Metal Fatigue: What It Is, Why It Matters (Solid Mechanics and Its Applications)

METAL FATIGUE SOLID MECHANICS AND ITS APPLICATIONS Volume 145 Series Editor: G.M.L. GLADWELL Department of Civil Eng...

Author: Les Pook

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METAL FATIGUE

SOLID MECHANICS AND ITS APPLICATIONS Volume 145 Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages.

Metal Fatigue What It Is, Why It Matters

by

LES POOK University College London, UK

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-5596-6 (HB) ISBN 978-1-4020-5597-3 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

The author’s ﬁrst car, a 1932 Riley 9 Falcon. Its many ailments included metal fatigue failures in a gearbox selector fork, timing gear teeth, a half shaft, and a suspension spring. Modern cars are much better in that metal fatigue failures are very unusual.

Leslie Philip (Les) Pook was born in Middlesex, England in 1935. He obtained a BSc in metallurgy from the University of London in 1956. He started his career at Hawker Siddeley Aviation Ltd, Coventry in 1956. In 1963 he moved to the National Engineering Laboratory, East Kilbride, Glasgow. In 1969, while at the National Engineering Laboratory, he obtained a PhD in mechanical engineering from the University of Strathclyde. Dr Pook moved to University College London in 1998. He retired formally in 1998 but remained afﬁliated to University College London as a visiting professor. He is a Fellow of the Institution of Mechanical Engineers and a Fellow of the Institute of Materials, Minerals and Mining. Les married his wife Ann in 1960. They have a daughter, Stephanie, and a son, Adrian.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Fatigue Testing of Components and Structures . . . . . . . 2.2.2 Fatigue Testing of Laboratory Specimens . . . . . . . . . . . 2.2.3 Investigation of the Mechanisms of Metal Fatigue . . . . 2.2.4 Investigation of Fatigue Crack Paths . . . . . . . . . . . . . . . . 2.2.5 Fatigue Crack Propagation Rate Testing . . . . . . . . . . . . . 2.3 The Modern Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Servohydraulic Testing Equipment . . . . . . . . . . . . . . . . . 2.3.3 Inﬂuence of Computers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Standardisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 9 10 10 11 12 12 12 13 13

3

Constant Amplitude Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 S/N (Wöhler) Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Scatter in Fatigue Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mechanisms of Metal Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Fatigue Crack Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Fatigue Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Ratchetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 15 20 24 25 26 30

viii

Contents

3.5 Effect of Mechanisms on S/N Curves . . . . . . . . . . . . . . . . . . . . 31 4

Variable Amplitude and Multiaxial Fatigue . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Description of Fatigue Loadings . . . . . . . . . . . . . 4.2.1 Constant Amplitude Fatigue Loadings . . . . . . . . . . . . . . 4.2.2 Gaussian Random Loadings . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Block Fatigue Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Miner’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Weighted Average Stress Range . . . . . . . . . . . . . . . . . . . 4.3.2 Damage Density Functions . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Cycle Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Standard Load Histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Applications of Standard Load Histories . . . . . . . . . . . . 4.4.2 Development of Standard Load Histories . . . . . . . . . . . . 4.5 Multiaxial Fatigue in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Multiaxial Fatigue Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Multiaxial Failure Criteria . . . . . . . . . . . . . . . . . . . . . . . .

35 35 37 39 39 42 44 47 49 50 53 54 55 59 59 62

5

Fatigue Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Failure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Situations, Philosophies and Approaches . . . . . . . . . . . . . . . . . . 5.3.1 Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Philosophies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Product Liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Consumer Protection Act . . . . . . . . . . . . . . . . . . . . . 5.4.2 Enforcement Authorities . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Safety Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The General Product Safety Regulations . . . . . . . . . . . .

67 67 68 69 69 70 72 77 78 79 79 80

6

The Uncracked Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Effect of Surface Finish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Effect of Mean Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Effect of Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Effect of Multiaxial Fatigue Loading . . . . . . . . . . . . . . . . . . . . . 6.5 Effect of Notches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Bicycle Crankshaft Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 84 85 88 88 89 93

Contents

ix

6.7 Scatter under Variable Amplitude Fatigue Loading . . . . . . . . . 96 6.8 Improvement of Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7

The Cracked Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Initial Crack Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Final Crack Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Constant Amplitude Fatigue Crack Propagation . . . . . . . . . . . . 7.4.1 Determination of Fatigue Crack Propagation Rates . . . 7.4.2 Factors Affecting Fatigue Crack Propagation Rates . . . 7.4.3 Threshold for Fatigue Crack Propagation . . . . . . . . . . . . 7.4.4 Overall Fatigue Crack Propagation Behaviour . . . . . . . . 7.4.5 Short Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.6 Fatigue Life in the Presence of a Crack . . . . . . . . . . . . . 7.5 Variable Amplitude Fatigue Crack Propagation . . . . . . . . . . . . 7.5.1 Weighted Average Stress Range . . . . . . . . . . . . . . . . . . . 7.5.2 Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Fatigue Crack Propagation in a Welded Joint . . . . . . . .

101 101 102 103 105 106 109 116 117 118 122 124 124 127 129 130

8

Fatigue Crack Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Classes of Fatigue Crack Propagation . . . . . . . . . . . . . . . 8.2 Crack Paths in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Mixed Mode Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Crack Path Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Crack Path Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Planar Crack Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Crack Path Development . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Leak before Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Crack Paths in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Lower Bound Mixed Mode Thresholds . . . . . . . . . . . . . 8.4.2 Crack Path Determination . . . . . . . . . . . . . . . . . . . . . . . .

135 135 137 138 138 142 148 151 151 154 155 157 158

9

Why Metal Fatigue Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Avoidance of Fatigue Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Research and Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Role of Non Destructive Testing . . . . . . . . . . . . . . . . . . . . . 9.5 Current Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 162 163 164 164

x

Contents

A

Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Notation for Stress and Displacement Fields . . . . . . . . . . . . . . . A.2.1 Crack Surface Displacement . . . . . . . . . . . . . . . . . . . . . . A.2.2 Volterra Distorsioni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Stress Intensity Factor Solutions . . . . . . . . . . . . . . . . . . . A.3.2 Validity of Stress Intensity Factors . . . . . . . . . . . . . . . . . A.3.3 Effects of Small Scale Yielding . . . . . . . . . . . . . . . . . . . . A.4 Corner Point Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 Crack Front Intersection Angle . . . . . . . . . . . . . . . . . . . . A.4.2 Derivation of Stress Intensity Factors . . . . . . . . . . . . . . . A.5 Stress Intensity Factors for Irregular Cracks . . . . . . . . . . . . . . . A.5.1 Use of Semi Ellipses and Ellipses . . . . . . . . . . . . . . . . . . A.5.2 Projection onto a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.3 Crack Front Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.4 Use of Crack Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.5 Interaction between Cracks . . . . . . . . . . . . . . . . . . . . . . .

167 167 168 168 170 172 174 180 182 186 189 191 195 196 197 199 199 200

B

Random Load Theory and RMS . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Basic Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Random Process Theory . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.2 Fatigue Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Some Sinusoidal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 Narrow Band Random Loading . . . . . . . . . . . . . . . . . . . . B.3.2 Two Parameter Weibull Distribution . . . . . . . . . . . . . . . . B.3.3 Incomplete Distributions and Processes . . . . . . . . . . . . . B.3.4 Non Stationary Narrow Band Random Loading . . . . . . B.4 Broad Band Random Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.1 Spectral Density Function . . . . . . . . . . . . . . . . . . . . . . . .

203 204 204 204 207 208 208 210 212 216 218 218

C

Non Destructive Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Visual Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Magnetic Particle Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Dye Penetrant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Radiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 223 223 226 228 229 231 234

Contents

C.7.1 Eddy Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7.2 Alternating Current Potential Drop . . . . . . . . . . . . . . . . . C.8 Probability of Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8.1 Determination of Probability of Detection . . . . . . . . . . . C.8.2 Probability of Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

235 236 238 240 242

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Preface

The ﬁrst major book in English on metal fatigue was Fatigue of Metals (Gough 1926). Since then numerous books have been published which are primarily on this important topic. These range from general surveys, for example Suresh (1998) to books on speciﬁc aspects of metal fatigue, for example Radaj and Sonsino (1999) on assessment of welded joints, and Murakami (2002) on the effects of small defects. Books on the related topic of fracture mechanics, for example Broek (1988), usually include extensive treatment of fatigue crack propagation. There are many other books where metal fatigue is a major theme. These range from undergraduate texts, for example Dowling (1993), to a 10 volume compilation on structural integrity edited by Milne et al. (2003). There do not appear to be any recent books on metal fatigue which are presented in a format that appeals to engineers, and which can be recommended to newcomers to the topic. Metal fatigue has both metallurgical and engineering aspects, and also its own specialised jargon which newcomers often ﬁnd confusing. In the last few decades treatment of most aspects of metal fatigue has become much more mathematical, with extensive use of computers. This book aims to present the important ideas in metal fatigue in as straightforward a manner as possible for the beneﬁt of readers who need to be able understand more advanced documents on a wide range of metal fatigue topics. Indications on how metal fatigue problems are solved in engineering practice are included. It is based on 50 years experience of metal fatigue, and on 15 years experience of introducing engineering undergraduates to its basic ideas. The prerequisite knowledge required for readers is a basic understanding of stress analysis and mathematics covered in engineering undergraduate courses. No prior knowledge of metal fatigue is assumed. The objectives of the book are: to explain the terminology used in metal fatigue, to provide a

xiv

Preface

brief description of the mechanisms of metal fatigue, to describe the basis of design against the fatigue failure of components and structures, to provide examples illustrating various points made, and hence to provide a ﬁrm foundation for readers who wish to study various aspects of metal fatigue in more detail, including ﬁnding solutions to speciﬁc fatigue problems. The symbols used have been chosen to represent widespread practice but it has not been possible to be entirely consistent. Other symbols are used by some authors. Les Pook August 2006

Notation

I, II, III A A, m a af ai δa B C, m c E E(k) F f K KI∗ KIc KB Kc Kf Kh Kmax Kmin Kop Kt KλA

modes of crack surface displacement crack area constants in Basquin equation crack length, half crack length of internal crack, crack radius, characteristic crack dimension, semi minor axis of ellipse ﬁnal value of a initial value of a increment of crack propagation specimen thickness, biaxiality ratio constants in Paris equation half crack surface length, semi major axis of ellipse Young’s modulus complete elliptical integral of the second kind force frequency stress intensity factor, subscripts I, II, III denote mode value of KI for a small Mode I branch crack plane strain fracture toughness bending mode stress intensity factor fracture toughness, plane stress fracture toughness fatigue strength reduction factor weighted average range of KI maximum value of KI in fatigue cycle minimum value of KI in fatigue cycle crack opening value of KI stress concentration factor antisymmetric mode stress intensity measure

xvi

KλS Kλ K Ki Keff Kth k M m N Ni n ni P P p(S) q R r rch r, θ r, θ, φ r(S/σ ) S Sa Sh Sm T TR TRc t V W x, y, z u, v, w Y

Notation

symmetric mode stress intensity measure stress intensity measure range of KI in fatigue cycle value of K for ith cycle effective value of K threshold value of K = 1 − a 2 /c2 bending moment dimensionless magnetic permeability parameter number of cycles life at ith load level number of applied cycles number of applied cycles at ith load level pressure cyclic pressure range probability density of S notch sensitivity index stress ratio (= σmin /σmax ), stress ratio (= ni /Ni ), relative damage region radius, radius of open crack tip, distance from crack tip characteristic value of r polar coordinates spherical coordinates damage density of S/σ stress, stress range (= 2σa = σmax − σmin ), reference stress, statistical variable alternating value of reference stress, peak stress in a Rayleigh process weighted average stress range mean value of reference stress time interval, temperature, torque, stress parallel to crack T -stress ratio critical value of TR time voltage specimen width Cartesian coordinates displacements in x, y, z directions geometric correction factor

Notation

α α, β, γ β βc γ δ δa ε θ λ µ µ0 ν ρ σ σa σe σm σmax σmin σt σx , σy , σz σY σ0 σ1 , σ2 , σ3 σ σi τxy , τyz , τzx

xvii

= a/W rotations about axes parallel to x, y, z axes crack front intersection angle critical crack front intersection angle curvature factor probe spacing skin depth increment of crack propagation strain, reference strain branch crack propagation angle coefﬁcient deﬁning singularity, biaxial loading ratio magnetic permeability magnetic permeability of a vacuum Poisson’s ratio equivalent stress tensile stress, root mean square value of S, electrical conductivity alternating stress equivalent stress mean stress maximum stress in fatigue cycle minimum stress in fatigue cycle tensile strength stresses in x, y, z directions yield stress fatigue limit principal stresses stress range value of σ for ith cycle shear stresses on xy, yz, zx planes

1 Introduction

The term metal fatigue refers to gradual degradation and eventual failure that occur under loads which vary with time, and which are lower than the static strength of the metallic specimen, component or structure concerned. The static strength is the load which causes failure in one application. The loads responsible are called fatigue loads. These loads are cyclic in nature, but the cycles are not necessarily all of the same size or clearly discernible. A fatigue load in which individual cycles can be distinguished is sometimes called a cyclic load. Metal fatigue is largely a descriptive subject, and as such it has accumulated an enormous literature (Pook 1983a). Nevertheless, the basic concepts needed for an understanding of the metal fatigue literature are reasonably straightforward, and these are described in this book. The descriptions can be divided into two groups, metallurgical and mechanical. Metallurgical descriptions are concerned with the state of the metal before, during and after the application of fatigue loads, and are usually taken to include the study of metal fatigue mechanisms. Mechanical descriptions are concerned with the mechanical response to a given set of loading conditions, for example the number of load cycles needed to cause failure. Mechanical descriptions are more useful from an engineering viewpoint, where service behaviour must be predicted, and are therefore given more emphasis in this book. Rigorous deﬁnition of exactly what is meant by metal fatigue is difﬁcult and not particularly helpful for its understanding. An early dictionary deﬁnition is: the condition of weakness in metal caused by repeated blows or longcontinued strain (Murray 1901). A more recent deﬁnition is: failure of a metal under a repeated or otherwise varying load which never reaches a level sufﬁcient to cause failure in a single application (Pook 1983a).

2

Chapter 1: Introduction

Figure 1.1. Car drive line component. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

From an engineering viewpoint, metal fatigue matters because it is a major potential cause of failure of components and structures, including load bearing consumer items. In a sense, the problem might appear to have been largely solved, in that catastrophic failures due to metal fatigue are now rare. Ofﬁcial inquiries into catastrophic metal fatigue failures, involving loss of life or major ﬁnancial loss, usually indicate a clear reason for the failure and often indicate apparent human negligence. An example is the failure of a fairground ride in Glasgow, which killed two people (Pook 1998). Lesser metal fatigue failures are still common and cause a great deal of inconvenience and expense. In other words, they are a nuisance rather than catastrophic. These lesser failures are often unrecognised as being due to metal fatigue unless they happen to be seen be an expert. Figures 1.1–1.4 show a selection of nuisance metal fatigue failures. They illustrate the point that what is meant by failure depends on the function of component. Figure 1.1 shows part of the car drive line component that had broken off due to metal fatigue, and this meant that power was no longer transmitted to the car wheels. The car had covered many miles in mountainous country, so many power on, power off cycles, probably of the order of 106 , had been applied. This particular failure led to some acrimonious correspondence with the manufacturer over whether or not the failure, following use of the car

Metal Fatigue

3

Figure 1.2. Cracking in aircraft jet engine nacelle.

Figure 1.3. Cracked ring spanner. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

under unusually severe operating conditions, could be regarded as fair wear and tear. Figure 1.2 shows cracking in the aircraft jet engine nacelle found during a routine inspection after 2000 hours ﬂying (Pook 2004). The cracks were in an area affected by jet noise. The nacelle was made from 0.56 mm thick clad aluminium 4% Cu alloy. Acoustic fatigue due to jet noise is well known. If it had been allowed to continue the nacelle would eventually have broken up and would no longer have provided protection to the engine. The cracking was due to jet noise so very many cycles had been applied. Failure analysis suggested that failure had been caused by biaxial bending fatigue loading. In 2000 hours ﬂying very many load cycles (of the order of 1010 ) would have been applied. Figure 1.3 shows a ring spanner with a fatigue crack that had propagated right through the ring. This caused severe loss of stiffness, so the ring sprung open and slipped when the user tried to tighten a nut. The spanner had been used a lot but even so only a relatively small number of cycles, probably of the order of thousands, could have been applied.

4

Chapter 1: Introduction

Figure 1.4. 9.5 mm long brass chain link from weight driven clock. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

The brass chain link from a weight driven clock, shown in Figure 1.4, was bent into shape from a piece of brass wire, but the ends of the wire were not joined. When a fatigue crack had propagated about half way through one of the links, the section was so reduced that the link distorted due to plastic deformation. The chain link still withstood the load due to the clock driving weight, but the distortion meant that it would no longer pass through the clock mechanism. The clock was wound daily, so one load cycle was applied per day, with the total number of cycles of the order of 103 . These examples show that failure can mean that a component no longer performs its intended function because it has broken into two (or more) parts, or because of loss of stiffness or distortion. They also illustrate the very wide variation in the number of load cycles applied during the life of a component. Fatigue also affects non metallic materials. For example, Figure 1.5 shows a plastic domestic tap. It was observed to be leaking where it was screwed into a ﬁtting on the supply pipe. The tap had a ﬁtting for a hose pipe and appeared to be a replacement for the original brass tap which did not have a hose ﬁtting. When an attempt was made to unscrew the tap, it failed completely. The two parts of the broken tap are shown in Figure 1.5. The dark area is where fatigue crack propagation took place and the light area where the ﬁnal static failure took place. The age of the tap at the time of failure was unknown but as one

Metal Fatigue

5

Figure 1.5. Plastic domestic tap.

fatigue cycle is applied each time a tap is turned on and off the number of load cycles applied was probably of the order of thousands. Pressure containing components where safety is an issue, such as parts of hydraulic systems, are often designed to leak before break in order to avoid catastrophic failure. It is fortunate that the tap did leak otherwise the utility room in which it was installed would probably have been ﬂooded. The failed tap was replaced with a brass tap, which has given satisfactory service. Comparison of the plastic and brass taps showed that the detail design in the vicinity of the failure was exactly the same. In retrospect the detail design of the plastic tap should gave been changed to allow for the different mechanical properties of the plastic. The episode is an example of the danger of using a different material for a component, which is subject to fatigue loads, without making appropriate changes to detail design.

2 Historical Background

2.1 Introduction Early in the history of engineering design, there was a recognition of the need to know the different ways in which a material or component could fail. Failure was usually associated with fracture, or with excessive deformation. Failure under static loads, tensile, compressive and shear, became widely known. Much early design aimed at making a component or a structure that would last indeﬁnitely. Metal fatigue has been of interest for about 170 years. This interest dates back to the development of the steam engine, mechanical transport, and the more extensive use of mechanical devices. This mechanisation meant that many components were subjected to fatigue loads, and fatigue failure was beginning to become a common occurrence. The history of metal fatigue, from an engineering viewpoint, is well documented but early references are often difﬁcult to locate. Most books on metal fatigue include a historical summary, usually concentrating on mechanical descriptions. The best recent history, on mechanical descriptions of metal fatigue, is by Schütz (1996). It includes over 500 references, mostly in English and German. The ﬁrst use of the term fatigue in print appears to be by Braithwaite (1854), although in his paper Braithwaite states that it was coined by a Mr Field. The general opinion had developed (Frost et al. 1974) that the material had tired of carrying the load, or that the continual re-application of a load had in some way exhausted the ability of the material to carry load. The use of the term has survived to this day. Since fatigue failures also occur in many non metallic materials the term metal fatigue is often used, as in this book, to

8

Chapter 2: Historical Background

denote the particular kind of fatigue that occurs in metallic materials, components and structures. The ﬁrst known catastrophic fatigue failure, involving major loss of life, was the Versailles (France) railway accident in 1842 (Smith 1990). The train was unusually long, with 17 carriages hauled by two steam engines. The front axle of the leading, four wheeled engine failed due to metal fatigue and the body of the leading engine fell to the ground. The second engine smashed it to pieces. Following carriages passed over the wreck and some were set on ﬁre. This, and numerous other railway axle failures, led to extensive investigations into the nature of metal fatigue (Parsons 1947, Smith 1990, Schütz 1996). The ﬁrst known, reasonably well documented, metal fatigue failures were in clock mainsprings (Wayman et al. 2000). The use of uncoiling springs, rather than descending weights, as a driving force was an important factor in the development of clocks for general use, and appears to have started in the early ﬁfteenth century. By the late eighteenth century the technology for the manufacture of durable watch and clock mainsprings was well established; a detailed description of the state of the art of making watch springs was published by Blakey (1780). Even so, high quality watches and clocks were designed (and still are) so that a broken mainspring could easily be replaced. This shows that metal fatigue failures were indeed a problem,

2.2 Experimental Work Understanding of metal fatigue is based largely on the results of laboratory experiments and on service behaviour. Much information was obtained before mechanisms of metal fatigue were understood, and there was no underlying central theme to correlate the data. Thus isolated sets of data became almost metal fatigue folklore, and their relationship to other sets of data was not obvious (Frost et al. 1974). Numerous theories, some seriously misleading (Schütz 1996), have been developed in attempts to explain experimental results. There are ﬁve main themes in experimental work on metal fatigue, which started at various times. Partly for convenience, and partly because it is often representative of service conditions, most experiments are carried out in air at ambient temperatures. 2.2.1 FATIGUE T ESTING OF C OMPONENTS AND S TRUCTURES Fatigue testing of components and structures started in the 1830s. The ﬁrst known fatigue test results were published in German by Albert in 1837. These

Metal Fatigue

9

results were cited by Schütz (1996) and by Frost et al. (1974). Albert constructed a machine for repeatedly proof-loading welded mine hoist chains, continuing some tests up to 105 cycles. The ﬁrst fatigue test results published in English appear to be those by Fairbairn (1864) on repeated bending fatigue tests on beams. He used a mechanism actuated by a water wheel to apply a load repeatedly to the centres of 6.7 m long wrought iron built up girders. His apparatus is illustrated in Gough (1926), Timoshenko (1953) and Marsh (1988). The calculated static failure load of the beam under a central load was 120 kN. Fairbairn found that repeated loads of 30 kN were insufﬁcient to cause failure in 3 × 106 cycles, while if a greater load was applied failure did occur at a lower number of repeated loads. He concluded that there was a safe repeated load which could be applied to such a structure. This load would either be sustained indeﬁnitely, or the number of repetitions to failure would be so large as to exceed the normal life of a bridge. Fairbairn estimated 12 × 106 repetitions as being equivalent to a bridge life of 328 years, assuming that the loading is applied 100 times per day. 2.2.2 FATIGUE T ESTING OF L ABORATORY S PECIMENS Fatigue testing of specially designed laboratory specimens started in the 1850s. It is generally accepted that the ﬁrst fatigue tests on laboratory specimens were carried out by Wöhler. He published the results of his classical experiments on metal fatigue, using both notched and unnotched specimens, in a series of papers in German starting in 1858. These results are cited by Schütz (1996). An account in English is given in Anon. (1871). Wöhler constructed various types of fatigue testing machines, and carried out the ﬁrst fatigue tests in which strict attention was paid to the magnitude of the applied fatigue loads. An important point is that appropriate stress analysis had been developed (Timoshenko 1953) and was available to Wöhler. This enabled him to relate his experimental results to service stresses in railway axles, and hence produce design rules which were incorporated in Technical Regulations for German Railways (Schütz 1996). This did not require detailed knowledge of the mechanisms of metal fatigue. In 1870 Wöhler published a ﬁnal report. This is cited in Schütz (1996) and contains the following conclusions, known as Wöhler’s laws. (a) Materials can be induced to fail by many repetitions of stress, all of which are lower than the static strength.

10

Chapter 2: Historical Background

(b) The stress amplitudes are decisive for the destruction of the cohesion of the material. (In modern terminology this is the stress range, see Section 3.1.) (c) The maximum stress is of inﬂuence only in so far as the higher it is, the lower are the stress amplitudes which lead to failure. (In modern terminology this means that increasing the mean stress decreases the number of cycles to failure, see Section 6.3.) 2.2.3 I NVESTIGATION OF THE M ECHANISMS OF M ETAL FATIGUE Investigation of the mechanisms of metal fatigue started in earnest in the 1900s. During the nineteenth century there was much speculation on the nature of metal fatigue ‘damage’ (Parsons 1947, Smith 1990). With the beneﬁt of hindsight, some of the theories advanced now appear bizarre. Extensive metallurgical investigations, starting with Ewing and Humphrey (1903) showed that metal fatigue damage in a plain specimen is a surface phenomenon. They showed that, in a ductile metallic material, bands of slip lines form on the surface of grains under fatigue loading. These slip lines eventually turn into cracks. In general only one crack develops to any considerable depth, and this crack propagates across the material until the section is so reduced that static failure takes place. Detailed investigation of the mechanisms of fatigue crack propagation in metals started in the 1960s. One of the ﬁrst signiﬁcant papers was published by Forsyth (1961). 2.2.4 I NVESTIGATION OF FATIGUE C RACK PATHS The complete solution of a fatigue crack propagation problem includes determination of the crack path. At the present state of the art the factors controlling the path taken by a propagating fatigue crack are not completely understood. In general, fatigue crack paths are difﬁcult to predict (Pook 2002a). Macroscopic aspects of fatigue crack paths have been of industrial interest since the earliest fatigue investigations (Parsons 1947, Smith 1990). The book by Cazaud published in 1948 includes an analysis of fatigue crack paths in both laboratory specimens and industrial components. An English translation of his book, including additional material, was published in 1953 (Cazaud 1953). In the last four decades there have been substantial advances in the understanding and prediction of fatigue crack paths, largely through developments in fracture mechanics and in the application of modern computers. Despite recent theoretical advances, fatigue crack paths in structures are still often determined by large scale structural tests (Pook 2000a). At macroscopic

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11

scales differential geometry has been used in the interpretation of some features of crack paths (Pook 2002a). The microscopic examination of fatigue fracture surfaces started in the 1950s. The appearance of fatigue fracture surfaces in metals, at low magniﬁcation, had been of interest since the early days of service failure analysis and of fatigue testing. The study of fracture surfaces is known as fractography. This term appears to have been coined by Zapffe and Clogg (1945). The optical metallurgical microscope was ﬁrst used for the examination, at high magniﬁcation, of fatigue fracture surfaces in metals by Zapffe and Worden (1951). The use of fractography in metal fatigue research and development rapidly became routine, for example Forsyth et al. (1959). By 1962 the use of quantitative fractography in the reconstruction of crack path information was well developed (Pook 1962) (see Section 7.6). At microscopic scales there has been interest in the use of fractals (Mandelbrot 1983) and random process theory (Pook 1976a, 2002a) in the characterisation of fatigue crack paths. 2.2.5 FATIGUE C RACK P ROPAGATION R ATE T ESTING Fatigue crack propagation rate testing, in both laboratory specimens and structures, also started in the 1950s. The importance of fatigue crack propagation in metallic materials had been appreciated since about 1870. However, the lack of an appropriate applied mechanics framework meant that little effort had been devoted to experimental determination of the laws governing the rate of propagation of fatigue cracks in metals and structures. In fact no accurate quantitative data had been published until around the time that Head (1953, 1956) published his theoretically derived relationship between crack length and number of stress cycles. Following the catastrophic failures in Comet aircraft in the 1950s, major structural fatigue tests were carried out on Comet aircraft structures (Anon. 1954) during which the propagation of fatigue cracks was monitored. At that time it was not possible to relate such data on structures directly to the results of fatigue crack propagation rate tests on laboratory specimens. However, it was possible to correlate fatigue crack propagation data, obtained during fatigue tests on structures, with crack propagation data obtained fractographically from the fatigue fracture surfaces (Pook 1960, 1962, Troughton et al. 1963). At about this time it also became clear that fatigue crack propagation does not take place if the applied fatigue load is too low. That is, there is a threshold for fatigue crack propagation. This threshold exists because a fatigue crack propagation rate of less than one lattice spacing per cycle is

12

Chapter 2: Historical Background

not possible on physical grounds (Frost et al. 1971). However, because crack propagation is not necessarily continuous along the whole crack front, average crack propagation rates of less than one lattice spacing per cycle are sometimes observed.

2.3 The Modern Era The modern era in metal fatigue research started around 1970 due to several major developments (Pook 1983a). By this time the mechanisms of metal fatigue were understood in general, but not necessarily in detail. A very large amount of data had been accumulated, and there was a good general understanding of how to avoid fatigue failures in service. The state of the art of metal fatigue in the early 1970s was summarised by Frost et al. (1974). At the time, the book was unusual in that it was written in SI units, and a fracture mechanics approach to fatigue crack propagation was used; both are now universal. 2.3.1 F RACTURE M ECHANICS The analysis and application of fatigue crack propagation rate and threshold data, in both laboratory specimens and structures, became much easier with the development of fracture mechanics through the pioneering work of Irwin (Anon. 1965a, Rossmanith 1997). The fracture mechanics parameter stress intensity factor provides a convenient single parameter description of the elastic stress ﬁeld in the vicinity of a crack tip (see Section A.3). One of the ﬁrst collections of fatigue crack propagation rate and threshold data for a wide range of metallic materials, analysed in terms of stress intensity factors, was published by Frost et al. (1971). Despite the apparent simplicity of laboratory fatigue crack propagation rate and threshold testing it was another ten years before standard test methods started to appear (Anon. 1981a). 2.3.2 S ERVOHYDRAULIC T ESTING E QUIPMENT The introduction of closed loop servohydraulic testing equipment (Marsh 1988, Schijve 2001) meant that almost any desired load history could be applied to a component or structure. Service load simulation testing, using appropriate load histories, is often the most cost effective means of determining service life and is sometimes a requirement of regulatory authorities. It ﬁrst became common in the aircraft industry, but is now used for critical

Metal Fatigue

13

components in a wide range of industries. Much effort has been devoted to the development of standard load histories for fatigue testing purposes (Schütz and Pook 1987, Schütz 1996)). A load history is sometimes called a load spectrum. 2.3.3 I NFLUENCE OF C OMPUTERS For a long time metal fatigue was largely a descriptive subject (Pook 1983a). This has changed in the last four decades. The increasing power and sophistication of now ubiquitous desktop computers has meant an increase in the application of numerical methods, sometimes based on sophisticated mathematics, to metal fatigue research and development. Much current work on metal fatigue simply would not be possible without computers. In the Preface to his book Statistics of Extremes (Gumbel 1958), which includes metal fatigue applications, Gumbel states ‘Graphical procedures are preferred to tedious calculations.’ This is not surprising since at that time a typical desktop calculator was an electro mechanical device which would not even extract square roots automatically, and mainframe computers were in their infancy. In the 1960s mainframe computers became more accessible, and started to be used for metal fatigue related calculations such as Miner’s rule summations (see Section 4.3). Around 1970 user friendly programmable desk top calculators, such as the HP 9100B, started to become available. As is well known, computer development has continued apace and has now reached the stage where entry level desk top computers are as powerful as many of the mainframe computers in use in the 1980s. One of the consequences is that approaches to metal fatigue have become much more mathematical. 2.3.4 S TANDARDISATION Standard procedures of various degrees of formality are the most satisfactory way of carrying out a metal fatigue assessment of a proposed design (Pook 1983a). These range from established good practice in a particular design ofﬁce to elaborate published procedures. These may be based on analytical procedures, service experience, the results of fatigue tests, or some combination of these, and need not have a theoretically sound basis, provided that they give sufﬁciently accurate answers. Very simple methods are often used in the early stages of design. The development of standard procedures, which can be a lengthy process, is probably the best way of assimilating the results of metal fatigue research into engineering practice. Most metal fatigue

14

Chapter 2: Historical Background

assessments are based on simpliﬁed standard procedures, which despite their apparent lack of physical validity, are known to give conservative answers. In the modern era, the use of formal standards in many aspects of metal fatigue has become more and more widespread. Such standards are often incorporated in software and are sometimes insisted upon by regulatory authorities. An attempt to list all the documents issued by the British Standards Institution which contained fatigue clauses included more than 300 items (Pook 1989a). Lists of metal fatigue standards and related software are not included in this book because such lists would date rapidly.

3 Constant Amplitude Fatigue

3.1 Notation A constant amplitude fatigue loading (or constant amplitude loading) is a fatigue loading in which all the load cycles are identical. A load cycle is usually called either just a cycle or a fatigue cycle. The notation used is shown in Figure 3.1. A cycle is the smallest unit of the stress history which repeats exactly. Cycles are often, but not always, sinusoidal. σa (sometimes σ , as in the ﬁgure) is the alternating stress, σm is the mean stress, σmax is the maximum stress in the load cycle, and σmin is the minimum stress in the load cycle. Mathematically a load cycle (or stress cycle) is expressed as σm ± σa . Compressive stresses are taken as negative. It follows that σmax = σm + σa ,

(3.1)

σmin = σm − σa ,

(3.2)

σm =

σmax + σmin . 2

(3.3)

The stress range is S = 2σa = σmax − σmin , and the stress ratio, R = σmin /σmax . The term stress ratio and its symbol, R, are very well established, and both are often used in the metal fatigue literature without explanation.

3.2 S/N (Wöhler) Curves Very large numbers of constant amplitude fatigue tests have been carried out on plain (unnotched) metallic specimens. Test results are sensitive to the surface ﬁnish. Compilations of results are available, for example Shiozawa and Sakai (1996). Many tests were on specimens of circular cross section tested

16

Chapter 3: Constant Amplitude Fatigue

Figure 3.1. Notation for constant amplitude fatigue loading.

in rotating bending. In rotating bending the mean stress is zero (stress ratio, R = −1), and cycles are always sinusoidal. Results are presented in terms of the nominal surface stress, which is calculated assuming that a specimen remains wholly elastic. Figure 3.2 shows a typical specimen. This is designed for testing in four point bending so that stress conditions are uniform along the parallel portion. In his original tests Wöhler (see Section 2.2.2) used cantilever bending as shown in Figure 3.3. Two specimens were tested simultaneously and the load was applied by springs. In modern machines only one specimen is tested and loads are usually applied by weights. For tests at other stress ratios tests are usually carried out either on circular cross section specimens tested in direct stress or on rectangular cross section specimens tested in plane bending. Sinusoidal load cycles are normally used. Figure 3.4 shows a specimen with threaded ends for testing under direct stress. Techniques are now being developed for the fatigue testing of very small specimens (Connolley et al. 2005). Conventionally, results are presented as S/N curves. These are plots of alternating Stress versus Number of cycles to failure, with an appropriate curve ﬁtted through the individual data points. Sometimes, stress range is used; care is needed when using data to check which convention has been used. Failure is usually deﬁned as the separation of a specimen into two parts, but other

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17

Figure 3.2. Plain fatigue specimen for testing in four point rotating bending.

Figure 3.3. Schematic diagram of Wöhler’s rotating cantilever bending fatigue testing machine.

Figure 3.4. Plain fatigue specimen for testing in direct stress.

deﬁnitions are sometimes used. For example, loss of a speciﬁed amount of stiffness or the appearance of a crack of a speciﬁed size. S/N curves are sometimes called Wöhler curves after August Wöhler (see Section 2.2.2). The number of cycles to failure is sometimes called the life or the endurance. It is usually plotted on a logarithmic scale, but the alternating stress may be plotted on either a linear or a logarithmic scale. Typical S/N curves are shown in Figures 3.5 and 3.6. As used to be conventional (Frost et al. 1974) these S/N curves are for endurances of less than 108 cycles. The region where failure takes place in less than about 104 cycles is called low cycle fatigue, and the region for longer endurances high cycle fatigue. Arrows are attached to symbols to indicate that a specimen was un-

18

Chapter 3: Constant Amplitude Fatigue

Figure 3.5. S/N curve for carbon steel specimens tested in rotating bending. Arrows attached to symbols denote specimen unbroken (Frost et al. 1974).

broken when a test was discontinued. The reason why metal fatigue data are conventionally presented in terms of numbers of cycles, rather than in terms of time, is that for many metallic materials tested in air at room temperature the number of cycles to failure is independent of the test frequency (Frost et al. 1974). These materials are called frequency independent. Frequency independence implies that the number of cycles to failure is also independent of the waveform that connects positive and negative peaks. It was usually found for steels having tensile strengths up to about 700 MPa that, if a specimen has not broken after 107 cycles, it was most unlikely to break if tests were continued to longer endurances; tests on these materials were therefore terminated at about 2 × 107 cycles. Tests on other materials were often continued up to about 108 cycles. Figure 3.5 shows an S/N curve obtained from a batch of carbon steel specimens tested in rotating bending. The data showed that specimens either had a life less than 5 × 106 cycles or were still unbroken when the tests were stopped at 2–4 ×107 cycles, and suggested that the line through the points became horizontal. When this is so, the stress corresponding to the horizontal line is called the fatigue limit. It was implied that specimens tested at below the fatigue limit would never break no matter how many stress cycles were applied. If stress levels are plotted on a logarithmic scale, it is often found that the ﬁnite life portion of the S/N curve can be represented as a straight line given by the Basquin equation (Basquin 1910) N = Aσ m ,

(3.4)

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19

Figure 3.6. S/N curve for high-strength aluminium alloy specimens tested in rotating bending (Frost et al. 1974).

where A and m are empirically determined constants. The intersection between this straight line and the horizontal straight line is often referred to as the knee. It is now widely known (Murakami 2002) that the S/N does not become quite horizontal and that failures can occur in the region of 109 cycles. This region is known as gigacycle fatigue, and was ﬁrst observed by Shabalin in 1958 (Romanovskaya and Botvina 2003). The importance of gigacycle fatigue in practical engineering was pointed out by Gough (1926). He listed estimated numbers of fatigue cycles applied in the lives of various structures and components. These ranged up to 1.5 × 1010 cycles for a steam turbine shaft. The suspension springs of clock pendulums are subjected to very large numbers of cycles. For a typical long case clock pendulum, 10 years corresponds to about 1.6 × 108 cycles (Matthys 2004); many long case clocks are still in use after more than two centuries, corresponding to more than 3 × 109 cycles. Not all metals gave S/N curves which exhibited a deﬁnite knee, even when tests were continued to very long endurances (Frost et al. 1974). An S/N curve for a high strength 4.5% Cu aluminium alloy is shown in Figure 3.6; failures were seen to occur at approaching 108 cycles. It is usual in such cases to specify the fatigue strength of the material at a given endurance, that is, specimens tested at higher stress levels will break before the stipulated

20

Chapter 3: Constant Amplitude Fatigue

life, whereas those tested at lower stress levels will be unbroken after a life of at least the stipulated value. The fatigue strength at the stipulated value is usually called the endurance limit. The slope of an S/N curve such as that shown in Figure 3.6 is generally small at endurances in the region of 108 cycles, and the stress for this life is typically used as an endurance limit. Fatigue testing is very time consuming. For example, if a testing machine operates at 30 Hz then 108 cycles takes more than a month. With the development of very high speed fatigue testing machines, operating at up to 20 kHz, testing in the gigacycle region became practical (Bathias 2001, Stanzl-Tschegg 2002). It has been found, for a wide of metallic materials, that fatigue failures can occur at up to around 1010 cycles, although as 1010 cycles is approached the slope of an S/N curve is small. At high stress levels, yielding takes place, and the S/N ﬂattens, as shown schematically in Figure 3.7(a). It may then be necessary to plot results in terms of strain, ε, rather than stress (Figure 3.19(b). Other conventions are sometimes used; for example, results may be plotted in terms of the maximum stress in the fatigue cycle, σmax , rather than the alternating stress. For tests on components, it is sometimes convenient to use the applied fatigue load rather than a stress.

3.3 Scatter in Fatigue Data By its very nature, metal fatigue is a random process, and the consequent scatter of results, even in carefully controlled experiments, complicates both the analysis of experimental data and their subsequent application to practical problems. The amount of scatter in data, such as those shown in Figures 3.5 and 3.6, is greater than can be accounted for by experimental error. The determination of an S/N curve involves subjective judgements when ﬁtting a curve to the individual data points. Statistical methods provide a rational approach to this type of problem, but do not of themselves either avoid the need for subjective judgement at some stage or increase the amount of information present in a given set of data. Statistical theory provides information on the most efﬁcient use of a limited number of test specimens, and on the number of test specimens required to give a speciﬁed degree of conﬁdence in test results (Anon. 2003a). If a number of nominally identical fatigue specimens are tested at the same stress amplitude and the lives tabulated, a histogram may be plotted by dividing lives into groups, of a ﬁxed width, distributed about the mean (average) value. If a large enough number of specimens is tested, the groups

Metal Fatigue

21

Figure 3.7. Schematic diagrams showing S/N curves plotted in terms of (a) stress and (b) strain.

may be made sufﬁciently narrow for the histogram to be represented by a smooth curve. One function that has been found to give an adequate ﬁt to certain data is the Gaussian distribution or (Normal distribution), which is given by 2 −S 1 exp , (3.5) p(S) = √ 2σ 2 σ 2π where S is the variable involved, p(S) is the probability density, and σ is the root mean square (RMS) value for all values of S, measured from the mean, usually called the standard deviation. This is a measure of the width of the scatter band. Distributions of fatigue lives of metallic specimens do not conform to the Normal distribution, but it has been shown that, in many cases, the Normal distribution is a good approximation to the distribution of the logarithms of lives of metallic specimens (Bastenaire 1963, Frost et al. 1974). This is called

22

Chapter 3: Constant Amplitude Fatigue

Figure 3.8. Distribution of fatigue lives of plain copper specimens at various stress levels (Frost et al. 1974). (a) 150 specimens, ±88 MPa. (b) 148 specimens, ±90 MPa. (c) 133 specimens, ±97 MPa. (d) 200 specimens, ±109 MPa. (e) 350 specimens, ±116 MPa. (f) 100 specimens, ±131 MPa.

the log Normal distribution. Other distributions are sometimes used in modern work, for example in Murakami (2002) and Schijve (2005). Large numbers of specimens have sometimes been tested in attempts to ﬁnd the precise form of the probability density function in given circumstances. Some typical results are shown in Figure 3.8 (Frost et al. 1974). A basic concept of statistics is that a group of one or more specimens is merely a sample taken from a large body or population. Such a sample is considered to be just one of a number of samples that could be tested. The results obtained from tests on a random sample from the population can be used to estimate the characteristics of the whole population and to measure

Metal Fatigue

23

Figure 3.9. Schematic P-S-N diagram for three probabilities of failure and log Normal distributions of lives.

the reliability of the estimates. The values of the parameters of the population can only be estimated from tests on the sample. To obtain exact values would require the whole population to be tested. These estimates of the behaviour of the population from tests on a sample, and the conﬁdence that can be placed on them, are the essence of statistical analysis in general and the statistical analysis of metal fatigue specimens in particular (Anon. 2003a). If batches of specimens of a particular metallic material are tested at different stress levels, S/N curves for different probabilities of failure can be drawn; 50 per cent probability corresponds to the median, that is the middlemost life, and is the conventional S/N curve. These curves are sometimes referred to as P-S-N curves. Figure 3.9 shows schematic P-S-N curves for failure probabilities of 1, 50 and 99 per cent. Scatter in log endurance increases as the fatigue limit is approached, as indicated in the ﬁgure. Experimental data on the scatter of the applied alternating stress required to cause failure at a speciﬁed life cannot be obtained directly. This is because the applied alternating stress is the independent variable and specimen life is the dependent variable. However, indirect methods suggest that for a speciﬁed life the distribution of applied alternating stresses is approximately Normal (Bastenaire 1963, Ford et al. 1963). Further, when alternating stresses are normalised by the alternating stress for a probability of failure of 50 per cent their standard deviation is approximately independent of the speciﬁed life. Table 3.1 shows normalised stresses for various probabilities of failure assuming a Normal distribution and a standard deviation for normal-

24

Chapter 3: Constant Amplitude Fatigue Table 3.1. Normalised stresses for various probabilities of failure. Probability of failure (Per cent) 0.01 0.1 1 5 10 50 90 95 99

Normalised stress 0.702 0.752 0.814 0.864 0.897 1 1.103 1.132 1.186

Table 3.2. Metallurgical feature scales. Scale (mm) 10−6 10−5 10−4 10−3 10−2

Metallurgical feature Ions, electron cloud Dislocations Subgrain boundary precipitates Subgrain slip band Grains, inclusions, voids

ised stresses of 0.08, which is a typical value. In design against metal fatigue very low probabilities of failure are of interest. The important point is that the normalised stress is much lower for low probabilities of failure.

