Multiaxial Notch Fatigue

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Multiaxial notch fatigue

Related titles: Weld cracking in ferrous alloys (ISBN 978-1-84569-300-8) This book covers the processes of weld cracking in different ferrous alloys and for different welding technologies. It also covers methods for testing and avoiding weld cracks. Fatigue assessment of welded joints by local approaches (Second edition) (ISBN 978-1-85573-948-2) ‘. . . a standard reference for designers, structural analysts and testing engineers.’ Welding and Cutting Cumulative damage of welded joints (ISBN 978-1-85573-938-3) A standard work from one of the world’s leading authorities. The book reviews damage caused by different levels of load, different stress cycles and variable amplitude loading. Details of these and other Woodhead Publishing materials books can be obtained by: • •

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Multiaxial notch fatigue From nominal to local stress/strain quantities Luca Susmel

Oxford

Cambridge

New Delhi

Published by Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK www.woodheadpublishing.com Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi-110002, India Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2009, Woodhead Publishing Limited and CRC Press LLC © 2009, Woodhead Publishing Limited The author has asserted his moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publishers cannot assume responsibility for the validity of all materials. Neither the author nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 978-1-84569-582-8 (book) Woodhead Publishing ISBN 978-1-84569-566-8 (e-book) CRC Press ISBN 978-1-4398-0301-1 CRC Press order number: N10057 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by SNP Best-set Typesetter Ltd., Hong Kong Printed by TJ International Limited, Padstow, Cornwall, UK

Contents

Nomenclature Foreword Introduction

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2 2.1 2.2 2.3 2.4 2.5 2.6

Useful stress quantities used in fatigue problems Introduction Stress state Mohr’s circles under plane stress Amplitude, mean value, range and load ratio, R, under uniaxial cyclic loading Stress components relative to a generic material plane: the tridimensional problem Amplitude, mean and maximum value of the stress normal to a given material plane Amplitude and mean value of the shear stress relative to a given material plane Stress concentration factor, Kt Singular stress fields References Fundamentals of fatigue assessment Introduction Fatigue strength and Wöhler curves The mean stress effect under uniaxial fatigue loading The notch effect in fatigue Linear Elastic Fracture Mechanics to predict fatigue damage in cracked bodies Different links between Linear Elastic Fracture Mechanics and continuum mechanics

ix xv xvii

1 1 2 3 6 7 9 10 24 25 30 33 33 34 37 40 50 53 v

vi

Contents

2.7 2.8 2.9 2.10

The Theory of Critical Distances Fatigue assessment under torsional loading Fatigue damage under multiaxial fatigue loading References

3

The Modified Wöhler Curve Method in fatigue assessment Introduction Fatigue damage model Degree of multiaxiality of the stress field damaging the fatigue process zone according to the MWCM The Modified Wöhler Curve Method Use of the MWCM to estimate high-cycle multiaxial fatigue strength Use of the MWCM to estimate finite life under multiaxial fatigue loading References

3.1 3.2 3.3 3.4 3.5 3.6 3.7 4 4.1 4.2 4.3 4.4 4.5 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6 6.1 6.2 6.3

Fatigue assessment of notched components according to the Modified Wöhler Curve Method Introduction Inherent and external multiaxiality The MWCM applied in terms of nominal stresses The MWCM applied along with the TCD to estimate notch fatigue strength References Multiaxial fatigue assessment of welded structures Introduction Stress quantities used to assess welded structures Preliminary assumptions MWCM and nominal stresses MWCM and hot-spot stresses The MWCM applied in terms of the PM to perform the fatigue assessment of weldments References The Modified Wöhler Curve Method and cracking behaviour of metallic materials under fatigue loading Introduction Crack initiation in single crystals Stage I and Stage II in polycrystals subjected to uniaxial fatigue loading

61 73 82 91

98 98 98 101 106 111 119 123

125 125 126 127 134 149 152 152 153 158 159 162 169 186

189 189 190 192

Contents 6.4 6.5 6.6 6.7 6.8 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8 8.1 8.2 8.3 8.4 8.5 8.6

Mesoscopic cracking behaviour of metallic materials under multiaxial fatigue loading The MWCM and structural volume The problem of estimating fatigue limits Concluding remarks References The Modified Manson–Coffin Curve Method in fatigue assessment Introduction Strain quantities used in low-cycle fatigue problems Stress–strain behaviour of metallic materials Uniaxial and torsional fatigue assessment according to Manson and Coffin’s idea The Modified Manson–Coffin Curve Method Low-cycle fatigue assessment of notched components References Multiaxial fatigue of composite materials Introduction Stress quantities used to assess composite materials subjected to cyclic loading Design parameters affecting multiaxial fatigue strength of composites Multiaxial fatigue assessment of composite materials Concluding remarks References

vii

194 198 202 206 207

210 210 211 213 221 224 234 236 240 240 240 243 255 263 263

Appendix A Experimental values of the material characteristic length, L A.1 Introduction A.2 Steel A.3 Cast iron A.4 Aluminium alloy A.5 Other materials A.6 References

266 266 267 272 273 275 276

Appendix B Experimental results generated under multiaxial fatigue loading B.1 Introduction B.2 Adopted symbolism and nomenclature B.3 High-cycle fatigue strength of plain specimens B.4 High-cycle fatigue strength of notched specimens

280 280 282 286 310

viii

Contents

B.5

Fatigue results generated by testing plain and notched specimens under strain control Low/medium-cycle fatigue results generated under stress control Fatigue results generated by testing steel and aluminium welded specimens under multiaxial fatigue loading Multiaxial fatigue strength of composite materials Geometries of the notched/welded samples References

B.6 B.7 B.8 B.9 B.10

Index

319 385 483 505 548 557

564

Nomenclature

a a a* a, b, a, b a0 an, bn, an, bn ax, ay, az aH, CH aN aP 2a b b b0 bx, by, bz c c0 d k kI, kII kt l l0 m n n n′ n nx, ny, nz

Unit vector defining the orientation of axis a Notch depth Notch depth defining the blunt notch regime in Atzori and Lazzarin’s diagram Constants in the governing equations of the MWCM El Haddad’s short crack constant Constants of the MWCM calculated in terms of nominal net stresses Components of unit vector a Constants in Heywood’s formula Constant in Neuber’s formula Constant in Peterson’s formula Crack length Unit vector defining the orientation of axis b Fatigue strength exponent Fatigue strength exponent under shear strain Components of unit vector b Fatigue ductility exponent Fatigue ductility exponent under shear strain Hole diameter Negative inverse slope Non-dimensional quantities depending on the welded joint geometry Negative inverse slope of the modified Wöhler curve Actual length Gauge length Mean stress sensitivity index Number of cycles Strain hardening exponent Cyclic strain hardening exponent Unit vector normal to plane D Components of unit vector n ix

x

Nomenclature

Oxyz Oabn Orfq q q qx, qy, qz rn t t, L t tx, ty, tz z A A0 A, B C, m D E E (%) F Fa F(t) I1s, I2s, I3s K K′ K1, K2, K3 Kf Kft Kf,e Ki(Nf) Kt Ktt Kt,gross K*t,gross Kt,net KC KIC KI, KII, KIII KI,max

Frame of reference Frame of reference relative to plane D Polar coordinates Notch sensitivity factor Unit vector defining the orientation of direction q Components of unit vector q Notch root radius Time Welded plate thickness Total stress vector Components of vector t Weld bead height Constant in Kuhn and Hardraht’s formula Initial value of the cross-sectional area Constants in the LM vs. Nf relationship Constants in Paris’ law Notch depth Young’s modulus Error LEFM geometrical factor Amplitude of the axial force Axial force First, second and third stress invariant Strength coefficient Cyclic strength coefficient Notch stress intensity factors due to Mode I, II and III loading Fatigue strength reduction factor Fatigue strength reduction factor under torsional loading Estimated fatigue strength reduction factor Calibration functions for assessing composite materials (i = 1, 2, . . . , 6) Stress concentration factor Stress concentration factor under torsional loading Stress concentration factor referred to the gross area Kt value defining the transition from the blunt to the sharp notch regime Stress concentration factor referred to the net area Fracture toughness Plane strain fracture toughness Stress intensity factors due to Mode I, II and III loading Maximum value of the stress intensity factors due to Mode I loading

Nomenclature L LM LT Mt M-DV N0 Nf Nf,e NA ND NS PS R RCP Re T Ts UF,a UF,A 2a ga g ′f g ij,a g eij,a g pij,a g xy, g xz, g yz d 2,1 d 6,1 d xy,x d y,x e– e– a e– ea e– pa e′f e1, e2, e3 eeng e ei,a

xi

Material characteristic length Material characteristic length in the medium-cycle fatigue regime Material characteristic length under Mode III loading Torque Multiaxial critical distance value Number of cycles to failure defining the position of the knee point Number of cycles to failure Estimated number of cycles to failure Reference number of cycles to failure Reference number of cycles to failure Reference number of cycles to failure in the low cycle fatigue regime Probability of survival Load ratio (R = si,min /si,max where i = x, y, z) Load ratio relative to the critical plane Strain ratio Period of the cyclic load history Scatter ratio of reference shear stress amplitude for 90% and 10% probabilities of survival Amplitude of the strain energy density Reference amplitude of the strain energy density extrapolated at NA cycles to failure Opening angle Maximum shear strain amplitude Fatigue ductility coefficient under shear strain Amplitude of total shear strains g ij (i, j = x, y, z) Amplitude of the elastic part of shear strain g ij (i, j = x, y, z) Amplitude of the plastic part of shear strain g ij (i, j = x, y, z) Total shear strains Phase angle between stress components s2(t) and s1(t) Phase angle between stress components s6(t) and s1(t) Phase angle between stress components txy(t) and sx(t) Phase angle between stress components sy(t) and sx(t) Von Mises’ equivalent strain Amplitude of Von Mises’ equivalent strain Amplitude of Von Mises’ elastic equivalent strain Amplitude of Von Mises’ plastic equivalent strain Fatigue ductility coefficient Principal strains Engineering normal strain Amplitude of the elastic part of normal strain ei (i = x, y, z)

xii

Nomenclature

e pi,a emax emin etrue ex, ey, ez q l1, l2, χ1, χ2 l1, l2 l C, l T μsn μt νe np n12, n21 r reff rlim rw rw,lim s– s–a s0 s 0,R=−1 s 1, s 2, s 3 seng sep s ′f s gross s gross,max s i,a s i,m s i,max s i,min s max s min sn s n,a s n,m s n,max

Amplitude of the plastic part of normal strain ei (i = x, y, z) Maximum normal strain Minimum normal strain True normal strain Normal strains Off-axis angle Constants in Lazzarin and Tovo’s equations depending on the notch opening angle Biaxiality ratios calculated with respect to the material principal stresses Biaxiality ratios calculated with respect to the geometrical stresses Microscopic normal stress Microscopic shear stress Poisson’s ratio for elastic strain Poisson’s ratio for plastic strain Poisson’s ratios in composites Stress ratio relative to the critical plane Effective value of the stress ratio relative to the critical plane Limit value of the stress ratio relative to the critical plane Stress ratio relative to the critical plane used to estimate fatigue damage in weldments Limit value of the stress ratio rw Von Mises’ equivalent stress Amplitude of Von Mises’ equivalent stress Plain fatigue limit Fully reversed plain fatigue limit Principal stresses or material principal stresses in composites Engineering normal stress Linear elastic peak stress Fatigue strength coefficient Nominal stress referred to the gross area Maximum value of the nominal gross stress Amplitude of stress component si (i = x, y, z or i = 1, 2, 3) Mean value of stress component si (i = x, y, z or i = 1, 2, 3) Maximum value of stress component si (i = x, y, z or i = 1, 2, 3) Minimum value of stress component si (i = x, y, z or i = 1, 2, 3) Maximum normal stress Minimum normal stress Normal stress parallel to unit vector n Amplitude of stress component sn Mean value of stress component sn Maximum value of stress component sn

Nomenclature s n,min s net s nom s true sx, sy, sz sA sAn s HS sS sUTS sY sq, sr t0 t0,R=−1 t0n ta tij,a tij,m t ′f tmax tmin tn tn,a tn,m tna, tnb tq trq, tqz, trz txy, txz, tyz ty tA tAn tA,Ref tHS w D ΔKI ΔKI,50%

xiii

Minimum value of stress component sn Nominal stress referred to the net area Nominal stress Engineering normal stress Normal stresses Endurance limit extrapolated at NA cycles to failure Notch endurance limit extrapolated at NA cycles to failure Hot-spot stress Reference stress in the low-cycle fatigue regime Ultimate tensile stress Yield stresses Normal stresses calculated with respect to polar coordinates Orfq Plain torsional fatigue limit Fully reversed plain torsional fatigue limit Notch torsional fatigue limit Maximum shear stress amplitude Amplitude of shear stress component tij (i, j = x, y, z) Mean value of stress component tij (i, j = x, y, z) Fatigue strength coefficient under shear strain Maximum value of the shear stress during the load cycle Minimum value of the shear stress during the load cycle Shear stress relative to plane D Amplitude of the shear stress relative to plane D Mean value of the shear stress relative to plane D Shear stresses calculated with respect to axes a and b, respectively Shear stress resolved along direction q Shear stress components calculated with respect to polar coordinates r, q, z Shear stresses Yield shear stress Plain torsional endurance limit extrapolated at NA cycles to failure Notch torsional endurance limit extrapolated at NA cycles to failure Reference shear stress amplitude at NA cycles to failure Hot-spot shear stress Angular velocity Material plane Range of the stress intensity factors due to Mode I loading Mode I notch stress intensity factor range calculated at ND = 5× 106 cycles to failure for a probability of survival equal to 50%

xiv

Nomenclature

ΔKIII,th ΔKth Δσ0 Δs0n Δs0n,e Δseff Δsgross Δsi Δsn Δsnom ΔsA ΔsHS Δt Δteff Δtij Δtnom ΔtA ΔtAn ΔtHS Y Y′ Acronyms AM FE LEFM LM MMCCM MWCM NPC N-SIF PM PSB SED SIF TCD VM

Range of the threshold value of the stress intensity factor under Mode III loading Threshold value of the stress intensity factor range Range of the plain fatigue limit Range of the notch fatigue limit Estimated range of the notch fatigue limit Range of the effective stress Range of the nominal gross stress Range of stress component si (i = x, y, z or i = 1, 2, 3) Normal stress range relative to the critical plane Range of the nominal stress Range of the nominal normal stress at NA = 2 ×106 cycles to failure Range of the hot-spot stress Shear stress range relative to the critical plane Range of effective shear stress Range of shear stress component tij (i = x, y, z and j = x, y, z) Range of the nominal shear stress Range of the plain torsional endurance limit extrapolated at NA cycles to failure Range of the notch torsional endurance limit extrapolated at NA cycles to failure Range of the hot-spot shear stress Curve plotted by the tip of shear stress tn(t) Polygon describing close curve Y

Area Method Finite Element Linear Elastic Fracture Mechanics Line Method Modified Manson–Coffin Curve Method Modified Wöhler Curve Method Non-propagating crack Notch-stress intensity factor Point Method Persistent slip band Strain energy density Stress intensity factor Theory of Critical Distances Volume Method

Foreword

In this book, Luca Susmel has brought together the results of his work on notch fatigue under complex loading. This body of work, the fruit of almost a decade of research, has been previously presented in many journal articles and conferences. Having made significant progress and having come to some conclusions which are so strongly supported by the data as to be undeniable, he decided to present the entire body of work in the form of a book. What I find rather surprising is that so few researchers in the field of science and engineering today are writing books of this kind. One excellent example which comes to mind is Prof. Murakami’s Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Such books are rare – perhaps the modern system of promotions in universities discourages their production – but when they come along they are very useful. For an expert in the field, reading Luca’s book is a convenient way to appreciate his approach in its totality, in a unified form which tells the story more clearly than is ever possible in a series of individual short articles. For the nonexpert, for example a design engineer in a company who wishes to make assessments of multiaxial fatigue at stress concentration features but who is not, as such, much concerned with the science of the subject, the book presents the underlying methodology and shows how it can be used in practice, via closed-form solutions or numerical analysis. This book discusses an important problem and one which has not yet been solved, despite many decades of research activity. In the history of our field – the strength and fracture of materials – there has been one huge breakthrough: the development of fracture mechanics by Griffith, Irwin and others in the early part of the twentieth century. The definition of the stress intensity parameter was a revolution, not just in science but in industrial development. It allowed us to measure in an unequivocal manner the toughness of a material and to predict precisely its response to the presence of a crack. Many aspects of the modern world – vehicles, energy xv

xvi

Foreword

production, manufactured goods of all kinds – would be impossible without this knowledge. But fracture mechanics is, strictly speaking, only applicable to cracks. Many other stress concentration features exist, and cause failure by cracking processes such as fatigue and brittle fracture, but we have no equivalent of the stress intensity parameter to use for these features, to characterise materials and to predict failure. This is despite the fact that research into notches goes back just as far as research into cracks, through the work of Neuber, Peterson and colleagues. The other main aspect of Luca’s book is the consideration of multiaxial stress states. An understanding of multiaxial loading is clearly necessary in order to have a general approach which is applicable to real industrial components, which are not loaded in the nice convenient ways which we choose to apply to laboratory specimens. But the need to understand multiaxial behaviour goes deeper than that. As Luca points out, once we introduce a crack or notch of any kind into a body, then even simple uniaxial loading will generate multiaxial stress states near the feature. So, if one really wishes to understand what is going on, a multiaxial approach is simply not an option. Another important feature of this book is the large amount of test data, obtained by the author from his own experiments and collected from the published literature. A point on which Luca and I agree strongly is that, in developing a new theory, it is vital to show that it is capable of predicting the data. Only by this test can science advance; however elegant and inspired a theory may be, it will not be adopted unless it can be shown to coincide with reality, to predict the results of controlled experiments and to concur with observations of real-world phenomena. The work described in this book is not a complete solution to the problem of notches or of multiaxiality. I suspect that much more work and further crucial insights will be needed before we can claim to have a general understanding of the type expressed by the stress intensity parameter for cracks. What this book does is to present an approach which has had considerable success, which can already be recommended for industrial use and which merits further investigation by the research community. David Taylor Professor of Materials Engineering Trinity College Dublin Ireland

Introduction

In 1997, when I was an undergraduate student in the Department of Mechanical Engineering of the University of Padova, Italy, I called in to Professor Bruno Atzori’s office to ask him to supervise my final project. After a quick chat, he proposed that I should do some theoretical work on something strange called ‘multiaxial fatigue’. After getting my degree in Mechanical Engineering, I started doing my PhD in the University of Padova, continuing to work on the problem of estimating fatigue damage in plain metals subjected to multiaxial loading paths. In 2000, having formalised the Modified Wöhler Curve Method, I sent a long e-mail to Professor David Taylor at Trinity College, Dublin, to ask whether I could spend a few months in Dublin investigating, under his supervision, whether it was possible for my multiaxial fatigue criterion to be applied in conjunction with the Theory of Critical Distances. Professor Taylor replied with a very enthusiastic e-mail, and on 12 July 2001 I caught a plane at Venice airport to fly to Ireland: this moment undoubtedly represents, from both a professional and a personal point of view, the most important point of no return in my life. Since 2001 I have spent at least 6 months every year lecturing and doing research in the Department of Mechanical and Manufacturing Engineering at Trinity College. This is just the beginning of the story. Over the last decade I have been working at the University of Padova, the University of Ferrara and Trinity College investigating the different aspects related to the fatigue assessment of engineering materials damaged by multiaxial fatigue loading. Having done so much work in this particular ambit of the structural integrity discipline, I realised that I needed to coherently reorganise all my ideas about such a subject, so I decided to write a book about it. Thus the present book attempts to summarise my understanding of those phenomena damaging materials when constant-amplitude xvii

xviii

Introduction

multiaxial cyclic loadings are involved, as well as of the most efficient way to perform the fatigue assessment of real components subjected to complex in-service loading paths. In more detail, this book begins by considering those definitions to be used to calculate the stress/strain quantities suitable for estimating multiaxial fatigue strength of real mechanical components. Subsequently, those pivotal ideas which should always be borne in mind when designing components against fatigue are briefly summarised, considering mainly uniaxial and torsional situations. By taking the above concepts as a starting point, the way of using the Modified Wöhler Curve Method is then investigated in depth, by focusing attention on both the high- and the mediumcycle fatigue regimes. An attempt is also made to show the existing links between the Modified Wöhler Curve Method and physical reality. Subsequently, the procedure suitable for employing this approach to estimate fatigue damage in both notched and welded components is explained in detail. Since the Modified Wöhler Curve Method as it stands cannot be applied to estimate multiaxial fatigue damage in the low-cycle fatigue regime, the main ingredients of the so-called Modified Manson–Coffin Curve Method are investigated in depth, by also reviewing those concepts that play a fundamental role in the strain-based approach. Finally, the problem of performing fatigue assessment of composite materials is addressed by considering the most efficient strategies suitable for designing real components against multiaxial fatigue. The book also contains two appendices summarising, in a systematic and organised way, several experimental results taken from the technical literature. In particular, Appendix B reports about 4500 experimental fatigue results generated by testing plain and notched specimens under constantamplitude multiaxial fatigue loading, and lists results obtained by testing, under proportional and non-proportional loading paths, both welded details and composite materials. I hope that the present book will interest a wide range of readers: academic researchers working on the theoretical basis of the problem of predicting multiaxial fatigue damage in engineering materials weakened by stress concentration phenomena; materials scientists interested in the cracking behaviour of metallic materials under complex cyclic stress states; industrial engineers who need a sound theory suitable for assessing real mechanical components by taking full advantage of modern Finite Element software packages; and, finally, students who want to study the subject in depth. To conclude, even though I take full responsibility for the contents of this book, I would like to thank several people for their invaluable contributions. First of all and above all, I wish to extend my heartfelt thanks to David

Introduction

xix

Taylor: not only a first-class mentor but also ‘un vero amico’. I wish to thank Roberto Tovo and Paolo Livieri for covering me when abroad as well as for helping me in making my career in the University of Ferrara. I am also very grateful to the Fatigue Team of the University of Padova: Bruno Atzori, Paolo Lazzarin, Marino Quaresimin, Giovanni Meneghetti and Nicola Petrone . . . and, last but not least, . . . thank you Cinzia for being always so supportive, and I’m sorry for the time that I left you alone because I was too distracted by my boring equations.

