the sites of fatigue crack initiation are different. In the HCF (>107 cycles) regime, the initiation sites were found at non-metallic inclusions located in the interior of the specimen. The initiation sites were found at the surface for higher stress low cycle fatigue. A modified Murakami model, which evaluates the effects of non-metallic inclusions and small defects as well as the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Vickers hardness for a specified number of cycles, was proposed in this study. This model predicts the fatigue strength of high strength steels more accurately (see Sections 4.2.1 and 7.5.2). 4.2.1. High Strength Steel Two low-alloy–high-strength steels, 4240U and 4240R, have been studied (Wang, 1998). Specimens are characterized by the differences in the S-content, viz 0.024 wt% for 4240U, and 0.087 wt% for 4240R, and in the tempering temperature, viz 600 C for group B and 425 C for group C. The details are listed in Tables 4.1 through 4.3. The specimens were tested at ultrasonic fatigue frequency 20 kHz with a stress ratio R ¼ 1 under load control. The samples were polished by using #500, 1200, 2400, and 4000 papers. The central part of the specimen was cooled by compressed air and the temperature was kept at about 70 C. The fatigue results obtained are presented in Figure 4.2 and 4.3. Table 4.1

Chemical Compositions of Materials (wt%)

C

Mn

P

S

Si

Al

Ni

Cr

Cu

Mo

4240U 0.428 0.827 0.012 0.024 0.254 0.023 0.173 1.026 0.21 0.224 4240R 0.412 0.836 0.015 0.087 0.242 0.023 0.186 1.032 0.209 0.164

Table 4.2

Heat Treatments Austentization: 950 C; Oil quenching; Temper: 600 C Austentization: 950 C; Oil quenching; Temper: 425 C

Rep B Rep C

Table 4.3

Mechanical Properties

4240U-Rep B 4240R-Rep B1 4240U-Rep C 4240R-Rep C1

E (Gpa)

r (Kg=m3)

sm (MPa)

HV (30)

211 211 216 216

7820 7820 7870 7870

1100 1040 1530 1485

345 320 465 450

sm ¼ yield strength corresponding to upper yield point.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.2

S-N curves of 4240U.

The experimental results show that the fatigue failure of high strength steels may occur beyond 107 cycles. There is apparently no horizontal asymptote between 106 and 109 cycles, where the fatigue limit decreases by 60 MPa. However, the scatter of the results seems very large. In fact, two distinct initiation mechanisms are acting: up to 106 cycles the crack initiation occurs at the surface; beyond 108 cycles, the initiation takes place at the interior. The fatigue life can be substantially different depending on the mechanisms (Wang, 1998). A typical subsurface crack initiation site in the 4240 low alloy steel is shown in Figure 4.4. The stages of crack initiation, stable crack propagation, unstable crack propagation, and final failure are well defined. Fracture surfaces at all the subsurface crack initiation sites appeared flat and smooth. The fracture origin was identified by use of energy Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.3

S-N curves of 4240R.

dispersive analysis. In the high-cycle regime (>107 cycles), all the initiation sites were found at non-metallic inclusions located in the interior of the specimen. The chemical composition of most inclusions was sulphide. The sizes of the inclusions range from 10 to 40 mm.

Figure 4.4

Subsurface initiation in 4240 steel.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

There are few models that can predict the effect of nonmetallic inclusions on fatigue strength. This may be because adequate and reliable quantitative data on non-metallic inclusions are hard to obtain. Murakami and co-workers (1994, 2002) have investigated the effects of defects, inclusions, and inhomogeneities on fatigue strength of high strength steels, and expressed the fatigue limit as a function of Vickers hardness and the square root of the projection area of an inclusion or small defect. Their formula is   CðHv þ 120Þ 1  R a sw ¼ pffiffiffiffiffiffiffiffiffiffi 1=6 2 ð areaÞ pffiffiffiffiffiffiffiffiffiffi where Hv is Vickers hardness (in Kgf=mm2), area in mm, C ¼ 1.43 for surface inclusion or defect, 1.56 for interior inclusion or defect, 1.41 for inclusion or defect just below surface, and a ¼ 0.026 þ Hv104. This model does not specify the effect of number of cycles. Some materials, such as most non-ferrous and aluminium alloys, do not exhibit a fatigue limit. Instead their S-N curves continue to drop at a slow rate at a high number of cycles. For these materials, the fatigue strength rather than the fatigue limit should be reported. We propose below an empirical formula that does include the number of cycle to estimate high cycles fatigue life of high strength steels   bðHv þ 120Þ 1  R a sw ¼ pffiffiffiffiffiffiffiffiffiffi 1=6 2 ð areaÞ where b ¼ 3.09–0.12 ln Nf for interior inclusion or defect and b ¼ 2.79–0.108 ln Nf for surface inclusion or defect. Table 4.4 compares the fatigue limits predicted by Murakami model and the modified model with experimental data for high strength steels and nickel base alloys. 4.2.2. Spring Steels Two spring steels are tested to obtain their gigacycle fatigue strength. Their chemical compositions and mechanical properties are listed in Tables 4.5 and 4.6. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.4

Hv Nfffiffiffiffiffiffiffiffiffiffi p area h (mm) sexp sw (1) sw (2) Err(1)% Err(2)%

Comparison of Predicted Fatigue Strength with Experimental Data

4240C-5

4240C-3

4240C-11

4240B-10

4240C1-7

SUP9TM1 [8]

SUP10M 3[8]

SUP10M 6[8]

N18 [9]

Cr–Si

465 5.75e8 20 900 760 555 724 27 4.7

465 8.76e7 16 135 740 575 787 22 6.4

465 7.12e8 13 25 750 595 775 21 3.3

345 4.92e5 25 0 630 390 592 38 6.0

450 2.59e5 20 0 760 495 763 35 0.4

445 4.5e5 60.1 0 588 408 621 30.6 5.6

550 2.0e7 14.1 0 862 673 862 21.9 0

554 1.63e6 28.9 240 883 589 902 33.3 2.2

445 1.45e7 53 350 550 417 588 26 6.9

500 1.7e8 25 650 780 566 762 27 2.3

sexp ¼ experimental fatigue strength (MPa); sw(1), sw(2) ¼ fatigue strengths estimated by Murakami model and modified model; Err% ¼ (sw  sexp)=sexp.

Table 4.5

Chemical Compositions of Spring Steels (wt%) C

Si

Cr

V

Mn

S

P

Cr–V 0.48–0.53 0.1–0.4 0.8–1.1 0.15 min 0.7–1.0 0.04 max 0.035 max Cr–Si 0.51–0.58 1.2–1.6 0.6–0.8 0.6–0.8 0.04 max 0.035 max

Table 4.6

Mechanical Properties of Spring Steels UTS (MPa)

E (Gpa)

r (kg=m3)

Ef %

Hv

1800 1800

210 210

7850 7850

35

435 500

Cr–V Cr–Si

Specimens are in the form of normalized hot-cooled 6.5 mm diameter wire. The Cr-V and Cr-Si wires are suitable for service under shock loads at moderately elevated temperatures, and the latter has better relaxation resistance and can work at temperatures as high as 245 C. Tests are performed at 20 kHz frequency with a stress ratio R ¼ 1 under load control and at ambient temperature. During testing, the middle section of the specimen is cooled by compressed air and the temperature is kept at about 70 C. The fatigue strength is determined in the life range of 106  109 cycles as shown in Figures 4.5 and 4.6.

Figure 4.5

S-N data of Cr-V steel at R ¼ 1.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.6

S-N data of a spring steel Cr-Si steel at R ¼ 1.

Fatigue crack initiations are observed at the sites of internal defects for fatigue life beyond 107 cycles. Using the Paris law, the number of cycles to propagate a crack from the defect to the surface (Np) can be estimated, assuming the stress intensity factor to be: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi DK ¼ 0:5Ds p area around an internal non-circular defect and 2 pffiffiffiffiffiffi DK ¼ Ds pa p for a circular crack, the number of cycles to the initiation, Ni, is estimated as: Ni ¼ Nf  N p The portion of fatigue life contributing to crack initiation is estimated to be greater than 90% in the high cycle regime for these steels (Wang, 1998). Fatigue fractures beyond 107 cycles were observed. The fatigue strength at 107 cycles is 860 MPa for Cr–Si steel, and better than 810 MPa for Cr–V steel. At 109 cycles, the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

strength is about 800 MPa for Cr–V steel and 770 MPa for Cr–Si steel. Fatigue life experiments on ferrous materials show an important difference in crack initiation between the gigacyclic fatigue and the low cycle fatigue. In the latter, crack initiation is the result of local plastification around surface discontinuities.The local plastic deformation brings about the multiplication of dislocations due to cyclic hardening of the material. Contrary to this, in the gigacyclic fatigue, the site of crack initiation is observed at the interior rather than at the surface. An explanation may be that, in the gigacyclic fatigue, the applied stress is too small to provoke a cyclic plastic zone localized at the surface, and the internal defects are likely to become the main sources of crack initiation. In the case of ferrous materials, the crack initiation from inclusions takes place 50 mm  1000 mm under the surface, which is followed by nucleation and circular propagation of micro-cracks. The diameter of the initiation zone is about 50 mm  100 mm. A macro-crack is formed and propagates circularly until this zone approaches the specimen surface. From these initial observations, we concluded that the mechanism of gigacyclic fatigue needs to be further explored. For additional comparison of test results, the fatigue life experiments of other ferrous alloy (i.e., steel 17-4PH and 12%Cr) are discussed in the next section. 4.2.3. Martensitic Stainless Steels Steel 17-4PH is also tested at 20 kHz and R ¼ 1, with the maximum strain controlled constant in the middle section of the specimens (Bathias, 2001). Some samples are used for conventional fatigue tests with a loading frequency of 20 Hz  50 Hz under a push–pull stress alternating cycle. Another martensitic stainless steel, 12% Cr steel, is tested with the same conditions in order to compare the results with 17-4PH steel data. Chemical composition and mechanical properties after quenching and tempering (620 C for 2H) are given in Tables 4.7 and 4.8. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.7 C 0.11

Chemical Composition of 12% Cr Steel (wt%)

Si

Mn

P

S

Cr

Mo

Ni

Al

0.11

0.77

0.01

0.002

11.4

1.47

2.53

0.064

Table 4.8

Mechanical Properties 12% Cr Steel

E (Gpa)

UTS (MPa)

sy (MPa)

216

958

829

Ef

Ra

64%

18%

Figure 4.7 gives the gigacycle fatigue S-N curves for these two martensitic stainless steels. It is shown that failure can occur between 109 and 1010 cycles in both of these materials. The location of the initiation is on the surface up to 107 cycles and in the subsurface beyond this life. The results obtained here do seem to suggest that there is no lower limit of fatigue strength, even for ferrous alloys, and that they do

Figure 4.7

Experimental results of 17-4PH and 12% Cr steels.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.8

Gigacycle initiation in 17-4 PH steel.

undergo rupture even after a large number of cycles at low fatigue stresses. 4.2.4. Bearing Steels Two bearing steels were tested in gigacycle fatigue; their chemical compositions and mechanicals properties are given in Tables 4.9 and 4.10. Specimens were tested at 20 kHz with a load ratio R ¼ 1. The number of maximum testing cycles is limited to 1010 so that, when cycles pass 1010, the testing machine will stop automatically. Since heat is induced in the specimen by absorption of ultrasonic energy, during the testing process

Table 4.9 C

Chemical Compositions of SUJ2 and 100C6 (wt%) Si

Mn

P

S

Cr

Cu

Ni

Mo

SUJ2 1.01 0.23 0.96 0.012 0.007 1.45 0.06 0.04 0.02 100C6 0.35–1.1 0.15–0.35 0.20–0.4 20.025 20.015 1.35–1.60 20.10

Table 4.10 Mechanical Properties of SUJ2 and 100C6

SUJ2 100C6

E

UTS (MPa)

r

Hv30

210 213

2316 2500

7.86 7.45

778

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Figure 4.9 Fatigue results for 100C6 and SUJ2 bearing steels up to 1010 cycles.

the middle of the specimen is cooled by compressed air and the temperature is kept at 70 C in the megacycle regime and at 25 C in the gigacycle regime. The stress concentration effect was studied with notched specimens. ANSYS was used to calculate the stress field at the root of the notch. The diameters of the notch specimen are 9.2 mm and 6.4 mm. The results are given in Figure 4.9. Several specimens failed between 109 and 1010 cycles with crack initiation at an internal inclusion. A notch effect is observed in the gigacycle regime. The fatigue strength of 100C6 steel at 1010 cycles is 800 MPa without notch and 600 MPa with notch. Also a large scatter of the data is observed at these high cycles. 4.2.5. Low Carbon Ferritic Steel Thin Sheets Our piezo-electric fatigue machine was adapted to test specimens of a small thickness, less than 1 millimeter. Results for a low carbon steel are given here (Wang, 1993). The chemical composition and the mechanical properties of this low carbon steel is given in Tables 4.11 and 4.12. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.11 Chemical Composition of Low Carbon Steel (wt%) Co 0.14

C

Mn

Si

P

S

Al

0.08

0.4

0.1

0.025

0.025

0.02

Table 4.12 Mechanical Properties of Low Carbon Steels E (GPa) 203

Figure 4.10

sy (MPa)

UTS (MPa)

Ef

r

Hv5

225

340

36

7.83

95

S-N curve for a low carbon steel at 20 kHz, R ¼ 0.1.

The specimens, after polishing, are tested with a load ratio R ¼ 0.1. The temperature of the specimen during the test is kept between 50 C and 70 C. The results in Figure 4.10 show that a low carbon fenitic steel can fail at 5  108 cycles and do not confirm an asymptotic nature of S-N curve beyond 109 cycles. The fatigue strength at 109 cycles is close to 220 MPa. 4.2.6. Austenitic Stainless Steel In addition to a ferritic steel, an austenitic 304 stainless steel was tested in the gigacycle regime. In the test, the specimens were cooled with tap water to prevent temperature elevation. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.11 Quasi-asymptotic S-N data for 304 stainless steel.

The results presented in Figure 4.11 show that there is an apparent asymptote between 106 and 109 cycles. Some specimens failed beyond 108. However, since the surface of the specimens was not polished, the scatter is large. The fatigue strength at 109 cycles is 198 MPa. For this steel, the initiation was always observed at the surface. 4.2.7. Spheroidal Graphite Cast Iron In the automotive industry, the designed fatigue life of components often exceeds 109 cycles. Spheroidal graphite iron or ductile cast iron is a favored material for fabrication of some of these components because of its exceptional combination of high strength and ductility. In the literature, few data on the S-N curve of spheroidal graphite cast iron have been obtained beyond 107 cycles, as the test time and cost to perform fatigue tests of over 108 cycles using a conventional fatigue machine are extremely high (Wang, 1998). The chemical composition and mechanical properties are listed in Tables 4.13 and 4.14. High cycle fatigue S-N data of SGI52 at R ¼ 1 and R ¼ 0 are presented in Figures 4.12 and 4.13. For zero mean stress R ¼ 1, the results show no noticeable frequency effect on the fatigue behavior between 25 Hz and 20 kHz. The ultrasonic fatigue data closely match the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.13 Chemical Composition of Spheroidal Graphite Cast Iron SGI52 (wt%) C

Si

Mn

S

P

Mg

Cu

Ni

Mo

Ti

Cr

Sn

3.45 3.21 0.13 0.019 0.031 0.031 0.024 0.59 0.013 0.043 0.02 0.030

Table 4.14 Mechanical Properties of Spheroidal Graphite Cast Iron SGI52 E (GPa)

sy (MPa)

UTS (MPa)

Ra

r (kg=m3)

Hv

380

510

14.5

7100

184

179

sy ¼ yield strength corresponding to 0.2% offset; Ra ¼ fraction of reduction in area from a tensile test.

conventional fatigue data. At R ¼ 0, however, fatigue strength in ultrasonic fatigue tests seems to be slightly higher than that in conventional fatigue tests. It is evident that fatigue failure can occur over 107 cycles, and the maximum fatigue stress smax continues to drop with the increasing number of cycles between 106 and 109. It is also interesting to investigate the temperature of the specimens during the test. Heating in the specimens is caused by absorption of ultrasonic energy. During the testing process, the temperature in the middle section of the specimen

Figure 4.12

S-N data of SGI52 (R ¼ 1).

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.13 S-N data of SGI52 (R ¼ 0).

is controlled between 50 C and 90 C for tests at R ¼ 0, and 70 C and 120 C for tests at R ¼ 1. Temperature evolution (Figure 4.14) at R ¼ 0 consists of two periods: Period 1: There is a steep rise near 106 cycles and the maximum temperature depends on the amplitude of ultrasonic fatigue loading. Period 2: There is a horizontal curve over certain numbers of cycles. This may be interpreted as an equilibrium state between the dissipation of ultrasonic energy due to interior crack nucleation and the heat induced by interior friction of the material. After the maximum temperature is reached, no significant decrease is observed over certain numbers of cycles when the initiation of crack takes place on the surface. When the initiation occurs in the interior of the specimen, the maximum temperature probably corresponds to the nucleation; this is an illustration of gigacycle fatigue. Compare the temperature changes for smax ¼ 305 and 360 MPa, in Figure 4.14. 4.3. ALUMINIUM MATRIX COMPOSITE Generally, for fatigue life up to 107 cycles, the aluminium matrix composite materials reinforced by fibers have a higher Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.14

Temperature effect.

fatigue limit than those reinforced by particles. Comparing composites with alloys, the resistance gained can reach 20% when the processing is done with powder metallurgy. Some results (Bathias, 1996) show that the composite Copyright © 2005 by Marcel Dekker. All Rights Reserved.

2080=15%SiCp has a better fatigue resistance than alloy 7075T73 used for the manufacturing of helicopter cyclic trays; this is illustrated in Figure 4.15. However, when the S-N curves of Figure 4.15 are compared, we would expect that these curves cross each other in the range of 106 to 107 cycles; it would be interesting to explore the gigacyclic fatigue life. Particular attention must be given to the form of fatigue curves. A lot of fatigue tests are limited to 106 cycles because machining specimens of Metal Matrix Composites (MMC) are very expensive. Some tests by Jones (1991) carried out up to 107 cycles show the evidence that the S-N curve of 7075 is much flatter than that of its alloy. The quasi-hyperbolic shape in the semi-logarithmic plot of the S-N curve for aluminium alloys does not fit the S-N curve of the MMC. In order to verify the forms of the S-N curves, Japanese researchers performed fatigue tests on some MMCs until 108 cycles in rotating bending (Masuda, 1994). They showed that no horizontal asymptote could be determined between

Figure 4.15 Comparison of S-N curves: (A) 7075 alloy (full curve); (B) 2080=15%SiCp composite (filled circles). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

106 and 108 cycles, and that some ruptures occured between 107 and 108 cycles. Our laboratory performed experiments on some aluminium composites reinforced with 17% of SiC by squeeze casting (Bathias, 1994). The compositions of the matrices and mechanical properties of the materials are listed in Tables 4.15 and 4.16. The fatigue tests were carried out on prismatic bars in three-point bending at 20 kHz frequency, ambient temperature, and R ¼ 0.1. The results presented in Figure 4.16 show that some specimens were broken in fatigue between 108 and 109 cycles. Evidently, it is not possible to draw a horizontal asymptote between 106 and 109 cycles. The fatigue limit defined between 106 and 107 in the conventional standard does not seem to exist in gigacyclic fatigue tests. When the S-N curve of alloys is compared to the S-N curve of composites, it is observed again that the resistance of alloys is higher in low cycle fatigue and lower in gigacyclic fatigue. From these observations we propose an empirical formula for a good estimate of the equation of the S-N curves for R ¼ 0.1 in the (following) form smax ¼ UTS  Nfc

Table 4.15

AC4CH AC8C

Compositions of the Matrices of 17%SiC Cu

Si

Mg

Zn

Mn

Ti

0.113 2.98

7.124 10.048

0.338 1.126

0.007 0.023

0.008 0.017

0.132 0.002

Table 4.16 Mechanical Properties of 17% SiC

AC4CH AC8C

UTS (MPa)

E (GPa)

762–789 814–840

101 110

Hv 150–160 210–226

UTS ¼ ultimate tensile strength; Hv ¼ Vickers hardness.

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Figure 4.16 Fatigue curves for MMC, R ¼ 0.1 (Masuda, 1994; Bathias, 1996).

4.4. NON-FERROUS ALLOYS To explore the gigacyclic fatigue behaviors of other metallic alloys, several titanium and nickel alloys were selected as examples. 4.4.1. Titanium Alloys Titanium alloys play an important role in the aerospace industry. It is generally accepted that titanium alloys behave like steels in gigacyclic fatigue. This section examines structure–fatigue properties in titanium alloys (Bathias, 1994; Jago, 1996, 1998) in which high cycle fatigue behaviors have been shown to be significantly affected by microstructure. Microstructures that have a small probability of low-stress crack initiation, as in b-processed microstructures, generally yield the best high cycle fatigue limit and tensile resistance. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In this section we examine the effects of four thermo-mechanical processes on stage I fracture mode in the Ti6246 (Ti-6Al2Sn-4Zr-6Mo) alloy—an a þ b titanium alloy used for compressor disks and blades. The fatigue test programs are so arranged that fatigue properties are examined in the range of 107  109 cycles with R ¼ 1. The characteristics of fracture mechanism in this alloy are examined by SEM (scanning electron microscope) observations of fracture surface of each broken fatigue-limit specimen. Microstructure features are characterized by quantitative examination in the maximum stress plane and the plane perpendicular to it. Chemical compositions and microstructures obtained by different thermal processes are given in Tables 4.17 and 4.18. Quantification of the morphological aspects has been performed to provide a comprehensive description of various microstructures. Two orthogonal metallographic surfaces are examined. The number of whole particles detected is more than 2000. A global image analysis measures primary a-phase volume fraction, total a-phase volume fraction, thickness of primary a platelets, and mode distance between coarser particles. Size and shape measurements are analyzed individually. This procedure provides, for example, the perimeter, the area, and the longest dimension of each particle. Tensile tests have been performed in each TP condition. The strain rate is equal to 8.4  105 s1 for all tests. Finally, the yield strength, ultimate tensile strength, elongation, and reduction of area have been measured.

Table 4.17 Chemical Composition of Ti6246 TP Al Sn Zr Mo C Cu Si Fe O H N (No.) (wt%) (wt%) (wt%) (wt%) (ppm) (ppm) (ppm) (ppm) (ppm) (ppm) (ppm) 1(1) 5.76 1.97 4.08 3.97 1(2), 2 and 3 5.68 1.96 4.08 3.92

90

<50

<50

400

930

44

80

83

<50

<50

300 1100

28

70

TP ¼ Thermo-mechanical process.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.18 Ti6246

Thermo-mechanical Processes and Microstructures of

TP Cast number Forging (No.) (RMI) condition 1(1) 972886 and 1(2) 2 982560

3

982560

Thermal treatment

Final microstructure type

955 C=WQ 935 C=2H=WQ þ a platelets and (Tb þ 10 C) 905 C=1H=Air þ b-transformed 595 C=8H=Air matrix 955 C=WQ 935 C=2H= Coarse platelets (Tb þ 10 C) Slow cool Room T. and b-transformed þ 595 C=8H=Air matrix Bi-modal structure 905 C=Air 935 C=2H=WQ (a platelets and (Tb  40 C) þ 905 C=1H=Air a nodular) and þ 595 C=8H=Air b-transformed matrix

WQ ¼ Water quench.

All histograms of data exhibit the typical lognormal distribution. Average and standard deviations for each microstructure feature are given in Table 4.19. Despite standard deviations, it is seen, on one hand, that primary a platelets are significantly coarser in TP2 specimens than in TP1(1) and TP1(2) specimens. That is, the average particle thickness, longest dimension, and area are larger in TP2 microstructure. In addition, the TP2 process increases the primary a volume fraction and consequently decreases the mean length between primary a particles. On the other hand, the mean particle perimeter is longer in the TP1(2) microstructure because of an a platelet shape difference with TP1(1) and TP2 in lamellar phases. The bi-modal TP3 microstructure corresponds to a mixing of coarse nodular primary a-phase (mean width, 3.1 mm; volume fraction, 20%) and fine lamellar primary a-phase in b-transformed matrix (Figure 4.17). As for high cycle fatigue resistance at room temperature, the role of the secondary a volume fraction seems to manifest as an influence on yield strength without any differences between lamellar and duplex structures in contrast with the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.19

Quantitative Microstructure Features of Ti6246 Alloy fap(%)

TP condition (No.)

lam

1(1)

dap (mm)

Pap=Sap (mm1)

db (mm)

fas(%)

lam

54  2

13  4

1.7  0.2

0.8  0.2

2.9  0.9

Axial view: 1(1)

37  2

15.8  4

1.7  0.2

2.5  0.2

2.6  0.8

Radial view: 1(1)

42  4

14.8  8

1.7  0.4

1.9  0.4

2.7  1.7

14.1  10 7.3  10 20  10

1.9  0.4 2.2  0.4 1.0  0.4

2.4  0.4 0.74  0.4 1.6  0.4

2.2  1.5 1.7  1.6 1.16  0.8

Volume: 1(2) 2 3

43.9  4 66.2  4 20  4

glob

27.3  4

glob

3.1  0.4

lam

glob

4.3  0.4

lam

glob

4.5  3.2

fap ¼ primary a-phase volume fraction; fas ¼ secondary a-phase volume fraction; dap ¼ thickness of primary a platelets; db ¼ mode distance between coarser particles; Pap ¼ perimeter of particle; Sap ¼ area of particle; lam ¼ lamellar microstructure; glob ¼ globular microstructure.