3.4 Mechanisms of Metal Fatigue Mechanisms responsible for the failure of metals, and associated metallurgical features, can be viewed at a wide range of scales (McClintock and Irwin 1965) as listed in Table 3.2. Metal fatigue mechanisms are relatively simple when viewed at the largest scale in the table, but are more complex when viewed at smaller scales. This is typical of a wide range of physical phenomena, where different results are obtained by viewing at different scales (Mandelbrot 1977). In essence, the mechanisms involved in the fatigue failure of a plain metallic specimen with a polished surface are simple (Frost et al. 1974), although the details may be complex, especially when viewed at smaller scales, and may vary between different metals. Firstly, a fatigue crack is initiated at the specimen surface, secondly the crack propagates slowly across the specimen and ﬁnally, when the net cross section has been sufﬁciently reduced, a static failure takes place on the ﬁnal load cycle. In the presence of cracks or crack

Metal Fatigue

25 Table 3.3. Stress analysis feature scales. Scale (mm) 10−1 1 10 100

Stress analysis feature Large plastic strains Elastic-plastic ﬁeld Stress intensity factor Component or specimen

like defects, such as some types of inclusions, the crack initiation phase is largely absent. Much of the scatter observed in fatigue behaviour arises because on a microscopic scale metal fatigue is a very irregular process. In considering mechanisms in detail a metallic material cannot be regarded as a homogeneous continuum. However, from an engineering viewpoint, a simple approach to mechanisms is sufﬁcient for the understanding of metal fatigue. Associated stress analyses, used as an applied mechanics framework for the study of metal fatigue mechanisms, are usually at the larger scales shown in Table 3.3. 3.4.1 FATIGUE C RACK I NITIATION Fatigue crack initiation in a ductile metal is a consequence of reversed plasticity within a grain on a scale of 10−3 mm (Table 3.2). Surface grains are weakest, they deform plastically at the lowest stress and this leads to the production of a microcrack within a grain. Such microplasticity, due to slip within grains, can occur at stresses much lower than the tensile yield stress (Cottrell 1964). Slip can take place only on certain crystallographic planes within a grain. Resistance to crack initiation depends strongly on surface roughness, residual stress and environment, all of which are difﬁcult to control. Under a uniaxial tensile stress a plane of maximum shear stress is at 45◦ to the surface, so slip takes place on favourably oriented slip planes at about this angle. Under a unidirectional stress this leads to steps on the surface, as shown schematically in Figure 3.10. An important point is that slip takes place on a number of parallel planes. Slip which takes place when a load is applied is not simply reversed when the load is removed. Under fatigue loading reverse slip takes place on nearby planes leading to the development of intrusions and extrusions (Figure 3.10). Eventually, surface cracks are produced under these small amounts of reverse plastic strain (Forsyth 1969). Because slip lines traverse a grain, the cracking is transgranular rather than intergranular.

26

Chapter 3: Constant Amplitude Fatigue

Figure 3.10. Formation of surface cracks by slip (Pook 1983). Reproduced under the terms of the Click-Use Licence.

Figure 3.11. Schematic section through a fatigue fracture showing the three stages of crack propagation.

3.4.2 FATIGUE C RACK P ROPAGATION In the metal fatigue literature the terms fatigue crack propagation and fatigue crack growth are both used for the increase in size of a fatigue crack. In this book fatigue crack propagation is used. In the 1950s it became clear that fatigue crack propagation in metals is a two stage process (Forsyth 1961, 1969), as shown schematically in Figure 3.11. In Forsyth’s notation, Stage I crack propagation is an extension of the initiated microcrack without change of direction (see previous section). Hence a Stage I crack is a crack propagating within a slip band, which is on a plane of high shear stress. In fracture mechanics terms it is a mixed mode crack (see Section A.2.1). Stage I crack propagation is encouraged by plasticity (Pook 2002a). For convenience, the term fatigue crack initiation sometimes includes Stage I fatigue crack propagation. A Stage I crack becomes a Stage II crack

Metal Fatigue

27

Figure 3.12. Striations on the fracture surface of a Al-4%Cu alloy, direction of propagation left to right, spacing approximately 2 µm.

when it reaches a critical length, changes direction and propagates normal to the maximum principal tensile stress. In fracture mechanics terms it is Mode I (see Section A.2.1). The critical length is strongly dependent on microstructural features and on stress conditions, and varies widely. It is usually less than 1/4 mm and typically around 0.02 mm. After the transition a Stage II crack propagates through the majority of the cross section (see Sections 7.1, 8.1 and A.2.1). More descriptive terms for Stage I and II, microcrack and macrocrack respectively, are sometimes used. In current metal fatigue literature the terms Stage I crack, Stage II crack, microcrack and macrocrack are usually used without explanation. Similarly, unqualiﬁed references to fatigue crack propagation in this book and (usually) elsewhere are to macrocrack propagation. Finally, the cross section is so reduced that conditions for failure in one load cycle are satisﬁed. This is sometimes called Stage III and can be due to crack propagation by brittle fracture or due to ductile collapse. Under microscopic examination the most striking feature of many of the fracture surfaces created by a propagating Stage II fatigue crack in a metallic material is the presence of distinct line markings, parallel to each other, and normal to the local direction of crack propagation. These lines are called striations (Pook 1983a, Broek 1988). Figure 3.12 shows striations on the fracture surface of a Al-4%Cu alloy. Each striation corresponds to one stress cycle, hence the distance between striations is the amount the crack front has moved forward during one cycle. This one to one correspondence between striations and applied stress cycles has been demonstrated experimentally many times (Pook and Smith 1979). On a microscopic scale fatigue crack propagation in a metallic material is a very irregular process. The presence of striations is evidence that a crack was created by fatigue crack propagation. Striations are not always observed, so their absence does not necessarily imply that fatigue crack propagation did not take place.

28

Chapter 3: Constant Amplitude Fatigue

Figure 3.13. Repeated sequence of crack opening and closing under a fatigue loading of 0 to σmax . The sequence (a) to (e) is repeated for each successive cycle (Pook and Frost 1973). Reproduced under the terms of the Click-Use Licence.

The mechanism of Stage II fatigue crack propagation, and striation formation involves ductile deformation at the crack tip as it opens and closes. This mechanism is shown schematically in Figure 3.13 for a crack of initial length a, and a fatigue loading of 0 to σmax . When a crack is loaded the tip blunts, and fatigue crack propagation takes place in a ductile metal because unloading resharpens the crack tip on each cycle, leading to an increment of crack propagation δa, as shown schematically in the ﬁgure. In other words, fatigue crack propagation is a consequence of irreversible plastic deformation at the crack tip, and this basic mechanism is responsible for the production of striations. The increment of propagation in each cycle is related to this mechanism (see Section 7.4.2.1). The philosophical basis for theoretical analysis of crack propagation is that for a crack growth to propagate two conditions need to be satisﬁed (Cottrell 1964). Firstly, sufﬁcient energy needs to be available to operate a crack propagation mechanism (thermodynamic criterion), and secondly crack tip stresses must be high enough to operate the mechanism (stress criterion). The mechanism shown in Figure 3.13 satisﬁes both these conditions. The nature of the crack tip stress ﬁeld ensures that adequate stresses are available to operate the mechanism (see Section A.3). The energy absorbed by

Metal Fatigue

29

Figure 3.14. Schematic appearance of metal fatigue fracture surfaces for lives of around (a) 104 cycles, (b) 106 cycles and (c) 109 cycles.

plastic deformation at the crack tip, and by creation of new fracture surface, is provided by the cyclic loading. These criteria at not satisﬁed at below the threshold for fatigue crack propagation (see Section 7.4.3). It is often easy to recognise a metal fatigue failure from the macroscopic appearance of the fracture surface, and it is important to be able to do so. General features of metal fatigue fracture surfaces are shown schematically in Figure 3.14 for a plain specimen and depend upon the number of cycles to failure (Bathias 2001). The white areas represent fatigue crack propagation, and these have a smooth appearance. The shaded areas represent the ﬁnal static failures; these have a rough appearance and are usually darker. At high stress levels, with lives of around 104 cycles, fatigue cracks initiate at several points on the specimen surface, and the individual fatigue cracks eventually merge (Figure 3.14(a)). At low stress levels, with lives of around 106 cycles, there is usually just one dominant fatigue crack that initiated at the surface (Figure 3.14(b)). In the gigacycle region, with lives of around 109 cycles, the fatigue crack may propagate from an internal defect (Figure 3.14(c)). A small circular area around the initial defect usually has a dark appearance. This is called a ﬁsh eye (Murakami 2002). Some examples of metal fatigue service failures are shown in Figures 1.1 and 3.15, and in laboratory tests in Figures 3.16 and 3.17. These illustrate various possibilities. The fatigue crack in the drive line component (Figure 1.1) initiated at a stress concentration (see Section 6.5). Failure in the turbine blade was at mild stress concentration (Figure 3.15). The aircraft spar boom failed inside the fastening bolt hole on both sides (Figure 3.16). The rough areas are static fractures from where the spar boom was broken open for examination after testing. The fatigue crack in the cast steel specimen initiated internally at a deliberately introduced shrinkage defect (Figure 3.17). The fracture surface has a darker appearance after the propagating crack intersects the specimen surface.

30

Chapter 3: Constant Amplitude Fatigue

Figure 3.15. Marine steam turbine blade, width 18 mm. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

Figure 3.16. Al-Cu alloy aircraft spar boom, width 85 mm.

Figure 3.17. Cast steel specimen, diameter of light area 8 mm. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

3.4.3 R ATCHETTING At high stress levels, with a mean stress of above zero, a phenomenon known as ratchetting may occur (Dowling 1993, Suresh 1998). What happens is that

Metal Fatigue

31

plastic deformation occurring during loading is not fully reversed during unloading and this ratchetting may be repeated on successive cycles. Depending on circumstances ratchetting may slow down and stop, continue at a constant rate, or accelerate. If ratchetting slows down and stops it is known as shakedown. Ratchetting is sometimes observed in watch and clock mainsprings which, after a long period of use, take a permanent set and no longer provide sufﬁcient power to drive the clock work (Camm 1941). Ratchetting can also occur under thermal loading where local heating and cooling results in fatigue stresses due to expansion and contraction of the metal. Ratchetting due to thermal loading is often observed in cooking utensils, which sometimes become unusable because of excessive deformation.

3.5 Effect of Mechanisms on S/N Curves The total fatigue life is the sum of the number of cycles for fatigue crack initiation, taken as including Stage I fatigue crack propagation, plus the number of cycles for Stage II fatigue crack propagation (see Section 3.4.2). It is generally accepted that most of the scatter in the lives of metallic specimens tested at a given stress amplitude is associated with crack initiation (Frost et al. 1974). This apparently contradicts the statement that Stage II fatigue crack propagation is a very irregular process (see Sections 3.4.2, 7.4.1 and 7.6.1). In tests conducted near the fatigue limit on small specimens, up to about 90 per cent of the fatigue life is occupied by fatigue crack initiation. In tests at higher stresses, or on large specimens and structures the importance of Stage II fatigue crack propagation increases. Since the scatter associated with Stage II fatigue crack propagation is less than that associated with crack initiation, the overall scatter is reduced. The distribution of fatigue lives of plain specimens is an additive distribution made up from the distributions associated with fatigue crack initiation and with Stage II fatigue crack propagation. These two distributions are statistically independent. Data such as those shown in Figure 3.8 must therefore be interpreted in this light. At high stress levels, the number of cycles needed to initiate a crack is negligible, so that the S/N curve has the relatively narrow scatter in life associated with Stage II fatigue crack propagation. At low stress levels, the number of cycles needed to propagate the crack is negligible, so that the S/N curve shows the wide scatter in fatigue life associated with fatigue crack initiation. At intermediate stresses, crack initiation and crack propagation are of roughly equal importance and the resultant additive distribution can be bimodal (two-humped). Bimodal distributions are often asso-

32

Chapter 3: Constant Amplitude Fatigue

Figure 3.18. Schematic diagram showing the S/N diagram for a combination of S/N curves for fatigue crack initiation and for Stage II crack propagation.

ciated with two different mechanisms. Thus, the data may be interpreted as shown schematically in Figure 3.18. Unfortunately, it is not possible to separate an additive distribution into its components without prior knowledge of the components (Papoulis 1965). Metallurgical defects are inevitably present in plain specimens, and sometimes Stage II fatigue crack propagation starts directly from a small internal defect. Figure 3.17 shows an extreme example. This effect is most noticeable in high strength metallic materials at long endurances (Bathias 2001, Murakami 2002). When this happens, there may be separate S/N curves associated with fatigue crack initiation, followed by Stage II fatigue crack propagation, and with Stage II fatigue crack propagation directly from a defect. Fatigue crack initiation followed by Stage II fatigue crack propagation is associated with high stresses, and Stage II fatigue crack propagation from a defect with low stresses. This is shown schematically in Figure 3.19, where the underlying distributions are assumed to be log Normal. At stress levels where the scatter bands do not overlap, failure is entirely by the mechanism corresponding to the shorter life. In the vicinity of the crossover point the scatter bands overlap, and a proportion of the specimens will survive, and fail by the mechanism corresponding to longer life. As only a proportion of the specimens survive to fail by the longer life mechanism, the two distributions corresponding to the two mechanisms are not statistically independent. Hence the longer life mechanism has a conditional distribution (Papoulis 1965). The

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33

Figure 3.19. Schematic diagram showing S/N curves associated with fatigue crack initiation followed by Stage II Fatigue crack propagation, and with Stage II fatigue crack propagation from a small defect.

resulting distribution is therefore skewed, as in the ﬁgure, rather than bimodal. The overall S/N curve has a discontinuity at the crossover point. In welded joints, which contain small crack-like slag inclusions, the whole fatigue life may be occupied by Stage II fatigue crack propagation (see Sections 7.4.6.1 and 7.6.1). In testing structures it is sometimes found that one fracture mode predominates at high stresses and another at low stresses, and an effect similar to that shown in Figure 3.18 is observed (Frost et al. 1974). For example, in the testing of riveted sheet structures, rivet failure is associated with high stresses and sheet failure with low stresses (Ryman 1962).

4 Variable Amplitude and Multiaxial Fatigue

4.1 Introduction Early fatigue tests on structures, components and test specimens were all carried out using constant amplitude fatigue loading. However, many structures and components are subjected in service to variable amplitude fatigue loading (or variable amplitude loading), with variations following either a regular or a random pattern (Frost et al. 1974). Variable amplitude fatigue loadings can divided into two broad classes (Pook 1979): those in which individual load cycles can be distinguished, such as narrow band random loading (Figure 4.1), and those in which individual load cycles cannot be distinguished, such as broad band random loading (Figure 4.2). Modern fatigue testing equipment makes it possible to apply virtually any desired load history, and narrow and broad band random loading have been used extensively for some time, especially in structural testing (Pook 1983a, Marsh 1988). The use of broad band random loading can make the analysis of test results difﬁcult (Smith 1965). Concepts taken from random process theory (Papoulis 1965, Bendat and Piersol 1971, 2000) are used in the characterisation of random load histories (see Section B.2.1). Conventions used in the fatigue testing literature sometimes differ from those usual in random process theory. Although referred to as random, load histories used in tests are usually pseudo random in that they repeat exactly after a return period. Many fatigue tests on test specimens, components and structures are carried out using a simple, uniaxial loading. More complex fatigue loadings are known as multiaxial fatigue loadings (or multiaxial loadings). Many components and structures are subjected to multiaxial fatigue loading in service. A multiaxial fatigue loading may be deﬁned as one in which more than one stress system is produced at speciﬁc points in the object being loaded. This

36

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.1. Narrow band random loading, frequency ∼100 Hz, irregularity factor ∼0.99 (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

Figure 4.2. Broad band random loading, irregularity factor 0.410 (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

may be because more than one fatigue load is applied, or because of the nature of the load. Where more than one fatigue load is applied these are not necessarily proportional, or in phase. The crankshaft in a bicycle is an example of a component subjected to multiaxial fatigue loading. Figure 4.3 shows a crankshaft, bearing and housing assembly removed from a bicycle. The bearings are not visible in the photograph. When the rider pushes on a pedal, bending moments and torques are produced in the crankshaft. When the left hand pedal is pushed, a torque

Metal Fatigue

37

Figure 4.3. Bicycle crankshaft assembly, length 155 mm.

Figure 4.4. Bicycle chain wheel and crank.

passes through the crankshaft into the chain wheel. However, when the right hand pedal is pushed bending moments and torques are produced, but torques pass directly into the chain because the crank is attached to the chain wheel (Figure 4.4), and there is no torque in the crankshaft. The term biaxial fatigue loading (or biaxial loading) is often used in connection with sheets. It is a special case of multiaxial fatigue loading. Figure 4.5 shows a schematic diagram of a rig for carrying out biaxial fatigue tests on sheet specimens. The two actuators are controlled independently.

4.2 Mathematical Description of Fatigue Loadings A precise mathematical description of a fatigue loading is sometimes needed, especially for some types of variable amplitude fatigue loading. One deﬁnition of metal fatigue is: Failure of a metal under a repeated or otherwise

38

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.5. Schematic diagram of biaxial fatigue testing rig for thin sheet specimens, actuators are controlled independently. National Engineering Laboratory diagram. Reproduced under the terms of the Click-Use Licence.

varying load which never reaches a level sufﬁcient to cause failure in a single application (Pook 1983a). This deﬁnition was intended to exclude quasi static tests such as tensile and creep tests, although the boundary between fatigue tests and other types of test is not always clear (see Chapter 1). Mathematically, a fatigue loading is an example of a stochastic process (Papoulis 1965, Bendat and Piersol 1971, 2000), and may be deﬁned in terms of a varying force, F , which is applied as a function of time, F (t) in some time interval. Restricting the varying force to cases where there is at least one maximum and one minimum in the intervals considered excludes most quasi static tests. Alternatively, a fatigue loading can be expressed in terms of a reference stress, S, at a reference point or region such as the test section of a plain test specimen (Figures 3.2 and 3.4). A fatigue loading may be deﬁned in terms of the time function S(t). In many metal fatigue applications, material behaviour can be considered to be linearly elastic so that S is proportional to F , and the two approaches are equivalent (Frost et al. 1974).

Metal Fatigue

39

4.2.1 C ONSTANT A MPLITUDE FATIGUE L OADINGS In a constant amplitude fatigue loading individual cycles can be distinguished (see Figure 3.1) and S varies in a regular cyclic manner. Cycles are often sinusoidal so that S(t) = Sm + Sa sin(2πf t),

(4.1)

where Sm is mean reference stress, Sa alternating reference stress, f frequency of load application and t time. The process is statistically stationary in that the coefﬁcients Sm , Sa and f are unaffected by a shift of the origin along the time axis, that is they are independent of time. As the process is cyclic the continuous function S(t) can be replaced by the continuous function S(N), where N is the number of cycles and N = t/f . In metal fatigue, it is the maxima and minima, and the number of cycles (rather than time) which are the main controlling parameters. The shape of the intervening curve between a maximum and a minimum is of little importance (Frost et al. 1974). It is usual to regard a cycle as the interval between two successive minima, and to consider only integral numbers of cycles. Hence S(N) may be written as the discontinuous function S(N) = Sm ± Sa ,

(4.2)

which is understood to give successive maxima and minima. 4.2.2 G AUSSIAN R ANDOM L OADINGS Many of the random loadings encountered in fatigue work are statistically stationary, at least in the short term, and instantaneous values are a close approximation to the Gaussian distribution or Normal distribution. Characterisation of a Gaussian random loading using random process theory is well established (Bendat and Piersol 2000). Random process theory is also applicable where two or more different Gaussian random loadings are applied in multiaxial fatigue loadings. A Gaussian random process, such as that shown in Figure 4.2 may be described by the function S(t), where S is a random process and t is time. In metal fatigue, S will be a quantity such as stress or load. Considering the time interval 0 to T , the root mean square (RMS) value of the process is given by 1 T 2 S (t) dt . (4.3) σ = lim(T → ∞) T 0 In random process theory, values of S are measured from zero. In metal fatigue values of S are often measured from the mean value of the process.

40

Chapter 4: Variable Amplitude and Multiaxial Fatigue

When the mean value is not zero the two conventions give different answers. For example the RMS of 2 and 14 is (22 + 142 )/2 = 10. However, if one starts from the mean value of 2 and 14, which is 8, and takes the RMS of (2 − 8 = −6) and (14 − 8 = 6) the answer is 6. In this section, values of S are measured from its mean value. In other words, without loss of generality, mean values are taken as zero. This is often done in the metal fatigue literature because it simpliﬁes equations. When a mean value is non zero its value is stated in accompanying text and equations are not rewritten. Other deﬁnitions of RMS are sometimes used in the metal fatigue literature, which can make it difﬁcult to compare data produced by different authors. Some of these deﬁnitions of RMS are given in Sections B.2.1 and B.3.3.1, and various practical points which arise in the calculation of RMS values are discussed there. Numerical methods are usually required (Kreyszig 1983). The form of a Gaussian random process is usually characterised by the probability distribution of its instantaneous values, and also by the spectral density function, which is a measure of the frequency content, or bandwidth, of the process (see Section B.4.1). The irregularity factor, which is the ratio of zero crossings to peaks, and lies in the range 0 to 1, is also used. The irregularity factor can be obtained directly from a time history, this is usually done by counting the upward going zero crossings and positive peaks. It can also be derived from the spectral density function (see Section B.4.1). In metal fatigue the irregularity factor is often used to characterise bandwidth (Pook 1978). Probability distributions of Gaussian random processes may be described by the exceedance, P (S/σ ), which is the probability that it exceeds S/σ . The cumulative probability is 1 − P (S/σ ), and the probability density, p(S/σ ), is the derivative of 1 − P (S/σ ). The probability density of a Gaussian distribution is given by 2 −S 1 S = √ exp , (4.4) p σ 2σ 2 2π which is similar in form to Equation (3.5). Its bell shaped form is well known (Figure 4.6(a)). Integrating Equation (4.4) gives the exceedance ∞ 2 −S S S =√ , (4.5) exp d P 2 σ 2σ σ 2π S/σ which does not have a closed form. The values shown in Figure 4.6(b) are for the positive half of a Gaussian distribution, and are therefore twice those given by Equation (4.5). It can be seen that P (S/σ ) is the area under the

Metal Fatigue

41

Figure 4.6. Gaussian distribution. (a) Probability density; (b) Exceedance.

curve of p(S/σ ) between (S/σ ) and inﬁnity, as indicated by the shaded area in Figure 4.6(a). A Gaussian narrow band random process (Figure 4.1) occurs when a broad band random signal is applied to a sharply tuned resonant system. Individual sinusoidal cycles appear, whose frequency corresponds to the natural frequency of the resonant system. The cycles have a slowly varying random amplitude and the irregularity factor tends to one as the system is increasingly sharply tuned. There is no precise generally agreed deﬁnition of what constitutes narrow band random loading (Pook 1978), although it is sometimes taken as a process where the irregularity factor is at least 0.99. In metal fatigue, the distribution of peak values is usually of interest. For a Gaussian narrow band random process the probability density function of the occurrence of a positive peak of amplitude S is shown in Figure 4.7(a), and is given by the Rayleigh distribution

42

Chapter 4: Variable Amplitude and Multiaxial Fatigue

2 S −S S = exp p . σ σ 2σ 2

(4.6)

This equation is an approximation, which becomes exact as the irregularity factor tends to one. It is usually taken as applicable in practical situations provided that the irregularity factor is at least 0.99. Theoretically the Rayleigh distribution extends to inﬁnity, but in practice peaks do not exceed a cut off value of S/σ , know as the clipping ratio. Clipping implies that higher peaks are reduced to the level given by the clipping ratio; truncation that they are omitted altogether. The clipping ratio does not usually exceed four or ﬁve. The narrow band random process is symmetrical (Figure 4.1) so there are corresponding negative peaks, also known as troughs, although this is not necessarily true for the general case of a Rayleigh distribution. The corresponding exceedance (Figure 4.7(b)) is 2 −S S = exp . (4.7) P σ σ2 If the resonant system is not sharply tuned, the peak amplitudes vary more rapidly and their distribution deviates from Equation (4.6) (Cartwright and Longuet-Higgins 1965). In the limit as the bandwidth tends to inﬁnity, and the irregularity factor tends to zero (white noise), the distribution of peaks tends to the Gaussian distribution (Equations (4.4) and (4.5)). 4.2.3 B LOCK FATIGUE L OADINGS As applied in a fatigue test, a block fatigue loading (or block loading) is one in which the loading parameters vary stepwise with time. A regularly repeated block fatigue loading is sometimes called a programme loading. There is a very wide range of possibilities, and block fatigue loadings have been used extensively, both in basic metal fatigue research and in service simulation testing (Schütz 1996). Loading details are sometimes tabulated (Table B.1), but a graphical approach is more informative. Figure 4.8 shows, in general terms, a simple two level block programme in which each block is a constant amplitude fatigue loading. The return period is n cycles. At one time block programmes based on constant amplitude fatigue loading, and sometimes with large numbers of blocks, were widely used in variable amplitude fatigue testing (Frost et al. 1974). They have now largely been superseded by more sophisticated block programmes based on random loadings. Many service loadings are non stationary random processes, in which statistical coefﬁcients vary with time, and their characterisation can be difﬁcult (Bendat and Piersol 2000). An example is the fatigue loading of offshore

Metal Fatigue

43

Figure 4.7. Rayleigh distribution. (a) Probability density; (b) Exceedance.

Figure 4.8. A two level block programme.

structures due to wave action, where sea states vary with time (Pook and Dover 1989). What is often done is to model a non stationary random loading as blocks of stationary random loadings. An example is shown in Figure 4.9

44

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.9. A load history (C/12/20) for fatigue testing relevant to offshore structures, σ is its overall RMS (Pook 1976b). Reproduced under the terms of the Click-Use Licence.

(Pook 1976b, 1978). This load history was developed for fatigue testing relevant to offshore structures, and became known as C/12/20. In it, wave loading is modelled as blocks of narrow band random loading, at four different levels, arranged as an approximate representation of calms and storms. The return period is 100,000 cycles, which corresponds to approximately one week in real time.

4.3 Miner’s Rule For many years the vast majority of fatigue tests were carried out under constant amplitude fatigue loading (see Chapter 3). Designers are faced with the problem of how to use constant amplitude fatigue data in the prediction of fatigue lives under the wide range of variable amplitude load histories encountered in service. A variable amplitude load history is sometimes called a load spectrum. The investigation of metal fatigue under variable amplitude fatigue loading came to be known as the study of cumulative damage (Frost et al. 1974), although this term is not now in general use. This was because of early interest in how metal fatigue damage accumulated at various stress levels. Many attempts have been made to predict the life of a metallic specimen or component under variable amplitude fatigue loading. Much data has been accumulated for the purpose of either deriving empirical relationships, or of testing theoretical predictions. Although mechanisms of fatigue are the same as under constant amplitude fatigue loading (see Section 3.4), in detail the problem is extremely complicated. In general, only empirical solutions to particular problems are possible (Schütz 1979).

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45

Figure 4.10. Schematic S/N curve.

Palmgren (1924), in predicting the life of ball bearings, assumed that damage accumulated linearly with the number of revolutions. Similarly, Langer (1937) and Miner (1945), suggested that fatigue damage at a given stress level could be considered to accumulate linearly with the number of stress cycles. This idea is known as the Palmgren–Miner rule or, more usually, as Miner’s rule. It is also sometimes known as the Palmgren–Miner law or Miner’s law. From Miner’s rule, if a specimen or component stressed at S1 has a life of N1 cycles, as shown schematically in Figure 4.10, then the damage after n1 cycles will be n1 /N1 , and the damage per cycle will be 1/N1 . Similarly, at S2 the damage per cycle is 1/N2 . Hence, for the two level block test shown in Figure 4.8, at failure n2 n1 + = 1. (4.8) N1 N2 Similarly, for a multi level fatigue loading ni n2 n3 n4 n1 + + + + ... = = 1, (4.9) N1 N2 N3 N4 Ni where the ratio of the number of cycles at a given stress level to the expected life at the same stress level, ni /Ni , is sometimes called the stress ratio. This should not be confused with the ratio of minimum to maximum load in a fatigue cycle (see Section 3.1). The mathematical expression of Miner’s rule ni =1 (4.10) Ni is sometimes called the linear damage rule. The physical reality of the fatigue damage at any instant during the fatigue life is not deﬁned. However, in crack

46

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.11. Miner’s rule summation values for medium strength steel cruciform welded joints tested under non stationary narrow band random loading (Pook 1983b). Reproduced under the terms of the Click-Use Licence.

propagation dominated situations, damage is associated with the propagation of a fatigue crack, and it is sometimes possible to derive Miner’s rule from fatigue crack propagation data (see Section 7.5.1). Numerous authors, including Frost et al. (1974) and Schütz (1996), have pointed out that experimental data lead to a wide range of (ni /Ni ) values. These are typically in the range 0.3 to 3, but the range can be as much as 0.1 to 10. Figure 4.11 shows some typical data. These are for structural steel cruciform welded joints tested under non stationary narrow band loadingusing the C/12/20 load history shown in Figure 4.9 (Pook 1983b). Values of (ni /Ni ) less than one are unacceptable since they lead to unsafe predictions if used in fatigue assessments (Pook 1983b). The problem can be avoided by using a hypothetical S/N curve (rather than an experimental S/N curve), which is adjusted so that, for a particular set of circumstances, (ni /Ni ) is always at least one. This approach was used in the aircraft industry by around 1960 (Troughton et al. 1963), and was later used in standards that specify the use of Miner’s rule. A schematic hypothetical S/N curve for welded joints, developed by Gurney (1979), is shown in Figure 4.12, plotted using logarithmic scales. The straight line intermediate portion of the experimental S/N curve has a

Metal Fatigue

47

Figure 4.12. Schematic hypothetical S/N curve for welded joints. National Engineering Laboratory diagram. Reproduced under the terms of the Click-Use Licence.

slope of −1/m (see Equation (3.4)). There is a horizontal portion for static limitations at short lives, and another for the fatigue limit at long lives. In the hypothetical S/N curve the fatigue limit is replaced by a line with a slope of 1/(m + 2) for lives greater than 107 . A series of Miner’s rule summations showed that (ni /Ni ) was always at least one. This idea is used in various standards, including BS 5400 (Anon. 1980a). 4.3.1 W EIGHTED AVERAGE S TRESS R ANGE If all the applied stress ranges in a variable amplitude fatigue loading are within the region on an S/N curve which is a straight line when plotted on logarithmic scales (Figure 4.12), then a weighted average stress range (or equivalent constant amplitude stress range) can be calculated (Paris 1962, Etube 2001). This weighted average stress range may then be used to read the corresponding life from an S/N curve, and its use is equivalent to a Miner’s rule summation (see Section 7.5.1). For example, consider the annual fatigue loading shown in Table 4.1 where m is 4.13 (Figure 4.12). The weighted average stress range, Sh , is given by NS m 1/m , (4.11) Sh = N where m is as shown in Figure 4.12. NS 4.13 = 3.208 × Calculating NS 4.13 and summing (Table 4.2) gives 11 8 10 and N = 1.857 × 10 , hence

48

Chapter 4: Variable Amplitude and Multiaxial Fatigue Table 4.1. Annual fatigue loading. Number of cycles, N per year 4 × 103 5 × 103 8 × 104 6 × 105 1.5 × 107 1.7 × 108

Stress range, S (MPa) 59.0 46.5 26.3 14.4 6.8 4.2

Table 4.2. Annual fatigue loading calculations. Number of cycles, N per year 4 × 103 5 × 103 8 × 104 6 × 105 1.5 × 107 1.7 × 108 1.857 × 108

Sh =

3.208 × 1011 1.857 × 108

Stress range, S (MPa) 59.0 46.5 26.3 14.4 6.8 4.2

NS 4.13 8.235 × 1010 3.851 × 1010 5.855 × 1010 3.649 × 1010 4.115 × 1010 6.375 × 1010 3.208 × 1011

1/m = 6.080 MPa.

If any of the stress ranges are below the fatigue limit then they will be non damaging. For example if the lowest stress range below the fatigue limit, is 4.13 = 2.705 × 1011 and then ignoring damage due to this stress range NS 8 N = 1.857 × 10 , hence 1/m 2.705 × 1011 = 5.762 MPa. Sh = 1.875 × 108 Alternatively, ignoring the lowest stress range altogether NS 4.13 = 11 7 2.705 × 10 , and N = 1.569 × 10 , hence 1/m 2.705 × 1011 Sh = = 10.48 MPa. 1.569 × 107 The two methods give different values for Sh , but the same ﬁnal answer for the life in years provided, that the appropriate value for N is used when converting from the life in cycles. This is an example where, for a variable amplitude fatigue loading, quoting fatigue lives in numbers of cycles is potentially misleading. It is sometimes better to quote fatigue lives in time.

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Figure 4.13. Damage density for three distributions (Pook 1976b). Reproduced under the terms of the Click-Use Licence.

4.3.2 DAMAGE D ENSITY F UNCTIONS In a sinusoidal variable amplitude process, from Miner’s rule, the fatigue damage caused by each cycle is proportional to (S/σ )m , where S is stress range, σ the root mean square (RMS) value of the process, and m is as in Figure 4.12. The damage√relative to a constant amplitude sinusoidal loading of the same RMS is [S/( 2σ )]m . Deﬁne a damage density function as (Pook 1976b) m S S S p r = √ , (4.12) σ σ 2σ where r(S/σ ) is the damage density relative to peaks of (S/σ ). Damage density curves for m = 3 are shown in Figure 4.13 for three distributions: the Rayleigh distribution (Figure 4.7), the C/12/20 load history (Figure 4.9) and the Laplace distribution. The Laplace distribution is given by −S S S = exp =p . (4.13) P σ σ σ It is a straight line when plotted on log linear coordinates. Figure 4.13 shows that peaks in the region of S/σ of 2 to 3 cause the most fatigue damage, whereas peaks of S/σ of less than 1/2 cause very little damage. In practice some small peaks would be below the fatigue limit, and

50

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.14. Damage density curve showing effect of gust velocity on a typical aircraft wing component.

hence would not cause any fatigue damage. Large peaks also cause very little fatigue damage, but a rare very large peak could be above the static limitations shown in Figure 4.12, and hence cause immediate failure. Arrows on Figure 4.13 show values of S/σ for which the exceedance, P (S/σ ), is 10−5 . The area under a damage density curve gives the relative damage R, which is the ratio of the life of a constant amplitude fatigue loading to that of a variable amplitude fatigue loading of the same root mean square (RMS) value. Values of R are 1.33, 1.45 and 2.12 for the Rayleigh, C/12/20 and Laplace distributions respectively. An early application of damage density curves was to assess the fatigue damage due to gust loading of aircraft wings (Heywood 1962). Figure 4.14 shows a typical damage density curve; the load at which most fatigue damage is produced corresponds to a gust velocity of 2.75 m/s. It was argued that this load is appropriate for fatigue tests since it provided the best constant amplitude representation of the gust load distribution. 4.3.3 C YCLE C OUNTING In using Miner’s rule, the situation is further complicated when individual cycles cannot be distinguished, as in broad band random loading (Figure 4.2). A cycle counting method is then needed to reduce the process to discrete cycles, and hence permit the application of Miner’s rule. Equivalent cycles extracted from a process may be characterised by using the notation for constant amplitude cycles (see Section 3.1). A number of empirical cycle counting methods have been developed (Watson and Dabell 1976, Etube 2001, Steph-

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51

Figure 4.15. A simple process in which cycles cannot be distinguished. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

ens et al. 2001). Detailed descriptions of some of these methods are given in the references. Development of cycle counting methods is largely on a trial and error basis, and all have shortcomings. The success of a proposed method is assessed by comparing the results of Miner’s rule summations, based on the method, with experimental data. Results obtained using different cycle counting methods vary widely (Stephens et al. 2001). This is illustrated by cycle counts for the simple process shown in Figure 4.15 (Pook 2000a). Stresses are 45 MPa at peaks 2 and 6, 60 MPa at peak 4, and 30 MPa at troughs 3 and 5. A simple approach is the peak counting method in which peaks are counted from zero (or some other reference) load. This has no physical basis in terms of fatigue crack propagation mechanisms (see Section 3.4). Peak counting gives stress ranges, measured from zero, of 45 MPa (peak 2), 60 MPa (peak 4) and 45 MPa (peak 6). The corresponding weighted average stress range, Sh (see Section 4.3.1), with m taken as 3, is 51.01 MPa. A more realistic alternative is range counting in which positive going ranges are counted (Dover 1979). This gives stress ranges of 45 MPa (range 1 to 2), 30 MPa (range 3 to 4) and 15 MPa (range 5 to 6). It is assumed that there are corresponding negative going ranges nearby, that is ranges 6 to 7, 4 to 5 and 2 to 3). A justiﬁcation for this approach is that it is primarily the positive going half cycles that cause crack propagation, as indicated by the model shown in Figure 3.13, and it is widely used. The corresponding value of Sh is 34.34 MPa. Perhaps the most logically defensible and widely used cycle counting method is rainﬂow counting (Murakami 1992a). In this approach individual hysteresis loops in the plastically deformed material at the tip of a station-

52

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.16. Rainﬂow method of cycle counting (Pook 1983a). Reproduced under the terms of the Click-Use Licence.

ary crack are identiﬁed. It derives its name from the ﬁrst practical algorithm, developed by Endo, in which water is imagined to ﬂow down the load time history, with the axes reversed (Figure 4.16). Flow starts at the beginning of the record, then at the inside of each peak in the order in which peaks are applied. It stops when it either meets ﬂow from a higher level, or a point opposite a peak which is arithmetically greater (or equal to) the point from which it started, or when the end of the record is reached. Each separate ﬂow is counted as a half cycle. There is always a complimentary half cycle of opposite sign, except perhaps for a ﬂow which either starts at the beginning of the record or reaches the end. The method is fully discussed in Murakami (1992a), which includes a reprint of Endo’s original Japanese language paper. There are many variants of the rainﬂow method (Socie 1992) often differing in the treatment of the beginning and end of a record. Rainﬂow counting is now a generic term for any cycle counting procedure in which hysteresis loops are identiﬁed (Etube 2001). A typical practical algorithm is given by Anzai (1992). In Figure 4.15 rainﬂow counting gives stress ranges of 60 MPa (cycle 1-4-7), 15 MPa (cycle 3 -2-3) and 15 MPa (cycle 5-6-5 ). The corresponding value of Sh is 42.03 MPa, which is smaller than is given by peak counting but larger than is given by range counting.

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In these cycle counting methods the number of equivalent cycles is equal to the number of peaks in the process. Hence analysis of data in terms of the number of equivalent cycles would appear to be appropriate. However, if small cycles such as 3 -2-3 and 5-6-5 (Figure 4.15) are neglected, perhaps on the grounds that they are non damaging, then the number of equivalent cycles changes, and analysis in terms of equivalent cycles becomes ambiguous. In these circumstances analysis in terms of time may become appropriate (cf. Section 4.3.1). The disadvantage of both range and rainﬂow counting is that in general counting has to be done numerically using a sequential listing of peaks and troughs. Some analytical progress has been made using random process theory. For example, Bishop and Sherratt (1990) have developed a numerical method of estimating rainﬂow range statistics, to any desired degree of accuracy, directly from the spectral density function of a Gaussian process. Earlier, Chaudhury and Dover (1985) developed an approximate method for calculating the weighted average stress range without the need for a cycle count. The need for a cycle count is sometimes avoided by carrying out tests using an appropriate standard load history representative of the structure of interest, for example Austin (1994).

4.4 Standard Load Histories Standard load histories for fatigue testing, as opposed to ad hoc sequences used for speciﬁc purposes, have quite a long history (Schütz and Pook 1987). As early as 1939, Gassner (1954) described his eight level block programme, which was used from about 1948 onwards in Germany and later elsewhere (cf Figure 4.8). At that time limitations of available fatigue testing equipment meant that nothing more complex was attainable. The peak distribution used in Gassner’s programme became known as the LBF Normal distribution. It has been used as the basis of a very large number of fatigue tests simulating service loading conditions in which a continuous load (or stress) distribution is replaced by an eight level block programme. It has been claimed that the results of such tests have a very good correlation with the service lives of a wide range of components (Frost et al. 1974). Computer controlled servohydraulic fatigue test machines ﬁrst became available in the early 1960s and within ten years they were in widespread use, together with servohydraulic actuators in rigs for fatigue testing of structures. They had the big advantage that there was no limitation whatsoever on the load histories which could be applied; but that was also their main dis-

54

Chapter 4: Variable Amplitude and Multiaxial Fatigue

advantage (Schütz 1989). Many different load histories have been employed indiscriminately, sometimes without an adequate documentation of exactly what had been done. Consequently, fatigue test results were often not usable by anybody except the author. Moreover, the results of test programmes carried out by different laboratories were then not comparable. This may not be of importance in ad hoc tests for particular situations, but for general fatigue investigations it can produce a confusing situation or, worse, it may result in even qualitatively incorrect conclusions. It is therefore not surprising that complex standard load histories have been developed for fatigue test purposes in a wide range of industries, and by the late 1980s were in widespread use (Watanabe and Potter 1989). Ideally, a standard load history should cover several different types of structure or component, and the number of variants should be minimised. Inevitably, compromises are needed, but some success has been achieved. For example, FALSTAFF (Fighter Aircraft Loading Standard For Fatigue) covers several types of tactical aircraft, when ﬂown for peace time training missions, but is restricted to wing lower surface stresses near the wing to fuselage joint (Schütz 1989). On the other hand, during the development of WASH (Wave Action Standard History) it became clear that variants would be needed to cover different types of offshore structures installed in various locations. Software was therefore written using a modular architecture so that variations could easily be incorporated, but different versions of WASH would have a family resemblance (Pook and Dover 1989). Later the same modular approach was used in the successful development of JOSH (Jack up Offshore Standard load History) (Etube 2001). 4.4.1 A PPLICATIONS OF S TANDARD L OAD H ISTORIES As noted by Edwards and Darts (1984), standard load histories are needed when available life prediction methods are not sufﬁciently accurate to predict fatigue lives under variable amplitude service loadings. When making a fatigue assessment of, for example, a new detail, fastening system or fatigue life improvement method, variable amplitude fatigue loading has to be used. Frequently, such tests are of general application rather than in support of a particular project. If an appropriate standard load history exists then this will usually be the best choice. Any resulting data can be used as design data, and can also be compared directly with any other data obtained using the same standard load history. Experience has shown that, once a standard load history becomes available, relevant data accumulate quickly. This can greatly increase the

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55

technical value of individual test results, and can also reduce the need for expensive fatigue testing. When standard load histories are used, extensive fatigue testing programmes can easily be shared between different organisations. For example, the load sequence shown in Figure 4.9 was developed speciﬁcally for use in the fatigue testing of tubular welded joints under the United Kingdom Offshore Steels Research Project (Crisp 1974). Testing was carried out at several different laboratories. Hence there are numerous applications for which standard loads histories can be advantageously used. As summarised by Schütz (1989), these include: (a) Evaluation of the fatigue strength of specimens and components made from different materials. (b) Evaluation of data for the preliminary fatigue design of components. (c) Investigation of the scatter of fatigue lives under variable amplitude fatigue loading, which could well differ from that under constant amplitude fatigue loading. (d) Assessment of models, such as Miner’s rule, for the prediction of fatigue lives. (e) Round Robin programmes on general metal fatigue problems, under variable amplitude fatigue loading, in which several laboratories participate. 4.4.2 D EVELOPMENT OF S TANDARD L OAD H ISTORIES By the late 1980s, methods of constructing standard load histories were well established (Schütz 1989). Sophisticated computer methods for their implementation are now available (Heuler and Klätsche 2005). The development of a new standard load history for a particular application can be a lengthy process. It is best carried out as a collaborative effort between several organisations, preferably from different countries. Ideally, the basis of a standard load history should be strain or load measurements made in service, preferably from a number of similar structures, for example several types of transport aircraft. There are two distinct sources of fatigue loads, human actions and environmental effects. In a ﬁghter aircraft, fatigue loads are largely the result of the pilot’s actions, whereas in an offshore structure fatigue loads are largely the result of wave loading (Schütz and Pook 1987). Once data have been collected, common features then need to be extracted from the measurements. In particular, similar load probability distribution shapes need to be sought. What constitutes similarity in this respect is a difﬁcult question, which needs careful consideration (Pook and Dover 1989). Once similarity is found, then a representative load distribution must be selected, and a logical sequence of individual cycles must be decided upon.

56

Chapter 4: Variable Amplitude and Multiaxial Fatigue

For example, an aircraft’s ﬂight always begins with taxiing, followed by the ground to air cycle (transfer of the aircraft’s weight from the undercarriage to the wings), and so on. Only if the position and size of each and every cycle is ﬁxed in the sequence will the results obtained by different investigators be satisfactorily comparable. If just the load probability distribution were ﬁxed, an inﬁnite number of load histories could be synthesised, that is reconstituted, from one given distribution, and different versions could possibly result in different fatigue lives. This is a weakness of the C/12/20 load history for fatigue testing relevant to offshore structures shown in Figure 4.9 (Pook 1976b). This was originally developed for use in one laboratory (Ewing 1986). The root mean square (RMS) of each block is speciﬁed, but the user is left to generate a narrow band random loading sequence for each block. Two different implementations were used in the laboratory in which the load history was developed, but full details of these implementations were not included in their descriptions (Holmes and Kerr 1982). The later standard load histories for fatigue testing relevant to offshore structures, WASH, did specify load sequences unambiguously (Pook and Dover 1989, Schütz et al. 1990). By the late 1980s it had become clear that stress distributions due to wave loading could not necessarily be regarded as narrow band random loading, and WASH took this into account (Pook and Dover 1989). For example, Figure 4.17 shows spectral density functions (SDF) obtained for a 0.76 m diameter horizontal member, immersed 10.8 m, on a tubular welded tall platform in the North Sea (see Section B.4.1). In practice, although water waves have a dominant wave passing frequency, they are not particularly narrow band so energy may be available to excite resonances (Pook 1987b). The SDF for the water surface elevation (Figure 4.17(a)) shows this, with a clearly deﬁned peak corresponding to the dominant wave passing frequency, but with signiﬁcant energy at other frequencies. To avoid resonances, platforms are designed so that resonant frequencies are substantially higher than the dominant wave passing frequency. This has been successful for the axial stresses since, as might be expected, the SDF (Figure 4.17(b)) is of similar form, with no resonances excited. However, the SDF for the bending stress (Figure 4.17(c)) does show two peaks corresponding to structural resonances. 4.4.2.1 Truncation and Omission Levels The most difﬁcult aspects of the development of a standard load history are appropriate selection of a truncation level (see Sections 4.2.2 and B.3.1), an

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57

Figure 4.17. Spectral density functions for a 0.76 diameter horizontal member immersed 10.8 m, signiﬁcant wave height 4.75 m. (a) Water surface elevation, (b) bending stress, (c) axial stress (Pook 1989b). Reproduced under the terms of the Click-Use Licence.