(On the train, in the middle of nowhere between Udine and Venice)

This book is dedicated to my mum, Viviana, my daddy, Piera, and my sister, Sabina

1 Useful stress quantities used in fatigue problems

Abstract: The aim of this chapter is to review the procedures commonly used in practice to determine the stress quantities most often employed in fatigue problems. In particular, in order to properly deal with multiaxial fatigue situations, the problem of determining the stress components relative to the so-called critical plane is addressed in great detail. Key words: stresses, critical plane, singular stress fields.

1.1

Introduction

Assessing mechanical components against fatigue is a complex problem that has to be properly addressed during the design process in order to avoid catastrophic failures. In more detail, when engineering materials are subjected to time-variable loading, they can fail due to fatigue even without any evident large-scale plastic deformation altering the process zone (and this holds true especially in the medium/high-cycle fatigue regime). Moreover, the complex geometries of real mechanical assemblies favour the initiation of fatigue cracks due to stress concentration phenomena arising from the geometrical features of the components themselves. This is the reason why, as a general guideline, it is always advisable to design stress raisers having a notch root radius as large as possible. Unfortunately, very often this is not possible, so that safe engineering methods are needed to properly take into account the detrimental effect of such geometrical features. Independently of the approach adopted to perform the fatigue assessment, the starting point of any fatigue life estimation technique is the location of the components’ weakest points. At such points the stress/strain states have to be determined using either classical continuum mechanics or numerical approaches, typically the Finite Element (FE) method. Subsequently, such stress/strain states have to be recalculated in terms of quantities suitable for being used along with the adopted fatigue criterion. In other words, stress/strain states have to be expressed in terms of amplitudes, mean values, ranges, etc., to correctly estimate fatigue damage. The aim of this chapter is to review the procedures commonly used in practice to determine the above quantities. In particular, in order to properly deal with multiaxial fatigue situations, the problem of determining the stress 1

2

Multiaxial notch fatigue

components relative to the so-called critical plane will be addressed in great detail. Subsequent chapters of this book will focus on means of using such quantities to properly assess mechanical components. Finally, the present chapter also reviews some analytical tools suitable for performing the stress analysis in the presence of stress concentration phenomena.

1.2

Stress state

Consider a body loaded by an external system of forces. In order to constrain the required degrees of freedom, the appropriate boundary conditions are also applied to such a body (Fig. 1.1). Consider now a generic material point O, which is also the origin of a convenient frame of reference, Oxyz. The stress state at this material point is fully described by the following tensor: ⎡σ x

τ xy τ xz ⎤

[σ ] = ⎢τ xy σ y τ yz ⎥

1.1

⎥ σ z ⎥⎦

⎢ ⎢⎣τ xz τ yz

where sx, sy and sz are the normal and txy, txz and tyz are the shear stress components. It is important to remember here that in a body having complex geometry, in general, the stress state changes from point to point, whereas at any given point the numerical values of the normal and shear stress components vary as the orientation of the frame of reference changes. It is possible to demonstrate that there always exists a frame of reference, whose axes are called principal axes, for which the shear stress components are invariably equal to zero. In particular, if the stress state at the considered point is triaxial, the stress tensor becomes diagonal and can be rewritten as follows:

Fj

Fk z y O x Fi

1.1 Body subjected to an external system of forces and definition of the frame of reference at point O.

Useful stress quantities used in fatigue problems ⎡σ 1 0 0 ⎤ [σ ] = ⎢ 0 σ 2 0 ⎥ ⎢ ⎥ ⎣⎢ 0 0 σ 3 ⎥⎦

3

1.2

where s1, s2 and s3 are the so-called principal stresses (in general, the principal stresses are ordered so that s1 ≥ s2 ≥ s3). To determine the above stress components, assume that unit vector n(nx, ny, nz) defining the orientation of a principal direction is known. The principal stress, sn, parallel to vector n can then be calculated by imposing that the determinant of the first matrix on the left-hand side of the following identity is equal to zero (Viola, 1992):

τ xy τ xz ⎤ ⎛ nx ⎞ ⎛ 0⎞ ⎡(σ x − σ n ) ⎢ τ τ yz ⎥ ⎜ ny ⎟ = ⎜ 0⎟ (σ y − σ n ) xy ⎥⎜ ⎟ ⎜ ⎟ ⎢ (σ z − σ n ) ⎥⎦ ⎝ nz ⎠ ⎝ 0⎠ ⎢⎣ τ xz τ yz

1.3

σ n3 − I1σ σ n2 − I 2σ σ n − I 3σ = 0

1.4

that is,

where the first, second and third stress invariants are defined, respectively, as: I1σ = σ x + σ y + σ z I 2σ = σ x2 + σ y2 + σ z2 − (σ xσ y + σ xσ z + σ yσ z )

1.5

I 3σ = det [σ ] In order to correctly calculate the principal stresses, also the following condition must be assured: nx2 + ny2 + nz2 = 1

1.6

The three solutions obtained by solving Eq. (1.4) are the eigenvalues of matrix [s], i.e., the principal stresses, whereas the corresponding eigenvectors give the orientation of the principal directions.

1.3

Mohr’s circles under plane stress

Suppose that point O belonging to the body sketched in Fig. 1.1 is subjected to plane stress. According to the adopted absolute frame of reference, and considering an elementary volume centred at point O, the stress state at such a point is the one depicted in Fig. 1.2a. Due to the fact that the surface perpendicular to the z-axis experiences no shear stress, by definition such an axis is a principal direction. Using Mohr’s circles it is possible to fully

4

Multiaxial notch fatigue t z sx > sy

X

txy a

txy

sy

txy

sx

sy

–txy

y

x f = a/2

Y

Rc s c sx

s

(b)

(a)

1.2 (a) Elementary volume loaded in plane stress; (b) resulting Mohr’s circle.

describe the stress state relative to any material plane perpendicular to the x–y plane. In more detail, consider the chart of Fig. 1.2b which has in the abscissa the normal stress, s, and in the ordinate the shear stress, t. In such a diagram the stress state relative to the plane perpendicular to the x-axis is represented by point X(sx, txy), whereas the stress state relative to the plane perpendicular to the y-axis is represented by point Y(sy, −txy). It is worth remembering here that the shear stress is assumed to be positive when it tends to rotate the elementary volume clockwise. The stress state, in terms of normal and tangential stress, of every material plane parallel to the z-axis and passing through point O is then described by the circle having diameter equal to segment X–Y (Fig. 1.2b). The centre, C, of such a circumference has the following coordinates:

σx +σy 2 whereas its radius, Rc, is equal to: C (σ c, 0) where σ c =

1.7

σx −σy ⎞ 2 Rc = ⎛ + τ xy ⎝ 2 ⎠

1.8

2

From Fig. 1.2b it is possible to observe also that, according to Mohr’s schematisation, the angle a between axes X and Y is equal to π, whereas the corresponding angle, f, in the conventional representation (Fig. 1.2a) is π equal to . This means that the relationship between angles in Mohr’s 2 representation and angles in the conventional representation is: f = a/2 (anti-clockwise in Fig. 1.2). Mohr’s circle sketched in Fig. 1.2b then allows the stress state relative to any plane parallel to the z-axis and passing through point O to be easily determined. For instance, as sketched in Fig. 1.3, the maximum shear stress,

Useful stress quantities used in fatigue problems

5

t tmax

sc

X a

Rc

y

tmax

sc

sc f = a/2 x

s

tmax

(a) Y

(b)

1.3 Orientation of the plane experiencing the maximum shear stress, tmax.

t y X

s2

a

s3 = 0 s2

Rc

sc

s1

s x f = a/2

(a)

s1

Y (b)

1.4 Orientation of principal stresses s1 and s2.

tmax, is equal to Rc, whereas the stress perpendicular to such a plane is equal to sc. On the contrary, the principal stresses s1 and s2 can be calculated as: s1 = sc + Rc and s2 = sc − Rc. Figure 1.4 shows the procedure to determine these principal stresses from Mohr’s circle as well as the orientation of the principal axes. Finally, it can be useful to remember that, under plane stress, the normal, sn(f), and the shear stress, tn(f), relative to a generic material plane perpendicular to the component’s surface and having normal unit vector n at angle f to the x-axis can be calculated as follows (Socie and Marquis, 2000):

σ n (φ ) =

σx +σy σx −σy + cos ( 2φ ) + τ xy sin ( 2φ ) 2 2

τ n (φ ) =

σx −σy sin ( 2φ ) − τ xy cos ( 2φ ) 2

1.9 1.10

6

Multiaxial notch fatigue

1.4

Amplitude, mean value, range and load ratio, R, under uniaxial cyclic loading

Assume that the body of Fig. 1.1 is subjected to an external system of forces resulting, at point O, in the following uniaxial stress state: ⎡σ x( t ) 0 0 ⎤ ( ) 0 0⎥ σ t = [ ] ⎢ 0 ⎥ ⎢ ⎢⎣ 0 0 0 ⎥⎦

1.11

where sx(t) = sx,m + sx,a sin(wt). In this definition t is time, w is the angular velocity, and sx,m and sx,a are the mean value and the amplitude of stress component sx(t), respectively. It is worth remembering here also that, by definition, a cycle is a sequence of changing stress states that, upon completion, produces a final stress state which is identical to the initial one (see the example reported in Fig. 1.5). The maximum and minimum values of the stress component in tensor (1.11) turn out to be:

σ x,max = σ x,m + σ x,a

1.12

σ x,min = σ x,m − σ x,a

1.13

whereas its range is equal to: Δσ x = σ x,max − σ x,min = 2σ x,a

1.14

Finally, the load ratio, R, which is a very important parameter to be accounted for when performing the fatigue assessment, is defined as follows: R=

σ x,min σ x,max

1.15

s(t) s(t)

R>0

cycle t

smax sm

sa sa

smin

s(t)

R=0

Δs t

t s(t)

R = –1 t

1.5 Definition of the stress quantities used to predict fatigue damage under uniaxial fatigue loading.

Useful stress quantities used in fatigue problems

7

Figure 1.5 summarises all the definitions reported above, together with three examples of load histories characterised by three different values of the load ratio, R. It is important to highlight here that, for the sake of simplicity, only a sinusoidal loading path was considered above, but definitions (1.12)–(1.15) remain always valid independently of the wave form, provided that the applied loading is cyclic. To conclude, it is worth pointing out that, when dealing with fatigue problems, the stress parameters suitable for estimating fatigue damage are, in general, expressed either in terms of amplitude or in terms of range. In the present book we will use both notations, according to the symbolism commonly adopted in the technical literature to investigate the different aspects of the fatigue assessment problem.

1.5

Stress components relative to a generic material plane: the tridimensional problem

In Section 1.3 the problem of determining the stress components relative to a given material plane was addressed by considering a biaxial stress state due to an external static loading. Even if Mohr’s circles can be used to investigate also triaxial situations, in the present section the same problem will be addressed in a more general way. In more detail, the equations suitable for determining the stresses relative to a generic plane will be presented in their general form, formalising them by assuming that the investigated mechanical component is subjected to a cyclic load history. Consider then a material point O belonging to the body of Fig. 1.1. As above, such a point is also the centre of the absolute frame of reference, Oxyz. The orientation of a material plane Δ having normal unit vector n(nx, ny, nz) can be located by using spherical coordinates f and q (Fig. 1.6): the former is the angle between the projection of unit vector n on plane x–y z n

q

Δ b

y a

O

f

x

1.6 Definition of angles f and q and unit vectors defining the frame of reference relative to a generic material plane Δ.

8

Multiaxial notch fatigue

and the x-axis; the latter instead is the angle between the normal n and the z-axis. According to the above schematisation, all the material planes passing through point O can be investigated by making angles f and q vary between 0 and 2π and 0 and π, respectively. In the most general case, the stress state damaging point O is triaxial. In other words, at any instant t of the applied cyclic load history (having period T) the stress tensor at O can be expressed as: ⎡ σ x( t ) τ xy( t ) τ xz( t ) ⎤ [σ ( t )] = ⎢τ xy( t ) σ y( t ) τ yz( t ) ⎥ ⎢ ⎥ ⎣⎢τ xz( t ) τ yz( t ) σ z( t ) ⎥⎦

1.16

where t ∈ T. Consider now a generic plane Δ where n is its normal unit vector. In order to make all the calculations easier, another system of coordinates, Oabn, can be introduced. In more detail, axis n is parallel to unit vector n, whose components are as follows: nx = sin (θ ) ⋅ cos (φ ) ny = sin (θ ) ⋅ sin (φ )

1.17

nz = cos (θ ) whereas axes a and b are parallel to plane Δ and the components of the corresponding unit vectors, a and b in Fig. 1.6, turn out to be (Papadopoulos et al., 1997): ax = − sin (φ ) ay = cos (φ )

1.18

az = 0 bx = − cos (θ ) ⋅ cos (φ ) by = − cos (θ ) ⋅ sin (φ )

1.19

bz = sin (θ ) At any instant t ∈ T it is possible now to calculate the normal stress, sn(t), and the shear stress, tn(t), relative to plane Δ, provided that the total stress vector t(t) is known. In particular, the components of such a vector depend on tensor [s(t)], that is: t x( t ) = σ x( t ) ⋅ nx + τ xy( t ) ⋅ ny + τ xz( t ) ⋅ nz t y( t ) = τ xy( t ) ⋅ nx + σ y( t ) ⋅ ny + τ yz( t ) ⋅ nz

1.20

tz( t ) = τ xz( t ) ⋅ nx + τ yz( t ) ⋅ ny + σ z( t ) ⋅ nz The above relationships can finally be used to determine the stresses relative to the investigated material plane, obtaining:

Useful stress quantities used in fatigue problems

σ n( t ) = t x( t ) ⋅ nx + t y( t ) ⋅ ny + tz( t ) ⋅ nz

9 1.21

τ na(t ) = t x(t ) ⋅ ax + t y(t ) ⋅ ay + tz(t ) ⋅ az

1.22

τ nb(t ) = t x(t ) ⋅ bx + t y(t ) ⋅ by + tz(t ) ⋅ bz 2 2 (t ) + τ nb (t ) τ n (t ) = τ na

1.23

Equations (1.21)–(1.23) represent the mathematical tools the socalled critical plane approach is based on (Matake, 1977; McDiarmid, 1991, 1994).

1.6

Amplitude, mean and maximum value of the stress normal to a given material plane

As briefly said above, in order to properly estimate fatigue damage, stress quantities have to be expressed in terms of amplitude, mean and maximum value. Consider then a material plane Δ, whose orientation is defined by angles f and q. During the load cycle, the normal stress, sn(t), varies its magnitude by remaining always parallel to n (Fig. 1.7), so that its amplitude and mean value can simply be calculated as follows:

σ n ,a =

1⎡ max σ n (t1 ) − min σ n (t 2 )⎤ ⎦ t2 ∈T 2 ⎣ t1 ∈T

1.24

σ n,m =

1⎡ max σ n (t1 ) + min σ n (t 2 )⎤ ⎦ t2 ∈T 2 ⎣ t1 ∈T

1.25

where t1 and t2 are two instants of the cyclic load history having period T.

n s n(t)

Δ

Ψ′ O

a

t n(t)

Ψ b

1.7 Normal, sn(t), and shear stress, tn(t), relative to a given plane Δ.

10

Multiaxial notch fatigue

Finally, by definition, the maximum and minimum values of sn turn out to be:

1.7

σ n,max = σ n,m + σ n,a

1.26

σ n,min = σ n,m − σ n,a

1.27

Amplitude and mean value of the shear stress relative to a given material plane

Contrary to the normal stress, the definition of the amplitude, tn,a, and the mean value, tn,m, of the shear stress relative to a generic plane is much more complex, mainly because vector tn(t) changes its magnitude and direction during the loading cycle. Due to the complexity of the problem, many different definitions have been proposed and validated. As to this aspect, it has to be said that, as far as the writer is aware, no universally accepted definition exists yet, so that in the next subsections the most interesting proposals available in the technical literature will be reviewed in brief, trying, when possible, to give tn,a and tn,m in a simple and explicit form. Before addressing the above problem in detail, it is worth introducing here the schematisation that is usually adopted in order to simplify the complexity of calculation, reducing, at the same time, the numerical effort needed to obtain the final result (Papadopoulos, 1998). Consider then the material plane, Δ, sketched in Fig. 1.7. During the load cycle, the shear stress, tn(t), varies in both direction and magnitude, so that the tip of such a stress vector draws on Δ a close curve, Y. If the shear stress relative to the considered plane is determined at n different instants of the load history, then curve Y can be replaced with polygon Y ′ having n vertices. Figure. 1.7 should make it evident that the higher the number of consideredinstants, n, the better curve Y is described by polygon Y ′, that is, lim Ψ ′ = Ψ . n→∞

As will be explained in Chapter 3, the so-called critical plane approach postulates that the crack initiation phenomenon occurs on the plane experiencing the maximum shear stress amplitude, ta. The definitions that will be reviewed in the following subsections allow the shear stress amplitude, tn,a, to be determined on any material plane passing through the assumed critical point. Strictly speaking, the critical plane approach can then be applied only if all the material planes are explored in order to determine the orientation of the one experiencing the maximum shear stress amplitude. Even if this problem can easily be addressed by simple numerical procedures, sometimes the determination of the critical plane becomes extremely time-consuming, especially when long and complex load histories are considered. In order to reduce the time needed to calculate tn,a, different

Useful stress quantities used in fatigue problems

11

numerical techniques have been proposed in the technical literature and, amongst them, certainly the one due to Weber et al. (1999) deserves to be mentioned.