Figure 4.17 Microstructure of 6246 Ti alloy.

results of 77 K tests (that will be presented in Section 4.2.4.1). At low temperatures, the presence of a coarse nodular phase significantly reduces the fatigue limit resistance. Thus, at cryogenic temperature, fas cannot be considered to be the sole influence in a þ b processed materials. Figure 4.18 presents high cycle fatigue results of Ti6246 at room temperature, which demonstrates a significant difference in S-N curves among different TP conditions. The TP3 material has the highest fatigue resistance of 510 MPa at 109 cycles, while the TP1(1) shows a slightly lower fatigue limit that is estimated as 490 MPa, and TP1(2) samples give a much lower limit of only 400 MPa . TP2 material exhibits the lowest fatigue limit of 325 MPa. At room temperature, the 0.2% offset yield stress and the fatigue limit stress mainly depend on the secondary a-phase volume fraction. The final aging process, producing the secondary a precipitation, is the same for all TP conditions and gives an identical average width for the small needles. The experimental results also show a greater frictional stress for the primary a-phase than for b-phase. In comparison with Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.18 S-N curves of Ti6246 at room temperature for different TP conditions.

the yield stress, precipitation strengthening has a more significant effect on fatigue limit resistance. In addition to the above conclusions, the Ti6246 test data, like gigacyclic fatigue results of other materials, show that:
Some fatigue ruptures occur at least up to 109 cycles
There is no asymptote
Fatigue limit at 109 cycles is much lower than the conventional fatigue limit at 106 cycles. Another titanium alloy T6A4V was also tested at 20 kHz and R ¼ 1 with the maximum strain kept constant in the middle section of the specimens (Bathias, 1994). Chemical compositions, heat treatments, and mechanical properties of this material are included in Tables 4.20 through 4.22. Some samples were used for conventional fatigue tests Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.20 C U500

Compositions of U500 and T6A4V (wt%) Si

Mn

Al

Cr

Cu

Fe Mo Ti

Zr

Co

Ni

0.12 0.75 0.75 3.0

18

0.7

4.0

4.2 3.0 0.06 19 Remains

V

H

Y

C

Al

Fe

T6A4V 3.5 4.5

O

Ni

Ti

0.25 0.13 0.05 0.01 0.005 0.08 5.5 6.5

Table 4.21

Remains

Heat Treatments of U500 and T6A4V

U500

Annealing 5 mn to 1080 C, quenching in oil, aging 24 h to 845 C, cooling in air 16 h to 760 C, cooling in air

T6A4V

Quenching 1 h to 950  970 C, cooling in water, annealing 2 h to 700  10 C, cooling in air, stablizing (during machining) 2 h to 700  10 C

under push–pull stress alternating cycles at 20 Hz  50 Hz. Figure 4.19 gives the ultrasonic fatigue test results in comparison with test data obtained by other laboratories. We notice that for 17-4PH and T6A4V fatigue resistance behavior in ultrasonic fatigue regime seems to be slightly better than that observed in traditional fatigue loading. That is, materials seem to respond differently to ultrasonic fatigue and conventional fatigue. Similar observations have been mentioned in the literature (Bathias, 1994) for other materials. At the time of this investigation, the fatigue behavior of materials was considered to depend upon both the loading regime and mechanical characteristics of the materials. In ultrasonic fatigue tests, the specimen is excited at constant

Table 4.22

Mechanical Properties of U500 and T6A4V

sy UTS E Ed (MPa) (MPa) (MPa) (GPa) U500 T6A4V

826 989

1780 1190

192 108

214 110

ef

Ra

r (kg=m3)

23.5 19 15 42.5

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8020 4420

n

ef

0.29 0.2107 0.3 0.5516

Figure 4.19

Experimental results of T6A4V.

strain amplitude. As is well known, most engineering materials exhibit their maximum cyclic hysteresis properties when excited in push–pull fatigue. There will be softening or hardening according to the mechanical properties of the material. One discovery in the literature (Laird, 1982) is that, in most cases, softening occurs for materials in which the ratio of yield strength sm to the elastic limit s0.2 is less than 1.2, and hardening occurs when this ratio is greater than 1.4. Here we have sm=s0.2 < 1.2 for alloy T6A4V. We may assume that materials loaded in an ultrasonic fatigue regime exhibit the same cyclic hardening characteristics as in conventional fatigue excitation. We will find, in the next section, that alloy U500 with sm=s0.2 > 1.4 hardens under constant strain amplitude and the peak stress level increases. For 17-4PH and T6A4V, the inverse effect occurs; that is, the peak stress level decreases under constant strain cycling. We then examined the fracture surfaces of specimens that failed in ultrasonic and conventional fatigue tests by means of SEM. Figure 4.20 shows some examples of crack initiation and propagation configurations obtained in ultrasonic fatigue loading. It is observed that cracks initiate from either a single Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.20 Crack initiation in ultrasonic fatigue loading.

surface point or a mass defect beneath the surface, then propagate in a fan-shaped fatigue failure configuration. It is evident that the surface condition of specimens tested in ultrasonic fatigue loading influences experimental results due to high exciting frequency and high strain rate. In this study, the specimens used in high frequency tests had been polished mechanically to eliminate micro-cracks on the surface. 4.4.2. Udimet 500 A large family of nickel alloys used in turbine engines is no exception to the rule: some fatigue ruptures occur in the gigacyclic domain and the drop between the megacyclic and gigacyclic fatigue limits is large. Udimet 500 is a typical nickel super alloy whose chemical composition, heat treatment, and mechanical properties are listed in Tables 4.20 through 4.22. (These tables also include the data for T6A4V discussed earlier.) The experimental results (Ni, 1991) at 20 kHz and R ¼ 1 are illustrated in Figure 4.21. The data indicate a decrease of fatigue limit down to 200 MPa. 4.4.3. Aluminium Alloys To close the review of the gigacycle fatigue behaviors of the main alloy families, the cases of cast and wrought light alloys Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.21

S-N data of alloy Udimet 500 (Dentsoras, 1983).

are presented here. As has long been assumed, there is no asymptotic S-N curve endurance limit for aluminium alloys. For 2219 and 6061 alloys, there exists about 100 MPa difference between the two fatigue strengths at 106 cycles and 108 cycles (Tao, 1996) for R ¼ 1. The difference is as large as 200 MPa for 2024–T351 alloy (Stanzl et al., 1993b) at 106 cycles and 109 cycles. It is of interest to note that for cast aluminium and magnesium, this difference is smaller and less than 100 MPa (Figure 4.22) in spite of the presence of porosity in casting. In the gigacycle regime, the initiation site is located inside the specimens for titanium, nickel alloys, and high strength steels. However, the initiation takes place at the surface of specimens in aluminium and magnesium alloys, especially in cast alloys where the void formed near or at the surface is the main location for crack formation. 4.5. ALLOYS AT CRYOGENIC TEMPERATURES We now examine the influence of temperature on fatigue properties of turbine engines or cryogenic pumps. In this Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.22 Aluminium and magnesium alloys (A291) S-N data with R ¼ 1 at 20 kHz (Ds ¼ 50 to 100 MPa) (Stanzl, 1996).

section, some alloys used for manufacturing engines of Ariane pumps are discussed in the domain of the gigacyclic fatigue at the temperatures of 77 K and 20 K. 4.5.1. Titanium Alloys Titanium alloy Ti6246 is tested at 77 K, after the base test data is obtained at room temperature (Jago, 1996; Tao, 1996). In these experiments, four microstructures are tested to determine the effect of microstructure on the fatigue limit at cryogenic temperature. Chemical compositions and different microstructures obtained by different thermal processes are given in Tables 4.17 through 4.19. In liquid nitrogen, high cycle fatigue resistance depends mainly on the primary a grain size and decreases greatly with the presence of coarse globular primary a-phase. The resistance of gigacyclic fatigue is distinctly higher at 77 K than that at room temperature: 640 MPa vs. 490 MPa for the best microstructure (Figure 4.23). There appears to be a significant effect of thermal processing. The lowest fatigue limit of the material TP2 is Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.23 S-N curves of Ti6246 in liquid nitrogen until 109 cycles (T ¼ 77 K, R ¼ 1).

explained by the presence of large primary a platelets due to slow cooling after solution treatment. The highest fatigue limit at 77 K is obtained with a fine microstructure. But it is difficult to explain the difference between materials TP1(1) and TP1(2). TP3 process which yields a duplex structure, does not greatly increase the high cycle fatigue resistance; the difference is only 50 MPa between those at room and liquid nitrogen temperatures. However, we must emphasize that the S-N curves do not have a horizontal asymptote at cryogenic temperatures; some fatigue ruptures are observed beyond 108 cycles depending on the alloy microstructure. We observe similar results in the T6A4V alloy. (Tao, 1996) gives S-N curves of T6A4V in gigacyclic range at 300 K, 77 K, and 20 K. Some fatigue tests in liquid hydrogen Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.24 20 K.

Fatigue results of T6A4V in liquid H2 and He at

carried out up to 108 cycles were extremely difficult. Regardless of the temperature, the environment (air, nitrogen, helium, or hydrogen) and the load ratio (R ¼
1 or R ¼
0.1), the fatigue ruptures can occur between 106 and 109 cycles (Figure 4.24) (Tao, 1996). 4.5.2. Aluminium-Lithium Alloys The main reason for the replacement of conventional aluminium alloys by aluminium-lithium alloys is to obtain the lightweight structures demanded by the aeronautical and space industries. For example, aluminium-lithium alloys have been used in space vehicles for reservoirs of liquid oxygen and hydrogen fuel. The material studied is Al-Li8090. Tables 4.23 and 4.24 list the chemical composition and mechanical properties of this alloy. Table 4.23 Chemical Composition of AL-Li8090 (wt%) Li

Cu

Fe

Si

Mn

Cr

Zn

Ti

Mg

Zr

Al

2.2  2.7 1.0  1.6 0.30 0.20 0.10 0.10 0.25 0.10 0.6  1.3 0.04  0.16 Remains

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 4.24 T ( K) 300 77

Mechanical Properties of Al-Li8090

E (GPa)

sy (MPa)

UTS (MPa)

ef

r (kg=m3)

81 86

455

500

7

2350 2350

Tests were carried out at 77 K in liquid nitrogen and at 20 kHz (R ¼  1) (Tao, 1996). Figure 4.25 gives the S-N curve in comparison with some literature data. There exists a difference greater than 100 MPa between the fatigue strengths at 106 and 108 cycles. The crack initiation at 20 kHz and 77 K takes place mainly under the surface, at the internal flaws of the material. The initiation occurs at the surface as usual if the stress is very high and the fatigue life is less than 106 cycles. The failure surface of an Al-Li8090 specimen tested at high frequency and low temperature shows a very brittle rupture by cleavage, which results from the cryogenic

Figure 4.25 results.

S-N curve of Al-Li8090 in comparison with other

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temperature and elongated microstructure developed by tension during heat treatment. This forms very significant textures.

4.6. N18 ALLOY AT HIGH TEMPERATURE As the final part of this survey, we discuss the case of nickel alloys chosen for manufacturing turbine disks that must be resistant to the low cycle and megacyclic fatigue on different portions of the disks. The N18 high temperature fatigue seems to indicate that nickel base alloys can crack at 109 cycles (Bonis, 1997). Powder N18 is tried—with and without seeding of inclusions—to determine the effect of the inclusions in the gigacyclic domain. The microstructure of turbine disks in powder N18 is very homogeneous in the center with grain size of 7  10 mm. In the periphery the grain is coarser (10  15 mm). The powder is produced by atomization in argon, then sifted to 75 mm. Strengthening is achieved by extruding the powder container. This technique guarantees the high quality, reproducibility and stability of the desired properties of the material. The isothermal forging at 1120 C follows as the formation process. The inclusions in seeded (or polluted) N18 are ceramic particles. There are 30,000 ceramic inclusions of Al2O3 and MgO sifted to 70  80 mm in one kilogram of the alloy. In comparison, the standard N18 has fewer than 20 inclusions per kilogram. The chemical composition, mechanical properties, and heat treatments of the superalloy N18 are given in Tables 4.25 through 4.27.

Table 4.25

Composition of Superalloy N18

Cr (%)

Co (%)

Mo (%)

Al (%)

Ti (%)

Nb (%)

Hf (%)

C (ppm)

B (ppm)

Zr (ppm)

11.5

15.5

6.5

4.3

4.3

<1.5

0.5

200

150

300

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Table 4.26 T ( C) 0 800

Mechanical Properties of Superalloy N18 E (GPa)

n

r (kg=m3)

220 170

0.31 0.29

8000 8000

Table 4.27 Heat Treatments of Superalloy N18 Solution 4 h at 1165 C; quenching in oil 90 s; aging 24 h at 700 C, cooling in air; stabilizing 4 h at 800 C, cooling in air

From the test data in Figure 4.26, several points must be emphasized to characterize the fatigue between 106 and 109 cycles.
The effect of inclusions is sometimes obscured by the effect of porosity when R ¼ –1 or R ¼ 0. The scatter of the results for R ¼ 0 in N18 seeded with inclusions is more pronounced than that for standard N18.
It is very remarkable that, when R ¼ 0 or R ¼ –1, the resistance to the gigacyclic fatigue at 450 C is 250 MPa for the N18 with or without inclusions.

Figure 4.26 S-N data of N18 nickel based alloy at 450 C between 106 and 1010 cycles. Copyright © 2005 by Marcel Dekker. All Rights Reserved.


On the other hand, when the static strain of the fatigue cycle is very high, (R ¼ 0.8), the effect of inclusions becomes significant. Without inclusions, the N18 fatigue limit at 450 C is 155 MPa at 109 cycles. However, with inclusions it is 125 MPa . In practice, such fatigue cycles occur in turbine disks.
Finally, between 106 and 109 cycles, the fatigue limit can decrease by 30% depending on the load ratio R.

4.7. ROTATING-BENDING INTERNAL CRACK STRESS CORRECTION At this point, it is of interest to compare the fatigue curves in rotating-bending and in tension-compression. In Japanese literature, many results had been given for JIS SUJ2, by Sakai (1999), Murakami (2002), and others. It has been found by these Japanese researchers that the occurrence of the internal initiation follows the appearance of a plateau in rotating-bending loads (Figure 4.27). Similar curves have been observed by Nishijima (1999) in other steels tested in rotating-bending loads.

Figure 4.27 S-N curve of SUJ2 bearing steel (R ¼ 1) rotatingbending test (Sakai, 1999). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Although the SUJ2 is not equivalent to 100C6 bearing steel, it seems that the step wise S-N curve is related more to the rotating-bending behavior than to the steel itself. In fact, when the SUJ2 was tested in Bathias’s laboratory in tension-compression at 20 kHz, no step was observed in the S-N curve (Figure 4.28). Thus, it may be assumed that the appearance of this step results from the use of the nominal maximal stress in the S-N plot for the subsurface initiation data. Since the maximum stress in rotating-bending is located at the surface of the specimen, a correction of the stress should be made to the data for the specimens with internal crack initiation accounting for its radial distance to the surface (Figure 4.29). Figure 4.27 shows the curve originally obtained by Sakai (1999), where two straight lines may be drawn: one in the conventional fatigue regime and one in the high cycle fatigue regime, separated by a plateau. A correction to the stress is made for the distance from the inclusion to the surface, which yields a continually decreasing curve. The magnitudes of corrections are observed in the comparison

Figure 4.28 (R ¼ 1).

SUJ2 tested in tension-compression at 20 kHz

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Figure 4.29 Relationships between dinc and inclusion stress correction Ds.

shown in Figure 4.30, where we see corrections up to 110 MPa. Finally, the corrected values may be considered to form part of the S-N curve in tension-compression for the same steel. The aim of this correction is to demonstrate that, when correctly analyzed and plotted, the fatigue data would yield essentially the same S-N curves regardless of the difference in the loading method (tension-compression or rotatingbending, for example). 4.8. Ti-Al INTERMETALLIC ALLOYS The last example given is the gigacycle bending fatigue of Ti-Al intermetallic. A piezo-electric fatigue machine working in three point bending is useful when the metals are not available in large amounts, are expensive, or are difficult to machine. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.30 S-N curve for JIS SUJ2 steel tested at rotating bending fatigue system [plot of original (Sakai, 1999) results with corrected results].

4.8.1. Material The material tested in the investigation is cast Ti-Al alloy. Since Ti-Al alloy has excellent castability, studies have now been undertaken to replace the existing machined or forged parts with cast products of the Ti-Al alloy. However, the most serious problem in using cast Ti-Al alloy parts is its poor reliability and deterioration in mechanical properties. In order to reduce the effect of pores, Hot Isostatic Pressing (HIP) technology has been applied to the material. In the study, the Ti-Al alloy have been treated by HIP at 1200 C, 100 MPa for 4 h, followed by homogenization treatment in vacuum at Copyright © 2005 by Marcel Dekker. All Rights Reserved.

1000 C for 20–24 h. The composite of Ti-Al alloy is shown in Table 4.28. The specimens were machined out from a 100 mm diameter bar by electro discharge machining (EDM) followed by grinding. The specimens were mechanically polished to remove surface scratches before testing. The dimensions of the rectangular specimens were 36  8  4 mm, and the surface roughness was below 0.4 mm. 4.8.2. Mechanical Properties To determine the mechanical properties of the Ti-Al alloy, static tension tests and static bending tests were conducted; Tables 4.29 and 4.30 show the results of these tests. 4.8.3. Fatigue Test High-cycle three-point bending fatigue tests were performed on a resonance fatigue testing machine at a frequency of Table 4.28 Chemical Composition (wt%) of the Material

At(%)

Al

Nb

W

Mo

B

45

1.5

0.05

0.1

0.05

Table 4.29

Static Tensile Test Results

Stress (max) (MPa)

Modulus (MPa) E

Load (kN)

160668.578

5.835

500.544

Table 4.30

Static Bending Test Results

Load (kN)

Stress (MPa)

3702

851.61

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20 kHz and stress ratios of 0.1 and 0.5. All the tests were carried out at an ambient temperature, so as to determine the fatigue strength at the life of 1010 cycles. The static preload of fatigue tests was controlled on an INSTRON machine. The dynamic stress applied to the specimen is controlled by the maximum vibration displacement in the center of the specimen; that is, the vibration displacement at the end of the horn must agree with the value of input from the computer. Before testing, the specimen must be calibrated, to ensure that the error between the actual displacement amplitude and that given by the control unit is under 1%. The microstructure of the Ti-Al alloy specimens was observed and the microstructural characterization performed on materials from fractured test bars with samples prepared through standard grinding, polishing, and etching methods. Microstructure of the samples was analyzed in order to determine whether the fatigue test had altered the structure in any way. The fracture surface of fatigue specimens was examined by SEM to determine the microscopic fracture mode. 4.8.4. High-Cycle Fatigue Properties The fatigue S-N properties are shown in Figure 4.31 (Xue, 2004), which plots the elastic nominal peak stress at the outer fiber of the beam versus the number of cycles to failure. Specimens which did not fail are marked with open circles. When R ¼ 0.1, higher dynamic loads were applied, so it has lower peak stress compared to R ¼ 0.5. From the S-N plot we observe that fracture can occur between 107 and 1010 cycles. The asymptote of the S-N curve is inclined slightly, but it is not horizontal. The fatigue strength of the Ti-Al alloy was calculated at about 400 MPa for R ¼ 0.1 and 580 MPa for R ¼ 0.5. The S-N curve for R ¼ 0.5 has a higher slope. From the S-N curve we see that fatigue fracture occurred between 107 and 108 cycles for R ¼ 0.1, and many specimens cracked beyond 108 cycles for R ¼ 0.5. The asymptote of S-N curve is inclined gently and no fatigue life limit is observed for materials with R ¼ 0.5. Under the same nominal peak Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.31 Three-point bending high cycle fatigue test results for Ti-Al intermetallic alloy.

stress, the scatter in fatigue lifetimes remain large for two different test conditions (R ¼ 0.1 and R ¼ 0.5). The large scatter may be related to the fatigue crack mode for the Ti-Al alloy with the nearly lamellar structure. From the S-N curves obtained with R ¼ 0.1 and R ¼ 0.5, interesting observations can be made. Most specimens failed beyond 107 cycles when stress ratio is R ¼ 0.5; in fact, many of them failed beyond 108 cycles. In contrast, few of the specimens survived 108 cycles with R ¼ 0.1. The fatigue life is mainly decided by the crack initiation. Failure is initiated at weak spots such as large lamellar colonies oriented perpendicular to the loading direction, and large g-grains near the surface of the tested specimen. There is no apparent stable crack growth for the near lamellar material. Most fatigue crack initiations are from the interlamellas or the boundaries between lamellar colonies on the surface of the specimen. However, for specimens with very high fatigue life (over 107 cycles), crack initiation is often from the subsurface of the specimen and, even if crack initiation takes place at the surface, it may remain as a very small surface fatigue crack. More than one crack initiation can be found in many specimens. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 4.32 Fatigue fracture surface (smax ¼ 630 MPa; R ¼ 0.5; Nf ¼ 1.8  109): (a) surface fatigue initiation; (b) intergranular fatigue initiation on surface.

The heterogeneity and anisotropy of the lamellar colonies may account for the large scatter of the high fatigue test results (Figure 4.32).

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5 Crack Growth and Threshold

Using a piezo-electric fatigue machine, it is possible to determine the fatigue crack growth curve, DK versus da=dN, including the threshold regime to growth as low as 10
12 m=cycle. It will be noted that the fatigue thresholds are basically the same in both conventional and resonant fatigue if the computation of the stress intensity factor K is correct. One objective of using the piezo-electric fatigue machine to determine DKth is to save time and money, but it is also of interest to note that a piezo-electric fatigue machine is very effective both for a wide range of cyclic load ratios—R ¼
1 in tension-compression and R > 0.8 with a high mean stress. Prior to developing the data by piezo-electric fatigue machine methods, it is appropriate to review the history of the analysis of crack growth and its threshold. Mechanical fatigue testing machines (McEvily, 1958) developed the first very wide range of crack growth rates in aluminium alloy 7075T-6 (and 2024T-3) shown in Figure 5.1. This plot is the first (Paris, 1962) to employ the range of the crack tip stress intensity factor, DK, vs. the crack growth rate, da=dN, as the primary parameters on a double logarithmic basis. At that

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.1 Fatigue crack growth data (McEvily, 1958) for 7075 T6 (R ¼ 0)

time a straight line on that plot was noted to illustrate the broad trend as: da ¼ CðDKÞm dN with the constant, C, dependent on the load ratio, R. It was understood at the time that, for cyclic loading, a cyclic plastic zone occurs caused by DK at the crack tip. Within a static maximum load, Kmax, a plastic zone about 4 times larger, depending upon R and cyclic hardening or softening, results (Figure 5.2) (Paris, 1963). It is acknowledged that this static=cyclic plastic zone concept originated with McClintock. McEvily’s data (Figure 5.1) were noted to be close to one Burger’s vector, b, per cycle at the low rate end, so it was natural to wish for even slower data to see how far that straight line extended. Therefore (Lindner, 1965), using an Amsler Vibrofore machine which could produce 150 Hz load rates, developed the data given on Figure 5.3 with several months of continuous testing. The flattening of the curve in Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.2

Succession of zones of plasticity near a crack tip.