58

Chapter 4: Variable Amplitude and Multiaxial Fatigue

omission level (see Section B.3.3.2), and a return period, after which the load history repeats exactly (see Section B.3.3.4) (Schütz and Pook 1987). The retention of large but infrequent tensile stresses, due to a high truncation level, may actually prolong fatigue life due to the beneﬁcial residual stresses they cause (see Section 7.5.2). Thus, if a variable amplitude fatigue test is carried out on a structure or component with too high a truncation level the test result may lead to an optimistic prediction of service life. This is the truncation dilemma. One suggested rule of thumb is that the highest load should occur at least 10 times during a test (Schijve 1985). The position of high loads in a sequence is important since failure will normally occur at one of them. Long life structures may have fatigue lives up to the gigacycle fatigue region (see Section 3.2). For example, an offshore structure in the North Sea has approximately 108 wave loading cycles applied during a service life of 20 years. This is too many for any economically feasible standard load history. A typical service load sequence contains a large number of small loads, so omission of loads that are too small to cause fatigue damage can have a dramatic effect on the total number of loads in a standard load history. For example, omitting loads below 50 per cent of the fatigue limit may reduce the number of cycles by 90 per cent (Heuler and Seeger 1986). If the omission level is too high then optimistic lives may be obtained in tests on structures and components. This is the omission dilemma. Unfortunately, the only safe way of determining the maximum permissible omission level, for a particular set of circumstances, is to carry out comparative fatigue tests at various omission levels. In the C/12/20 load history (Figure 4.9) each block is a narrow band random loading, so some of the cycles in each of the three level lower blocks will be higher than some cycles in a higher level block. Omitting the two lower level blocks would reduce the number of cycles by 66 per cent. In this case low level cycles would not be omitted altogether, but their proportion would be reduced. The length of the return period is critical, and it has to be chosen in conjunction with truncation and omission levels. On one hand, the sequence needs to be repeated at least several times before failure occurs, otherwise the various loads would not be included in their correct proportions, and the fatigue life would not be accurately deﬁned. On the other hand, too short a return period means that infrequent but high loads would not be included in the load history, whereas they do occur in service and could well affect fatigue life. This is the truncation dilemma in reverse.

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The practicalities of fatigue testing can be another constraint. The C/12/20 load history (Figure 4.9) was designed for use on tests carried out in sea water, so tests had to be carried out in real time using a typical wave passing frequency of 1/6 Hz. For convenience the return period chosen is 105 cycles, which is approximately one week in real time. The clipping ratio is the ratio between the highest load in a sequence of its root mean square value (see Sections 4.2.2 and B.3.1). In a narrow band random process the expected clipping ratio depends on the length of the sample (see Section B.3.3.4). From Equation (B.20) the expected clipping ratio for the 4000 cycles at 1.960σ is 2.98, which gives an expected overall clipping ratio for C/12/20 of 5.84 whereas the desired clipping ratio for the underlying distribution corresponding to P (S/σ ) = 10−5 is 7.7 (see Section B.3.2). In practice, for the original implementation, a computer generated sample of narrow band random loading was selected such that the overall clipping ratio was 8 (Holmes and Kerr 1982). This is typical of the compromises needed when constructing load histories with speciﬁed properties.

4.5 Multiaxial Fatigue in Metals 4.5.1 M ULTIAXIAL FATIGUE T ESTS The fatigue test specimen shown in Figure 3.4 is designed so that, under a uniaxial fatigue loading, a uniaxial stress is produced in the 8.3 mm diameter test section. More generally, a uniaxial loading can result in multiaxial stresses in regions of interest. For example in the cylindrical specimen with a circumferential notch there is a biaxial stress system in an element at the notch root (Figure 4.18). When considering material response it is convenient, albeit not strictly accurate, to refer to such situations as biaxial loading or, more generally, as multiaxial loading. Another example is the cabin of an aircraft which, when pressurised, is subjected to a biaxial loading consisting of hoop stresses and longitudinal stresses. The rig shown in Figure 4.5 is designed so that biaxial loading produces biaxial stresses in the central portion of a thin sheet specimen (Pook and Holmes 1976). Since the two actuators are at right angles principal stress directions are always the same whatever loads are applied. In a proportional fatigue loading (or proportional loading) the ratio of the loads applied by the two actuators is always the same whereas in a non proportional fatigue loading (or non proportional loading) it is not. Non proportional fatigue loadings are of two types: those in which principal stress directions change with time, and those in which they do not.

60

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Figure 4.18. Biaxial stress system at a notch root element in a cylindrical specimen with a circumferential notch under uniaxial loading.

Figure 4.19. Bicycle crankshaft, length 155 mm.

In publications on multiaxial fatigue the introduction usually states correctly that nearly all components and structures are subject to multiaxial fatigue loading in service. However, in practice one stress component often dominates. Hence, for analysis purposes, a multiaxial loading can be reduced to a uniaxial stress state (Zenner 2004). For example, in an internal combustion engine crankshaft there are several critical regions, in each of which fatigue failure could occur, and each of these regions can be associated with a stress component. One of the purposes of service simulation fatigue testing (Marsh 1988) is to identify critical regions. Multiaxial stress states in service can be very complex, even for apparently simple situations. For example, consider the crankshaft in a bicycle (see Sections 4.1 and 6.6). Figure 4.3 shows a typical bicycle crankshaft assembly, and Figure 4.19 the crankshaft removed from its housing. The 16.65 mm diameter shaft is supported by 6.35 mm diameter ball bearings which run in the outer radii of the two shoulders, as indicated by the arrow in Figure 4.20. Forces applied by the rider to the pedals are transferred through the cranks to the squared ends of the crankshaft (Figure 4.3). The most signiﬁcant stresses are those due to bending and torsion of the shaft, and the most highly stressed regions are where the outer radii of the shoulders blend into the crankshaft. The stress concentration factors (SCF) for these regions are 1.32 for bending and 1.14 for torsion.

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Figure 4.20. Close up of bicycle crankshaft.

Figure 4.21. Schematic diagram showing torques and bending moments at crankshaft outer radii. (a) Left radius. (b) Right radius.

Figure 4.21 is a schematic diagram showing the fatigue loading (torques and bending moments) at the outer radii of the crankshaft. It is assumed that a force equal to the rider’s weight (typically 700 N) is applied to the centre of a pedal during the downward half cycle, and that the force on the pedal is zero during the upward half cycle. The bending moment is constant during the downward half cycle and zero during the upward half cycle. Its value is equal to the pedal force times the axial distance from the bearing to the centre

62

Chapter 4: Variable Amplitude and Multiaxial Fatigue

of the pedal. For a bicycle with a single front chain wheel (Figure 4.4) this is typically 120 mm on both sides. For a bicycle with a double or triple front chain wheel the distance is greater on the right hand side and is typically 140 mm. The torque varies sinusoidally during the downward half cycle. It is zero at the beginning and end of the half cycle and is maximum when the crank is horizontal; its value is then the pedal force times the crank length, typically 170 mm. The torque is zero during the upward half cycle. In practice pedal forces will vary with road and trafﬁc conditions, and with the rider’s inclinations. At the left hand radius (Figure 4.21(a)) the torques and bending moments are due to the force on the left hand pedal. The torques and bending moments are in phase but the fatigue loading is non proportional because of the different waveforms. At the right hand radius (Figure 4.21(b)) the torques are due to the force on the left hand pedal. Torques from this pedal pass through the crankshaft into the chain wheel, whereas torques from the right hand pedal pass directly into the chain wheel (Figure 4.4). Bending moments are due to the force on the right hand pedal. If the pedal to radius distance is greater on the right hand side, then the bending moments will be greater, as is shown schematically in the ﬁgure. The torques and bending moments are 180◦ out of phase. Overall, the fatigue loading on the crankshaft is non proportional in that the forces on the two pedals are applied 180◦ out of phase. Development of standard load histories for multiaxial fatigue testing is less advanced than for uniaxial fatigue testing. The necessary mathematical background needed for the application of non proportional random loading is well understood (Kam and Dover 1989), and some multiaxial non proportional random standard load histories have been developed for use in the motor industry (Bruder et al. 2004). Petrone and Susmel (2003) have proposed a simpliﬁed standard load history for the acceptance fatigue testing of welded handlebar stems used in mountain bikes. This consists of 105 cycles of constant amplitude torsion followed by 105 cycles of constant amplitude bending. This is equivalent to a bicycle life of 40,000 km. The simpliﬁed standard load history was veriﬁed for several different types of handlebar stem by comparative tests using realistic non proportional random loading. Use of the simpliﬁed load history shortened test times by a factor of 450 compared with non proportional random loading. 4.5.2 M ULTIAXIAL FAILURE C RITERIA The usual purpose of a multiaxial failure criterion is to calculate an equivalent tensile stress, ρ, which can then be compared with the uniaxial mechanical

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properties of a metal. For example, in a yield criterion, ρ is compared with the tensile yield stress in order to predict the onset of yielding under a multiaxial stress state. The equivalent tensile stress can be either scalar or a vector. If it is a vector then the maximum value deﬁnes a critical plane, normal to the maximum value of the vector, on which failure is predicted to take place. Equivalent stresses are sometimes shear stresses, in which case the plane of the maximum value is a critical plane. Multiaxial failure criteria are used extensively in the analysis of multiaxial fatigue loadings, and numerous criteria have been developed (Suresh 1998, Stephens et al. 2001, Sonsino et al. 2004). Most of these have been used successfully in the analysis of various experimental data sets. Some criteria require extensive numerical calculations for their implementation, and have been made possible by the widespread availability of powerful modern computers. Multiaxial failure criteria, as applied to multiaxial fatigue loadings are a rapidly developing ﬁeld, but despite extensive work it is not yet possible to make deﬁnitive statements on which criteria are most appropriate in particular circumstances. Some criteria used for multiaxial fatigue loadings include empirical material constants determined experimentally (Suresh 1998), which means that they are not of general application. A particular problem is that, even when the same failure criterion is used, software written by different authors sometimes leads to different results (Zenner 2004). 4.5.2.1 Scalar Criteria Scalar criteria for the onset of yielding have been of interest for about 150 years, and a number of different criteria are available (Dowling 1993). Most of the yield criteria which have been developed are now only of historical interest because they contain fundamental ﬂaws (Hill 1950). Two which are still in general use are the Tresca criterion and the von Mises criterion (or Mises criterion). Both are based on the principal stresses, σ1 , σ2 and σ3 . The Tresca criterion was developed in 1864 on the basis of a long series of experiments in which Tresca measured the loads required to extrude metals through dies of various shapes (Hill 1950). It states that yielding takes place when σ1 − σ3 = σY ,

(4.14)

where σ1 ≥ σ2 ≥ σ3 , and σY is the yield stress. That is, the equivalent stress, σe is given by σe = σ1 − σ3 .

(4.15)

The von Mises yield criterion was developed in 1913, but the underlying idea can be traced back to 1904 (Hill 1950). Mathematical derivations of the von

64

Chapter 4: Variable Amplitude and Multiaxial Fatigue

Mises criterion tend to be unconvincing, for example that given by Derby et al. (1992), but its use is justiﬁed by its widespread success in predicting the onset of yielding. It can be written as σ12 + σ22 + σ32 − σ1 σ2 − σ2 σ3 − σ3 σ1 = σY . (4.16) That is σe =

σ12 + σ22 + σ32 − σ1 σ2 − σ2 σ3 − σ3 σ1 ,

(4.17)

where σe is now the von Mises equivalent stress. Expressed in terms of Cartesian stress ﬁeld components Equation (4.17) becomes (σx − σy )2 + (σy − σz )2 + (σz − σx )2 2 + τ2 + τ2 ) . + 3(τxy (4.18) σe = yz zx 2 This is a convenient form when most of the stress components are zero. For example, for a combination of a uniaxial stress and a shear stress it becomes 2 . (4.19) σe = σx2 + 3τxy The von Mises equivalent stress is sometimes used in the analysis of proportional fatigue loadings, but it cannot be used for non proportional fatigue loadings (Suresh 1998, Zenner 2004). The Tresca criterion leads to the approximately correct prediction that the fatigue limit (at zero mean stress) in torsion (that is, under shear) is approximately half that under uniaxial stress. In some modern scalar criteria for analysis of multiaxial fatigue loadings, an integral approach is used in which σe is calculated as an average value of some stress feature for all possible directions (Zenner 2004). 4.5.2.2 Critical Plane Approaches Failure criteria in which the equivalent stress is a vector are usually known as critical plane approaches. The ﬁrst application of a critical plane approach to metal fatigue appears to have been by Bacon (1930). The simplest critical plane approach is the fracture criterion proposed by Lamé in 1831 (Zenner 2004). This maximum normal stress criterion states that failure is expected when the maximum principal stress reaches the uniaxial tensile strength of the material. That is σe = σ1 ,

(4.20)

where σ1 ≥ σ2 ≥ σ3 . This has had its greatest success in predicting the fracture of brittle materials (Dowling 1993). It can be adapted to multiaxial

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fatigue loadings by ﬁnding the critical plane on which the normal stress range is a maximum. This would work for the load history shown in Figure 4.21(a) where the maxima and minima of the torques and bending moments are in phase. For the load history shown in Figure 4.21(b) maxima are out of phase. Hence, values of σe at maxima and associated normal plane orientations are signiﬁcantly different for torsion and bending fatigue loadings. In this sort of situation an appropriately weighted averaging procedure has to be used to identify both an effective value of σe and the critical plane orientation (Carpinteri et al. 2004). The situation is further complicated when two (or more) applied fatigue loadings are random. The difﬁculty is that, in general, detailed procedures have to be justiﬁed by reference to appropriate experimental data and hence are not of general applicability. For low cycle fatigue, where analysis in terms of strain is appropriate, critical plane approaches are based on strain rather than stress (Suresh 1998). The KoNoS hypothesis is a typical critical plane approach which was used for aluminum tube to plate welded joints tested under combined constant amplitude fatigue bending and torsion loading (Kueppers and Sonsino 2004). Tests were carried out at zero mean load, with the bending and torsion fatigue loads in phase or 90◦ out of phase. Values of σe were calculated for all possible planes using the normal and shear stresses on each plane using the von Mises criterion (Equation (4.19)). A maximum value of σe was obtained for in phase fatigue loading and a different maximum value for out of phase fatigue loading. These maxima identiﬁed critical planes which had different orientations for the two cases. Values of σe were then recalculated. For the in phase case σe was recalculated from the stress components in a plane perpendicular to the weld using Equation (4.19). For the out of phase case, this value was then multiplied by the ratio of the maxima (out of phase to in phase) obtained earlier. This gave a satisfactory prediction for the out of phase experimental results. It was pointed out that the method did not work for steels.

5 Fatigue Design

5.1 Introduction The term fatigue design is a misnomer, what is meant is assessment of an existing design against the possibility of fatigue failure. The possibility of failure due to metal fatigue is only one of the factors that have to be considered during the design of an engineering component or structure. Usually, fatigue behaviour is checked only after basic design decisions have been taken. There are many successful structures which have been designed without any speciﬁc attention to the possibility of fatigue failure. A fatigue assessment sometimes shows that an initial design has to be modiﬁed in order to ensure that the product performs as required for its intended lifetime. Where the possibility of fatigue failure is a major preoccupation, design and fatigue assessments are integrated in an iterative approach. When a fatigue failure does occur, under product liability legislation, anyone connected with the design, manufacture, supply, servicing, repair or operation of the product might ﬁnd themselves in the position of being held legally responsible for personal injury and material damage caused by the failure (Anon. 2001a). Modiﬁcations carried out without consulting the original designer are a frequent cause of fatigue failure, for example Pook (1998). Many authors, for example Jones (2003), have drawn attention to the importance of analysing fatigue failures so that future designs may be modiﬁed to avoid similar failures. In broad terms, if not in detail, the mechanisms of metal fatigue are not very complex (see Section 3.4), and a century and a half of metal fatigue testing means that large amounts of experimental data are available. Despite this, even rough estimates of fatigue strength at the preliminary design stage can present problems. As pointed out by Frost (1975) this is the inevitable result

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of the complexity and variety of realistic engineering design situations, together with the difﬁculty of generalising from known fatigue data and then applying these generalisations to speciﬁc engineering problems. Unfortunately, there is in general no simple relationship between a metal’s fatigue resistance and its tensile strength, so simple substitution of a stronger material is seldom the answer to a fatigue problem. At times there seems to be little correlation between what appear to be ﬂaws of one sort or another and service performance (Pook 1983a). Most books on metal fatigue contain advice on its alleviation, for example Stephens et al. (2001). Much advice amounts to hints and tips on how to reduce the level of stresses due to fatigue loads at actual or potential failure sites. Traditionally, many metal fatigue problems have been solved by taking some action to improve fatigue performance on what amounts to a trial and error basis (Whyte 1975). However, knowledge and experience are required in order to avoid apparent solutions which actually make matters worse, especially when welding is involved (Radaj and Sonsino 1999). In the last 30 years or so much effort has been devoted to the analysis of experimental metal fatigue data so that it can be incorporated in standards such as BS 5400 (Anon. 1980a) (see Section 2.3.4).

5.2 Failure Analysis There are numerous publications which include examples of failure analysis involving metal fatigue. These include Parsons (1947), Anon. (1954, 1958), Pugsley (1966), Whyte (1975), Pook (1983a), Schijve (2001) and Jones (2003). One of the earliest documented examples of metal fatigue failure analysis is in the records of the Institution of Mechanical Engineers (Anon. 1958). In 1850 members were concerned with the fracture of wrought iron railway axles. John Ramsbottom, with his characteristic good sense, pointed out that all axles failed (by fatigue) at the shoulder on the axle used to locate the wheels. He rightly stated that it was more important to redesign the axle to eliminate this shoulder and so reduce stresses, than to argue about the mechanisms of metal fatigue. From a practical design viewpoint his advice still holds good. A century later, the Proceedings of the Institution of Mechanical Engineers’ Conference on Fatigue, held in 1956, includes about 100 examples of fatigue failures, all intended to illustrate some point on the alleviation of fatigue (Anon. 1958). Many metal fatigue failures, perhaps the majority, are never diagnosed. Failures such as those shown in Figures 1.1, 1.3 and 1.4 are recognised as

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being due to metal fatigue only if they happen to be seen by a specialist. It is at times difﬁcult to establish the precise sequence of events leading up to a particular failure (Pook 1998). The reasons range from simple forgetfulness, through lack of adequate documentation, to a human desire to avoid blame. The results of failure analyses can leave designers with difﬁcult decisions (Pook 1983a). Using defective in the legal (see Section 5.4) rather than the intuitive sense, defects leading to metal fatigue failures may be divide into two broad classes. Firstly, design defects where the original design was inadequate for the envisaged use, including cases where the loading was more severe than anticipated by the designer; and secondly, manufacturing defects where the failure occurred because the designer’s intentions were not followed during manufacture, including faulty repairs and the use of material that is incorrect or below speciﬁcation. Relatively few fatigue failures are found to be due to the use of faulty or incorrect materials (Pook 1983a).

5.3 Situations, Philosophies and Approaches In approaching a mechanical engineering design problem in general, or a fatigue assessment in particular, a designer must ﬁrst select, perhaps unconsciously, the approach to be used (Pook 1983a, Zerbst et al. 2003). It was pointed out some time ago by Bignall et al. (1977) that failure to adopt an appropriate overall strategy, including allowance for human fallibility, is the one theme common to diverse catastrophic failures. The topic is frequently discussed in the literature, for example by Grandt (2004). Arguments over the selection of an appropriate overall strategy are at the heart of the controversy over the safety of nuclear power stations (Steele and Stahlkopf 1980, Anon. 2005a). In the UK the Nuclear Installations Inspectorate’s Safety Assessment Principles were ﬁrst published in 1979, and are currently being revised (Anon. 2004). At a more mundane level, much time and expense can often be saved by choosing an appropriate strategy early in the job. The planning and modiﬁcation and repairs should be taken as seriously as the original design. Too often it is left to skilled tradesmen who lack the necessary expertise (Pook 1983a). 5.3.1 S ITUATIONS There are four basic situations in mechanical engineering design which will often include a fatigue assessment (Fuchs 1980). The ﬁrst is the in house

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tool. A tool engineer given the job of designing a welding ﬁxture for some new device will probably think about it, sketch several possible solutions, select one by intuition or with the help of some quick analysis, and proceed to draw the tool which will do the desired job. In effect, the design is based on previous experience of broadly similar jobs, together with the assumptions that any failure will not be catastrophic, and also that any problem which arises when the tool is ﬁrst used can be resolved quickly. Second is the mass product. The product engineer in charge of a new digital door lock will probably build several prototypes, test them extensively, and analyse and optimise the design as much as possible. Reliability is important from both customer and manufacturer viewpoints. Wherever possible standards will be used in metal fatigue assessments (Pook 1997). The cost of design and development can be amortised over a large number of products, so it is worth evaluating several ideas thoroughly. The eventual solution will be documented in detail. Figures 1.1, 1.3, 1.5, 3.15, 4.4 and 4.19 are all examples of components from mass products. Thirdly, there is the major project. The designer of a steel offshore structure cannot afford prototypes. As much as possible must be learnt from existing ﬁeld experience, and much analysis be carried out, even though this is complex and expensive (Etube 2001). The cost of design and analysis will normally be small compared with rectiﬁcation of faults at a later stage. The fourth is the code design, which uses a standard procedure. Standards whose use in design are a legal requirement are often referred to as codes. The designer of a boiler will usually have to conform to a code, in contrast to the other designers mentioned, and hence is concerned with legalistic study of the code. Ideally, the designer ought to be able to assume that code requirements are the real requirements, and will lead to a satisfactory design. In practice, this does not always happen, and a designer may need to ensure that a design meets the real requirements, as well as the code requirements. In the past 25 years the imposition of codes through legislation has become much more widespread in many countries, not always with satisfactory results. Most mechanical design jobs fall somewhere between the four extreme situations, for example a motor car is a mass product, but is subject to various codes. 5.3.2 P HILOSOPHIES The two basic philosophies of design against metal fatigue are safe life and fail safe (Anon. 1986). They were originally developed in the aircraft industry (Troughton 1960). In a structure designed on safe life principles, the objective

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is to ensure that the structure will not fail within the design life, which may be indeﬁnite. When the design life is reached, the component or structure concerned is discarded. The design life can be expressed either as the fatigue loading service, or in time based on the maximum likely rate of usage. For example, lift cables are replaced at speciﬁed time intervals. The safe life may be extended by periodic inspection for cracks. If one is found, the component is discarded or repaired. In this case, a fatigue assessment must have demonstrated that the largest crack likely to be missed by the inspection technique used must not lead to failure before the next inspection. Steps to ensure that adequate inspection is feasible need to be taken at the design stage. If the part concerned is cheap, such as the rotating head of a centrifuge, The cost of inspection may well be greater than the cost of replacing the part when the deign life is reached. In a fail safe structure, alternative load paths are provided, and the structure is said to be redundant. This means that failure of any one member does not lead to failure of the whole structure. Once the structure has lost redundancy owing to failure of a member, it is less safe. Provision must be made, perhaps by regular inspection, for any loss of redundancy to become known. In practice, many structures are designed using a combination of fail safe and safe life philosophies, for example tubular offshore structures (Kallaby and Price 1978). A variation on fail safe design is to ensure that failed parts are constrained before they can cause injury or excessive damage. For example, failure of the rotating head of a centrifuge, if its contents are innocuous, is merely an expensive nuisance, provided that the broken parts are contained within the machine. The incorporation of fail safe features makes the fatigue assessment of safe life less crucial, but too much reliance should not be placed on fail safe features. The metal fatigue failure of a cross bracing member on the semi submersible accommodation rig Alexander L. Kielland resulted in a catastrophic capsize because, in the event, the structure proved to be non redundant (Anon. 1981b). Sister rigs were later modiﬁed by additional cross bracing in order to ensure redundancy. The damage tolerance approach to aircraft design was developed as a supplement to the safe life and fail safe philosophies (Anon. 1974a, Kirkby et al. 1980). It requires that fatigue assessments be carried out using the assumption that cracks of certain speciﬁed sizes, are present at all critical locations such a rivet holes. It therefore directs particular attention to the determination of fatigue crack propagation rates in structures.

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5.3.3 A PPROACHES There are three possible approaches to fatigue assessment; an analytical approach, service loading testing, and reliance on satisfactory experience with previous similar designs. In practice, some combination of these is normally used. Often, analytical methods are best used to extrapolate experimental data to broadly similar situations. Where innovation is involved, ensuring the integrity of structures subject to fatigue loading can be an extremely expensive, time consuming business. The United Kingdom Offshore Steels Research Project is an example (Anon. 1974b). It was set up to collect fatigue and fracture information relevant to tubular structures in the North Sea where wave loading is very severe. Twenty-ﬁve years later it was still an active research and development area (Etube 2001). Whichever approach is used, allowance must be made for the inevitable scatter in the fatigue life of specimens and structures (see Sections 3.3, 3.4 and 4.4.1), and factors such as uncertainty in service load history and stress analysis (Pugsley 1966). Methods range from a safety factor based on experience (Pook 1983a, Dowling 1993), through the statistical methods used to develop the design curves in BS 5400 (Gurney 1979, Anon. 1980a), to the elaborate procedures that have been used in the assessment of nuclear pressure vessels (Smith 1979). The degree of conﬁdence required obviously depends on the consequences of a failure. For example, the failure of the rotating head of a centrifuge would be more serious if it were being used in a medical laboratory and dangerous organisms escaped. 5.3.3.1 Analytical Approach An analytical approach makes use of information on service loads, material properties and applied mechanics. Such an approach requires expert knowledge. The fatigue crack initiation and fatigue propagation phases must be considered separately, although in practice one or the other normally predominates. This is usually the crack propagation phase. The exceptions are generally carefully made, highly stressed components where stress concentrations are kept to a minimum (Stephens et al. 2001). Typically, elaborate calculations are necessary, and the approach may fail because not all the large amount of detailed information required is available (Schütz 1979). The desire to put fatigue assessment on an analytical basis has stimulated much of the enormous amount of academic work which has been carried out on metal fatigue. Sometimes simpliﬁcations are possible, provided that they can be shown to give conservative estimates of fatigue lives. For example, the fatigue life

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of the bicycle shaft shown in Figures 4.3 and 4.19 could be assessed using estimates of the number of revolutions in the design life of the bicycle, the fatigue loading shown schematically in Figure 4.21, the stress concentration factors given in Section 4.5.1, and S/N data for the shaft material together with an allowance for scatter taken from Table 3.1. In practice, the number of revolutions is of the order of millions, so design would be based on the fatigue limit of the material (see Section 3.2). From the designer’s viewpoint, standardised analytical procedures of various degrees of formality are undoubtedly the most satisfactory. These range from informally established good practice in a particular design ofﬁce to elaborate published codes, often imposed by regulatory authorities. If they give sufﬁciently accurate answers they need only have an empirical basis. Differences in detail design requirements, and in the stringency of quality control required, usually account for the different stresses permitted for similar structures by different codes (Gerlach 1980, Zerbst et al. 2003). Standardised methods have the advantage that little or no expert knowledge is required, and they can be conveniently incorporated in software packages for computers. This facilitates the assessment of complex structures, for example (Sumi 2005). Available packages for fatigue design should be carefully examined to ensure that the basis on which they operate is suitable for the intended application (Sonsino et al. 2004). Most metal fatigue assessments are based on simpliﬁed, standard procedures which, despite their apparent lack of physical validity, are known to give conservative answers. From time to time assessments are revised using later versions of standards. The need to carry out a fatigue re-assessment of a structure sometimes arises when it has reached the end of its design life and an extension is desired. This can cause problems when the structure was constructed using materials which are not permitted by current standards. 5.3.3.2 A Fatigue Re-assessment Flow measurement faculties at the National Engineering Laboratory included a unique high pressure gas ﬂow primary standard test rig (Anon. 1979). The main air storage vessel for this rig was a 1.8 m diameter welded steel cylinder 6.3 m long which had a volume of 11.5 m3 , and was intended to hold air pressures up to 138 bar. The consequences of failure were therefore potentially catastrophic. A fatigue re-assessment was needed because the fatigue loading used during its design differed from the estimated fatigue loading. The vessel was manufactured BS 1515 (Anon. 1965b). This was written for Imperial units and these are therefore given with conversions to ISO

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units added. The design life for the vessel was 15 years and the estimated fatigue loading in terms of internal pressure was 1500 full pressure cycles of 0–2000 lbf/in2 (0–137.9 bar), 15000 ﬂuctuating cycles of 1400–2000 lbf/in2 (96.5–137.9 bar). Perusal of the manufacturer’s drawings and calculations during commissioning showed that the vessel met the requirements of BS 1515, except that the full pressure cycles had not been taken into account, and also that the number of ﬂuctuating cycles was taken as 21900 cycles. The manufacturer had carried out a simpliﬁed fatigue assessment in accordance with BS 1515 Appendix B. In this the permitted number of fatigue cycles is quoted as

1160(3000 − T ) 2 , (5.1) N= 4S − 14500 where T is the temperature in degrees C, and S is the fatigue stress range. The value of S is based on the nominal hoop stress. Stress concentrations due to penetrations, etc. are allowed for in the code, and a corrosion allowance is deducted from the actual vessel thickness. For this particular vessel the internal diameter was 72 in (1.83 m) and the net thickness 2.75 in (69.9 mm). Hence S = 13.614P , where P is the cyclic pressure range. For T = 50◦ C and the ﬂuctuating loading, Equation (5.1) gives N = 35430. As this was greater than 21900 the manufacturer had concluded, in accordance with Appendix B, that the fatigue strength of the vessel was satisfactory. To check this conclusion, further calculations were carried out to BS 1515 using the estimated fatigue loading. Where more than one fatigue loading is involved, Appendix B requires that a Miner’s rule summation (see Section 4.3) in the form ni ≤ 1.0 (5.2) Ni be satisﬁed, where ni is the expected number of cycles for the ith load level, and Ni is the permitted number of cycles for the ith load level calculated using Equation (5.1). For the ﬂuctuating cycles ni /Ni = 0.423 and for the full pressure cycles ni /Ni = 1.42, showing immediately that the vessel did not meet the requirements of Appendix B. A calculation for the expected fatigue loading using Equations (5.1) and (5.2) showed that the vessel would have had a safe life of only 9.6 years. This was not acceptable. However, keeping the ﬂuctuating pressure range (600 lbf/in2 , 41.4 bar) the same, but reducing the maximum working pressure, provided a potentially acceptable solution. A further calculation showed that the desired 15 year design life could be achieved by reducing the maximum working pressure to 1499 lbf/in2 (103 bar).

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BS 1515 was obsolescent, so a further re-assessment was carried out using the more recent BS 5500 Appendix C (Anon. 1976). Appendix C contains a simpliﬁed fatigue assessment procedure, which was deliberately intended to be very conservative. A calculation in accordance with Appendix C, using the same ﬂuctuating pressure range as before, showed that the maximum working pressure should be 75.7 bar. Detail differences in the general requirements of the two standards mean that this is not directly comparable with the higher ﬁgure obtained from BS 1515, to which the vessel was designed. Nevertheless, it was decided that the maximum working pressure should be limited to 76 bar. This reduced maximum working pressure was not a serious operating constraint. Both standards allowed the fatigue clauses to be disregarded provided that evidence of adequate fatigue life, such as more detailed calculations or practical experience of usage, was available. Details of the methods to be used were not speciﬁed in the standards, and were left to agreement between the manufacturer and the purchaser. Such escape clauses were at one time common in British Standards, but are not now permitted (Anon. 2005b). 5.3.3.3 Service Loading Testing Modern servo hydraulic equipment permits the application of virtually any load history (see Section 4.4) and can therefore be used to determine service life where an analytical approach fails. Service loading testing of prototypes has for some time been widely used for critical structures in various branches of engineering (Marsh 1988). It is sometimes a requirement of regulatory authorities. Service loading testing has the advantage that its basis is easily understood by laymen. Acquiring the necessary information on service loads may be a major problem, and expert judgement is needed on how representative the loading applied to the structure needs to be. Actual load histories in service cover a very wide range of possibilities. They can be difﬁcult to characterise because they are usually statistically non stationary (see Sections 4.2.3 and 4.3). Service loading testing also has the advantage that weak points in a design can be identiﬁed and rectiﬁed at the prototype stage. It is sometimes the only way that fatigue crack paths and crack propagation rates can be determined (Etube 2001). Simple comparative tests are sometimes useful in establishing whether a proposed modiﬁcation does result in a worthwhile increase in fatigue strength.

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5.3.3.4 Acceptance Testing Because of cost and time constraints, simpliﬁed service loading testing is used as much as possible for acceptance testing purposes, in which prototypes are required to withstand a speciﬁed fatigue loading without failure (Petrone and Susmel 2003). A striking feature of an analysis of documents issued by the British Standards Institution is the large number which include clauses requiring acceptance fatigue testing of components (Pook 1997). Most of them are product standards. Some examples are given below. BS 2A 241: 2005. General requirements for steel protruding head bolts of tensile strength 1250 MPa (180000 lbf/in2 ) or greater (Anon. 2005c). This gives requirements for high strength steel bolts for aerospace use. A bolt is a ubiquitous engineering component. It is not surprising that a long established standard for high quality high strength steel bolts includes an acceptance fatigue test. Sophisticated statistical criteria are used to determine the acceptability of a particular batch of bolts. The constant amplitude direct stress fatigue loading speciﬁed is obviously not intended to represent any particular service loading. BS ISO 10771-1: 2002. Hydraulic ﬂuid power. Fatigue pressure testing of metal pressure containing envelopes (Anon. 2002). This was prepared as the result of research, during which it became apparent that both the frequency and the waveform have a pronounced effect on the internal pressure fatigue life of metal hydraulic ﬂuid power components made from materials that are normally frequency independent (see Sections 3.2 and 7.4.2.3). The waveform of the pressure cycle, the maximum permissible viscosity of the pressurizing medium, and the maximum permissible pressure cycling rate are all speciﬁed. Hydraulic system components are examples of safety related industrial components. The constant amplitude fatigue loading speciﬁed does not represent any particular service loading. The standard uses fatigue tests to deﬁne a fatigue pressure rating rather than precise acceptance criteria. BS AU 50-2c: 1996. Tyres and wheels. Wheels and rims. Speciﬁcation for road wheels manufactured wholly or partly of cast light alloy for passenger cars (Anon. 1996). This includes a radial fatigue acceptance test. For this test the wheel is ﬁtted with an appropriate tyre and a constant radial force, which rotates around the wheel, is applied. A car wheel is an example of a safety related vehicle component. It is not surprising that a long established standard for car wheels includes an acceptance fatigue test. A constant amplitude fatigue loading is representative only of a vehicle running at constant speed on a straight and level road. The number of cycles speciﬁed is small compared

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with the number of service cycles, but to compensate for this a high fatigue load is speciﬁed. BS EN 60669-1: 2000, BS 3676-1: 2000. Switches for household and similar ﬁxed electrical installations. General requirements (Anon. 2000a). This dual numbered standard includes a normal operation acceptance test. In this test switches make and break a resistive load equal to their rated current, at their rated voltage, in a substantially non inductive alternating current circuit. Switches are operated for a speciﬁed number of operations. The acceptance criteria are that a switch must remain operational, and mechanically and electrically sound, throughout a test. An electrical switch is an example of an electrical component with mechanical parts which might fail in fatigue due to repeated operation. The test speciﬁed is deﬁned in terms of a number of normal operations. It is effectively a constant amplitude fatigue test on the mechanical components of the switch. BS EN 12983-1: 2000 Cookware. Domestic cookware for use on top of a stove, cooker or hob (Anon. 2000b). This was originally prepared in response to accident statistics which demonstrated that serious accidents can occur as the result of the premature failure of handles of domestic cookware. It sets levels of performance for cookware for use on top of a stove, cooker or hob by the accelerated simulation of hazards experienced in normal use. The acceptance fatigue test speciﬁed involves continuously raising and lowering a loaded item of cookware from a level surface once per minute by means of its handle; this is a constant amplitude fatigue test. A cookware handle is an example of a safety related item in domestic use. The acceptance criterion is that there must be no permanent distortion or loosening of the handle or its ﬁxing system.

5.4 Product Liability In general the law on product liability is complex and varies widely from country to country. Anyone likely to be adversely affected by product liability legislation should seek legal advice, and perhaps take out appropriate insurance cover. Unfortunately, there are no precise, internationally accepted deﬁnitions of the various legal terms involved. For product liability to be established a product has to be defective. A product that is not defective is sometimes called a safe product. There is usually a distinction between criminal liability and civil liability. For criminal liability to be established, that is for a person to be held to have committed a criminal offence for which he (or

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she) may be punished, it is usually necessary to prove that he (or she) was negligent in some way in his (or her) association with the product. For civil liability to be established, that is for compensation to be payable to a person who is injured or suffers loss, it is sometimes necessary to prove negligence. However, under strict liability it is merely necessary to show that the product was defective, but some defences are possible. One possible defence is that a product conforms with relevant safety regulations (see Section 5.5). The concept of strict liability now covers many situations, especially in personal injury cases. It is therefore of importance to those concerned with all types of products. A variant of strict liability is absolute liability (or no fault liability) where there is no defence. Misuse of a product which is not defective can also lead to legal liability. For example, the operator of a fairground ride who deliberately exceeded the manufacturer’s recommended maximum speed had to pay compensation when this resulted in catastrophic metal fatigue failure (Pook 1983a). From the viewpoint of a designer the meaning of defective is obviously of crucial importance. In the special case of metal fatigue, it can be helpful to differentiate between design defects and manufacturing defects where the designer’s intentions have not been followed during manufacture (see Section 5.2). Once a metal fatigue failure has occurred, with the beneﬁt of hindsight it is always possible to point to some action which would have prevented the failure. However, it is difﬁcult to reconcile the essentially random nature of metal fatigue with any deﬁnition of defective which requires precise prediction of future events (see Sections 3.3, 3.4, 4.4.1 and 5.3.3). This emphasizes the importance of selecting an appropriate strategy in design against metal fatigue (see Section 5.3). If necessary, a designer must be able to show that care had been taken. 5.4.1 T HE C ONSUMER P ROTECTION ACT The UK Consumer Protection Act 1987 is a typical example of product liability legislation (Anon. 2001a). The Act implements the European Community directive on product liability, which provides a similar degree of protection for people throughout the European Union. The aim of the Act is to help safeguard the consumer from products that do not reach a reasonable level of safety. Liability under the Act applies to all consumer goods, and to goods used at a place of work. Under the Act, people injured by defective products may have the right to sue for damages. The Act provides the same rights to anyone injured by a defective product, whether or not the product was sold to them. People can also sue for damage to private property. For civil liability

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to be established the plaintiff must be able to show that, on the balance of probabilities, the defect in the product caused the harm (injury or damage). For criminal liability this must be proved beyond reasonable doubt. The Consumer Protection Act imposes strict liability on harm caused by defective products. Hence the plaintiff does not have to prove negligence on anyone’s part. Under the Act a defective product is deﬁned as one where the safety of the product is not such as persons are generally entitled to expect. This deﬁnition provides an objective test of defectiveness and refers neither to the harm caused by the product nor to the particular product. When deciding whether a product is defective, a court takes into account all relevant circumstances, including the manner in which a product is marketed, any instructions or warnings that are given with it, and what might reasonably be expected to be done with it. Clearly the deﬁnition of defective product in the Consumer Protection Act, and similar deﬁnitions in other legislation, include products where catastrophic metal fatigue failures have occurred during normal use. It is obviously acceptable from the consumer viewpoint. However, from the point of view of those responsible for a product it is unsatisfactory, in that it could be very difﬁcult indeed to decide in advance whether a particular product might be regarded as defective. 5.4.2 E NFORCEMENT AUTHORITIES Product liability legislation makes provision for enforcement authorities who enforce the legislation. Complaints about defective products can be made to the relevant enforcement authority. In most of the UK enforcement of the Consumer Protection Act is primarily the responsibility of trading standards ofﬁcers of local authorities, who act as enforcement ofﬁcers (Anon. 2001a). Under the act, enforcement ofﬁcers have, inter alia, authority to make test purchases, seize goods, enter premises for the purpose of ascertaining whether there has been a breach of the Consumer Protection Act, and bring prosecutions.

5.5 Safety Regulations Products may be subject to safety regulations whose purpose is to ensure that products are safe under normal or reasonably foreseeable conditions of use. The aim of safety regulations is to ensure that only safe products are marketed. This is in contrast to product liability legislation which aims to

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identify defective products (see Section 5.4). Design guidance in safety regulations must be followed by designers whenever this is a legal requirement, a contractual requirement, or both. It is sometimes very detailed. Preventative measures in safety regulations often form part of quality systems such as those speciﬁed in the BS EN ISO 9000 family of standards (Anon. 2005e). Safety regulations are overseen by regulatory authorities. For example, a passenger carrying aircraft is not allowed to ﬂy without a certiﬁcate of airworthiness from an appropriate regulatory authority. Another example is the periodic inspection of motor vehicles by a regulatory authority that is a legal requirement in many countries. Unfortunately, categorising products as either safe or unsafe means that safe products are sometimes wrongly categorised as unsafe products, when what is meant is that they have not been proved to be safe. The resulting expense causes much resentment among consumers. 5.5.1 T HE G ENERAL P RODUCT S AFETY R EGULATIONS The UK General Product Safety Regulations 2005 are a typical example of product safety regulations (Anon. 2005d). They cover both new and second hand goods intended for, or likely to be used, by consumers. The General Product Safety Regulations were made under the European Communities Act 1972, and transpose a European General Product Safety Directive into UK law. The Directive pursues its principal objective of ensuring consumer product safety by specifying that products must be safe, deﬁning a safe product, imposing obligations on producers and distributors consistent with marketing safe products, laying down a framework for assessing safety, and requiring that enforcement authorities be empowered to take action to protect consumers from unsafe products. Under the Regulations it is an offence for a producer to place a product on the market, or supply a product, unless it is safe. Where speciﬁc product legislation covers exactly the same ground as the General Product Safety Regulations the speciﬁc legislation applies. The UK Department of Trade and Industry has overall policy responsibility for the Regulations, but responsibility for the safety of some consumer products rests with other Departments. For example, the Vehicle Operator Services Agency leads on motor vehicle safety. In the General Product Safety Regulations a safe product is deﬁned as any product which under normal, or reasonably foreseeable conditions of use, presents no risk, or only minimum risk compatible with the product’s use,

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and which is compatible with a high level of protection for consumers. Under the Regulations, the safety of a product is assessed using various considerations including the product’s characteristics, packaging, instructions for assembly, maintenance, use and disposal, labelling and other information provided for the consumer, and the categories of consumer at risk when using the product. The safety of some products depends on how they have been installed and maintained. These services are an essential feature of the safety of the product, and may form part of the contract to supply the product. As such they are taken into account when judging whether a product is a safe product.

6 The Uncracked Situation

6.1 Introduction In plain metallic specimens, such as those shown in Figures 3.2 and 3.4, or in the presence of a mild notch, such as that shown schematically in Figure 4.18, fatigue lives are dominated by fatigue crack initiation, rather than by fatigue crack propagation (Frost et al. 1974). If fatigue crack initiation is deﬁned as the development of a small crack, say 0.5 mm deep, then typically 80 per cent of the fatigue life is occupied by fatigue crack initiation. Very few components are of uniform cross section; most contain some form of change of cross section resulting from a discontinuity such as a ﬁllet, a hole, or an external groove or notch. The bicycle crankshaft shown in Figure 4.19 is an example. For convenience, in metal fatigue, any of these discontinuities is generally referred to as a notch, irrespective of its geometric shape. Failure analysis of components often shows that a fatigue crack had initiated at some point at a notch root. A crack initiates here because the fatigue stresses at, or near, the notch root are higher than the nominal fatigue stresses away from the notch. The car drive line component shown in Figure 1.1 and the marine steam turbine blade shown in Figure 3.15 are examples of fatigue failures originating at a notch. The brass chain link, Figure 1.4, is an example of a fatigue failure in a region of uniform cross section. All three are examples of crack initiation dominated situations, that is uncracked situations. In crack initiation dominated situations a fatigue assessment using an analytic approach is relatively straightforward, especially if an appropriate standard method is available (see Section 5.3.3.1). In this chapter some of the factors affecting the fatigue strength of metals in crack initiation dominated situations are discussed and illustrated by examples.

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Over the years much effort has been devoted to the analysis of experimental metal fatigue data for plain and mildly notched specimens with a view to their presentation in forms that can be used in analytic approaches to fatigue assessments (Gough 1926, Frost et al. 1974, Stephens et al. 2001).