1.7.1 Determination of curve Y under out-of-phase tension (or bending) and torsion As said above, the different ways of defining the shear stress amplitude relative to a material plane will be reviewed in the next subsections, but, before doing that, it is worth addressing in more detail the problem of determining curve Y on a generic material plane. In particular, in the present subsection the equations proposed by Papadopoulos et al. (1997) and suitable for the tension (or bending) and torsion case are briefly reviewed. Consider then the body sketched in Fig. 1.1 and assume that point O, which is also the centre of the absolute frame of reference, is subjected to the following synchronous sinusoidal stress components: ⎡ σ x(t ) τ xy(t ) 0 ⎤ 0 0⎥ [σ (t )] = ⎢τ xy(t ) ⎥ ⎢ ⎢⎣ 0 0 0 ⎥⎦

1.28

σ x(t ) = σ x,m + σ x,a sin (ω t ) τ xy(t ) = τ xy,m + τ xy,a sin (ω t − δ xy, x )

1.29

where

In the above definition dxy,x is the out-of-phase angle between the two applied stress components. If stress tensor (1.28) is introduced into Eq. (1.21), it is possible to calculate, at any instant of the load history, the normal stress relative to a generic material plane, that is: 2 ⎧[σ x ,a sin (ω t ) + σ x ,m ] cos (φ ) ⎫ σ n( t ) = sin 2(θ ) ⎨ ⎬ ( ) 2 + sin − + τ sin φ τ ω t δ ( ) [ ] xy,m xy,a xy, x ⎩ ⎭

1.30

According to Eqs (1.24) and (1.25), the mean value and the amplitude of the above normal stress turn out to be: 2 2 σ n,a = sin 2(θ ) cos (φ ) σ x2,a cos 2(φ ) + 4τ xy ,a sin (φ ) + 2σ x ,aτ xy ,a sin ( 2φ ) cos (δ xy , x )

1.31

σ n,m = sin 2(θ ) [σ x,m cos 2(φ ) + τ xy,m sin ( 2φ )]

1.32

Equations (1.22) can now be used to determine the shear stress components relative to the considered generic plane, Δ, and resolved along axes a and b, i.e.:

12

Multiaxial notch fatigue

σ τ na(t ) = f ⋅ sin (ω t ) + g ⋅ cos (ω t ) + ⎡ − x,m sin ( 2φ ) + τ xy,m cos ( 2φ )⎤ sin (θ ) ⎢⎣ 2 ⎥⎦

1.33

τ xy,m σ sin ( 2φ )⎥⎤ sin ( 2θ ) 1.34 τ nb(t ) = p ⋅ sin (ω t ) + q ⋅ cos (ω t ) + ⎡⎢ − x,m cos 2(φ ) − 2 ⎣ 2 ⎦ where functions f, g, p and q are defined as follows:

σ f = sin (θ ) ⎡ − x,a sin ( 2φ ) + τ xy,a cos ( 2φ ) cos (δ xy, x )⎤ ⎢⎣ 2 ⎥⎦ g = −τ xy,a sin (θ ) cos ( 2φ ) sin (δ xy, x ) 1 p = − sin ( 2θ ) [σ x,a cos 2 (φ ) + τ xy,a sin ( 2φ ) cos (δ xy, x )] 2 1 q = τ xy,a sin ( 2θ ) sin ( 2φ ) sin (δ xy, x ) 2

1.35

The curve plotted according to Eqs (1.33) and (1.34) on a generic plane, Δ, is then an ellipse (Fig. 1.8) whose centre has the following coordinates:

σ Ca = ⎡ − x,m sin ( 2φ ) + τ xy,m cos ( 2φ )⎤ sin (θ ) ⎢⎣ 2 ⎥⎦ τ xy,m σ sin ( 2φ )⎤⎥ sin ( 2θ ) Cb = ⎡⎢ − x,m sin 2(φ ) − 2 ⎣ 2 ⎦

1.36

whereas the semi-length of the major and minor axes can be determined as follows: 2

Amax =

2 2 2 2 f 2 + g 2 + p2 + q 2 ⎛ f +g + p +q ⎞ 2 + ⎜ ⎟⎠ − ( fq − gp) ⎝ 2 2

Amin =

2 2 2 2 f 2 + g 2 + p2 + q 2 ⎛ f +g + p +q ⎞ 2 − ⎜ ⎟⎠ − ( fq − gp) ⎝ 2 2

1.37

2

b

1.38

Ψ

Δ

Amax Cb

Amin t n(t)

O

Ca

a

1.8 Elliptic curve Y on a generic material plane Δ under sinusoidal tension (or bending) and torsion.

Useful stress quantities used in fatigue problems

13

The relationships reported above should make it evident that only when a mechanical component is subjected to simple load histories can the curve plotted on a generic plane by the tip of the shear stress be obtained in explicit form. On the contrary, in the presence of complex loading paths, only numerical approaches can be used to determine curve Y on any material plane passing through the assumed crack initiation point.

1.7.2 The Longest Chord Method The equations reported in the above subsection suggest that the calculation of the shear stress amplitude relative to a material plane is not trivial, due to the fact that the shear stress vector, tn(t), varies in both magnitude and direction during the load cycle. This means that different definitions can successfully be used to address the above problem. The first method considered in the present section is the so-called Longest Chord Method which was devised by Lemaitre and Chaboche (1990). In more detail, this method takes as a starting point the idea that the shear stress amplitude is equal to half of the longest chord amongst those linking together any two points belonging to curve Y (Fig. 1.9a). In other words, according to the longest chord method, tn,a has to be calculated as follows:

τ n,a =

1 max ⎡ max τ n ( t1 ) − τ n ( t2 ) ⎤ ⎦ 2 t1∈T ⎣ t2∈T

1.39

where again t1 and t2 are two instants of the cyclic load history having period equal to T. The mean shear stress, tn,m, instead is equal to the magnitude of the vector joining point O to the midpoint of the longest chord (Fig. 1.9a). Even if the Longest Chord Method is really effective in defining the amplitude of the shear stress relative to a generic material plane, its use in the field hides some theoretical problems. In particular, as shown in Fig. 1.9b, the mean value cannot be defined unequivocally when two or more reference chords having the same length can be defined. In any case, it has A

Δ

b

b t n,a

A Ψ

Ψ

M

M t n,m?

O

a (a)

C

B

M t n,m?

t n,m t n(t)

Δ

AB = AC

t n(t)

O

B a

(b)

1.9 tn,a and tn,m defined according to the Longest Chord Method.

14

Multiaxial notch fatigue

to be said that, from a practical point of view, the above possible ambiguity in defining tn,m does not affect the accuracy in estimating high-cycle fatigue damage, because, as will be discussed in Chapter 2, the torsional mean stress effect can be neglected as long as, during the load cycle, the maximum shear stress is lower than the material torsional yield strength (Sines, 1959; Davoli et al., 2003).

1.7.3 The Longest Projection Method The Longest Projection Method takes as a starting point the idea that the amplitude as well as the mean value of the shear stress relative to a given material plane depend on the projection of curve Y on a straight line passing through point O (Fig. 1.10a). According to Grubisic and Simbürger (1976), the reference line to be used to calculate tn,a and tn,m is the one on which the projection of Y reaches its maximum length: tn,a is then equal to half the length of such a projection, whereas tn,m is equal to the distance between the origin, O, and the midpoint of the longest projection (Fig. 1.10a). To conclude, it can be pointed out that, similarly to the Longest Chord Method, under particular circumstances some inconsistencies may arise, resulting in an ambiguous definition of tn,m (see, for instance, Fig. 1.10b).

1.7.4 The Minimum Circumscribed Circle Method The theoretical problems highlighted when reviewing the above two methods can be fully overcome by using the so-called Minimum Circumscribed Circle Method proposed by Papadopoulos (1998). According to such a method, the

b

b

Δ Ψ

Ψ

t n(t)

A

Δ

t n(t)

A t n,a

O t n,m M

a t n,a

(a)

O≡M t n,m = 0?

a B

(b)

B

1.10 tn,a and tn,m defined according to the Longest Projection Method.

Useful stress quantities used in fatigue problems

15

n

O t n,m

t n(t)

a t n,a Minimum circumscribed circle

Ψ

b

Δ

1.11 tn,a and tn,m defined according to the Minimum Circumscribed Circle Method.

shear stress amplitude is equal to the radius of the minimum circumference fully containing curve Y, whereas tn,m is equal to the magnitude of the vector joining the origin of the frame of reference to the centre of the above circumference (Fig. 1.11). The fact that the minimum circumscribed circle always exists and is unique makes it evident that the Minimum Circumscribed Circle Method allows both tn,a and tn,m to be defined unequivocally. In order to calculate the above stress quantities according to such an approach, as briefly mentioned above, initially curve Y has to be schematised as a polygon having n vertices (polygon Y′ in Fig. 1.7). Consider then two generic instants of the cyclic load history, so that the following two points can be defined: P1 ≡ [τ na( t1 ) , τ nb( t1 )] P2 ≡ [τ na( t2 ) , τ nb( t2 )]

where t1 ∈T where t2 ∈T and t1 ≠ t2

with tna(t) and tnb(t) calculated according to Eqs (1.22). Now all the nD circles passing through the above two points have to be determined, where nD =

n! 2 ! ( n − 2 )!

The centre, having coordinates (Ca, Cb), and the radius, RD, of any of the above nD circumferences are given by the following trivial relationships:

τ na(t1 ) + τ na(t 2 ) 2 τ nb(t1 ) + τ nb(t 2 ) Cb = 2 Ca =

1.40

16

Multiaxial notch fatigue RD =

1 2

[τ na(t1 ) − τ na(t2 )]2 − [τ nb(t1 ) − τ nb(t2 )]2

1.41

Now another series of nT circles passing through three points have to be determined, where nT =

n! 3!( n − 3 )!

The three points, P1, P2 and P3, needed to determine any of the above circumferences belong to curve Y′ and have the following coordinates: P1 ≡ [τ na( t1 ) , τ nb(τ 1 )] P2 ≡ [τ na( t2 ) , τ nb( t2 )] P3 ≡ [τ na( t3 ) , τ nb( t3 )]

where t1 ∈T where t2 ∈T and t2 ≠ t1 where t3 ∈T and t3 ≠ t2 ≠ t1

The centre of each circle, having coordinates (Ca, Cb), and the corresponding radius, RT, can easily be obtained by solving the following equations: 2 2 ⎧ RT2 = [Ca − τ na(t1 )] − [Cb − τ nb(t1 )] ⎪ 2 2 2 ⎨ RT = [Ca − τ na(t 2 )] − [Cb − τ nb(t 2 )] ⎪ 2 2 2 ⎩ RT = [Ca − τ na(t3 )] − [Cb − τ nb(t3 )]

1.42

Amongst the nD + nT circumferences determined as above, only those fully containing curve Y, that is, only those containing all the vertices of polygon Y′, have to be extracted, selecting the one having the minimum radius. If the minimum circumscribed circle has radius equal to Rmin and the coordinates of the centre are (Ca,min, Cb,min), then the Minimum Circumscribed Circle Method postulates that the amplitude and the mean value of the shear stress relative to a given material plane turn out to be:

τ n,a = Rmin

1.43

τ n,m = Ca2,min + Cb2,min

1.44

Even if the Minimum Circumscribed Circle Method allows the problems highlighted when reviewing the previous definitions to be overcome, its use in the field to determine the shear stress amplitude relative to a given material plane can be extremely time-consuming, especially when complex load histories are involved. In order to increase the efficiency of such a method, Bernasconi has recently devised some efficient algorithms allowing tn,a and tn,m to be determined by considerably reducing the required computational time (Bernasconi, 2002).

Useful stress quantities used in fatigue problems

17

1.7.5 The Minimum Circumscribed Ellipse Method The main advantage of the Minimum Circumscribed Circle Method is that it allows the relevant shear quantities relative to a generic material plane, Δ, to be determined unequivocally. Unfortunately, by carefully observing the way such a method works, one may argue that the above approach is not sensitive to the real shape of curve Y: all the shear stress loading paths having the same radius of the minimum circumscribed circumference are characterised by the same value of the amplitude, and this holds true independently of the complexity of the load history experienced by the component to be assessed. In order to overcome the above problem, different authors have recently suggested using the minimum circumscribed ellipse instead of the minimum circumscribed circle to define the shear stress amplitude relative to a material plane (Li et al., 2000, 2001; Zouain et al., 2006). In more detail, consider the closed curve, Y, plotted by the tip of the shear stress vector, tn(t), on a generic material plane Δ (Fig. 1.12). Assuming that the minimum ellipse fully containing curve Y is known, then according to the Minimum Circumscribed Ellipse Method, the mean value of the shear stress, tn,m, is equal to the magnitude of the vector joining point O to the centre of the ellipse, whereas the shear stress amplitude is defined as follows (Li et al., 2000): 2 2 τ n,a = Rmin + Rmax

1.45

In the above definition, Rmin and Rmax are the semi-lengths of the minor and major axes of the minimum circumscribed ellipse, respectively (Fig. 1.12). The main feature of such a definition is that the actual shape of curve Y is now better taken into account when defining the shear stress amplitude relative to a generic material plane. The main drawback instead

b

Δ

Minimum circumscribed ellipse Ψ Rmax

Rmin M

t n,m t n(t) O

a

1.12 tn,a and tn,m defined according to the Minimum Circumscribed Ellipse Method.

18

Multiaxial notch fatigue

is that the use of different numerical strategies to determine the minimum circumscribed ellipse may result in different values of both tn,a and tn,m (Zouain et al., 2006; Kumar and Alper Yildirim, 2005). As to the latter aspect, Zouain et al. (2006) have observed that the highest level of accuracy in estimating multiaxial high-cycle fatigue strength is obtained by calculating the minimum ellipse through its Frobenius norm, with the advantage over the other existing methods that the minimum ellipse calculated in this way is unique and does not depend on the adopted numerical algorithm. From a practical point of view, it is also important to observe that Araújo and co-workers (Gonçalves et al., 2005) have formalised a procedure suitable for directly determining the minimum ellipse under synchronous harmonic loadings, allowing those situations involving biaxial loading paths to be addressed by using simple analytical solutions. Finally, it is interesting to point out also that Zouain et al. (2006) have recently developed an efficient algorithm allowing the minimum ellipse (again through its Frobenius norm) to be determined independently of the complexity of the investigated multiaxial loading path. According to the above considerations, in order to correctly use the minimum ellipse concept, it is suggested that it should be applied by always following the strategies devised by our colleagues Professor Araújo and Zouain.

1.7.6 The Maximum Variance Method For the reasons briefly mentioned at the beginning of Section 1.7, the infield use of the definitions reviewed above could become extremely timeconsuming when employed to determine the plane experiencing the maximum shear stress amplitude, ta: strictly speaking, the shear stress amplitude relative to any plane passing through the considered material point should be calculated by searching for the plane on which tn,a reaches its maximum value. Contrary to the aforementioned methods, the approach presented in this subsection allows the critical plane to be defined directly through the direction along which the variance of the resolved shear stress is maximised. This approach was devised by our colleague Professor Roberto Tovo (Bel Knani et al., 2007) taking as a starting point the work done by Macha and coworkers (Łagoda et al., 1996). It is the writer’s opinion that such a statistical method is extremely promising not only because it is highly efficient from a computational point of view, but also because, when in-phase and out-ofphase uniaxial and torsional fatigue loading are considered, it allows the plane of maximum shear stress amplitude to be determined by using simple explicit formulas.

Useful stress quantities used in fatigue problems

19

In order to better understand the main features of such a method, it is worth remembering here that, in the presence of sinusoidal signals, the variance is always proportional to the amplitude and not to the mean value of the signal itself. This implies that, in such circumstances, the maximum variance method correctly defines the orientation of the critical plane by determining the plane containing the direction experiencing the maximum amplitude of the resolved shear stress. Similarly, also under general random uniaxial loading, fatigue damage is seen to be proportional to the variance of the load history and this holds true under both Gaussian (Benasciutti and Tovo, 2005a) and nonGaussian (Benasciutti and Tovo, 2005b) loading: the above considerations should make it evident that the maximum variance method is based on a sound theoretical background fully supported by the experimental evidence. Consider then a time-variable process s(t) having mean value m = E[s(t)] and defined over the time interval [0, T]. By definition, the variance of the above process turns out to be: var ( s (t )) =

1T ( s ( t ) − m )2 d t ∫ T0

1.46

As outlined above, the method reviewed in the present section assumes that the critical plane orientation is unequivocally located by the direction experiencing the maximum variance of the resolved shear stress. Consider now a generic direction q, defined by unit vector q, on a given material plane Δ (Fig. 1.13): the orientation of q is fully defined by angle a, which is the angle between unit vector q and the a-axis. The components of vector q can then be written as follows: qx = cos (α ) sin (φ ) + sin (φ ) cos (θ ) cos (φ ) qy = − cos (α ) cos (φ ) + sin (α ) cos (θ ) sin (φ )

1.47

qz = − sin (α ) sin (θ ) b

Δ

Ψ t n(t)

q t q(t) q O

a a

1.13 Definition of direction q and corresponding resolved shear stress.

20

Multiaxial notch fatigue

At any given instant, t, of the load history, the shear stress, tq(t), resolved along direction q, can then be calculated as:

τ q ( t ) = [ qx

⎡ σ x(t ) τ xy(t ) τ xz(t )⎤ ⎡ nx ⎤ qz ] ⎢τ xy(t ) σ y(t ) τ yz(t )⎥ ⎢ ny ⎥ ⎢ ⎥⎢ ⎥ ⎣⎢τ xz(t ) τ yz(t ) σ z(t ) ⎥⎦ ⎢⎣ nz ⎥⎦

qy

1.48

In order to formalise the maximum variance method, stress tensor [s(t)] at point O can now be expressed as a six-dimensional vector process, [s(t)]:

[ s (t )] = [σ x(t ) σ y(t ) σ z(t ) τ xy(t ) τ xz(t ) τ yz(t )]

1.49

Using the above vector, the resolved shear stress, tq(t), can be rewritten as the scalar product between [s(t)] and the vector of the direction cosines, [d]:

τ q( t ) = [ d ]⋅ [ s ( t )]

1.50

where ⎡ 12 (sin (θ ) sin ( 2φ ) cos (α ) + sin (α ) sin ( 2θ ) cos (φ ) )⎤ ⎢1 2 ⎥ ⎢ 2 (sin (θ ) sin ( 2φ ) cos (α ) + sin (α ) sin ( 2θ ) sin (φ ) ) ⎥ ⎢ − 1 sin (α ) sin ( 2θ ) ⎥ [d ] = ⎢ 2 ⎥ 1 ⎢ 2 sin (α ) sin ( 2φ ) sin ( 2θ ) − cos (α ) cos ( 2φ ) sin (θ ) ⎥ ⎢sin (α ) cos (φ ) cos ( 2θ ) + cos (α ) sin (φ ) cos (θ ) ⎥ ⎢ ⎥ ⎣⎢sin (α ) sin (φ ) cos ( 2θ ) − cos (α ) cos (φ ) cos (θ ) ⎦⎥ 2

1.51

According to the above relationships, the problem of determining the variance of the shear stress resolved along direction q can be explicitly addressed by using the following relationship: var (τ q(t )) = var

(∑ k

)

dk sk (t ) = ∑ ∑ di dj cov ( si (t ) , s j (t )) i

1.52

j

where, for i = j, Eq. (1.52) results in the variance terms, since cov(si(t), si(t)) = var(si(t)), whereas when i ≠ j it results in the covariance terms. Identity (1.52) can also be rewritten as: var (τ q(t )) = [d ] [C ][d ] T

1.53

In the above definition matrix [C] is a symmetric squared matrix of order six, in which the diagonal elements are the stress variance terms, whereas the other elements are the stress covariance terms. Therefore, by assuming that the critical plane is the one containing the direction along which the variance of the resolved shear stress, tq(t), reaches its maximum value, the problem of determining the critical plane orientation becomes very easy to