Figure 5.3 was initially thought to be a threshold for fatigue crack growth for rates below one Burger’s vector per cycle. However, it was felt that more observations should be made prior to accepting that the threshold indeed did exist. (Paris, 1970), via tests of 9310 gear steel in a servohydraulic test machine modified to produce 200 Hz by R. Churchill and H. R. Hartmann, verified the threshold concept and showed the strong effect of load ratio, R (Figure 5.4). At about this time, (Elber, 1970, 1971) demonstrated that crack closure occurs during cyclic loading. Shortly thereafter, (Schmidt, 1973) produced the first extensive data for load ratio effects on threshold using an electro-dynamic shaker to produce up to 700 Hz cyclic loads. Some of his data are plotted in Figures 5.5 and 5.6. He noted that the load ratio effects can be explained with Elber’s crack closure concept. The data demonstrate that at high load ratio, R, no crack Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.3 First fatigue crack threshold data on 7075-T6 (Lindner, 1965) (R ¼ 0).

closure occurs, hence in that range the DKth for threshold is constant. However, at low load ratio, R, crack closure occurs at a K-level, which is quite independent of the minimum load and results in a range in which the Kmax for the threshold is basically constant. This is because it was assumed that the effective stress intensity range, DKeff, extended from the crack opening load to the maximum load. This general form of behavior of threshold is shown in Figure 5.7. More recently (Donald, 1997 and later) has provided extensive data for a wide variety of load ratios near and above threshold and analyzed these data using various concepts such as closure to explore load ratio effects. Typical results are presented in Figures 5.8 through 5.10 to illustrate these concepts as simply as possible. All three figures plot the same data on 7055 aluminum alloy for load ratios, R ¼
1, 0.1, 0.3, 0.5, and 0.7, where in Figure 5.8, the actual applied loads are used to compute DK with no accounting for crack closure. Figure 5.9, the crack Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.4 Verification of threshold behavior in 9310 gear steel for R ¼ 0 and 0.9 (Paris, 1970).

opening load is determined from the load displacement records becoming linear (by the ASTM method) and is designated as: DKeff ¼ DKop ¼ Kmax
Kopen in Figure 5.10 the partial closure model (Paris, 1999) is used where, for simplicity, the form adopted is: 2 DKeff ¼ Kmax
Kopen p Incidentally, when the minimum load is above the crack opening load the applied load range should be used since there is no closure. For R ¼ 0.7 and higher this is usually the case. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.5 Threshold data on T-1 steel with load ratio effects (Schmidt, 1973).

Figure 5.6

Threshold data for A533 steel (Schmidt, 1973).

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.7

Form of closure effects on threshold (Schmidt, 1973).

The partial closure model assumes that closure does not occur all the way to the crack tip and it is noted to improve the data correlation in the near-threshold regime, as well as above threshold. The adjusted compliance ratio of Donald (1997) denoted ACR is equally able to improve the data correlation here, as well as other methods he has developed. This ability to better correlate the data shows that more than simply first crack closure is of influence, especially in the near-threshold regime. Further, for the high load ratio regime where closure does not occur, Donald (1997, 1999) has demonstrated a sensitivity of the data to Kmax, as is shown in Figures 5.11 and 5.12 by improving the data correlation further using the 0:125 . These correlations correlating parameter: DK 0:875  Kmax Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.8 Near-threshold data of Donald (1997) using applied stress intensity for 7055 aluminum (Paris, 1999).

Figure 5.9 The same 7055 data using the opening load method (Paris, 1999). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.10 The same 7055 data using the partial closure 2=Pi0 method (Paris, 1999).

Figure 5.11 High maximum stress intensity data comparison on 2024T-3 (with no closure) (Donald, 1997,1999). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.12 The same 2024T-3 data given in Figure 5.11 adjusted by including the maximum stress intensity in the load parameter (Donald, 1997, 1999).

are presented to show that there is some understanding of load ratio effects and maximum load effects. However, for practical circumstances, measurements or predictions of opening loads etc. are not possible. Therefore data for practical use simply are developed in terms of the applied stress intensity factor range disregarding closure for the load ratios involved. The piezo-electric ultra-sonic data to be presented later will be given on that basis, using simply the DK applied. Before going on to the ultra-sonic data, another strong data trend of a different nature is worthy of discussion here. In the early 1960s, W. E. Anderson (Donaldson, 1961) observed that the basic data on fatigue crack growth of many alloy base materials and alloys of different strength levels of each base material can be normalized by dividing DK by the elastic modulus, E. This was observed after analyzing the data in Figure 5.13. In addition (Hertzberg, 1997) had noted in the 1960s that Lindner’s first data on threshold flattened at a growth rate of one Burger’s vector, b, per cycle. He also noted that the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.13 The comparison of data for a variety of base metals (Donaldson, 1961).

normalization with modulus is dimensional and tried the pffiffiffi non-dimensional parameter, DK=E b with good success. Further Paris (1999) noted that this parameter was improved further by using the effective stress intensity from the preceding discussion. Consistent with Hertzberg’s analysis plus our modification, the threshold corner is predicted to be at da ¼b dN

and

DKeff pffiffiffi ¼ 1 E b

and this can be used as the starting point to predict above threshold growth rates for all metal alloys that all follow the same relationship   Keff 3 da ¼ b pffiffiffi dN E b This threshold corner and line are plotted in Figures 5.14 and 5.15 to illustrate its effectiveness as a predictor of the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.14 The predicted threshold corner and slope using the partial closure model for 2024T-3 (Paris, 1999); see also (Hertzberg, 1996, 1997).

Figure 5.15 The partial closure predicted threshold corner for 6061 (data provided by Donald, 1997). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

data. Hertzberg (1996,1997) presents considerable supporting data for this prediction formula on many different metal alloys. Small crack behavior, which differs from long cracks by being absent of crack closure, is known to be bounded in growth rates by high load ratio long crack behavior. The high load ratio long crack behavior is also closure free and consequently is also at a higher maximum K, which accounts for the bounding effect. Therefore, in the ultrasonic data that follows, the high load ratio data with other considerations will suffice to cover this area of small crack behavior in a minimal way. Let us indicate that the computation of load ratio R is different in ultrasonic fatigue and in conventional fatigue. In conventional fatigue tests, force can be measured. The nominal force and nominal stress are constant at each cross section of the specimen and the stress intensity factor is proportional to the nominal stress. So, the ratio R will be easily determined from R¼

Fmin smin Kmin ¼ ¼ Fmax smax Kmax

where F is the applied force, s the nominal stress, and K the stress intensity factor. In ultrasonic fatigue, however, the total stress is the superposition of static stress and dynamic stress. Because there exists an inertia force in vibration, the nominal stress will not be constant in the cross sections of the specimen. Since the nominal force is impossible to measure, the nominal stress cannot be determined during tests. In this case, R cannot be defined by using F and s and the following formula is perhaps the only choice R¼

Kmin Kmax

and it is noted : DK ¼ Kmax
Kmin

Therefore in ultrasonic fatigue, the calculation of K is necessary not only to determine DK, the crack growth rate curve, but also to obtain R. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

5.1. TITANIUM ALLOYS The material to be tested is T6A4V, whose chemical composition, heat treatment, and mechanical properties are listed in Tables 4.20 to 4.22. (The calculation of stress intensity factor K has been discussed elsewhere.) 5.1.1. Fatigue Crack Growth Rate The experiments are performed for FCG (fatigue crack growth) rates between 10
7 m=cycle and 10
11 m=cycle and with stress ratios R ¼
1 to 0.9. The fatigue threshold DKth is determined for a rate near 10
11 m=cycle by increasing progressively the vibration amplitude from 5% to 10% and the static load about 10%. The experimental results (Bathias, 1997) are given with curves da=dN versus stress intensity amplitude DK ¼ Kmax
Kmin. Recall that Figure 5.1 showed that the crack propagation rate da=dN increases with R, when the latter is positive.

Figure 5.16

Stress ratio influence on T6A4V fatigue crack growth.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In conventional fatigue, when R becomes relatively high, the crack closure effect is suppressed because the cycle minimum load is above the opening load according to (Elber, 1971). This model has already explained the influence of ratio R on the effect of the loading cycle and on the FCG rate. In ultrasonic fatigue with mean load equal to zero (R ¼
1), the closure effect is very small when we consider only the tensile part of the cycle (Kmin ¼ 0). Consequently, the results obtained for R ¼
1, 0.7, and 0.9 are close to one another (Schmidt, 1973). However, the tests show that between R ¼ 0.7 and R ¼ 0.9, there is a very small DKth variation. Therefore, the closure effect seems to remain even for the highest values of ratio R but to the contrary this is Kmax effect. pffiffiffiffiffi For R ¼ 0, the threshold equals 5.66 MPa m, a value significantly higher than the others. This result is comparable to that obtained at conventional fatigue machine speed with at frequency of 2 Hz, and with a stress ratio R equal to 0.03. 5.1.2. Fractography Fractographic observations have been made for specimens tested in ultrasonic fatigue at 20 kHz, at room temperature and with R ¼ 0,
1, 0.8, and 0.9. The observation can be summarized as follows:  For R ¼ 0.8 and 0.9, the fracture surfaces are the same: Near the threshold, the fracture mode is very crystallographic with a strong influence of the microstructure. When FCG rate da=dN becomes greater than 10
8 m=cycle (intermediate regime), the fracture surface becomes smoother and less crystallographic.  For R ¼
1, as for R ¼ 0.8 and 0.9, the propagation mode is very crystallographic at the threshold because of the low value of DK. When da=dN becomes considerably higher, some fatigue striations have been identipffiffiffiffi ffi fied with some difficulty, for DK¼ 8MPa m and da=dN ¼ 1.510
8 m=cycle.  For R ¼ 0, the main observation that characterizes the fracture surface is the presence of fatigue striations Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.17

Striations (T6A4V, 20 kHz).

pffiffiffiffiffi for DK ¼ 13.62 MPa m and da=dN ¼ 210
8 m=cycle (Figure 5.17). These striations are clear and distributed in a regular way, and sometimes separated by secondary cracks. The results of ultrasonic fatigue show that for high load ratio R (0.8 and 0.9), striations are absent whereas for R ¼ 0 with a significant crack closure effect, the striation mechanism is the same as in conventional fatigue. However, at da=dN rates higher than 10
8 m=cycle, in both cases, striations are absent. To summarize, the ultrasonic fatigue data of T6A4V show that:  DKth decreases with the increase of R from 0 to 0.9. Thus, the effect of ratio R exists in ultrasonic fatigue as in conventional fatigue, and the closure effect remains despite the possibility of a reduction of residual stress by vibration.  At room temperature and when the environment is not aggressive, the crack propagation mechanisms are apparently the same in ultrasonic fatigue and in conventional fatigue (Bathias, 1997). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

5.1.3. Cumulation of Ultrasonic Fatigue with Slow Fatigue This part studies the damage cumulation during crack propagation of specimen T6A4V when a 20 kHz vibration combines with a low fatigue cycle. Thus, the aim is to determine the conditions under which the vibration becomes significantly damaging with regard to the low cycle. During the test, a crack pffiffiffiffiffi is first propagated at low cyclic rates with Kmax ¼ 20 MPa m and R ¼ 0. Then, the specimen is subjected to a superimposed 20 kHz loading. When the static load reaches its maximum value, a small amplitude vibration is superposed at a frequency of 20 kHz. This sequence was repeated threeptimes, while the same static ffiffiffiffiffi SIF Kmax was kept at 20 MPa m (Figure 5.18). In particular after applying 2122ppropagation cycles in ffiffiffiffiffi slow cyclic fatigue with Kmax ¼ 20 MPa m, a 20 kHz vibration 6 is superposed during three pffiffiffiffiffi additional blocks (5.4  10 small cycles) at DK ¼ 2.6 MPa m and with R ¼ 0.89. Analysis shows that this vibration has no significant effect on the behavior of low fatigue crack propagation. With the vibration pffiffiffiffiffi amplitude having slightly been increased (DK ¼ 2.8 MPa m), the crack propagated by 0.2 mm (after three cumulation blocks). Therefore, there is a threshold for which a superposed vibration is damaging when it is applied to a slow cyclic fatigue. Note that the crack propagation rate per block (6  10
5 m=cycle) under combined charge is much more important than that in slow

Figure 5.18 Cumulation fatigue test with a superposed SIF value. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.19 Fracture aspects of cumulative fatigue: (a) Three sequences corresponding to three cumulation blocks; (b) transition zone from combined fatigue to slow fatigue; (c) transition zone from slow fatigue to combined fatigue.

fatigue without superimposed vibration (2  10
7 m=cycle) although the vibration amplitude is very small (Bathias, 1997). When the experiment was completed, we performed a metallographic and microfractographic analysis in the scanning electron microscope. Figure 5.19 allows us to identify the three blocks of cumulation corresponding to the three superposition sequences. It shows that between two consecuCopyright © 2005 by Marcel Dekker. All Rights Reserved.

tive sequences of ultrasonic fatigue (two blocks) a second crack appears, playing the role of fatigue striation, which allows one to identify the low fatigue cycle. Passing from slow fatigue to ultrasonic fatigue, we notice a transition zone of a few microns in length depending on the plastic zone radius at the crack tip as shown in Figure 5.19(c). This zone does not exist in the passage from ultrasonic fatigue to low fatigue Figure 5.19(b). Therefore, we think that the passage from ultrasonic fatigue to slow fatigue is different than the passage from slow fatigue to ultrasonic fatigue. The plastic zone radius is proportional to (DK=sy)2 where of the alloy. In slow fatigue, sy is the yield strength pffiffiffiffiffi MPa m whereas in ultrasonic fatigue DK ¼ Kmax ¼ 20 pffiffiffiffiffi DK ¼ 2.8 MPa m. Hence, the plastic zone radius at the crack tip is more significant in the passage from slow fatigue to ultrasonic fatigue than vice versa. That explains why the combined fatigue threshold is slightly higher than the one obtained in ultrasonic fatigue alone with R ¼ 0.9 (Figure 5.16). The cumulation tests show that a light vibration at high frequency superimposed on slow fatigue may accelerate the propagation of a crack. This could be very dangerous for mechanical structures such as turbine blades. 5.2. NICKEL-BASED ALLOYS A series of nickel based alloys have been tested at 20 kHz in the CNAM=ITMA Laboratory for many years. This section will deal with three of them: Astroloy (Wu, 1992), N18 (Bonis, 1997), and Inconel 706 (Bonis, 1997). 5.2.1. Astroloy The chemical composition and mechanical properties of Astroloy are listed in Tables 5.1 through 5.3. Figure 5.20 gives the two FCG rate curves, the threshold corresponds to a growth rate of 10
9 mm=cycle at R ¼
1 where DK is taken to be Kmax. Again, it is necessary to emphasize that in ultrasonic fatigue, a special method has to be Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 5.1

Composition of Astroloy (%wt)

C

S

O

Mn

Si

Cr

Zr Mo Co

Ti

Al

Fe

Ni

0.022 0.013 0.002 <0.02 <0.1 14.9 0.05 5.0 17.0 3.51 4.02 <0.1 Remainder

Table 5.2

Heat Treatment of Astroloy

Isothermal forged thread (70 C=min), oven vacuum, 1100 C – 4 h; quenching in air; 650 C – 24 h – air, 760 C – 8 h – air

Table 5.3

Mechanical Properties of Astroloy

Temperature ( C) 20 400

E (GPa)

r (kg=m3)

n

214 200

8000 8000

0.30 0.30

n ¼ Poisson’s ratio.

used to obtain the FCG rate curves. The appearance of an initial crack from the notch requires a high load. In this case, a large plastic zone exists at crack tip and it is necessary to decrease the test load, i.e., the vibration amplitude. The starter crack length of between 2 and 3 mm is generated and then the FCG rate curves is obtain by increasing DK tests. These curves at R ¼
1 are analogous to conventional crack propagation curves at R  0.7, that is, without a crack closure effect. 5.2.2. N18 The chemical composition, mechanical properties, and heat treatment of alloy N18 were given in Tables 4.25 through 4.27. FCG tests of superalloy N18 are performed at high temperatures. At 450 C, three values of R are used: R ¼
1, 0, and 0.8. At 700 C, there is one value of R: R ¼ 0. Starter cracks have a minimum length of 1.5 mm after pre-cracking at the notch of the specimen. The specimen’s Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.20 Crack growth rate of Astroloy at 20 kHz.

surface is finely polished to make observations of FCG easy at high temperature. Tests at R ¼
1 do not need the addition of static traction; this permits ultrasonic excitation to continue during crack propagation. For other values of R > 0, it is occasionally necessary to stop excitation in order to keep the R ratio constant. In this case of R > 0, the stopping time, even if minimized, brings some oxidation at the crack tip. This phenomenon is also observed when a static traction keeps the crack open continuously. This is, of course, one of the difficulties for tests at R > 0 because of the persistent oxidation at the crack tip during the test process. Figure 5.21 groups experimental results of FCG of N18 at  450 C for R ¼
1, 0, and 0.8. The thresholds pffiffiffiffiffi for these pffiffiffiffiffi three values of R are respectively 5.5 MPa m , 8 MPa m, and pffiffiffiffiffi 4.5 MPa m. For R ¼
1 and 0.8, the effect of crack closure is notably weak, which explains why the thresholds are similar. Figure 5.22 illustrates the results of tests done in air at 400 , 450 , and 700 C and in vacuum at a temperature of 650 C. These results are also compared with the tests performed at low frequency of 0.5 Hz and with an R ratio approaching zero. We notice that the curve at 650 C in ambient air is above that of 700 C tested at 20 kHz. This difference can be Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.21 Crack growth rate of N18 at 450 C and different R values.

Figure 5.22 700 C.

Mixed comparison of FCG rate curves of N18 up to

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

explained by the fact that very low testing frequency facilitates oxidation in the crack between cycles. This conclusion about oxidation is also confirmed by the good agreement of the curves obtained at low frequency in a vacuum with the results at ultrasonic fatigue. Another difference is observed between the curves of conventional fatigue at 400 C and that of ultrasonic fatigue at 450 C. At very high frequency, the air has much less time to arrive at the crack tip, which limits the oxidation. On the other hand, the increase of temperature between 450 C and 700 C diminishes the threshold for cracking. We believe the reason may be the thermal activation of oxidation at the crack tip. This demonstrates that a test at 20 kHz is not always similar to the one at low frequency. 5.2.3. Inconel 706 Inconel 706 is used to manufacture jet engine disks and turbines blades. It is a superalloy mainly composed of nickel and iron, with a precipitation hardening structure and large grains of about 125 mm, and it is easy to weld. The precipitates are g-Ni3 (Ti, Al) and b-Ni3Nb. Tests were carried out at 20 kHz with a load ratio R ¼ 0.1 and at two temperatures of 20 C and 400 C. The chemical composition, heat treatment, and mechanical properties are listed in Tables 5.4 through 5.6. With alloy N18, we also find an increase in crack growth rate induced by oxidation at the crack tip when the Table 5.4

Chemical Composition of Inconel 706 (wt%)

C

Mn

Si

Cr

Al

Nb

Ti

Ni

Fe

0.03

0.18

0.18

16

0.2

2.9

1.75

41.5

37.44

Table 5.5

Heat Treatment of Inconel 706

982 C 1 h, cooling in air; 843 C 3 h, cooling in air; 718 C 8 h, cooling to 621 C (38 C=min) and stable 8 h, cooling in air

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 5.6

Mechanical Properties of Inconel 706

s0.2 (MPa)

E (GPa)

r (kg=m3)

su (MPa)

931

210

8050

1138

temperature is increased. However with Inconel 706 for a lower FCG rate, an oxide film props the crack faces open and reduces the cyclic effects. This side effect is particularly notable with Inconel 706 and prevents the exploration of the very slow speeds of propagation (Figure 5.23). The results of ultrasonic fatigue tests are consistent with those obtained at 20 Hz. We observe in Figure 5.24 that the slopes of the upper FCG regime are parallel. At 20 kHz, the environment has less time to interact with the metal at the crack tip than at 20 Hz, which may explain the fact that the curve of 20 kHz is slightly below that of 20 Hz. The following test procedures were designed to better distinguish the propagation acceleration effect and side effect mentioned earlier that are both due to oxidation. A specimen

Figure 5.23

FCG rate of Inconel 706 at 20 C and 400 C.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.24 FCG rate of Inconel 706 at 20 C: Comparison of 20 kHz and 20 Hz.

is tested at 20 C keeping the crack in propagation, and progressively decreasing the DK value until the complete arrest of propagation. Always at ambient temperature, we impose a DK value less than the threshold of propagation at 400 C. Once the excitation is well established, the temperature rises rapidly. Then the progression of the crack takes a much higher speed than the one that can be extrapolated from the curve of da=dN at 400 C. After a stop of several minutes at 400 C, the propagation does not recover the same value of DK. However, by increasing the excitation, we return to the curve of da=dN at 400 C. The following experiment demonstrates the significance of the side effect due to oxidation and explains its blockage effect on crack propagation as the threshold is approached. Because the crack is swept by a gaseous nitrogen jet, the cracking speed is slightly lower and blockage to the propagation when approaching the threshold is delayed. This can be seen in Figure 5.25. Curves of FCG in ambient air with and without nitrogen sweep do not differ greatly although the paths of cracking and the crack surfaces are noticeably different. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.25 FCG rate of Inconel 706 at 400 C with and without nitrogen sweep.

A comparison of da=dN curves of N18 at 450 C with that of Inconel 706 at 400 C in Figure 5.26 indicates that N18 has a slightly better behavior in cracking. Since N18 is a little less sensitive to the phenomenon of blockage due to the side effect of oxidation, the threshold is higher.

Figure 5.26

Comparison of FCG rate of N18 and Inconel 706.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

5.3. ALUMINIUM ALLOYS Stanzl’s group and co-researchers in Vienna, Austria have studied a series of aluminium alloys. The testing conditions cover a variety of loading: rapid load reduction, low amplitude two-step loading, in-service loading, and so on. Results regarding the properties of crack growth and threshold mainly in gigacyclic regime can be found in (Mayer, 1991, 1992) and (Stanzl, 1991, 1993).

5.3.1. Al2024 Alloy After Rapid Load Reduction (Mayer, 1991) One effect of FCG with variable cyclic stress intensities is crack growth retardation after overloads. This causes the crack growth rate to be generally lower than the sum of the single crack propagation rates expected from the crack propagation curve obtained with constant load amplitude. The retarded crack growth is due to several crack closure effects. For multi-stage loading of a homogeneous metallic material, plastic deformation and crack surface roughness, as well as oxidation, are regarded as the main causes. Retardation effects are important in designing structures because, in general, service load amplitudes are not constant. Knowledge of the influence of previous load history on crack propagation rate is therefore important. A comprehensive study of crack closure effect after load reduction in the threshold regime has not yet been presented. This regime is important to study since crack propagation behavior and the crack closure effects cannot be extrapolated to low Kmax values from measurements at higher stress intensity values. The objective of this study was to begin to analyze quantitatively crack retardation effects in the threshold regime. The material tested was aluminium alloy Al2024 whose chemical composition and mechanical properties are listed in Tables 5.7 and 5.8. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 5.7

Chemical Composition of Al2024 (wt%)

Cu

Mg

Si

Fe

Mn

Cr

Zn

Ti

Al

4.5

1.5

0.1

0.2

0.7

< 0.05

< 0.05

< 0.05

Remainder

Table 5.8 E (GPa) 72.5

Mechanical Properties of Al2024 sm (MPa)

s0.2 (MPa)

A%

pffiffiffiffiffi KIC (MPa m)

460

352

18

35

The material was delivered as rolled sheets of 20 mm thickness. The heat treatment procedures consisted of solution heat treatment at 495
C for 2 h, quenching, cold work and age hardening at room temperature for more than 4 days. For fatigue crack propagation measurements, rectangular specimens with a circular tapered cross section in the center and a single edge notch were used. As stated before, this type of specimen has been commonly used for ultrasonic fatigue testing and allows the strain and stress to form a standing wave with the maximum of strain in the middle of the reduced section. Loading was performed with a 20 kHz ultrasonic fatigue machine at R ¼ 1. The tests were carried out in two different environments: ambient air and vacuum. The curves of da=dN versus Kmax without load reductions are shown in Figure 5.27 (Stanzl, 1996; Mayer, 1999). Before rapid reduction of the stress intensity factor value, a fatigue crack is initially grown pffiffiffiffiffiapproximately 1 mm at the high cyclic SIF of K1 ¼ 7 MPa m and takes approximately 5  105 cycles. This fatigue crack extension at the high stress intensity values K1 leads to well-defined starting conditions for the rapid load reduction test. The crack surface roughness is typical for K1; both the plastic zone in front of the crack tip according to K1, and the oxide layer of characteristic thickness on the crack surface are the same at beginning of each load reduction test. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.27 FCG curves of 2024-T3 in a vacuum and in humid air (Stanzl, 1996).

After rapid load reduction, cycling with a defined lower stress intensity value K2 follows until the crack has grown 0.10  0.15 mm. If no crack growth could be detected after 1109  1.5109 cycles after rapid load reduction, the experiment is stopped and it is assumed that the threshold for this type of loading is obtained. This stress intensity value is called reduction threshold. Figure 5.28 (Mayer, 1992) Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.28 Test sequence for delay time cycle measurements after growing the p crack ffiffiffiffiffi by approximately 1 mm with a cyclic stress intensity of 7MPa m ( approximately 5  105 cycles).

presents the test sequence; Figure 5.29 (Mayer, 1992) gives crack propagation after constant amplitude cycling at pffiffiffiffiffi Kmax ¼ 7 MPa m and rapid load reduction. Loading is performed at the reduced stress intensity value until crack growth of 0.1–0.15 mm is detected. If no further crack extension is observed, loading at the low stress intensity is continued for at least 1  109 cycles (Stanz, 1996).

Figure 5.29 Delayed crack growth after rapid reduction of the stress intensity value (Mayer, 1991). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.30 Dependence of delay cycles after rapid reduction of stress intensity to the lower stress intensity level (Mayer, 1991).