6.2 Effect of Surface Finish Fatigue crack initiation in a metal is, in general, associated with a free surface (see Sections 3.4.1 and 3.4.2). Hence, the fatigue strength of plain specimens, especially at long endurances depends on the surface ﬁnish, including the condition and roughness of specimen surfaces. This in turn depends on the techniques used for specimen manufacture such as grinding, turning, forging and extrusion. There are three main reasons why manufacturing techniques used to prepare specimen surfaces may affect fatigue strength. First, notch like surface irregularities may have been created, for example machining marks. Secondly, the condition of the material at the surface may have been changed, for example it might have been hardened by cold work. Finally, residual stresses might have been introduced into the surface layers. An estimate of the magnitude of the surface irregularities created by a particular machining process can be obtained from a proﬁle record of the specimen surface. However, these records do not necessarily provide an accurate record of the surface proﬁle because the probe cannot explore a groove narrower than itself. The roughness is often expressed by a single ﬁgure; this is the centre line average roughness (CLA) over the distance traversed by the probe. The higher the ﬁgure, the rougher the surface. The CLA is frequently used to specify surface ﬁnishes on workshop drawings. Surfaces are normally much smoother when measured parallel to the ﬁnal direction of machining than they are when measured perpendicular to this direction. Measured perpendicular to the direction of machining a mechanically ﬁne turned or ground ﬁnish may have a CLA value of 0.125 µm, and a rough ground or turned ﬁnish a value of 0.75–1.25 µm (Frost et al. 1974). Unfortunately, the CLA is not an adequate parameter for evaluating the effect of surface ﬁnish on fatigue strength; the pitch. that is the distance between successive grooves must also be taken into account (Murakami 2002). Nevertheless, experimental data do provide qualitative guidance on the magnitude of effects. In evaluating the effects of surface ﬁnish on fatigue strength under constant amplitude fatigue loading the concept of intrinsic fatigue strength is sometimes used (Frost et al. 1974). This is the fatigue strength for carefully polished specimens in which care has been taken to avoid the introduction of

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Table 6.1. Surface factors for a 0.4% C steel, tensile strength 570 MPa. Condition Polished, 0000 emery Buffed with red lead Polished, 0 emery Ground, 120 wheel (ﬁne) Ground, 46 wheel (medium) Ground, 30 wheel (coarse)

Fatigue limit (MPa) ±280 ±276 ±272 ±268 ±258 ±232

Surface factor 1 0.99 0.97 0.96 0.92 0.83

residual stresses, or hardening or softening the surface layers. Final polishing is in the longitudinal direction. Specimens are sometimes electropolished to remove surface layers that might have been modiﬁed during specimen manufacture. The effect of surface ﬁnish can conveniently be characterised by the surface factor. This is the ratio of the fatigue strength for a particular surface ﬁnish to the intrinsic fatigue strength, and is normally, as would be expected, less than 1. Some typical surface factors for constant amplitude fatigue loading at zero mean stress are shown in Table 6.1 (Frost et al. 1974). Dowling (1993) gives surface factors for the fatigue limit of a range of steels. These are shown as plots of surface factor against tensile strength for several qualitatively described surface conditions. In general the surface factor decreases as the tensile strength increases.

6.3 Effect of Mean Stress Figure 3.1 shows the constant amplitude fatigue loading σm ± σa , where σm is the mean stress and σa is the alternating stress. The fatigue strength of a metallic material, both at given endurances and at the fatigue limit, depends on the mean stress. In general for a tensile mean stress, the fatigue strength, expressed in terms of the alternating stress, decreases as the mean stress is increased. In order to avoid the need to carry out large numbers of tests, attempts have been made, starting in the 19th century, to ﬁnd relationships between fatigue strength and mean stress (Frost et al. 1974). In his pioneering experiments Wöhler investigated the effects of mean stress on fatigue strength (Anon. 1871). A contemporary analysis showed that Wöhler’s tensile mean stress data for fatigue limits conformed to a parabola, having as end points the fatigue limit at zero mean stress, ±σ0 , and the tensile strength of the material, σt (Gerber 1874). The Gerber parabola is given by

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Figure 6.1. Gerber diagram.

Figure 6.2. Goodman diagram.

±σ = ±σ0 1 −

σm σt

2 .

(6.1)

The corresponding Gerber diagram is shown in Figure 6.1. The Goodman diagram, shown in Figure 6.2, is a straight line given by

σm (6.2) ±σ = ±σ0 1 − σt is now the most widely used relationship for both fatigue limits and fatigue strengths at given endurances. In its original form σ0 in Equation (6.2) was taken as σt /3 (Goodman 1899). Consequently, Figure 6.2 is sometimes called the modiﬁed Goodman diagram. The Goodman line is more conservative than the Gerber line (cf. Figures 6.1 and 6.2). In practice most experimental data lie between the Goodman and Gerber lines, and some typical data are shown in Figures 6.3 and 6.4. This means that the Goodman diagram is usually conservative. The Goodman

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Figure 6.3. Effect of mean stress on the fatigue strength at 5 × 108 cycles of two wrought aluminium alloys (Frost et al. 1974).

Figure 6.4. Effect of mean stress on the fatigue strength at 5 × 107 cycles of a wrought 5.5% Zn-magnesium alloy (Frost et al. 1974).

diagram has been criticised, for example by Cazaud (1953), as being insufﬁciently accurate, and it should not be used unless it has been established as satisfactory for a given set of circumstances. To ensure that neither yielding nor fatigue failure occurs, a diagram of similar form to the Goodman diagram has been proposed in which the criterion of failure at zero alternating stress is taken as the yield stress (or 0.1

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per cent proof stress) instead of the tensile strength. The straight line joining this to σ0 is sometimes called the Soderberg line (Frost et al. 1974). More elaborate expressions are sometimes used. For example, Heywood (1962) derived an experimental relationship from analysis of available experimental data which can be written in the form

σm σ0 σ0 σ = 1− +γ 1− , (6.3) σt σt σt σt where for steels σm σm 2+ (6.4) γ = 3σt σt and for aluminium alloys −1 σt log N 4 σm 1+ γ = , (6.5) σt 2200 where stresses are in MPa and N is the number of cycles at which the fatigue strength is estimated. The parameter γ can be regarded as characterising the curvature on an alternating stress versus mean stress diagram. For γ = 0 Equation (6.5) reduces to Equation (6.1) but with ± signs omitted. 6.3.1 E FFECT OF R ESIDUAL S TRESSES When residual stresses are present in a metallic specimen, component or structure they are superimposed on applied fatigue loads. Hence, in general, residual stresses affect fatigue behaviour in the same way as an applied mean stress. It follows that in crack initiation dominated situations residual stresses at a surface are favourable if compressive, but detrimental if tensile. In principle assessment of the effect of residual stresses is therefore straightforward, but in practice the very wide range of possibilities, together with the difﬁculty of determining the levels of residual stress, mean that empirical methods have to be used. The deliberate introduction of surface compressive residual stresses by shot peening is a well known and widely used method of improving fatigue strength (see Section 6.8).

6.4 Effect of Multiaxial Fatigue Loading A number of criteria have been proposed for failure under in phase multiaxial fatigue loading (see Section 4.5.2). Experience has shown that, in crack initiation dominated situations, the von Mises criterion (Equation (4.17)) sometimes works well for metallic materials. In situations where fatigue cracks

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Figure 6.5. Fatigue limit of 3.5% Ni steel under multiaxial loading (Frost et al. 1974).

are initiated at a free surface only the biaxial surface stresses need to be considered, and the von Mises criterion (Equation (4.18)) becomes 2 , (6.6) σe = σx2 − σx σy + σy2 + 3τxy where σe is the von Mises equivalent stress, and σx , σy and τxy are Cartesian stress ﬁeld components. The von Mises criterion predicts that the ratio of the fatigue limit in torsion √ (shear) to that in tension is 1/3 (≈ 0.577). Data for 89 metallic materials gave an average value for this ratio of 0.559 (Frost et al. 1974). Fatigue limit data for two steels tested under combined in phase (proportional) tension and torsion are shown in Figures 6.5 and 6.6. Scales are chosen so that the von Mises criterion (solid lines) becomes a straight line. The dashed straight lines are ﬁts to the experimental points. Agreement with the von Mises criterion is good for both steels. Some experimental data for proportional multiaxial fatigue loading do not conform to the von Mises criterion, and it should only be used in situations where it is known to give satisfactory results. The von Mises criterion cannot be used for non proportional fatigue loadings (see Section 4.5.2.1).

6.5 Effect of Notches The effect of a notch on metal fatigue behaviour may be illustrated by considering the cylindrical specimen with a circumferential notch, under uniaxial

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Figure 6.6. Fatigue limit of Cr-V steel under multiaxial loading (Frost et al. 1974).

loading, shown in Figure 4.18. If it is subjected to a static tensile load which gives rise to a nominal stress on the net cross section at the notch root of, say, 80 per cent of the yield stress, it will not break, even though the metal in the highly stressed region in the vicinity of the notch root will have deformed plastically. However, if the specimen is subjected to a fatigue load in which the nominal stress range on the net section is equal to 80 per cent of the plain fatigue limit fatigue, cracks will be initiated at the notch root because the local stress range is greater than the fatigue limit of the material. Once initiated, fatigue cracks then propagate across the net section and cause complete failure. The need to provide design data has led to widespread constant amplitude fatigue testing of notched specimens. Just as it is possible to estimate fatigue strengths (fatigue limit or strengths at given endurances) for plain specimens, it is also possible to estimate fatigue strengths for notched specimens. The ratio of a fatigue strength for a plain specimen, to the corresponding fatigue strength in the presence of a given notch, based on nominal stresses rather than stresses at the notch root, is called the fatigue strength reduction factor, Kf . Figure 6.7 shows the S/N curve for notched mild steel cylindrical specimens tested in direct stress at zero mean stress. The circumferential V-notch was 5 mm deep with an included angle of 55◦ , and a root radius of 0.13 mm. Nominal stresses are based on the net cross section. The results show that the notched specimen fatigue limit is about ±58 MPa. It would have a different

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Figure 6.7. S/N curve for notched mild steel specimens tested in direct stress at zero mean stress. Arrows indicate unbroken specimens (Frost et al. 1974).

value if specimens containing a notch of a different geometry were tested, or if stresses were based on the gross cross section rather than the net cross section. That is, the notched fatigue limit depends on specimen geometry and method of analysis, as well as on the material. The severity of a notch is usually characterised by the stress concentration factor, Kt . This is the ratio of the maximum elastic stress at a notch root and the nominal elastic stress. Compilations of stress intensity factors for a wide range of geometries are available, for example Pilkey (1997). In using stress concentration factor data it is important to check exactly how the nominal stress was calculated. Traditionally, stress concentration factors were presented graphically for a range of related geometries (Peterson 1953, 1974). This is convenient for preliminary calculations. While the fatigue behaviour of any particular notched conﬁguration can always, given sufﬁcient effort, be analysed in detail, extracting generalisations of reasonably wide applicability is very difﬁcult. It might appear obvious that if failure due to metal fatigue is to be avoided then the alternating stress at a notch root must not exceed the plain specimen fatigue strength of the material, in other words Kf = Kt . However, experimental data show

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that Kf may be less than Kt , and sometimes very much less. This has led to the description of materials as notch sensitive and notch insensitive. This is sometimes expressed numerically by the notch sensitivity index, q, given by (Peterson 1974) Kf − 1 q= . (6.7) Kt − 1 If Kf = Kt then q = 1, and a material is said to be fully notch sensitive, whereas if Kf = 1, q = 0, and a material is said to be fully notch insensitive. Unfortunately, q is not a material constant, and this classiﬁcation can be seriously misleading. This has been demonstrated by the analysis of a large amount of experimental data for the fatigue limits of metallic materials, which may be summarised as follows (Frost et al. 1974). (a) For low values of Kt , Kf may equal Kt , but in general it is somewhat less. (b) Different geometries producing the same Kt may give differing Kf values. (c) For high values of Kt , Kf is often very much less than Kt , (d) For a given material and certain notch geometries, there appears to be a particular value of Kt at which Kf reaches a maximum value; higher values of Kt result in no further increase in Kt . There are two major factors which affect metal fatigue behaviour in the presence of a notch. First, a stress concentration factor gives only the maximum stress at a notch root, it does not describe the stress ﬁeld or its degree of biaxiality. For the cylindrical specimen with a circumferential notch under uniaxial loading shown in Figure 4.18, there is a biaxial stress system at the notch root. According to the von Mises criterion (Equation (6.6)) the hoop stress at the notch root can increase the fatigue stress required for failure by up to about 13 per cent, depending on the degree of constraint (Frost et al. 1974). Differences in stress gradients around notches can also have a signiﬁcant effect on material behaviour (Schijve 1980). The nature and extent of any yielding at a notch is strongly dependent, both on the notch root stress ﬁeld, and also on material properties (Dowling 1993). Secondly, the fatigue life of a notched specimen is the sum of the number of cycles required to initiate a fatigue crack, and the number of cycles required to propagate the crack to failure. For a blunt notch (small Kt ) fatigue life is, as for plain specimens, crack initiation dominated (see Sections 6.1 and 6.4). By contrast, for a sharp notch (large Kt ) cracks may initiate quickly at the notch root, even at low nominal stress levels, and their propagation across the specimen may then occupy the major part of the fatigue life. The fatigue life is then crack propagation dominated. In extreme situations the phenomenon of non

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propagating cracks appears. That is, for sufﬁciently sharp notches, initiated fatigue cracks propagate for a short distance, of the order of a millimetre, and then stop (see Sections 7.4.3 and 7.4.5). The fatigue limit data shown in Figure 6.8 were obtained from 43 mm diameter cylindrical mild steel specimens containing circumferentional Vnotches, 1.3 mm deep, tested in rotating bending, that is at zero mean stress. The notched specimen fatigue limits, based on nominal gross cross section stresses, are shown plotted against the stress concentration factor Kt . The curved line represents the plain specimen fatigue limit of the material (260 MPa) divided by Kt . When Kt exceeds a value of about 3 the notched specimen fatigue limit, based on complete failure, is constant at about 90 MPa, as shown by the horizontal line. The junction between the two lines is sometimes called the branch point. Above the branch point, the necessary and sufﬁcient criterion for complete failure is the initiation of a fatigue crack, and this is correctly predicted by the notch root fatigue stresses. Below the branch point, fatigue crack initiation is correctly predicted by notch root fatigue stresses, but this is not a sufﬁcient condition for complete failure, and non propagating cracks may be present in unbroken specimens after testing to very long endurances. Most books on metal fatigue, starting with Gough (1926), draw attention to the need to avoid inadvertent stress raising notches, such as machining marks and accidental scratches. Gough also remarks ‘A number of sudden failures in connecting rods and the valve springs of Diesel engines have resulted from the practice – which cannot be too strongly condemned – of stamping an inspector’s mark upon these components.’ This advice still holds good. Deep machining marks, transverse to fatigue stresses, are signiﬁcant notches (Murakami 2002). As an example, Figure 6.9 shows the fatigue failure of a hammer head that originated at transverse machining marks. In a hammer head, tensile fatigue loads arise from reﬂected shock waves at impact. This particular hammer was a favourite tool that had been used by a mechanic for many years.

6.6 Bicycle Crankshaft Analysis Simpliﬁed fatigue assessments are often helpful at the preliminary design stage in order to check that a proposed design is feasible from the metal fatigue viewpoint. Analysis of the bicycle crankshaft shown in Figures 4.3 and 4.19 provides an example. The 16.65 mm diameter crankshaft is supported by 6.35 mm diameter ball bearings, which run in the outer radii of the two

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Figure 6.8. Nominal stress at fatigue limit versus stress concentration factor for notched mild steel specimens tested in rotating bending (Frost et al. 1974).

shoulders, as indicated by the arrow in Figure 4.20. Forces applied by the rider to the pedals are transferred through the cranks to the squared ends of the crankshaft. The crank length is 170 mm, and the distance from the centre of a pedal to a bearing is 120 mm. The most signiﬁcant stresses are those due to bending and torsion of the crankshaft, and the most highly stressed regions are where the outer radii blend into the crankshaft (see Section 4.5.1). A convenient and conservative simplifying assumption for the fatigue loading of the crankshaft is that a vertical force equal to the rider’s weight is applied to the centre of a pedal during the downward half cycle of the pedal, and that the force is zero during the upward half cycle (see Section 4.5.1). The resulting bending moments and torques at the outer radii are shown schematically in Figure 4.21. The bending moment, M, at the nearer outer radius on the crankshaft is constant during a downward half cycle of a pedal. Its value is equal to the pedal force (700 N) times the axial distance from an outer radius to the centre

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Figure 6.9. Fatigue failure of a hammer head from machining marks. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

of a pedal (120 mm), so is 84 Nm. The bending moment is zero during an upward half cycle, and is zero throughout a cycle at the outer radius further from the pedal. The direct stress, σx , due to the bending moment, M, is given by 32M , (6.8) π d3 where d is the crankshaft diameter, and is 185.37 MPa. Multiplying by the stress concentration factor (1.32) gives σx = 244.69 MPa. A force on the left hand pedal produces a torque during a downward half cycle. This torque passes through the crankshaft to the chain wheel. Hence, it appears at both outer radii on the crankshaft. The torque varies sinusoidally. It is zero when the crank is vertical, and has its maximum value when the crank is horizontal. The maximum value is equal to the pedal force (700 N) times the crank length (170 mm) so is 119 Nm. The torque is zero during the upward half cycle. The right hand crank is attached directly to the chain wheel (Figure 4.4) so torque due to a force on the right hand pedal passes directly into the chain wheel, and the torque in the crankshaft is zero throughout a cycle. At the left hand outer radius the bending moment and torque are σx =

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in phase, whereas at the right hand outer radius they are 180◦ out of phase (Figure 4.21). The shear stress, τxy , due to the torque, T , is given by 16T (6.9) π d3 and is 131.30 MPa. Multiplying by the stress concentration factor (1.14) gives τxy = 149.69 MPa. Taking the design life of the bicycle as 40000 km, and the average distance travelled for each turn of the pedals as 8 m, the design fatigue life for the crankshaft is 5 × 106 cycles. This large number of cycles means that fatigue limits may be used to select an appropriate material for the crankshaft. It is difﬁcult to generalise on whether in phase or out of phase biaxial fatigue loadings are the more severe. However, for fatigue limit design uncracked situations fatigue crack initiation must be avoided, and it is reasonable to assume that the in phase fatigue loading is the more severe. Hence σx and τxy may be combined using Equation (4.19) to obtain the von Mises equivalent stress, which is 356.5 MPa. Since the minimum bending moment and torque are both zero the design values of the alternating stress, σa , and the mean stress, σm , are both 356.5/2 = 178.25 MPa (Figure 3.1). Assume that a steel with a tensile strength of 700 MPa and a fatigue limit, at zero mean stress (stress ratio R = −1), of 300 MPa, is being considered for the crankshaft. The fatigue limit when the mean and alternating stresses are equal may be found by using a Goodman diagram (Figure 6.2). Entering values in Equation (6.2), which is the equation for a Goodman diagram, gives the fatigue limit as 210 MPa. This is greater than 178.25 MPa, so the proposed design appears to be satisfactory. However, conventional fatigue limits are for a probability of failure of 50 per cent. Table 3.1 lists normalised stresses for various probabilities of failure. The present calculations give a normalised stress of 178.25/210 = 0.849. Comparing this with Table 3.1 gives an estimated probability of failure of about 5 per cent, which might not be acceptable. It may therefore be concluded that the proposed design is probably satisfactory, but that a more detailed fatigue assessment is needed. τxy =

6.7 Scatter under Variable Amplitude Fatigue Loading Scatter of fatigue lives under constant amplitude fatigue loading of metallic specimens has been studied extensively for many years (see Section 3.3). Much experimental data has been accumulated, and methods of analysis are well established. The situation for variable amplitude fatigue loading is quite

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different. Very little experimental data is available and, in general, it is possible to make only qualitative observations on the amount of scatter in various situations, for example Schijve (2005). This situation is unexpected in view of the practical importance of the variable amplitude fatigue loading of metallic components and structures. However, there does seem to be a consensus that the amount of scatter under variable amplitude fatigue loading is usually less than for constant amplitude fatigue loading. This would imply that carrying out a Miner’s rule summation (see Section 4.3) using a P -S-N curve for a low probability of failure could be expected to lead to a safe result. Miner’s rule summations are usually based on conventional S/N curves, for which the probability of failure is 50 per cent, and it is assumed that 1 (Equation (4.10)). In practice experimental data lead to a (ni /Ni ) = wide range of (ni /Ni ) values (see Section 4.3). These values are typically in the range 0.3 to 3, but the range can be as much as 0.1 to 10. Although it is not usually done, these data could simply be regarded as scatter and analysed statistically. Implicitly, the use of a hypothetical S/N curve does take this scatter into account (see Section 4.3). Analysing data in terms of stress, instead of life, a range of 0.3 to 3 becomes a range of around 0.7 to 1.4. Exact values depend on the shape of the S/N curve. This range is still wide, but nevertheless comparable with variations in service load histories that can occur. The car drive line component failure shown in Figure 1.1 was associated with an unusually severe service load history. As pointed out earlier (see Section 5.1), many practical metal fatigue problems have been solved by taking some action to improve fatigue performance on what amounts to a trial and error basis. What seems to have happened over scatter under variable amplitude fatigue loading is that accumulated lore has been taken into account when writing standards and regulations on design against metal fatigue, but that the underlying data used by the writers has not been made readily available. This sometimes happens when conﬁdential information, (whose source cannot be acknowledged) is used by standards writers. Requirements to use particular methods in design against metal fatigue thus mean that, in effect, scatter under variable amplitude fatigue loading is indeed taken into account (see Section 4.4.1).

6.8 Improvement of Fatigue Life The results of fatigue assessments and failure analyses often reveal the need to improve the fatigue lives of existing, or proposed, components and structures. Basically, there are only two ways in which fatigue lives can be im-

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proved. The ﬁrst is to reduce fatigue stresses at critical locations, and the second is to improve the fatigue strength of the material used. The ﬁrst is the province of the engineer and the second the province of the metallurgist. The development of materials with improved fatigue strength is a popular topic in the metal fatigue literature. If a material with better fatigue strength is available, then changing the material can be a straightforward way of improving the fatigue strength of components and structures, but will usually involve increased material costs, and often involve increased manufacturing costs. Fatigue stresses can be reduced by reducing the applied fatigue loading. For example, ﬂying an aircraft around severe turbulence, rather than through it, reduces gust loading of the aircraft’s wings, as well as improving the comfort of passengers and crew. Sometimes, fatigue loadings are reduced by imposing operational constraints (see Section 5.3.3.2). Much published advice on the alleviation of metal fatigue amounts to hints and tips on how to reduce the level of fatigue stresses at actual or potential failure sites (see Section 5.1). This is often easier said than done. For example, consider the bicycle crankshaft shown in Figures 4.19 and 4.20 (see Section 6.6). Fatigue stresses at the outer radii of the shoulders could be reduced by increasing the diameter of the crankshaft, but this would impose a cost and weight penalty, which might not be acceptable. One way to reduce fatigue stresses is to increase radii at shoulders on crankshafts. Made up ball bearings are sometimes used to support bicycle crankshafts. For these bearings to ﬁt snugly against the shoulders there is a maximum permissible radius at a shoulder. This means that there may be a limit to the extent to which fatigue stresses can be reduced by increasing a radius. In crack initiation dominated situations, metal fatigue is usually a surface phenomenon (see Section 3.4). Hence surface hardening is sometimes used to improve overall fatigue strength by making fatigue crack initiation more difﬁcult. Reﬁnements of surface hardening techniques to optimise them for particular situations is a popular topic in the metal fatigue literature. The introduction of compressive residual stresses at the surface, hence reducing the severity of the applied fatigue stresses is another method of improving fatigue strength (see Section 6.3.1). The surface compressive residual stresses are, of course, balanced by tensile residual stresses elsewhere. There are three main metallurgical techniques commonly used to produce a hardened surface layer on steel (Frost et al. 1974): induction hardening or ﬂame hardening, carburising, and nitriding. Induction and ﬂame hardening consist of heating the surface above the critical temperature of the steel, and then quenching to produce a hard martensitic surface layer. Both carburising

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and nitriding produce a hardened layer on the surface of suitable steels. Carburising consists of heating the steel in a carbon bearing environment and then quenching. Nitriding consists of heating in an ammonia environment so that nitrogen combines with certain elements in the steel. It has the advantage that a quench is not needed, so minimising any distortion, but it is beneﬁcial to lap or hone the ﬁnally hardened surface in order to remove a brittle surface skin. All three processes have the additional beneﬁt of introducing compressive residual stresses at the surface. Shot peening is a widely used and low cost method of introducing biaxial compressive residual stresses into the surface of a metallic material, and hence improving its fatigue strength. In shot peening the surface is bombarded by small, hard spheres, typically 0.1 to 1 mm in diameter. The mechanisms by which the residual stresses are introduced are surprisingly complex (Kobayashi et al. 1998). Typically, shot peening produces biaxial compressive residual stresses in a layer 0.1 to 0.5 mm deep. It has been shown to signiﬁcantly increase fatigue strength in a wide range of metals (Frost et al. 1974). A method of measuring the intensity of shot peening for quality control purposes is well established (Almen and Black 1963, Anon. 2001b). This is based on shot peening a thin steel Almen strip (3 in (76 mm) long ×0.75 in (19 mm) wide), which is bolted to a stiff support. After the bolts have been removed, the Almen strip is curved due to the residual stresses induced by the shot peening. The amount of curvature is taken as a measure of the intensity of the shot peening. Other methods of introducing surface compressive residual stresses can be effective in increasing fatigue strength. If a notched cylindrical specimen (Figure 4.18) is loaded in tension at a load high enough to deform the material plastically at the notch root, then compressive residual stresses are induced at the notch root on unloading. For example, applying a static tensile stress of 390 MPa to 4% Cu aluminium alloy specimens containing a circumferential V-notch 1 mm deep and 0.1 mm root radius, increased the rotating bending fatigue strength, at 5 × 106 , cycles from ±58 MPa to ±123 MPa (Frost et al. 1974). This effect is sometimes exploited by the proof loading of safety critical components, such as chains used for lifting purposes, for example as speciﬁed in Anon. (1997). In proof loading, a speciﬁed load higher than the maximum working load is applied before a component is put into service. Conversely, applying a static compressive stress can introduce tensile residual stresses, and hence decrease the fatigue strength. In a thick walled cylinder, internal pressurisation leads to higher stresses at the bore than elsewhere. In autofrettage a pressure high enough to cause

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yielding at the cylinder bore is applied (Hill 1950). On removal of the pressure there are residual compressive stresses at the bore. Autofrettage is used to improve the fatigue strength of internally pressurised cylinders such as large gun barrels (Newhall 1999).

7 The Cracked Situation

7.1 Introduction Because of the nature of metal fatigue mechanisms, the fatigue lives of specimens, components and structures are sometimes dominated by fatigue crack initiation, and sometimes by fatigue crack propagation (see Sections 3.4, 6.1 and 6.5). From a practical viewpoint, probably the most signiﬁcant advance in the understanding of metal fatigue behaviour was the general realisation, some 30 years ago, that many components and structures are crack propagation dominated (Frost 1975). Cracks, or crack like ﬂaws, may be introduced during manufacture, especially if welding or casting is used, or cracks may form early on during service (Pook 2000a, Murakami 2002). As examples of cracked situations, Figure 7.1 shows multiple fatigue cracking from crack like ﬂaws in a 25 mm thick structural steel cruciform welded joint, and Figure 7.2 fatigue cracking from shrinkage in a 30 × 35 mm cast steel bar. In the presence of an actual or postulated crack, determining the fatigue life requires ﬁnding the number of cycles needed to propagate a fatigue crack from the initial crack size, ai , to the ﬁnal crack size, af , at which static failure takes place. This is usually at or near the maximum load in a fatigue cycle. Methods of fatigue life determination in the presence of cracks and crack like ﬂaws are now well established, for example Anon. (2005f). Fracture mechanics, especially the concept of stress intensity factor, provides the necessary applied mechanics framework (see Appendix A). In this chapter, some of the factors affecting the fatigue strength of metals in crack propagation dominated situations are discussed and illustrated. It is assumed that the crack is a Stage II crack propagating in Mode I (see Sections 3.4.2, 8.1 and A.2.1). It is also assumed that the crack path is known. If

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Figure 7.1. Fatigue cracking in a 25 mm thick structural steel cruciform welded joint showing programme markings, specimen A2/5 (Pook 1983b). Reproduced under the terms of the ClickUse Licence.

Figure 7.2. Fatigue cracking from shrinkage in 30 × 35 mm cast steel bar. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

the crack path is not known a priori then it has to be determined as part of the solution (see Chapter 8).

7.2 Initial Crack Size The initial crack size is an important variable in the determination of fatigue crack propagation life (see Section 7.4.6). The initial crack size can either be for an actual crack or for a postulated crack. In a failure analysis the position, dimensions and shape of an initial crack can usually be determined directly from a fracture surface of a broken specimen or component. However, secondary damage to fracture surfaces sometimes makes this difﬁcult or impossible. Uncertainties are relatively small.

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Information on cracks that may be present, either at the start of service or at periodic inspections during service, can be obtained by a variety of non destructive testing (NDT) techniques (see Appendix C). There is always some uncertainty, and this has to be taken into account in fatigue assessments. Accurate sizing of cracks using NDT is expensive, so cost versus beneﬁt analysis is sometimes needed. Limitations of NDT techniques for the detection and sizing of cracks mean that postulated cracks are sometimes used in fatigue assessments. If a specimen or component is inspected using a particular NDT technique, it is assumed that cracks of the minimum size that can be reliably detected are present. The damage tolerance approach to fatigue assessments does not rely on NDT. It is assumed that cracks of certain speciﬁed sizes are present at critical locations (see Section 5.3.2). Because of the important role of non destructive testing in the determination of initial crack size, NDT and fracture mechanics are complementary disciplines that inﬂuence each other (Pook 1992). Both disciplines are used in the measurement of material properties in the presence of a crack, and both are used in fatigue assessments of components and structures in the presence of actual or postulated cracks. Taken together the two disciplines answer questions such as: ‘Is the minimum crack size that can be found by an NDT technique acceptable?’, ‘Is a more sophisticated (and expensive) NDT technique needed?’, and ‘Should the part be redesigned (perhaps lowering its structural efﬁciency) so that a larger initial crack is acceptable?’. It is widely recognised that, for critical structures, allowance for appropriate NDT must be made at the design stage.

7.3 Final Crack Size Final crack sizes can found by analysis of laboratory and service static failures. However, in general ﬁnal crack sizes have to be determined analytically. Consequently, various practical procedures have been devised and standardised. These procedures are usually presented as the calculation of static strength in the presence of a crack of a given size. There are two extreme situations. One extreme situation is failure by plastic collapse at a limit load. A limit load is normally higher than the load for general yielding. Limit load compilations for various conﬁgurations are available, for example (Miller 1988). The other extreme situation is brittle fracture, where failure occurs by rapid crack extension at a nominal stress below the yield stress, that is under essentially elastic conditions. Modern ana-

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Figure 7.3. Effect of specimen thickness on fracture toughness of Ti6 Al6 V2.5 Sn, σY = 1200 MPa. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

lytical procedures are essentially interpolations between these two extremes (Zerbst et al. 2003, Anon. 2005f). In the presence of a Mode I crack, brittle fracture takes place when the Mode I stress intensity factor, KI , exceeds the fracture toughness, Kc , of the material (see Sections A.2.1 and A.3). In general Kc is a function of material thickness. Under plane strain conditions, Kc has a minimum value, KIc , called the plane strain fracture toughness, which is a material property (see Section A.3.3.1). Figure 7.3 shows an example of the effect of thickness on fracture toughness (Srawley et al. 1967). In practice, failure is not completely abrupt, and in standard test methods the fracture toughness is deﬁned as the value of KI to cause a small, speciﬁed amount of crack propagation (Anon. 2005g). Because of this, fracture toughness values calculated from the crack size at the end of fatigue life, and the maximum load in the fatigue cycle, are usually slightly different from those measured using a standard test method. End of life values are sometimes called the fatigue fracture toughness.

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7.4 Constant Amplitude Fatigue Crack Propagation For fatigue crack propagation under constant amplitude fatigue loading the fatigue cycle is usually described by the stress intensity factor range, K (Pook 2000a). This is given by K = Kmax − Kmin ,

(7.1)

where Kmax and Kmin are the maximum and minimum values of the Mode I stress intensity factor, KI , calculated from the corresponding values of σmax and σmin (Figure 3.1). Compressive stresses simply close a crack so if Kmin is negative it is taken as zero in the calculation of K (see Section A.3). It has been shown experimentally that, in general, K has the major inﬂuence on fatigue crack propagation rates in metallic materials (Frost et al. 1974). In particular, if K is constant then the fatigue crack propagation rate is constant. Limitations on the validity of stress intensity factors apply to Kmax (see Section A.3.2). In particular, yielding at the crack tip must be small scale. Corrections to Kmax for the effect of the crack tip plastic zone may be made (see Section A.3.3.2). However, these corrections are usually very small so are not normally used. If σmax is such that extensive yielding occurs, it may still be possible to use stress intensity factors in the analysis in the analysis of fatigue crack propagation data. If the stress range, σ , is not too large then unloading from σmax to σmin is essentially elastic, and for some metallic materials it is possible to calculate meaningful values of K, even though meaningful values of Kmax cannot be calculated (Frost et al. 1971). The Paris equation (Paris 1962) is often used to characterise fatigue crack propagation rates in metallic materials. It is given by da = C(K)m , (7.2) dN where a is crack length, N is number of cycles, and C and m are empirically determined constants. Strictly speaking, N is a discontinuous variable but it is usually large, and it is conventionally regarded as continuous. The Paris equation can be integrated to give the number of cycles for a crack to propagate from an initial size to a ﬁnal size. The constant C in Equation (7.2) has peculiar dimensions, which depend on the value of the exponent m. To avoid confusion, it is usually best to work in MN-m units, that is with a in metres √ and K in MPa m. The Paris equation is sometimes called the Paris law, but because of its empirical basis the latter term should be avoided.

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Figure 7.4. Centre cracked tension specimen, width 2W .

7.4.1 D ETERMINATION OF FATIGUE C RACK P ROPAGATION R ATES In principle, determination of fatigue crack propagation rates under constant amplitude fatigue loading is straightforward. Crack length, a, versus number of cycles, N, data are obtained for a fatigue crack propagating in a convenient specimen, such as the pin loaded centre cracked tension specimen shown in Figure 7.4 (see Section A.3.1.1). Some typical data for mild steel are shown in Figure 7.5. In the ﬁgure the individual data points are averages for both sides of the sheet and for both ends of the crack. The slope of the resulting curve is then taken, at convenient intervals, to obtain values of the fatigue crack propagation rate, da/dN, and corresponding values of K are calculated. As in Figure 7.6 it is conventional to plot da/dN versus K on logarithmic scales. Data from several specimens may be combined to give a scatter band for the material. Figure 7.6 shows data for mild steel. These include data derived from Figure 7.5. Most of the scatter band is straight, corresponding to Equation (7.2). For the centre line of the band C in the equation is 2.4×10−12 , and m is 3.3. The dashed lines at the top of the scatter band in Figure 7.6 indicate a region where, because of extensive yielding, it is not possible to

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Figure 7.5. Fatigue crack propagation curve for a centre crack, length 2a, in 0.76 m wide × 2.5 mm thick mild steel sheet specimen, nominal stress 108 ± 31 MPa (Frost et al. 1974).

calculate meaningful values of Kmax , but it is possible to calculate meaningful values of K (see previous section). Despite the apparent simplicity of determining fatigue crack propagation rates in metallic materials, difﬁculties arise in practice, and it took a long time for standard test methods to appear (see Section 2.3.1). Problems arise because, on a microscopic scale, fatigue crack propagation in metallic materials is a very irregular process (see Section 3.4.2). Further, fatigue crack propagation rates of down to the order of one lattice spacing per cycle (about 3 × 10−10 m/cycle) are of interest (see Section 7.4.3). This means, in general, that the amount of crack propagation per cycle is very much less than the sensitivity of available crack size measurement systems (see Appendix C). In consequence, the apparent amount of scatter in fatigue crack propagation rate data, such as those shown in Figure 7.6, is a strong function both of the test method used, and also of the data reduction technique used. A particular practical problem is that accurate numerical differentiation of a curve deﬁned by discrete data points is difﬁcult (Kreyszig 1983). The amount of scatter in experimental da/dN versus K data can be very large, especially when viewed on the da/dN axis. The scatter band in Figure 7.6 includes about 90 per cent of the data points. Unfortunately, it is not possible to separate inherent scatter in da/dN data from that due to experimental inaccuracies. It can be shown that, neglecting measurement errors,

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Figure 7.6. Fatigue crack propagation rates for mild steel, 56 tests, stress ratio R = 0.06−0.74 (Frost et al. 1974).

the amount of scatter in da/dN data is approximately inversely proportional to the square root of the crack length increment over which an average crack propagation rate is calculated, and also approximately inversely proportional to the square root of specimen thickness (Pook 1976a). In standard fatigue crack propagation rate test methods, the intervals at which crack length measurements are taken, and methods of numerical differentiation, are both carefully speciﬁed. This means that it is possible to make qualitative judgments about the relative amount of scatter in fatigue crack propagation rate data for different metallic materials. In some standards for fatigue crack propagation rate determination, for example Anon. (2003b), there is also a requirement that da/dN versus K data be presented in tabular form, as well as graphically. The point is that ﬁtting a curve through the data points could give a misleading impression of the amount of scatter in the data.

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Frost et al. (1974) include a compilation of fatigue crack propagation rate data for a range of metallic materials. This has been reproduced in some later publications, for example Derby et al. (1992). However, extensive recent compilations of fatigue crack propagation rate data do not appear to be available. What appears to have happened is that carefully validated upper bound fatigue crack propagation rate data for relevant materials are included in fatigue assessment standards such as Anon. (2005f) (see Section 7.4.3). An upper bound to a scatter band is appropriate for fatigue design purposes, whereas the mean values quoted by Frost et al. (1974) are appropriate for failure analysis. 7.4.2 FACTORS A FFECTING FATIGUE C RACK P ROPAGATION R ATES The mechanism of fatigue crack propagation in ductile metallic materials shows that it is primarily a deformation controlled, rather than a stress controlled, process (see Section 3.4.2). This has important consequences for fatigue crack propagation behaviour. In particular, fatigue crack propagation in metallic materials tends to be insensitive to a material’s strength, and also to metallurgical factors. Some of the factors affecting fatigue crack propagation rates are described below. 7.4.2.1 Effect of Young’s Modulus Fatigue crack propagation takes place because unloading resharpens the crack tip on each cycle, as shown schematically in Figure 3.13, and Young’s modulus, E, is an important factor. Fatigue crack propagation is a consequence of irreversible plastic deformation at the crack tip, and this was used as the basis of a continuum mechanics deformation theory of fatigue crack propagation (Pook and Frost 1973). The increment of crack propagation on each cycle, for plane stress and a zero to maximum load (σmax ), can be estimated by calculating the crack proﬁle at the maximum σmax using Equation (A.3). In their theory Pook and Frost assumed that the part of the crack proﬁle subjected to tensile stresses greater than the yield stress, σY , retains its length on unloading, as shown in Figure 3.13. For ductile metals, E/σY is of the order of 103 , and the assumption leads to 9 K 2 da = . (7.3) dN π E For plane strain, the equation becomes

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7 da = dN π

K E

2 .

(7.4)

Equations (7.3) and (7.4) were derived for the fatigue cycle of 0 to σmax , that is the stress ratio, R = 0 (see Section 3.1). However, the resharpening mechanism ensures that they also apply to R > 0. It had been shown that any continuum mechanics deformation theory of fatigue crack propagation leads to a (K/E)2 dependence, unless a characteristic dimension is introduced (Rice 1967). Hence reﬁnements to Pook and Frost’s theory would simply alter the numerical factors in Equations (7.3) and (7.4). The exponent, m, in the Paris equation (Equation (7.2)) is usually greater than the theoretical value of 2 in Equations (7.3) and (7.4). This is largely because fatigue crack propagation mechanisms differ in detail at different levels of K. Despite its limitations, Pook and Frost’s theory does correctly predict general features of fatigue crack propagation behaviour in metals, especially the dependence on Young’s modulus rather than strength. Data for materials of widely different Young’s moduli may be collapsed onto one scatter band by plotting da/dN against K/E rather than K. For example, Figure 7.7 shows data for tungsten (E = 390 GPa) and dispersion strengthened lead (E = 16 GPa) (Speidel 1982). The stress ratio, R, is zero. The theory also correctly predicts that fatigue crack propagation rates in mild steel are largely independent of the stress ratio. Figure 7.6 includes data from 56 test specimens with R values ranging from 0.06 to 0.74. Such materials are called mean stress insensitive. However, some materials are mean stress sensitive and crack propagation rates increase signiﬁcantly as R increases. This is because fatigue crack propagation mechanisms differ in detail at different values of R (Frost et al. 1974). As an example, Figure 7.8 shows data for a 5.5% Zn aluminium alloy. Mean stress sensitivity appears at high crack propagation rates as static failure is approached. This is simply because short burst of brittle fracture, controlled by the maximum load in the fatigue cycle, occur. 7.4.2.2 Effect of Thickness Fatigue crack propagation rates in metallic materials sometimes depend on sheet or plate thickness, so tests should be carried out using material of the thickness of interest. There are two effects of thickness that can affect fatigue crack propagation rates. Firstly, the fracture toughness, Kc , in general decreases as the thickness increases (see Section 7.3). Secondly, the transition from square to slant fatigue crack propagation, that sometimes occurs

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Figure 7.7. Fatigue crack propagation rates for tungsten (E = 390 GPa) and dispersion strengthened lead (E = 16 GPa), R = 0. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

in thin sheets, but this does not happen in thick plates (see Section A.3.3.3). Figure 7.9 shows the main features of the transition. Fatigue crack propagation rates tend to increase as the maximum stress intensity factor in the fatigue cycle, Kmax , approaches Kc (Pook 2000a). Consequently, at high values of stress intensity factor range in the fatigue cycle, K, fatigue crack propagation rates increase as thickness increases as shown, for example, in Figure 7.10. Equation (7.2) is sometimes modiﬁed to allow for this effect. Fatigue crack propagation rates usually decrease following the transition to slant fatigue crack propagation. The mechanisms involved in the transition, and in slant fatigue crack propagation, are not clear, and only qualitative theoretical explanations appear to be possible (Pook 2000a, 2002a). However, a large amount of fatigue crack propagation data has been accumulated, and for practical engineering purposes the phenomenon is well understood (Zuidema 1995). The transition takes place only in thin sheets. In the transition region the transition starts with the appearance of shear lips at the sheet surfaces (Figure 7.9). The initiation of shear lips is sensitively dependent on precise initial conditions, and cannot be predicted theor-

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Figure 7.8. Effect of stress ratio R on fatigue crack propagation rates for 5.5% Zn aluminium alloy (Frost et al. 1974).

etically, but it does appear to be associated with the crack tip plastic zone (see Section A.3.3.2). The start of the transition appears to be associated with the attainment of a minimum value of the range of stress intensity factor in the fatigue cycle, K, or equivalently a minimum value of the fatigue crack propagation rate, da/dN). Hence this is a necessary condition for the transition to take place. However, the transition is sometimes suppressed, so it is not a sufﬁcient condition. The shear lips increase in size as fatigue crack propagation proceeds until they either reach a maximum size, or if the specimen is sufﬁciently thin, they meet, completing the transition. After the transition the crack propagation surface is ﬂat, and its inclination to the sheet surface is approximately 45◦ . Crack fronts are roughly straight. It is possible to reverse the transition by reducing the fatigue loading (Schijve 1974).

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Figure 7.9. Transition from square to slant fatigue crack propagation in thin sheets. The arrow shows the direction of fatigue crack propagation (Pook 1983a). Reproduced under the terms of the Click-Use Licence.

7.4.2.3 Crack Closure Conventionally, a Mode I crack is assumed to be open when the applied stress, σ , is tensile, and closed when σ is zero or compressive. However, this convention is correct only for the linearly elastic situation (see Section A.3). In particular, it is implicit in the sequence shown in Figure 3.13, that for a minimum stress in the fatigue cycle, σmin , which is greater than zero, a fatigue crack is open when σ > σmin and closed when σ ≤ σmin . In terms of the Mode I stress intensity factor, KI , this means that, for a propagating fatigue crack, the crack is open when KI > Kmin , where Kmin is the minimum value of KI in the fatigue cycle. In a ductile metal the presence of a crack tip plastic zone means that a plastic wake of plastically deformed material is left adjacent to the surfaces of a propagating fatigue crack (see Section A.3.3.2). It follows that the crack does not open unless KI > Kop where Kop is the crack opening value of KI where Kop > Kmin . In consequence the effective value of the range of KI in the fatigue cycle, Keff , is given by Keff = Kmax − Kop ,

(7.5)

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Figure 7.10. Effect of thickness on fatigue crack propagation rates for RR 58 aluminium alloy (Frost et al. 1974).

where Kmax is the minimum value of KI in the fatigue cycle. The value of Keff given by Equation (7.5) is less than the value of K given by Equation (7.1), as shown schematically in Figure 7.11. This phenomenon is known as crack closure. Its existence was demonstrated experimentally by Elber (1970), but it had been noted much earlier (Paris et al. 2006). Elber showed that fatigue crack propagation data for mean stress sensitive metallic materials, for example as is shown in Figure 7.8, could be collapsed into a single scatter band by plotting in terms of Keff , instead of K. Various empirical expressions have been proposed for the calculation of Keff (Suresh 1998). However, because behaviour depends on numerous factors, these expressions are all of limited applicability. Fortunately, fatigue crack propagation rate behaviour for a particular metallic material is usually sufﬁciently consistent for K (rather than Keff ) to form a satisfactory basis for the analysis of fatigue crack propagation rate data and their subsequent application to the solution of practical problems. The development of a plastic wake means that as an initially stress free crack propagates, the value of Kop increases, with concomitant reductions in Keff , until a stable state is reached. This is illustrated by some ﬁnite element calculations for constant K, which are shown in Figure 7.12 (Nakagaki and

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Figure 7.11. Deﬁnition of Keff in the presence of crack closure at KI = Kop .

Atluri 1979). In the ﬁgure the crack length is normalised as the ratio of current crack length to initial crack length (a/ai ). The development of a plastic wake does not always account for crack closure. It can also be induced by several other mechanisms. Suresh (1998) summarises these: oxide induced crack closure, roughness induced crack closure, transformation induced crack closure, and viscous ﬂuid induced crack closure. Oxide induced crack closure occurs when corrosion products (usually oxides) form on the crack surfaces; these have a greater volume than the original metal. Roughness induced crack closure occurs when microscopic irregularities on opposite crack surfaces interfere with each other as a result of complex deformation patterns. Transformation induced crack closure occurs when stress or strain induced phase transformations in material in the vicinity of the crack tip lead to a net increase in the volume of the transformed material. These mechanisms are all material dependent. They are sometimes deliberately invoked in the development of fatigue crack propagation resistant metallic materials.