Useful stress quantities used in fatigue problems

21

handle by using conventional and very well-established numerical algorithms. In particular, from a computational point of view, the main advantage of such an approach is that in Eq. (1.53) the orientation of direction q is defined by vector [d], which contains only simple trigonometric quantities: this results in an analytical problem whose solution is simple and straightforward. Finally, after calculating the values of angles f and q determining the orientation of the critical plane, the shear stress amplitude, ta, relative to such a plane can directly be estimated by using one of the definitions reviewed in the previous subsections. Critical plane determined according to the Maximum Variance Method under tension (or bending) and torsion Consider a cylindrical bar loaded in combined tension (or bending) and torsion (Fig. 1.14). By defining the appropriate system of coordinates, in a generic instant t of the load history the stress tensor at point O turns out to be: ⎡ σ x(t ) τ xy(t ) 0 ⎤ 0 0⎥ [σ (t )] = ⎢τ xy(t ) ⎢ ⎢⎣ 0

0

1.54

⎥ 0 ⎥⎦

The corresponding stress vector [s(t)] is then:

[ s (t )] = [σ x(t ) 0 0 τ xy(t ) 0 0 ]

1.55

According to Eq. (1.48), the shear stress, tq(t), resolved along a generic direction q can be calculated as:

τ q(t ) = ⎡⎣ 12 (sin (θ ) sin ( 2φ ) cos (α ) + sin (α ) sin ( 2θ ) cos (φ )2 )⎤⎦ σ x(t )

+ [ 12 sin (α ) sin ( 2φ ) sin ( 2θ ) − cos (α ) cos ( 2φ ) sin (θ )]τ xy(t )

1.56

Mt(t) F(t) z y O

x Mt(t)

F(t)

1.14 Cylindrical specimen subjected to combined tension and torsion.

22

Multiaxial notch fatigue

Observing now that under tension (or bending) and torsion the critical plane is always perpendicular to the surface of the component, it is evident that the direction along which tq(t) reaches its maximum value depends only on angle f, that is:

τ q( t ) =

sin ( 2φ ) σ x(t ) − cos ( 2φ ) τ xy(t ) 2

1.57

For the considered loading path, the covariance matrix, [C], can easily be rewritten as: ⎡ Vσ ⎢ 0 ⎢ ⎢ 0 [C ] = ⎢ ⎢Cσ ,τ ⎢ 0 ⎢ ⎣ 0

0 0 0 0 0 0

0 Cσ ,τ 0 0 0 0 0 Vτ 0 0 0 0

0 0 0 0 0 0

0⎤ 0⎥ ⎥ 0⎥ 0 ⎥⎥ 0⎥ ⎥ 0⎦

1.58

where Vσ = var [σ x(t )] Vτ = var [τ xy(t )]

1.59

Cσ ,τ = cov [σ x(t ) , τ xy(t )] Finally, the variance of the resolved shear stress, tq(t), turns out to be: var [τ q(t )] = f (φ ) =

sin ( 2φ ) 2 Vσ + cos ( 2φ ) Vτ − sin ( 2φ ) cos ( 2φ ) Cσ ,τ 4 2

1.60

The values of angle f defining the orientation of the planes of maximum variance can be obtained by simply solving the following equation: f ′(φ ) = (Vσ − 4Vτ ) sin ( 4φ ) − 4Cσ ,τ cos ( 4φ ) = 0

1.61

whereas maxima and minima can be determined from the second derivative: f ′′(φ ) = 4 (Vσ − 4Vτ ) cos ( 4φ ) + 16Cσ ,τ sin ( 4φ )

1.62

Table 1.1 summarises the possible solutions that are obtained following the procedure described above. If the cylindrical bar considered at the beginning of this section is subjected to synchronous fully-reversed sinusoidal loading, then the stress components of tensor (1.54) can be expressed as follows:

σ x(t ) = σ x,a sin (ω t ) τ xy(t ) = τ xy,a sin (ω t + δ xy, x )

1.63

Useful stress quantities used in fatigue problems

23

Table 1.1 Values of angle f defining the orientation of the plane containing the direction experiencing, under tension (or bending) and torsion, the maximum variance of the resolved shear stress* Condition

Cs,t ≠ 0

Vs − 4Vt < 0

φ0 + i

Vs − 4Vt = 0 Vs − 4Vt > 0

π 2 3π π +i 8 2

(φ + 4π ) + i π2 0

* φ0 =

Cs,t = 0 π 2 Any f i

π π +i 4 2

1 ⎛ 4Cσ ,τ ⎞ arctan ⎜ ; i = 0, 1, 2, . . . ⎝ Vσ − 4Vτ ⎟⎠ 4

where dxy,x is again the out-of-phase angle between the above two stress components, and the period, T, of the cyclic load history is equal to 2π/w, w being the angular velocity. The variance and covariance terms over T turn out to be: Vσ =

1T 2 σ2 2 σ x ,a sin (ω t ) dt = x ,a ∫ T0 2

Vτ =

τ2 1T 2 2 τ xy,a sin (ω t + δ xy, x ) dt = xy,a ∫ T0 2

Cσ ,τ =

1.64

σ τ cos (δ xy, x ) 1T σ x ,aτ xy,a sin (ω t ) sin (ω t + δ xy, x ) dt = xa xy,a ∫ T0 2

Finally, starting from the covariance matrix [C] and following the same procedure as the one summarised at the beginning of the present subsection, it is trivial to obtain the solutions reported in Table 1.2 which allow the critical plane to be defined under fully-reversed sinusoidal tension (or bending) and torsion in a simple and direct way.

1.7.7 Concluding remarks The above sections should make it evident that, for a given shear stress load history acting on a fixed material plane, the use of the aforementioned strategies may result in different values of the calculated shear stress amplitude, especially when complex loading paths are involved. This implies that, apart from the philosophical problems hidden behind any of the above methods, it is very difficult to express a final verdict indicating which strategy is the most efficient to be used in situations of practical interest. In any

24

Multiaxial notch fatigue Table 1.2 Values of angle f defining the orientation of the plane containing the direction experiencing, under fully-reversed sinusoidal tension (or bending) and torsion, the maximum variance of the resolved shear stress* Condition

dx,xy ≠ π/2

sx,a − 2txy,a < 0

φ0 + i

sx,a − 2txy,a = 0

3π π +i 8 2

sx,a − 2txy,a > 0

(φ + 4π ) + i π2

π 2

π 2 Any f i

0

* φ0 =

cos δ 1 ⎛ 4σ τ arctan ⎜ x ,a 2 xy ,a 2 xy , x σ x,a − 4τ xy ,a 4 ⎝

dx,xy = π/2

π π +i 4 2

⎞ ⎟ ; i = 0, 1, 2, . . . ⎠

case, it is the writer’s opinion that the Longest Chord Method is not only very simple to use but also very effective when applied in conjunction with the critical plane concept. Accordingly, in the next chapters it is always intended that the maximum shear stress amplitude is to be determined using the Longest Chord Method.

1.8

Stress concentration factor, Kt

Real mechanical components contain a variety of geometrical features resulting in stress concentration phenomena. Such phenomena must always be taken into account during the design process due to their detrimental effect on the material fatigue strength. The problem of accounting for the influence of stress raisers will be addressed extensively in the next chapters, considering both uniaxial and multiaxial fatigue situations. In the present section the definition of the stress concentration factor as proposed by Peterson (1974) is briefly reviewed. Consider a body of homogeneous, isotropic, linear-elastic material containing stress raisers. Initially it is useful to remember here that, when dealing with simple geometries, it is always possible to define two different types of reference stress, i.e., the net (snet) and the gross (sgross) nominal stress, where the first is calculated with respect to the net cross-sectional area and the latter with respect to the gross section. In order to correctly define the stress concentrator factor, Kt, consider now the U-notched plate loaded in tension that is sketched in Fig. 1.15a.

Useful stress quantities used in fatigue problems F

25

s1 s ep

s gross s net

Distance

(a)

F

(b)

1.15 Definition of (a) net and gross nominal stress, and (b) linearelastic peak stress at the tip of a notch.

Owing to the geometrical features of this plate, the maximum linear-elastic principal stress, s1, gradually decays along the bisector from the apex of the notch to the mid-section of the specimen itself (Fig. 1.15b). It is also worth remembering that, if the net width of the sample is large enough, the above stress field tends to its nominal net value. At the tip of the notch instead the maximum principal stress reaches its maximum value, i.e. the so-called linear-elastic peak stress, sep. The above stress quantities allow the stress concentration factor, Kt, to be defined as follows: Kt ,net =

σ ep σ net

Kt ,gross =

σ ep σ gross

1.65 1.66

where definitions (1.65) and (1.66) refer to the net and gross nominal section, respectively (Fig. 1.15a). The stress concentration factor is a stress parameter widely used in notch fatigue problems. Its values, for a given loading configuration, depend only on the shape of the component to be assessed and not on its absolute dimensions (Peterson, 1974).

1.9

Singular stress fields

As explained briefly in the previous section, the effect of notches having a finite tip radius on the linear-elastic stress field distributions is usually quantified through the stress concentration factor, Kt. On the contrary, when linear-elastic materials are weakened by sharp geometrical

26

Multiaxial notch fatigue

discontinuities (such as cracks), the stress analysis problem becomes much more complex, because the linear-elastic stress fields in the vicinity of the stress raiser apices become singular, resulting in values of the peak stress tending to infinity. Over the last century several analytical tools have been devised in order to efficiently and concisely describe stress fields in the presence of geometrical singularities, without missing the doubtless advantages of linear-elastic analysis. Before briefly reviewing the equations mentioned above, it is useful to remember here that geometrical discontinuities (like cracks or notches) can be subjected to three different types of loading. In more detail, when a stress raiser is loaded in tension it experiences Mode I loading, when subjected to in-plane shear stress it experiences Mode II loading and, finally, under anti-plane shear the stress concentrator is said to be subjected to Mode III loading (Fig. 1.16). Consider now a plate loaded in tension and containing a central throughcrack (Fig. 1.17). The width, w, of the plate is initially assumed to be much larger than the crack length, 2a. According to Irwin, the linear-elastic stress field damaging the material in the vicinity of the above crack tip can be described by using the following well-known relationships:

θ θ 3θ KI cos ⎡1 − sin sin ⎤ 2 ⎣⎢ 2 2 ⎦⎥ 2 πr θ θ 3θ KI σy = cos ⎡1 + sin sin ⎤ 2 ⎢⎣ 2 2 ⎥⎦ 2 πr 3θ KI θ θ sin cos cos τ xy = 2 2 2 2 πr σx =

1.67

and in plane strain (ne = Poisson’s ratio for elastic strain):

σz =

2ν e KI θ cos 2 2 πr

whereas in plane stress sz is invariably equal to zero.

Mode I loading

Mode II loading

Mode III loading

1.16 Definition of the three fundamental loading modes.

Useful stress quantities used in fatigue problems

27

F

sy y r

t xy sx

q x

2a

w

F

1.17 Central through-crack in a plate loaded in tension.

In Eqs (1.67) stress components sx, sy and txy are calculated with respect to the frame of reference as defined in Fig. 1.17 and KI is the so-called stress intensity factor (SIF). The above equations can always be used to describe linear-elastic stress fields in cracked bodies, provided that the considered component is subjected to Mode I loading and the distance, r, from the crack tip is lower than about a/10. Stress intensity factors are widely used to assess real cracked components subjected to both static and fatigue loading and their values depend on the specific geometry/load configuration of the problem to be addressed. For instance, the SIF for the flat cracked specimen sketched in Fig. 1.17 can be calculated as follows: KI = σ gross π a

1.68

On the contrary, if in the above sample the length of the crack cannot be assumed to be small compared to the plate width, then the influence of the actual geometry has to be taken directly into account through a shape factor F. Considering then such a shape factor, the above equation can be rewritten as: KI = F ⋅ σ gross π a

1.69

where, according to Irwin, F can be calculated as: F=

( )

w πa tan w πa

1.70

28

Multiaxial notch fatigue

It is important to remember here that, apart from the case of a central through-crack in an infinite plate loaded in tension, in general F is always larger than unity and, for a given geometrical configuration, its value depends not only on the shape of the considered component but also on the type of applied loading and on other factors (Tada et al., 2000). Apart from the aforementioned classical relationships, in general both KI and F can be determined either by using numerical/analytical methods or by taking advantage of many handbooks reporting their values for those situations which are commonly encountered when addressing practical problems (Murakami, 1987; Tada et al., 2000). Owing to the important role played by singular stress fields in local approaches, it also worth reviewing here some analytical tools suitable for describing the linear-elastic fields in the presence of sharp re-entrant corners. Consider then the V-notched sample sketched in Fig. 1.18 and subjected to Mode I, II and III loading. Thanks to the geometry of the investigated stress raiser, the stress components due to the three fundamental modes are always uncoupled along the notch bisector. According to the frame of reference defined in the above figure, the stress fields due to Mode I and II loading can concisely be described by the following relationships devised by Lazzarin and Tovo (1996): ⎧σ θ ⎫ ⎪ ⎪ ⎨σ r ⎬ = ⎪⎩τ ⎪⎭ rθ

⎧σ θ ⎫ ⎪ ⎪ ⎨σ r ⎬ = ⎪⎩τ ⎪⎭ rθ

⎡ ⎧(1 + λ1 ) cos (1 − λ1 ) θ ⎫ 1 r λ1 −1 K1 ⎪ ⎪ ⋅ ⎢ ⎨( 3 − λ1 ) cos (1 − λ1 ) θ ⎬ ⎢ 2 π (1 + λ1 ) + χ1(1 − λ1 ) ⎪ ⎢⎣ ⎩(1 − λ1 ) sin (1 − λ1 ) θ ⎪⎭ ⎧cos (1 + λ1 ) θ ⎫⎤ ⎪ ⎪ + χ1(1 − λ1 ) ⎨ − cos (1 + λ1 ) θ ⎬⎥ ⎥ ⎪⎩sin (1 + λ ) θ ⎪⎭⎦⎥ 1 ⎡ ⎧ − (1 + λ 2 ) sin (1 − λ 2 ) θ ⎫ 1 r λ2 −1 K 2 ⎪ ⎪ ⋅ ⎢ ⎨ − ( 3 − λ 2 ) sin (1 − λ 2 ) θ ⎬ 2 π (1 − λ 2 ) + χ 2 (1 + λ 2 ) ⎢ ⎪ ⎢⎣ ⎩(1 − λ 2 ) cos (1 − λ 2 ) θ ⎪⎭ ⎧ − sin (1 + λ 2 ) θ ⎫⎤ ⎪ ⎪ + χ 2(1 + λ 2 ) ⎨sin (1 + λ 2 ) θ ⎬⎥ ⎥ ⎪⎩cos (1 + λ ) θ ⎪⎭⎥⎦ 2

1.71

1.72

where, as shown in Table 1.3, constants l1, l2, c1 and c2 depend on the opening angle value, 2a (Atzori et al., 1999; Lazzarin and Tovo, 1996; Williams, 1952). Similarly, when a cracked body is subjected to anti-plane loading (Fig. 1.18), the resulting linear-elastic stress field can be described as (Dunn et al., 1997; Qian and Hasebe, 1997):

Useful stress quantities used in fatigue problems

y

sq

t rq

t qz

sr

r 2a

29

q

sz

t rz x

z

1.18 V-notched plate subjected to Mode I, II and III loading. Table 1.3 Values of constants l1, l2, c1 and c2 in Lazzarin and Tovo’s equations 2a (rad)

λ1

c1

λ2

c2

0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6

0.500 0.501 0.505 0.512 0.544 0.616 0.674 0.752

1.000 1.071 1.166 1.312 1.841 3.003 4.153 6.362

0.500 0.598 0.660 0.731 0.909 1.149 1.302 1.486

1.000 0.921 0.814 0.658 0.219 −0.314 −0.569 0.787

{ }

τθz = τ rz

{

}

cos ( λ3θ ) 1 λ3 − 1 r K3 sin ( λ3θ ) 2π

1.73

where, again, l3 depends on the notch-opening angle. In Eqs (1.71)–(1.73) K1, K2 and K3 are now the so-called notch-stress intensity factors (N-SIF) due to Mode I, Mode II and Mode III loading, respectively. These factors can be calculated directly by applying the following definitions (Gross and Mendelson, 1972; Dunn et al., 1997; Qian and Hasebe, 1997): K1 = 2 π lim (σ θ )θ = 0 r 1− λ1 r→0

K 2 = 2 π lim (τ rθ )θ = 0 r 1− λ2 r→0

K3 = 2 π lim (τ θ z )θ = 0 r 1− λ3 r→0

1.74

30

Multiaxial notch fatigue

It is important to highlight that, when the notch-opening angle, 2a, is equal to zero (i.e. in the presence of a crack) and the specimen of Fig. 1.18 is subjected to Mode I loading, the use of Eq. (1.71) results in the same stress field as the one given by Irwin’s equations (1.67). As will shortly be described in the next chapters, when the stress fields damaging the fatigue process zones can be assumed to be singular, both SIFs and N-SIFs are powerful stress parameters which can successfully be used to perform the fatigue assessment. Therefore, it is worth concluding this section by noting that, like other quantities commonly used in fatigue problems, SIFs and N-SIFs are generally employed in terms of their range, that is: ΔK = K max − K min

1.75

where Kmax and Kmin are the maximum and minimum values of K during the load cycle, respectively.

1.10

References

Atzori, B., Lazzarin, P., Tovo, R. (1999) Stress field parameter to predict the fatigue strength of notched components. Journal of Strain Analysis for Engineering Design 34, 437–453. DOI: 10.1243/0309324991513876. Bel Knani, K., Benasciutti, D., Signorini, A., Tovo R. (2007) Fatigue damage assessment of a car body-in-white using a frequency-domain approach. International Journal of Materials and Product Technology 30, 172–198. DOI: 10.1504/IJMPT.2007.013113. Benasciutti, D., Tovo, R. (2005a) Spectral methods for lifetime prediction under wide-band stationary random processes. International Journal of Fatigue 27, 867–877. DOI: 10.1016/j.ijfatigue.2004.10.007. Benasciutti, D., Tovo, R. (2005b) Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes. Probabilistic Engineering Mechanics 20, 115–127. DOI: 10.1016/j.probengmech.2004.11.001. Bernasconi, A. (2002) Efficient algorithms for calculation of shear stress amplitude and amplitude of the second invariant of the stress deviator in fatigue criteria applications. International Journal of Fatigue 24, 649–657. DOI: 10.1016/S01421123(01)00181-5. Davoli, P., Bernasconi, A., Filippini, M., Foletti, S., Papadopoulos, I. V. (2003) Independence of the torsional fatigue limit upon a mean shear stress. International Journal of Fatigue 25, 471–480. DOI: 10.1016/S0142-1123(02)00174-3. Dunn, M. L., Suwito, W., Cunningham, S. (1997) Fracture initiation at sharp notches: correlation using critical stress intensities. International Journal of Solids and Structures 34, 3873–3883. DOI: 10.1016/S0020-7683(96)00236-3. Gonçalves, C. A., Araújo, J. A., Mamiya, E. N. (2005) Multiaxial fatigue: a stress based criterion for hard metals. International Journal of Fatigue 27, 177–187. DOI: 10.1016/j.ijfatigue.2004.05.006.