As a quantitative measure for retarded crack propagation after reduction of the Kmax value, the so-called delaycycles are determined. These are defined as the number of cycles necessary for a crack to grow approximately 0.10  0.15 mm at the lower SIF value after rapid load reduction, minus the number of cycles needed for the same crack increment at low stress intensity amplitude without previous loading at high stress intensity level. Figure 5.30 (Mayer, 1991) shows the delay cycles for rapid load reduction pffiffiffiffiof ffi the cyclic stress intensity amplitude from Kmax ¼ 7 MPa m to the value indicated at the ordinate. The square dots show the measurements in ambient air, and the circles characterize the measurements in vacuum. The phenomena observed are as follows. The crack propagation rate is rapidly slowed after rapid reduction of SIF. The retardation is characterized by delay time cycles as defined above. If Kmax is rapidly reduced, load reduction threshold SIF amplitude is required to continue crack growth. This reducpffiffiffiffiffi tion threshold is approximately 4.3 MPa m in ambient air pffiffiffiffiffi and 3.5 MPa m in vacuum. In both environments, this SIF value is higher than the threshold determined in a usual Copyright © 2005 by Marcel Dekker. All Rights Reserved.

crack propagation test with a SIF reduction of, at most, 5%  7%. If the crack continues to grow after rapid reduction of the SIF, the retardation number of cycles is lower in air than in a vacuum. The reduction threshold, however, is higher than that in a vacuum. The results show less scattering for the vacuum than for air. In a vacuum, crackppropagation can be ffiffiffiffiffi observed at an SIF amplitude of 4 MPa m, although crack arrest (delay) is observed during more than 5  107 cycles. Detailed environmental influence on FCG and threshold properties of Al2024 will be further discussed. 5.3.2. Al2024 Alloy Under Low Amplitude Two-Step Loading As indicated earlier, the increased threshold SIF value after a reduction of Kmax is referred to as the reduction threshold. The significance of such a reduction threshold for crack propagation under variable amplitude loading is not clear, because other considerable interaction effects can apparently occur from a load amplitude reduction. This discussion deals with the same material studied previously, but under low amplitude two-step loading, as shown in Figure 5.31 (Mayer, 1992). Particular interest is placed on whether cycles at or below the previous reduction threshold will also affect crack growth in such two-step load sequences. For practical reasons it is worthwhile to know if numerous small cycles of an ultrasonic fatigue load spectrum contribute to fatigue crack growth. Tests were performed at room temperature, at 20 kHz and R ¼
1. All p tests ffiffiffiffiffi begin with a constant amplitude load at Kmax ¼ 7 MPa m with a pre-crack length increase of about 1 mm. The plastic zone size for this Kmax value is about 40 mm. This procedure guarantees well-defined starting conditions for subsequent two-step loading tests. In the two-step tests, the load level alternates periodically between the starting level of Kmax and some lower level. The two-step loading continues until a stationary crack growth velocity is attained. Then constant amplitude loading is applied again at the high Kmax value. The fatigue load is essentially K-controlled. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.31 Tabulation of load sequences. For all tests Kmax,1 is pffiffiffiffi ffi 7 MPa m and N1 is 500 cycles (Mayer, 1992).

Each two-step pffiffiffiffiffi test period consists of 500 cycles at Kmax,1 ¼ 7 MPa m and a 100 times or 1000 times larger number of cycles at the lower level. Some of the crack growth results are illustrated on Figure pffiffiffiffiffi 5.32 (Mayer, 1992) with Kmax,2 ¼ 3.5 MPa m, N2 ¼ 50,000 cycles. The two predicted reference levels for the average crack growth rate are also shown. The crack growth exhibits the following trends: Some crack extension does occur during the two-step period before a stabilized average crack growth rate is reached. After the two-step loading period is completed, again some crack acceleration occurs during the subsequent constant higher amplitude loading before the original crack growth rate stablizes. In all cases, according to Miner’s prediction the average crack growth rate during the two-step loading starts at Miner’s prediction with increased threshold. However, the crack growth rate immediately increases, Copyright © 2005 by Marcel Dekker. All Rights Reserved.

pffiffiffiffiffi Figure 5.32 Crack propagation behavior, Kmax,2 ¼ 3.5 MPa m, N2 ¼ 50000 cycles. (A) Miner’s law crack propagation rate without increased threshold; (B) Miner’s crack propagation rate, but with increased threshold value (Mayer, 1992).

which implies that the increased threshold level does not fully apply. In other words, either crack extension does occur at the lower Kmax level, or the higher Kmax level has become more damaging than in the constant amplitude tests (or both apply). On the other hand, crack growth rate does not reach the Miner’s prediction level without increased threshold. Apparently, some retardation effect remains. Quantitatively, differences between the test results are evident. Crack growth rate during the two-step loading increases by a factor of 1.8, well below Miner’s non-interaction prediction of a factor of 15. Other results (Mayer, 1992) indicate that for the higher Kmax,2 value, the crack growth rate increased by factors of 15 and 300. This result comes much closer to the non-interaction Miner prediction. For N2 ¼ 50,000 it remains 2.5 times below that level, while for N2 ¼ 500,000 Copyright © 2005 by Marcel Dekker. All Rights Reserved.

the crack growth rate becomes equal to the non-interaction level. In conclusion, ultrasonic fatigue results show significant retardation effects. Low SIF level does contribute to crack growth. Delayed retardation explains this observation. Low amplitude cycles may induce a smoother fracture surface and increase crack growth rate at higher amplitude. Accurate predictions are therefore problematic. For similar reasons, omitting numerous low amplitude cycles in random load experiments should be done with great care. 5.3.3. AlSi Alloy Under In-service Loading (Stanzl, 1993) Fatigue properties of structural materials are normally reported for loads with constant amplitude. Simulations of in-service loading conditions are sometimes attempted by the use of randomly distributed amplitudes. The resulting alternative S-N and lifetime curves are then often used to select a material for a specific purpose. The study presented here concerns the fatigue properties of two AlSi cast alloys used for automobile wheels. Specimens are tested in the regime of 108  1010 cycles at 20 kHz and R ¼
1. Table 5.9 gives the chemical compositions of the two materials. These materials are nominally identical. In material 1, the H2 gas content of the melt is artificially enhanced before casting, so that its microporosity is also influenced. Material 2 is cast under the same conditions as in normal production. Both alloys are used without additional heat treatment.

Table 5.9 Si

Chemical Compositions of AlSi Alloys (wt%) Fe

Cu

Mg

Zn

Ti

Mn

Al

Remainder 10.0 7 11.8 <0.18 <0.03 0.001  0.4 <0.07 <0.15 <0.05 for mat. 1; 0.12  0.20 for mat. 2

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

The stress amplitudes are applied as pulses. In-service loading is simulated according to a cumulative frequency curve obtained from actual load measurements. The stress amplitudes of successive pulses are changed according to this distribution during high frequency simulation of random loading. Because of the resonance type loading, single cycles within the pulses cannot be chosen individually, but instead pulses of 1000 cycles are used. The sequence length is 106 pulses; thus, the return period consists of 109 cycles. The maximum applied stress amplitude smax occurs once during 106 pulses and is 100 MPa. This value for smax is chosen so that the expected number of cycles to failure is 108  109. The load sequence is stationary, which means that the almost constant root mean square values of applied smax prevail during the whole process. If a specimen has not failed at the end of a full sequence, the experiments are stopped. Pauses between pulses allow for the dissipation of the heat produced by internal friction in specimens during fatigue loading. The pause length is chosen so that the heating of the specimens is inhibited. In addition, a fan is used for cooling. Figures 5.33 and 5.34 (Stanzl, 1993) give the results of FCG experiments in the threshold regime. The threshold values for R ¼
1 obtained by reducing pffiffiffiffiffi the SIF values pffiffiffiffiffi in the steps by 5%  7% are 2.8 MPa m and 3.0 MPa m for materials 1 and 2, respectively. After determination of the threshold value, loading is increased stepwise. The specimens are loaded for at least 2  107 cycles at each amplitude level if no crack growth can be observed. The maximum cyclic p SIF, ffiffiffiffiffi for which no crack extension is detected, is 3.15 MPa m for material 1 and pffiffiffiffiffi 3.7 MPa m for material 2. In this near-threshold region, with pffiffiffiffiffi R ¼
1 and Kmax between 3.7 and 5.5 MPa m, the FCG rates are generally three times lower for material 2 than for material 1. Another important result is that cracks remain stopped when Kmax is somewhat increased continuously after the threshold value has been obtained in the tests. This may be partially explained by crack closure effects. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.33

FCG curve of AlSi cast alloy, material 1 (Stanzl, 1993).

To summarize, the ultrasonic fatigue tests demonstrate the superiority of material 2 over material 1. This is attributed to higher mean crack initiation times due to the absence of the large cast voids.

Figure 5.34 FCG curve of AlSi cast alloy, material 2 (Stanzl, 1993). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

5.3.4. Al6061-T6 Alloy Reinforced by Al2O3 Particle (Papakyriacou, 1995) Higher stiffness than that of conventional aluminium alloys, improved wear resistance, and superior high temperature characteristics make aluminium alloys reinforced with SiC or Al2O3 particles attractive as engineering material. They are now used in the automobile industry for pistons or sleeves. Various other applications are possible since production and machining of such isotropic composite materials is possible without too many problems. Knowledge of crack initiation and propagation processes is necessary to guarantee safe in-service behavior of components. Of special interest are the fatigue properties at high numbers of cycles and the FCG behavior at very low crack growth rates since, in automotive applications, components often must endure lifetimes of up to 108 cycles. Tested materials were 6061-T6 aluminium alloys with different contents and size distribution of Al2O3 particles. Specifically, an alloy with a content of 15.0% volume of fine particles (6061=Al2O3=15p), an alloy with a content of 21.1% volume of more coarse Al2O3 particles (6061=Al2O3= 21p) and, for comparison, 6061-T6 without reinforcement (6061) were tested. Table 5.10 lists the chemical compositions of these three materials. The production procedures for the particle reinforced aluminium alloys are as follows. After extrusion of rods with cross-sections of 17  17 mm, the reinforced materials are solution heat treated at 560 C for 0.5 h, and then water quenched and aged at 160 C for 8 h (heat treatment T6). The unreinforced aluminium alloy is solution heat treated

Table 5.10 Chemical Compositions of Al6061 Alloys (volume %) Material

Mg

Si

Cu

Fe

Mn

Cr

Zn

Ti

6061 6061=Al2O3=15p 6061=Al2O3=21p

0.88 0.96 0.95

0.69 0.64 0.63

0.43 0.26 0.27

0.45 0.15 0.08

0.13 0.004 0.004

0.17 0.105 0.10

0.01 0.012 0.009

0.07 0.01 0.01

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

at 525 C for 0.5 h after extrusion of rods with the same crosssection dimensions, followed by water quenching and aging at 160 C for 24 h. A quantitative evaluation of mean particle size and density is listed in Table 5.11, together with mechanical properties of the materials. Tests were performed at a frequency about 20 kHz and load ratio R ¼
1. Cyclic loading is applied in pulses with a pulse length of 1000 cycles. Pauses between the pulses help to dissipate the heat caused by internal friction in the specimen during loading. The length of the pauses was chosen to be approximately 20  30 ms to avoid increase of specimen temperature. To control cyclic loading the displacement amplitude U0 at the specimen end was used. The maximum SIF Kmax is obtained from the formula Kmax ¼

U0 E pffiffiffi af ða=wÞ w

To obtain da=dN versus Kmax curves, the load is lowered in steps of 7% (close to the threshold, 5%) and then increased. Crack increments of about 300 mm are evaluated for each data point. At least 2  107 cycles are applied to the specimen in the threshold regime if no crack advance can be detected. From this and the optical resolution of the assembly of 7 mm, a crack growth rate below 3.5  10
13 m=cycle is guaranteed for characterization of the threshold in the crack growth curve. Test results are plotted in Figure 5.35 (Papakyriacou, 1995). The FCG properties of the reinforced alloys are

Table 5.11 Mechanical Properties of 6061 Alloys and Mean Particle Size Material

E (GPa)

sy (MPa)

UTS (MPa)

e%

Dcircle (mm)

Dmax (mm)

Particles per mm2

60612 6061=Al2O3=15p 6061=Al2O3=21p

70.4 90.5 96.3

335 340 365

375 385 405

14 6 4

8.0 11.1

13.3 16.8

3500 2100

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.35 Fatigue crack growth rates vs. Kmax of: (a) unreinforced alloy 6061-T6; (b) 6061=Al2O3=15p; and (c) 6061=Al2O3=21p. A comparison of fatigue crack growth rates vs. Kmax for all three alloys is given in (d) [Stanzl as reported in (Papakyriacou, 1995)].

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

pffiffiffiffiffi superior to pure 6061 for Kmax below 5 MPa m for pffiffiffiffivalues ffi 6061=Al2O3=21p and 6.5 MPa m for 6061=Al2O3=15p, but worse for high Kmax values. Alloy 6061=Al 2O3=15p shows the pffiffiffiffiffi highest threshold SIF of 5.4 MPa m. The threshold stress pffiffiffiffiffi intensity of 6061=Al2O3=21p is lower (4.7 MPa m) than that of 6061=Al2O3=15p, but the difference between the crack propagation behavior of the two particle reinforced alloys becomes smaller with increasing load. Unreinforced 6061 alloy has a pffiffiffiffiffi threshold value Kmax of 3.9 MPa m. The slope of the crack propagation curve of 6061, however, is smaller than for both reinforced alloys, thus leading to superior crack growth properties of the unreinforced material at higher cyclic loads. To conclude (Papakyriacou, 1995): The threshold values of Kmax for fatigue crack propagation at load ratio R ¼
1 are increased by the addition of 15% volume fine as well as 21% volume coarse Al2O3 particles to the aluminium alloy 6061. These hard particles improve FCG properties by acting as obstacles to crack propagation. Firstly, impediment of crack growth by these obstacles, changes of the crack propagation direction, and branching of the crack path restrain fatigue crack growth, thereby, increasing fracture surface roughness and raising the modulus of elasticity, which is beneficial. Secondly, Al2O3 particle reinforcement improves the FCG properties more efficiently in the threshold regime than at higher Kmax values. At higher loads, particle and interface fracturing is more frequent, which results in higher crack growth rates. Finally, the alloy with finer particles shows better FCG properties than the alloy with coarser particles. Particle fracture and cracking of the interface between particles and matrix are both more frequent for coarse reinforcing particles, which may explain this result.

5.4. MATERIALS OF B.C.C. AND F.C.C. CRYSTALLINE STRUCTURE (TSCHEGG, 1981) (Tschegg, 1981) presents an early study at low temperature. The aim is to investigate crack growth behavior of b.c.c. (body Copyright © 2005 by Marcel Dekker. All Rights Reserved.

centered cubic) and f.c.c. (face centered cubic) metals in the threshold regime and to correlate their fracture appearance with crack growth rates in order to explain the differences in low and high temperature behaviors. Mild steel is chosen as an example for b.c.c. metals. Rods are cold drawn from 3–7 mm diameter and cold rolled to bands with the cross section 1 mm  4 mm and a length of 125 mm. For f.c.c. metals, 99.9% commercial copper and austenitic AISI type 304 stainless steel are used. At 77 K, steel 304 exhibits rather large amounts of strain induced martensite. Both materials are cold rolled to bands of cross section 1 mm  10 mm. The chemical compositions, heat treatments, and mechanical properties of these materials are listed in Tables 5.12 through 5.14. To study the influence of the surrounding temperature on crack propagation the samples are immersed in silicone oil at 273 K (20 C) and liquid nitrogen at 77 K, respectively. Silicone oil is a noncorrosive liquid and mainly considered as replacing vacuum conditions. Experiments are repeated to compare measurements in vacuum and silicon oil extensively. In order to avoid temperature rises due to damping effects at very high frequency, the stressing is done in a pulsed manner. Pulse packets and intervals are chosen so that the temperature rises at the sample surface are less than 5 C. For DK calculation, only the tensile part of the stress is used, assuming that the compressive load does not contribute Table 5.12 Chemical Compositions of Mild Steel, Copper, and Steel 304 (wt%) Mild steel

C 0.036

Copper

Cu 99.9

Steel 304

C 0.036

Si 0.01

Mn 0.08

P 0.012

S 0.008

Al 0.002

Cr 19.04

Ni 10.40

Si 0.75

Mn 1.18

Mo 0.48

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

N 0.005

O 0.015

Table 5.13 Heat Treatments of Mild Steel, Copper, and Steel 304 Heat treatment Mild steel Copper Steel 304

Table 5.14 Steel 304

273 K

77 K

700 C=90 min furnace-cooling 630 C=90 min furnace-cooling 1050 C=30 min water-quenched

Microstructure and grain size Ferrite: 15 lm Grain size: 20 lm Austenite þ small amounts ferrite

Mechanical Properties of Mild Steel, Copper, and

Mild steel Copper Steel 304 Mild steel Copper Steel 304

sy (MPa)

UTS (MPa)

eu (%)

275 69 230

325 235 700 840 365 1500

35 48 55 6 60 39

83 420

to crack propagation. This means that an eventual contribution of the compressive cycle toward DK in the near threshold growth is not taken into account. Tests are performed at about 20 kHz and symmetric load, R ¼
1. The dependence of the FCG rate on stress intensities at 293 K and 77 K are compared in Figures 5.36, 5.37, and 5.38 for f.c.c. copper, steel 304, and b.c.c. mild steel, respectively. It is emphasized that the scattering of the data points is caused partly by the method of measuring; i.e., the stressing amplitude is not kept constant during the tests but lowered and raised repeatedly so as to attain very low Kmax values. Therefore, crack growth rates are obtained from single crack increments and not from a smoothed curve. A second reason for the scatter of the results is discontinuous crack growth. For calculating da=dN curves, crack increments of about Copyright © 2005 by Marcel Dekker. All Rights Reserved.

XXXX-0 Bathias Ch05 R3 081204

Figure 5.36 Influence of temperature (293 K silicone oil and 77 K liquid nitrogen) on fatigue crack growth rates in polycrystalline copper (21 kHz; R ¼
1) (Tschegg, 1981).

100 mm are used. Crack growth rates below about 10
10 m= cycle are mean values of higher crack propagation rates and crack stops. Thus it is understandable that crack propagation rates of less than one Burgers vector per cycle on average can occur. Data points with arrows refer to cracks that did not propagate 10 mm of measurement accuracy at minimum. The crack growth rates of these points are calculated by dividing 10 mm by the applied number of cycles. The main result shown in Figure 5.36 for polycrystalline copper is that the crack growth curve is shifted to somewhat higher stress intensities at 77 K compared to 293 K; that is, the crack growth rates are a bit lower for identical stress intensities. A threshold stress intensity seems to exist for Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.37 Influence of temperature (293 K silicone oil and 77 K liquid nitrogen) on fatigue crack growth rates in steel 304 (21 kHz; R ¼
1) (Tschegg, 1981).

pffiffiffiffiffi pffiffiffiffiffi both temperatures; it is 2.7 MPa m and about 3 MPa m at 293 K and 77 K, respectively. Crack propagation curves of steel 304 in Figure 5.37 show essentially the same result as copper. At 77 K somewhat higher stress intensities Kmax are necessary to gain the same crack growth rates. Again, a threshold stressp intenffiffiffiffiffi sity seems to exist p for both temperatures; it is 7 MPa m at ffiffiffiffi ffi 293 K and 8.5 MPa m at 77 K, respectively; i.e., it is about 20% higher at 77 K for steel 304. For crack propagation curves of mild steel in Figure 5.38, the differences of crack growth rates at 77 K and 293 K are more pronounced than in f.c.c. copper p and The ffiffiffiffiffi steel 304.  threshold intensity is 3.8 MPa m at 293 K and pffiffiffiffistress ffi 5.3 MPa m at 77 K, respectively. The increase is about 40%. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.38 Influence of temperature (293 K silicone oil and 77 Kelvin liquid nitrogen) on fatigue crack growth rates in mild steel (21 kHz; R ¼
1) (Tschegg, 1981).

To summarize (Tschegg, 1981), low temperature causes a shift of FCG curves to higher stress intensity Kmax. This is found for polycrystalline f.c.c. copper and steel 304 as well as b.c.c. mild steel and might be explained by the increased tensile properties. The fracture mode of mild steel pisffiffiffiffifficleavaged completely at stress intensities Kmax  20 MPa m and cleavage at crack growth rates da=dN > 10
8 m=cycle, but by reversed plasticity at lower values. It p isffiffiffifficoncluded that ffi sometimes below intensity Kmax ¼ 20 MPa m the stress is obviously too low to cause cleavage, but then crack propagation is still possible in this case by a reversed plasticity mechanism. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

5.5. LOW CARBON STEEL SHEET The cold-rolled low carbon steel sheet is produced from pickled hot rolled by cold reduction to the desired thickness. Owing to its advantages of hardness and much thinner thickness, cold rolled steel sheets are widely used in the automotive industry, which in turn leads to a reduction in weight. In the automotive industry, most components are required to have a very long fatigue lifetime both for safety and economic reasons, hence it is necessary to well describe the fatigue crack growth behaviors in the near-threshold regime. We discussed some special testing measures adapted to this type of thin specimen in Chapter 3. The test is conducted at a stress ratio R ¼ 0.1 at a frequency of 20 KHz and at room temperature. The thickness of the cold-rolled steel sheet is 0.75 mm. The chemical composition and mechanical properties obtained in static tests were listed in Tables 5.15 and 5.16, respectively; the threshold of this material is determined at a rate of 10
11 m=cycle. In the test, the specimen is first pre-cracked without static load, R ¼
1. Then the experiments are performed with a constant stess ratio of R ¼ Kmin=Kmax ¼ 0.1 by decreasing progressively the static load and the vibration amplitude from 5% to 10%. When the threshold is reached, experiments continue by increasing the stress intensity factor. Table 5.15 Composition of Cold-Rolled Low Carbon Steel Sheet (%wt) Cu 0.14

C

Mn

Si

P

S

Al

0.08

0.4

0.1

0.025

0.025

0.02

Table 5.16 Mechanical Properties of Cold-Rolled Low Carbon Steel Sheet sy (MPa) 225

UTS (MPa)

eu %

Hv5

E(GPa)

r (kg=m3)

340

36

95

203

7830

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

The results of the tests are shown in Figure 5.39. The threshold p value is found at a stress intensity factor of 5.0 MPa m with a propagation rate of 7  10
12 m=cycle. In the study, temperature is measured at the crack tip by the infrared thermograph method. It was found that the rise of temperature depended upon the material microstructure and the amplitude of vibration loading. For the fatigue crack growth test in the near-threshold regime, the loading amplitude is very small and the temperature rise is not very significant with the flow of compressed air (at most 60 C; Figure 5.40). In this case, the absorption of ultrasonic vibration energy has no considerable effect on the fatigue crack growth behavior. Fractographic analyses of the fracture surface are grouped in Figure 5.41. No evident fatigue striation is observed on the whole fracture surface. This is probably because of the small amplitude of vibration. At high DK, the fraction of transgranular rupture is observed in Figure 5.41a. For the moderate values of DK, the fracture surface exhibits a mixed mode of transgranular and ductile intergranular rupture, and several secondary cracks can be seen at

Figure 5.39

FCG rate of the thin steel sheet (R ¼ 0.1).

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.40 fatigue test.

Temperature rise in the specimen of an ultrasonic

intergranular locations, (Figure 5.41b). In the near threshold region, the ductile intergranular rupture becomes predominant and the fracture surface is smoother (Figure 5.41c). 5.6. AUSTENITIC STAINLESS STEEL (SUN, 1999) Test results for 304 steel are presented on Figure 5.42 as a plot of fatigue crack growth rate da=dN vs stress intensity factor range, DK ¼ Kmax
Kmin. The propagation threshold was determined at a very low growth rate down to 10
12 mm=cycle. For R ¼
1, Kmin is taken to be zero. 5.6.1. Fatigue Crack Growth Rates and the Threshold Ultrasonic fatigue endurance experiments have shown that, for a great number of alloys, fatigue failure can still occur at 109 cycles or beyond, and the difference of fatigue resistance between 106 and 109 can reach 100 or even 200 MPa Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.41

SEM of fracture surface of low carbon steel at 20 kHz.