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Figure 7.12. Change in Keff due to development of a plastic wake. Constant amplitude fatigue loading, R = 0. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

Fluid induced crack closure is a special case in that it is of mechanical rather than metallurgical origin. In a viscous ﬂuid environment, the ﬂuid cannot ﬂow freely into and out of the crack as it opens and closes during a fatigue cycle, and this leads to crack closure. Under internal pressure fatigue this can lead to frequency dependence in materials that are normally frequency independent. This is because Keff decreases as the frequency increases (Pook and Short 1988, Davis and Ellison 1989) (see Sections 3.2 and 5.3.3.4). 7.4.3 T HRESHOLD FOR FATIGUE C RACK P ROPAGATION The Paris equation (Equation (7.2)) implies that any value of the range of Mode I stress intensity factor, K, no matter how small, will result in a positive value of fatigue crack propagation rate, da/dN. However, experimental data show that the crack propagation rate tends to zero at some critical value of K, which is the threshold value, Kth . The existence of a threshold for fatigue crack propagation is responsible for the phenomenon of non propagating cracks (see Sections 6.5, 7.4.5 and 8.2.1). The threshold is not necessarily sharply deﬁned (Pook 1983a). Unlike fatigue crack propagation rates, thresholds are sensitive to metallurgical factors, and also to the stress ratio, R. Kth tends to decrease as R increases. Compilations of Kth values for vari-

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ous metallic materials are available (Frost et al. 1974, Taylor 1989). However, extensive, recent compilations do not appear to be available. What appears to have happened is that carefully validated lower bound fatigue crack threshold data for relevant materials are included in fatigue assessment standards such as Anon. (2005f). This is similar to the situation for fatigue crack propagation rate data (see Section 7.4.1). There are several possible formal deﬁnitions of the fatigue crack propagation threshold (Pook and Greenan 1976, Pook 1989c). Thresholds can be obtained using several techniques. Usually, all give essentially the same result (Pippan et al. 1994). The most obvious technique is simply to follow downwards a curve of fatigue crack propagation rate (da/dN) versus range of Mode I stress intensity (K). However, unless this is done very carefully, the threshold can be seriously overestimated (Pook 2000a). Some standards specify what is essentially a reﬁned version of this method, for example Anon. (2003b). Figure 7.13 shows some data for ASTM A517 Grade F (T-1) steel, √ R = 0.1 (Paris et al. 1972). The threshold is about 4.4 MPa m. The threshold exists because fatigue crack propagation rates of less than about one lattice spacing per cycle are not possible on physical grounds. If the lattice spacing is taken as a typical value of 3 × 10−10 m, then experimental values of the threshold, Kth , correspond fairly well with values of K calculated from the Paris equation (Equation (7.2)), with da/dN taken as one lattice spacing per cycle (Frost et al. 1971). This means that the stress criterion and the thermodynamic criterion are not both satisﬁed at K < Kth (see Section 3.4.2). Fatigue crack propagation at average rates of much less than one lattice spacing per cycle is sometimes observed. When this happens the crack cannot be propagating along the whole crack front on every cycle (Pook 1983a). 7.4.4 OVERALL FATIGUE C RACK P ROPAGATION B EHAVIOUR Overall fatigue crack propagation behaviour for metallic materials under constant amplitude fatigue loading is summarised schematically in Figure 7.14 (Pook 2000a). When plotted on logarithmic scales the overall curve of fatigue crack propagation rate, da/dN, versus range of Mode I stress intensity factor, K, is usually sigmoidal. In the intermediate Paris region the curve becomes a straight line and fatigue crack propagation rates are given by the Paris equation (Equation (7.2)). In general, the microstructure and the stress ratio, R, have little inﬂuence. In the threshold region fatigue crack propagation rates become vanishingly small as the threshold for fatigue crack propagation, Kth , is approached. In this region fatigue crack propagation rates may

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Figure 7.13. Fatigue crack propagation data for ASTM A517 Grade F (T-1) steel, stress ratio R = 0.1. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

be strongly inﬂuenced by metallurgical factors, and also by the stress ratio. In the static failure region, fatigue crack propagation rates increase rapidly as the conditions for static failure, at the maximum load in the fatigue cycle, are approached. Fatigue crack propagation rates are strongly inﬂuenced by the stress ratio. They may be strongly inﬂuenced by metallurgical factors, and also by specimen thickness (see Section 7.4.2.2). 7.4.5 S HORT C RACKS Conventionally, fatigue crack propagation data, such as those shown in Figures 7.5–7.8, 7.10 and 7.13 are obtained from specimens containing relatively long cracks (a > 10 mm). Short cracks (say a less than about 1 mm) are often of practical importance. For some time there has been consider-

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Figure 7.14. Overall fatigue crack propagation behaviour for metallic materials under constant amplitude fatigue loading. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

able interest in the acquisition and application of fatigue crack propagation data for short cracks (Miller 1982, Miller and de los Rios 1992). In a given metallic material the threshold for fatigue crack propagation from a short crack can be lower than for a long crack (Murakami 2002). Similarly, fatigue crack propagation rates can be greater. Unfortunately, behaviour varies in detail from material to material. In general, it is not possible to extract useful generalisations, and practical problems have to be solved on an ad hoc basis. Another source of difﬁculty with short cracks can be the failure of stress intensity factors to provide an adequate description of the crack tip stress ﬁeld (see Section A.3.2). Test methods for short cracks have not yet been standardised. Short crack effects can have several different origins (Miller 1982, Suresh 1998). These origins include the irregular nature of many short cracks, and also the inﬂuence of metallurgical features such as inclusions and grain boundaries, whose scale is comparable with the crack size (see Section 3.4). The plastic wake left by a propagating fatigue crack is sometimes responsible for crack closure (see Section 7.4.2.3). A plastic wake can only reach a stable state if the range of the Mode I stress intensity factor, K, for each cycle differs little from its predecessor, that is d(K)/da is small (Pook 2000a).

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Differentiating Equation (A.4) with respect to the crack length a, and normalising with respect to the Mode I stress intensity factor, KI , leads to 1 1 dKI = . (7.6) KI da 2a Hence, for a short crack, d(K)/da may not be small. The plastic wake may then never reach a stable state. This corresponds to the minimum value of the effective value of K, Keff , which is shown in Figure 7.12. The amount of crack closure would then be reduced when compared with that for a long crack, leading to an increase in Keff , and a concomitant increase in fatigue crack propagation rates. What can happen with short cracks propagating in fatigue under constant amplitude fatigue loading is shown schematically in Figure 7.15. For a speciﬁc specimen, and a constant amplitude fatigue loading, the range of Mode I stress intensity factor, K, is a measure of the crack length, a. If the crack is sufﬁciently long then overall fatigue crack propagation behaviour is the same as in Figure 7.14. This is shown by the curve marked ‘long crack’. For a short crack there are two possibilities. Firstly, fatigue crack propagation can slow down and arrest, as shown by the solid line marked ‘short crack’, leading to a non propagating crack (see Sections 6.5 and 7.4.3). Alternatively, a propagating fatigue crack can slow down and then accelerate, blending with long crack behaviour, as shown by the dashed line. In a cracked metallic specimen the fatigue limit (see Section 3.2) is sometimes controlled by the crack length, together with the threshold for fatigue crack propagation. The relationship, for a metallic material, between the fatigue limits of specimens containing cracks of various sizes and the fatigue limit of an uncracked (plain) specimen may be summarised by means of a Kitagawa diagram (Kitagawa and Takahashi 1976). In a Kitagawa diagram, shown schematically in Figure 7.16, fatigue limits are plotted against crack length, both on logarithmic scales. In the diagram the fatigue limit of a plain specimen is shown by the horizontal line. If cracks are below a critical size then they have no effect on the fatigue limit. The other line, which has a slope of −0.5, shows fatigue limits calculated from the crack length and the long crack fatigue crack propagation threshold using an appropriate expression for the Mode I stress intensity factor, such as Equation (A.6). Actual material behaviour, shown by the dashed line, is a smooth blend between the two straight lines. This blend can be interpreted as summarising the threshold behaviour of short cracks. As an example of the relationship between crack length and the fatigue crack propagation threshold, Kth , some data for NiCr steel specimens con-

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Figure 7.15. Short crack and long crack fatigue crack propagation behaviours for metallic materials under constant amplitude fatigue loading.

Figure 7.16. Kitagawa diagram showing relationship between crack length and fatigue limits. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

taining edge cracks between 0.25 and 6 mm long tested were ﬁtted by an empirical expression (Frost et al. 1974). This empirical expression implies that Kth ∝ a 1/6 , where a is crack length. The point of a Kitagawa diagram is that the fatigue crack propagation threshold, expressed in terms of stress, decreases as crack length decreases, even though Kth increases.

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7.4.6 FATIGUE L IFE IN THE P RESENCE OF A C RACK If the initial crack size, ai , the ﬁnal crack size, af , and the crack path are all known then calculation of the number of cycles, N, for a crack to propagate from ai to af is straightforward. If, at the initial crack size, the range of Mode I stress intensity factor, K, is less than the threshold for fatigue crack propagation then the fatigue life is indeﬁnite (see Section 7.4.3). Otherwise, an expression for the stress intensity factor is substituted into a fatigue crack propagation equation and integrated. In general, integration has to be carried out numerically (Anon. 2005f). Some stress intensity expressions for common fatigue crack propagation specimens, which permit analytical integration, in conjunction with the Paris equation, are available (Jones and James 1996). The simple approach below, based on the Paris equation (Equation (7.2)), is adequate for some purposes. It illustrates the general approach to fatigue crack propagation life calculations. In carrying out fatigue crack propagation life calculations it is sometimes assumed, as an approximation. that the Paris region, shown in Figure 7.14, extends from the threshold region to the static failure region (Pook 2000a). This simpliﬁcation, shown schematically in Figure 7.17, is safe in the vicinity of the threshold region, where it leads to an underestimate of fatigue crack propagation life. However, it is unsafe in the vicinity of the static failure region, where it leads to an overestimate of fatigue crack propagation life. This overestimate may be unimportant because in this region fatigue crack propagation rates are high, and relatively few cycles are involved. Scatter in fatigue crack propagation rate data has to be taken into account in calculations (see Section 7.4.1). This is often done by taking the upper bound to a fatigue crack propagation data scatter band, such as those shown in Figures 7.6 and 7.8. For failure analysis calculations mean values are used. For a constant amplitude fatigue loading the general expression for the stress intensity factor (Equation (A.6)) may be written in the form √ (7.7) K = SY π a , where S is stress range in the fatigue cycle (see Section 3.1)), Y is a geometric correction factor (of the order of one), and a is the crack length. Inverting the Paris equation (Equation (7.2)) gives 1 dN = . (7.8) da C(K)m Combining Equations (7.7) and (7.8) 1 dN = . (7.9) da CY m S m π m/2 a m/2

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Figure 7.17. Simpliﬁed overall fatigue crack propagation behaviour for metallic materials under constant amplitude fatigue loading. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

Assuming that Y is constant then the number of cycles, N, for a crack to propagate from its initial length, ai , to its ﬁnal length, af , is given by af 1 1 . (7.10) N= m m m/2 m/2 CY S π ai a Hence 2 N= (m − 2) × CY m S m π m/2

1 (m−2)/2

ai

−

1

(m−2)/2

af

.

(7.11)

For example, for S = 119 MPa, ai = 1.3 mm, af = 51 mm, Y = 1.12, m = 3.23, and C = 2.88 × 10−12 . Equation (7.11) becomes 2 1.23 × 2.88 × × 1.123.23 × 1193.23 π 1.615 1 1 × − . 0.00130.615 0.0510.615

N=

10−12

(7.12)

C is in MN-m units so crack lengths are entered in metres, and N = 649,700 cycles. If af is very large compared with ai then the second term in the square brackets may be neglected and Equation (7.11) becomes

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N=

2

(7.13) (m − 2) × and N becomes 725,700 cycles. This is an increase of 3.2 per cent. For comparison, decreasing S by 1 per cent also increases N by 3.2 per cent. CY m S m π m/2 a (m−2)/2

7.4.6.1 Welded Joints In a welded joint small crack like ﬂaws may be present at the toe of a weld. These are often slag inclusions. In consequence virtually the whole fatigue life may be occupied by fatigue crack propagation from these ﬂaws (Gurney 1979). In a welded joint, in the as welded condition, tensile residual stresses of yield point magnitude may be present in the vicinity of the weld (Maddox 1991). A stress relief heat treatment is often used to relieve residual stresses in welded joints. In unstressrelieved (as welded), welded joints residual stresses are superimposed on any applied fatigue loading. Detailed information on the distributions of residual stresses is available for some types of welded joints, and this facilitates calculation of fatigue crack propagation life in the presence of residual stresses (Lee et al. 2005). In the absence of detailed information, it is conventional to assume that the effective stress cycle is from the yield stress downwards. This means that a crack may remain open throughout the stress cycle, even when the minimum applied stress is compressive. Hence, for unstressrelieved welded joints it has become conventional to use the whole stress range in the calculation of K (Pook 2000a). Stress relieving a welded joint may result in a substantial improvement in its fatigue life.

7.5 Variable Amplitude Fatigue Crack Propagation 7.5.1 W EIGHTED AVERAGE S TRESS R ANGE The simplest approach to estimating fatigue crack propagation rates under variable amplitude fatigue loading is to assume that each cycle causes the same amount of fatigue crack propagation as if it were part of a constant amplitude fatigue loading. This approach leads to the use of the concept of a weighted average stress range (see Section 4.3.1). Its use to characterise the level of a variable amplitude fatigue loading, was ﬁrst proposed by Paris (1962). It is deﬁned so as to have physical signiﬁcance for fatigue crack propagation, and also in such a way that it can be used in fatigue crack propagation life calculations as if it were a constant amplitude stress range. If necessary, equivalent cycles are extracted from a load history in which

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125

individual cycles cannot be distinguished by using cycle counting (see Section 4.3.3). Interaction effects due to changes in load levels which affect the crack tip plastic zone size and the plastic wake are not take into account (see Section 7.5.2). The implicit assumption that interaction effects are negligible is sometimes justiﬁed, for example for some offshore structures (Pook and Dover 1989). To derive the weighted average stress range ﬁrst write a generalised expression for the Mode I stress intensity factor, KI in the form KI = σ f(a),

(7.14)

where σ is stress and a crack size. Hence if the stress range for the ith cycle is σi , then the corresponding stress intensity factor range, Ki , is given by Ki = σi f(a).

(7.15)

Assuming that fatigue crack propagation rates are given by the Paris equation (Equation (7.2)), then the increment of crack propagation, δa, corresponding to Ki is given by δa = C(Ki )m and the average increment of crack propagation over N cycles is C(Ki )m . δa = N If the weighted average range of KI , Kh , is deﬁned by C(Ki )m 1/m , Kh = N

(7.16)

(7.17)

(7.18)

then the Paris equation becomes da = CKhm . (7.19) dN This means that it is possible to calculate fatigue crack propagation rates for variable amplitude fatigue loading by replacing K in the Paris equation by Kh . It follows that for constant amplitude fatigue loading Kh = K. The Paris equation and Equation (7.14) are both nonlinear, which at ﬁrst sight appears to invalidate the argument. However, Eastabrook’s theorem (Eastabrook 1981) ensures that, within wide limits, it is possible to deﬁne a corresponding weighted average stress range as (σi )m 1/m . (7.20) σh = N

126

Chapter 7: The Cracked Situation Table 7.1. Stress cycles during each operation. Number of cycles 1 4 10

Stress range (MPa) 285 195 95

This equation provides a direct means of calculating Kh from an appropriate expression for the stress intensity factor, and hence fatigue crack propagation rates. It is mathematically equivalent to the Miner’s rule summation given by Equation (4.11) (see Section 4.3.1). In effect it is a derivation of Miner’s rule in which fatigue damage is taken as increments of crack propagation (see Section 4.3). The approach can be extended to more elaborate fatigue crack propagation expressions, but calculations become less straightforward (Austin 1994). It has been found experimentally, for example by Hibberd and Dover (1977), that for situations where interaction effects are not signiﬁcant, the use of Kh collapses fatigue crack propagation rate data for constant amplitude and variable amplitude fatigue loading onto a single scatter band. For a variable amplitude fatigue loading Equation (7.11) becomes 1 1 2 − , (7.21) N= (m − 2) × CY m Shm π m/2 ai(m−2)/2 af(m−2)/2 where Sh is the weighted average stress range. For the loading shown in Table 7.1 the number of cycles per operation is 15 and Sh , calculated using Equation (4.11), is 162.9 MPa. If the initial crack depth ai = 1 mm, the ﬁnal crack depth af = 3.98 mm, the geometric correction factor Y = 1.12, and the Paris equation constants are m = 3.23 and C = 2.88 × 10−12 MN-m units, then Equation (7.21) becomes 2 N= −12 1.23 × 2.88 × 10 × 1.123.23 × 162.93.23 × π 1.615 1 1 × − (7.22) 0.0010.615 0.0039880.615 and N = 177,023 cycles, hence the number of operations to failure is 177,023/15 which is approximately 11,800. √ If the threshold for fatigue crack propagation, Kth = 9 MPa m, and this is taken into account, then a different result is obtained. First calculate the crack depth at which each stress range becomes damaging by entering Kth and stress ranges into Equation (7.7). This leads to the results shown in Table 7.2.

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127 Table 7.2. Damaging crack depths. Stress range (MPa) 285 195 95

Damaging crack depth (mm) 0.25 0.54 2.28

Since 0.25 mm and 0.54 mm are both less than ai the 285 MPa and 195 MPa stress ranges are damaging throughout the fatigue crack propagation life. The 95 MPa stress range becomes damaging at a crack depth of 2.28 mm. The fatigue crack propagation life integration is carried therefore out in two increments. For the ﬁrst increment calculate the weighted average stress range, Sh , assuming that the lowest stress range is non damaging (see Section 4.3.1). Thus Sh = 156.7 MPa, ai = 0.001 m and af = 0.0028 m and N = 139,398 cycles. For the second increment all three stress ranges are damaging and Sh = 162.9 MPa, ai = 0.00228 m, af = 0.00398 m and N = 54,045 cycles. Hence the number of operations to failure = (139,398 + 54,045)/15 = 12,900, a signiﬁcant increase of about 9 per cent. 7.5.2 I NTERACTION E FFECTS Interaction effects affecting fatigue crack propagation rates in metallic materials can occur when load levels change during a block fatigue loading (see Section 4.2.3). Within a block fatigue loading each block consists of a constant amplitude fatigue loading. At the new load level the fatigue crack propagation rate may not be the same as it would be for the same fatigue load under constant amplitude fatigue loading. Numerous tests have been carried in attempts to quantify interaction effects, including tests using the simple two level load histories shown schematically in Figures 7.18–7.21. Some generalisations are possible (Pook 1983a, 2000a). It is often found that a crack will propagate, at the rate for constant amplitude fatigue loading, if the load is increased, as shown in Figure 7.18, but will be temporarily retarded to below the expected rate if the load is reduced (Figure 7.19) or following a single overload (Figure 7.20). These effects can be explained by considering changes in crack closure due to changes in the plastic wake (Nakagaki and Atluri 1979) (see Section 7.4.2.3). On the other hand a single underload, as shown in Figure 7.21, can temporarily increase crack propagation rates. A practical load history will often start and ﬁnish at zero, as shown schematically in Figure 7.22. When this load history is repeated there is effectively a periodic underload.

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Chapter 7: The Cracked Situation

Figure 7.18. Two level load history, load increased.

Figure 7.19. Two level load history, load decreased.

In detail, interaction behaviour is too complicated for the results of simple tests, such as those based on Figures 7.18–7.22, to be used to predict fatigue crack propagation behaviour under more complicated load histories such as that shown in Figure 4.9 (Schütz 1979, Pook 1983a, 2000a). A number of models to account for interaction effects have been proposed. The best known is Wheeler’s model (Wheeler 1972). He considered that fatigue crack propagation would be retarded if the calculated plastic zone for the current cycle was within that due to some previous cycle (see Section A.3.3.2). All the models, including Wheeler’s, make use of empirically determined constants. The models are therefore restricted to making predictions for situations similar to those used to determine the constants. In practice, calculations have to be carried out using computer programs. Sophisticated software, validated for various situations, is now available.

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Figure 7.20. Two level load history, single overload.

Figure 7.21. Two level load history, single underload.

7.6 Fractography The study of fracture surfaces is known as fractography (see Section 2.2.4). For more than 40 years it has been used to reconstruct fatigue crack propagation data from the fracture surfaces of metallic specimens and components (see Section 2.2.5). Striations, such as those shown in Figure 3.12, can be used to reconstruct crack length versus number of cycles curves by the measurement of striation spacing. This was an early application of fractography (Pook 1962). Under programme loading, programme markings such as those visible in Figure 7.1, may appear (see Sections 4.2.3 and 4.4). These programme markings can also be used to reconstruct fatigue crack propagation curves (Ryman 1962). Further, programme markings provide information on crack front shape development during fatigue crack propagation.

130

Chapter 7: The Cracked Situation

Figure 7.22. Load history starting and ﬁnishing at zero load.

Fractography is important in the failure analysis of both laboratory and service failures in metallic materials. It is particularly valuable when, for any reason, fatigue crack propagation has not been monitored during the fatigue life. A practical difﬁculty is that striations and programme markings cannot always be observed, and even when they are present, it can be difﬁcult to resolve them. They are sometimes damaged mechanically during subsequent fatigue crack propagation, and are also sometimes damaged by corrosion. 7.6.1 FATIGUE C RACK P ROPAGATION IN A W ELDED J OINT A fractographic and fracture mechanics analysis was carried out on some medium strength structural steel cruciform welded joints (Pook 1982b, 1983b. Fatigue tests were carried out as part of the United Kingdom Offshore Steels Research Project, which was set up to obtain fatigue and fracture data relevant to tubular structures in the North Sea (Anon. 1974b). Details of the specimens, test techniques used, and fatigue test results are given by Holmes (1980a, 1980b) and by Holmes and Kerr (1982). The specimens were full penetration welded joints of the type shown in Figure 7.23. They were made from BS 4360: 1979 grade 50D steel (modiﬁed), assembled using manual metal arc welding, and were not stress relieved after welding (see Section 7.4.6.1). Tests were carried out joints under direct stress (axial) fatigue loading at zero mean stress, using the load history shown in Figure 4.9. In all specimens several individual fatigue cracks were nucleated at the weld toe on both sides of the specimen, as shown in Figure 7.1. There was usually more fatigue crack propagation on one side of the specimen than on the other. As fatigue crack propagation proceeded individual cracks merged

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Figure 7.23. 25 mm thick structural steel cruciform welded joint (Pook 1983b). Reproduced under the terms of the Click-Use Licence.

into a single fatigue crack of large aspect ratio (ratio of crack surface length to depth. (In the literature aspect ratio is sometimes taken as the ratio of crack depth to surface length). Fatigue crack propagation data were obtained from measurements of the programme marking spacing made with a travelling microscope capable of being read to 0.01 mm. Measurements were made along a line passing through one of the multiple origins, and at or near the greatest fatigue crack depth. The programme markings were sometimes difﬁcult to resolve and illumination had to be carefully adjusted. Only limited data were obtained for some specimens. Programme markings below a certain size were not resolvable, even if the magniﬁcation was increased. In addition, few data were obtained for crack depths of less than 1 mm because of mechanical damage to the fracture surfaces. This apparently occurred during the later stages of fatigue tests. Figure 7.24 shows the extensive data which were obtained for the larger fatigue crack in Specimen A2/5 (Figure 7.1 bottom). The programme markings on this crack surface were unusually clear. The nominal root mean square (RMS) stress applied during the fatigue test was 25.6 MPa, and the fatigue life was 4.798 × 106 cycles (47.98 programmes). The programme

132

Chapter 7: The Cracked Situation

Figure 7.24. Fatigue crack propagation in a 25 mm thick structural steel cruciform welded joint, specimen A2/5 (Pook 1983b). Reproduced under the terms of the Click-Use Licence.

markings suggest that fatigue crack initiation took place at about 1.1 × 106 cycles (11 programmes). Data not shown in Figure 7.24 suggest that for the shallower crack (Figure 7.1 top) initiation was at about 3.7 × 106 cycles (37 programmes). For comparison with the fractographic results theoretical predictions were carried out using stress intensity factor solutions and fatigue crack propagation data recommended by Det Norske Veritas (Anon. 1977). This information was in N-mm units, so these units were used in calculations rather than the more usual MN-m units. The Mode I stress intensity factor, KI , for the joint design used is given by KI = 3.41σ a 0.3 ,

(7.23)

where σ is nominal stress and a is crack depth. The fatigue crack propagation rate is given by the Paris equation (Equation (7.2)) with C = 3.1 × 10−13 and m = 3.1. The Paris equation was integrated numerically (see Section 7.5.1). Det Norske Veritas did nor recommend a value for the threshold for fatigue √ crack propagation so a typical value of 200 N/mm3/2 (6.32 MPa m) was used (Frost et al. 1974). Two sets of calculations were carried out and the results are shown in Figure 7.24. In one set it was assumed that whole fatigue cycles are damaging, and in the other set only the tensile half cycles were assumed to be damaging. For shallow cracks (<1.5 mm deep) the results shown in Figure 7.24 approach the prediction in which it was assumed that whole load cycle are dam-

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133

aging, in accordance with the conventional view of the effect of tensile residual stresses in welded joints (see Section 7.4.6.1). The prediction assuming that only the tensile half cycles were damaging provides a good ﬁt at medium crack depths. This implies that the effect of tensile residual stresses has died away, in agreement with residual stress measurements on welded joints (Lee et al. 2005). Fatigue crack propagation rates are underestimated for deep cracks (>5 mm deep). This appears to be because Equation (7.22) underestimates stress intensity factors for crack depths greater than one ﬁfth of the plate thickness (Pook 1983b). Similar results were obtained from other welded joints analysed (Pook 1982b, 1983b). Cruciform welded joints are notorious for scatter in fatigue test results (Gurney 1976). From the above results it is clear that this is associated with fatigue crack propagation events within 0.5 mm of the weld surface. This emphasises the importance of the precise weld quality achieved. However, the established view that, for unstressrelieved welded joints, fatigue life is controlled by stress range irrespective of mean stress is, for many purposes, a convenient generalization (Gurney 1979).

8 Fatigue Crack Paths

8.1 Introduction Macroscopic aspects of fatigue crack paths, corresponding to the top two scales in Table 3.3, have been of industrial interest since the earliest fatigue investigations (see Sections 2.2.4 and 5.3.3.3). Microscopic aspects are of interest in the development of fatigue crack propagation resistant materials. Macroscopic aspects are emphasised in this chapter. Much current interest in fatigue crack paths is in the collection of data for the solution of speciﬁc engineering problems. The fatigue crack path in a critical structure in aerospace, automotive, offshore and other industries can determine whether a fatigue failure is benign or catastrophic (Grandt 2004). There is relatively little information on how to ensure that a fatigue crack path is benign. Knowledge of potential crack paths is also needed for the selection of appropriate non destructive testing procedures. In the last four decades there have been substantial advances in the understanding of macroscopic aspects of fatigue crack paths and in their prediction. This is largely through developments in fracture mechanics and computer software (Carpinteri and Pook 2005). Theoretical prediction of fatigue crack paths in three dimensions is an active area of research. Nevertheless, despite recent theoretical advances, fatigue crack paths in structures are still often determined by large scale structural tests. For example, Figure 8.1 shows an experimentally determined fatigue crack path in a ﬁghter aircraft centre section. It was tested using a load history representative of service loads. The times shown are equivalent ﬂying hours. Where two times are shown the ﬁrst is when the crack was observed to reach a rivet, and the second when it was observed to emerge from the other side of the rivet.

136

Chapter 8: Fatigue Crack Paths

Figure 8.1. Fatigue crack path in a ﬁghter aircraft centre section. Times are in ﬂying hours.

It has been pointed out by numerous authors that on a macroscopic scale, and under essentially elastic conditions, most fatigue cracks in isotropic metallic materials tend to propagate in Mode I (Figure A.2). These authors include Broberg (1987), Cotterell (1966), Knauss (1970), Minoshima et al. (2000), Parton (1992), Pook (1983a, 2000a, 2002a) and Vinas-Pich et al. (1996). Hence attention is often conﬁned to this mode (see Sections 7.1 and A.2.1). In metal fatigue this applies to macrocracks, which are Stage II in Forsyth’s notation (see Section 3.4.2). A Mode I crack is not necessarily straight, and crack paths are not readily determined. It is usually assumed in the literature that criteria developed for the path taken by a Mode I brittle fracture crack are also applicable to a Mode I fatigue crack, and vice versa. The tendency to Mode I crack propagation, for both brittle fracture and fatigue cracks, can at best be regarded as a useful generalisation based on observation. It appears in the literature in a number of forms, usually in a brittle fracture context, but does not appear to be capable

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of proof in any strict sense of the word (Ayer 1956). For example, in Cotterell (1965) it appears as a self evident axiom for a perfectly isotropic material, with a statistical argument to justify its use in practical situations where crack direction and isotropy can only be deﬁned in a macroscopic sense. An equivalent form is the criterion of local symmetry (Goldstein and Salganik 1974). This takes as self evident that a crack tends to propagate such that the elastic crack tip stress ﬁeld tends to become symmetrical, and hence Mode I. The tendency to Mode I crack propagation cannot be justiﬁed by a thermodynamic criterion alone because satisfaction of an appropriate thermodynamic criterion is a necessary, but not a sufﬁcient, criterion for crack propagation (see Section 3.4.2). 8.1.1 C LASSES OF FATIGUE C RACK P ROPAGATION There are two fundamentally distinct classes of crack propagation (Miller and McDowell 1999a). The ﬁrst is maximum principal stress dominated crack propagation, which is in Mode I (Figure A.2), and is the class most often observed in metal fatigue. The fatigue cracks shown in Figures 1.1–1.4, 3.15– 3.17, 6.9, 7.1, 7.2, 8.2, 8.7, 8.8, 8.12 and 8.13 are all maximum principal stress dominated. A Mode I crack is usually, but not always, approximately perpendicular to the maximum principal stress in the uncracked situation, hence the term maximum principal stress dominated. As a generalisation, it is sometimes stated that fatigue cracks tend to propagate perpendicular to the maximum principal tensile stress (Pook 1983a). The second class is shear dominated crack propagation. It usually takes place on planes of maximum shear stress and is an important exception to the tendency to Mode I fatigue crack propagation. Shear dominated crack propagation is often observed when the crack tip plastic zone becomes large (Miller and McDowell 1999b). In metal fatigue this applies to microcracks, that is to Stage I cracks in Forsyth’s notation (see Section 3.4.2). It also applies in some special situations such as spot welded joints at high fatigue loads (Pook 2002a). A shear dominated fatigue crack propagates in either Mode II or in Mode III, or in a combination of the two. A transition from square (Mode I) to slant crack propagation is sometimes observed in thin sheets (see Section A.3.3.3). It is a third class of fatigue crack propagation, and is another exception to the tendency to Mode I fatigue crack propagation. Slant crack propagation is sometimes stated to be mixed Modes I and III, but this is true only for the sheet centre line. Away from the centre line it is mixed Modes I, II and III (see Section A.3.3.3). It is sometimes

138

Chapter 8: Fatigue Crack Paths

called shear crack propagation, on the grounds that it takes place on planes of maximum shear stress in an uncracked sheet, but this is a misnomer.

8.2 Crack Paths in Two Dimensions Two dimensional linearly elastic analyses were used in early theoretical work on fatigue crack paths. In two dimensions only Modes I and II stress intensity factors, K and KII , can have non zero values. The Mode III stress intensity factor, KIII , is zero. It is assumed here, as is usual in the literature, that crack path criteria originally developed for crack propagation under a static loading (brittle fracture) are also applicable to fatigue loadings. Associated experimental work was on thin sheets of constant thickness, which were regarded as quasi two dimensional. In the analysis of sheets and plates of constant thickness the use of two dimensional stress intensity factor solutions is usually satisfactory for fatigue crack propagation from an initial Mode I crack (see Sections A.3.1 and A.3.3.3). If, in a sheet or plate the initial crack front is curved, as in Figure 8.2, then this curvature merely leads to uncertainties in values of K. However, for an initial nominal mixed Modes I and II crack, crack front curvature (Figure 8.3) not only introduces uncertainty into values of K and KII , but also introduces, usually unwanted, non zero values of KIII . In addition in a two dimensional approach the inﬂuence of corner point singularities is not taken into account (see Section A.4). Neglect of these three dimensional effects means that the use of two dimensional stress intensity factors is sometimes unsatisfactory for mixed mode fatigue loadings. 8.2.1 M IXED M ODE T HRESHOLDS For a Mode I fatigue loading the threshold for fatigue crack propagation, Kth , is the value of the range of Mode I stress intensity factor, K, below which fatigue crack propagation does not take place (see Section 7.4.3). More precisely, it is the value of K below which Stage II crack propagation does not take place (see Section 3.4.2). Under mixed mode fatigue loading the situation is more complicated (see Section A.2.1). There are at least three events which could, in principle, be used to deﬁne a mixed mode threshold for fatigue crack propagation (Pook 1994b). (a) Crack propagation in the plane of the initial crack. (b) Mode I branch crack formation at or near the tip of the initial crack. (c) Mode I branch crack propagation.

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Figure 8.2. Fracture surface of 19 mm thick aluminium alloy fracture toughness test specimen (Pook 1968). Reproduced under the terms of the Click-Use Licence.

Figure 8.3. Fracture surface of DTD 5050 aluminium 5.5 Zn aluminium alloy fracture toughness test specimen. The fatigue precrack fronts are slightly curved (Pook 1971). Reproduced under the terms of the Click-Use Licence.

The precise deﬁnition of a mixed mode threshold for fatigue crack propagation can have a signiﬁcant effect on numerical values, and there is some semantic confusion in the literature on what should be regarded as a ‘true’ mixed mode threshold. This is easily resolved through the use of failure mechanism maps such as Figure 8.4. In two dimensions, only Modes I and II are possible. For Mode I (Stage II) fatigue crack propagation to take place from an initial mixed mode crack, a small Mode I branch crack must ﬁrst be formed at the tip of the initial crack, (or main crack) as shown schematically in Figure 8.5. Mode I branch crack formation is a necessary, but not sufﬁcient condition, for Mode I fatigue crack propagation to take place. Occasionally, Mode I branch cracks form behind the initial crack tip. Sometimes, Mode I branch crack formation is preceded

140

Chapter 8: Fatigue Crack Paths

Figure 8.4. Failure mechanism map for mixed Modes I and II (Pook 1985a). Reproduced under the terms of the Click-Use Licence.

by a limited amount of mixed mode Stage I crack propagation (up to about 0.5 mm) in the initial crack plane (Suresh 1998). A Mode I branch crack will not propagate unless K, for the branch crack, exceeds Kth . In a metallic material, mixed mode fatigue crack propagation threshold behaviour is sometimes controlled by Mode I branch crack formation, and sometimes by branch crack propagation. Failure to make this distinction has led to some confusion in the literature. For a combination of Modes I and II on the initial crack (Figure 8.5) the Mode I stress intensity factor for the branch crack, KI∗ , is given approximately by (Pook 1994b)

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Figure 8.5. Quasi two dimensional mixed Modes I and II initial crack with a small Mode I branch crack (Pook 1989a). Reproduced under the terms of the Click-Use Licence.

KI∗ = cos

3 θ θ K cos2 − KII sin θ , 2 2 2

(8.1)

where K and KII are Modes I and II stress intensity factors for the initial crack, and θ is the branch crack propagation angle given by KI sin θ = KII (3 cos θ − 1)

(70.50◦ ≤ θ ≤ −70.50◦ ).

(8.2)

In the derivation of Equations (8.1) and (8.2) it is assumed that a branch crack is within the core region of a K-dominated region (Figure A.13). Experimental mixed Modes I and II fatigue crack propagation threshold data were used to construct the failure mechanism map shown in Figure 8.4 (Pook 1994b). Ranges of Modes I and II stress intensity factors, K and KII , are normalised by Kth . In region C Stage II fatigue crack propagation is possible, but does not necessarily take place. Once Stage II fatigue crack propagation starts it usually continues to complete failure. Experimental data show considerable scatter. It has been demonstrated that this is due to differences in Mode I branch crack formation. The conditions for Mode I branch crack formation are unclear, but it appears to be facilitated by precrack front curvature, as in the fracture toughness test specimen shown in Figure 8.3, and also by metallurgical discontinuities (Pook 1982a). In chaos theory terms, Mode I branch crack formation is a chaotic event which is strongly dependent on initial conditions (Hall 1992). The theoretical lower bound shown in Figure 8.4 is based on Equation (8.1), and the theoretical upper bound on the assumption that when K for the initial crack exceeds Kth a Mode I branch crack must form and propagate (region D). In

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Figure 8.6. Directionally stable fatigue crack propagation. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

regions C and D Stage II fatigue crack propagation is sometimes preceded by a limited amount of Stage I fatigue crack propagation. In region B, Stage I fatigue crack propagation has been observed, but fatigue cracks usually become non propagating. The mechanism by which these non propagating cracks develop is not clear (cf. Section 7.4.5). In region A, fatigue crack propagation does not take place. Any of the boundaries between regions could be used to deﬁne a fatigue crack propagation threshold. 8.2.2 C RACK PATH S TABILITY Two dimensional linearly elastic analyses are normally used in the consideration of crack path stability. Related experimental work is usually carried out on sheets or plates of constant thickness, which are regarded as quasi two dimensional. From this viewpoint a fatigue crack propagating in Mode I may be regarded as directionally stable if, after a small random deviation, it returns to its expected, ideal crack path, as shown in Figure 8.6 (Pook 2002a). A directionally unstable crack does not return to the ideal path following a small random deviation; its path is a random walk, which cannot be predetermined. These ideas are not easily given rigorous mathematical form (Pook 2002a). For example, arbitrary limits have to be placed on what is regarded as returning to the ideal crack path. Deciding whether or not a particular crack path is stable is a practical difﬁculty in the analysis of experimental crack path stability results. A practical deﬁnition of crack path stability needs to be associated with a ﬁnite amount of crack propagation. The British Standard for fatigue crack propagation rate testing, which uses directionally stable specimens, states that a crack path is acceptable only if it lies within a validity corridor deﬁned by planes 0.05W on either side of the plane of symmetry containing the crack starter notch root (Anon. 2003b). Here, W is the specimen width, or half width for a specimen containing an internal crack. This criterion may be adapted as a crack path

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stability criterion by deﬁning a stable crack as one which remains within the validity corridor. This criterion is easy to apply, but has the disadvantage that it does not take into account changes in stability as a crack propagates. The T -stress criterion is based on a two dimensional linearly elastic analysis of a Mode I crack (see Sections A.2.1 and A.3). In consequence, crack tip plasticity and three dimensional effects are not taken into account (see previous section). The elastic stress ﬁeld in a cracked body may be expanded as a series. The ﬁrst term is the stress intensity factor, which dominates the crack tip stress ﬁeld, and is a singularity. Its coefﬁcient is the Mode I stress intensity factor, K. Other terms are non singular. For a Mode I crack, the second term is a stress parallel to the crack, usually called the T -stress. Values of the T -stress are available for a range of conﬁgurations (Sherry et al. 1995). The third and higher terms can usually be neglected in the vicinity of a crack tip. It has been argued that the directional stability of a Mode I crack in an isotropic material under essentially elastic conditions is governed by the T stress (Cotterell 1966). If the T -stress is compressive and there is a small random crack deviation, perhaps due to microstructural irregularity, then the direction of Mode I crack propagation is towards the initial crack line (Figure 8.6). The analysis is a special case of crack propagation from a mixed Modes I and II initial crack (see next section). A Mode I crack in a centre cracked sheet loaded in uniaxial tension (Figure 7.4) is directionally stable in this sense. Repeated random deviations mean that the crack follows a zigzag path about the ideal crack path. In nonlinear dynamics terms, the ideal crack path is an attractor (Pook 2002a). When the T -stress is tensile a crack is directionally unstable, and following a small random deviation, it does not return to the ideal crack path path. A fatigue crack propagating in a double cantilever beam specimen is directionally unstable in this sense. Typical crack path behaviour is shown schematically in Figure 8.7 (Pook 2000a). An initial random deviation can be either above or below the centre line, so there are two possible crack paths. These are shown as solid and dashed lines in the ﬁgure. The directional stability of a crack may change as it propagates, and a stable Mode I crack may follow a curved path. Cracks tend to be attracted by boundaries, as in Figure 8.7, and are increasingly stable as a boundary is approached. The biaxiality ratio, B, is a non dimensional function of the T -stress, which is widely used (Sherry et al. 1995). It is given by √ T πa , (8.3) B= K

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Figure 8.7. Crack paths in a double cantilever beam specimen. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 185312703-5, 2000.

where a is the crack length (half crack length for an internal crack), and K is the Mode I stress intensity factor. In practice, it is sometimes found that cracks are directionally stable even when the T -stress is tensile (or the biaxiality ratio is positive). In particular, the T -stress is tensile for the widely used compact tensile specimen. This is speciﬁed in fracture mechanics based Mode I testing standards such as Anon. (2003b) and in practice fatigue cracks in compact tension specimens are usually directionally stable. The crack path shown in Figure 8.8 is an exception. Initially, the fatigue crack propagated in the plane of the crack starter notch but then turned upwards, and its path left the validity corridor. The light area on the left is static fracture where the specimen was broken open for examination. An alternative approach is to consider the direct stresses, parallel to the crack, and near the crack tip. That due to the T -stress is simply T . The stress, σx , due to the Mode I stress intensity factor, on the crack line and ahead of the crack, is given by (cf. Equation (A.2)) KI , (8.4) σx = √ 2π r where r is the distance from the crack tip. The T -stress ratio, TR , may now be deﬁned as the ratio of the T -stress to σx , given by Equation (8.4), at some characteristic value of r, rch . Provided that rch is small TR may be regarded as a crack tip parameter which is within the K-dominated region (Figure A.13). Since the T -stress criterion is based on the idea of random crack path perturbations due to microstructural irregularities, rch should be of the same order of size as microstructural features. Taking rch = 0.0159 . . . mm leads, using MN-m units, to the convenient expression

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Figure 8.8. Unstable fatigue crack path in a high strength aluminium alloy compact tension specimen.

0.01B . TR = √ πa

(8.5)

For a particular material, there should be a critical value of TR , TRc , below which a fatigue crack path is directionally stable. As an example of the calculation of TR consider an inﬁnite panel containing a centre crack, length 2a, with a uniaxial tensile stress, σ , perpendicular to the crack (Figure 8.9). From the Westergaard solution for this conﬁguration (Westergaard 1939) (see Equation (A.4)), the Mode I stress intensity factor, K, is given by √ (8.6) K = σ πa and T = −σ.

(8.7)

Substituting Equations (8.6) and (8.7) into Equations (8.3) and (8.5) gives B = −1 and −0.01 . (8.8) TR = √ πa

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Figure 8.9. Centre crack in an inﬁnite sheet under uniaxial tension (Frost et al. 1974).

This equation shows that there is a size effect. That is for geometrically similar conﬁgurations, TR decreases in absolute value as the crack size increases. Taking a as a typical value of 25 mm, Equation (8.8) gives TR = −0.0357. A biaxially loaded square sheet, edge 2W , containing a centre crack, length 2a, (Figure 8.10) is sometimes used for crack path studies (Pook 2000a). The biaxial loading ratio, λ, is the ratio of the stress parallel to the crack to the stress perpendicular to the crack. Figure 8.11 shows a typical specimen, it was tested using the rig shown in Figure 4.5. The biaxiality ratio is given by (Pook 2002a) B = −[1 + 0.085(a/W )] + λ[1.029 + 0.115(a/W ) − 2.87(a/W )2 + 4.829(a/W )3 − 3.125(a/W )4 ]

(0.1 ≤ a/W ≤ 0.6).

(8.9)

Table 8.1 shows values of the biaxiality ratio for λ values of −1, 0, 1, 2 and 3 calculated using Equation (8.9). Also shown are exact limiting values for a/W = 0. For λ values of 0 and 1, B has the correct limiting values, but for other values there appear to errors of the order of a few per cent. Values of TR , taking W as 50 mm are shown in Table 8.2.