Useful stress quantities used in fatigue problems

31

Gross, R., Mendelson, A. (1972) Plane elastostatic analysis of V-notched plates. International Journal of Fracture 8, 267–276. DOI: 10.1007/BF00186126. Grubisic, V., Simbürger, A. (1976) Fatigue under combined out of phase multiaxial stresses. In Proceedings of International Conference on Fatigue Testing and Design, Society of Environmental Engineers, London, pp. 27.1–27.8. Kumar, P., Alper Yildirim, E. (2005) Minimum value enclosing ellipsoids and core sets. Journal of Optimization Theory and Applications 126, 1–21. DOI: 10.1007/ s10957-005-2653-6. Łagoda, T., Macha, E., Dragon, A., Petit, J. (1996) Influence of correlations between stresses on calculated fatigue life of machine elements. International Journal of Fatigue 18, 547–555. DOI: 10.1016/S0142-1123(96)00025-4. Lazzarin, P., Tovo, R. (1996) A unified approach to the evaluation of linear elastic stress fields in the neighbourhood of cracks and notches. International Journal of Fracture 78, 3–19. DOI: 10.1007/BF00018497. Lemaitre, J., Chaboche, J. L. (1990) Mechanics of Solid Materials. Cambridge University Press, Cambridge. Li, B., Santos, J. L. T., de Freitas, M. (2000) A unified numerical approach for multiaxial fatigue limit evaluation. Mechanics of Structures and Machines 28, 85–103. Li, B., Santos, J. L. T., de Freitas, M. (2001) A computerized procedure for long-life fatigue assessment under multiaxial loading. Fatigue and Fracture of Engineering Materials and Structures 24, 165–177. DOI: 10.1046/j.1460-2695.2001.00389.x. Matake, T. (1977) An explanation on fatigue limit under combined stress. Bulletin of JSME, 20, No. 141, 257–263. McDiarmid, D. L. (1991) A general criterion for high cycle multiaxial fatigue failure. Fatigue and Fracture of Engineering Materials and Structures 14, 429–453. DOI: 10.1111/j.1460-2695.1991.tb00673.x. McDiarmid, D. L. (1994) A shear-stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction. Fatigue and Fracture of Engineering Materials and Structures 17, 1475–1484. DOI: 10.1111/ j.1460-2695.1994.tb00789.x. Murakami, Y. (1987) Stress Intensity Factors Handbook. Vol. 1/2. Pergamon Press, Oxford. Papadopoulos, I. V. (1998) Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane. Fatigue and Fracture of Engineering Materials and Structures 21, 269–285. DOI: 10.1046/j.1460-2695.1998.00459.x. Papadopoulos, I. V., Davoli, P., Gorla, C., Filippini, M., Bernasconi, A. (1997) A comparative study of multiaxial high-cycle fatigue criteria for metals. International Journal of Fatigue 19, 219–235. DOI: 10.1016/S0142-1123(96)00064-3. Peterson, R. E. (1974) Stress Concentration Factors. John Wiley & Sons, New York. Qian, J., Hasebe, H. (1997) Property of eigenvalues and eigenfunctions for an interface V-notch in antiplane elasticity. Engineering Fracture Mechanics 56, 729– 734. DOI: 10.1016/S0013-7944(97)00004-0. Sines, G. (1959) Behaviour of metals under complex static and alternating stresses. In: Metal Fatigue, edited by G. Sines and J.L. Waisman, McGraw-Hill, New York, pp. 145–169.

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Socie, D. F., Marquis, G. B. (2000) Multiaxial Fatigue, Society of Automotive Engineers, Warrendale, PA. Tada, H., Paris, P. C., Irwin G. R. (2000) Stress Analysis of Cracks Handbook. 3rd edition. ASME, New York. Viola, E. (1992) Scienza delle Costruzioni 1. Teoria dell’Elasticità. Pitagora Editrice, Bologna (in Italian). Weber, B., Kenmeugne, B., Clement, J. C., Robert, J. L. (1999) Improvements of multiaxial fatigue criteria computation for a strong reduction of calculation duration. Computational Materials Science 15, 381–399. DOI: 10.1016/S09270256(98)00129-3. Williams, M. L. (1952) Stress singularities resulting from various boundary conditions in angular corners of plate in extension. Journal of Applied Mechanics 19, 526–528. Zouain, N., Mamiya, E. N., Comes, F. (2006) Using enclosing ellipsoids in multiaxial fatigue strength criteria. European Journal of Mechanics A/Solids 25, 51–71. DOI: 10.1016/j.euromechsol.2005.07.006.

2 Fundamentals of fatigue assessment

Abstract: The present chapter attempts to review the most important concepts that should always be borne in mind when performing the fatigue assessment. In particular, the uniaxial fatigue problem is initially addressed by considering those parameters affecting the overall fatigue strength of engineering materials. Subsequently, some aspects related to the torsional as well as to the multiaxial fatigue problem are briefly investigated. Key words: fatigue, mean stress effect, notches, torsion, Linear Elastic Fracture Mechanics.

2.1

Introduction

Over the last century a big effort has been made by researchers engaged in studying the fatigue problem in order to propose sound engineering tools suitable for assessing mechanical assemblies subjected to timevariable loading. Thanks to such extensive and systematic investigations, nowadays, when assessing real components, engineers can take full advantage of many well-established methods as well as of many experimental findings. The present chapter attempts to review the most important concepts that should always be borne in mind when performing the fatigue assessment. In particular, the uniaxial fatigue problem will initially be addressed by considering those parameters affecting the overall fatigue strength of engineering materials. Subsequently, some aspects related to the torsional as well as to the multiaxial fatigue problem will be investigated. It is worth noting also that these basic concepts will be reviewed considering not only the theories based on continuum mechanics but also those based on Linear-Elastic Fracture Mechanics (LEFM). To conclude, it is important to highlight that there exist many textbooks exhaustively addressing the fatigue assessment problem, so that, if necessary, the reader is referred to such books in order to deepen the basic ideas which will only be reviewed in the present chapter – see, for instance, Frost et al. (1974), Hertzberg (1996), Pook (2000), Socie and Marquis (2000), Stephens et al. (2000) and Taylor (2007). 33

34

Multiaxial notch fatigue

2.2

Fatigue strength and Wöhler curves

Consider a conventional plain specimen (Fig. 2.1a) loaded in tension– compression under a load ratio, R, equal to −1 (that is, during the test the mean stress is kept constant and equal to zero). After a certain number of cycles, which depends on the fatigue properties of the material the specimen is made of, fatigue breakage occurs. It is well known that by reducing the amplitude of the applied loading, the number of cycles to failure increases. By testing different samples at different stress levels, it is possible to build the so-called Wöhler diagram of the tested material (Fig. 2.1b). Such a loglog diagram has in the abscissa the number of cycles to failure, Nf, and in the ordinate the amplitude of the applied stress, sx,a (Sedeckyj, 2001). As an example, Fig. 2.2 reports the results we generated in our laboratory by testing, under fully reversed uniaxial loading, plain specimens of En3B, a commercial cold-rolled low-carbon steel. Figure 2.2 clearly shows also that fatigue results are always characterised by a physiological scattering, so that experimental data have to be postprocessed carefully in order to determine a reliable reference fatigue curve (the so-called S–N curve or Wöhler curve). In particular, Fig. 2.2 reports three different straight trend lines which have been calculated by considering three different values of the probability of survival, PS, that is, 10%, 50% and 90% (Hertzberg, 1996). Such curves have been determined under the hypothesis of a log-normal distribution of the number of cycles to failure for each stress level and assuming a confidence level equal to 95% (Spindel and Haibach, 1981). When designing real components it is always advisable to perform the fatigue assessment by using fatigue curves characterised by high values of PS: for instance, Eurocode 3 suggests assessing steel welded details by using reference fatigue curves having PS no lower than 97.7% (Anon, 1988). In order to mathematically express the Wöhler curve of a given plain material, the concepts of fatigue limit as well as of endurance limit deserve log sx,a

y

Wöhler curve

x F(t)

F(t)

R = –1 (a) log Nf (b)

2.1 (a) Plain fatigue specimen loaded in tension–compression and (b) Wöhler diagram.

Fundamentals of fatigue assessment

35

1000 R = –1 sUTS = 638.5 MPa sx,a (MPa) PS = 50%

PS = 10%

PS = 90% s0 = 206.3 MPa

sA = 197.5 MPa 100 10 000

100 000

1 000 000

10 000 000 Nf (cycles)

2.2 Fatigue results generated by testing plain flat specimens of En3B under fully reversed uniaxial loading (Susmel and Taylor, 2007).

log sx,a

log sx,a Knee point

1 s0

log sx,a

k

1 sA

1 k

1 NA

N0 log Nf (a)

k1

log Nf (b)

k2

log Nf (c)

2.3 Different fatigue behaviours in terms of Wöhler curves and definition of fatigue limit and endurance limit.

to be discussed in detail. Figure 2.3a depicts the typical fatigue behaviour of ferrous metal specimens: if the applied stress is lower than a certain stress amplitude, s0, then failure should not occur up to a number of cycles to failure theoretically equal to infinity; such a reference threshold is named the fatigue limit. It is worth noting that, from a scientific point of view, the fatigue limit in metallic materials results in the formation of nonpropagating cracks whose growth is arrested either by the first grain boundary or by the first microstructural barrier (Miller, 1993; Akiniwa et al., 2001). On the contrary, non-ferrous metals, such as aluminium alloys, do not exhibit a fatigue limit, so that they always have to be designed for finite life (Figs 2.3b and 2.3c). In this case it is common practice to define the so-called

36

Multiaxial notch fatigue

endurance limit, sA, which is the stress amplitude extrapolated, in the highcycle fatigue regime, at a given number of cycles to failure, NA (typically, NA ranges from 106 to 108 cycles to failure). It is interesting to observe also that sometimes the experimental behaviour of those materials which do not exhibit any fatigue limit cannot be summarised by using a unique straight trend line: in the high-cycle fatigue regime the experimental points tend to distribute themselves along a line which is less steep than the one describing the medium-cycle fatigue behaviour of the same material (Fig. 2.3c). When dealing with materials exhibiting the fatigue limit, there exist many different procedures suitable for determining such a fatigue property – see, for instance, Lin et al. (2001) and the references reported therein. Unfortunately, it has to be said that, in practice, many different causes can lead to its elimination (Miller and O’Donnell, 1999), so that engineers engaged in designing real components too often perform the high-cycle fatigue assessment by using suitable endurance limits also for those materials which potentially have a fatigue limit. As an example, Fig. 2.2 shows not only the fatigue limit determined by using the accelerated staircase method (Lin et al., 2001) but also the endurance limit, sA, extrapolated at 2 × 106 cycles to failure. For the sake of simplicity, consider now the Wöhler curve depicted in Figure 2.3a. According to the symbolism adopted in such a schematic chart, the above S–N curve can be described by using the following well-known relationship (the so-called Wöhler equation):

σ xk,a ⋅ N f = σ 0k ⋅ N 0 = constant

2.1

where k is the negative inverse slope as defined in Fig. 2.3. Typically, for conventional plain engineering materials, k ranges between 8 and 12. Another important aspect which deserves to be mentioned here is the influence of the adopted failure criterion as defining the finite life region in Wöhler diagrams. Initially, it is worth remembering that fatigue life can be divided into three different phases (Hertzberg, 1996): initiation, propagation and final breakage. Even if, theoretically speaking, any of the above phases could be used to define a suitable failure criterion, the experimental points are seen to move in Wöhler’s diagrams as the adopted failure criterion changes, resulting in different fatigue curves. This problem is clearly highlighted by the tests carried out by Socie and co-workers: as the length of the crack defining the final failure increased, the corresponding number of cycles to failure changed, resulting in an apparent increase of fatigue strength (Socie et al., 1985, 1989; Socie, 1987). In order to overcome the above problem, the final breakage can also be determined by adopting other strategies. For instance, failures could be defined by using the stiffness decrease criterion (Atzori et al., 2006a; Susmel and Taylor,

Fundamentals of fatigue assessment

37

2007). Unfortunately, the main drawback in using the above method is that it is seen to be sensitive to the magnitude of the applied loading, making it difficult for the number of cycles to failure to be defined coherently. The considerations reported above should make it evident that the definition of a suitable failure criterion is never a simple task: as far as the writer is aware, there exists no universally accepted definition, so that the failure criterion always has to be carefully chosen according to the type of tests which have to be carried out. In the above paragraphs, only the fully reversed uniaxial fatigue behaviour of plain materials was considered. It has to be pointed out that the response of engineering materials to fatigue is highly sensitive to many other parameters, including mean stress effect, geometrical features, size effect, surface treatments and roughness, metallurgical parameters and many others. All the above variables must always be taken into account during the design process, because, in some cases, they could lead to a decrease of the overall fatigue strength of the material from which the component to be assessed is made (Frost et al., 1974; Hertzberg, 1996; Pook, 2000; Stephens et al., 2000). Among the above parameters and due to their detrimental effect, certainly non-zero superimposed static stresses as well as stress raisers must always be taken into account very carefully when performing the fatigue assessment of real mechanical components, so the problems related to such aspects will extensively be discussed in the next sections.

2.3

The mean stress effect under uniaxial fatigue loading

As mentioned at the end of the previous section, the mean stress effect plays an important role in the overall fatigue strength of engineering materials. In particular, under uniaxial fatigue loading, it is seen that fatigue damage increases as the applied tensile superimposed static stress, sx,m, increases (Fig. 2.4a). Similarly, by decreasing the load ratio, R, fatigue curves are seen to move upward in Wöhler’s diagrams, that is, an increase of R lowers the material fatigue (or endurance) limit (Fig. 2.4b). It is also important to highlight that, for those materials exhibiting a fatigue limit, the number of cycles to failure, N0, corresponding to the fatigue curve knee point can change as either the mean stress value or the load ratio varies. As an example, Fig. 2.5 shows the experimental results we obtained by testing plain specimens of En3B under two values of the R ratio: this Wöhler diagram clearly shows the detrimental effect of a load ratio larger than −1. From such a chart it can be seen also that the experimental number of cycles to failure, N0, defining the position of the two knee

38

Multiaxial notch fatigue log sx,a

Increasing sx,m

log sx,a

Decreasing R

sx,m = 0

R = –1

log Nf

log Nf

(a)

(b)

2.4 (a) Mean stress and (b) load ratio, R, effect under uniaxial fatigue loading.

1000 R = –1 sUTS = 638.5 MPa

R = 0.1

sx,a (MPa)

Knee point at R = –1: s0 = 206.3 MPa N0 = 1331079 cycles

PS = 50% PS=50% Knee point at R = 0.1: s0 = 163.8 MPa N0 = 827570 cycles

100 10 000

100 000

1 000 000

10 000 000 Nf (cycles)

2.5 Fatigue results generated by testing plain flat specimens of En3B under uniaxial loading with a load ratio, R, equal to both −1 and 0.1 (Susmel and Taylor, 2007).

points varies as R increases (the two fatigue limits reported in Fig. 2.5 were determined by using the accelerated staircase method). The state of the art shows that many attempts have been made to propose sound engineering rules suitable for predicting the mean stress effect in fatigue. Almost all the proposed criteria can concisely be summarised by using Marin’s general equation (Marin, 1956): n

⎛ σ 0 ⎞ ⎛ σ x,m ⎞ +⎜ f ⎟ =1 ⎜⎝ σ 0, R =−1 ⎟⎠ ⎝ σ UTS ⎠ m

2.2

Fundamentals of fatigue assessment

39

where the symbol s0,R=−1 is used here to denote the fatigue limit generated under fully reversed loading. In the above relationship sUTS is the ultimate tensile strength, whereas f, m and n are constants which take on different values according to the considered fatigue damage model. In more detail, the most adopted criteria can be derived as follows (Iurzolla, 1991; Susmel et al., 2005): •

Soderberg’s relationship is obtained by imposing n = 1, m = 1 and f = sUTS/sY (where sY is the material yield stress):

σ0 σ 0,R=−1 •

+

σ x ,m σ = 1 ⇒ σ 0 = σ 0,R=−1 ⎛ 1 − x ,m ⎞ ⎝ σY σY ⎠

2.3

Goodman’s relationship is obtained by imposing n = 1, m = 1 and f = 1:

σ0 σ σ + x ,m = 1 ⇒ σ 0 = σ 0,R=−1 ⎛⎜ 1 − x ,m ⎞⎟ ⎝ σ 0,R=−1 σ UTS σ UTS ⎠ •

Gerber’s parabola is obtained by imposing n = 1, m = 2 and f = 1:

σ0 σ 0, R =−1 •

2.4

σ x,m ⎞ ⎤ σ x,m ⎞ ⎡ + ⎛⎜ = 1 ⇒ σ 0 = σ 0, R =−1 ⎢1 − ⎛⎜ ⎟ ⎝ σ UTS ⎟⎠ ⎝ σ UTS ⎠ ⎥⎦ ⎣ 2

2

2.5

Dietman’s parabola is obtained by imposing n = 2, m = 1 and f = 1: 2

σ x,m σ x,m ⎛ σ0 ⎞ = 1 ⇒ σ 0 = σ 0, R =−1 1 − ⎜⎝ ⎟⎠ + σ UTS σ 0, R =−1 σ UTS •

2.6

The so-called ‘elliptical relationship’ is obtained by imposing n = 2, m = 2 and f = 1: 2

⎛ σ 0 ⎞ ⎛ σ x,m ⎞ ⎛ σ x,m ⎞ ⎟ ⎟⎠ = 1 ⇒ σ 0 = σ 0, R =−1 1 − ⎜⎝ ⎜⎝ ⎟⎠ + ⎜⎝ σ UTS ⎠ σ 0, R =−1 σ UTS 2

2

2.7

Criteria (2.4)–(2.7) can easily be plotted together in a unique nondimensional chart having in the abscissa the sx,m to sUTS ratio and in the ordinate the s0 to s0,R=−1 ratio: the diagram reported in Fig. 2.6 clearly shows that the accuracy of any criterion considered above in predicting the effect of superimposed tensile static stresses varies from material to material. In other words, the experimental evidence suggests that different materials show different sensitivities to the presence of non-zero mean stresses, so that it is very difficult to predict behaviours which are so different by using a unique formula. Figure 2.6 also proves that the experimental points are mainly located within the region delimited by the elliptical relationship and by Goodman’s straight line. According to such an experimental outcome, when the response to non-zero mean stresses of a particular metallic material cannot be investigated by running appropriate tests, the use of Goodman’s criterion is always suggested because it allows the mean stress

40

Multiaxial notch fatigue 1.2

Mild steel

s0/s0,R = –1

NiCr alloy steel

1.0

SAE 4130 24S-T3

0.8

75S-T6 2014-T6

0.6

2024-T4

0.4

0.2

Goodman

6061-T6

Dietman Gerber

7075-T6 24S-T4 75S-T6

Elliptical relationship

14S-T6 0.0

2.5%Zn 0.65% Zr 0.0

0.2

0.4

0.6

0.8

1.0 1.2 sx,m/sUTS

5.6%Zn 0.66%Zr

2.6 Accuracy of the considered criteria in predicting the mean stress effect under uniaxial fatigue loading (data from Frost et al., 1974).

effect under uniaxial fatigue loading to be accounted for always with an adequate margin of safety. Moreover, due to the fact that Soderberg’s straight line, Eq. (2.3), uses the yield stress as second calibration information instead of the ultimate tensile strength, such a criterion is often preferred when addressing problems of practical interest, because it allows a larger degree of safety to be obtained. Lastly, it is important to highlight that the criteria reviewed above can be applied not only in terms of fatigue limits but also in terms of endurance limits. It is worth concluding this section by mentioning the fact that the effect of compressive mean stresses has not been investigated as much as the one due to superimposed tensile stresses. In any case, it is generally believed that compressive mean stresses have either no effect at all or a beneficial effect on the overall material fatigue strength (Sines, 1959). This is the reason why the presence of negative mean stresses is usually neglected, so that the fatigue assessment under these particular loading conditions is performed by using material fatigue properties simply generated under fully reversed loading.

2.4

The notch effect in fatigue

Owing to their well-known detrimental effect on the overall fatigue strength of real mechanical components, stress concentration phenomena are always a matter of concern to structural engineers engaged in performing the fatigue assessment. This is the reason why the notch fatigue problem has

Fundamentals of fatigue assessment

41

log sx,a y

sS x

Plain fatigue curve

s0 Kf

s0n Fa

(Net stress) (Gross stress)

Fa N0

NS (a)

log Nf

(b)

2.7 Wöhler fatigue curves generated by testing both plain and notched specimens and definition of the fatigue strength reduction factor, Kf.

extensively been investigated over the last century, so that nowadays there exists a variety of criteria suitable for successfully taking into account the damaging effect of stress raisers when subjected to fatigue loading. Before considering in detail some of the most accurate and efficient methods specifically devised to assess mechanical components containing different kinds of geometrical features, it is initially useful to review some basic concepts which are needed to correctly address the notch fatigue problem. Consider then the V-notched cylindrical bar sketched in Fig. 2.7a and subjected to a cyclic force of amplitude equal to Fa. According to what was briefly said in Section 1.8, it is always possible to define two different types of nominal stress, that is, the net and the gross nominal stress. The fact that the notched sample of Fig. 2.7a is loaded by a cyclic force makes it evident that nominal stress amplitudes can be calculate with respect to either the net or the gross cross-sectional area. In general, the classical approaches, i.e. those based on continuum mechanics, use the nominal net stresses to perform the fatigue assessment. In contrast, and as will be discussed below, LEFM usually refers to gross nominal stresses. In the present section, due to the type of problem addressed, all the nominal stresses are meant to be calculated with respect to the net cross-sectional area.1 In order to better clarify the above concepts, consider the Wöhler diagram sketched in Fig. 2.7b. In such a schematic chart three different fatigue curves are plotted. In particular, the upper one is supposed to be generated by testing plain specimens of a given material, whereas the other two are supposed to be determined by testing specimens of the same material as above and containing a known geometrical feature. The difference between the two notch curves is that the higher one is plotted in terms of amplitude of the nominal net stress, whereas the dashed one is plotted in terms of amplitude of the nominal gross stress: the dashed curve is simply scaled with respect to the other curve by a factor proportional to the ratio between the net and the gross cross-sectional area. 1

It is obvious that in plain specimens net and gross nominal stresses are coincident.