(Bathias, 2003). But the ultrasonic fatigue crack growth test results show that there is very small difference between the thresholds determined at the rates of 10
9 and 10
7 mm=cycle. In this study, this difference in DK threshold is only about pffiffiffiffiffi 0.2 MPa m. Some other studies on different alloys, such as nickel alloys (Bathias, 1993), titanium alloys (Bathias, 1994, 1997), aluminium alloys (Sun, 2001), and steels (Sun, 2001), have arrived at much the same conclusions. This means that the fatigue crack growth threshold determined by conventional fatigue testing is reliable for the engineering design. It is obvious that ultrasonic fatigue is time-saving and practical; for example, threshold measurements can be obtained within a few hours of testing at 20 kHz. Moreover, the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.42 Ultrasonic fatigue crack propagation test results for 304 steel at 20 kHz.

ultrasonic vibration fatigue technique is especially useful for tests at very high stress ratios. 5.6.2. Effect of Stress Ratio, R The effect of the stress ratio on the fatigue crack propagation behavior at ultrasonic frequency is illustrated in Figures 5.42 and 5.43. As expected in conventional fatigue tests, high load ratio induced lower threshold values and faster growth rates at a given applied DK. This relation can be described as: with

DKth ¼ f(R)DKth 0 f(R) ¼ 1
R f(R) ¼ R0 f(R) ¼ 1 þ 0.3 R

R0 ¼ Kopen=DK0 for 0 < R < R0 for R0 < R < 1 for
1 < R < 0

The influence of R-ratio on the fatigue crack growth behavior can generally be explained by the crack closure effects, as proCopyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.43 Dependence of threshold on R ratio (see also Figures 5.5 to 5.7).

posed by (Elber, 1971) and (Schmidt, 1973). However, in ultrasonic fatigue with mean load equal to zero (R ¼
1), the closure effect becomes very small when we consider only the tensile part of the cycle load (Kmin ¼ 0). Figure 5.43 shows that the results for R ¼
1 are close to those for R ¼ 0.5 and R ¼ 0.7. As originally suggested by (Schmidt, 1973), if the variation of threshold with load ratio is simply due to the crack closure, and if the requirement to induce crack closure is independent of load ratio, a transition behavior from the threshold at low R ratio to that at high R ratio could be expected at Kmin,th ¼ Kcl. From pffiffiffiffiffi Figure 5.44 this value was estimated as Kcl ¼ 3–4 MPa m. (See also Figures 5.5 to 5.7.) 5.6.3. Fatigue Crack Growth Mechanisms at Ultrasonic Frequency The SEM observations of fracture surfaces permit one to note the fatigue crack growth mechanisms for ultrasonic frequency at different R ratios. For this stainless steel, at threshold regime, the rupture appears transgranular and crystallographic, but the fracture surfaces are smoother and finer at R ¼ 0 and 0.1 because of the crack closure effect (Figure 5.45). For a higher rate of propagation to 2 or 3  10
6 mm=cycle, some quasi-cleavage has been found Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.44 Combination of (a) Kmax,th vs. R; (b) DKth vs. R; and (c) DKth vs. Kmax,th

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.45

SEM observations at near threshold regime.

locally at high R ratios (Figure 5.46). For a even higher propagation rate of 2  10
5 mm=cycle, fatigue striations can be found for R ¼
1, 0.5, 0.7 (Figure 5.47) but they are difficult to identify for R ¼ 0 and R ¼ 0.1. 5.7. SPHEROIDAL GRAPHITE CAST IRON (SGI) Figure 5.48 shows the SGI results of FCP rates vs. stress intensity factor. For R ¼
1 in ultrasonic fatigue, DK ¼ Kmax

Figure 5.46 SEM observations of quasi-cleavage at R ¼ 0.7; da=dN ¼ 2  10
6 mm=cycle. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.47 Fatigue striations at higher crack propagation rate.

Figure 5.48 Fatigue crack propagation test results at ultrasonic frequency in cast iron. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

is assumed because only the tension part of the cycle contributes to the propagation of crack. The threshold stress p intenp sity is 3.8 MPa m for R ¼
1 and DKth ¼ 6.3 MPa m for R ¼ 0.1. On the other hand, it can be seen from Figure 5.48 that there is very little difference for the rates above threshold beyond 3  10
10 m=cycle. The effect of frequency on FCP behavior has been studied by a number of researchers for various materials, showing that no significant effect of frequency was observed on the threshold values tested in ambient environment. For this SGI, a conventional FCP test has been conducted at a frequency of 35 Hz by Nadot (1999) in both ambient environment and vacuum conditions. The test results were compared in Figure 5.49 with the ultrasonic fatigue tests. It is shown that p the threshold atp35 Hz (8.6 MPa m) is higher than that at 20 kHz (6.3 MPa m). In contrast, in the high propagation rate regime, crack propagation rate is higher at 35 Hz than at 20 kHz for a given DK value.

Figure 5.49 Effect of test frequency on fatigue crack growth in cast iron. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

After fatigue testing, some specimens were broken and carefully cleaned in alcohol in an ultrasonic bath before taken to the SEM examination. Other specimens were cut in the mid-thickness location parallel to the specimen sides. The crack profile, after etching in 3% nital, was observed under SEM in that condition. The fractographic observations indicated that at a high propagation rate (da=dN > 4  10
9 m=cycle), this SGI specimen tested at ultrasonic frequency propagated in a transgranular mode with some quasi-cleavage facets (Figure 5.50). At an intermediate propagation rate, the quasi-cleavage facets disappeared and the rupture mode is mainly transgranular (Figure 5.51). In the threshold regime (below 3  10
10 m=cycle), intergranular failure was observed. However, the transgranular rupture was always present on the fracture surface, even at very low propagation rates (Figure 5.52). 5.8. DATABASE OF THRESHOLD SIF DKth Table 5.17 contains values of threshold SIF for some materials and alloys for practical and industrial use obtained by 20 kHz ultrasonic fatigue techniques. Test conditions are also reported

Figure 5.50 Transgranular quasi-cleavage at high propagation p rate DK ¼ 15.5 MPa m; da=dN ¼ 4  10
9 m=cycle. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

p Figure 5.51 Transgranular rupture DK ¼ 8 MPa m; da=dN ¼ 9.2  10
10 m=cycle.

where available. The data in Table 5.17 include the results discussed in this book and other sources. Some rather early data (Stanzl, 1996) are also listed; all sources are cited and footnoted. Readers should be aware that the data given Table 5.17 depend strongly on the material properties, heat treatment,

Figure 5.52 pMix rupture of transgranular and intergranular, DK ¼ 6.5 MPa m; da=dN ¼ 2.0  10
10 m=cycle. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table p 5.17 MPa m)

Material T6A4V Astroloy N18

Inconel706

AlSiII-1 AlSiII-2 6061 6061=Al2O3=15p 6061=Al2O3=21p f.c.c. Cu f.c.c.steel 304 b.c.c. mild steel Al

Threshold Stress Intensity Factors Obtained by Ultrasonic Fatigue Techniques (DKth in

R 0
1
1
1
1 0 0 0.8 0.1 0.1 0.1
1
1
1
1
1
1
1
1
1
1
1
1
1

Environment Temperaturea Air Air Air Air Air Air Air nitrogen sweep

silicon oil liquid N2 silicon oil nitrogen silicon oil nitrogen oil air

20 C 400 C 450 C 450 C 700 C 450 C 400 C 400 C 20 C

20 C 77 K 20 C 77 K 20 C 77 K 20 C 20 C

DKth 5.66 3.2 6.0 5.0 5.5 8 7 4.5 6.4 5.5 7 2.8 3.0 3.9 5.4 4.7 2.7 3 7 8.5 3.8 5.3 1.0 1.33

Tables (chemical composition, heat treatment, da=dN Source da=dN and mechanical properties) (m=cycle) (see footnote) Figure
11

9  10 5  10
11 1  10
9 1  10
9 1  10
9 1  10
9 1  10
7 1  10
9 1  10
9 1  10
9 5  10
10 2  10
13 2  10
13 3  10
13 3  10
13 3  10
13 7.4  10
14 9  10
14 8.5  10
13 1  10
12 8  10
14 1  10
13 8  10
13 1  10
13

(1) (1) (2) (2) (3) (3) (3) (3) (3) (3) (3) (4) (4) (5) (5) (5) (6) (6) (6) (6) (6) (7) (7)

4.22 4.22 4.26 4.26 4.27 4.27 4.28 4.27 4.29 4.31 4.29 4.39 4.40 4.41 4.41 4.41 4.42 4.42 4.43 4.43 4.44 4.44

Remarks

4.14  4.16 4.22  4.24 4.19  4.21

4.25  4.27

4.30 4.30 4.31 4.32 4.33  4.35 4.33  4.35 4.33  4.35

Grain size and cold work effects (Continued)

a

Material

R

AlMg5 AlZnMg1


1
1

oil oil

20 C 20 C

1.4 1.9

3  10
13 7  10
13

(7) (7)

Cu


1
1
1

Oil liquid N2 air

20 C 77 K 20 C

2.7 2.5 1.4  2.0 1.4  2.6 1.8  2.3

4  10
14 4  10
14 1  10
13 1  10
15 1  10
13

(7) (7) (7) (7) (7)


1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

oil liquid N2 liquid N2 oil liquid N2 oil humid air humid air air air air air air air air air

20 C 77 K 20 K 20 C 77 K 23 C 23 C 23 C 20 C 20 C 20 C 20 C 20 C 20 C 40 C 40 C 20 C

3.8 5.3 7.2 7.0 8.5 6.7 3.6 5.7 2.45 2.44 4.8  5.2 5.8 6.6 7.0 1.8 3.7 13.0

6  10
14 9  10
14 1  10
10 5  10
13 9  10
13 6  10
14 4.4  10
13 4.4  10
13 1  10
12 1  10
12 1  10
13 1  10
13 1  10
13 1  10
13 1  10
13 1  10
13 3  10
12

(7) (7) (7) (7) (7) (7) (8) (8) (7) (7) (7) (7) (7)

Low C Steel

AISI304 X10Cr13 X20Cr13 GGG 100-B 34CrMo4 Ck60 PM-Mo PM-Mo-0.8W PM-Mo-1.5W PM-Mo-Ti-Zr PM-Mb PM-Ta A286

Environment Temperature

Tables (chemical composition, heat treatment, da=dN Source da=dN and mechanical properties) (m=cycle) (see footnote) Figure

DKth

(7) (7) (7)

Remarks

Comparison with NaCl-solutions Grain size effect; cold work effect; single crystals Comparison with NaCl-solutions pffiffiffiffiffi 4.1 MPa m for 70 Hz Comparison With NaCl-Solutions

Grain size effect

XXXX-0 Bathias Ch05 R3 081204

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 5.17 (Continued )

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

IN-738 IN-792


1
1

air air

20 C 20 C

U-700 s.c.


1

air

20 C

INCO.800

Transv. 0.1  0.3

air

20 C

Transv. 0.1 Transv. 0.3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

air air water argon argon argon argon argon water argon argon argon argon argon air air air air vacuum humid air dry air air air air

20 C 20 C 20 C 200 C 250 C 300 C 400 C 500 C 20 C 200 C 250 C 300 C 400 C 500 C 20 C 20 C 20 C 20 C

HASR.X IN 600 CSN 412013 Steel

CSN 415313 Steel

Cr–Si U500 Al-Li8090 Al2024

Mo-3 Mo-0.8W Mo–Ti–Zr

25 C 25 C 25 C

3.13 1  10
12 4.48 1  10
12 no threshold down to 10–11 4 no threshold down to 10–10 5 7.7 6 5 5 4.2 3.05 5.3 4.7 4.4 4.3 3.85 3.4 7 13 8 7 3.3 2.1 2.3 5.2  0.5 5.9  0.4 7.0  0.5

1  10
13

1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 1  10
11 2  10
6 3  10
8 2  10
8 2  10
13 2  10
13 2  10
13 < 10
13 < 10
13 < 10
13

(7)

(7) (7) (7) (7) (7) (9) (9) (9) (9) (9) (9) (9) (9) (9)

(10) (11) (11) (11) (12) (12) (13) (14) (14) (14)

4.33 4.33

4.17, 4.18, 4.38 4.28 4.29 Ed ¼ 322 GPa Ed ¼ 318 GPa Ed ¼ 314 GPa

a Ambient temperature where not indicated. References: 1: Jago, 1993; 2: Wu, 1994; 3: Bonis, 1997; 4: Stanzl, 1995; 5: Papakyriacou, 1995; 6: Tschegg, 1981; 7: Stickler, 1982; 8: Mayer, 1995; 9: Sun, 2001; 10: Kong, 1991; 11: Mayer, 1992; 12: Stanzl, 1991; 13: Weiss, 1982. References: 1: Jago, 1993; 2: Wu, 1994; 3: Bonis, 1997; etc.

test temperature, and environment. Therefore, before using these ultrasonic fatigue threshold data, suitable references should be consulted. For R ¼
1, the compression part of the load cycle was neglected in computing the threshold. 5.9. OTHER APPLICATIONS: FRETTING FATIGUE 5.9.1. Fretting Fatigue of Aluminium–Lithium Alloy and Titanium Fretting fatigue experiments have been performed on aluminium–lithium alloy Al-Li8090 and titanium alloy T6A4V (Tao, 1996). Chemical composition, heat treatment, and mechanical properties of T6A4V can be found in Tables 4.20 through 4.22, while Tables 4.23 and 5.18 list those of AlLi8090. Some mechanical properties differ from those given in Table 5.18, as the heat treatments are not the same. Aluminium alloys of high strength, such as Al-Li8090, are widely used in aeronautic industries. At the same time the menace of fretting to the safety of aeroplane structures is of great concern (Kuzmenko, 1984). More than twenty years ago it was estimated that the existence of a fretting zone was responsible for 90% of fatigue damage (Tein, 1975). As has been indicated, titanium and its alloys like T6A4V, are ideal materials for aeronautical and aerospace equipment. But the sensitivity of these materials to fretting is so high it is an obstacle to their utilization, particularly the latter. Fretting can reduce the fatigue strength of titanium alloys by 40% to 70% (Li, 1992; Lutynski, 1982). Specimens and pads of the two materials are used in order to form the following three types of experiments:

Table 5.18 Heat Treatment and Mechanical Properties of Al-Li8090 Heat treatment

sm (MPa)

s0.2 (MPa)

E (GPa)

r (kg=m3)

2h30 in 106 C

434

352

80

2530

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

1. Specimen of Al-Li8090, pads of T6A4V 2. Specimen of Al-Li8090, pads of AL-Li8090 3. Specimen of T6A4V, pads of T6A4V The geometry of the pads and specimens is given in Figure 3.24 on p. 79. All the tests are carried out at 20 kHz, ambient temperature of 25 C, and humidity of about 60%. Before each test the position of pads is chosen and adjusted; after the test the position of pads is measured. The relative slip amplitude and stress si are re-calculated. With the above three types of specimen–pad combinations, we designed five groups of tests with conditions and results listed in Table 5.19. Test 1. After 1.2  107 cycles, we observe typical signs of fretting. On the specimen surface the platelets and small cracks due to fretting are evident. The debris and the surface are oxidized. At the same time fretting also takes place on the surface of pads of T6A4V; there are a lot of platelets and debris. The possibility of fretting at very high frequency is confirmed by these results, and alloy T6A4V seems sensitive to fretting at this high frequency of 20 kHz. Test 2. After 6  106 cycles, fretting occurs in the specimens (Figure 5.53). The signs of fretting are larger and more distinctive, because the amplitude is higher. From Ai ¼ 0.2 mm to Ai ¼ 0.5 mm, in the border of the friction zone we can distinguish the signs of successive stages of fretting: stripes, platelets, and small cracks. In some stripes we find thin platelets. When Ai ¼ 1.2 mm, the fretting is significant and the surface is covered with platelets and ejected oxidised debris. The relative slip amplitude constitutes an important factor in vibration fretting as in classical fretting. Test 3. Fatigue lives of the fretting fatigue specimens are Nf ¼ 4.8  106 and 3  106 cycles. The two specimens were cracked in fatigue after the fretting where the stress si ¼ 158 MPa and 236 MPa. The specimens for comparison without fretting were not broken after Ni ¼ 5  107 cycles for smax ¼ 161 MPa Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Conditions and Results of the Tests

Test

Material (specimen–pads)

FN (N)

A0 (mm)

1

8090–T6A4V

30

15

2-1 2-2 2-3 2-4

8090–8090

15 15 15 15

3 3 3 3

3-1 3-2

8090–8090

15 15

90 135

3-3 3-4

196

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 5.19

90 135

xi (mm)

Ai (mm)

smax (MPa)

35

7

26.8

3 5 7.5 18.5

0.2 0.34 0.5 1.2

9 9

18 27

5.36 5.36 5.36 5.36 161 241

161 241

si (MPa) 19

5.35 5.35 5.29 4.91 158 236

Ni or Nf (107) Ni ¼ 1.2

Ni ¼ 0.6 Ni ¼ 0.6 Ni ¼ 0.6 Ni ¼ 0.6 Nf ¼ 0.48 Nf ¼ 0.3

Ni ¼ 5 Ni ¼ 5

Notes Fretting only, in order to confirm the possibility of fretting at very high frequency. Fretting only, in order to observe the influence of the relative slip amplitude in the case of Al-Li 8090. Weak influence of stress is not considered. 3-1 and 3-2: Fretting–fatigue, in order to study the influence of fretting at very high frequency on the endurance in the case of Al-Li 8090. 3-3 and 3-4: Same tests with same parameters of vibration but without the contact of pads, in order to determine the importance of the fretting effect. Chapter 5

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

4-1 4-2 4-3 4-4 4-5 5-1

5-2

T6A4V–T6A4V

T6A4V–T6A4V

15 15 15 15 15 15

10 10 10 10 20 100

100

1.2 3.2 8.0 19.5 29.5 2.0

0.3 0.8 2 4.7 13.5 5

27.7 27.7 27.7 27.7 54.4 277

277

27.69 27.6 27.15 24.3 40.8 276

Nf ¼ 0.6 Nf ¼ 0.6 Ni ¼ 0.6 Ni ¼ 0.6 Ni ¼ 0.6 Nf ¼ 0.24

Nf ¼ 0.6

Fretting only, in order to observe the influence of the relative slip amplitude in the case of T6A4V. Weak influence of stress is not considered. 5-1: Fretting–fatigue, in order to study the influence of fretting at very high frequency on the endurance in the case of T6A4V. 5-2: Same tests with same parameters of vibration but without the contact of pads, in order to determine the importance of the fretting effect.

FN ¼ normal load on the pads; Ni ¼ number of cycles on the testing wear; Nf ¼ number of cycles on the life wear. See also Figure 3.24.

Figure 5.53

Fretting scar on T6A4V (N ¼ 6106; Ai ¼ 0.2 m).

and 241 MPa, much higher than that of fretting fatigue tests. The conjugation of the platelets and small cracks form large cracks that develop in the specimens and finally provoke a rupture. Around the rupture it is seen that fretting develops to a crack. In this way, at high frequency, the vibration fretting initiates a crack earlier and reduces the fatigue strength of the material, as in the case of classical fretting. Test 4. The results in Figure 5.54 (Sun, 2001) are similar to those of Test 2. Again, the relative slip amplitude influences fretting. Alloy T6A4V exhibits a sensitivity to fretting at high frequency. Even for low relative slip amplitude of 0.3 mm, platelets are produced. As the relative slip amplitude becomes higher than 0.8 mm, fretting was much more pronounced. Test 5. For the fretting fatigue specimen with fatigue life Nf ¼ 2.4  106 cycles, crack initiation happens at the point of fretting. Figure 5.55 (Sun, 2001) shows that fretting appears in the crack. The fatigue life of the comparison specimen without fretting was Nf ¼ 6  106 cycles. This specimen was broken at point xi ¼ 0 where the stress has the highest value. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.54 Fretting scar on T6A4V (Ai ¼ 0.8 mm).

These preliminary results demonstrate that vibration fretting can substantially decrease the fatigue strength in the case of T6A4V. This material should be considered as very sensitive to classical fretting.

Figure 5.55 Fretting scar and fatigue crack on T6A4V. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

In conclusion to results discussed above, the following remarks are made:  Fretting occurs under vibration friction at very high frequency of 20 kHz.  The relative slip amplitude is an important factor. The larger the amplitude, the more typical and more pronounced the signs of fretting become. In alloy Al-Li 8090, the fretting is very detrimental when the amplitude exceeds 1 mm.  Fretting due to high frequency triggers earlier cracks and leads to rupture, if the stress is high enough. Fatigue strength is reduced by vibration fretting as well as by conventional fretting.  Alloy T6A4V is very sensitive to vibration fretting as well as to conventional fretting.  The test method of vibration fretting is very rapid, and can be performed with variable displacement (0.1 m to a few dozen microns), and with or without fatigue stress. With this method, studies on very high cycle fretting fatigue strength can be carried out. 5.9.2. Fretting Fatigue of High Strength Steel The material studied was low alloy and high strength steel 42CrMo4U or 4240. (For chemical composition, heat treatment, and mechanical properties, see Tables 4.1 through 4.3.) The specimen for ultrasonic fretting fatigue had a cylindrical profile with different diameters and was asymmetrical to amplify the fatigue stress in the gauge length. (See the distribution of the vibration displacement and stress presented in Figure 3.24.) The specific length, L, is determined for a specimen with a resonance frequency of the first longitudinal vibration mode of 20 kHZ (Sun, 1999). L ¼ X1 þ X2   p L1 L2 ¼
S þ k S1 S2 Copyright © 2005 by Marcel Dekker. All Rights Reserved.

where k is a constant rffiffiffiffiffiffi r k ¼ 2pf Ed and S is the section area of the cylinder. In the test, a maximum displacement is achieved at free ends while the maximum strain (stress) is obtained in the gauge length of the specimen (Figure 5.56). In this test system, the fretting slip amplitude and the fatigue stress are the vibration displacement and vibration stress, respectively, at the point on the gauge length of the specimen where the pads are placed. They depend upon the position of the pad and the maximum vibration amplitude of the specimen. The latter is determined by the power of the generator and the amplification of the horn. In our experiments, this can vary from 3 mm to 95 mm. By regulating the position of the pads along the specimen and by changing the power of the generator, either the slip amplitude or the fatigue stress or both can be changed. As a result, these two parameters are dissociated (Sun, 1999). The fretting pad also has a cylindrical surface (Figure 5.57); it consists of the same materials as the specimen. The pads are held on the specimen by two springs. The sides normal contact force is Fn ¼ 30 N and the slip amplitude is about 17 mm. The conventional method to establish the important variables that can affect fretting fatigue is to generate S-N curves with and without fretting, allowing fretting fatigue strength

Figure 5.56 Ultrasonic fatigue and the distribution of vibration displacement and stress. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.57

Contact pad.

reduction factors to be evaluated. Such a curve is given for two 4240 steels (B and C in Figure 5.58), which reveals that fatigue strength is significantly reduced by fretting fatigue, and the factor of reduction is of the order of 3 but varies with the logarithm of the number of cycles in a linear relation (Figure 5.59). Moreover it is found that the reduction is greater on Rep-C (compared with Figure 4.2) with a higher tensile strength. These tests results agree well with others studies (Lindley, 1997; Li, 1992; Nakazawa, 1992). Many studies have shown that fatigue initiation is likely to occur at the surface. In ultrasonic fatigue, crack initiations

Figure 5.58

Fatigue S-N curve with and without fretting.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.59

Fatigue strength reduction factor caused by fretting.

were observed at the surface when the number of cycles was less than 107; otherwise, crack initiation occurs on the internal defects. In fretting fatigue, however, the initiation of fretting fatigue cracks is caused by the surface stress resulting from frictional forces, together with the bulk stress; they always occurs at the contact surface (Figure 5.60). Therefore, there is a consequent sharp reduction of fatigue strength in the high cycle regime. In conventional fretting fatigue tests it has been shown that cracks initiate at a very early stage (5–10%) of fretting fatigue life (Lindley, 1997; Li, 1992; Nakazawa, 1992). To define well fretting fatigue crack initiation and propagation conditions at ultrasonic frequency, a two-stage test is performed in which the fretting contact pads are applied on the specimen for a certain number of cycles in the fretting fatigue test and then removed, and subsequently the specimen tested in plain fatigue (fatigue without further fretting). The results are shown in Table 5.20. In the first test with a very low fatigue stress, no crack was observed at the very early stage. This Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.60

SEM of fretting fracture surface.

means that the fretting is a combined effect of the tangential friction force and the fatigue bulk stress. Both of these two parameters contribute to the crack initiation in fretting fatigue. On the other hand, from the results in Table 5.20, it can be seen that when the number of cycles in the fretting period is lower than a certain limit, failure does not occur in the following plain fatigue, even after a very long time. In ultrasonic fretting fatigue, this limit is higher (more than 50%), which implies that fretting fatigue crack initiation takes a great fraction of the whole fatigue life, as in the ultrasonic plain fatigue (Sun, 2001). In conventional fretting fatigue, some analyses and experiments show that the crack initiates at the contact edge Table 5.20 Two-stage Fretting Fatigue Test for 4240 Rep-B Fatigue stress Specimen (MPa) No. 1 No. 2 No. 3

172.9 198.3 221.9

N1

N2

Nf

4.4693  106 2.7782  109 2.2563  109 6.5792  107 2.5710  109 2.8659  108 2.0006  107 1.3757  109 3.9383  107

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

N1=Nf (%) Observation 0.20 23.0 50.8

Nonrupture Nonrupture Nonrupture

where the tangential force and the slip amplitude are maximum (Wright, 1971; Realigns, 1972). But others (Waterhouse, 1971; Nakazawa, 1992) observed that a crack was initiated at the boundary between an area of slip and an area of non-slip. A possible interpretation is that if relative slip is large enough to induce severe fretting wear, the rigidity of the surface contact layer in the slip region will be significantly reduced (Mutoh, 1995). Therefore in the whole slip region the contact force and hence the tangential force are decreased. However, a very high concentration of contact and tangential forces will occur in the slip region near the boundary between slip and non-slip regions. It is thought that the crack initiation point depends upon the condition of contact. In this ultrasonic fretting fatigue test where a particular contact of cylinder-oncylinder was used, the contact width is increased with the number of cycles, and the crack initiation occurs at the edge region of the initial contact (Figure 5.61). At the early stage of fatigue life, a crack was initiated on a plane inclined to the surface and then the propagation direction changed to approximately perpendicular. This type of crack path is very common in conventional fretting fatigue (Mutoh, 1995). The experimental results in Figure 5.59 show that fatigue failure can occur after more than 107 cycles, and even over 108 cycles, which reveals that for the fatigue design, the fatigue limit usually determined at 107 cycles is not always appropriate. This phenomenon indicates again the importance of a high frequency fatigue test technique. Fretting not only accelerates crack initiation but, also increases the rate of crack propagation. Hence, fretting fatigue life is likely to be determined by the continued propagation of small fretting cracks. But there exists a threshold of stress intensity factor in fretting fatigue below which a fretting crack does not propagate. In this case, fretting scars are larger in the form of an ellipse. Considerable fretting wear is encountered over the entire contact area, at the surface of both the specimen and the pad (Figure 5.60). The contact surface increases with the number of stress cycles. Red oxide debris is observed at the contact surface and the examination of Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 5.61 tion path.