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Figure 8.10. Biaxially loaded square sheet. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

Figure 8.11. Fatigue crack path in a Waspaloy sheet under biaxial fatigue load, specimen MFGT 7. The grid is 0.1 inch (2.54 mm). National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

As an example, some tests were carried out at room temperature on Waspaloy, a nickel based gas turbine material, in order to determine the conditions under which a fatigue crack path became unstable under biaxial fa-

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Chapter 8: Fatigue Crack Paths Table 8.1. Values of biaxiality ratio, B, for square, centre cracked sheet. a/W 0 0.1 0.2 0.3 0.4 0.5 0.6

λ = −1 –2 –2.025 –1.988 –1.936 –1.879 –1.820 –1.754

λ=0 –1 –1.009 –1.017 –1.026 –1.034 –1.043 –1.051

λ=1 0 0.008 –0.046 –0.115 –0.189 –0.265 –0.348

λ=2 1 1.024 0.925 0.795 0.656 0.512 0.355

λ=3 2 2.040 1.895 1.705 1.501 1.289 1.058

Table 8.2. Values of the T -stress ratio TR , for square, centre cracked sheet, W = 50 mm. a/W 0 0.1 0.2 0.3 0.4 0.5 0.6

λ = −1 –∞ –0.1616 –0.1122 –0.0892 –0.0750 –0.0649 –0.0571

λ=0 –∞ –0.0805 –0.0574 –0.0472 –0.0413 –0.0372 –0.0342

λ=1 0 0.0006 –0.0026 –0.0053 –0.0075 –0.0095 –0.0113

λ=2 ∞ 0.0817 0.0522 0.0366 0.0262 0.0183 0.0116

λ=3 ∞ 0.1628 0.1069 0.0786 0.0599 0.0460 0.0344

tigue loading (Pook and Holmes 1976). The specimens were 254 mm square and 2.6 mm thick. The material had been cross rolled during production to ensure that its properties were reasonably isotropic. Tests were carried out using sinusoidal constant amplitude fatigue loading at a stress ratio (ratio of minimum to maximum load in fatigue cycle), R, of 0.1. In each test the fatigue load perpendicular to the crack was kept constant. Cracks were ﬁrst propagated from each end of an initial slit under uniaxial fatigue loading (λ = 0). Then λ was increased by applying an in phase fatigue load parallel to the crack, and crack path behaviour observed. Figure 8.11 shows the fatigue crack path in one of the specimens. The results obtained were re-analysed, and the re-analysis indicated that the critical value of TR , TRc , at which a fatigue crack path became unstable in Waspaloy is about 0.022 (Pook 2002a). 8.2.3 C RACK PATH P REDICTION A centre cracked sheet with an inclined crack, loaded in uniaxial tension (Figure 8.12), provides an example of Mode I fatigue crack path behaviour under mixed Modes I and II fatigue loading on the initial crack (Pook 2002a). A Mode I fatigue crack is Stage II crack in Forsyth’s notation (see Section 3.4.2). Some experimental results for the conﬁguration shown in the ﬁgure, obtained using thin sheets, show that branch crack propagation angles from the initial crack vary widely (Lam 1993). This arises because Mode I

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Figure 8.12. Centre cracked sheet loaded in uniaxial tension with mixed Modes I and II initial crack. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

branch crack propagation angles given by expressions such as Equation (8.2) are not sharply deﬁned (Pook 2002a). Consequently, under fatigue loading, minor deviations from isotropy due to microstructural irregularities can be expected to have a signiﬁcant inﬂuence on the branch crack propagation angle. After branch crack formation a Mode I crack follows a curved path which tends towards a line of symmetry, as shown schematically in Figure 8.3, irrespective of the initial direction of Mode I fatigue crack propagation. This line of symmetry is an attractor (see previous section). Behaviour is in accordance with the well known observation that fatigue cracks tend to propagate perpendicular to the maximum principal tensile stress (see Section 8.1.1). In two dimensions, the crack tip is a point and the crack path is a line. Two dimensional Mode I crack path predictions have been carried out by a number of authors, using the same general scheme, for example Portela (1993). Calculations are carried out numerically using small increments of

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straight crack propagation. The direction taken by each increment is selected using the criterion that the increment is pure Mode I, that is the Mode II stress intensity factor, KII , is zero. Care is needed to ensure that the crack path approaches the correct limit as the increment size is reduced. In general, the limit is a curve. Agreement between theoretical predictions and experimental data obtained using thin sheets is variable (Pook 1995). This is not surprising, since neither the effects of the crack tip plastic zone (see Sections 7.4, 7.4.2.2, 7.4.2.3 and 8.1.1), nor three dimensional effects are taken into account (see Section 8.2). Two dimensional predictions have now largely been superseded by three dimensional predictions (see Section 8.4.2). Experimentally observed fatigue crack paths are either straight lines or smooth curves, and are often remarkably reproducible (Broberg 1987). Calculation of stress intensity factors for observed Stage II crack paths usually conﬁrms that fatigue crack propagation is indeed in Mode I. Occasionally, non zero values of KII appear to be present, but this is an artefact due to crack surface interference (Smith 1984). It is sometimes assumed, incorrectly, that a curved surface implies that Mode II displacements must have been present (Pook 1994b). A complication is that interference between crack surfaces in the presence of Mode II displacements can increase the Mode I stress intensity factor, K, and decrease KII . Geometric features can have a strong inﬂuence on fatigue crack paths, for example the fatigue crack path shown in Figure 8.1 is inﬂuenced by rivet holes. As another example, during routine maintenance in 2002, one of the two burners in the gas ﬁred domestic central heating boiler in the author’s house was found to be cracked due to thermal loading (see Section 3.4.3). The fatigue crack is shown in Figure 8.13. The boiler was about 12 years old so, assuming it ﬁred about 10 times per day, about 44,000 thermal fatigue cycles had been applied. The burner consists of a steel box with a series of small and large holes on top to distribute the gas to the ﬂame above the box. The larger holes have reinforced perimeters. An internal wire mesh, just visible in Figure 8.13, helped to distribute the gas evenly. Fatigue cracks appear to have initiated at three places on the perimeter of a smaller hole, propagated into two larger holes, with a small triangular piece becoming detached. Then two fatigue cracks propagated across most of the width of the box, resulting in improper combustion. The designer did not appear to have appreciated the point that stress concentration factors are largely independent of hole size. The reinforcement had prevented fatigue crack initiation at the large holes, but its absence had allowed crack initiation at a small hole. Annual inspection

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Figure 8.13. Fatigue crack in burner from domestic central heating boiler.

was recommended by the boiler manufacturer. This ensured that the cracking was detected before it became dangerous, and the burner was replaced.

8.3 Planar Crack Paths It is well known that fatigue cracks in metallic materials often tend to propagate in Mode I, that is approximately perpendicular to the maximum principal tensile stress (see Sections 8.1.1 and 8.2.3). This means that fatigue crack propagation is, on a macroscopic scale, often conﬁned to a particular plane, or in some complex geometries, to a particular slightly curved surface. Figure 7.1 is an example of the latter. 8.3.1 C RACK PATH D EVELOPMENT Fatigue crack path prediction for a planar crack is essentially two dimensional. It reduces to making the assumption that crack propagation is perpendicular to the crack front, and then calculating increments of crack propagation along the crack front, using an appropriate fatigue crack propagation expression such as Equation (7.2). Some highly reﬁned numerical algorithms have now been developed (Dhondt 2005, Kolk and Kuhn 2005). These take into account the change in the nature of the crack tip singularity at a corner point, shown in Figure 8.14, where a crack front intersects a surface (see Section A.4). Surface cracks are sometimes modelled as semi elliptical cracks

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Figure 8.14. Semi elliptical surface crack under uniaxial tension (Frost et al. 1974).

(see Section A.5.1). Surface cracks are sometimes called part through cracks. If it is assumed that the crack remains semi elliptical as it propagates then crack path predictions may be made by calculating increments of fatigue crack propagation for the deepest point of the crack and for the corner points. This simple approach is sometimes sufﬁcient. A crack has some analogies with a crystal dislocation (see Section A.5.3). In particular, the elastic stress ﬁelds associated with a crack front and a dislocation are both singularities. The associated energy means that a dislocation has a line tension which controls its shape under an applied stress ﬁeld. Similarly, a crack front may be regarded as having a line tension which controls its shape, but with the important difference that the motion of a crack front is, in general, irreversible; that is a crack can only propagate, not contract. Secondly, the crack tip stress ﬁeld, and hence the line tension, may vary along a crack front. At a corner point the corner point singularity provides a point force which balances the line tension in a direction corresponding to the crack front intersection angle, as shown in Figure 8.15 (Pook 2000a). The line tension concept may be used qualitatively to account for two well known aspects of fatigue crack behaviour (Pook 2002a). First, on a macroscopic scale, a crack front is smooth, and any initial sharp corners rapidly disappear (see Section A.5.3). This is not true if the scale of observation is reduced to a level at which microstructural effects are important; the crack front

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Figure 8.15. Crack front intersection angle. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

Figure 8.16. Convergence to stable crack front shape for a plate under pure bending. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

then shows local irregularities within the overall smooth shape. Secondly, in a particular set of circumstances a crack front tends to a particular stable shape. As an example, Figure 8.16 shows some theoretical results based on ﬁnite element analysis, for plates containing part through cracks loaded in fatigue under pure bending (Lin and Smith 1999). The aspect ratio a/c where a is the crack depth and 2c is the crack surface length (Figure 8.17) is plotted against a/B, where B is plate thickness (Pook 2002a). (This deﬁnition of aspect ratio differs from that in Section 7.6.1.) Irrespective of the initial aspect ratio, as fatigue crack propagation proceeds, values converge to the same trend line.

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Figure 8.17. Semi elliptical surface crack.

8.3.2 L EAK BEFORE B REAK If a pressure vessel is subjected to fatigue loading, and a fatigue crack propagates through its wall, it is important to know whether or not this will lead to a catastrophic failure. There are two extreme situations. In the ﬁrst, the vessel disintegrates catastrophically due to brittle fracture before the fatigue crack has propagated right through the wall, with complete depressurisation and loss of the contained ﬂuid. In the second, the fatigue crack propagates right through the wall, and leakage of the contained ﬂuid takes place without disintegration of the vessel. In the short term there is no signiﬁcant loss of pressure or of the contained ﬂuid. Further fatigue crack propagation is needed before the vessel fails by brittle fracture, or the net section is so reduced that it fails by plastic collapse. This is known as leak before break, and is an example of the fail safe philosophy of design (see Section 5.3.2). Pressure containing components, which are critical from a safety viewpoint, are often designed to leak before break in order to avoid catastrophic failure (Darlaston and Harrison 1977, Pook 2000a, Zerbst et al. 2003). Figure 1.5 shows a domestic tap which did, fortuitously, leak before break. The propagation of an initial crack from the inside of a pressure vessel wall is shown schematically in Figure 8.18. When the initial crack penetrates the vessel wall it rapidly becomes a through the thickness crack with approximately straight crack fronts The initial crack is in plane strain (see Section A.3.3.1). The stress state of the through the thickness crack (plane strain, plane stress, or an intermediate stress state) depends on the wall thickness, the applied stress, and the yield stress of the material. There are three main possible sequences of events as a fatigue crack, originating at the inner surface, propagates through the pressure vessel wall. In the ﬁrst the plane strain fracture toughness, KIc , is reached before the fatigue crack has propagated right through the wall, and the vessel fails catastrophically (see Sections A.3 and A.3.3.1). In the second sequence KIc is reached before the fatigue crack has propagated right through the wall, but the through the thickness crack is in plane stress. The plane stress fracture toughness, Kc , is greater than KIc so it is possible for the brittle fracture crack to stop. Hence

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Figure 8.18. Fatigue crack propagating from inner surface of pressure vessel wall.

depending on circumstances, the pressure vessel may either leak before break or fail catastrophically. In the third sequence the fatigue crack propagates right through the wall before KIc is reached and, if the stress intensity factor for the through the thickness crack is less than the fracture toughness, the vessel will leak before break. The margin of safety introduced by leak before break depends on the difference between the through crack length at which leakage can be detected, and that at which the pressure vessel fails catastrophically. Determining these crack lengths is therefore a key factor in carrying out a leak before break assessment, together with determining the length of the through the thickness crack which results when a crack has propagated right through the vessel wall. Because of the numerous variables involved, none of these are readily determined. This makes it difﬁcult to establish whether or not leak before break can be relied on. Ensuring that an aircraft pressure cabin will leak before break is a major preoccupation of aircraft designers (Wanhill 2003, Grandt 2004).

8.4 Crack Paths in Three Dimensions It is well known that fatigue cracks in metallic materials often tend to propagate in Mode I, that is approximately perpendicular to the maximum principal tensile stress (see Sections 8.1.1 and 8.2.3). This creates a fundamental difﬁculty in the study of the general three dimensional case of Mode I fatigue crack propagation from an initial mixed mode crack. Differential geometry considerations show that the crack propagation surface for a Mode I crack must be smooth (Pook 2002a). However, in the presence of Mode III on the initial crack an element of a Mode I branch crack can intersect the initial crack front at only one point (Figure 8.19).

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Figure 8.19. Mixed Modes I, II and III initial crack with a smooth branch crack and an element of a Mode I branch crack (Pook 1989a). Reproduced under the terms of the Click-Use Licence.

What happens when a fatigue loading is applied in the presence of Mode III on the initial crack is illustrated by the fracture surface of the mild steel angle notch specimen shown in Figure 8.20 (Pook 1985b). The rough area is static fracture where the specimen was broken open for examination. The specimen used was tested in three point bending, and is shown schematically in Figure 8.21. The expected tendency to Mode I crack propagation was observed on two distinct scales. On a scale of 1 mm crack fronts were approximately straight, and initially crack propagation was mixed mode, as shown schematically by the smooth branch crack in Figure 8.19. As the fatigue crack propagated the crack front rotated until it was perpendicular to the specimen surfaces, and crack propagation was in Mode I. On this scale the crack follows a curved path which tends towards a plane of symmetry in the uncracked specimen. Differential geometry considerations show that this curved path is not a permissible Mode I crack path (Pook 2002a). The plane of symmetry is an attractor (see Sections 8.2.2 and 8.2.3). On a smaller scale of 0.1 mm the tendency to Mode I fatigue crack propagation results in the production of what is known as a twist crack (Lawn and Wilshaw 1975). A twist crack consists of narrow Mode I facets connected by irregular, predominantly Mode III cliffs as shown schematically in Figure 8.22 (Pook 2002a). The Mode I facets gradually merge as, when viewed

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Figure 8.20. Twist crack fracture surface of mild steel angle notch specimen MFSH 35. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

Figure 8.21. Mild steel angle notch specimen (Pook 1985a). Reproduced under the terms of the Click-Use Licence.

on the larger scale, the crack propagation surface becomes perpendicular to the specimen surfaces (Figure 8.20). 8.4.1 L OWER B OUND M IXED M ODE T HRESHOLDS The theoretical lower bound for the Stage II fatigue crack propagation threshold under mixed Modes I and II fatigue loading, shown in Figure 8.4, can be extended to the general three dimensional case in which all three modes are present (Pook 1994b). For a combination of Modes I, II and III

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Figure 8.22. Schematic section through a twist crack. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

on the initial crack the stress intensity factor, KI∗ , for an element of a Mode I branch crack (Figure 8.19) is given approximately by 2 K(1 + 2ν) + K 2 (1 − 2ν)2 + 4KIII ∗ , (8.10) KI = 2 where K is the value of KI∗ from Equation (8.1), KIII is the Mode III stress intensity factor for the initial crack and ν is Poisson’s ratio. The presence of Mode III does not affect the value of θ given by Equation (8.2) (Figure 8.5). The lower bound threshold for fatigue crack propagation, shown in Figure 8.4, is extended to the general case of mixed Modes I. II and III loading in Figure 8.23. The ﬁgure is based on Equation (8.10) with ν = −1/3. Values of the ranges of Modes I, II and III stress intensity factors, K, KII , and KIII , are normalised by the Stage II fatigue crack propagation threshold, Kth . Experimental data conﬁrm this theoretical lower bound for pure Modes II and III fatigue loadings on the initial crack, for mixed Modes I and II fatigue loadings, and mixed Modes I and III fatigue loadings (Pook 1994b). For a long initial crack, Stage I crack propagation is sometimes observed at below the lower bound, but this does not lead to complete failure. Hence it is possible to use the envelope for design purposes, although at times it would be very conservative. 8.4.2 C RACK PATH D ETERMINATION The development of twist cracks (Figure 8.20) in the presence of Mode III on the initial crack complicates the numerical determination of fatigue crack paths in the general three dimensional case. At the present state of the art the best strategy appears to be to consider the crack path at a scale of the order of 1 mm (see Section 8.4). At this scale the crack path is smooth, but, in general, the crack path is mixed mode. This strategy was used in the development of an advanced ﬁnite element fatigue crack path prediction program (Schöllmann et al. 2003).

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Figure 8.23. Theoretical lower bound for Stage II fatigue crack propagation threshold (Pook 1989a). Reproduced under the terms of the Click-Use Licence.

In the program, increments of fatigue crack propagation are calculated using the range of an equivalent stress intensity factor, Keq , and a fatigue crack propagation equation such as Equation (7.2). The directions of these increments are determined for points along the crack front, using an appropriate expression such as Equation (8.2). The increments then deﬁne a ruled surface. Overall, the increments of fatigue crack propagation, provided that they are sufﬁciently small, can be taken as deﬁning a smooth crack path. Using this sort of approach good agreement has been obtained between theoretical predictions and crack paths observed on service (Richard et al. 2006). The following useful approximation functions have been developed (Richard et al. 2005). 2 |KII | |KII | − 70 , (8.11) θ = ∓ 140 KI + |KII | + |KIII | KI + |KII | + |KIII | where θ is in degrees and K, KII , and KIII are the Modes I, II and III stress intensity factors. 1 K + K 2 + 4(1.155KII )2 + 4KIII , (8.12) Keq = 2 2 where K, KII , and KIII are the Modes I, II and III stress intensity factor ranges.

9 Why Metal Fatigue Matters

9.1 Introduction In the past century and a half much has been written on the problem of metal fatigue, and it has only been possible to describe some of the more important metal fatigue topics in this book. One might argue from the infrequency of catastrophic metal fatigue failures in structures and components that the problem is no longer serious. Clear explanations are usually found for the major failures that do occur, with human error often being involved. However, lesser metal fatigue failures, often unrecognised unless they happen to be seen by a fatigue specialist, are a common and expensive nuisance. General awareness of the dangers of metal fatigue has greatly increased. By and large the problem can be contained, if not solved; but the price is eternal vigilance. Metal fatigue is very much a descriptive subject. Metallurgical descriptions are concerned with the effect of fatigue loading on the state of the material. Mechanical descriptions are concerned with matters such as the number of cycles to failure, or the rate of propagation of a fatigue crack, and are the more useful from an engineering viewpoint. The multi volume work on structural integrity, edited by Milne et al. (2003) includes encyclopaedic coverage of many metal fatigue topics. These are mostly written as state of the art reviews, and inevitably some of the material has already been outdated. It is probably impossible to write a comprehensive standard text on metal fatigue. Perhaps the closest approach is the book by Suresh (1998). On well established topics, his exposition is in straightforward textbook style. In more controversial areas, sections are written as state of the art reviews, with opposing views given equal prominence. Where the author expresses a preference, this is usually the conventional wisdom rather than a minority view.

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It must always be remembered that fatigue is only one of many, often conﬂicting requirements with which a designer must contend. Innovation frequently brings the risk of fatigue failure, but because metal fatigue is not a suddenly occurring phenomenon, and also because of the need to pursue detail, it is only possible to progress slowly as experience is acquired in fresh areas. The transfer of metal fatigue technology from producer to user is often a frustrating and expensive process.

9.2 Avoidance of Fatigue Failure Even rough estimation of fatigue strength at the preliminary design stage presents problems. At ﬁrst sight this is surprising, since large amounts of experimental data are available and, in general, the basic mechanisms of metal fatigue failure are relatively well understood and documented. Problems in estimating fatigue strength arise because of the complexity and variety of realistic engineering design situations. This makes it difﬁcult to generalise on known metal fatigue data and then apply these generalisations to speciﬁc engineering problems. In practice, many advances in design against fatigue have been made on what amounts to a trial and error basis. A considerable research and development effort on metal fatigue is usually needed whenever the service life of a fundamentally new type of structure or component needs to be predicted. Long-term background work on metal fatigue is needed as it is likely to decrease the effort needed for a particular new situation, and also means that a pool of expertise is kept available. In principle, an analytical approach, based on materials data and applied mechanics, can be used to predict the service life of a structure or component. The fatigue crack initiation and fatigue crack propagation phases must be treated separately, although in practice one or the other usually predominates. An analytical approach requires expert knowledge, and often fails because of lack of the detailed information required. Simpliﬁed calculations can be useful when an approximate answer is adequate. Simpliﬁed fracture mechanics calculations have proved particularly valuable in failure analysis. As new information becomes available, the range of situations that can be analysed is increasing. Standard procedures of various degrees of formality are perhaps the most satisfactory from the designer’s viewpoint. They may be based on analytical procedures, service experience, the results of structural tests, or on some combination of these. They need not have a theoretically sound basis, provided they give sufﬁciently accurate answers, and ideally no expert knowledge

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is required. These methods fail where no appropriate standard procedure is sufﬁciently accurate for the situation being considered. The development of standard procedures, which can be a lengthy process, is probably the best way of assimilating the results of research into engineering practice. In practice, most metal fatigue assessments are based on simpliﬁed standard procedures, which despite their apparent lack of physical validity, are known to give conservative answers. For critical structures and components the use of speciﬁc standard procedures is often enforced by legislation. Service loading testing is sometimes the most cost effective method of determining the service life of a component or structure. Modern servohydraulic equipment permits the application of virtually any desired load history. This kind of testing, which ﬁrst became established in the aircraft industry, is now used for critical structures and components in a wide range of industries. Such tests are often used to calibrate approximate analytical methods so that these methods can be used to extrapolate the results of service loading tests to broadly similar situations. Much effort has been devoted to the development of appropriate standard load histories for service loading testing. When a structural failure due to metal fatigue (or other causes) leads to catastrophe, society as a whole takes an interest through its institutions. In a typical catastrophic metal fatigue failure, the immediate cause of the failure is usually easily ascertained, but it is more difﬁcult to discover how procedures designed to ensure safety came to fail. Recommendations of ofﬁcial enquiries into catastrophic failures are often mainly concerned with plugging procedural loopholes.

9.3 Research and Development Basic research on metal fatigue has always had a strong practical bias in that the motivation is usually to acquire data which will help to avoid service failures. In academic circles, it is sometimes assumed that an analytical approach is the ideal to be aimed for. Much early metal fatigue work was concerned with the experimental determination of the fatigue strength of plain, or relatively mildly notched, specimens. Numerous variables can affect the fatigue behaviour of a particular metallic material, including mean stress, surface ﬁnish, the environment, and special situations such as biaxiality of loading. Further complications arise when realistic load histories are introduced. Consequently, no data collection, however large, can be comprehensive in the sense that the need for fatigue testing during product development can definitely be eliminated. Fatigue testing is time consuming and expensive, but

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claimed shortcuts to the acquisition of data should be treated with caution, as they are sometimes based on empirical correlations of limited applicability, or on an oversimpliﬁed view of the nature of metal fatigue. Perhaps the most signiﬁcant advance in metal fatigue was the general realisation that many structures and components contain crack like ﬂaws which are either introduced during manufacture, especially if welding is used, or form early in service. Virtually the whole service life is then occupied by fatigue crack propagation from these ﬂaws. An understanding of fatigue crack propagation is therefore essential for the understanding and prediction of the fatigue behaviour of many structures and components. One consequence is that design often has to be for a speciﬁc, limited life, rather than for an indefinite life. Another consequence is that the presence of a crack does not necessarily mean that a structure or component is defective in a legal sense. The widespread use of the fracture mechanics parameter stress intensity factor to analyse fatigue crack propagation data means that behaviour can be predicted for any cracked body for which a stress intensity factor expression is available. Hence a designer is not limited to situations similar to those used to generate experimental fatigue crack propagation data.

9.4 The Role of Non Destructive Testing In the context of metal fatigue the usual meaning of non destructive testing (NDT) is the detection and sizing of cracks and crack like ﬂaws in components, structures and laboratory specimens. NDT and fracture mechanics (the applied mechanics of crack propagation) are complementary disciplines that inﬂuence each other. Both are used in the measurement of material properties in the presence of a crack, and both are used in fatigue assessments of components and structures in the presence of actual or postulated cracks. Taken together they answer questions such as ‘Is the minimum crack size that can be found by an NDT technique acceptable?’, ‘Is a more sophisticated (and expensive) NDT technique needed?’, and ‘Should the part be redesigned (perhaps lowering its structural efﬁciency) so that a larger crack is acceptable?’. It is widely recognised that, for critical structures and components, allowance for appropriate NDT must be made at the design stage.

9.5 Current Trends Despite increasing use of non metallic materials in various components and structures, the extensive use of metallic materials will continue indeﬁnitely.

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Hence, research and development aimed at the resolution of speciﬁc practical metal fatigue problems will also continue. Many critical structures and components are subject to legislation, which often means that designers are constrained in the approaches to fatigue assessment that can be used. This trend will continue, with further codiﬁcation of existing theoretical and experimental data. Another legislative trend is to ban potentially harmful metals such as lead. This is leading to the development of lead free solders, with the concomitant need to asses their fatigue properties. In the 1980s work on metal fatigue as an academic discipline, appeared to have passed its peak. Much research work on fatigue and related topics was concerned with obsessive pursuit of detail, involving much labour, but yielding little in the way of deeper understanding. Metallurgical descriptions of metal fatigue mechanisms were usually at scales of the same order as sizes of metallurgical features, that is 10−4 to 10−2 mm. These mechanisms were often referred to as micromechanisms. The descriptions were usually qualitative, rather than quantitative, in nature. The applied mechanics framework, including fracture mechanics, which formed the basis of mechanical descriptions dealt with macroscopic aspects at scales of 10−1 mm and above. Fracture mechanics descriptions of fatigue crack propagation at these scales was highly developed. What has happened, in the intervening two decades, is that developments in instrumentation, computers and software have made it possible to extend metallurgical and mechanical descriptions to smaller scales, with as associated increase in understanding. This has led to remarkable improvements in the fatigue resistance of some metallic materials. Finally, a very welcome trend is the increased international exchange of information on metal fatigue that has followed political changes in a number of countries.

A Fracture Mechanics

A.1 Introduction The applied mechanics framework for study of the behaviour of cracked bodies under load is known as fracture mechanics. The application of fracture mechanics to fatigue crack propagation is well established, and most modern books on metal fatigue include an introduction to the topic. The account below is based on Frost et al. (1974) and Pook (2000a, 2002a). These books include numerous references. The book by Frost et al. was the ﬁrst on metal fatigue in which a fracture mechanics approach was used throughout. It was reprinted in 1999. Fracture mechanics does not provide any information about the processes involved in fatigue crack propagation. It does provide the descriptive and analytic framework needed for their characterisation, and for the application of fatigue crack propagation data to practical engineering problems. Simplifying assumptions have become conventional in much present day fracture mechanics, and these are satisfactory for many purposes. The material is assumed to be a homogeneous isotropic continuum, and its behaviour is assumed to be linearly elastic. Crack surfaces are assumed to be smooth, although on a microscopic scale they are generally very irregular. Modiﬁcations are made to basic linear elastic fracture mechanics theory to allow for the actual behaviour of real materials. The basic ideas in linear elastic fracture mechanics are straightforward. The mathematics involved is often formidable, but does lead to the useful and easily applied key concept of stress intensity factor, which describes the elastic stress and displacement ﬁelds in the vicinity of a crack tip. A stress intensity factor has the dimensions of √ √ stress × length. The most widely used units are MPa m. These units ap√ pear in many standards and are therefore to be preferred. The use of MPa m

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Figure A.1. Notation for crack tip stress ﬁeld (Frost et al. 1974).

is not particularly convenient since crack sizes are normally √ measured in mm, √ and N/mm3/2 units are sometimes used. 1 MPa m = 1000 N/mm3/2 ≈ 31.62 N/mm3/2 . Some ﬁgures which appear in the main text are repeated in order to make the appendix self contained.

A.2 Notation for Stress and Displacement Fields The conventional notation for the position of a point relative to the crack tip, and for the stresses at this point, is shown in Figure A.1. The point on the crack tip is the origin of the coordinate system and the z axis lies along the crack tip. Displacements of points within the cracked body when the body is loaded are u, v, w in the x, y, z directions. The terms crack tip and crack front are synonymous. Crack tip tends to be used for two dimensional situations and crack front in three dimensions. A.2.1 C RACK S URFACE D ISPLACEMENT A fundamental fracture mechanics concept is that of crack surface displacement. In fracture mechanics the interest is in what happens in the vicinity

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Figure A.2. Notation for modes of crack surface displacement (Frost et al. 1974).

of the crack tip, so it is sometimes referred to as crack tip surface displacement. If a load is applied to a cracked body, then the crack surfaces move relative to each other. For points on opposing crack surfaces that were initially in contact there are three possible modes of crack surface displacement (Figure A.2); Mode I where opposing crack surfaces move directly apart in directions parallel to the y axis; Mode II where crack surfaces move over each along the x axis, that is, perpendicular to the crack tip; and Mode III where crack surfaces move over each other in directions parallel to the z axis, that is, parallel to the crack tip. By superimposing the three modes, it is possible to describe the most general case of crack surface displacement. The terms Mode I, Mode II and Mode III are usually capitalised, and are often used in the metal fatigue literature without explanation. The descriptive terms; opening mode, edge sliding mode, and shear mode are sometimes used for Modes I, II and III respectively. The term mixed mode means that at least one mode, other than Mode I, is present. The modes of crack surface displacement may also be used to characterise crack propagation. A particular type of elastic crack tip stress ﬁeld is associated with each mode of crack surface displacement (Paris and Sih 1965). These stress ﬁelds are characterised by stress intensity factors, symbol K. Subscripts I, II and III are used to denote mode. Where there is no subscript, Mode I is usually implied; this convention is sometimes used in the text. Corresponding displacement ﬁelds permit calculation of crack tip surface displacements. It is matter of observation that, when viewed on a macroscopic scale, and under essentially elastic conditions, cracks in metals tend to propagate in Mode I, so attention is largely conﬁned to this mode (see Sections 3.4.2,

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Figure A.3. Square section ring element around crack tip. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312703-5, 2000.

7.1 and 8.1). At smaller scales, cracks are generally very irregular and different modes of crack surface displacement may be observed. A.2.2 VOLTERRA D ISTORSIONI If a crack surface is considered as consisting of points then the three modes of crack surface displacement (Figure A.2) provide an adequate description of the movements of crack surfaces when a load is applied. However, if the surface is regarded as consisting of inﬁnitesimal elements, then element rotations must also be described, and Volterra distorsioni (distortions) are appropriate. Volterra distorsioni are also applicable in the theory of crystal dislocations (Nabarro 1967). In his analysis of distorsioni Volterra considered the simplest multiply connected body, that is a cylinder with a central hole. An account in English is given in Zastrow (1985). The cylinder, free of body forces, surface forces, or any initial stress, is made simply connected by a cut along a radial plane. The cut surfaces are regarded as completely rigid, but the remainder of the cylinder is elastic. The cut surfaces may be moved relative to each other in six different ways, so there are six distinct Volterra distorsioni. The conventional approach to Volterra distorsioni needs to be modiﬁed for the description of crack tip surface movements (Pook 2000a). Consider a pair of inﬁnitesimal elements, A and B, which are in the xy plane and are situated on the upper and lower surface of an unloaded crack respectively (Figure A.3). Their initial coordinates are (r, 180◦ ) and (r, −180◦ ). The elements A and B are connected by a ring element of inﬁnitesimal width. For clarity, this ring element is shown as having a square cross section and the elements are separated in the ﬁgure. Such ring elements correspond to Volterra’s cylinders. Three of the six Volterra distorsioni, correspond to the three modes of crack tip surface displacement, Figure A.4 (Pook 2002a). Under a Mode I

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Figure A.4. Volterra distorsioni, modes of crack surface dislocation. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

loading element A moves a distance v in the y direction, and element B a distance −v (Figure A.4 left). For consistency with crack tip surface displacement notation this is called a Mode I dislocation. The ring element remains within the xy plane, that is, initially plane sections, perpendicular to the crack tip remain plane. Distortion of the ring element is symmetrical about the xz plane. Under a Mode II loading element A moves a distance u in the x direction, and element B a distance −u, and this is a Mode II dislocation (Figure A.4 centre). The ring element remains within the xy plane, that is plane sections remain plane, but its distortion is not symmetrical about the xz plane. Under a Mode III loading element A moves a distance −w in the z direction, and element B a distance w (Figure A.4 right). This is a Mode III dislocation. The ring element does not remain within the xy plane, that is plane sections perpendicular to the crack front do not remain plane. Distortion of the ring element has rotational symmetry about the x axis. The remaining three Volterra distorsioni involve rotation of the elements A and B, and are usually called disclinations. The notation used here is based on the idea that each mode of dislocation and disclination is associated with the same coordinate axis (Pook 2002a). Hence in a Mode I disclination, element A rotates through an angle β about an axis parallel to the y axis, and element B rotates through an angle −β (Figure A.5 right). In a Mode II disclination, element A rotates through an angle α about an axis parallel to the x axis and element B rotates through an angle −α (Figure A.5 centre). In a Mode III disclination, element A rotates through an angle γ about an axis parallel to the z axis and element B rotates through an angle −γ (Figure A.5 left). Modes II and III disclinations cannot exist in isolation because of interference between elements A and B. Any dislocation mode may be decomposed into a dipole of equal and opposite disclinations of either of the other two modes.

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Figure A.5. Volterra distorsioni, modes of crack surface disclination. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

If movements of the elements A and B are identical, then a dislocation becomes a translation and a disclination a rotation. For convenience, continuous distributions of dislocations are usually referred to as dislocations, and similarly for disclinations, translations and rotations.

A.3 Stress Intensity Factors It is intuitively obvious that a large crack is more severe than a small crack. A basic requirement for the study of the behaviour of cracked bodies is to express this numerically. The presence of a crack dominates the stress ﬁeld in the vicinity of the crack tip, and some results are not intuitively obvious. The key concept of stress intensity factor, for Mode I and for Mode II, arises from a two dimensional linearly elastic analysis for a straight crack. This follows the usual methods of elastic stress analysis in which strains and distortions are assumed to be small, and conditions of equilibrium and compatibility must be satisﬁed (Gere and Timoshenko 1991). The use of a two dimensional analysis simpliﬁes the mathematics, and also simpliﬁes some descriptions. For example, what is meant by crack length is unambiguous. Mode III is not possible in two dimensions, so for this mode a quasi two dimensional anti plane analysis is used. Stress intensity factors may be used to characterise the mechanical properties of cracked specimens in just the same way that stresses are used to characterise the mechanical properties of uncracked specimens. They help to quantify the rather elusive concept of a material’s toughness. One convenient deﬁnition is this: resistance to crack propagation (including fatigue crack propagation). For example the fracture toughness, Kc , of a metallic material may be deﬁned as the value of the Mode I stress intensity factor, KI , for

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failure under a static load. In practice, failure is not abrupt, and the fracture toughness is deﬁned for a small, speciﬁed amount of crack propagation (Anon. 2005g). The stress ﬁeld in vicinity of a crack tip is dominated by the leading term of a series expansion of the stress ﬁeld. For a particular mode of crack surface displacement this leading term is always of the same general form. Individual √ stress components are proportional to K/ r where K is the stress intensity factor and r is the distance from the crack tip (Figure A.1). A stress intensity factor is a singularity of order −1/2; as the crack tip is approached, stresses tend to inﬁnity. A formal deﬁnition of the Mode I stress intensity factor is √ (A.1) KI = lim(r → 0)σy 2π r , where σy is √ the stress perpendicular to the crack along the x axis. The numerical factor, 2π , in the equation is the usual convention. Other conventions are occasionally encountered, especially in early work, leading to numerically different values for stress intensity factors. There are corresponding equations, in terms of shear stresses, for the other two modes. Stress intensity factors of the same mode may be combined by algebraic addition. Once K is known, stress and displacement ﬁelds in the vicinity of the crack tip are given by standard equations (Pook 2000a). In Modes I and II, stresses are independent of the stress state, but displacements are a factor (1 − ν 2 ) less for plane strain than for plane stress (ν is Poisson’s ratio). For example, for Mode I on the x axis in front of the crack KI . (A.2) σy = √ 2π r Also, for the upper crack surface 2KI 2r (plane stress), (A.3) v= E π where v is the displacement in the y direction, that is, perpendicular to the crack, and E is Young’s modulus. The equation implies that for Mode I a crack opens up into a parabola (see Section A.4). Displacements are also parabolic for Modes II and III. Equation (A.3) also implies that Mode I crack surface displacement is a combination of a Mode I dislocation and a Mode III disclination (Figures A.4 and A.5). Similarly, Mode III crack surface displacement is a combination of a Mode III dislocation and a Mode I disclination. However, Mode II crack surface displacement is just a Mode II dislocation. KI must be positive, since a compressive load simply holds a crack closed. However, the Modes II and III stress intensity factors, KII and KIII can be

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either positive or negative, and the sign used is a matter of convention. It is usually taken as positive, but careful attention to sign is needed when calculating stresses and displacements. It is possible to obtain stress ﬁeld equations, using Kirchoff plate bending theory, in terms of a bending mode stress intensity factor, KB (Paris and Sih 1965). At a plate surface KB is equivalent to KI . Corresponding crack surface displacements are a Mode II disclination (Figure A.5 centre). Because of crack surface interference this is not physically realistic so KB is not now used. A.3.1 S TRESS I NTENSITY FACTOR S OLUTIONS Stress intensity factors are available for numerous conﬁgurations, for example Murakami (1987, 1992b, 2001) and this facilitates practical applications. Solutions for test specimens are included in appropriate standards, for example Anon. (2003a). A solution for a particular conﬁguration is sometimes called a K-calibration or a compliance function. Where a solution is presented as an equation ﬁtted to numerical results, care must be taken not to use it outside its speciﬁed range. Conventionally, two dimensional solutions are used for sheets and plates of constant thickness subjected to in plane loads. This is usually satisfactory. To illustrate the general form of solutions for the Mode I stress intensity factor, KI , some examples are given below. These are all for loads perpendicular to the crack. Stresses parallel to a crack have no effect. A.3.1.1 Two Dimensional Solutions For a centre crack, length 2a (it is conventional to take the length of an internal crack as 2a) under a remote uniaxial tension σ (Figure A.6) √ (A.4) KI = σ π a . For a small edge crack, length a, in a sheet under uniaxial tension, shown in Figure A.7 (Pook 2000a). √ (A.5) KI = 1.12σ π a . Equation (A.5) also applies to a crack at a blunt notch if the local stress is used (Figure A.8), and to a crack at a sharp notch if a is taken as the crack length plus the notch depth shown in Figure A.9 (Pook 2000a). Solutions are sometimes presented in the form √ (A.6) KI = σ Y π a ,

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Figure A.6. Centre crack in an inﬁnite sheet under uniaxial tension (Frost et al. 1974).

where Y is a geometric correction factor, usually of the order of 1, and a is a characteristic crack dimension. Hence in Equation (A.5) Y = 1.12. To facilitate calculations other deﬁnitions of Y are sometimes used as in Equation (A.7), below. Test specimens of standard design are included in various fracture mechanics based standards. For example, for the three point bend single edge notch specimen shown in Figure A.10 (Anon. 2003b), FY (A.7) √ , B W where F is force, B specimen thickness and W specimen width. Y is given by √ 6 α [1.99 − α(1 − α)(2.15 − 3.93α + 2.7α 2 )] , (A.8) Y = (1 − 2α)(1 − α)3/2 where α = a/W and the equation is valid for 0 ≤ α ≤ 1. In Anon. (2003b) Equation (A.7) includes a factor 101.5 so that with force measured in kN, and √ specimen dimensions measured in mm, KI values are in MPa m. KI =

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Figure A.7. Small edge crack in a sheet under uniaxial tension. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 185312-703-5, 2000.

For a centre crack, length 2a, with point forces F per unit thickness on the crack faces (Figure A.11) F . (A.9) KI = √ πa KI usually increases with increasing crack length. This is one of the few cases in which KI decreases with increasing crack length. A.3.1.2 Three Dimensional Solutions A circular crack in an inﬁnite body is often called a penny shaped crack. Under uniaxial tension, σ , perpendicular to the crack a , (A.10) KI = 2σ π where a is the crack radius. Hence Y in Equation (A.6) is 2/π . With central point loads F on the crack faces (cf. Figure A.11) F . (A.11) KI = (π a)3/2 For an elliptical crack in an inﬁnite body under uniaxial tension the maximum stress intensity factor is at the ends of the minor axis and is given by

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Figure A.8. Small edge crack at a blunt notch in a sheet under uniaxial tension. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

√ σ πa , (A.12) KI = E(k) where a is the semi minor axis of the elliptical crack and E(k) is the complete elliptic integral of the second kind. This is given by π/2 1 − k 2 sin2 d, (A.13) E(k) = 0

where

a2 (A.14) c2 and c is the semi major axis of the ellipse. An approximation for E(k) is a 1.65 (0 ≤ a/c ≤ 1). (A.15) E(k) = 1 + 1.464 c For c = a Equation (A.12) reduces to Equation (A.10) and for c a to Equation (A.4), this is also the solution for a tunnel crack, width 2a, in an inﬁnite body. Many of the cracks observed in service are surface cracks (Figure A.12). They are often called part through cracks. The aspect ratio of a surface crack is the ratio of crack surface length to crack depth. Other deﬁnitions of aspect ratio are sometimes used (see Sections 7.6.1 and 8.3.1). Surface cracks with k=

1−

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Figure A.9. Small edge crack at a sharp notch in a sheet under uniaxial tension. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

Figure A.10. Three point single edge notch bend specimen (Frost et al. 1974).

an aspect ratio of more than two are usually approximated as semi elliptical cracks (see Section A.5.1). For such a semi elliptical surface crack in a semi inﬁnite body under uniaxial tension the maximum stress intensity factor is at the deepest point of the crack, and its value is dominated by the crack depth rather than the crack surface length. For a semi inﬁnite body approximate values of the stress intensity factor at the deepest point are given by

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Figure A.11. Centre crack in an inﬁnite sheet with point forces on crack surfaces (Frost et al. 1974).

Figure A.12. Semi elliptical surface crack in a semi inﬁnite body under uniaxial tension (Frost et al. 1974).

√ 1.12σ π a . (A.16) KI = E(k) For c a Equation (A.16) reduces to Equation (A.5), for c = 2.89a to Equation (A.4), and for a semi circular surface crack (a = c) it becomes √ (A.17) KI = 0.713σ π a .

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A.3.1.3 Effect of Residual Stresses The effect of introducing a crack into a body containing large scale residual stresses is to relieve the residual stresses on the crack plane. Provided that the crack is not too large the corresponding stress intensity factor is the same as if stresses, equal in magnitude but opposite in sign, were applied to the crack surfaces. For example Equation (A.5), for a small edge crack (Figure A.7) becomes √ (A.18) KI = 1.12P π a , where P is the pressure on the crack surfaces. The presence of residual stresses of unknown magnitude can be a serious limitation in the practical application of fracture mechanics. A.3.2 VALIDITY OF S TRESS I NTENSITY FACTORS The application of stress intensity factors to practical engineering problems involving cracks has been a spectacular success over the past 40 years or so. Nevertheless there are some limitations on their validity that arise from the linearly elastic stress analyses on which they are based. Accumulated experience has shown how these limitations can be managed, and what steps need to be taken to mitigate their effects. A stress intensity factor provides a reasonable description of the crack tip stress ﬁeld in a K-dominated region at the crack tip, radius r ≈ a/10, where a is the crack length, Figure A.13 (Pook 2000a). An apparent objection to the use of the stress intensity factor approach is the violation, in the immediate vicinity of the crack tip, of the initial linearly elastic assumptions, in that strains and displacements are not small. However, as the assumptions are violated only in a small core region, radius r, the general character of the K-dominated region is, to a reasonable approximation, unaffected. Similarly, by this small scale argument, small scale nonlinear effects due to crack tip yielding, microstructural irregularities, internal stresses, irregularities in the crack surface, the actual fracture process, etc., may be regarded as within the core region. If a crack is too short then it may not be possible to use this small scale argument (see Section 7.4.5). Elastic stress ﬁelds at the tips of sharp notches are, with due attention to detail, similar to those for cracks. By the small scale argument, stress intensity factors for cracks can be used to describe the elastic stress ﬁeld at the tips of sharp notches. The elastic stress at the tip of a sharp V-notch can be described by stress intensity factors, provided that the included angle does not exceed

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Figure A.13. K-dominated and core regions at a crack tip. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 185312-703-5, 2000.

30◦ . Negative values of KI are possible for sharp notches and also for open cracks, that is, cracks where the surfaces are separated by a small amount. When yielding is not small scale, stress intensity factors do not provide a reasonable description of the crack tip stress ﬁeld, and other less versatile fracture mechanics parameters become appropriate. Failure to check whether large scale yielding might be occurring is the commonest error in the practical application of stress intensity factors. One early approach was to ensure that the nominal net section stress did not exceed 80 per cent of the yield stress, σY , and this is still a useful check. In stress intensity factor based metallic material test standards, minimum acceptable test specimen dimensions are speciﬁed in order to avoid large scale yielding. Actual minimum values depend on values of the stress intensity factors and the material’s yield stress, for example Anon. (2003b, 2005g). Use of a two dimensional stress intensity factor solution for plates and sheets implicitly assumes that the crack front is straight through the thickness and also perpendicular to the surfaces. In practice, crack fronts are usually curved. For example Figure A.14 shows the fracture surface of a 19 mm thick aluminium alloy fracture toughness test specimen, the fatigue precrack front is clearly curved. Standards place limits on the permissible amount of fatigue crack front curvature, so that a two dimensional stress intensity factor solution can be used, for example Anon. (2003b, 2005g).

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Figure A.14. Fracture surface of 19 mm thick aluminium alloy fracture toughness test specimen (Pook 1968). Reproduced under the terms of the Click-Use Licence.

Extension of the essentially two dimensional concept of stress intensity factor to three dimensions makes two implicit assumptions. The ﬁrst is that the line deﬁning a crack front is smooth so that stress intensity factors do not change abruptly along the crack front. This is generally, but not always, true for cracks observed in practice when these are viewed on a macroscopic scale, and does not usually cause difﬁculties. The second assumption is that a crack front is continuous, that is, it is a closed curve. This is sometimes true, for example for an internal elliptical crack. It is not true at a corner point where a crack front intersects a free surface (Figure A.15). The nature of the crack tip singularity changes in the vicinity of a corner point. This can usually be neglected on the basis of the small scale argument, but sometimes has to be taken into account (see Section A.4). A.3.3 E FFECTS OF S MALL S CALE Y IELDING Small scale yielding in the high stress region at a crack tip has two main practical consequences. First, it leads to a practical deﬁnition of plane strain in the presence of a crack which differs from that usual in the theory of elasticity. Secondly, the relaxation of stresses within the crack tip plastic zone means that, to maintain equilibrium, stresses outside the plastic zone increase, and the effective crack length is increased.