42

Multiaxial notch fatigue

If the problem is addressed in terms of nominal net stresses, according to Fig. 2.7b, the detrimental effect of the considered notch on the investigated material can be quantified through the so-called fatigue strength reduction factor, Kf (Peterson, 1959): Kf =

σ0 σ 0n

2.8

In order to coherently determine Kf, the two fatigue limits used in the above definition must always be determined under the same experimental conditions (and, in particular, either under the same load ratio or by imposing the same mean stress value). It has to be pointed out that the fatigue strength reduction factor can be calculated not only by considering plain and notch fatigue limits, but also by using plain and notch endurance limits. For the sake of simplicity, in what follows we will consider only fatigue limits, but the same ideas can successfully be used also when Kf is determined through suitable endurance limits, provided that they are again determined under the same experimental conditions and, above all, extrapolated at the same reference number of cycles to failure, NA. As an example, Fig. 2.8 reports the fatigue curves we generated by testing, under rotating bending, plain and V-notched cylindrical specimens of S690, a commercial high-strength steel (Susmel, 2001). Such a Wöhler diagram clearly shows the detrimental effect of the considered stress raiser: the notch fatigue curve is steeper and is characterised by a lower value of the endurance limit than the corresponding plain S–N curve. According to

1000 Endurance limits at NA = 2·106 cycles (PS = 50%):

PS = 10%

sA = 309.1 MPa (k = 5.6) sAn = 172.6 MPa (k = 4.3)

sx,a (MPa)

PS = 90%

y

60° x R0.6

PS = 10% PS = 90%

6.5

9

Plain Notched 100 10 000

R = –1 100 000

1 000 000

10 000 000 Nf (cycles)

2.8 Fatigue results generated by testing plain and V-notched cylindrical specimens of S690 steel under rotating bending (Susmel, 2001).

Fundamentals of fatigue assessment 1000

sUTS = 638.5 MPa

sx,a (MPa)

Knee point at R = –1: s0 = 206.3 MPa N0 = 1331079 cycles

PS = 50%

Plain, R = –1 Notched, R = –1 Notched, R = 0.1

60°

x R0.12 PS = 50%

100

y

43

4

PS = 50%

25 Knee point at R = –1: s0n = 61.6 MPa N0 = 1518396 cycles

10 10 000

Knee point at R = 0.1: s0n = 41.6 MPa N0 = 1875947 cycles

100 000

1 000 000

10 000 000 Nf (cycles)

2.9 Fatigue results generated by testing plain and single V-notched flat specimens of En3B under uniaxial loading (Susmel and Taylor, 2007).

the above experimental results, the strength reduction factor, Kf, calculated at 2 × 106 cycles to failure, is equal to 1.79. As a second example, Fig. 2.9 reports the Wöhler curves we obtained by testing, under uniaxial fatigue loading, plain and single V-notched specimens of En3B (Susmel and Taylor, 2007). This chart clearly shows that, under fully reversed loading, the notch lowers the fatigue limit, resulting in a Kf factor equal to 3.35. Moreover, it is evident that an increase of the load ratio, R, results in a further reduction of the notch fatigue limit: under fully reversed loading, s0n is equal to 61.6 MPa, whereas under a load ratio equal to 0.1, s0n drops down to 41.6 MPa. Finally, it is important to highlight that, by using only the pieces of information summarised in Fig. 2.9, the strength reduction factor cannot be calculated for the lowest curve: the reported plain fatigue limit refers to fully reversed conditions, whereas the lowest notch fatigue curve was generated by setting R equal to 0.1. It is evident that the most accurate way to determine the fatigue strength reduction factor is by running appropriate experiments. Unfortunately, this is not always possible in practice, so that many formulas suitable for estimating Kf have been proposed and validated by carrying out extensive experimental investigations – for an exhaustive review of the formulas available in the technical literature, see, for instance, Yao et al. (1995) and Ciavarella and Meneghetti (2004) and references reported therein. In order to understand the way the classical formulas work, it is initially useful to introduce the so-called notch sensitivity factor, q. In particular, according to Peterson (1959), for a given geometrical feature, the fatigue strength reduction factor

44

Multiaxial notch fatigue

can be expressed as a function of both q and the corresponding linearelastic stress concentration factor, Kt (calculated with respect to the net nominal stress): K f = 1 + q ( Kt − 1)

2.9

where q ranges between 0 and 1. It is interesting to observe that when q equals zero, then the presence of the considered stress raiser does not result in any reduction of the plain fatigue limit (non-damaging notches). On the contrary, when q is equal to unity (full notch sensitivity), then Kf is equal to Kt, so that fatigue damage can successfully be predicted by simply using the linear-elastic stress calculated at the tip of the stress concentrator to be assessed. Among the different formulas devised ad hoc to estimate the fatigue strength reduction factor, the most commonly adopted ones are certainly those proposed by Neuber (1936, 1958) and Peterson (1959), respectively. In particular, Neuber suggested estimating Kf by using the following wellknown relationship: Kf = 1 +

Kt − 1 1+

aN rn

2.10

where aN is a constant depending on the ultimate tensile strength and rn is the root radius of the notch contained by the component to be assessed. Equation (2.10) makes it evident that, according to Neuber, Kf depends on the notch root radius, which is assumed to be the most important parameter controlling the linear-elastic stress field distribution along the notch bisector, as well as on a critical distance parameter. In more detail, formula (2.10) is based on the idea that the elastic stresses in the vicinity of stress raisers cannot reach values as high as those predicted according to continuum mechanics, so that Neuber suggested averaging the stress close to the notch apex over materials units (that is, crystals or structural particles) to calculate an engineering quantity representative of the real stress damaging the fatigue process zone. In other words, according to Neuber’s idea, a reliable reference stress to be used to assess notched components has to be calculated by considering finite volumes and not infinitesimal volumes as postulated by the classical theory. As to constant aN, different formulae have been proposed and validated. For instance, Dowling (1998) suggests using the following empirical relationship: aN = 10

σ − 134 − UTS 586

[ mm ]

2.11

It is important to highlight that the above formula can be used only to assess steels having ultimate tensile strength lower than 1520 MPa.

Fundamentals of fatigue assessment

45

As briefly said above, Neuber’s formula assumes that the most important parameter controlling the distribution of the linear-elastic stress damaging the fatigue process zone is the notch root radius. It is well known that in conventional notches also the opening angle is an important parameter to be considered in order to correctly calculate the resulting linear-elastic stress fields, especially when opening angles are larger than 90° (Lazzarin and Tovo, 1996). In order to correctly take into account also this important geometrical variable, Kuhn and Hardraht (1952) devised the following formula suitable for estimating the fatigue strength reduction factor, Kf, in the presence of open notches: Kf = 1 +

Kt − 1 π 1+ π − 2α

A rn

2.12

In the above relationship 2a is the notch opening angle, whereas A is a material constant depending on the ultimate tensile strength and ranging from 0.025 mm up to 0.51 mm. Starting from a strategy similar to the one adopted by Neuber, Peterson (1959) suggested using as reference strength suitable for determining Kf the linear elastic stress determined, along the notch bisector, at a given distance from the stress raiser tip. In other words, Peterson formalised the so-called Point Method. By assuming again that the notch root radius, rn, is the most important geometrical parameter defining the profile of the stress field along the notch bisector, he suggested using the following formula: Kf = 1 +

Kt − 1 a 1+ P rn

2.13

By a systematic experimental investigation, Peterson determined a set of values for constant aP, which was assumed to be a material parameter. According to Dowling (1998), the constant aP for steels having sUTS larger than 550 MPa can be estimated by using the following empirical relationship: ⎛ 270 ⎞ aP = ⎜ ⎝ σ UTS ⎟⎠

1.8

[ mm ]

2.14

It is important to point out here that, as observed by Peterson himself, Eq. (2.13) is not recommended to be used to perform the fatigue assessment of mechanical components containing very sharp notches. Another empirical relationship that certainly deserves to be mentioned is the one reported by Heywood (1962) which is suitable for estimating the

46

Multiaxial notch fatigue

detrimental effect of notches in cylindrical bars of steel. In particular, he suggested using the following formula: Kt

Kf =

1+ 2

2.15

aH rn

where the constant aH is a function of the material static properties as well as of the geometrical feature to be assessed. In more detail, aH can be calculated as: C aH = ⎛⎜ H ⎞⎟ ⎝ σ UTS ⎠

2

2.16

where CH is equal to 3 when the cylindrical bar to be assessed contains grooves, equal to 4 in the presence of shoulders, and equal to 5 when the considered component is weakened by a transversal hole. In order to briefly investigate the reliability in estimating notch fatigue limits of the methods reviewed above, Figs 2.10 and 2.11 show the accuracy of both Neuber’s and Peterson’s formulae when applied to notched specimens made of the materials summarised in Table 2.1. In particular, the data considered in these figures were generated by testing V-notched cylindrical samples as well as flat specimens containing different geome-

Table 2.1 Summary of the static and fatigue properties of the materials used to check the accuracy in estimating notch fatigue limits of the formulas proposed by Neuber and Peterson, respectively (see Figs 2.10 and 2.11) s0 (MPa) aN (mm) aP (mm) Reference

Material

sUTS (MPa) R

C45

632

−1

291

0.141

0.216

Nisitani and Endo, 1988

C36

565

−1

223

0.184

0.265

Nisitani and Endo, 1988

SAE 1045

745

0 −1

224 303

0.91

0.161

DuQuesnay et al., 1988

NiCr steel

957

−1

500

0.039

0.103

Frost, 1957

304 stainless 685 steel

−1

360

0.115

0.187

Ting and Lawrence, 1993

En3B

0.1 142* −1 197*

0.138

0.213

Susmel and Taylor, 2007

638

* Endurance limit at 2 × 106 cycles to failure.

Fundamentals of fatigue assessment 10

47

Neuber's formula C45, R = –1

Non-conservative

C36, R = –1 E = +20%

E = –20%

SAE 1045, R = 0 SAE 1045, R = –1

Kf

NiCr steel, R = –1 Conservative Error [%] =

AISI 304, R = –1

Kf,e − Kf

En3B, R = 0.1

Kf

En3B, R = –1

1 1

Kf,e

10 (a)

10

Peterson's formula Non-conservative

C45, R = –1

E = –20%

C36, R = –1 E = +20%

SAE 1045, R = 0 SAE 1045, R = –1

Kf

NiCr steel, R = –1 Conservative Error [%] =

AISI 304, R = –1

Kf,e − Kf

En3B, R = 0.1

Kf

En3B, R = –1

1 1

Kf,e

10 (b)

2.10 Experimental (Kf) vs. estimated (Kf,e) fatigue strength reduction factor diagrams obtained by using (a) Neuber’s and (b) Peterson’s formula – see Table 2.1 for a summary of the static and fatigue properties of the considered materials.

trical features. The investigated notches had root radius ranging between 0.01 mm and 4 mm, resulting in Kt values varying from 1.6 up to 20. In Fig. 2.10 the experimental fatigue strength reduction factors, Kf, are plotted against the corresponding ones, Kf,e, obtained by using Eqs (2.10) and (2.13). Constants aN and aP were estimated according to relationships (2.11) and (2.14), respectively. Figure 2.10 clearly shows that, for the materials considered and for the geometrical features investigated, the use of

48

Multiaxial notch fatigue 1000

Neuber's formula C45, R = –1

E = +20%

C36, R = –1

Conservative

SAE 1045, R = 0 Ds0n (MPa)

SAE 1045, R = –1

E = –20%

NiCr steel, R = –1 AISI 304, R = –1 Non-conservative

En3B, R = 0.1 En3B, R = –1

100 100

Ds0n,e (MPa)

1000

(a) 1000

Peterson's formula C45, R = –1 Conservative

C36, R = –1

E = +20%

SAE 1045, R = 0 E = –20%

Ds0n (MPa)

SAE 1045, R = –1 NiCr steel, R = –1 AISI 304, R = –1

Non-conservative 100 100

Ds0n,e (MPa)

En3B, R = 0.1 En3B, R = –1

1000

(b)

2.11 Experimental (Δs0n) vs. estimated (Δs0n,e) notch fatigue limit diagrams obtained by using (a) Neuber’s and (b) Peterson’s formula – see Table 2.1 for a summary of the static and fatigue properties of the considered materials.

Neuber’s formula results in more accurate estimates than those obtained by applying Peterson’s equation. In order to address the same problem from a different perspective, the experimental (Δs0n) vs. estimate (Δs0n,e) notch fatigue limit diagrams obtained by applying, Neuber’s as well as Peterson’s relationship to the above data are sketched in Fig. 2.11. To plot the considered experimental points more clearly, in these charts experimental and estimated notch fatigue limits (calculated with respect to the net section) are reported in

Fundamentals of fatigue assessment

49

terms of ranges. Figure 2.11 confirms that the most accurate estimates are obtained by applying the formula devised by Neuber. It is also interesting to observe that such an equation allows the considered experimental points to fall mainly within an error interval equal to about ±20%: this aspect is very interesting, especially in light of the fact that the most modern techniques suitable for estimating fatigue limits are characterised, from a statistical point of view, by the same accuracy level (Taylor and Wang, 2000; Susmel and Taylor, 2003). The considerations reported above should make it evident that, even if with a certain level of uncertainty, there exist many engineering tools suitable for estimating the fatigue strength reduction factor, Kf. This means that the estimated notch fatigue limits, corrected by adopting an adequate safety factor, can be used directly to perform the high-cycle fatigue assessment of real mechanical components. It is worth noting also that the estimated values of Kf can be used not only to evaluate high-cycle fatigue damage but also to predict fatigue lifetime of notched components, provided that an ad hoc fatigue curve is assumed. Consider then the cylindrical notched bar sketched in Fig. 2.7a and the corresponding schematic Wöhler diagram (Fig. 2.7b). By using one of the formulas reviewed above, and according to both the static properties of the material used and the sharpness of the geometrical feature considered, it is possible to estimate the corresponding notch fatigue limit. If the plain material fatigue curve is known from the experiments, the hypothesis can be formed that the notch fatigue curve to be estimated has its knee point positioned at the same number of cycles to failure, N0, as the knee point of the reference plain fatigue curve (Fig. 2.7b). On the contrary, if the position of the plain fatigue curve knee point is not known, then N0 can be taken to be equal to 2 × 106 cycles to failure (Atzori, 2000). In order to define the second point needed to build the required notch fatigue curve, the number of cycles to failure, NS, at which the component is supposed to fail when the maximum stress, calculated with respect to the net area, equals the material ultimate tensile stress has to be estimated (Fig. 2.7b). The experimental evidence suggests that NS is different for different materials and different load ratios, by ranging from 1/4 up to about 104 cycles to failure. This fact results in a difficult estimation of such a reference number of cycles to failure. Moreover, the problem is further complicated by the fact that stress-based approaches are not that adequate to model engineering material behaviour in the low-cycle fatigue regime. In spite of the well-known difficulties mentioned above, 103 cycles to failure are, in general, considered to be a reliable value to be used to estimate NS (Atzori, 2000). To conclude, it is important to highlight that the value of the amplitude of the stress, sS, at NS cycles to failure depends on the material ultimate tensile stress as well as on the load ratio (or the mean stress value)

50

Multiaxial notch fatigue

for which the notch fatigue curve has to be estimated (Fig. 2.7b). In particular, for a given value of R, sS results in the following trivial relationship:

σS =

1− R σ UTS 2

2.17

whereas if the mean stress is kept constant, sS has to be determined as follows:

σ S = σ UTS − σ x,m

2.5

2.18

Linear Elastic Fracture Mechanics to predict fatigue damage in cracked bodies

In the above section, the problem of accounting for the detrimental effect of notches was addressed by considering only stress concentrators having tip radius, rn, larger than zero, that is, components characterised by a finite value of Kt. On the contrary, when rn approaches zero, the formulas summarised above cannot be used to estimate Kf any more because the linear elastic peak stresses tend to infinity, resulting in stress concentration factors also tending to infinity. These considerations should make it evident that, in the presence of singular stress fields, different strategies have to be followed in order to estimate fatigue damage without missing the doubtless advantages of linear-elastic approaches. Such methods will be discussed in the next sections in great detail. The present section instead attempts to briefly summarise the basic concepts suitable for performing, in accordance with LEFM, the fatigue assessment of mechanical components containing cracks. It is important to highlight that in what follows only the fundamental ideas which will be used to address the notch fatigue problem are considered, so that to have a better overview of the way this powerful method can successfully be used in practice, the reader is referred to those exhaustive books treating the LEFM theory in great detail – see, for instance, Gdoutos (1993), Anderson (1995), Miannay (1998) and Jansen et al. (2002). Consider then the cracked plate subjected to uniaxial fatigue loading sketched in Fig. 2.12. According to Irwin (Section 1.9), the entire stress field damaging the material in the vicinity of the crack tip is a function of the stress intensity factor (SIF) range, ΔKI: 3θ θ θ cos ⎛ ⎞ ⎡1 − sin ⎛ ⎞ sin ⎛ ⎞ ⎤ ⎢ ⎝ ⎠ ⎝ ⎠ ⎝ 2 ⎣ 2 2 ⎠ ⎥⎦ 2 πr θ θ 3θ ΔKI coss ⎛ ⎞ ⎡1 + sin ⎛ ⎞ sin ⎛ ⎞ ⎤ Δσ y = ⎢ ⎝ ⎠ ⎝ ⎠ ⎝ 2 ⎣ 2 2 ⎠ ⎥⎦ 2 πr ΔKI θ θ 3θ Δτ xy = sin ⎛ ⎞ cos ⎛ ⎞ cos ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ ⎝ 2 2 2⎠ 2 πr Δσ x =

ΔKI

2.19

Fundamentals of fatigue assessment

51

Δsgross

Δsy y r

q

Δtxy Δsx x

2a

w

Δsgross

2.12 Central through-crack in a plate subjected to uniaxial fatigue loading.

where ΔKI = F ⋅ Δσ gross ⋅ πa

2.20

Equations (2.19) make it clear that ΔKI is a powerful stress quantity which can be used directly to estimate fatigue damage in cracked bodies subjected to cyclic loading, because it gives a direct measure of the magnitude of the entire stress field damaging the material close to the crack propagation front. In order to better understand the way ΔKI can be used to address the crack growth problem, consider again the plate sketched in Fig. 2.12. If the initial semi-length of the crack, ai, is large enough and the range of the applied loading allows the crack itself to propagate, it is possible to plot the resulting crack growth curve in a semi-length, a, vs. number of cycles, n, diagram (Fig. 2.13a). The schematic curves sketched in Fig. 2.13a show that the crack length increases as the number of cycles increases and the profile of any a vs. n curve depends on the range of the applied loading. When the crack reaches a certain critical value, af, the sample fails statically under the maximum value of the applied nominal loading. Starting from the above curves, the cracking behaviour of the investigated material can be described in a more concise way by simply plotting the crack growth rate, da/dn, against the corresponding stress intensity factor range, ΔKI. In particular, it is easy to determine from the crack growth curve (Fig. 2.13a) the slope of the straight tangent line at any semi-length of the crack, determining the crack growth rate, da/dn. Such a rate, together with the corresponding ΔKI calculated according to Eq. (2.20), can then be plotted in a log-log da/dn

52

Multiaxial notch fatigue

a

Δsa>Δsb

log da/dn

Region III

Region I

af,b Δsb

af,a aj

da = C⋅(ΔKI)m dn

Δsa

KI,max = KC

Static failures da dn aj,Δsb

ai n (a)

m Region II

Experimental trend log ΔKI

ΔKth (b)

2.13 (a) Fatigue crack growth curves and (b) fatigue crack growth rate curve.

vs. ΔKI diagram, obtaining the so-called fatigue crack growth rate curve relative to the tested material. Figure 2.13b shows the typical shape of a crack growth rate curve. It is useful to remember here that this diagram is classically subdivided into three different zones (Fig. 2.13b). In more detail, in Region I cracks can propagate only when ΔKI is larger than the corresponding threshold value of the SIF range, ΔKth. Such a threshold value is a material property which can be determined by following one of the procedures stated by the available standard codes (see, for instance, Standard Test Method for measurement of Fatigue Crack Growth Rates, ASTM E647, 2000). It is important to highlight here also that the cracking behaviour of engineering materials in Region I is strongly influenced by the material’s microstructure, mean stress and environment, but does not depend on the thickness of the specimen used to determine the threshold value of the SIF range. This implies that ΔKth is a material property which is different for different materials and different load ratios. In Region II instead the crack propagation curve is usually described by using the well-known law devised by Paris, which is a relationship between da/dn and ΔKI: da m = C ⋅ ( ΔKI ) dn

2.21

In the above equation, C and m are material constants to be determined experimentally and their values vary as the load ratio, R, changes. Finally, in Region III the da/dn vs. ΔKI curve rises very rapidly by approaching the vertical straight asymptotic line at which the maximum value of the SIF equals, during the loading cycle, the material fracture toughness, KC. In other words, the condition resulting in the static failure in Region III can be expressed as follows:

Fundamentals of fatigue assessment KI,max = F ⋅ σ gross,max ⋅ πaf = KC

53 2.22

where af is the final length of the crack. It is important to remember here that, contrary to Region I, the material cracking behaviour in Region III is strongly influenced by the thickness of the specimens used to determine KC itself. This is the reason why, when KC cannot be determined by running appropriate experiments, the static assessment of cracked components is always suggested to be performed by referring to the plane-strain fracture toughness, KIC, of the considered material.