Fretting fatigue crack initiation point and propaga-

fretting scars demonstrates some fine, but non-propagating, cracks at the surface. In conclusion, an ultrasonic fretting fatigue test performed on high strength steel demonstrates that there is a significant reduction in fatigue strength caused by fretting. The factor of reduction is of the order of three times and increases with the number of cycles. Fretting fatigue failure is a combination of fretting slip and fretting stress. In ultrasonic fretting fatigue, crack initiation occurs at the edge region of the initial contact area, and is present over a greater fraction of the whole fatigue life. There is a minimum fatigue stress below which fretting cracks do not propagate. In this case, fretting scars are larger and fretting wear is considerable. For more information about fretting fatigue of high strength steel at ultrasonic frequency, see (Sun, 2001).

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

6 Frequency and Environmental Effects

6.1. FREQUENCY EFFECT The ultrasonic fatigue technique is an accelerated testing method with a frequency far beyond that of conventional fatigue experiments and applications. A problem is naturally posed: Is there an important frequency influence on the experimental results? If the frequency effect is too great, the usability of the test data might be in question. Fortunately, results in the literature and our own experience indicate that for ultrasonic fatigue tests under low displacement amplitude and small deformation conditions, the frequency effect is small in most cases. Quite a few similarities between ultrasonic and conventional fatigue data have already been noted in previous chapters. The frequency effects are discussed in this chapter in detail. We will show that the frequency effect is much more significant between 10 Hz and 0.001 Hz than between 10 Hz and 20 kHz.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

6.1.1. S-N Curve and Fatigue Limit There have been numerous studies of different materials and microstructures considering the frequency effect on S-N curve and fatigue limit. Some materials respond differently to ultrasonic fatigue loading than others. For example, for the nickelbased alloy Udimet 500, the ultrasonic fatigue data closely match the conventional fatigue tests results between 105 and 107 cycles. For this material, the fatigue limit diminished continuously beyond 107 to 109 cycles when tested at 20 kHz. (see Figure 4.21). Some materials even present slightly better fatigue resistance behavior in ultrasonic fatigue regime as shown in Figures 4.19 and 4.20 for titanium alloy T6A4V at room temperature. (Kuzmenko, 1984) found that, for a number of steel and other alloys [a carbon steel–steel 45 (0.5C0.8Mn-0.25Si), an aluminium alloy D16T (4.5Cu-1.5Mg0.6Mn), and the titanium alloys OT4-1 (Ti-2Al-1.5Mn) and VT22 (Ti-7.5Mo-2.5Al-1.0Cr-1.0Fe), for example], fatigue strength increases monotonically as the loading frequency is raised from 10 Hz to 20 kHz. But for a chromium-nickel alloy E1612 (35Ni-15Cr-3.0W-1.5Mn-1.2Ti) and a steel X18N10T (10Ni-18Cr-1.5Mn-0.6Si-0.5Ti), no noticeable frequency effect on the fatigue behaviors and on the S-N curves has been found for 20 kHz and 20 Hz tests. (Weiss, 1982) claimed that the loading frequency effect diminishes as the strength of the material becomes greater, and concluded quite correctly that insufficient cooling may be the reason for these results. That means that the reported differences are not intrinsic but essentially temperature effects. (Yeske, 1982) reported experimental fatigue life data on a nickel based super-alloy MAR-M-246, tested over a frequency range of 3.2 Hz to 23.2 kHz, and a cast copper-nickel alloy tested at 30 Hz and 13.6 kHz. No appreciable frequency effect on fatigue properties was noted for those two materials. (Tschegg and Stanzl, 1981) found that, in general, f.c.c. materials are only ‘‘mildly’’ frequency sensitive but b.c.c. materials are highly frequency dependent. The explanations of this phenomenon recalled the fact that the dislocation sources and the slip systems for f.c.c. materials remain active up to ultrasonic frequency, Copyright © 2005 by Marcel Dekker. All Rights Reserved.

while for b.c.c. materials, the frequency effect results from the increase in material strength as the strain rate or loading frequency increases. Applying this logic to the fatigue situation, higher frequency displacement controlled fatigue test data should then tend to shift toward higher fatigue strength and longer fatigue lifetime when plotted on an S-N diagram. From a comparison of the S-N curves of a high purity copper in the high cycle regime, a reasonable similarity of fatigue data between bending tests of normal frequency and ultrasonic tests is found, but the axial load control fatigue data give a lower fatigue strength. To sum up the effect of high frequency on the S-N curve it appears that the data obtained are only slightly or not at all affected between 10 Hz and 30 kHz. Figure 6.1 gives the best examples. According to published investigations and our own observations (Marines, 2003), no appreciable difference could be found in the fatigue crack growing mechanism between ultrasonic and conventional loading regimes. Nevertheless, the matting effect and the area reduction effect are hardly important in facets of the specimens damaged in ultrasonic fatigue tests, even with a cyclic stress ratio of R ¼
1. Figures 6.2 (for T6A4V) and 6.3 (for 17-4PH) make a comparison of the rupture feature of specimens loaded in the two different regimes (tests are performed at room temperature). This is because in ultrasonic fatigue tests, the specimen vibrates at longitudinal resonant frequency and the vibrational energy is transmitted by the remaining ligament and reflected on the crack facets, resulting in a smaller closure effect at the crack tip and a smaller plastic zone dimension by comparison with that obtained in conventional fatigue loading. Additionally, it is shown that for most engineering materials, the crack grows at a lower rate da=dN, corresponding to a lower stress intensity factor DK applied when loaded at ultrasonic frequency in resonant vibration. It seems that at cryogenic temperature, the frequency effect decreases. This is the case for material Ti-6A4V. At 77 K, the results of vibratory fatigue and conventional Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 6.1

Comparison of S-N curves at low and high frequency.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 6.2 Fatigue crack growth with striation formation for T6A4V (R ¼
1) at 20 kHz.

fatigue are quite similar as shown in Figure 6.4 (Jago, 1998). Investigations in Stanzl’s laboratory on a steel alloy and several aluminum and magnesium alloys at different testing frequencies also confirm that a real intrinsic frequency effect cannot be detected, as long as elasticity dominates the mechanical fields of the tested materials (Stanzl, 1996). It must be pointed out that the old high-frequency results must be considered with caution because it is always difficult to control a high frequency fatigue machine. The reader must know that a piezo-electric machine, working at 20 kHz, cannot be controlled without a computer processor type 486 or better. This means the older data before 1990 are suspected to have errors. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 6.3 Fatigue crack growth with striation formation at high and low frequency for 17-4PH.

6.1.2. FCG Rate and Threshold Since crack propagation involves cyclic deformation at the crack tip, similar frequency effects may be expected for crack propagation itself. Therefore, one might expect a weak dependence or none at all due to strain rate in f.c.c. materials and a rather complex response of b.c.c. materials, resulting in the possibility of inter-granular or cleavage fracture (Laird, 1982). According to (Stanzl, 1996), the following experimental results available thus far are:  No influence of the testing frequency is observed for 13% chromium steel in tests in inert environment at testing frequencies between 0.1 Hz and 20 kHz (Speidel, 1980). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

213

Figure 6.4 Comparison of fatigue strengths of T6A4V at 20 kHz and 33 Hz in liquid nitrogen.

 No influence is also observed for grey cast irons over a frequency range of 5 Hz to 21 kHz, as well as for mild steel, tested at 100 Hz and 20 kHz in inert oil at 20 C (Stanzl, 1986).  No measurable influence of frequency on FCG rates between 2.3 Hz and 20 kHz at room temperature is reported for three nickel alloys (Hastelloy-X, RA-333, and IN800-H) (Hoffelner, 1982). It should be mentioned, however, that the range of scatter of the ultrasonic results is up to approximately 30% in all cases, mainly due to lack of precise amplitude control.  More recent measurements (Hoffelner, 1982) on Armco iron at frequencies of 100 Hz and 0.1 Hz at room temperature in air confirm the frequency independence of the threshold value. However, pronounced frequency effects have been found for a carbon steel by (Puskar, 1986) with threshold values increased by a factor of 1.8 but the crack propagation rates decreased by a factor of 16 at 22 kHz from 70 Hz, and by (Kuzmenko, 1984) for a titanium alloy VT-1 and an aluminium alloy AK4. One might suspect, however, that their DK values have not been accurately determined. Meanwhile, recent experimental data on FCG rates and threshold at ultrasonic frequency in our laboratory have little Copyright © 2005 by Marcel Dekker. All Rights Reserved.

214

Chapter 6

scatter and show only some weak frequency effect. For nickel based alloy N18, the influence of frequency on FCG curves at high temperature is due undoubtedly to different environmental influences at ultrasonic and conventional frequencies (see Section 5.2). A small frequency effect on FCG rate of another nickel based alloy (Inconel 706) may be caused by the difference in the interaction time between-environment and the metal at crack tip. That is to say, an intrinsic frequency effect on FCG rate and threshold is not observed. 6.1.3. Dislocation Structures and Plateau Stress In this section we mainly cite remarks made by (Stanzl, 1996) concerning the frequency effect on dislocation structures and plateau stress. In a fundamental consideration, (Laird, 1982) compares the strain rate dependence of cyclic deformation and the dislocation movement in f.c.c. and b.c.c. materials. In f.c.c. metals, where the cyclic flow stress is weakly dependent on strain rate, little frequency effect on fracture mechanism is expected. In b.c.c. materials that are sensitive to high strain rate, strain aging, dislocation glide, and shape changes can be affected as well as the failure mechanism, by favoring inter-granular failure during crack initiation and cleavage during crack propagation. Several extensive studies have been performed during the last ten years on the dislocation structures and the stress-strain response of single crystals. TEM studies on the dislocation structure of ultrasonically loaded Cu single crystals showed no significant differences in the dislocation structures below as well as at plateau stress compared with those established under conventional fatigue (Stanzl, 1986). The plateau stress, where PSBs (persistent slip bands) are formed, is found at 26 MPa at room temperature, which is close to the value of 28 MPa obtained under plastic strain controlled testing conditions at conventional frequencies. This coincidence is remarkable, since ultrasonic load control was not as accurate in 1984 as it is today (about 95% accuracy). In addition, the ladder spacing of the PSBs is approximately the same as Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

215

observed for conventional frequencies. In (Stanzl, 1996), a replica of a PSB is observed, which clearly shows how a micro crack is formed in a PSB. In a similar study (Tschegg, 1981), at 77 K the samples of copper single crystals of 99.9999% purity show a plateau stress of 47  2 MPa at 20 kHz, while the value for the same material tested at low frequency is 48 MPa. Ultrasonic fatigue loading at 77 K (Buchinger, 1984) likewise causes similar dislocation arrangements as with conventional frequencies. As an example, (Stanzl, 1996) shows a typical fully developed LP (loop patch) structure. Furthermore, the ultrasonic fatigue of a polycrystalline Cu-16% Al alloy also generates a dislocation structure similar to the one caused by conventional frequencies in the monocrystalline alloy (Laird, 1986). However, one does find in (Laird, 1986) that, for high frequency testing and in dense dislocation structures that are essentially dipolar, the dipole width is considerably smaller than that observed in low frequency testing. (Mayer, 1994) presents an experimental study on polycrystalline copper at different frequencies between 0.5 Hz and 8 Hz under load and strain control, and finds that under load control the formation of PSBs depends on the loading frequency, but that no such influence is present for strain controlled loading. As ultrasonic fatigue loading is essentially strain controlled, this result explains very well the above mentioned frequency independent formation of dislocation structures, and is another verification of the non-existence of a real inherent frequency effect. Furthermore, a frequency change from 0.5 Hz to 2 Hz generally produces more significant effects than one from 2 Hz to 8 Hz. This observation is helpful in explaining little frequency effect at high frequency for most materials. The above results agree with another basic study on the deformation mechanism, i.e., the so-called Blaha effect (influence of superposition of cyclic ultrasonic loading on unidirectional loading) (Kirchner, 1984). There is no frequency effect in the load response of several aluminium alloys at room temperature. For specimens tested at frequencies of 0.5 Hz, 10 Hz, 50 Hz, and 20 kHz, the stress drop is caused by elastic Copyright © 2005 by Marcel Dekker. All Rights Reserved.

216

Chapter 6

relaxation, not by high frequency. On the other hand, discussion of this effect was initiated again by (Kobayashi, 1991) who did find a frequency influence. It may be suspected that this might be caused more by the experimental procedure than by frequency effect. 6.1.4. Fretting Wear (Soderberg, 1986) observed a similar effect of frequency in the fretting of mild steel at ultrasonic frequencies and at low frequencies, with respect to both wear rates and wear mechanisms. 6.2. HEAT EFFECT High temperature ultrasonic fatigue testing of materials that are designed for high temperature fatigue applications has been developed by our group in recent years, after several works on this subject had been published in the past. The testing system is presented in Chapter 3. The machine is designed to have 20 kHz resonant frequency. The displacement amplitude of the specimen extremity is measured by means of a capacitive transducer, which permits evaluating the stress intensity factor with aid of FEM. A heating solenoidal inductor is intalled at the specimen center. The maximum temperature is at the center section with a crack and the minimum temperature is at the ends; this gradient must be taken into account in numerical calculation. So, the resonance length L2 is less in elevated temperature. A system of video-camera television has been used for the detection of crack initiation and propagation, the system resolves events to 1=25th of a second and magnifies the picture 200 times to obtain the threshold at 10
12 m=cycle during 1 hour. The performing of comparative fatigue crack growth measurements on the nickel alloy Astroloy at 20 C and 400 C in an induction furnace by (Wu, 1994) shows that the fatigue crack growth resistance decreased by about 30% at 400 C (Figure 6.5). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

Figure 6.5

217

FCGR of Astroloy.

It is seen that the threshold is determined for a low rate: 10
12 m=cycle for ultrasonic fatigue and 10
10 m=cycle for conventional fatigue with CT specimen. At 20 kHz, the threshold is lower at 400 C than at room temperature. (Hoffelner, 1982) believes that there is no frequency influence between 20 kHz and conventional fatigue. In Figure 6.5, two curves at 20 Hz at room temperature for R ¼ 0.05 and R ¼ 0.7 are found. The first can be considered as the Kmax curve because of Kmin ¼ 0. The second can be considered as the Keff curve because for R ¼ 0.7 there is no closure effect. Between them is the ultrasonic curve at 20 C, which is near the curve at R ¼ 0.7. In the study by Hoffelner (1982), we find the same phenomenon: The threshold at 20 kHz is less than at 60 Hz. Figure 6.6 shows FCGR as a function of DK for N18 at different temperatures in vibratory fatigue. At 20 kHz, the threshold values are also determined at very low FCGR, and they are lower at elevated temperatures than at room temperature. However at 400 C, the threshold is lower than Copyright © 2005 by Marcel Dekker. All Rights Reserved.

218

Figure 6.6

Chapter 6

FCGR of N18 at high temperature.

at 650 C and 700 C. It is well known that oxidation can decrease crack propagation under certain conditions. So, the threshold increase at 400 C can be explained by oxidation for N18. In fractographical observation, oxidation phenomenon is found in the fractures at 650 C and 700 C. (Hudak, 1988) has studied Astroloy in conventional fatigue. He also found that the threshold was lower at 200 C than at 600 C. Obviously, oxidation has a similar effect at 20 kHz and low frequency. Heat effect relates closely to frequency effect, because under high frequency with alternating loading the internal friction, or damping, of materials results in an elevation of the temperature in the specimen, most notably at the crack tip. This, in turn, changes the mechanical properties of the tested material. At 20 kHz for high damping materials, the specific energy input (heat source due to thermodynamic coupling) at high stress may reach as high as several hundred watts per cubic centimeter. This is another aspect of heating effects that must be considered in some cases. Generally, suitable cooling measures are required. The coolant may be air or water, depending on the damping of material and=or Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

219

the geometry of the specimen. For example, a compressed air cooling system can maintain the temperature at the center of a T6A4V specimen at 35 C, which effectively eliminates the temperature rise caused by the ultrasonic frequency effect. On the other hand, if cooling is introduced, environmental effects such as corrosion fatigue must be considered in some cases. Figure 6.7 shows the temperature rise at the center of a cast iron endurance specimen, tested at different stress levels (Wang, 1998a). If there were no cooling due to the compressed air, the temperature rise would be considerable even for low stress levels. For this kind of high damping material, a thin sheet specimen is required. On the other hand, titanium alloys exhibit little heat sensitivity at 20 kHz and at modest stresses, so there is no restriction in the specimen geometry within temperature rise limitations for these alloys. Another possibility to avoid the heating of the specimens is to perform loading in a pulsed manner with periodic interruptions. According to (Stanzl, 1981), usually a pulse length of 500 to 1000 cycles is suitable with pauses between 50 and 1000 ms.

Figure 6.7

Temperature rise with stress levels for cast iron.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

220

Figure 6.8

Chapter 6

Variation of Young’s modulus with temperature.

For highly heat sensitive materials, one must take the variation of Young’s modulus into account in order to obtain a correct mechanical field. Figure 6.8 gives the decrease of Young’s modulus of Astroloy with increasing temperature (Wu, 1992). 6.3. CRYOGENIC TEMPERATURE Ultrasonic fatigue tests have been performed on three different thermomechanical processes (TP) of the Ti-6246 alloy usually used in high temperature conditions. Two b-forged titanium alloys (TP1: fine lamellar primary a structure; TP2: coarse lamellar primary a structure) and one a þ b forged processed (TP3: coarse equiaxed and fine lamellar primary a) have been studied (Jago, 1995) to compare fatigue-limit differences at 109 cycles. Results in liquid nitrogen are grouped in Figure 6.9. For all conditions, fatigue-limit stress resistance is higher at low temperature than at room temperature. There is a cross-over of TP1 and TP3 S-N curves: At cryogenic temperature, TP1(1) fatigue-limit value (sd) is higher than that for TP3; this behavior is reversed at Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

Figure 6.9 and 77 K.

221

S-N curves for TP conditions of Ti-6246 alloy at 300 K

room temperature. In addition, TP1(1) curves are significantly above those for TP1(2). For both test temperatures, the TP2 condition has the lowest fatigue-limit strength properties at 109 cycles. Observations on the effect of cryogenic temperature at 20 kHz follow. Figure 6.10 presents a comparison of S-N curves of alloy Ti-6A4V tested in liquid hydrogen and liquid helium with conditions of 20 K and 20 kHz. These

Figure 6.10 S-N curves of Ti-6A4VPQ at liquid hydrogen and liquid helium (20 K; R ¼
1). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

222

Chapter 6

experiments were performed as described in (Tao, 1996). The difference in fatigue strengths for the two environments is of the order of several tens MPa. It is believed that this difference is caused by the difference in real temperatures around the specimens. The temperature of a specimen in liquid hydrogen is about 20 K, but for the tests in liquid helium, the situation is different. Considering the place of entry of the liquid helium fluid, the specimen, and the temperature sensor, we find there is a temperature gradient in the cryostat (Figure 6.11). The real temperature of the gas of liquid helium around the specimen is lower than 20 K (between 10 K and 20 K). This drop of local temperature increases the fatigue life of the specimen. Another interesting result is given in Figure 6.12 which shows the S-N curves obtained in liquid hydrogen for aluminium alloys 6061 and 2219, tested at 20 kHz. For those alloys the fatigue strength is much higher at 20 K than at 300 K.

Figure 6.11

S-N temperature gradient in the cryostat.

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Frequency and Environmental Effects

223

Figure 6.12 S-N curves for Al alloys in liquid hydrogen.

Thus, the effect of cryogenic temperature is the same for titanium and aluminium alloys, which is considered an effect of microstructure on fatigue strength in the gigacycle fatigue regime. For the Ti-6246 alloys the fatigue strength at 109 cycles is more than 200 Mpa higher at 200 K, depending on the microstructure (Jago, 1996; Tao, 1996). 6.4. ENVIRONMENTAL EFFECTS As indicated (Stanzl, 1996), the question of the influence of frequency on the results obtained in ultrasonic fatigue experiments is especially relevant when environmental influences (i.e., time dependent processes) become effective. After numerous research in the past, corrosion fatigue at ultrasonic frequencies was further investigated mainly by (Ebara, 1994, 1987), where the influence of aggressive environments, such as aqueous NaCl solutions on stainless steel 13Cr and NaOH solutions on alloy Ti6A4V, was found. The fatigue strength of 13Cr at 1010 cycles in an aerated 3% NaCl aqueous solution Copyright © 2005 by Marcel Dekker. All Rights Reserved.

224

Chapter 6

was 74% lower than that in the atmosphere. This reduction is almost equal to that of 75% at 3.7  107 cycles in the result of the conventional rotating bending fatigue test at 60 Hz. This observation confirms again the suitability of the ultrasonic corrosion fatigue for this material. Besides the expected reduction of fatigue strength, the experiments with Ti6A4V in an NaOH solution revealed a pronounced additional decay of fatigue strength at around 1010 cycles. The degradation of the specimen due to corrosion pits is assumed to be most likely responsible for this phenomenon. Because of the very high number of cycles, this can only be investigated by ultrasonic fatigue techniques. Stanzl’s group has explored the influence of humid and dry air, as well as vacuum on the FCG behavior of the aluminium alloy 2024-T3 (Stanzl, 1991). In humid air, a plateaulike crack growth regime (Figure 5.12a in Stanzl, 1991) at crack growth rates between 10
10 and 10
9 m=cycle is observed, which is attributed to hydrogen effects. The results show unambiguously that the plateau in the curve of threshold is mainly determined by the environment and not by micro-structural features. The increase in crack growth rates depends on the water vapor content in air. Dry air experiments lead to crack growth rates between those obtained in humid air and those in vacuum. The plateau-like regime occurs at fatigue crack growth rates of about an order of magnitude lower than that reported in the literature for a loading frequency of 35 Hz. This difference may be explained by the mechanism of surface diffusion of water vapor molecules to the crack tip being the time governing process. With the help of SEM studies of the fracture surfaces, relevant crack growth mechanisms are discussed by (Stanzl, 1991). Figure 6.13 shows another approach to the environmental fatigue limit at 109 cycles and at ultrasonic frequency for 174PH and Ti6A4V, and stainless steel 403 in an aggressive environment (Willertz, 1982). Comparison of high and low frequency corrosion fatigue tests by (Willertz, 1982) shows that adjoining and overlapping fatigue curves are produced for these materials in both aggressive and non-aggressive environments over the range of frequencies examined (40 Hz to 20 kHz). Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

225

Figure 6.13 Ranking of fatigue strength of three engineering alloys tested at ultrasonic and conventional frequencies (Willertz, 1982).

(Roth, 1985) indicates that the degradation in fatigue properties due to environment–fatigue interaction is obvious by comparing fatigue limits in pure water with those in any other environment. For example, the 109 cycle fatigue limit of stainless steel 403 in air-saturated 22% sodium chloride solution is 1=8 of the fatigue limit in pure water. Varying degrees of degradation of the 109 cycle fatigue limit are observed for the other combinations of materials and environments (Figure 6.14). It is also remarked by (Roth, 1985) that the fatigue limit at ultrasonic frequencies for some engineering alloys exhibits sensitivity to the imposition of electrochemical potential during testing similar to that observed at lower frequencies. (Whitlow, 1982) investigated the frequency effect on corrosion fatigue of Udimet 720, a nickel based alloy, at high temperature of 704 C and at the frequencies of 48 Hz and 20 kHz. The tests are carried out in both air and an aggressive molten salt environment. The conventional and ultrasonic frequency S-N data in each environment are shown to be essentially the same. In air, it is suggested that the alloy Copyright © 2005 by Marcel Dekker. All Rights Reserved.