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Figure A.15. Semi elliptical surface crack under uniaxial tension showing corner points (Frost et al. 1974).

A.3.3.1 Deﬁnition of Plane Strain From a theory of elasticity viewpoint an uncracked plate loaded in uniaxial tension is in a state of plane stress, that is the stress in the thickness direction is zero. This is still so in the bulk of a plate when a Mode I crack is introduced, but highly stressed material adjacent to the crack tip is constrained by the less highly stressed surrounding material, and stresses are induced in the thickness direction in the interior of the plate in the vicinity of the crack tip. This situation is often referred to as plane strain in fracture mechanics. It must not be confused with the conventional theory of elasticity deﬁnition, which is used in the theoretical analysis of crack tip surface displacements (see Sections A.2.1 and A.3). Limited plastic ﬂow due to yielding of the material adjacent to the crack tip does not affect the situation in the interior of a thick plate (Figure A.12). In this context a plate is said to be thick if the thickness is at least 2.5(KI /σY )2 where KI is the Mode I stress intensity factor and σY is the yield stress. This practical deﬁnition of plane strain arose from consideration of the results of tests to determine the fracture toughness, Kc . It was observed that, in general, Kc decreased as the thickness increased, but that for most metallic materials it reached a constant minimum value if the specimen thickness was at least 2.5(Kc /σY )2 (see Section 7.4.2.2). This minimum value can be regarded as a material constant, and is known as the

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Figure A.16. Plane strain, plate thickness ≥ 2.5(KI /σY )2 (Frost et al. 1974).

plane strain fracture toughness, KIc . In the special case of this symbol, the subscript I denotes both a Mode I crack and plane strain. The minimum specimen thickness requirement of 2.5(Kc /σY )2 is included in plane strain fracture toughness test standards, for example Anon. (2005g). When a crack front is curved, as in a semi elliptical surface crack (Figure A.12) there is a high degree of constraint along the crack front, except at the corner points, and such cracks can usually be regarded as being in plane strain. When the plate thickness is very much less than 2.5(K/σY )2 , then the crack tip plastic zone size becomes comparable with the thickness, and yielding can take place on 45◦ planes. This relaxes the through thickness stresses, so that the whole plate is in a state of plane stress (Figure A.17). The symbol Kc is sometimes reserved for the plane stress fracture toughness which is then regarded as a material constant. At intermediate thicknesses the stress state is uncertain and Kc is a function of thickness. A.3.3.2 Effective Crack Length The increase in effective crack length over the physical crack length due to yielding at the crack tip is shown schematically in Figure A.18. A ﬁrst estimate of the plastic zone size may be obtained by substituting von Mises’ criterion of yielding (see Section 4.5.2.1) into the elastic crack tip stress ﬁelds. For Mode I, plane stress this leads to

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Figure A.17. Plane stress, plate thickness 2.5(KI /σY )2 (Frost et al. 1974).

rY =

1 2π

KI σY

2 (plane stress),

(A.19)

where rY is the plastic zone size measured in the crack direction, KI is the Mode I stress intensity factor, and σY is the yield stress. The actual size and shape of a crack tip plastic zone depends on the ﬂow properties of the metal, but its dimensions are always proportional to (KI /σY )2 . Typically, a plastic zone size is about twice that given by Equation (A.19), so rY is interpreted as the plastic zone radius. The effective crack length becomes a + rY , as indicated in Figure A.18, and the corresponding stress intensity factor is calculated iteratively. Under plane strain conditions the plastic zone radius is about one third of that given by Equation (A.19), and 1 KI 2 (plane strain). (A.20) rY = 6π σY The plastic zone corrections given by Equations (A.19) and (A.20) are often very small and hence unnecessary. At one time they were quite popular, but are now rarely used. Plastic zone corrections do not appear to be speciﬁed in any standards.

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Figure A.18. Physical and effective crack length, rY is the plastic zone radius.

A.3.3.3 Slant Crack Propagation in Thin Sheets The transition from square (Mode I) to slant crack propagation sometimes observed in thin sheets under both static and fatigue loading, as shown schematically in Figure A.19, is an exception to the observation that fatigue cracks in metals tend to propagate in Mode I (see Sections 7.4.2.2, 8.1.1 and A.2.1). Slant crack propagation is sometimes stated to be mixed Mode I and Mode III, but this is true only for the sheet centre line. Away from the centre line, it is mixed Mode I, Mode II and Mode III. This has been conﬁrmed by ﬁnite element analysis (Pook 1993). It is sometimes called shear crack propagation, on the grounds that it takes place on planes of maximum shear stress in an uncracked sheet, but this is a misnomer. In the calculation of stress intensity factors it is usual to treat slant crack propagation, and crack propagation in the transition region, as if they were Mode I crack propagation, and to use a two dimensional stress intensity factor solution. This is difﬁcult to justify by the small scale argument (see Section A.3.2), but it does not cause difﬁculties in practice (see Section A.5.2).

A.4 Corner Point Singularities The analyses on which the concept of stress intensity factor is based are essentially two dimensional in nature, and the crack front is a point (see Section A.3.2). When analysis is extended to three dimensions, the crack front

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Figure A.19. Transition from square to slant crack propagation in thin sheets. The arrow shows the direction of fatigue crack propagation (Pook 1983a). Reproduced under the terms of the Click-Use Licence.

becomes a line. Derivations then include the implicit, and usually unstated, assumption that a crack front is continuous. This is not the case at a corner point, where a crack front intersects a free surface. The crack front shown by the dashed line in Figure A.15 intersects the surface at two corner points. As is well known, the nature of the crack tip singularity changes in the vicinity of a corner point. For corner point singularities, the polar coordinates (r, θ) in Figure A.1 are replaced by spherical coordinates (r, θ, φ) with origin at the corner point. The angle φ is measured from the crack front. The stress intensity measure, Kλ , is used to characterise corner point singularities, where λ is an exponent deﬁning the corner point singularity. Stresses are proportional to Kλ /r λ and displacements to Kλ r 1−λ , where r is measured from the corner point. For a crack surface intersection angle, γ of 90◦ , deﬁned as in Figure A.20, there are two modes of stress intensity measure. These are the symmetric mode, KλS , where crack tip surface displacements are Mode I (Figure A.2), and the antisymmetric mode, KλA , which is a combination of Modes II and III displacements. In other words, the presence of one of these modes of crack tip surface displacement always induces the other. For the special case of λ = 0.5, stress intensity factors are recovered. KλS becomes KI , and KλA a combination of Modes II and III stress intensity factors, KII and KIII . For the symmetric mode, and Poisson’s ratio, ν = 0.3,

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Figure A.20. Deﬁnition of crack surface intersection angle, γ . Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

Figure A.21. Deﬁnition of the crack front intersection angle, γ . Reprinted from Pook (1994a), Copyright 1994, with permission from Elsevier.

the theoretical value of λ is 0.452 for a crack front intersection angle of 90◦ , Figure A.21 (Pook 2002a), whereas for the antisymmetric mode λ is 0.598. For λ = 0.5, Mode I crack surface displacements are given, for plane stress, by Equation (A.3), and the crack opens up into a parabola (Figure A.22 centre). The radius at the tip of the loaded crack, r, is given by 4KI2 , (A.21) π E2 where KI is the Mode I stress intensity factor and E is Young’s modulus. When λ = 1/2 stresses and displacements cannot, in general, be calculated in detail because of lack of information. However, when λ < 0.5 r = 0 (Figure A.22 top) and, from Equation (A.21), stress intensity factors tend to zero as a corner point is approached. Conversely, when λ > 0.5 r = ∞ and stress intensity factors tend to inﬁnity (Figure A.22 bottom). There is only limited information available on the size of the corner region (boundary layer) in which the crack tip stress ﬁeld is dominated by the stress intensity measure, although it must be associated with some characteristic dimension, such as sheet thickness. As with stress intensity factors, an apparent objection to the use of the stress intensity measure approach is the violation, in the vicinity of the crack r=

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Figure A.22. Crack proﬁles for loaded Mode I crack. Crack tip radius r = 0 for λ < 0.5. For λ = 0.5 r is ﬁnite and for λ > 0.5 r = ∞. Reprinted from Pook (1994a), Copyright 1994, with permission from Elsevier.

tip, of the initial assumption on which linearly elastic analyses are based (see Section A.3.2). However, as the assumptions are violated only in a small core region, the general character of the corner point singularity dominated region in the vicinity of the crack tip is unaffected, as is shown for stress intensity factors in Figure A.13. Similarly, small scale nonlinear effects may be regarded as within the core region inside a corner point singularity dominated region. In turn the corner point singularity dominates only within a limited region, so in some circumstances a corner point singularity dominated region may lie within a K-dominated region, as shown schematically for a surface plane in Figure A.23 (Pook 2002a). For practical engineering purposes the use of stress intensity measures is usually unnecessary. They do not appear in standards which make use of stress intensity factors, for example Anon. (2005f). However, they are two situations in which corner point singularities have an important inﬂuence, and these are discussed in the next two sections. A.4.1 C RACK F RONT I NTERSECTION A NGLE The coefﬁcient deﬁning a stress intensity measure, λ, is a function of Poisson’s ratio, ν, and the crack front intersection angle, β (Figure A.21). At a

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Figure A.23. K-dominated, corner point singularity dominated and core regions at a surface plane. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

critical crack front intersection angle, βc , λ = 0.5 and stress intensity factors then have ﬁnite values in the corner point region. For β < βc , λ < 0.5, and for β > βc , λ > 0.5. For the symmetric mode βc is given approximately by (Pook 1994a) −1 (ν − 2) , (A.22) βc = tan ν where ν is Poisson’s ratio. For ν = 0.3, βc = 100.4◦ . It has been argued, from energy and other considerations, that the crack front intersection angle must be βc . Intersection angles of about this value may be observed for Mode I cracks, and in consequence crack fronts in plates of constant thickness are often curved, for example Figure A.14. For the antisymmetric mode βc is given approximately by −1 (1 − ν) . (A.23) βc = tan ν For ν = 0.3, βc = 67.0◦ . When the crack surface intersection angle γ = 90◦ , a crack at a corner point is always in a combination of Modes I, II and III crack tip surface displacements. For any given value of γ there are two possible values of βc . For γ = 45◦ and ν = 0.3 these are 108◦ and 60◦ . There is a corresponding value

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of the ratio KI : KII : KIII for each value of βc . KI , KII and KIII are the Modes I, II and III stress intensity factors. A.4.2 D ERIVATION OF S TRESS I NTENSITY FACTORS Numerical schemes for the calculation of stress intensity factors for cracks in three dimensional bodies, explicitly make the assumption that stress intensity factors provide a description of the crack tip stress ﬁeld. This is true only for a continuous crack front, for example an internal crack, and for a crack front which intersects the surface at the critical crack front intersection angle, βc . In consequence, such schemes cannot adequately reproduce the behaviour of stress intensity factors in the vicinity of a corner point. In practice values obtained are ﬁnite, and usually of the same order as elsewhere along the crack front. At a corner point, values are, in effect, extrapolations which depend on details of the numerical scheme used (Pook 1994a, 2000c). At a corner point Mode II and Mode III displacements cannot exist in isolation. The presence of one of these modes always induces the other (see Section A.4). Numerical calculations usually show induced values of the Mode II stress intensity factor, KII , or the Mode III stress intensity factor, KIII , as would be expected, in the vicinity of a corner point. Two dimensional numerical schemes are widely used in the determination of stress intensity factors for quasi two dimensional specimens of constant thickness. Only Mode I and Mode II stress intensity factors are possible in two dimensions, so such schemes cannot reveal induced KIII values at corner points. Many three dimensional ﬁnite element calculations are carried out for a Poisson’s ratio of about 0.3, and crack front and crack surface intersection angles of 90◦ (Figures A.20 and A.21). Hence theoretically KI should tend to zero, and KII and KIII should tend to inﬁnity, as corner points are approached. As the corner point is approached, the ratio KIII /KII tends to a ﬁnite limiting value, which is a function of the crack front inclination angle, β, and √ Pois◦ son’s ratio, ν. For β = 90 and ν = 0.3 it is 0.5 and for ν = 0.5 it is 0.5. From an engineering viewpoint the ﬁnite stress intensity factor values actually obtained at corner points do not matter provided both that results are reasonably consistent, and also that difﬁculties do not arise in practical applications. In practice, the situation for Mode I is indeed satisfactory, and numerous standards make use of Mode I stress intensity factors without any mention of corner point singularities, for example Anon. (2005f). The situation appears to be reasonably satisfactory for Mode II, but it is not satisfactory for Mode III where there are inconsistencies in reported KIII values.

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Figure A.24. Through thickness variation of KI for 20 mm square models under Mode I loading (Pook 2000c).

A.4.2.1 Stress Intensity Factors for Square Models Figures A.24–A.28 show some typical results for through thickness distributions of stress intensity factors for Mode I, Mode II and Mode III loadings (Pook 2000c). These were obtained from ﬁnite element analysis of a 20 mm square model, with a crack extending from the middle of one side of the square to its centre. Loadings and boundary conditions were chosen to give large K-dominated regions, and Poisson’s ratio was taken as 0.3. For Mode I and Mode II loadings, the results are normalised by stress intensity factors obtained from two dimensional calculations, whereas for the Mode III loading they are normalised by centre line values. The results for the Mode I loading of a 4 mm thick plate and a 40 mm long bar (Figure A.24) show the well known increase in the Mode I stress intensity factor, KI , at the centre line compared with the corresponding two dimensional solution. The decrease in KI towards the surface is also well known, and suggests the existence of a corner point singularity dominated region. The presence of a corner point singularity dominated region was conﬁrmed by estimating values of λ from crack surface displacements at the model surface. These estimates gave λ = 0.452 for both the plate and the bar, which agrees with the theoretical value (see Section A.4). The displacements indic-

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Figure A.25. Through thickness variation of KII for 20 mm square models under Mode II loading (Pook 2000c).

ated that the size of the corner point singularity dominated region (boundary layer) was about 0.2 mm for the plate, and about 1 mm for the bar. Through thickness distributions of KII for Mode II loading are shown in Figure A.25. At the centre line, KII for the 40 mm long bar is lower than for the corresponding two dimensional solution. It increases towards the surface, again suggesting the existence of a corner point singularity dominated region. This was conﬁrmed by the Mode II crack surface displacements at the surface, which gave λ = 0.560 compared with the theoretical value of 0.598. They also indicated a corner point singularity dominated region size of about 3 mm. The results for the 4 mm thick plate show a different trend. KII is nearly constant through the thickness, except for a sharp decrease at the surface, the corner point singularity dominated region size is about 3 mm, and λ = 0.535. Figure A.26 shows the distribution of induced values of KIII . These are zero at the centre line (by symmetry) for both the plate and the bar, and increase towards the surface, except that for the plate there is a sharp decrease at the surface. The Mode III crack surface displacements at the surface did not deﬁne corner point singularity dominated regions, and it was not possible to

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Figure A.26. Through thickness variation of KIII for 20 mm square models under Mode II loading (Pook 2000c).

Figure A.27. Through thickness variation of KIII for 20 mm square bar under Mode III loading (Pook 2000c).

derive values of λ. Also, it was not possible to derive a limiting value for the ratio KIII /KII . The through thickness distribution of KIII for Mode III loading of the 40 mm long bar is shown in Figure A.27. KIII is constant over the central part of the bar, but there is a marked decrease as the surface is approached,

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Figure A.28. Through thickness variation of KII for 20 mm square bar under Mode III loading (Pook 2000c).

rather than the theoretical increase towards inﬁnity. The Mode III crack surface displacements at the surface indicated a corner point singularity dominated region size of about 0.2 mm, but it was not possible to derive a value for λ. Induced values of KII are zero at the centre line, decrease slightly, and then increase towards the surface (Figure A.28). The Mode II crack surface displacements at the surface indicate a corner point singularity region size of about 0.3 mm, and λ = 0.570. The inconsistencies in the results for Modes II and III loadings (Figures A.25–A.28) in the vicinity of corner points are typical of results obtained from ﬁnite element analyses. Scatter increases in the vicinity of a corner point, and what must be regarded as nominal values of KII and KIII are strongly dependent on details of numerical calculation methods. The inconsistencies arise partly from the use of linearly elastic ﬁnite element analyses. In principle, in a linearly elastic analysis KIII at a surface must be zero because shear stresses perpendicular to a free surface must be zero. This effect shows up clearly in Figure A.27, but not in Figure A.26.

A.5 Stress Intensity Factors for Irregular Cracks In three dimensions there are numerous possible crack conﬁgurations (Pook 1986). In general, cracks three dimensional cracks observed in service cannot be approximated by two dimensional stress intensity factor solutions. Naturally occurring cracks, and crack like ﬂaws, are often irregular in shape, for example the casting defects in Figures 3.17 and 7.2. The three

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Figure A.29. Crack types.

main types of cracks encountered in practice are shown schematically in Figure A.29. These are surface cracks, internal cracks, and through the thickness cracks. Methods of estimating stress intensity factors for irregular cracks have been of interest for many years (Paris and Sih 1965, Anon. 1980b, Chang 1982, Pook 1982a, Murakami 2002). Normally, stress intensity factors vary along a crack front, and it is the largest value of the Mode I stress intensity factor, KI , that is usually of interest. Adjacent cracks, as in Figure 7.1, can increase stress intensity factors through interaction effects. These are difﬁcult to assess when cracks are not coplanar. In the analysis of laboratory and service failures, the size and shape of cracks can usually be ascertained in detail by the examination of fracture surfaces. However, the results of non destructive testing will usually supply only a limited amount of information. For example, only the surface length and maximum depth of a surface crack might be available (see Appendix C). Initially, an intuitive approach was used by various authors to estimate stress intensity factors for irregular cracks. This approach was based on general knowledge of stress intensity factors for similarly shaped regular cracks (Paris and Sih 1965). Fifteen years later standardised estimation procedures for various situations started to appear (Anon. 1980b), and are now in widespread use (Anon. 2005f). Standardised procedures should be used whenever possible. Some of the ideas used in estimating stress intensity factors for irregular cracks are described below. The ideas sometimes have a sound theoretical basis, but the main justiﬁcation for their use is that they have been found to work well in practice, and also that they do not lead to unconservative results. A.5.1 U SE OF S EMI E LLIPSES AND E LLIPSES A ﬂat irregular surface crack under Mode I loading is nearly always modelled by a semi ellipse of the same surface length and depth. The Mode I stress intensity factor, KI , is greatest at the deepest point of the semi ellipse (see Section A.3.1.2). This intuitive approach is satisfactory for cracks that

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Figure A.30. Modelling of an irregular surface crack as a semi ellipse.

Figure A.31. Modelling of a more irregular surface crack as a semi ellipse.

Figure A.32. Modelling of a more irregular surface crack as a semi ellipse inscribed in a containment rectangle.

are close to this shape, as shown schematically in Figure A.29 left, and in Figure A.30. The approach does not work well for some more irregular surface cracks such as shown in Figure A.31. What is sometimes done is to ﬁrst construct a containment rectangle around the crack, and then inscribe a semi ellipse in the rectangle (Figure A.32). This results in a longer semi ellipse with a concomitantly higher value of KI at the deepest point. A similar idea is sometimes used for internal cracks in bodies of rectangular cross section, as shown in Figure A.33. The containment rectangle sides are parallel to the body surfaces. A containment rectangle may be used as it stands for a through the thickness crack (Figure A.29 right). A.5.2 P ROJECTION ONTO A P LANE Cracks are not necessarily ﬂat, and are not necessarily oriented so that they are in Mode I. One approach is to project them onto a plane so that they become equivalent Mode I cracks. The plane chosen is a plane of maximum principal tensile stress in the uncracked body. After projection, a crack can then be modelled as in the previous section. As an example, the method works quite well for the specimen shown in Figure A.20 when this is loaded in three

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Figure A.33. Modelling of an internal crack as an ellipse inscribed in a containment rectangle.

Figure A.34. Quasi two dimensional mixed Modes I and II crack with a small Mode I branch crack (Pook 1989a). Reproduced under the terms of the Click-Use Licence.

point bending (Pook and Crawford 1990). Another example is treating slant crack propagation in thin sheets (Figure A.19) as if it were Mode I (see Section A.3.3.3). In effect, the slant crack and the transition region are projected onto the plane indicated by dashed lines in the ﬁgure. One theoretical justiﬁcation is that under mixed mode loading a small Mode I branch crack may form at the initial crack tip, as shown in Figure A.34 (see Section 8.2.1). Projection of the initial mixed mode crack onto an appropriate plane can provide a method of estimating KI for such a branch crack (Pook 1989c).

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Figure A.35. Crack front smoothing.

A.5.3 C RACK F RONT S MOOTHING A crack has some analogies with a crystal dislocation (Pook 2002a). In particular, the elastic stress ﬁelds associated with a crack front and with a dislocation are both singularities. The associated energy means that a dislocation has a line tension, which controls its shape under an applied stress ﬁeld (Cottrell 1964). Similarly, a crack front may be regarded as having a line tension which controls its shape, but with the important difference that the motion of a crack front is irreversible; that is a crack can propagate, but in general cannot contract. The line tension concept explains why, on a macroscopic scale, a fatigue crack front is smooth and any initial sharp corners rapidly disappear as the crack propagates. Overall, an initially irregular Mode I crack rapidly becomes convex, as shown by the dashed line in Figure A.35: at a re-entrant region (x on the ﬁgure) the Mode I stress intensity factor, KI , is much higher than elsewhere on the crack front, leading to rapid fatigue crack propagation towards a convex shape. Under fatigue loadings, stress intensity factors for initially irregular cracks may be approximated by ﬁrst enclosing them by a convex outline, as in the ﬁgure, and then using the methods in the previous section. A.5.4 U SE OF C RACK A REA For irregular cracks that are small compared with other dimensions it is possible to use the crack area, A, as a characteristic crack dimension (Chang 1982, Murakami 2002). A is calculated after projection onto a plane, followed by crack front smoothing (see Sections A.5.2 and A.5.3). For very slender cracks the crack length is truncated to 10 times the width before calculating A.

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Figure A.36. Interaction between two semi circular surface cracks. (a) No interaction. (b) Imaginary third crack inserted.

For an internal crack (Figure A.29 centre) the maximum Mode I stress intensity factor along the crack front, KI , is given approximately by √ (A.24) KI ∼ = 0.5σ π A , where σ is the stress perpendicular to the crack. The equation applies to cracks whose length is up to about 5 times the width. For a surface crack √ π A. (A.25) 0.65σ KI ∼ = For a very shallow surface crack the crack surface length is truncated at 10 times the crack depth before calculating A, and for a deep surface crack the crack depth is truncated at 2.5 times the crack surface length. For a very shallow surface crack Equation (A.25) becomes √ (A.26) KI ∼ = 1.16σ π a , where a is crack depth, and it is close to Equation (A.5), which is the equivalent two dimensional solution. A.5.5 I NTERACTION BETWEEN C RACKS When two cracks are close to each other the interaction between them increases their stress intensity factors compared with those for isolated cracks. Unfortunately, this interaction effect cannot be expressed by a simple equation, partly because of the numerous possible conﬁgurations. Various approximations have been proposed for a wide range of conﬁgurations but these tend

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to be inconsistent, partly because different authors introduce different degrees of conservatism. The usual approach is ﬁrst to deﬁne the crack separation between two cracks below which interaction occurs, and then in some way to deﬁne an equivalent single crack for which stress intensity factors are calculated. For example, the following rules are suggested for two adjacent semi circular surface cracks of different sizes (Murakami 2002). If there is enough space between the cracks to insert an additional crack of the same size as the smaller crack (Figure A.36(a)) then the interaction effect is negligibly small, and A in Equation (A.25) is taken as A1 . If the space between the two cracks is too small to insert a crack of the same size as the smaller crack then under fatigue loading the cracks coalesce rapidly. An imaginary semi circular crack is inserted between the two cracks and the areas of all three cracks are summed. That is insert in Equation (A.25) A = A1 + A2 + A3 (Figure A.36(b)).

B Random Load Theory and RMS

Notation A separate notation is included because many of the symbols listed are used only in this appendix. a, b f G(f ) H H1/3 I m0 , m2 , m4 N P (H1/3) P (S) P (S/σ ) p(S) p(S/σ ) R(τ ) S s ¯ S/σ Sc /σ Sm So T t γ

constants in two parameter Weibull distribution frequency power spectral density wave height signiﬁcant wave height irregularity factor moments of spectral density function number of cycles, return period exceedance of H1/3 exceedance of S exceedance of S/s probability density of S probability density of S/σ autocorrelation function random process instantaneous value of S expected value of S/σ clipping ratio mean value of S value of S below which peaks are omitted total time, wave period time Euler’s constant = 0.5772 . . .

204

ε σ σp σp,c σp,o σp,t σr σ2 τ φ

Appendix B: Random Load Theory and RMS

spectral bandwidth standard deviation (random process theory), root mean square (fatigue) root mean square of peaks root mean square of peaks after clipping of high peaks root mean square of peaks after omission of low peaks root mean square of peaks after truncation of high peaks root mean square of ranges variance time interval root mean square (random process theory)

B.1 Introduction In this appendix the application of random process theory (Papoulis 1965, Bendat and Piersol 2000) to fatigue loading is discussed. From an engineering viewpoint, it might appear that some of the points made are unimportant and pedantic. However, lack of attention to detail can result in difﬁculty in interpreting fatigue test data. In reporting random loading fatigue data it is important that the precise conventions used in calculations be clearly stated. No one set of conventions is of universal applicability. Some equations and ﬁgures which appear in the main text are repeated in order to make the appendix self contained.

B.2 Basic Deﬁnitions B.2.1 R ANDOM P ROCESS T HEORY Figure B.1(a) shows a random process in which load is plotted against time. This may be described by the function S(t), where S is a random process and t is time. In metal fatigue S will be a quantity such as stress or load. Assume that S(t) is statistically stationary and ergodic. Stationary means that statistical parameters characterising the process are independent of time. Ergodic means, broadly, that different samples of the same process yield the same values for statistical parameters. Only stationary random processes can be ergodic, and in practice most are. Considering the time interval 0 to T the mean value of S, Sm is given by 1 T S(t) dt (B.1) Sm = lim(T → ∞) T 0

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Figure B.1. Broad band random process, irregularity factor 0.410, spectral bandwidth 0.912. (a) Time history. (b) Spectral density function (Pook 1987). Reproduced under the terms of the Click-Use Licence.

and the mean square value φ 2 by 1 T 2 2 S (t) dt. φ = lim(T → ∞) T 0

(B.2)

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Hence the root mean square (RMS) value, φ, is given by 1 T 2 φ = lim(T → ∞) S (t) dt . T 0

(B.3)

The positive square root is understood in Equation (B.3) and subsequent equations. The RMS can equally well be calculated for periodic processes such as a sine wave. The use of RMS ﬁrst became popular in electrical engineering because it can be used directly in calculations involving power. For convenience it is sometimes used in metal fatigue (Pook 1987a). Carrying out calculations from the mean rather than from zero gives the variance, σ 2 , where 1 T {S(t) − Sm }2 dt (B.4) σ 2 = lim(T → ∞) T 0 and the standard deviation, σ , is given by 1 T {S(t) − Sm }2 dt . (B.5) σ = lim(T → ∞) T 0 The quantities given by Equations (B.1)–(B.5) are related through the expression φ 2 = σ 2 + Sm2 .

(B.6)

Hence, for zero mean the RMS and standard deviation are numerically equal. Instantaneous values of S(t) may be characterized by probability distribution functions. The exceedance, P (S), is the probability that a value exceeds S. The cumulative probability, 1 − P (S), is the proportion of values up to S. The probability density, p(S), is the derivative of the cumulative probability. For convenience, S is often normalised by σ . The instantaneous values of many ‘naturally occurring’ random processes are statistically stationary, at least in the short term, and approximate to the Gaussian distribution (or Normal distribution), which theoretically extends from −∞ to +∞. The probability density of a Gaussian distribution (Figure B.2(a)) for a process with zero mean is given by 2 −S 1 S = √ exp (B.7) p σ 2σ 2 2π and the exceedance (Figure B.2(b)) by ∞ 2 −S S S =√ . exp d P 2 σ 2σ σ 2π S/σ

(B.8)

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Figure B.2. Gaussian distribution. (a) Probability density. (b) Exceedance (Frost et al. 1974).

This integral does not have an explicit solution. The values shown in Figure B.2(b) are for the positive half of a Gaussian distribution and are therefore twice those given by Equation (B.8). P (S/s) is the area under the curve of p(S/σ ) between (S/σ ) and inﬁnity, as indicated by the shaded area in Figure B.2(a). B.2.2 FATIGUE L OADING Conventions used in the metal fatigue literature sometimes differ from those used in random process theory. The random processes encountered in metal fatigue are usually symmetrical, in a statistical sense, about the mean value, Sm , and all calculations are then carried out using values of S measured from Sm . Mathematically, this is equivalent to treating only cases where Sm is zero. It follows that there is no numerical difference between root mean square (RMS) and standard deviation, and in metal fatigue the term standard devi-

208

Appendix B: Random Load Theory and RMS

ation is not normally used in the characterisation of random processes. (Alternatively it might be said that RMS is used where standard deviation is meant.) This convention is used in what follows unless the context indicates otherwise. In particular, references to zero are understood to include the mean value of a process with non zero mean. Retaining the symbol σ for what was called standard deviation and is now called RMS Equation (B.5) becomes T 1 σ = lim(T → ∞) S 2 (t) dt . (B.9) T 0 The apparent lack of rigour is justiﬁed because RMS, as given by Equation (B.9), has no physical signiﬁcance in metal fatigue (see Section 4.3.3).

B.3 Some Sinusoidal Processes B.3.1 NARROW BAND R ANDOM L OADING In general a narrow band random process (Figure B.3) results when a random input is applied to a sharply tuned resonant system (Papoulis 1965, Pook 1983b, 1984, Bendat and Piersol 2000). Individual sinusoidal cycles appear whose frequency corresponds to the centre frequency of the resonant system. They have a slowly varying random amplitude. The probability density function for the occurrence of a positive peak of amplitude S (Figure B.4(a)) is given by the Rayleigh distribution 2 S −S S = exp . (B.10) p σ σ 2σ 2 As the process is statistically symmetrical, corresponding negative peaks also appear. The exceedance (Figure B.4(b)) is given by 2 −S S = exp . (B.11) P σ σ2 Equations (B.10) and (B.11) become exact only as the bandwidth tends to zero (see Section 4.2.2). Used in its general sense Rayleigh distribution does not imply the existence of a corresponding narrow band random process, and parameters in Equations (B.10) and (B.11) may differ. A narrow band random process is Gaussian, so instantaneous values do follow the Gaussian distribution (Equations (B.7) and (B.8)). Conventionally, in discussion of the Rayleigh and related distributions, only positive peaks are described and shown in diagrams such as Figures A2.4, it being understood that the negative peaks, with due attention to sign,

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Figure B.3. Narrow band random process, frequency ≈100 Hz, irregularity factor ≈0.99 (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

Figure B.4. Rayleigh distribution. (a) Probability density. (b) Exceedance (Frost et al. 1974).

are also included. Negative peaks are sometimes called troughs. Theoretically the Rayleigh distribution extends to inﬁnity, but in practice peaks do not exceed a cut off value of S/σ , known as the clipping ratio. Clipping implies that

210

Appendix B: Random Load Theory and RMS

higher peaks are reduced to the level given by the clipping ratio; truncation that they are omitted altogether. The clipping ratio does not usually exceed four or ﬁve. As fatigue damage depends on the peak values of cycles, and is largely independent of waveform (see Section 3.2), the root mean square (RMS) value of √ peaks, σp , is sometimes used. For narrow band random loading: peaks, σp = 2 σ . The RMS of the ranges between positive and negative √ σr , is also in use for narrow band random loading, σr = 2σp = 2 2 σ . B.3.2 T WO PARAMETER W EIBULL D ISTRIBUTION The two parameter form of the Weibull distribution has a variety of engineering applications; some of its general properties are discussed by Lipson and Sheth (1975). For metal fatigue purposes it is convenient to write its exceedance in the form (Pook 1984)

a S −b S , (B.12) P = exp σ a σ where a and b are adjustable constants (parameters) used to ﬁt the equation as needed. A functional relationship between b and a can be obtained by assuming that Equation (B.12) gives the distribution of peaks of a sinusoidal process which is symmetrical about zero. There is no closed form relationship between b and a. Values of b for a in the range 0.5 to 3 are tabulated in Pook (1984). The expression b = (1 − 0.076(a 2 − 3a + 2)

(B.13)

provides a satisfactory ﬁt for a in the range 0.71 to 2.36. Putting a = 2, b = 1 and a = 1, b = 1 gives as special cases the Rayleigh distribution (Equation (B.11)) and the Laplace distribution (or Exponential distribution) −S S S = exp =p . (B.14) P σ σ σ Exceedances for a range of a values are shown in Figure B.5. A logarithmic scale is used for exceedances in order to emphasise detail at low values. As a result the curve for the Rayleigh distribution has a different appearance from Figure B.4(b) where a linear scale is used. The peaks of the C/12/20 load history, shown in Figure 4.9, can be ﬁtted approximately by the two parameter Weibull equation with a = 1.2715 (Pook 1987a). Differentiating Equation (B.12) gives the probability density of the two parameter Weibull distribution as

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211

Figure B.5. Exceedances for the two parameter Weibull distribution (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

a a−1

b S S S −b = p exp . σ a σ a σ

(B.15)

If only the distribution of peaks is speciﬁed by a probability distribution, then in a computer generated process both the order in which peaks are applied and the waveform connecting them need to be speciﬁed. Usually, a negative peak is made arithmetically equal to the preceding positive peak. The value of σ depends on the waveform used to connect the peaks, but both σp and σr are independent of waveform. When peaks are connected by sine waves to √ √ give a sinusoidal process, σp = 2 σ and σr = 2 2 σ . In general, computer generated processes are do not follow the Gaussian distribution. However, for the special case of a sinusoidal process with a Rayleigh distribution of peaks,

212

Appendix B: Random Load Theory and RMS

and with constant frequency, the process is Gaussian, irrespective of the order in which peaks are applied. B.3.3 I NCOMPLETE D ISTRIBUTIONS AND P ROCESSES Distributions and processes encountered in metal fatigue practice are always incomplete in some way. In consequence root mean square (RMS) values differ from those of theoretical forms which extend to inﬁnity. Provided that a difference in RMS values is less than about 1 per cent it can usually be neglected. Examples illustrating some of the issues involved are given below. B.3.3.1 Truncated and Clipped Distributions The term clipping ratio is used to cover both truncation and clipping. This is because, in a physical system involving a narrow band random process, it is nonlinearities rather than clipping or truncation that limit the peaks which appear, and it may not be possible to maintain a clear distinction between truncation and clipping (see Section B.3.1). However, in a process generated by ﬁrst generating positive, and corresponding negative peaks, which follow some distribution, then joining the positive peaks and adjacent negative peaks with an appropriate waveform, a distinction has to be made to avoid ambiguity. In any process used for fatigue testing the maximum load applied has to be limited to meet physical limitations. If the distribution of peaks is known in terms of the root mean square (RMS) value of the process, then the RMS of the peaks, σp , is given by ∞ 2 S S S d . (B.16) p σp = σ σ σ 0 Hence, if the process is truncated at a clipping ratio, Sc /σ , the RMS of the truncated distribution of peaks, σp,t , is given by Sc /σ S 2 p S d S σ σ Sc σ , (B.17) σp,t = 1 − P 0 σ where the term {1 − P (Sc /σ )} corrects for the reduction in the total number of peaks, and σ is the RMS of the complete process. If the process is clipped, then the RMS of the clipped distribution of peaks, σp,c , is given by

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213

σp,c = (B.18) 2 Sc −σ 2 S S S Sc S S c c P d , + 1−P p σ σ σ σ σ σ 0 where the ﬁrst term under the square root sign represents the peaks that that have been reduced to Sc /σ . Clipping has less effect on RMS than truncation. Equations (B.17) and (B.18) may be used to calculate the change in the RMS of peaks due to truncation and clipping of different versions of the two parameter Weibull distribution (Equation (B.12)). Results show that for the Rayleigh distribution (a = 2 in Equation (B.12)) the effects are negligible when the clipping ratio exceeds about 3.5, and for the Laplace distribution (a = 1) when it exceeds about 8. Further terms appear in Equations (B.17) and (B.18) if positive and negative peaks are not truncated or clipped symmetrically. As an example of what can happen, consider the construction of a sinusoidal process, whose peaks follow the two parameter Weibull distribution, with a taken as 0.5. Assume that the complete process will be truncated to give a desired clipping ratio of 5. From Equation (B.17), taking S/σ = 5, σp,t = 0.7704σ , and the clipping ratio for the truncated process is 5/0.7704 = 6.490. For the clipping ratio of the truncated process to be 5, the clipping ratio applied to the complete process would have to be 3.243. For a process to remain sinusoidal, clipping has to be carried out correctly, as shown in Figure B.6. Form (a) is an original unclipped half cycle. Reducing instantaneous values of the process to the clipping ratio results in form (b), which is not sinusoidal. For the process to be sinusoidal the half cycle has to be reshaped, as in form (c). Truncation or clipping of a process that follows the Gaussian distribution renders it non Gaussian. However, it can reasonably be regarded as Gaussian if the percentage change in RMS is negligibly small. B.3.3.2 Omission of Low Loads Peaks below an omission level, σo , are sometimes omitted to reduce fatigue testing times, on the grounds that they cause negligible fatigue damage. The root mean square (RMS) value of the remaining peaks, σp,o, is given by ∞ S 2p S d S σ σ σ . (B.19) σp,o = P Sσo So /σ In practice, large numbers of cycles are omitted, so there is always a signiﬁcant effect on RMS. Omission is always combined with truncation or clipping.

214

Appendix B: Random Load Theory and RMS

Figure B.6. Clipping a half cycle. (a) Original half cycle. (b) Clipped. (c) Clipped and reshaped (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

B.3.3.3 One Sided Narrow Band Random Loading Tests are sometimes carried out using a modiﬁed narrow band random loading from which negative peaks have been removed, to give a one sided process (Sherratt and Edwards 1974). Three ways of doing this are shown, for a constant amplitude sinusoidal process, in Figure B.7. Part (a) of the ﬁgure shows the original process. In Figure B.7(b) the negative half cycles have been reduced to zero height, whereas in Figure B.7(c) they have been removed altogether. In Figure B.7(d) the negative peaks have been removed altogether and the positive peaks joined to zero by sine waves. Usually, root mean square (RMS) values are calculated for the original complete process. Parameters can be calculated for a one sided process, but different results are sometimes obtained for√the three methods. For example, mean values (Sm ) are σ/π , 2σ/π and σ/ 2 respectively, where σ is the RMS of the original complete process The RMS of ranges, σr , is the same for all three methods of removal, and √ is equal to the RMS of peaks, σp , for the original complete process, that is 2 σ , and would appear to be a good choice. The original complete process is Gaussian, but instantaneous values of a one sided process do not follow the Gaussian distribution (Equations (B.7) and (B.8)). B.3.3.4 Finite Random Processes Any practical random sinusoidal process must be of ﬁnite length and contain a ﬁnite number of cycles, N. For a pseudo random process, N is the return period after which it repeats exactly. One consequence is that the maximum peak size, and hence the clipping ratio, are restricted (see Sections B.3.1 and B.3.3.1). An intuitive approach is to set the exceedance, P (S/σ ), equal to 1/N and then take the corresponding value of S/σ from Equation (B.11)

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Figure B.7. One sided constant amplitude sinusoidal processes. (a) Original cycle. (b) Negative half cycles reduced to zero height. (c) Negative half cycles removed. (d) Negative half cycles removed and positive half cycles reshaped (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

as the clipping ratio. However, in narrow band random loading, large cycles ¯ , which will occur in groups and the expected maximum value of S/σ , S/σ

216

Appendix B: Random Load Theory and RMS Table B.1. COLOS 7 level load history. Level number 7 6 5 4 3 2 1

Number of cycles in level 1,000 4,000 40,000 180,000 575,000 1,250,000 2,950,000

RMS of level 4.07σ 3.46σ 2.90σ 2.27σ 1.68σ 1.10σ 0.426σ

be the expected clipping ratio, is somewhat less. It is given approximately by (Pook 1978) √ γ 1 S¯ ≈ ln N + , (B.20) σ 2 ln N where γ is Euler’s constant = 0.5772. . . . For example, for N = 105 , P (S/σ ) = 10−5 and from Equation (B.11) S/σ = 4.80, whereas Equa¯ = 3.48. tion (B.20) gives S/σ B.3.4 N ON S TATIONARY NARROW BAND R ANDOM L OADING In service, random loadings are usually statistically non stationary so that root mean square (RMS) values, and perhaps other parameters, are a slowly varying function of time. In the short term they can usually be regarded as statistically stationary. For a succession of narrow band random loadings whose RMSs follow the positive half of a Gaussian distribution (Equation (B.7)) the peaks sum to the Laplace distribution (Equation (B.14)) (Pook 1983b). Corresponding load histories for fatigue testing also need to be non stationary. A procedure was developed (Pook 1984) which made it possible to approximate a wide range of probability distributions as the sum of several Rayleigh distributions and hence produce load histories which consist of a sequence of narrow band random loadings. In one example a 7-level approximation of the Laplace distribution was used as the basis of an agreed standard load history known as the COmmon LOad Sequence (COLOS) (Anon. 1985). The numbers of cycles and load levels are listed in Table B.1 in terms of the of the overall RMS, σ . The water surface elevations of ocean waves are an example of a process which often approximates to a non stationary narrow band random process (Pook and Dover 1989). In oceanography the primary parameter used in the characterisation of sea state is the wave height, H , which is measured peak to trough. Over a period of time short enough (conventionally 20 min) for

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217 Table B.2. Scatter diagram for M V Famita for full year.

H1/3 m

Zero crossing wave period, s 7.5 8.5 9.5 10.5 11.5 8 0 0 0 0 40 4 0 0 0 78 24 2 0 0 95 33 7 1 0 72 41 8 2 0

0.3 0.91 1.52 2.13 2.74

4.5 14 64 18 6 0

5.5 40 159 103 53 19

6.5 34 135 164 126 103

12.5 0 0 0 0 0

13.5 0 0 0 0 0

3.35 3.96 4.47 5.18 5.79

0 0 0 0 0

9 1 0 0 0

46 23 6 5 1

71 63 20 13 9

31 38 31 15 4

3 6 10 12 6

1 1 2 1 3

0 0 1 0 0

0 0 0 0 0

0 0 0 0 0

6.40 7.01 7.62 8.23 8.84

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

2 0 1 1 0

2 1 1 2 0

4 3 0 0 2

2 7 4 0 2

0 0 0 0 0

0 0 2 0 0

2 0 0 0 0

9.45 0 Parts per 1924.

0

0

0

0

0

0

0

0

2

a sea state to be regarded as statistically stationary, the usual measure of its severity is the signiﬁcant wave height, H1/3 , which is the average height of the highest one third portion of the waves, and is approximately equal to 4σ (Sarpkaya and Isaacson 1981). One of the ways in which wave height data can be usefully presented is by means of a scatter diagram that gives the relative occurrences of sea states within speciﬁed small intervals of H1/3 and wave period, T , which is the reciprocal of the wave passing frequency (Pook 1987b). Table B.2 is an example of a scatter diagram obtained by the M V Famita (Holmes and Tickell 1975). The M V Famita is one of a number of ships that have been stationed in the North Sea to collect oceanographic data. Long term records show that variations in sea state have the appearance of a random process (Anon. 1985), but examination of data from four different sources showed that the distribution of H1/3 is not Gaussian (Pook 1987b). A detailed examination of some data for ﬁve years (Pook and Dover 1989) showed that the distribution of H1/3 was more accurately represented by the Gumbel distribution (Gumbel 1958). In its simplest form the exceedance, P (S), of the Gumbel distribution is given by P (S) = 1 − exp{− exp(−S)}.

(B.21)

218

Appendix B: Random Load Theory and RMS

For S > 4, P (S) ≈ exp(−S), and for S < 2, P (S) ≈ 1. Taking S as the signiﬁcant wave height, H1/3 , the sea state data were well ﬁtted by the expression

1.9 − H1/3 , (B.22) P (H1/3 ) = 1 − exp − exp 1.06 where P (H1/3) is the exceedance of H1/3.

B.4 Broad Band Random Loading In a broad band random loading (Figure B.1) individual cycles cannot be distinguished (see Section 4.3.3). When such a process is encountered in metal fatigue it is usually characterised by σ , that is the root mean square (RMS) value of the whole process. A measure of bandwidth is also required; a common one in metal fatigue is the irregularity factor, I , which is the ratio of mean crossings to peaks (see Section 4.3.3). It lies in the range 0 to 1. The irregularity factor has the advantages that is easily understood, and is not restricted to processes whose instantaneous values follow the Gaussian distribution. B.4.1 S PECTRAL D ENSITY F UNCTION In random process theory a process which is statistically stationary and ergodic, and whose instantaneous values follow the Gaussian distribution, is usually regarded as adequately described statistically if its root mean square (RMS) value and spectral density function (SDF) are known. Mathematically, the SDF is obtained by ﬁrst calculating the autocorrelation function, R(τ ). This describes the relationship between the values of the random process S(τ ) at times t and t + τ , and is given by (Papoulis 1965, Bendat and Piersol 2000) 1 T S(t)S(t + τ ) dt. (B.23) R(τ ) = lim(T → ∞) T 0 The SDF, G(f ), where f is frequency, is the Fourier transform of R(τ ), and is given by ∞ R(τ ) exp(−iπf τ ) dτ G(f ) = 2 =4

−∞ ∞ −∞

R(τ ) cos(2πf τ ) dτ.