2.6

Different links between Linear Elastic Fracture Mechanics and continuum mechanics

In order to better understand how the most modern theories suitable for estimating the damaging effect of notches subjected to fatigue loading work, it is useful to review here some existing links between LEFM and continuum mechanics. Before explicitly addressing the above problem, it is worth remembering again that, generally speaking, continuum mechanics estimates fatigue damage in terms of nominal net stresses, whereas LEFM makes use of nominal gross stresses. Therefore, in order to coherently link the above two theories, in the present section fatigue limits are always assumed to be calculated by using the gross section of the considered samples. The first aspect to be addressed here is the so-called short-crack problem. Consider then the cracked plate subjected to uniaxial fatigue loading sketched in Fig. 2.12. Assume that the nominal fatigue limit, Δs0n, is known for different lengths, 2a, of the central crack. For the sake of simplicity, the hypothesis can be formed that, for every geometrical configuration considered, the crack length is in any case much smaller than the width, w, of the plate, so that the geometrical correction factor for the LEFM stress intensity factor, F, is invariably equal to unity. The experimental fatigue limits obtained by considering different values of 2a can then be plotted against the semi-crack length, a, obtaining the so-called Kitagawa–Takahashi diagram (Kitagawa and Takahashi, 1976). Figure 2.14 shows a schematic representation of such a diagram. This chart should make it evident that, for large values of a, LEFM is successful in predicting fatigue limits in the presence of cracks by simply using the following relationship: Δσ 0n =

ΔKth πa

In other words, LEFM describes very well the fatigue behaviour at the fatigue limit of the so-called long cracks. On the contrary, for very small

54

Multiaxial notch fatigue log Δs0n

Δs0

Experimental results

Δs0n =

ΔKth p⋅a

Short-crack region Long-crack region L

≈10L

log a

2.14 Schematic representation of Kitagawa and Takahashi’s diagram.

values of a, nominal fatigue limits estimated by applying the above equation are not only higher than the experimental results, but also higher than the material plain fatigue limit, Δs0. This makes it evident that when a is lower than L, where L=

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

2.23

the use of LEFM is no longer justified. Unfortunately, for crack lengths approaching L also the application of continuum mechanics results in nonconservative estimates, so that a different criterion is needed for correctly describing the short-crack behaviour (Fig. 2.14). The state of the art shows that, due to its relevance, a variety of methods have been devised to address the above problem – see, for instance, Miller (1982), Tanaka (1987), Usami (1987) and Akiniwa et al. (2001) – and, among them, certainly the one formalised by Topper and co-workers (El Haddad et al., 1979) deserves to be considered in detail. Such a method postulates that the short-crack behaviour can be predicted by simply using the following relationship: Δσ 0n =

ΔKth π ( a + L)

2.24

In the above equation L plays the role of an intrinsic imaginary defect. Even if such a method was seen to be capable of highly accurate estimates, it is very difficult to find a sound connection between Eq. (2.24) and physical reality (Taylor, 2007). In particular, when the crack length approaches L, the propagation phenomenon is strongly influenced by the actual elastoplastic behaviour of the grains as well as by the material morphology in the vicinity of the crack tip (Miller, 1993). On the contrary, the use of LEFM is justified as long as the material can be thought of as linear-elastic, homogeneous and isotropic. As observed by Taylor (2007), Eq. (2.24) simply states that the fatigue behaviour of a short crack can be predicted by considering an imaginary crack of length (a + L) which is forced to obey the

Fundamentals of fatigue assessment

55

laws of LEFM. The above considerations should make evident the empirical nature of Topper’s intrinsic defect. Apart from the above philosophical problems, it is important to highlight that the ideas summarised in the previous paragraphs can easily be extended to more complex geometries by simply incorporating in the model the geometrical factor, F. In particular, if the actual shape of the component to be assessed is taken into account, Eq. (2.23) takes on the following form: a0 =

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ F ⋅ Δσ 0 ⎠

2

2.25

whereas Eq. (2.24) can be rewritten as: Δσ 0 n =

ΔKth F ⋅ π ( a + a0 )

2.26

The diagrams reported in Fig. 2.15 show the accuracy of Topper’s approach in estimating fatigue limits in the short-crack region for two different metallic materials. It has to be pointed out here that the methods summarised above can be successfully applied not only to predict the short-crack fatigue behaviour but also to estimate the sensitivity of materials to defects as well as to the so-called short notches, that is, notches having relevant dimensions of the order of L (Taylor, 2001). In particular, according to Atzori and Lazzarin (2000), Kitagawa and Takahashi’s diagram can easily be modified to create an explicit link between the sensitivity of materials to notches and their sensitivity to defects. Consider then a plate containing a central slot of length 2a and root radius equal to rn (Fig. 2.16a). By keeping constant the 2a to rn ratio, i.e. by keeping constant the Kt,gross value, the length of the slot can be gradually reduced, obtaining the diagram sketched in Fig. 2.16b. It is worth noting that this schematic chart takes as its starting point the assumption that the geometrical factor, F, would be invariably equal to unity if this plate contained a through-crack of the same length, 2a, as that of the considered slot. In other words, the gross width of the plate is assumed to be much larger than the length of the slot, so that the problem is addressed by simply considering a geometrical feature weakening an infinite plate. Figure 2.16b shows that notches behave like short cracks when a approaches L, whereas when a is larger than a* (blunt notches), where (Atzori and Lazzarin, 2000) 2 a* = Kt,gross ⋅L

2.27

fatigue assessment can be performed by simply using the linear elastic stress determined at the notch tip. Finally, for values of a ranging between L and

56

Multiaxial notch fatigue 1000 Δs0 = 518 MPa

G40.11 Steel ΔKth = 13 MPa·m1/2 R = –1

Δs0n (MPa) 100

Eq. (2.24) LEFM L = 0.2 mm

10 0.01

0.1

1

10

100 a (mm)

1000 Δs0n (MPa)

2.25Cr-1Mo Steel R = –1

Δs0 = 500 MPa

ΔKth = 10 MPa·m1/2 Eq. (2.26)

LEFM a0 = 0.21 mm

100 0.01

0.1

1

a (mm) 10

2.15 Accuracy of Topper’s method in predicting fatigue behaviour of short cracks in specimens of G40.11 steel (El Haddad et al., 1979) and of 2.25Cr–1Mo steel (Lukas and Kunz, 1992).

Δsgross

rn

log Δs0n

Experimental trend Δs0

Short notches

Δs0n =

ΔKth p⋅a

sΔ0 Kt,gross

2a

Sharp notches Δsgross

(a)

Blunt notches a*

L

(b)

2.16 Schematisation of Atzori and Lazzarin’s diagram.

log a

Fundamentals of fatigue assessment 1000

Δs0 = 582 MPa

Ds0n (MPa)

100

ΔKth = 8.1 MPa·m1/2

57

C45 Steel R = –1

Kt,gross = 3.2 Kt,gross = 7.8

Eq. (2.28) Blunt notches

LEFM

L = 0.061 mm 10 0.001

0.01

0.1

1

F2a (mm)

100

2.17 Accuracy of Atzori and Lazzarin’s diagram in linking the defect to the notch sensitivity region (data from Nisitani and Endo, 1988).

a*, notch fatigue limits can be estimated by directly using the LEFM concepts: in this region notches behave like long cracks and are usually called sharp notches. Atzori and Lazzarin’s diagram can also be used successfully to predict notch fatigue limits of components characterised by a shape factor, F, larger than unity. In particular, the effective length (or depth) of the notch can be calculated by simply multiplying the actual length by F2 (Atzori et al., 2003). As an example, Fig. 2.17 shows the sound agreement between Atzori and Lazzarin’s schematisation and experimental reality. In particular, the data reported in this diagram were generated by testing, under rotating bending, V-notched cylindrical specimens of C45 steel (Nisitani and Endo, 1988). Such samples had notch depth, a, ranging between 0.005 mm and 1.5 mm, tip radius between 0.01 mm and 0.6 mm, gross diameter between 5.01 mm and 8 mm and net diameter equal to 5 mm, resulting in geometrical factors, F, varying from about 1.1 up to about 2 and Kt,gross values from 1.68 up to 50.4. Figure 2.17 shows that the two results generated by testing blunt notches clearly deviate from the LEFM line, approaching the corresponding straight horizontal lines characterised by a Kt,gross value equal to 3.2 and 7.8, respectively. The problem of estimating fatigue strength in the transition zone from sharp to blunt notches will shortly be addressed from a different perspective at the end of the present section. The diagram of Fig. 2.17 also proves that the experimental fatigue limits relative to sharp notches are very well predicted by the LEFM straight line. Finally, short notches behave like short cracks, so that predictions made according to LEFM as well as from continuum mechanics are non-conservative. In order to correctly estimate the fatigue strength of the above type of notches, Atzori and co-workers have suggested modifying Topper’s equation as follows (Atzori et al., 2003):

58

Multiaxial notch fatigue ΔKth = Δσ 0 n π ( F 2 a + L)

2.28

Figure 2.17 clearly proves that Eq. (2.28) is really effective in estimating high-cycle fatigue strength of short notches. The last schematisation linking the sharp- to blunt-notch behaviour which definitely deserves to be mentioned is the one proposed by Frost (1957, 1959) and subsequently reinterpreted in terms of LEFM by Smith and Miller (1978). For the sake of simplicity, consider a U-notched cylindrical bar subjected to uniaxial fatigue loading (Fig. 2.18a). Assume that, by keeping constant the notch depth, a, as well as the gross diameter, Kt,gross is increased by simply reducing the length of the tip radius, rn. For any value of the stress concentration factor investigated, the notch fatigue limit obtained can then be plotted against the corresponding Kt,gross. Following the above procedure it is possible to build diagrams similar to the one sketched in Fig. 2.18b. In such a chart also the curve estimated by using the peak stress criterion as well as the straight line determined according to the LEFM concepts are plotted. Such a schematic diagram makes it evident that the blunt-notch behaviour (Kt,gross < K*t,gross in Fig. 2.18b) can be successfully predicted by simply dividing the plain fatigue limit, Δs0, by the stress concentration factor. On the contrary, for Kt,gross values larger than K*t,gross, predictions made by using the peak stress criterion become too conservative, because in that region notches behave like long cracks. In other words, sharp-notch fatigue limits can be accurately estimated by simply using the following relationship (Smith and Miller, 1978): Δσ 0n =

ΔKth

2.29

F πa

It is interesting to highlight that Frost, Smith and Miller’s diagram allows also the non-propagating crack (NPC) zone to be easily located. In particular, for Kt,gross values larger than K*t,gross, where Δs0n

Δs0n =

Δs0

a rn

Δsgross

(a)

Δs0 Kt,gross

Δs0n =

ΔKth F p⋅a

Δsgross Blunt notches 1

Non-propagating cracks Sharp notches Kt,gross

K*t,gross (b)

2.18 Frost, Smith and Miller’s diagram.

Fundamentals of fatigue assessment * = Kt,gross

59

F Δσ 0 πa ΔKth

if the applied stress is between the LEFM straight line and the peak stress criterion curve (Fig. 2.18b) then meso/macro cracks are seen to grow from stress raiser apices and their propagation stops when they reach a length depending on the microstructure of the tested material as well as on the magnitude of the applied loading. In particular, according to Yates and Brown (1987), the maximum NPC length is equal to a0, Eq. (2.25), whereas according to Taylor (2001) it is equal to 2L, where L is given by Eq. (2.23). It is also worth noting that in the sharp notch region the curve obtained by applying the peak stress approach also represents the minimum stress at which the crack initiation phenomenon can take place. To conclude, Fig. 2.19 shows the accuracy of Frost, Smith and Miller’s schematisation in describing the transition from the blunt- to sharp-notch regime when V-notched cylindrical specimens of C45 steel having notch depth, a, equal to 0.5 mm and to 1.5 mm are considered. The remarks reported above should make it evident that, at the fatigue limit, there exists a close connection between the LEFM and the continuum mechanics concepts. In other words, even if at first such fatigue assessment philosophies may appear to be different, in practice they strongly interact with each other. This means that, if they worked correctly, the classical equations suitable for calculating Kf, and reviewed at the beginning of Section 2.4, should be capable of connecting the sharp- to the blunt-notch region by gradually linking the two asymptotic behaviours described by

Ds0n (MPa)

300 C45 Steel R = –1

250 a = 0.5 mm F = 1.24

200

Δs0 = 582 MPa ΔKth = 8.1 MPa·m1/2

150 a = 1.5 mm F = 1.97

100 50 0 0

10

20

30

40

50 60 Kt,gross

2.19 Accuracy of Frost, Smith and Miller’s diagram in predicting sharp- and blunt-notch fatigue limits (data from Nisitani and Endo, 1988).

60

Multiaxial notch fatigue Ds0n (MPa)

250 C45 Steel R = –1

a = 0.5 mm F = 1.24

200

Δs0 = 582 MPa ΔKth = 8.1 MPa·m1/2

150

a = 1.5 mm F = 1.97

100 50 Eq. (2.10)

Eq. (2.30)

Eq. (2.31)

30

50 60 Kt,gross

0 0

10

20

40

2.20 Accuracy of different formulas suitable for calculating Kf in describing the sharp- to blunt-notch regime transition (data from Nisitani and Endo, 1988).

LEFM and by the peak stress criterion, respectively. As an example, Fig. 2.20 reports in a Frost, Smith and Miller diagram the same data as those considered in Fig. 2.19, where the experimental points are initially attempted to be predicted by using Neuber’s formula, Eq. (2.10). Such a diagram shows that for the V-notched cylindrical specimens having notch depth, a, equal to 0.5 mm Neuber’s formula gives poor predictions in the crack-like notch region. On the contrary, for V-notches of depth, a, equal to 1.5 mm the use of Eq. (2.10) results in accurate predictions in both the blunt- and sharpnotch zones. Owing to the close connections existing between LEFM and continuum mechanics, in recent years many researchers have attempted to use the LEFM concepts in order to devise reliable formulas suitable for estimating Kf. In this scenario, it is interesting to highlight that the most successful theories are based on the use of distance L, Eq. (2.23). Among the above methods, certainly those devised by Lukas and Klesnil (1978) and Atzori et al. (2005) deserve to be mentioned. In particular, Lukas and Klesnil suggested predicting Kf through the following well-known relationship: Kf =

Kt,gross L 1 + 4.5 rn

2.30

Similarly to Neuber and Peterson’s idea, Eq. (2.30) postulates that highcycle fatigue damage in the presence of stress raisers depends on the stress concentration factor, the notch root radius and a critical distance material parameter.

Fundamentals of fatigue assessment

61

Atzori et al. (2005) instead argued that Kf can be successfully estimated by using the following relationship: Kf =

Kt,gross 4

⎛ Kt,gross L ⎞ 1+ ⎜ 2 ⎝ F a + L ⎟⎠ 2

2

2.31

Contrary to the other formulas reviewed above, Eq. (2.31) assumes that fatigue damage explicitly depends on the equivalent length of the notch, F2a, and not on the tip radius, rn. The diagram shown in Fig. 2.20 proves that both Eq. (2.30) and Eq. (2.31) are highly accurate in estimating the high-cycle fatigue strength of the V-notched cylindrical bars tested, under rotating bending, by Nisitani and Endo (1988). All the relationships reviewed in the present section and suitable for estimating fatigue damage in components containing different types of geometrical features were seen to be successful, even if characterised by different levels of accuracy. Unfortunately, the main difficulty in using the above methods to address problems of practical interest is that the definition of a reference section is always needed: this fact may limit the in-field application of such criteria when components having complex geometries have to be assessed. Moreover, it is evident that the type of stress concentrator (short, sharp or blunt) has to be recognised a priori in order to choose the correct fatigue assessment strategy. All the problems mentioned above can be overcome by using the so-called Theory of Critical Distances (TCD): thanks to its features, such a theory can be used to estimate fatigue damage by directly post-processing simple linear-elastic Finite Element (FE) models, without the need for classifying a priori the type of notch to be assessed as well as without the need for defining a reference nominal section to be used to calculate all the relevant stress quantities required to design stress raisers against fatigue. The peculiarities of the TCD will be reviewed in the next section in detail.

2.7

The Theory of Critical Distances

As briefly mentioned at the beginning of Section 2.4, Neuber’s formula, Eq. (2.10), represents the first attempt at using the so-called Theory of Critical Distances (TCD) to estimate high-cycle fatigue strength of mechanical components experiencing stress concentration phenomena (Neuber, 1936, 1958). In particular, by following a sophisticated reasoning, he suggested calculating an engineering quantity suitable for estimating notch fatigue damage by averaging over materials units (that is, crystals or structural particles) the linear-elastic maximum principal stress close to the

62

Multiaxial notch fatigue

notch apex. A few years later, Peterson (1959) argued that Neuber’s idea could be greatly simplified by determining the reference stress to be used to design notches against fatigue at a given distance from the apex of the stress raiser: this method is known as the Point Method (PM). It is important to highlight that at Neuber’s and Peterson’s time the above ideas could not easily be applied in practice, because of a lack of analytical tools suitable for describing linear-elastic stress fields in the vicinity of those geometrical features commonly contained by real components. This explains why Neuber and Peterson proposed using the two well-known formulas reviewed in Section 2.4 to perform the high-cycle fatigue assessment in the presence of stress concentration phenomena: both Eqs (2.10) and (2.13) simply take as their starting point the idea that the notch root radius, rn, is the most important parameter controlling the stress field distribution along the notch bisector. In recent years, the above pioneering ideas have been revised by different researchers by taking full advantage of both the LEFM concepts and the accuracy of the FE method in determining the linear-elastic stress fields damaging engineering materials in the vicinity of any type of stress concentrator. Thanks to such extensive theoretical and experimental work, nowadays we have a powerful theory which can be used to accurately assess mechanical components in situations of practical interest. Working in collaboration with Professor David Taylor (Trinity College, Dublin, Ireland), we also made a big effort in order to better investigate the peculiarities of such a theory as well as to systematically check its accuracy when applied to estimate notch fatigue strength under different loading conditions. In the next subsections the most important features of the TCD will be considered in detail, attempting to review the way such a theory works when used to predict both notch fatigue limits and fatigue lifetime of notched components failing in the medium-cycle fatigue regime under uniaxial cyclic loading. Finally, it is worth concluding by observing that the TCD proved to be highly accurate not only when applied to address the notch fatigue problem but also when used in other ambits of the structural integrity discipline: for an exhaustive overview of all the peculiarities of such a powerful theory, the reader is referred to the excellent book recently written by David Taylor (Taylor, 2007).