226

Chapter 6

Figure 6.14 Effect of cathodic polarization on ultrasonic and conventional-frequency corrosion fatigue of AISI 403 stainless steel in aqueous chloride solution (Roth, 1985).

behavior is independent of strain rate in the range studied. The observed degradation in life for the specimens tested in salt solution as compared to those in air suggests that fatigue resistance is reduced on a per cycle basis regardless of the cycle duration. Whitlow further concludes that the use of ultrasonic fatigue to extend the S-N curve to the gigacycle regime may produce reliable long-term data. (Yeske, 1982) reported the ultrasonic corrosion fatigue properties of some surgical implant materials (tantalum, niobium, and stainless steel) in the environments of pure water and aggressive modified Early solution. The results show that at the test frequency of 20 kHz, the fatigue life of stainless steel is significantly reduced by the corrosive environment Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Frequency and Environmental Effects

227

while no such effects could be observed for tantalum and niobium specimens. It is not easy to develop theories of mechanism for the environmental effects on fatigue at 20 kHz, because the cycle period is extremely small and the test data and environments vary considerably. One attempt was made in (Weiss, 1982) where a slip dissolution model for the 20 kHz corrosion fatigue behavior of stainless steel 403 in aqueous chloride solution was proposed. In that study, the authors indicate similarity of the corrosion fatigue characteristics observed for frequencies of 40 Hz and 20 kHz. The S-N data, while not representing a complete overlapping of results from the two frequencies, do in fact permit the data from both frequencies to be comfortably attributed to a single S-N curve. The morphology of the pit-initiated cracking appears to be the same for tests at both frequencies, despite the wide difference in the exposure times for tests with equivalent accumulated stress cycles. The responses to changes in the electrochemical conditions of the test environment were similar for the two frequencies investigated. However, in our opinion, these apparent similarities can hardly be extended to a general conclusion because of the different corrosion fatigue and cracking mechanisms of different materials. Consequently it is necessary to emphasize that developing a more general model for the environmental effects on ultrasonic fatigue is difficult. For example, experience indicates that for many materials the site of crack initiation in the gigacycle regime is internal rather than at the surface of the specimen.

6.5. S-N CURVE AT ROOM TEMPERATURE AND HIGH PRESSURE HYDROGEN FOR Ti-6A4V Some high frequency fatigue tests have been carried out in Industeel Industry Laboratory of Le Creusot with this piezo-electric system for Ti-6A4V alloy at 300 bars hydrogen pressure and room temperature for R ¼
1. At the same time, other tests have been conducted for the same material and Copyright © 2005 by Marcel Dekker. All Rights Reserved.

228

Chapter 6

Figure 6.15 Ti-6A4V life curve at 20 kHz at 300 bars H2 and atmospheric pressure for R ¼
1 20 kHz.

same conditions, at room temperature and air at atmospheric pressure. The results are shown in Figure 6.15 together with the results of fatigue performed at 20 kHz on the same batch of materials by SNECMA. The objective of this high frequency test is to reproduce the same condition of environment as in a hydrogen turbine, i.e, with the same fatigue stress level and exposure duration at 300 bars pressure. In fact, the duration of exposure to hydrogen, the temperature, the pressure, and the stress level are also important parameters. Hydrogen diffusion is a function of the metallurgical structure and texture and therefore has some effect.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

7 Microstructural Aspects and Damage to Materials in the Gigacycle Regime

At the microscopic level, it seems that the initiation of a fatigue crack in gigacycle fatigue can be generally described in terms of a local microstucturally irreversible portion of the cumulative plastic cycle strain. In one way, there is no difference between fatigue mechanisms in mega- and gigacycle regimes except for the surface strain localization in persistent slip bands in the megacycle case. However, many other specific mechanisms can occur especially at high cycles internally in the gigacycle fatigue regime.

7.1. GIGACYCLE S-N CURVE SHAPE Generally, it is assumed that S-N curves for steel are different from those for other materials. In order to obtain in a wide overview of the gigacycle behavior, many alloys including steels are considered here:

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230

Chapter 7


Steels: Low carbon steel, 4240, spring steel, bearing steels, 17-4 PH, rail steel, stainless steel 304
Sphero€
dal graphite cast iron
Ni based alloys: 718 and N18
Ti-alloys: Ti-6A4V, Ti-6246
Aluminium alloys: Al-Si and 2024
Magnesium alloy: AZ91 For fatigue S-N curves approaching 109 cycles, limited results are available in the literature, with more than one-half of the results from our laboratory. The other results are from Japanese researchers such as (Nishijima, 1999), (Murakami, 2002), and (Sakai, 1999), and are often limited to 108 cycles. Also, four S-N curves for light alloys are from the laboratory of S. Stanzl-Tschegg and H.R. Mayer (Stanzl, 2001); these go Table 7.1. Chemical Compositions of the Alloys Tested to Gigacycle Fatigue Alloy 4240U 4240R 54SC6 12Cr 17–4PH

C

Mn

P

0.428 0.412 0.535 0.11 0.07

0.827 0.836 0.629 0.77 1.0

0.012 0.015 0.006 0.011 0.04

Table 7.2

S

Si

Al

Ni

Cr

Cu

Mo

0.024 0.254 0.023 0.173 1.026 0.210 0.224 0.087 0.242 0.023 0.186 1.032 0.209 0.164 0.016 1.400 0.056 0.635 0.002 0.064 2.53 11.4 1.47 0.03 5 17.5 0.5

Mechanical Properties of the Alloys

Alloys 4240UrepB 4240RrepB1 4240UrepC 4240RrepC1 54SC6 12Cr 304 100C6 Low Carbon Steel SG Cast Iron 17-4PH

E (GPa)

r (kg=m3)

UTS (MPa)

Hv30

200 200 205 205 210 216 185 210 203 178 203

7820 7820 7870 7870 7850 7760 7978 7860 7830 7100 7800

1100 1040 1530 1485 1692 1026 599 2316 340 510 1420

345 320 465 450 510 360

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778 95 184

Microstructural Aspects and Damage

231

to 109 cycles. When available, the chemical composition and the mechanical properties of alloys are given (Tables 7.1 and 7.2) order to document the results. 7.2. MECHANICAL ASPECTS OF INITIATION BETWEEN 106 AND 109 CYCLES Safe-life design based on infinite-life criteria was initially developed from the Wo¨ehler approach, which is the stress-life or S-N curve related to the asymptotic behavior of steels. Some materials display a fatigue limit (or ‘‘endurance’’ limit) at a high number of cycles (typically >106). Most other materials do not exhibit this response, instead displaying a continuously decreasing stress-life response, even at a great number of cycles (106–109), which is more correctly described by a fatigue strength at a given number of cycles. Several examples are given in the following discussion. 7.2.1. Gigacycle Fatigue of Steels and Cast Iron Figures 7.1 to 7.10 present a large number of S-N curves for steels and iron loaded in tension-compression or in tensiontension where crack initiation can appear near and sometimes beyond 109 cycles. It is noted that the shape of the S-N curve is not the same from one steel to another. It is difficult to predict the shape of an S-N curve between 106 and 109 cycles— sometimes, the S-N curve is quite flat but the slope can be steep for some metals. However, it is clear that for a first class of alloys, the difference between the fatigue strength at 106 cycles and 109, denoted by DsD, is only few megapascals; that is to say, less than 50 MPa. Low carbon steel (Figure 7.2), stainless steel 304 (Figure 7.3), 12Cr steel (Figure 7.4), and also sphero€
dal graphite cast iron (Figure 7.5) present such behavior. In contrast, there is a second class of steels for which the difference between the fatigue strength DsD at 106 and 109 cycles ranges from 50 to 200 MPa. That means a higher slope of the S-N curve in the gigacycle regime. This domain includes Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 7.1 Low carbon steel sheet S-N curve R ¼ 1 at 20 kHz (difference between fatigue strength at 106 and 109, DsD ¼ 3 MPa).

Figure 7.2 Low carbon steel sheet S-N curve R ¼ 0.1; 20 kHz; DsD ¼ 20 MPa. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

233

Figure 7.3

Stainless steel 304 S-N curve R ¼ 1.

Figure 7.4

S-N data for 12% Cr steel R ¼ 1; 20 kHz; DsD ¼ 40 MPa.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

234

Chapter 7

Figure 7.5 Spheroidal graphite cast iron S-N curve R ¼ 1 (up) and R ¼ 0.1 (down); DsD ¼ 10 and 30 MPa.

4240 steel (Figure 7.6), bearing steel (Figure 7.7), rail steel (Figure 7.8), spring steels (Figure 7.9), and martensitic stainless steels such as 17–4 PH (Figure 7.10). For technical applications a gap of 100 or 200 MPa on the fatigue strength cannot be ignored. It seems that the higher the UTS of steels, the Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

235

Figure 7.6 High strength steel 4240 S-N curves R ¼ 1; 20 kHz; DsD ¼ 30 and 80 MPa.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

236

Chapter 7

Figure 7.7 Fatigue curve for 100C6 steel R ¼ 1; 20 kHz; DsD¼ 180 MPa.

Figure 7.8

Rail steel S-N curve R ¼ 1; 20 kHz; DsD ¼ 100 MPa.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

237

Figure 7.9 Spring steel S-N curves (55SiCr7; 54SiCr6) R ¼ 1; 20 kHz; DsD ¼ 200 and 150 MPa.

higher the S-N curve slope in the gigacycle fatigue regime. Several mechanisms are involved in explaining crack initiation in gigacycle fatigue as will be discussed in this chapter. It can be pointed out that a progressive transition of mechanisms around 107–108 cycles induces a step in the S-N curve as noted in (Murakami, 2002), (Nishijima, 1999), and (Sakai, 1999), and is shown in Figure 7.6 for 4240 steel. In this case Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Figure 7.10 Martensitic stainless steel S-N curves (17–4 PH and 13–8 Mo) R ¼ 1 20 kHz; DsD ¼ 200 MPa.

Figure 7.11

N18 nickel base alloy S-N curves at 450 C between

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Microstructural Aspects and Damage

239

the plateau is due to the large scatter of the results, or perhaps due to rotating bending loading. 7.2.2. Gigacycle Fatigue of Ni Base Alloys Several Ni base alloys were tested in the gigacycle regime (Ni, 1991) such as Udimet 500, Inco 718, and N18. For all these alloys, the fatigue strength decreases by 150 to 200 MPa from 106 to 109cycles. Figure 7.11 gives our results for N18 alloy tested at 450 C at R ¼ 0 and R ¼ 0.8. It is of interest to note that, when the R ratio increases, the slope of the S-N curve decreases. As expected, a high density of inclusions and pores induced a large scatter in the number of cycles for initiation. However, the fatigue strength at 109 cycles is only slightly affected by the density of inclusions, since the controlling feature is the size of the largest inclusion. 7.2.3. Gigacycle Fatigue of Titanium Alloys Titanium alloys behave in the gigacycle fatigue regime in a manner similar to Ni based alloys. For example, Figure 7.12 presents an S-N curve determined up to 109 cycles, at

Figure 7.12 Titanium alloy S-N curves between 106 and 109 cycles at Ti-6246 alloy. See Table 7.5 for thermal processing. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

240

Chapter 7

20 kHz, 300 K, and R ¼ 1 for a Ti-6246 titanium alloy forged with different heat treatments (Jago, 1998). Several features should be pointed out:
Some fatigue initiations near 109 cycles
There is no asymptote
The fatigue strength sD at 109 cycles is much smaller (by as much as 150 MPa) than the fatigue strength at 106 cycles
The forging process and the microstructure have a marked influence on the high cycle fatigue life (sD ranges from 325 to 490 MPa for 109 cycles) Not having any inclusions or pores in this alloy enhances the influence of processing and metallurgical transformation in the Ti-6246 alloy. The worst fatigue strength at 109 cycles for 6246 material is exhibited for heat treatments 1 and 2, which give a larger volume fraction of secondary a and small platelets of primary a. However, for heat treatment 3, the fatigue strength at 109 cycles reaches 490 MPa rather than only 325 MPa, and at 106 cycles it rises to 600 MPa. This means the study of gigacycle fatigue is an important way to characterize the alloys that are used for high technology applications such as jet engine turbine disks. 7.2.4. Gigacycle Fatigue of Aluminium Alloys In this book, several gigacycle S-N curves for cast and wrought aluminium alloys (2024, 2219, 6061, and A57) were presented. These results come from Stanzl’s group (Stanzl, 2001), (Mayer, 1999), or the Bathias group (Bathias, 1998, 1999, 2001). For wrought aluminium alloys, there exists a difference of 100 MPa between the fatigue strengths, sD at 106 cycles and at 108 cycles. It is notable that for cast aluminiums this difference is smaller and less than 100 MPa (Figure 7.13). The relative fatigue strength in the gigacycle regime decreases 30% compared to the standard fatigue limit at 107. Once again, it is shown in the Figure 7.13 that the effect of frequency is very small. It is pointed out that one fatigue failure of AS5U3G is at 51010 cycles! Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

241

Figure 7.13 Aluminium alloy AS5U3G S-N curve R ¼ 1; DsD ¼ 30 MPa.

In the gigacycle regime, the initiation site is located inside the specimens for titanium, nickel alloys, and high strength steels. However, fatigue initiation appears at the surface of specimens in aluminium and magnesium alloys, especially in castings where voids at the surface are the main locations for crack formation. 7.3. INITIATION ZONE FOR LOW CYCLE TO GIGACYCLE FAILURES According to our own observations and those of the literature, crack initiation in gigacycle fatigue seems usually to occur inside the sample and not at the surface, if there are inclusions or porosity inside the metal. Three types of crack initiation occur in cylindrical samples with a polished surface depending on whether it is low cycle (104 cycles), megacyclic (106 cycles), or gigacyclic (109 cycles) fatigue (Figure 7.14). For the smallest number of cycles to Copyright © 2005 by Marcel Dekker. All Rights Reserved.

242

Chapter 7

Figure 7.14

Mechanisms of fatigue initiation.

rupture, the initiation sites are multiple and on the surface, according to the normal observation. However, at 106 cycles, there is only one surface initiation site and, for a much higher number of cycles to rupture, the initiation is located in an internal zone. What remains is to understand how and why some fatigue cracks can initiate inside the metal in gigacycle fatigue. Generally, but not always, the crack initiates from a defect, inclusion, or pore. Sometimes the initiation is related to microstructure anomalies, for example, long platelletes or perlite colonies. 7.4. INITIATION MECHANISMS AT 109 CYCLES The full explanation of the gigacycle initiation phenomenon is not clear. It seems that the triggering cyclic plastic deformation becomes very small in the gigacycle regime, in which case, internal defects or large grain size can play a role creating plasticity, in competition with the surface damage. As the initiation requirements for cyclic strain and maximum strain to decrease, three main factors operate: 1.

2.

Anisotropy of metals: In the gigacycle regime, the plastic strain is very small. Thus, the plasticity appears only if the grain orientation and the grain size are in agreement with dislocations sliding at the surface or in the bulk of metals. Stress concentration: It is suspected that stress concentration due to metallurgical microstructure

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Microstructural Aspects and Damage

243

misfit becomes an important factor when the applied load is low. Inclusions, porosities, and grain size effect are among efficient concentrators. 3. Statistical conditions: Statistically, the probability of finding a sufficient stress concentration is more likely in the bulk than in the surface of the metals under high cycle low plastic strain requirement conditions. Thus, the probability of an offending stress concentrator in the bulk of the metal is the best explanation for the localization of the initiation of the crack in the gigacycle fatigue. 7.5. ROLE OF INCLUSIONS In steel and nickel base alloys, inclusions can be significant crack initiation sites especially if load ratio R is high. An example of N18 alloy at 450 C is discussed in Section 7.5.2. It is of great importance to understand and predict a fatigue life in terms of crack initiation and small crack propagation. It has been generally accepted that at high stress levels, fatigue life is determined primarily by crack growth, while at low stress levels, the life span is mainly consumed by the process of crack initiation. Several authors demonstrated that the portion of life attributed to crack nucleation is over 90% in the high cycle regime (106 to 107 cycles) for steel, aluminium, titanium, and nickel alloys. In the case when the crack nucleates from a defect, such as an inclusion or pore, it is said that a probable relation may exist between the fatigue limit and the crack growth threshold. However, the relation between crack growth and initiation is not obvious for many reasons. First of all, it is not sure that a fatigue crack grows at the very first cycle from a sharp defect. Secondly, when a defect is small, a short crack does not grow in the same manner as a long one. Thus, the relation between DKth and sD is still to be studied. Another important aspect is the concept of infinite fatigue life. It is understood that below DKth and below sD Copyright © 2005 by Marcel Dekker. All Rights Reserved.

244

Chapter 7

the fatigue life is infinite. In fact, the fatigue limit sD is usually determined for Nf ¼ 107 cycles. But fatigue failure can occur up to 109 cycles and perhaps beyond 109. The fatigue strength difference at 107 and 109 cycles can be more than 100 MPa. This means the relation sD versus DKth must be established in the gigacycle regime if any relation exists.

7.5.1. Estimation of Crack Growth Life from the da=dN Curve To predict the number of cycles to initiate a fatigue crack from an inclusion, several models are used with more or less success; the integration of the Paris law, Murakami formula, Tanaka model, and Kitagawa diagram are among the well known approaches on the megacycle regime. In gigacycle fatigue regime, the geometry of the fish eye initiation is a circle that collapses at the surface of the specimen. As discussed in the preceding section, the initiation of crack growth consumes a dominant portion of the life in the gigacycle fatigue range. The initiation from an inclusion or other defect itself must be close to the total life, perhaps much more than 99% of the life in many cases. This is made evident by integrating the fatigue crack growth rates for small cracks to estimate the possible extent of crack growth life. In order to do this, one should refer to the general behavior pattern of the crack growth rate curve as illustrated by the equations on page 143. It is noted that small cracks such as those growing from small inclusions do not exhibit crack closure, therefore, in terms of DKeff, these equations apply fairly well. They form the upper boundary on crack growth rates for small cracks in ‘‘fish eye’’ range for which crack closure is minimal. Estimating the life of a crack of this type beginning just above threshold, it is then appropriate to consider the growth law as   DKeff 3 da pffiffiffi ¼b dN E b Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

245

where, for the circular crack growing in a ‘‘fish eye,’’ the stress intensity factor formula is 2 pffiffiffiffiffiffi DK ¼ Ds pa p The integration to determine the crack growth life begins here, with the crack growth rate corner (as indicated by the star on Figure 5.14), which we denote as DK0 corresponding to an initial circular crack of radius a0. Substituting these into the first formula we obtain  3=2     da DK0 3 a 3=2 a ¼b ¼ b pffiffiffi dN a0 a0 E b This equation can be integrated from a0 to afinal, which gives Np ¼

Z

af a0

" # ða0 Þ3=2 da 2ða0 Þ3=2 1 2a0
¼  small ffi 3=2 1=2 b b b ðaÞ ða0 Þ

but it can be noted that pffiffiffiffiffi DK0 2Ds a0 1 ¼ pffiffiffi ¼ pffiffiffi pffiffiffi pE b E b

or a0 ¼

pE2 b 4ðDsÞ2

Then combining the last two expressions Np ¼

pE2 2ðDsÞ2

This is the result for the approximate number of cycles from threshold corner to failure from an initial crack size a0 as expressed above. Now, if the example of 4240 steel (high strength) is taken with a modulus, E, of about 215 GPa and a gigacycle fatigue strength Ds of 500 to 750 MPa (from Table 4.3 and Figures 4.2 and 4.3), the result for the number of cycles, Np, of crack Copyright © 2005 by Marcel Dekker. All Rights Reserved.

246

Chapter 7

growth is about 150,000 to 290,000 cycles for starting crack sizes, a0, of 37 to 72 mm. These numbers of cycles are not anywhere near the 109 cycles for these fatigue strengths for these initial crack sizes close to the ‘‘fish eyes’’ observed. Further, attempting to include integration of crack growth rates below the threshold corner for these very small initial cracks leads to smaller numbers of cycles added to the numbers above. Such calculations can be done in a manner similar to that given above. Consequently, there seems to be strong evidence that crack initiation is a dominant feature in gigacycle fatigue life where ‘‘fish eyes’’ are observed. 7.5.2. Prediction of the Fatigue Limit at 109 Cycles Using the Murakami Formula Another approach to predict the number of cycles at initiation from a defect is to understand the notch effect phenomenon on the fatigue strength. This approach is well known for the prediction of the megacycle fatigue of steels where an empirical relation is found between the ultimate strength (or hardness) and the standard fatigue limit. In this respect, (Murakami, 2002) has proposed a parametric formula to predict the fatigue limit of steel at 107cyles, depending on hardness and defect size. Murakami has confirmed that the fatigue limit concept of materials containing defects is essentially a crack problem. In his approach the size of defect is given by the expression (area)1=2, that is to say, the square root of the initial defect projected area. From a great number of experiments, Murakami and co-workers have determined a strong correlation between the apparent fatigue threshold at the tip of the defect and the size of the defect measured as indicated above. They found the following relationship DKth ¼ 3:3  103 ðHv þ 120Þ ðareaÞ1=6 According to Murakami, it seems the fatigue strength at 109 cycles can be predicted using this formula with few exceptions. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

247

From our present data, we have verified this relation: " #a CðHv þ 120Þ ð1  RÞ3 sw ¼ pffiffiffiffiffiffiffiffiffiffi 1=6 2 ð area Þ where sw ¼ fatigue strength at 109 cycles MPa C ¼ 1.78 for internal defects Hv ¼ Vickers hardness pffiffiffiffiffiffiffiffiffiffi area ¼ projected defect surface (mm) R ¼ load ratio a ¼ 0.878 þ Hv104 An example of the application of the Murakami formula in gigacycle fatigue of N18 nickel alloy is shown in Table 7.3. (Bonis, 1997). The prediction of the model is always better than 10% except in one case. It is of interest to note that inclusions are mixed, sometimes, with porosities in this powder metallurgy alloy. The role of the inclusion is sometime hidden by the role of porosities when the R load ratio is equal to –1 or to 0. Conversely, when the mean stress of the fatigue cycle is very high, for R ¼ 0.8, the role of inclusions becomes dominant. For all R ratios, the Murakami formula gives a reasonable prediction of gigacycle fatigue strength. In order to adapt the validity of the Murakami approach for different fatigue life Nf from 106 to 109, we have introduced a correction factor b depending on Nf, as shown in Figure 7.15 where the formula is applied for high strength Table 7.3 R

Application of Murakami Model to N18 Nickel Alloy 1

1

0

0

0.8

0.8

Defect Mixed Inclusion Porosity Inclusion Porosity Inclusion Localization Surface Internal Internal Internal Internal Internal pffiffiffiffiffiffiffiffiffiffi area (mm) 50 100 25 100 25 100 sw(MPa) 524 466 309 246 160 126 s experimental (MPa) 525 400 280 270 160 130 Error % 0 þ14 þ10 9 0 3

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248

Chapter 7

Figure 7.15

Murakami model for different fatigue lifes.

steels (Wang, 1998).   bðHv þ 120Þ ð1  RÞ a sw ¼ pffiffiffiffiffiffiffiffiffiffi 1=6 2 ð areaÞ 7.5.3. Kitagawa Diagram at 109 Cycles It is well known that the effect of inclusions is related to the tensile strength of the metals in the megacycle fatigue regime. According to the Kitagawa diagram (Kitagawa, 1976), the effect of inclusions in gigacycle fatigue depends also of the UTS of metals, especially for steels. From our own results, a Kitagawa diagram is drawn for the gigacycle regime (Figure 7.16). It is found that if the UTS of steels is high (1500 MPa and greater) a small inclusion, less than 10 m, can be the initiation site. Conversely, if the UTS of steels is low (500 MPa and less), an inclusion of 100 m or more is need to initiate a fatigue crack in the gigacycle cycle regime. This means that in low carbon steel, the initiation is related to the grain size itself. In this case, roughly speaking, sD =UTS ¼ 0:5. For high strength steels, this ratio is less than 0.5 at 109 cycles. 7.6. GIGACYCLE FATIGUE OF ALLOYS WITHOUT INCLUSIONS What happens in alloys without inclusions in the gigacycle fatigue regime? It is well known that in titanium alloys there Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

249

Figure 7.16 Relationship between the fatigue strength and defects size (Kitagawa diagram).

