Gigacycle Fatigue in Mechanical Practice Claude Bathias-Paul Paris

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Gigacycle Fatigue in

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DK3165_title 8/19/04 10:54 AM Page 1

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


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


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


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-


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


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.


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


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 ½K2 ; ½Mc ; ½Kg 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


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


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 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 r1=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.


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


(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.


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.


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


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

to the order of 109 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 1012 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.


Figure 3.21 Autoclave description.

Figure 3.22 pressure.

Wo¨hler Curve–INCONEL 718:Effect of hydrogen


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


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


Figure 3.24 Pad and specimen.


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.


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 UðxÞðsymmetryÞ

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


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


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 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.


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.


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.


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.


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.


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


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).


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.


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

Gigacycle Fatigue in Mechanical Practice Claude Bathias-Paul Paris

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