(B.24)

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219

Determining the SDF in this way is called transforming from the time domain to the frequency domain. The SDF is sometimes plotted on a logarithmic scale and sometimes on a linear scale. Figure B.1(b) shows the SDF for the broad band random loading shown in Figure B.1(a). In practice it is calculated using an algorithm known as the Fast Fourier Transform (FFT) (Bendat and Piersol 2000). Physically, the SDF gives the frequency content of the random process, and in the narrow band random process case is sharply peaked at the centre frequency (resonant frequency). An alternative method of determining the SDF is to pass the process of interest through a bandpass ﬁlter of very narrow bandwidth and plot the amplitude of the resulting signal against frequency. The area under the PSD is equal to the mean square value of the process, as given by Equation (B.2). In electrical engineering the SDF is usually known as the power spectral density (PSD) because it provides a measure of the electrical power, which may be ascribed to the various frequency components. Some useful results depend only on the spectral bandwidth, ε, which is a measure of the RMS width of the SDF (Pook 1978, Bendat and Piersol 2000). It lies in the range 0 to 1 and is given by m22 , (B.25) ε = 1− m0 m4 where m0 , m2 and m4 are the zeroth, second and fourth moments of the SDF about the origin. It is related to the irregularity factor by ε2 = 1 − I 2.

(B.26)

As ε → 0 the distribution of peaks tends to the Rayleigh distribution (Equations (B.10) and (B.11)) and as ε → 1 to the Gaussian distribution (Equations (B.7) and (B.8)). There is no generally accepted deﬁnition of what is meant by narrow band, partly because of the physical difﬁculties of measuring bandwidth as ε → 0. In metal fatigue a random process is usually called narrow band if the peak distribution approximates to the Rayleigh distribution; this is generally so, provided that I ≥ 0.99, corresponding to ε ≤ 0.14 (see Section 4.2.2). Difﬁculties in determining the irregularity factor for narrow band random process are illustrated by the example shown in Figure B.3. Inevitably, only a ﬁnite length process can be examined, so decisions are needed on how to deal with the beginning and end of the process. Also, the mean value of the process has to be determined. The horizontal line in Figure B.3 is intended to be the mean value. In the ﬁgure there are 44 upward going zero crossings of this line and 45 positive peaks, giving an irregularity factor of 44/45 ≈ 0.98.

220

Appendix B: Random Load Theory and RMS

Figure B.8. Spectral density functions for a 0.76 m diameter horizontal member immersed 10.8 m, signiﬁcant wave height 4.75 m. (a) Water surface elevation; (b) bending stress; (c) axial stress (Pook 1989b). Reproduced under the terms of the Click-Use Licence.

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221

It could be argued that the mean crossing at the end of the process should not be counted, giving an irregularity factor of 43/45 ≈ 0.96. However, if the horizontal line were slightly lower there would be 45 upward going crossings, and the irregularity factor would be 45/45 = 1. What should be taken as the correct value is not easily resolved. In principle, the irregularity factor could be determined by ﬁrst ﬁnding the spectral bandwidth and then using Equation (B.26) but there are corresponding difﬁculties in determining the spectral bandwidth of a narrow band random process. As an example of the sort of information that can be derived from spectral density functions, Figure B.8 shows data for a tubular welded tall platform in the North Sea (Pook 1989b). A 0.76 m diameter horizontal member, immersed 10.8 m, was strain gauged so that bending and axial stresses could be derived. In practice, although sea states have a dominant wave passing frequency, they are not particularly narrow band so energy may be available to excite structural resonances (Pook 1987b). The SDF for the water surface elevation (Figure B.8(a)) shows this. There is a clearly deﬁned peak corresponding to the dominant wave passing frequency, but there is signiﬁcant energy at other frequencies. To avoid structural resonances, offshore platforms are designed so that resonant frequencies are substantially greater than the dominant wave passing frequency. This has been successful for the axial stresses since, as might be expected, the SDF (Figure B.8(b)) is of similar form, with no structural resonances exited. However, the SDF for the bending stress (Figure B.8(c)) does show two peaks corresponding to structural resonances. The point of collecting data of this sort is to permit comparison of actual structural behaviour with theoretical calculations.

C Non Destructive Testing

C.1 Introduction Non destructive testing (NDT) is not a clearly deﬁned concept (Halmshaw 1991). NDT has a wide range of applications in the detection and evaluation of ﬂaws in materials. Many different methods are used, and a wide range of commercially available instruments has been developed. These are often automated under computer control. The key feature of NDT is that it has no deleterious effect on the item tested. In the context of metal fatigue the usual meaning of non destructive testing is the detection and sizing of cracks and crack like ﬂaws in components, structures and laboratory specimens. This includes monitoring of fatigue crack propagation in service and in laboratory specimens; the advantages and disadvantages of various methods are summarised by Richards (1980). The accuracy of crack sizing that can be achieved varies widely. Some of the non destructive testing techniques used in metal fatigue work are described brieﬂy in this appendix, together with the important statistical concepts of probability of detection and probability of sizing. In order to make the appendix more self contained some ﬁgures in the main text are repeated here.

C.2 Visual Inspection Visual inspection is the simplest method of detecting surface cracks, usually called surface breaking cracks in the non destructive testing literature. This term is used in this appendix. The utility and importance of visual inspection are often underestimated. Under good conditions fatigue cracks with a surface length of 3 mm can be detected by the naked eye, but in general 25 mm is a

224

Appendix C: Non Destructive Testing

Figure C.1. Fatigue cracks in an aircraft engine nacelle.

more realistic detection limit. A low power lens (say ×3) and additional portable lighting are useful. For a permanent record photographs may be taken or replicas of the surface made. If the surface is irregular, as in welds, surface breaking cracks are difﬁcult to detect by visual methods. Regulatory authorities often call for periodic visual inspection of structures for defects, including cracks. For example, Figure C.1 shows unexpected fatigue cracking found in an aircraft engine nacelle during a routine inspection (Pook 2004). Another example is the cracking in a burner from a domestic central heating boiler shown in Figure 8.13. Routine visual inspection is tedious, and fatigue cracks are sometimes missed. For example, one of the concerns at the ofﬁcial inquiry into the catastrophic fatigue failure of a fairground ride was why fatigue cracks, which should have been detected, were missed during routine visual inspections (Pook 1998). When fatigue crack propagation is being monitored visually, crack length measurement is often aided by markings etched or scribed onto the specimen surface, for example the grid shown in Figure C.2 (see Section 8.2.2). Visual methods have been widely used to collect data during fatigue crack propagation rate tests, scribed marks were used to collect the data shown in Figure C.3. The use of guide markings does not meet resolution accuracy requirements in modern fatigue crack propagation rate testing standards such as Anon. (2003b). A common technique, which does meet the requirements, is to use a micrometer thread travelling microscope with a magniﬁcation of ×20 to ×50. Visual methods of inspection have the advantage that the equipment needed is relatively inexpensive, but they are labour intensive when used to monitor fatigue crack propagation, and are not amenable to automation. The

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225

Figure C.2. Fatigue crack path in a Waspaloy sheet under biaxial fatigue load. The grid is 0.1 inch (2.54 mm). National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

Figure C.3. Fatigue crack propagation curve for a central crack, length 2a, in a 0.76 m wide × 2.5 mm thick mild steel specimen. Nominal stress 108 ± 31 MPa (Frost et al. 1974).

major disadvantage is that only crack surface lengths can be measured. In a plate of constant thickness a fatigue crack front of a through the thickness crack is often curved (Figure C.4). This curvature can affect the calculation of stress intensity factors (see Section A.3.2). With a surface breaking crack

226

Appendix C: Non Destructive Testing

Figure C.4. Fracture surface of 19 mm thick aluminium alloy fracture toughness test specimen (Pook 1968). Reproduced under the terms of the Click-Use Licence.

Figure C.5. Uniform alternating current on the surface of a plate containing a surface breaking crack.

(Figure C.5) it would not be possible to calculate stress intensity factors because these are largely dependent on crack depth (see Section A.3.1.2). In practice, the major use of visual inspection is to detect surface breaking cracks. Any cracks found are then sized using an appropriate technique such as ultrasonics (see Section C.6) or alternating current potential drop (see Section C.7.2).

C.3 Magnetic Particle Inspection Magnetic particle inspection (MPI) is a well established technique for the detection of surface breaking cracks. The method can be used only on ferromagnetic materials which can be strongly magnetised. These include irons

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227

Figure C.6. Principle of magnetic particle inspection.

and ferritic steels, but not all steels. Magnetic effects arise through electromagnetic ﬁelds. These can be represented as lines of magnetic force through space which form a magnetic ﬂux. The principal of magnetic particle inspection is shown in Figure C.6. A magnetic ﬂux is established in the material by placing the poles of a magnet (usually an electromagnet) in contact with the material. If the magnetic ﬂux encounters a transverse surface breaking crack the ﬂux becomes distorted. Some of the magnetic ﬂux passes through the crack, some passes around the crack tip, and some leakage ﬂux passes around the crack at the surface. This leakage ﬂux attracts ferromagnetic particles to the crack mouth, and the resulting visible concentration of particles marks the crack. The magnetic particles are applied as a suspension in a carrier liquid, such as light oil or water, at a concentration by volume of about 2 per cent. If water is used, a wetting agent and a corrosion inhibitor are incorporated. The suspension is normally supplied in an aerosol, and is sometimes called a magnetic ink. The particles are usually black iron oxide of around 1–25 µm in size, and may be dyed to improve visibility. Florescent dyes are sometimes used, and the particles are then viewed under ultra violet light Magnetic particle inspection is the most widely used non destructing testing method for detecting surface breaking cracks in welded joints. MPI is easily carried out using portable equipment, but expertise is needed for satisfactory results, and it can be a messy procedure. The major disadvantages are that only the surface length of a crack can be determined, and the accuracy of crack sizing is low. An advantage is that, with special equipment, magnetic particle inspection can be used under water.

228

Appendix C: Non Destructive Testing

Figure C.7. Schematic view of dye penetrant in crack after removal of excess penetrant from the surface.

C.4 Dye Penetrant The dye penetrant method is used to detect surface breaking cracks. The method can be applied to any material that has a non absorbent surface. Most of the cracks found by dye penetrants can be seen visually in good conditions, but dye penetrants make them much easier to detect. The principle of the method is shown in Figure C.7). After the surface has been cleaned a penetrant, which contains a dye in solution, is applied. The penetrant is chosen so that it wets the material being inspected, and it is drawn into cracks by capillary action. Excess penetrant is then removed from the surface, and a thin layer of a porous developer is applied. Penetrant is drawn out of cracks by the developer, thus making cracks visible. The dye and developer colours are chosen to provide good contrast. Penetrant dyes and developers are usually supplied in aerosols. A wide range of techniques is available, and for good results the technique chosen must be carefully matched to the intended application (Halmshaw 1991). Unfortunately, much published information on the results of dye penetrant non destructive testing is of little value because full details of techniques used are not included. The dye penetrant method is widely used for aluminium alloys and other metallic materials which cannot be magnetised so that magnetic particle inspection is impossible (see previous section). The main advantage of dye penetrant is that it is simple to use, and particularly suitable for ﬁeld work. The main disadvantage, as with visual inspection and magnetic particle, is that only the surface length of a crack can be determined (see Sections C.2 and C.3). If fatigue crack propagation is being monitored, a potential disadvantage is that dye penetrant remaining in a fatigue crack could affect its subsequent propagation behaviour.

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Figure C.8. Fatigue cracking from shrinkage cavity in 30 × 35 mm cast steel bar. National Engineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

Figure C.9. Schematic view of shrinkage cavity in 30 × 35 mm cast steel bar.

C.5 Radiography The main use of radiography in metal fatigue is the detection of voluminous defects including internal cavities, porosity and inclusions. An example is the shrinkage cavity in a cast steel bar shown in Figure C.8. This has an irregular boundary and can be regarded as a crack like ﬂaw. The general shape of the cavity was obtained by radiography, and it is shown schematically in Figure C.9 (Pook et al. 1981). The principle of radiography in its original form is shown in Figure C.10. X-rays are emitted from a small source, and travel in straight lines towards a sheet of photographic ﬁlm, which acts as a detector. The X-rays are partly absorbed as they pass through the specimen, and then strike the photographic ﬁlm to produce a radiograph. There is less absorption when the X-rays pass through a cavity, and a two dimensional image of the three dimensional cavity is formed on the ﬁlm. To get three dimensional information on the cavity,

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Figure C.10. Principle of radiography in its original form.

including its location within the specimen, radiographs have to be taken from more than one direction. This was done to produce the sketches shown in Figure C.9. As an alternative to using ﬁlm, X-rays can be captured electronically, and images displayed in real time on a monitor. X-rays are a form of electromagnetic radiation, similar to light, but with very much shorter wavelengths. They are produced when a beam of high energy electrons strikes a metal anode in a vacuum. Wavelengths of X-rays used in radiography range from about 10−4 nm to about 10 nm. Long wavelength X-rays are sometimes called soft X-rays and will penetrate only small distances. Short wavelength X-rays are sometimes called hard X-rays, and can penetrate up to about 50 cm thick steel. Gamma rays are sometimes used in radiography. They are also a form of electromagnetic radiation and are produced by the decay of a radioactive isotope such as cobalt-60. Radiography is a very versatile and easily used technique. It is probably the oldest non destructive testing technique used for the quality control of welded joints. Radiographs provide a convenient, permanent record. The major disadvantage of radiography is that stringent safety precautions have to be taken to protect operators, and the public at large, from radiation. From a metal fatigue viewpoint its major disadvantage is that it is difﬁcult to detect and size tight cracks, that is cracks whose opposite surfaces are either close together or touching.

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C.6 Ultrasonics In metal fatigue the main use of ultrasonics is crack sizing, and a wide range of techniques is available. The method is based on the propagation of sound waves through the material at frequencies above the audible range, hence the term ultrasonics. Sound waves are mechanical vibrations. Hence the velocity of propagation is different in different materials, and also depends on the type of wave. Frequencies used are of the order of a MHz, and the resulting wavelengths are of the order of a mm. The essential feature of the waves used in ultrasonics is that they propagate through the material in the same way that ocean waves move across the water surface. This contrasts with the standing waves observed in the vibrations of a tuning fork. Inadvertent standing waves can be a problem in the use of ultrasonics. Two main types of wave are used in ultrasonics. One is compressional waves, also known as longitudinal waves, where particles vibrate in the direction of wave propagation. The other is shear waves, also known as transverse waves, where vibration is at right angles to the propagation direction. Several other types of wave are used for special purposes (Halmshaw 1991). Ultrasonic waves used to interrogate the specimen under test are generated in pulses, not continuously. Ultrasonic wave pulses are generated by applying short electric pulses to a suitable probe. One type of probe uses a piezoelectric disc, which resonates at a selected frequency (Figure C.11). A couplant, such as thin layer of oil, is used to ensure good transmission of ultrasonic waves from the probe into the specimen under test. Ultrasonic waves emerging from the specimen are received by a suitably positioned probe, and processed to give information about defects within the specimen. The same probe is sometimes used for both transmission and reception. The arrangement for ultrasonic testing of a cracked specimen, using a compressional probe, is shown schematically in Figure C.11. Ultrasonic wave pulses are reﬂected from the crack, and also from the top and bottom surfaces of the specimen. These echoes are displayed on an oscilloscope using what is known as an A-scan, shown schematically in Figure C.12. The time base of the oscilloscope is triggered as each ultrasonic pulse is transmitted. Hence, the positions of the echoes on the time axis provide information on the crack location. In practice, dispersion effects within the specimen mean that subsidiary echoes also appear. These subsidiary echoes can complicate interpretation of the display. Other methods of display may be used when a probe is scanned across a specimen surface. A B-scan is obtained from a scan along a line on the

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Figure C.11. Arrangement for ultrasonic testing of a cracked specimen using a compressional probe.

Figure C.12. A-scan showing crack.

surface. The display is arranged so that it shows the sizes and positions of ﬂaws on a cross section perpendicular to the surface of the specimen. A Cscan produces a radiograph like display (see previous section). It is obtained from a series of scans along parallel lines, and the display shows a plan view of the specimen in which defects appear in their correct positions, but with no information on their through the thickness locations. A D-scan is similar to a B-scan, but is obtained from a series of scans along parallel lines. The terms A-scan, etc., are often used in the ultrasonics literature without explanation. Various ultrasonic techniques are used to size cracks and crack like ﬂaws. Two of these, which are used for surface breaking cracks, are shown schematically in Figures C.13 and C.14. In the end on technique a high intensity compressional probe, located at the opposite surface, is used to ﬁnd the tip

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Figure C.13. End on technique for measuring the depth of a surface crack.

of the crack (Figure C.13). The depth of the crack can then be determined directly from an A-scan. A different approach is used in the time of ﬂight diffraction (TOFD) technique, shown schematically in Figure A.14. In this technique there are separate transmitter and receiver compressional probes. The principle is that when a compressional ultrasonic beam meets a crack tip some of the energy is diffracted. The diffracted waves spread over a large angular range, and may be detected by a suitably placed receiver probe. If the transmitter and receive probes are symmetrically placed about the crack tip, then a simple calculation gives the position of the crack tip, and hence the crack depth. To ensure symmetry about the crack tip the two probes may be linked mechanically, and traversed across the crack. The probes are symmetrically positioned when the time of ﬂight is minimised. Shear waves are sometimes used in the TOFD technique; these are more suitable for deeper cracks. Both the end on technique and the TOFD technique are compatible with methods of approximation of stress intensity factors since, in effect, an oblique crack is projected onto a plane perpendicular to the surface (see Section A.5.2). Simple theory suggests that, for a ﬂaw of a given size, the height of the echo on an A-scan is inversely proportional to the square of the distance of the ﬂaw from the probe. A distance amplitude correction curve, usually known as a DAC curve, is used to correct for this effect. The term DAC level refers to the heights of echoes, relative to background noise, that are regarded as signiﬁcant. (Background noise is known as grass, because of its appearance on an oscilloscope screen.) Thus, 50 per cent DAC (level) means that only

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Figure C.14. Time of ﬂight diffraction technique for measuring the depth of a surface crack.

echo heights that are at least 50 per cent greater than the height of the grass are considered signiﬁcant. The choice of DAC level is important when the probability of detection and probability of sizing are being determined (Visser 2002) (see Section C.8). Ultrasonics is a very versatile and well established method of crack sizing, and a wide range of techniques is available. Technique details are readily adaptable to speciﬁc applications. The major disadvantages of ultrasonics are cost and that it is not suitable for small, thin specimens. It is also difﬁcult to use on austenitic steels.

C.7 Electromagnetic Fields In metal fatigue electromagnetic ﬁeld methods are used both for crack detection and for crack sizing. They are based on the injection of a uniform alternating current ﬁeld into the surface of a specimen, such as the plate shown in Figure C.5. Eddy current and alternating current potential drop (ACPD) methods are usually regarded as distinct methods of non destructive testing, but they are actually limiting cases of general electromagnetic ﬁeld methods . Due to the skin effect the alternating current density is greatest at the surface, and decreases exponentially with depth below the surface. The skin depth, δ, is usually deﬁned as the depth at which the alternating current density is 1/e (36.8%) of its surface value, and this depth is given by (Lewis et al. 1988) δ=√

1 , π µσf

(C.1)

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where µ is magnetic permeability, σ is electrical conductivity, and f is the frequency of the alternating current. Some typical skin depths shown in Table C.1. Magnetic permeability is sensitive to precise material composition, and it is also a function of current density, Hence data in the table should only be used as a guide to the selection of an appropriate frequency. Table C.1. Typical skin depths. Material 18/8 stainless steel Brass Aluminium Copper Mild steel

1 kHz 13.1 mm 3.90 mm 2.65 mm 2.00 mm 0.148 mm

Frequency 10 kHz 100 kHz 4.14 mm 1.31 mm 1.23 mm 0.390 mm 0.838 mm 0.265 mm 0.632 mm 0.200 mm 0.047 mm 0.015 mm

1 MHz 0.414 mm 0.123 mm 0.084 mm 0.063 mm 0.005 mm

For satisfactory results the skin depth must be small compared with the crack depth. Typically δ is about 0.1 mm so, in general, electromagnetic ﬁeld methods cannot be used for cracks less than about 1 mm deep. The response of a crack being interrogated by an alternating current depends on the value of the dimensionless parameter m, which is given by (Lewis et al. 1988) µ0 a , (C.2) m= µδ where µ0 is the magnetic permeability of a vacuum (= 4π ×10−7 H m−1 ). For non magnetic materials of high electrical conductivity, such as aluminium, µ ≈ µ0 , m is large because δ/a is small, and eddy current testing is appropriate. For magnetic materials, such as ferritic steels, µ µ0 , m is small, and ACPD testing is appropriate. Typical values of m are 12 for aluminium and 0.6 for mild steel (Lewis et al. 1988). C.7.1 E DDY C URRENT In metal fatigue the main uses of eddy current testing are the detection and sizing of cracks in non magnetic materials, especially aluminium alloys. Frequencies of the order of one MHz are used in order to ensure a small skin depth (Table C.1). The principle of eddy current testing is shown schematically in Figure C.15. A probe with a current carrying coil is scanned across the specimen at a small ﬁxed lift off distance. The alternating current in the coil produces

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Figure C.15. Principle of eddy current testing of a cracked specimen.

an alternating magnetic ﬂux, which induces eddy currents in the specimen. These eddy currents are sometimes called Foucault currents. The induced eddy currents in turn produce an alternating magnetic ﬂux. This opposes the ﬂux produced by the current carrying coil, and changes the impedance of the coil. As the probe passes over a crack the eddy currents are distorted and the presence of a crack is detected, using suitable instrumentation, by changes in the coil impedance. In some systems the effect of the magnetic ﬂux produced by the eddy currents is monitored by voltages induced in a second coil, similar to the current carrying coil. An important practical advantage of eddy current testing is that physical contact between the probe and the specimen is not necessary. Eddy current testing is a very versatile method of crack detection and sizing, and a wide range of techniques is available. It is possible to detect cracks as small as 10 µm deep. However, for good results the technique and instrumentation chosen must be carefully matched to the intended application. Its major disadvantages are cost and, at times, interpretational difﬁculties. It is possible to monitor fatigue crack propagation by using a system in which the probe is traversed by a motor, and locks onto the crack tip. When used on magnetic materials, eddy current testing is sometimes called alternating current ﬁeld measurement (Lewis et al. 1988). Lower frequencies are used, and methods of data reduction are different. C.7.2 A LTERNATING C URRENT P OTENTIAL D ROP In metal fatigue the main uses of alternating current potential drop (ACPD) techniques are the detection and sizing of cracks in magnetic materials, espe-

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Figure C.16. Principle of alternating current potential drop testing of a cracked specimen.

cially steel. Frequencies of the order of around 1–10 MHz are used in order to ensure a small skin depth (Table C.1). The principle of ACPD testing is shown in Figure C.16. The current is impressed through two contacts some distance apart. It ﬂows along the specimen skin from one contact to the other, passing down one side of the crack and up the other. The voltage (potential drop) is measured using a probe with known contact spacing, , placed across the crack. In automatic systems for monitoring fatigue crack propagation, current impression and potential drop measurement measuring wires are spot welded to the specimen (Austin 1999). Calibration is straightforward. A voltage, V1 , is ﬁrst measured by placing the probe near the crack, as shown in Figure C.17 (left). The voltage, V2 , across the crack is then measured (Figure C.17 (centre)). Assuming that the current is constant for the two measurements, then the crack depth, a, is given by V2 . (C.3) −1 a= V1 2 This one dimensional solution has to be modiﬁed for a two dimensional crack, such as that shown in Figure C.5. Modiﬁers are available for various conﬁgurations. Because the technique is self calibrating, the impressed current ﬁeld does not have to be completely uniform. This makes the technique suitable for irregularly shaped specimens such as welded joints. The technique is particularly suitable for automatic collection of fatigue crack propagation data during structural fatigue tests. For fatigue crack propagation rate testing on

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Figure C.17. Alternating current potential drop calibration.

plate specimens, in which a crack front may be curved (Figure C.4), contact points can be arranged so as to measure an average crack length through the thickness. The main advantages of the ACPD method are its versatility and its ease of calibration. Voltages are low, so no safety precautions are needed, and the method can be used under water. Care has to be taken in arranging the impressed current and probe leads so as to avoid spurious interactions, and also interference in electrically noisy environments. Crack bridging by metallic particles is not usually a serious problem. Various systems are available commercially. Techniques are still evolving. It is moderately expensive. A disadvantage is that for an oblique crack the method gives the crack length, not the depth of the crack tip below the surface (Figure C.17 right). The ACPD method is therefore not compatible with methods of approximation of stress intensity factors where an oblique crack is projected onto a plane (see Section A.5.2). In effect, the projected crack length is overestimated, as is the concomitant approximated stress intensity factor.

C.8 Probability of Detection Traditionally, it has been assumed that a particular non destructive testing technique is capable of detecting all cracks larger than a critical size, and that no cracks smaller than the critical size will be detected. What is meant by crack size depends on the application; for a surface breaking crack (Figure C.5) this is usually either the surface length or the maximum depth. In practice a critical crack size is not clearly deﬁned, and the probability of detection (POD) increases with the crack size as shown schematically in Figure C.18, which also shows the ideal situation where a critical crack size is clearly deﬁned.

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Figure C.18. Actual and ideal probability of detection (POD) curves.

Figure C.19. Schematic relative operating characteristic (ROC) curve.

Another possible outcome of an inspection is a false call where no crack is present but one is apparently detected. In a relative operating characteristic (ROC) curve the probability of detection is plotted against the false call probability (FCP) as shown schematically in Figure C.19. A good performance, shown by dashed lines on the ﬁgure, is deﬁned by Visser (2002) as a probability of detection of ay least 80 per cent combined with a false call probability of at most 20 per cent.

240

Appendix C: Non Destructive Testing Table C.2. Typical data obtained from a probability of detection trial. Crack size (mm) 2.2 2.5 3.0 3.4 3.8

Whether detected No No No Yes No

Crack size (mm) 8.8 9.5 10.7 11.5 13.4

Whether detected Yes No Yes Yes Yes

Crack size (mm) 21.5 21.7 21.8 22.2 22.4

Whether detected Yes Yes Yes No Yes

4.2 5.1 5.8 6.5 7.1

Yes Yes No Yes Yes

13.9 16.0 17.6 17.9 19.4

No Yes Yes No Yes

22.9 23.9 24.4 26.0 26.2

Yes Yes Yes Yes Yes

7.3 8.0

Yes No

19.7 19.9

Yes Yes

28.2 28.9

Yes Yes

C.8.1 D ETERMINATION OF P ROBABILITY OF D ETECTION Probability of detection (POD) curves are determined experimentally by blind trials on a set of specimens containing cracks of known sizes. The objective is to see what can be achieved by a skilled inspector using a particular non destructive testing technique, not to test the inspector. It is good practice to include some uncracked specimens so that data on false calls can be obtained. Sets of specimens kept for blind trials are sometimes called a library. Inspectors carrying out blind trials are not given any information on the cracks in the specimens, or on the results of the trials. There are two particular difﬁculties in carrying out POD trials. The ﬁrst is producing specimens with cracks of the desired size range. Cracks are usually produced by fatigue loading but fatigue cracks are, in general, difﬁcult to control. The second is ensuring that crack sizes, especially crack depths, are accurately known. The most satisfactory way of determining crack depths is to section specimens, but this destroys them so that they cannot be used for further trials. What is sometimes done is to size the cracks using an accurate method, such as time of ﬂight diffraction (see Section C.6) and, from time to time, section a few specimens in a library as a cross check. The sectioned specimens are replaced by new specimens containing similar cracks. Typical data obtained from a POD trial are shown in Table C.2. In order to obtain point estimates of POD, specimens are grouped according to crack size, as shown in Table C.3. It is good practice to have about the same number of groups as there are specimens in each group. The number of successful

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241 Table C.3. Point estimates of probability of detection (POD). Crack size range (mm) 0–5 5–10 10–15 15–20 20–25 25–30

Number in group 6 8 4 6 8 4

POD (per cent) 33.3 62.5 75 83.3 87.5 100

Figure C.20. Probability of detection point estimates.

detections in each group divided by the number of specimens in the group is a point estimate of POD, and is usually expressed as a percentage. The point estimates are plotted at the upper limit of each size range, and are usually joined by straight lines, as shown in Figure C.20. The probabilities of detection shown in Table C.3 are point estimates based on small samples of 4 to 8 cracks. However, point estimates obtained from small samples may not accurately reﬂect probabilities of detection obtained from large samples. Point estimates obtained from small samples are sometimes analysed statistically to provide lower bound estimates of probabilities of detection at, say, the 95 per cent conﬁdence level. However, the small sample sizes typically used in practice for cost reasons make rigorous statistical analysis difﬁcult, and lower bound estimates of probabilities of detection are not in general use (Visser 2002).

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Appendix C: Non Destructive Testing Table C.4. Typical data obtained from a probability of sizing trial. Actual crack size (mm) 1.5 1.9 2.3 2.7 3.1

Measured crack size (mm) Not detected 1.9 Not detected 3.7 Not detected

Actual crack size (mm) 13.1 13.5 14.7 16.1 16.2

Measured crack size (mm) 13.7 15.1 15.4 16.2 18.4

3.3 3.9 4.5 5.2 5.5

5.5 5.0 Not detected 4.7 7.9

16.5 17.7 17.9 18.1 18.3

17.3 19.7 17.8 19.9 Not detected

5.9 6.5 7.1 7.7 7.9

5.4 Not detected 7.4 9.6 Not detected

18.5 18.9 19.7 20.1 21.5

20.2 21.0 21.2 19.4 21.6

9.4 11.1 11.4

9.4 10.7 Not detected

21.6 23.3 23.9

22.0 23.6 26.1

Table C.5. Point estimates of probability of sizing (POS) using 90 per cent accuracy criterion compared with probability of detection (POD). Size range (mm) 0–5 5–10 10–15 15–20 20–25

Number in group 8 8 5 10 5

POS (per cent) 12.5 50 60 60 80

POD (per cent) 50 75 80 90 100

C.8.2 P ROBABILITY OF S IZING The probability of sizing (POS) is a reﬁnement of probability of detection in which a crack is counted as detected only if its size is measured to within an accuracy criterion. An accuracy criterion of x per cent means that the measured crack size is within ±(100 − x) per cent of the actual crack size. Table C.4 shows typical data obtained from a probability of sizing trial, and Table C.5 point estimates of probability of sizing, using a 90 per cent accuracy criterion. Point estimates of probability of detection are also shown in the

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Figure C.21. Comparison of probability of sizing (90 per cent accuracy criterion) with probability of detection.

table. These point estimates are plotted in Figure C.21; probabilities of sizing are signiﬁcantly lower than probabilities of detection.

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Index

K-calibration, 174 K-dominated region, 141, 144, 180, 192 S/N curve, 16 T -stress, 143 T -stress criterion, 143 T -stress ratio, 144 A-scan, 231 absolute liability, 78 acceptance testing, 76 accuracy criterion, 242 additive distribution, 31 Almen strip, 99 alternating current ﬁeld measurement, 236 alternating current potential drop, 226, 234, 236 alternating stress, 15, 16 analytical approach, 72, 75, 162 antisymmetric mode, 187, 190 aspect ratio, 153, 177 attractor, 143, 156 autocorrelation function, 218 autofrettage, 99 B-scan, 231 bandpass ﬁlter, 219 bandwidth, 40, 208, 218 basic situation, 69 Basquin equation, 18 bending mode, 174 biaxial fatigue loading, 37 biaxial loading, 37, 59 biaxiality ratio, 143 bimodal distribution, 31

blind trial, 240 block fatigue loading, 42, 127 block loading, 42 block programme, 53 boundary layer, 188, 193 branch crack, 138, 155, 198 branch crack formation, 140, 149 branch crack propagation, 140 branch crack propagation angle, 141, 148 branch point, 93 brittle fracture, 103, 110, 136, 138, 154 broad band random loading, 35, 50, 218, 219 C-scan, 232 carburising, 98 centre cracked tension specimen, 106 centre frequency, 208, 219 centre line average roughness, 84 chaos theory, 141 chaotic event, 141 characteristic crack dimension, 175, 199 civil liability, 78 cliff, 156 clipping, 209, 212, 213 clipping ratio, 42, 59, 209, 212, 214 code, 70, 73 code design, 70 COLOS, 216 compliance function, 174 compressional probe, 232 compressional wave, 231 conditional distribution, 32 constant amplitude loading, 15

Index

260 constant amplitude fatigue loading, 15, 39, 42, 96, 105, 106, 117, 120, 122, 124, 127 Consumer Protection Act, 78, 79 core region, 141, 180, 189 corner point, 151, 182, 187, 190, 191 corner point singularity, 152, 187 corner point singularity dominated region, 189, 192 corner region, 188 couplant, 231 crack area, 199 crack bridging, 238 crack closure, 114, 127 crack front, 168 crack front curvature, 138 crack front inclination angle, 191 crack front intersection angle, 152, 188, 189 crack front shape, 129 crack initiation dominated, 83, 88, 92, 98 crack mouth, 227 crack opening, 113 crack path prediction, 149, 151, 158 crack path stability, 142 crack propagation, 138, 172, 173, 186 crack propagation dominated, 92, 101 crack sizing, 223, 227, 231 crack surface displacement, 168, 170, 192 crack surface intersection angle, 187, 190, 191 crack tip, 168 crack tip plastic zone, 105, 112, 113, 125, 137, 150, 182 crack tip surface displacement, 169, 170, 183, 187, 190 cracked situation, 101 criminal liability, 77, 79 critical crack front intersection angle, 190, 191 critical length, 27 critical plane, 63 critical plane approach, 64 crystal dislocation, 170, 199 cumulative damage, 44 cumulative probability, 40, 206 cycle, 15 cycle counting, 50, 125 cyclic load, 1

D-scan, 232 DAC curve, 233 DAC level, 233 damage density curve, 49 damage density function, 49 damage tolerance, 71, 103 defective, 69, 77, 78, 80, 164 design defect, 69, 78 Det Norske Veritas, 132 detector, 229 developer, 228 differential geometry, 11, 155 diffracted wave, 233 directionally stable crack, 142 directionally unstable crack, 142 disclination, 171, 173 dislocation, 152, 171, 173, 199 dispersion effect, 231 distance amplitude correction curve, 233 dye penetrant, 228 Eastabrook’s theorem, 125 eddy current, 234, 235 edge sliding mode, 169 effective crack length, 182, 184 effective value, 113 effects of thickness, 110 electromagnetic ﬁeld method, 234 electromagnetic radiation, 230 end on technique, 232 endurance, 17 endurance limit, 20 enforcement authority, 79, 80 enforcement ofﬁcer, 79 equivalent Mode I crack, 197 equivalent constant amplitude stress range, 47 equivalent cycle, 50, 124 equivalent single crack, 201 equivalent stress, 63, 64, 89, 96 equivalent stress intensity factor, 159 equivalent tensile stress, 62 ergodic, 204, 218 escape clause, 75 exceedance, 40, 206, 208, 210, 217 exponential distribution, 210 facet, 156 fail safe, 70, 154

Metal Fatigue failure, 2, 4 failure analysis, 68, 83, 97, 102, 109, 122, 130, 162 failure mechanism map, 139 false call, 239 false call probability, 239 FALSTAFF, 54 Fast Fourier Transform, 219 fatigue, 7 fatigue assessment, 67, 69, 71–74, 83, 93, 97, 109, 117 fatigue crack growth, 26 fatigue crack initiation, 25, 26, 31, 83, 98, 132 fatigue crack path, 10, 75, 135, 138 fatigue crack propagation, 11, 26, 105, 109, 114, 117, 118, 122, 124, 128, 129, 135, 138, 155, 159, 164, 165, 167, 223, 224, 228, 236, 237 fatigue crack propagation life, 102, 122, 124 fatigue crack propagation rate, 105, 106, 109, 110, 114, 116, 119, 122, 124, 127, 133, 224, 237 fatigue cycle, 15, 105 fatigue design, 67, 109 fatigue fracture toughness, 104 fatigue life, 83, 122, 124, 130 fatigue limit, 18, 120 fatigue load, 1, 55 fatigue loading, 15, 38, 61, 94, 98, 138, 149, 154, 156, 199, 204, 240 fatigue strength, 90, 162 fatigue strength reduction factor, 90 fatigue testing, 8, 9 ﬁnal crack size, 101, 103, 122 ﬁsh eye, 29 ﬂame hardening, 98 Forsyth’s notation, 26, 137, 148 Foucault current, 236 fractals, 11 fractography, 11, 129 fracture criterion, 64 fracture mechanics, 12, 164, 167 fracture toughness, 104, 110, 155, 172, 183 frequency dependence, 116 frequency domain, 219 frequency independent, 18, 76, 116 gamma rays, 230

261 Gaussian distribution, 21, 39, 206, 208, 211, 213, 214, 216, 218 geometric correction factor, 175 Gerber diagram, 86 Gerber parabola, 85 gigacycle fatigue, 19, 58 Goodman diagram, 86, 96 good practice, 73 grass, 233 Gumbel distribution, 217 hard X-rays, 230 high cycle fatigue, 17 ideal crack path, 142 in house tool, 70 induction hardening, 98 initial crack, 139, 154, 155, 158, 198 initial crack size, 101, 102, 122 integral approach, 64 interaction effect, 125, 127, 196, 200 internal crack, 142, 196, 200 internal defect, 32 intrinsic fatigue strength, 84 irregular crack, 196 irregularity factor, 40, 218 JOSH, 54 Kirchoff plate bending theory, 174 Kitagawa diagram, 120 knee, 19 KoNoS hypothesis, 65 Laplace distribution, 49, 210, 213, 216 LBF Normal distribution, 53 leak before break, 5, 154 leakage ﬂux, 227 library, 240 life, 17 lift off, 235 limit load, 103 line tension, 152, 199 linear damage rule, 45 linear elastic fracture mechanics, 167 load cycle, 16 load history, 12, 35, 44, 49, 53, 56, 58, 65, 127, 130, 135 load spectrum, 13, 44 long crack, 118

Index

262 longitudinal wave, 231 low cycle fatigue, 17, 65 macrocrack, 27, 136 magnetic ﬂux, 227, 236 magnetic ink, 227 magnetic particle inspection, 226 magnetic permeability, 235 main crack, 139 major project, 70 manufacturing defect, 69, 78 mass product, 70 mathematical description, 37 maximum normal stress criterion, 64 maximum principal stress dominated crack propagation, 137 maximum stress, 15, 20 mean stress, 10, 15, 85, 88 mean stress insensitive, 110 mean stress sensitive, 110, 114 mechanical description, 1, 7, 161, 165 metal fatigue, 1, 7, 37, 161–163, 165, 204, 207, 210, 212, 218, 223, 229, 231, 235, 236 metal fatigue damage, 10 metal fatigue mechanism, 24 metallurgical description, 1, 161, 165 microcrack, 25, 27 micromechanisms, 165 Miner’s law, 45 Miner’s rule, 13, 45, 49, 50, 97, 126 minimum stress, 15 Mises criterion, 63 mixed mode, 26, 138, 155, 156, 158, 169, 198 mixed mode threshold for fatigue crack propagation, 139 Mode I, 169, 170, 172, 183, 184, 186, 190–192, 196, 197, 199 Mode II, 169, 171, 172, 186, 191, 192 Mode III, 169, 171, 172, 186, 191, 192 modiﬁed Goodman diagram, 86 multiaxial failure criterion, 62 multiaxial fatigue loading, 35, 39, 60, 63, 65, 88 multiaxial loading, 35, 59 narrow band random loading, 35, 56, 58, 210, 214–216

narrow band random process, 41, 59, 208, 219 negative peak, 42 negligence, 78, 79 nitriding, 98 no fault liability, 78 non destructive testing, 103, 164, 196, 223, 238 non metallic material, 4 non propagating crack, 93, 116, 120, 142 non proportional fatigue loading, 59, 64 non proportional loading, 59 non proportional random loading, 62 non stationary random processes, 42 nonlinear dynamics, 143 Normal distribution, 21, 39, 206 notch, 83, 89, 180 notch insensitive, 92 notch sensitive, 92 notch sensitivity index, 92 number of cycles, 16 ocean wave, 216, 231 omission dilemma, 58 omission level, 58, 213 one sided process, 214 open crack, 181 opening mode, 169 overload, 127 oxide induced crack closure, 115 P-S-N curves, 23 Palmgren–Miner law, 45 Palmgren–Miner rule, 45 Paris equation, 105 Paris law, 105 Paris region, 117, 122 part through crack, 152, 153, 177 peak counting, 51 penny shaped crack, 176 periodic processes, 206 philosophies of design, 70, 154 physical crack length, 184 plane strain, 109, 182, 183, 185 plane strain fracture toughness, 104, 154, 184 plane stress, 109, 154, 183, 184, 188 plane stress fracture toughness, 154, 184 plastic collapse, 103, 154

Metal Fatigue plastic wake, 113, 119, 125, 127 plastic zone, 128, 182, 184 plastic zone correction, 185 plastic zone radius, 185 point estimate, 240, 242 positive peak, 41 postulated cracks, 103 power spectral density, 219 probability density, 21, 40, 206, 208, 210 probability of detection, 223, 234, 238, 240, 242 probability of failure, 23, 96, 97 probability of sizing, 223, 234, 242 product liability, 67, 77, 78 programme loading, 42, 129 programme marking, 129 proof loading, 99 proportional fatigue loading, 59, 64 proportional loading, 59 pseudo random, 35, 214 quality system, 80 radiograph, 229 radiography, 229 rainﬂow counting, 51 random process, 204, 207 random process theory, 35, 39, 204, 207, 218 random walk, 142 range counting, 51 range of scales, 24 ratchetting, 30 Rayleigh distribution, 41, 208, 210, 213, 216 re-assessment, 73 redundant, 71 reference stress, 38 regulatory authority, 75, 80, 224 relative operating characteristic, 239 residual stress, 88, 98, 99, 124, 133, 180 resonant frequency, 219 return period, 35, 42, 58, 214 root mean square, 39, 131, 206, 207, 210, 212–214, 216, 218 rotating bending, 16 rotation, 172 roughness induced crack closure, 115 safe life, 70 safe product, 77, 79, 80

263 safety factor, 72 safety regulation, 78, 79, 80 scalar criterion, 63 scatter, 20, 31, 72, 96, 97, 107, 122, 133 scatter band, 106 scatter diagram, 217 sea state, 216, 221 service loading testing, 72, 75, 76, 163 shakedown, 31 shear crack propagation, 138, 186 shear dominated crack propagation, 137 shear lip, 111 shear mode, 169 shear wave, 231, 233 short crack, 118 shot peening, 88, 99 shrinkage, 101 sigmoidal, 117 signiﬁcant wave height, 217 skin depth, 234, 235 skin effect, 234 slant crack propagation, 137, 186, 198 slant fatigue crack propagation, 110 slip, 25 slip line, 10 small scale argument, 180, 186 Soderberg line, 88 soft X-rays, 230 spectral bandwidth, 219 spectral density function, 40, 56, 218 stable state, 114 Stage I crack, 26, 137, 140, 158 Stage II crack, 26, 101, 138, 148 Stage III, 27 standard deviation, 21, 206, 207 standard load history, 13, 53–56, 62, 216 standard procedure, 13, 70, 73, 162 standard test method, 12, 107 standing wave, 231 static failure, 101 static failure region, 118, 122 static strength, 1 statistically non stationary, 75, 216 statistically stationary, 39, 204, 216, 218 stochastic process, 38 stress concentration factor, 91, 150 stress criterion, 28, 117 stress cycle, 15, 124 stress history, 15

Index

264 stress intensity factor, 12, 104, 113, 116, 119, 122, 125, 132, 138, 143, 150, 155, 158, 159, 164, 167, 169, 172, 180, 186, 190–192, 195, 196, 199, 200, 225, 233, 238 stress intensity factor range, 105 stress intensity measure, 187, 189 stress range, 10, 15, 105 stress ratio, 15, 45, 110 stress relief, 124 stress state, 154 striation, 27, 129 strict liability, 78, 79 subsidiary echo, 231 surface breaking crack, 223, 226, 228, 232, 238 surface crack, 25, 151, 196, 200, 223 surface factor, 85 surface ﬁnish, 84 surface hardening, 98 surface irregularity, 84 symmetric mode, 187, 190 thermal loading, 31, 150 thermodynamic criterion, 28, 117, 137 threshold for fatigue crack propagation, 11, 29, 116, 117, 119, 122, 132, 138, 158 threshold region, 117, 122 through the thickness crack, 154, 196, 225 tight crack, 230 time domain, 219 time history, 40, 52 time of ﬂight diffraction, 233, 240 toughness, 172 trading standards ofﬁcer, 79 transformation induced crack closure, 115 transition region, 111, 186 translation, 172 transverse wave, 231 Tresca criterion, 63 trough, 42, 209 truncation, 42, 58, 210, 212, 213

truncation dilemma, 58 truncation level, 56 twist crack, 156, 158 ultrasonics, 226, 231 uncracked situation, 83, 96 underload, 127 uniaxial fatigue loading, 59 uniaxial loading, 35, 59, 90 unsafe product, 80 validity corridor, 142 variable amplitude fatigue loading, 35, 37, 96, 124 variable amplitude loading, 35 variance, 206 viscous ﬂuid induced crack closure, 115 visual inspection, 223 Volterra distorsioni, 170 von Mises criterion, 63, 88, 92, 184 Wöhler curves, 17 Wöhler’s laws, 9 WASH, 54 water surface elevation, 216, 221 wave height, 216 wave loading, 55, 56, 58 wave passing frequency, 56, 59, 217, 221 wave period, 217 waveform, 18, 76, 211 Weibull distribution, 210, 213 weighted average stress range, 47, 124 welded joint, 124, 130, 227, 230, 237 Wheeler’s model, 128 white noise, 42 X-rays, 229 yield criterion, 63 Young’s modulus, 109 zero crossing, 40