is not any inclusion or porosity. Therefore, to answer this question, titanium alloys were tested for initiation and propagation. Since fatigue cracks cannot nucleate from internal geometrical defects, the fatigue cracks must initiate from metallurgical discontinuities. 7.6.1. Thermomechanical Processing of Ti-6246 Alloys The chemical composition of a Ti-6246 alloy, as supplied by the RMI Company, is shown in Table 7.4. Four thermomechanical processes (TP)—denoted TP1(1) TP1(2) TP(2) and TP(3)—were used to produce forgings with different microstructures and attendant mechanical properties (Table 7.5). The b-processed microstructures present similar lamella aphase morphology with different primary a volume fraction and grain size in a transformed b matrix. The a þ b process is characteristic of a bi-modal structure with duplex lamella and globular primary alpha phase. Quantification of the morphological aspects has been performed to provide a complete description of various microstructures. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

250

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table 7.4

Chemical Compositions of Ti-6246 Alloy Investigated (in wt%)

TP number

Al

Sn

Zr

Mo

C (ppm)

Cu (ppm)

Si (ppm)

Fe (ppm)

O2 (ppm)

H2 (ppm)

N2 (ppm)

1(1) 1(2), 2 and 3

5.76 5.68

1.97 1.96

4.08 4.08

3.97 3.92

90 83

<50 <50

<50 <50

400 300

930 1100

44 28

80 70

Table 7.5

Thermomechanical Processes and Microstructures of Ti-6246

TP number

Forging condition

1(1) and 1(2)

955 C= WQ (Tb þ 10 C)

2

955 C=WQ (Tb þ 10 C)

3

905 C=Air (Tb  40 C)

Thermal treatment

Final microstructure type

935 C=2H=WQþ905 C=1H= Airþ595 C=8H=Air 935 C=2H=Slow cool: Room T. þ595 C=8H=Air  935 C=2H=WQþ905 C=1H= Airþ595 C=8H=Air

a platelets and b-transformed matrix Coarse a platelets and b-transformed matrix Bi-modal structure (a platelets and a nodular) and b-transformed matrix Chapter 7

Microstructural Aspects and Damage

251

7.6.2. Fatigue Initiation in Ti-6246 Alloys With this alloy and various thermomechanical processings it is found that crack initiation and failure can occur up to 109 cycles though there is not any inclusion or pore. Figure 7.12 presents S-N curves showing dependency on the thermal processing. At room temperature a significant differ ence can be observed in S-N curves between the different TP conditions. Thus, the TP3 material has comparatively the higher fatigue resistance (510 MPa); the TP1(1) and TP1(2) materials exhibit a lower response with a fatigue strength for 109 cycles estimated respectively at 490 MPa and 400 MPa; and the TP2 material has the lowest fatigue-limit resistance of only 325 MPa. In Figure 7.17, we notice that

Figure 7.17 Low fatigue crack growth rate of Ti-6246 alloy, for different TP, at 20 kHz. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

252

Chapter 7

the TP3 alloy has the lowest DK threshold with the best fatigue strength. The SEM fractographic observations indicate that all the TP1(1) broken samples have systematic near-surface initiation (less than 40 mm from the external surface), whereas TP1(2), TP2, and TP3 have systematic internal fatigue crack sites. In TP2 conditions, microstructure and more specifically colonies of primary alpha phase a(P) show on the fracture surface by backscattered electrons observations and form a sort of facet (Figure 7.21). It can be seen that the facets are oriented to the fracture plane, a feature common to all specimens. In conclusion, it is emphasized that gigacycle fatigue regime is not always correlated to defects such as inclusions or pores. For Ti-6246, the gigacycle fatigue strength is associated with transformed amount and secondary alpha volume fraction. Internal fatigue initiation with quasicleavage facets in primary alpha phase has been demonstrated. Under these conditions, it is very difficult to get a general relation between DKth and DsD. A nucleation process must exist. However, a linear relation is found between yield stress and sD in the gigacycle regime for 6246 titanium alloy. sD ¼ 1184 þ 1:6 sY 7.7. GENERAL DISCUSSION OF THE GIGACYCLE FATIGUE MECHANISMS With respect to the competition between surface and internal initiations sites, gigacycle fatigue initiation mechanisms are split into three cases, as follows. 7.7.1. A Critical Defect Exists at the Interior of the Alloys The gigacycle initiation mechanisms are strongly dependent on the stress concentration due to inhomogeneous Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

253

Figure 7.18 Surface-subsurface transition in crack initiation location in 4240 steel.

microstructure. They can be:
Mineral inclusions in high strength steels, martensitic stainless steels, bearing steels, and spring steel.
Perlite nodules in rail steels
Platelets in titanium alloys
Carbides in cast irons Copyright © 2005 by Marcel Dekker. All Rights Reserved.

254

Chapter 7

Figure 7.19 Internal initiation in gigacycle fatigue regime of 17–4 steel. Initiation is related to a mineral inclusion.

Figure 7.20 Internal initiation in rail steel in gigacycle regime. Initiation is related to a nodule of fine lamellar perlite shown on the right side.

For all these alloys it was found that the initiation site for fatigue cracks shifts from the external surface to subsurface at a particular lower stress range. The subsurface crack initiation is dominant in the gigacycle range. A surface– subsurface transition in crack initiation location has been established at approximately 107 cycles. Sometimes there is a plateau between 106 and 108 cycles on the S-N curve that is not fully understood. The effect of microstructural details Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

255

Figure 7.21 Fatigue crack initiation on a primary phase in Ti-6246 alloy.

is apparently very important in the gigacycle fatigue regime (Figures 7.18 to 7.20). 7.7.2. Surface Initiation in Gigacycle Fatigue Another special case is the gigacycle fatigue of Al-Si castaluminium alloys for which the initiation location is at the surface (Figure 7.22). It could be related to the high density of pores always present on the specimen surface. In this case, the S-N curve between 106 and 109 cycles is decreasing monotonically without transition at 107 cycles (Figure 7.13).

Figure 7.22 Initiation of a fatigue crack under the surface in cast aluminium alloy. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

256

Chapter 7

Figure 7.23 Internal initiation in N18 alloy related with both pores and inclusions.

7.7.3. Multiple Mechanisms It seems that several mechanisms can operate in gigacycle fatigue. This is the case of powder metallurgy alloys where inclusions and pores are co-existing (N18 alloy). When inclusions are present, the initiation site is in the interior. When there are pores, the initiation is in the surface or in the interior (Figure 7.23). For N18, the slope of the S-N curve is uniform between 6 10 and 1010 cycles but there is a transition in initiation sites around 107 cycles, after which a competition appears between the surface or interior initiation, depending on the load ratio R. Inclusions are more dominant for high R tending to induce initiation in the interior.

7.7.4. Conclusion It has been shown that, for a large number of alloys, fatigue crack initiation occurs beyond 107 cycles and that the difference in fatigue strength between 106 and 109 cycles often decreases by 50 to 200 MPa. Obviously, the concept of infinite fatigue life on an asymptotic S-N curve is not correct. Very often there is a surface-to-subsurface initiation site transition around 107 cycles when inclusions or microstrucCopyright © 2005 by Marcel Dekker. All Rights Reserved.

Microstructural Aspects and Damage

257

tural defects are present. This mechanism is not unique since, in the case of no critical damage in the interior, the initiation will appear at the surface.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Appendix 1 Stress Calibration

For the ultrasonic fatigue machines to work correctly, stress calibration is necessary. The first objective of the calibration is to make the test system vibrate in resonance at ultrasonic frequency of about 20 kHz. Among the important factors of concern is the variation of Young’s modulus of the material due to the high frequency or non-uniform temperatures. Although some measures for calibration have been mentioned in previous sections, it is necessary to present, in this appendix, the detailed calibration procedures for the low temperature test system described earlier. The principles for the calibration of machines operated at room temperature are the same; those procedures will be simpler. The mechanical system works in an elastic regime. The relationships between displacement, strain, and stress are therefore linear. The electrical voltage applied to the piezoceramic is also linear and proportional to the displacement. The electrical current density depends on the impedance, the dynamic mechanical resistance of the equipment fixed in the converter. For these reasons, we use a J2 type generator

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

260

Appendix 1

connector that permits us to know the displacement amplitude and the output power. The measures required at low temperatures are more involved. One solution may be to use an accelerometer installed at the end of the converter and to calibrate its signal with an optical displacement sensor. Before using this signal, we must convert the sinusoidal wave to a DC voltage. However, since we have the driving voltage to the piezo-ceramic that is already a signal proportional to the displacement amplitude, utilization of this signal will be a simpler and more practical solution. This solution is better than the introduction of an accelerometer, especially since the signal is proper and the additional installation of an apparatus (the accelerometer) will be avoided. Furthermore, for the accelerometer, the linear relation between this signal and the displacement in the specimen’s head, is unknown because of the electrical chain. Therefore, calibration is required. A1.1.

AMPLIFYING HORN

As the displacement amplitude of the converter is limited, the role of the cone shape (amplifying horn) is very important in order to increase the displacement of the specimen and thus to raise its stress to the required level. There are two antinodes and one node in the displacement distribution of the cone (Figure A1.1). The amplification factor Fampl is defined as


V1



V
2 Generally, a numerical analysis such as FEM is needed to obtain the correct geometry of the horn incorporating the effects of the temperature gradient and the variation of the elastic modulus. The temperature of the horn may be reasonably assumed to vary linearly between two end values of 20
K and T0, the temperatures of the cryostat and ambient air, respectively, as shown in Figure A1.2. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Stress Calibration

261

Figure A1.1 Displacement distribution of amplifying horn (horn 1).

Figure A1.2 Temperature field of amplifying horn. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

262

Appendix 1

Figure A1.3

Dimensions of the second horn (horn 2).

The amplification factor of the first horn (horn 1) shown in Figure A1.2 is 3.52 at room temperature, 3.42 in liquid nitrogen, and 3.40 in liquid helium. Horn 1 works well at ambient temperature and in liquid nitrogen with or without specimen. Meanwhile, the cone can only produce a maximum stress of about 500 MPa. A second horn design is shown in Figure A1.3, which is capable of developing stress between 650 MPa and 1300 MPa. Horn 2 gives an amplification factor Fampl ¼ 8.93 in nitrogen and 8.98 in helium. A1.2.

FIRST CALIBRATION

The driving electrical voltage and power is supplied by a Branson power source, and the first measurement is carried out with an IBM PC computer. Since horn 2 joined with the fatigue life specimen, does not vibrate at ambient temperature, horn 1 must be used. Figure A1.4 gives the measured signal in response to the input signal; i.e., the voltage in plug 8. The open points are the values measured from the optical sensor, and the full points are those from plug 9. The linearity is practically perfect. Figure A1.5 presents the relation between the signal of plug 9 in the connector J2 and the signal of the sensor for horn 1 at ambient temperature. This relation can be Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Stress Calibration

263

Figure A1.4 Response to input signal.

expressed by the equation Uc1 ¼ 2:57 þ 4:146 Vm

ðA1:1Þ

where Uc1 is the measured displacement of the first horn by the sensor, and Vm is the voltage of plug 9. The symmetry of the specimen, which implies identical displacement amplitudes on two sides, has been taken into account. The voltage applied to the ceramic imposes a displacement at the end of the converter. For a given temperature, this displacement must be constant. In a CNAM=ITMA test in liquid nitrogen,

Figure A1.5 Measured amplitude of signal. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

264

Appendix 1

the temperature of the converter is above 0
C, because this part does not freeze immediately after the removal of the cryostat. Therefore, we consider that the displacement in the converter is constant for a given electrical applied voltage, whatever the mechanical load may be. Based on the preceding analyses, the amplitudes of displacement and stress are determined for horn 2 by the simple calculation, as follows: Uc2 ¼

8:93 Uc1 3:52

s2 ¼ 16:2Uc2 ¼ 105:66 þ 170:54 Vm

ðA1:2aÞ ðA1:2bÞ

Here, the value 8.93 is the amplification factor of horn 2 at low temperatures, and 3.52 is that of horn 1 at room temperature. For material Ti6A4VPQ, one micron of displacement corresponds to a stress of 16.2 MPa. Therefore the stress is calculated by Eq. A1.2b using Eqs. A1.1 and A1.2. The constant 105.66 in this equation comes from the electronics or the adjustment of the A=D card. In other words, the initial voltage is about 0.5 volt without vibration. The curve of input stress-signal in Figure A1.6 is employed in the control program to guarantee good convergence.

Figure A1.6

Stress versus signal.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Stress Calibration

A1.3.

265

SECOND CALIBRATION

With a computer PC486DX33 and a high-performance Keithley card, we have carried out another calibration at ambient temperature. The optical sensor measures the displacement amplitude at the specimen’s head and a strain gauge mounted on the specimen’s center measures the strain there. The Ti6A4V specimen used is a uniform cylindrical bar with 124.72 mm in length. The two amplifying horns work at ambient temperature, which permits us to verify the stress for the two horns. With the strain values of the specimen’s center measured by the gauge and the displacement in the head measured by the optical sensor, the strain along the specimen can be calculated. The results are as follows. Horn 1. The amplification factor Fampl is 3.52. Displacement measured by the sensor Um ¼ 17:4 mm;

at 50% power

Um ¼ 35:4 mm;

at 100% power

Subscript m indicates measured value, and subscript c for calculated value in the following. Strain in specimen center (calculated value) 2pf ec ¼ pffiffiffiffiffiffiffiffiffi Um ¼ 25:19Um E=r

ðA1:3Þ

with f ¼ 20 kHz, E ¼ 110 GPa, and r ¼ 4420 kg=m3. Substituting measured values, we have ec ðUm ¼ 17:4 mmÞ ¼ 438:3 m;

at 50% power

ec ðUm ¼ 35:4 mmÞ ¼ 891:2 m;

at 100% power

Strain in specimen center (measured by the gauge) em ¼

2 1000 Vm ðmÞ K V1000

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

ðA1:4Þ

266

Appendix 1

where K ¼ 2.055 is the gauge factor, and V1000 ¼ 2.6 volts is the voltage of calibration at 1000 m of strain. Similarly, substituting measured values, we have em ðVm ¼ 1:2 VÞ ¼ 449:2 m;

at 50% power

em ðVm ¼ 2:4 VÞ ¼ 898:4 m;

at 100% power

Horn 2. The amplification factor Fampl is 9.16. Displacement measured by the sensor Um ¼ 43:4 mm at 50% power Um ¼ 84:4 mm at 100% power Strain in specimen center (calculated value) ec ðUm ¼ 43:4 mmÞ ¼ 1093 m; at 50% power ec ðUm ¼ 84:4 mmÞ ¼ 2126 m at 100% power Strain in specimen center (measured by the gauge and with V1000 ¼ 1.56 volts) em ðVm ¼ 1:8 VÞ ¼ 1123 m;

at 50% power

em ðVm ¼ 3:56 VÞ ¼ 2221 m;

at 100% power

The results are summarized in Table A1.1 This verification is satisfactory. Note that the difference of strains from the measurement of the gauge and the calculation, starting from the values measured by the optical sensor, is about 2%. This demonstrates that the optical displacement sensor and the strain gauges used are reliable in the vibration environment of 20 kHz, and the stress distribution in the antinodes is good. We can also see that the measurements at 100% power are close to twice those at 50% power. The strain gauge measurements give the best results. The measurements show that the ratio of vibration between horns 1 and 2 is 2.45 on the average, while the calculated ratio is 2.6. The difference between them is about 6%. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Stress Calibration

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Table A1.1

Results of Second Calibration Fampl

Um (50%)

Um (100%)

ec (50%)

ec (100%)

em (50%)

em (100%)

Horn 1

3.52

17.4 mm

35.4 mm

438.3 m

891.2 m

449.2 m

898.4 m

Horn 2

9.16

43.4 mm

84.4 mm

1093 m

2126 m

1123 m

2221 m

No. 2=No. 1

2.602

2.494

2.384

2.494

2.386

2.500

2.472

267

268

Appendix 1

From these measurements, the stress can be calibrated in accordance with the voltage of plug 9 in connector J2 that is 5 volts at 50% power and 10 volts at 100% power. This voltage is proportional to the displacement amplitude of the horn (at point C of Figure 3.11) and does not depend on load. Taking account of the linearity, we only consider the state at 100% power in the following discussion. The converter is always at ambient temperature; that is to say, the amplitude is constant at a given power. This displacement amplitude is denoted by UC. With the measurement of the gauge at room temperature, the displacement can be calculated at the head of the specimen (at point A of Figure 3.11). The values are 35.6 mm and 88.6 mm for horns 1 and 2, respectively, both at 100% power. Because the displacement of point B in Figure 3.11 is the same as that of point A, horn 1 also gives an amplitude value of 35.6 mm, and horn 2 gives 88.6 mm at 100% power. These values are independent of the specimens. At low temperatures, the amplifying horns are more rigid because their elastic moduli increases. Consequently the amplification factor decreases slightly. Since UC is constant, we have     UB UB ¼ ðA1:5Þ UC ¼ Fampl ambient Fampl cold Using a superscript c for cold, we obtain   UB c c UB ¼ Fampl Fampl ambient

ðA1:6Þ

Formula A1.6 can be used to include the influence of temperature of the horns. Having obtained UBc (identical to UAc ), we are able to determine the stress in the specimen. The stress depends on the geometry of the specimen and the elastic modulus of the material. The related formulas are available in Chapter 2, both for a cylindrical bar and an endurance (fatigue life) specimen with a longitudinal profile of hyperbolic cosine in the central portion. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Stress Calibration

A1.4.

269

THIRD CALIBRATION

In the above, the verification of the calculations has been made for ambient temperature. To know the behavior of the machine at low temperatures, several gauges are mounted on the cylindrical specimen of TA6VPQ. Theoretically, the strain is of a sinusoidal function with the maximum in the center and the minimum at two ends. For this reason, we mount five small gauges (gauge length, 1.57 mm) in the central region, and three long gauges (3.18 mm) in the zone of small strain. Another specimen of Ti-6A4VPQ is also tested where a gauge is used to compensate temperature effect. The resistance of the gauge is 350 O. During the measurement, the two specimens are soaked in liquid nitrogen. The compensation gauge is linked to the demi-bridge. The results of the calculations are compared with the measurements at low temperature to verify the vibration mode analysis. Material Ti-6A4VPQ was supplied by an industrial company. The static Young’s modulus at 77
K is 128 GPa. This yields a resonance length of 134.5 mm at 20 kHz. But the test machine did not vibrate in resonance under these conditions. Only when the specimen was shortened by 3 mm did the system vibrate in liquid nitrogen. This shows the real modulus is lower than 122.4 GPa. The measurements of strain start after the cooling of the two specimens. For horn 2, the signal from gauges is quite sinusoidal and the measurements give good results as shown in Figure A1.7. The differences between the results of calculations and the measurements by the four gauges are about 5%. The measurements verify that the antinodes of stress are situated within 0 to 5 mm from the specimen center. Although these tests are performed at 50% power from the ultrasonic generator, the gauges wires broke one after another because of acceleration. This is the reason that only a small number of results have been obtained. Figure A1.8 shows the strain values measured by gauges for two horns at various levels of power. (Note: This figure was shown in a simplified form as Figure 3.12.) The measurement points lie very close to the straight line, em ¼ ec. For Copyright © 2005 by Marcel Dekker. All Rights Reserved.

270

Figure A1.7

Appendix 1

Results measured by gauges.

horn 1, the largest error is 7.7% at high power. For horn 2, the error is 5.1% at 50 power, the same value as that of the previous test (Figure A1.7). Other points in Figure A1.8 are well aligned and the error is lower than 2%.

Figure A1.8 and 2.

Measurements and calculations of strain for horns 1

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Stress Calibration

271

From the calibration results, we can conclude that the concept and the design of the test system is adequate. The machine can be satisfactorily used to determine stress levels with about 95% accuracy. The test results will be presented in the other chapters. (Wu, 1992) presents calibration procedures of an ultrasonic fatigue machine working at high temperatures.

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Appendix 2 Remarks on the Statistical Prediction

As stated earlier, it is of interest to point out that many structural components are subjected to beyond 107 cycles. Normally the materials characterization and the fatigue prediction are carried out from data limited to between 106 and 107 cycles. The fatigue life requirement of a car engine is around 108 cycles. The big diesel engines for ships or highspeed trains work up to 109 cycles. An extreme example for the technical limit of fatigue life is about 1010 cycles for turbine engines. In principle, the fatigue limit is given for a number of cycles to failure. Using the staircase method, for example, the fatigue limit is given by the average alternating stress, sD, and the probability of fracture is given by the standard deviation of the scatter, s. A classical way to determine the infinite fatigue life is to use a Gaussian function. Roughly speaking, it is said that sD – 3 s gives a probability of fracture close to zero. Assuming standard deviation is equal to 10 MPa, the true infinite fatigue limit should be sD – 30 MPa. However, our experiments show that between sD for 106 and sD for 109, the difference is much greater than 30 MPa for many

Copyright © 2005 by Marcel Dekker. All Rights Reserved.

274

Appendix 2

Figure A2.1 S-N curves that typify fatigue test results for testing of medium-strength steels. (From ASM Atlas of Fatigue Curves, p. 28.).

alloys. The so-called SD approach to the average fatigue limit is certainly not the best way to reduce the risk of rupture in fatigue. Whereas one is conscious that it is a convenient approximation, only experience can remove the ambiguity by appealing to some tests of accelerated fatigue. The concept of a fatigue limit is bound to the hypothesis of the existence of a horizontal asymptote of the S-N curve between 106 and 107 cycles (Figure A2.1). A sample that reaches 107 cycles and is not broken is often considered to have an infinite life. This is not a rigorous approach. It is important to understand that if the staircase method is popular today to determine the fatigue limit at 107 cycles, it is because of convenience of the approximation. A main goal of this work is to introduce a new practical definition of the fatigue strength which extends into the gigacycle regime, i.e., 109 cycles or more. A2.1.

REMARKS ON STATISTICAL ANALYSIS IN THE MEGACYCLE REGIME

It has long been assumed that a log-normal relation takes into account the scatter of fatigue results up to the megacycle regime. The staircase method is based on this assumption. Thus, the prediction of a fatigue strength for the so-called infinite life depends on this assumption. In order to check this Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Remarks on the Statistical Prediction

275

point, an investigation of the distribution of the fatigue lifetime of a high strength aluminium alloy 2024T3 has been done in Bathias’s laboratory (Bathias, 2003). The specimen used in the study was a dog-bone type with thickness of 4 mm and length of 130 mm. The test machine was an INSTRON 8501. It was controlled by computer, for which load fluctuation is less than 0.1 KN in constant amplitude tests. More than 100 specimens were tested under constant amplitude loading at 13 stress levels (the stress levels were from 175 M to 400 MPa). For tests with maximum stress, smax, of 240 MPa, 260 MPa, 280 MPa, and 300 MPa, the dispersion of lifetimes was very large. About 20 specimens were run at each stress level to obtain enough samples to statistically analyze the distribution of logN for a given smax. The experimental results of fatigue total lifetimes are shown in Figure A2.2. The test data concentrated into two zones; the boundary between the zones is about at 106 to 2  106 cycles. It is clear that the data obtained in these experiments (see in Figure A2.2) forms a narrow scatter band when the stress level is high enough (smax > 320 MPa). When the

Figure A2.2 Results of fatigue lifetime of 2024-T3. Copyright © 2005 by Marcel Dekker. All Rights Reserved.

276

Figure A2.3

Appendix 2

Cumulative frequency of lifetime.

stress level is less than 300 MPa, the test points are not very concentrated. A statistical analysis has been done and the cumulative frequency of logarithms of lifetime on three stress levels is presented in Figure A2.3. The statistical results show that for smax ¼ 260 MPa, the dispersion of total fatigue lifetime is very large. The test points do not conform to a log-normal distribution (a steep straight line). It is noted that, in the stress range from 240 MPa to 300 MPa, it is impossible to draft an acceptable S-N curve with these test data. All failed specimens have been analyzed by microfractography methods. Some different types of inclusions were found in this alloy and one of them was very fragile. When the fatigue cracks initiated from these broken inclusions, they denoted mode B. For those failed specimens in the second zone, the slip bands were observed very clearly. Thus it could be supposed that the persistent slip bands initiate the cracks by mode A. An S-N curve is drawn separately for the test data for each different mode. These S-N curves with a new dual form, composed of two curves, are shown in Figure A2.4. This new dual S-N curve was also statistically analyzed. Based on the two mode hypothesis, the cumulative frequency of total fatigue lifetime at three stress levels is presented in Figure A2.5. The results showed the experimental data for each mode as a distribution in accordance with the log-normal Copyright © 2005 by Marcel Dekker. All Rights Reserved.

Remarks on the Statistical Prediction

277

Figure A2.4 The new S-N curve of aluminium alloy 2024-T3 noting two modes of initiation.

Figure A2.5 Cumulative frequency of lifetime in two modes.

behavior. A 13.5% and 21% level of significance is noted separately for mode B and mode A. As compared with the statistical results in Figures A2.3 and A2.5, the reduction of dispersion of this new S-N curve shows remarkable improvement.

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