2.7.1 The TCD to predict notch fatigue limits under uniaxial fatigue loading Thanks to its features, the TCD is capable of accurately estimating fatigue damage in components containing not only cracks but also any type of notch (i.e., short, sharp and blunt). In other words, such a theory was seen

Fundamentals of fatigue assessment

63

to be suitable for describing the transition from the short-crack/notch to the blunt notch regime (Susmel, 2008). Moreover, the TCD makes use of both the LEFM and continuum mechanics concepts, so that, for the same reasons as above, in the present section stresses are meant to be calculated with respect to the gross sectional area. Before considering in detail the different formalisations of the TCD, it is worth highlighting here that such a theory estimates fatigue damage by directly post-processing the linearelastic stress fields acting on the process zone. This aspect is very important, because the TCD allows real components to be assessed without the need for carrying out complex and time-consuming elasto-plastic analyses. The TCD takes as a starting point the assumption that fatigue damage in the presence of stress concentration phenomena has to be estimated by using a stress quantity which is representative of the entire linear-elastic stress field damaging the fatigue process zone. In particular, notches are assumed to be in their fatigue limit condition when a suitable effective stress, Δseff, equals the material plain fatigue limit, Δs0, that is: Δσ eff = Δσ 0

2.32

The above effective stress can be calculated in different ways by simply defining a convenient critical distance and a suitable integration domain. In particular, independently of the adopted definition for Δseff, all the most modern formalisations of the TCD assume that the critical distance is a material property which has to be calculated as (Tanaka, 1983; Atzori et al., 1992; Lazzarini et al., 1997; Taylor, 1999): L=

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

2.23

where, as usual, ΔKth is the range of the threshold value of the stress intensity factor and Δs0 is the plain fatigue limit (both determined under the same load ratio, R, as the one damaging the mechanical component to be assessed). According to definition (2.23), L depends on two material properties, so that it is in turn a material property which is different for different materials and different load ratios. Appendix A summarises several values of the material characteristic length, L, experimentally determined by considering steels, aluminium alloys, cast irons and other materials. As mentioned above, the TCD can be formalised in different ways by simply changing the definition of the integration domain used to calculate the range of the effective stress. In particular, if Δseff is calculated at a given distance from the apex of the stress concentrator, according to the so-called Point Method (PM) a notched component (as shown in Fig. 2.21a) is at its fatigue limit if the following condition is assured (Tanaka, 1983; Taylor, 1999):

64

Multiaxial notch fatigue Δsgross Point Method

r q

Δs1 Δseff

Δs1 Δseff

y

L/2

x

(a)

Line Method

r

Area Method Δseff L

2L r

Δseff = Δs0

Δseff = Δs0

(b)

(c)

Δseff = Δs0

(d)

Δsgross

2.21 Different formalisations of the Theory of Critical Distances.

L Δσ eff = Δσ 1 ⎛ θ = 0, r = ⎞ = Δσ 0 ⎝ 2⎠

2.33

Figure 2.21b is a schematic sketch showing the way the PM works when applied to a notched component subjected to uniaxial fatigue loading. According to Neuber’s idea, instead of determining Δseff at a fixed distance from the notch tip, it can also be calculated by averaging the range of the linear-elastic maximum principal stress, Δs1, along the notch bisector over a distance equal to 2L (Tanaka, 1983; Lazzarin et al., 1997; Taylor, 1999). In other words, the Line Method (LM) postulates that the fatigue limit condition for a notched component under cyclic loading can be expressed as follows (Fig. 2.21c): Δσ eff =

1 2L ∫ Δσ 1 (θ = 0, r ) dr = Δσ 0 2L 0

2.34

Moreover, according to Sheppard’s idea (Sheppard, 1991), Taylor (1999) argued that the range of the effective stress can also be calculated by averaging Δs1 over a semicircular area centred at the notch tip and having radius equal to L. This approach is known as the Area Method (AM) (Fig. 2.21d) and it postulates that a notched component is at its fatigue limit when: Δσ eff =

4 πL2

π 2L

∫ ∫ Δσ 1 (θ , r ) ⋅ r ⋅ dr ⋅ dθ ≅ Δσ 0

2.35

0 0

To conclude this brief review of the different ways to determine the range of the effective stress according to the TCD, it is also worth observing that Bellett et al. (2005) suggested calculating Δseff by averaging the linearelastic maximum principal stress over a hemisphere centred at the apex of

Fundamentals of fatigue assessment

65

the stress raiser and having radius equal to 1.54L: this is known as the Volume Method (VM). It is interesting to observe that Eqs (2.33)–(2.35) can be rigorously obtained by considering a through-crack in an infinite plate (F = 1) subjected to fatigue loading (Taylor, 1999). In particular, the linear-elastic stress field in the vicinity of the crack tip contained by the specimen sketched in Fig. 2.12 can be expressed as (Westergaard, 1939): Δσ 1 ( r, θ ) = Δσ gross

a 3θ θ θ cos ⎛ ⎞ ⎡1 + sin ⎛ ⎞ sin ⎛ ⎞ ⎤ ⎢ ⎝ ⎠ ⎝ ⎠ ⎝ 2r 2 ⎣ 2 2 ⎠ ⎥⎦

2.36

When the cracked sample of Fig. 2.12 is at its fatigue limit, then ΔKI = ΔKth: by combining Eq. (2.36) with definition (2.23), it is easy to see that at the point having polar coordinates q and r equal to 0 and L/2, respectively, the range of the maximum principal stress equals the material plain fatigue limit, obtaining identity (2.33). Similarly, if the above cracked specimen is again assumed to be at its fatigue limit (ΔKI = ΔKth), by averaging Eq. (2.36) over a distance equal to 2L it is possible to obtain identity (2.34), that is: Δσ eff =

1 2L a Δσ gross dr = Δσ 0 ∫ 2L 0 2r

2.37

Finally, by using a procedure similar to the one adopted above (Taylor, 1999), the numerical integration of Eq. (2.36) results in definition (2.35). The above considerations should make it evident that the use of the TCD is fully justified in the presence of cracks. In particular, such a method proved to be successful in predicting both short- and long-crack behaviour (Taylor, 1999). As to this aspect, it is interesting to observe that when applied to an infinite plate containing a through-central crack (F = 1), the use of the LM results in the same predictions as those obtained by applying Topper’s approach, Eq. (2.24). On the contrary, observing that, for F ≠ 1, the existing relationship between L, Eq. (2.23), and a0, Eq. (2.25), is as follows: L = F 2 a0

2.38

Taylor’s LM and Topper’s approach are slightly different when the crack length, a, approaches L, whereas they agree for the extreme cases given by the two straight asymptotic lines in Kitagawa and Takahashi’s diagram (Taylor, 2007). Unfortunately, contrary to the short/long-crack problem, as far as the writer is aware there exist no theoretical justifications explaining why the TCD is also so accurate in predicting fatigue limits of components containing conventional notches. As to the above aspect, it can be highlighted that Taylor (2001) has argued that the LM may work because it is capable

66

Multiaxial notch fatigue

of predicting the propagation (or non-propagation) of cracks emanating from the stress concentrator apices. According to the above remark, the use of the TCD would then be fully justified only in the presence of sharp notches, for which the fatigue limit condition results in the formation of NPCs. On the contrary, such an argument is not capable of explaining why the TCD is so successful also in predicting fatigue limits in the presence of blunt notches. In order to show the accuracy of the different formulations of the TCD reviewed above, several experimental results have been selected from the technical literature. The most important pieces of information concerning the considered tests are summarised in Table 2.2, whereas Fig. 2.22 plots the experimental plain fatigue limit, Δs0, vs. the effective stress, Δseff, obtained by systematically applying the PM, LM and AM, respectively, to the data reported in Table 2.2. It is important to highlight that notches have been classified into short, sharp and blunt according to the following rule proposed by Taylor (2001): short notches have length (or depth) lower than 3L, whereas for long notches the transition from blunt to sharp occurs Table 2.2 Summary of the experimental data used to show the accuracy of the different formalisations of the TCD L (mm) Spec. Load Reference ΔKth Δs0 type* type* (MPa) (MPa m1/2)

Material

R

AA356-T6

−1

231

4.4

0.116

CNB

RB

Atzori et al., 2004

C45

−1

582

8.1

0.061

CNB

RB

Nisitani and Endo, 1988

C36

−1

450

4.5

0.033

CNB

RB

Nisitani and Endo, 1988

6060-T6

0.1

110

6.1

0.999

DENP AX

Luise, 2001

SM41B

−1 0 0.4

326 274 244

12.4 8.4 6.4

0.458 0.296 0.218

CNP

AX

Tanaka and Nakai, 1983; Tanaka and Akiniwa, 1987

Mild steel

−1

420

12.8

0.296

CNB

AX

Frost, 1957, 1959

Mild steel

−1

420

12.8

0.296

DENP AX

Frost, 1957, 1959

Steel 15313

−1

440

12

0.237

CNB

AX

Lukas et al., 1986

G40.11

−1

464

15.9

0.374

CNP

AX

El Haddad, 1978

AISI 304

−1

720

12

0.088

CNB

AX

Frost, 1957

Ni-Cr steel

−1

1000

12.8

0.052

CNP

AX

Frost, 1957

440

8.1

0.108

DENP AX

EN-GJS-800–8

0.1

Gasparini, 2001

* CNB = circumferential notch in a cylindrical bar, CNP = central notch in a flat plate, DENP = double edge notch in a flat plate, RB = rotating bending, AX = push-pull.

Fundamentals of fatigue assessment 1000

Point Method

1000

Conservative

Ds0 (MPa)

Line Method Conservative

Ds0 (MPa) 100

100

E = +20%

E = +20%

E = –20% Non-conservative

E = –20% Non-conservative 10

10 100 Dseff (MPa) 1000

10

1000

67

10

100 Dseff (MPa) 1000

Area Method Conservative

Ds0 (MPa)

Blunt Sharp Short

100 E = +20% E = –20% Non-conservative 10 10

100 Dseff (MPa) 1000

2.22 Experimental plain fatigue limit (Δs0) vs. effective stress (Δseff) diagrams obtained by applying the PM, LM and AM, respectively.

at Kt*. The diagrams sketched in Fig. 2.22 clearly show that the different formalisations of the TCD are capable of estimates falling within an error interval of about ±20% and this holds true independently of material and type of geometrical feature (Taylor and Wang, 2000; Susmel and Taylor, 2003). It is important to highlight that errors of ±20% are, in general, considered to be acceptable, because this is the magnitude of the normal error arising both in the experimental work and in the numerical stress analysis (Taylor and Wang, 2000). In order to better show the accuracy of the different formalisations of the TCD reviewed in the present section, Fig. 2.23 summarises the error made when attempting to estimate high-cycle fatigue strength of open notches (see Table 2.3 for a summary of the considered experimental results). Such a type of stress concentrator represents an interesting testing ground to check the accuracy of the TCD, because, for a given geometry, opening angles influence the profile of the stress field in the vicinity of the stress raiser tip, especially when they are larger than 90° (Lazzarin and Tovo, 1996).

68

Multiaxial notch fatigue

1000

Point Method

1000

Line Method

Conservative E = +20%

Ds0 (MPa)

Conservative

E = –20%

Ds0 (MPa)

E = –20%

Non-conservative 100 100

1000

Dseff (MPa)

1000

E = +20%

Non-conservative 100 100

Dseff (MPa)

1000

Area Method 45°

Conservative E = +20%

90° 120°

Ds0 (MPa)

E = –20%

135° 150° 160°

Non-conservative 100 100

Dseff (MPa)

165°

1000

2.23 Experimental plain fatigue limit (Δs0) vs. effective stress (Δseff) diagrams obtained by applying the PM, LM and AM, respectively, to open notches. Table 2.3 Summary of the experimental data used to show the accuracy of the different formalisations of the TCD in predicting high-cycle fatigue strength of open notches Material

R

Opening angles Δσ0 (MPa) ΔKth (MPa m1/2)

Spec. type*

Reference

FeP04

0.1

247

10

45°, 135°, 160°

DENP

580

13

90°, 135°, 165° 135° 135° 90°, 120° 120° 135° 90°, 120° 120° 135°, 150°

DENP BT CT DENP SENP CT DENP SENP CT

Atzori et al., 2006b Usami, 1987

HT 60 (1) 0

SS41

0.05 231

6.4

HT 60 (2) 0.05 425

6.6

Kihara and Yoshii, 1991 Kihara and Yoshii, 1991

BT = butt type, CT = cruciform type, DENP = double edge notch in a flat plate, SENP = single edge notch in a flat plate.

Fundamentals of fatigue assessment

69

Figure 2.23 clearly proves that the TCD is successful also in predicting fatigue strength of open notches, resulting again in estimates falling within an error interval of about ±20%. With such a high accuracy level, it is not surprising that this theory was seen to be successful also in predicting the fatigue strength of welded connections, that is, in the presence of stress raisers having a notch opening angle equal to 135° (Taylor et al., 2002; Crupi et al., 2005). To conclude the present section, it is worth mentioning the fact that Bellett and Taylor (Bellett et al., 2005; Bellett and Taylor, 2006) have recently observed that the use of the TCD results in estimates which are too conservative when it is applied to tridimensional stress raisers. They argued that such a high degree of conservatism could be due to the different shapes of the cracks propagating in two-dimensional geometries to those growing in three-dimensional bodies. According to this idea, they attempted to correct the estimates made by the TCD through an ad-hoc correction factor: such a strategy proved to be successful, allowing them to increase the accuracy of the TCD when used to predict notch fatigue limits in the presence of tridimensional stress concentrators.

2.7.2 The TCD to estimate fatigue lifetime of notched components under uniaxial fatigue loading The linear-elastic TCD proved to be highly accurate not only in predicting notch fatigue limits but also in estimating fatigue lifetime of notched components failing in the medium-cycle fatigue regime (Susmel and Taylor, 2007). Such an extension of the TCD takes as its starting point the assumption that the critical distance value, LM, changes as the number of cycles to failure, Nf, decreases. In other words, fatigue lifetime of components containing stress raisers can be estimated provided that the LM vs. Nf relationship is known. According to the main feature of the TCD, the above relationship is assumed to be a fatigue property which is different for different materials and different load ratios. Observing that the behaviour of engineering materials in the medium-cycle fatigue regime is described by a power function, it is logical to presume that the above LM vs. Nf relationship results in turn in the following power law (Susmel and Taylor, 2007): LM ( N f ) = A ⋅ N fB

2.39

where A and B are material constants to be determined by running appropriate experiments.

70

Multiaxial notch fatigue

In particular, it is easy to observe that Eq. (2.39) can be calibrated by simply considering two experimental results generated at two different values of the number of cycles to failure. Strictly speaking, constants A and B could be determined by using the critical distance value determined under static loading, LS, and the critical distance value, L, determined according to Eq. (2.23), where LS is defined as follows (Taylor, 2007): LS =

1 ⎛ KIc ⎞ ⎜ ⎟ π ⎝ σS ⎠

2

2.40

In the above definition KIc is the plane-strain fracture toughness, whereas the inherent material strength, sS, is a static property to be determined experimentally (Taylor, 2007). It is evident that such a procedure can be applied to calibrate the LM vs. Nf relationship provided that the reference number of cycles to failure is known in both the low/medium, NS, and the high-cycle fatigue regime, N0 (Fig. 2.24). Even if there exists no theoretical inconsistency in following the above strategy to determine constants A and B, it is very difficult to use such a method in practice for two reasons: (i) because the stressbased approach is not adequate at describing the behaviour of engineering materials in the low-cycle fatigue regime, so that the determination of NS is in any case approximate; and (ii) because it is very difficult to coherently define N0 due to the fact that, for a given material, the position of the knee point can change by changing the geometry of the tested samples. In order to overcome the above problems, we have proposed an alternative procedure based on two calibration curves (Susmel and Taylor, 2007). In particular, by using the PM argument and according to Fig. 2.25, for a given value of Nf, that is, Nf = Nf,k, it is easy to determine the distance from the notch tip, LM(Nf,k)/2, at which the linear-elastic maximum principal

log sx,a

y

LS =

sUTS

F(t)

2

R = –1

Plain fatigue curve 1 ⎛ ΔKth L= ⎜ p ⎝ Δs0

x F(t)

1 ⎛ KIc ⎞ ⎜ ⎟ p ⎝ sUTS ⎠

s0

NS

N0

log Nf

2.24 Determination of the critical distance value in the low/ medium- as well as in the high-cycle fatigue regime.

⎞ ⎟ ⎠

2

Fundamentals of fatigue assessment

71

Δsgross Δsgross

Δs1 Plain fatigue curve

Δs1,k

Δs1,k Δsgross

Notch fatigue curve

Nf,k

LM (Nf,k) 2

r

Nf Δsgross

2.25 Determination of the critical distance value in the medium-cycle fatigue regime by using two calibration fatigue curves.

stress equals the stress to be applied to the plain specimens, Δs1,k, to generate a failure at Nf = Nf,k cycles. By calculating the critical distance value both in the low/medium and in the high-cycle fatigue regimes, it is then possible to directly determine constants A and B in Eq. (2.39). Such a simplified procedure allows one not only to overcome the problems mentioned above but also to correctly take into account the statistical dispersion of fatigue data: in order to coherently determine constants A and B in Eq. (2.39), the two calibration curves have to be calculated under the same statistical hypotheses and for the same probability of survival. Finally, after calibrating the LM vs. Nf relationship, by using a conventional recursive procedure the number of cycles to failure can be estimated in the presence of any geometrical feature, provided that the component to be assessed is made of the same material as the one for which constants A and B in Eq. (2.39) have previously been determined. For the sake of clarity, the simplest recursive procedure suitable for using the TCD to predict fatigue damage in the medium-cycle fatigue regime is briefly explained in Fig. 2.26. Consider then a notched specimen subjected to a given value of the nominal stress range, Δsgross (Fig. 2.26a). If an initial value of the number of cycles to failure, Nf,a, is assumed, then, according to Eq. (2.39), the corresponding critical distance results in the following value: B LM ( N f,a ) = A ⋅ N f,a

2.41

72

Multiaxial notch fatigue Δs1 Δsgross Δs1,a Notch tip

LM(Nf,a) 2

r

(b) Δsgross

(a)

2.26 In-field use of the TCD to estimate lifetime of notched components in the medium-cycle fatigue regime.

The linear-elastic stress field close to the notch tip (Fig. 2.26b) can now be used to calculate the maximum principal stress range, Δs1,a, at a distance from the notch tip equal to LM(Nf,a)/2. From the Wöhler equation of the parent material, the value of Δs1,a determined as above can then be used to recalculate the number of cycles to failure: ⎛ Δσ 0 ⎞ N f,b = N 0 ⎜ ⎝ Δσ 1,a ⎟⎠

k

2.42

If Nf,b is not equal to the assumed initial value, Nf,a, then the above procedure has to be reapplied by imposing Nf,a = Nf,b and has to be iterated until the problem has reached convergence. To conclude, it is worth saying that only the way of applying the TCD in terms of the PM was considered above, but similar strategies can be employed to estimate medium-cycle fatigue damage according to both the LM and the AM. The accuracy and reliability of the above reformulation of the TCD were systematically checked by using several experimental results generated by testing specimens containing different geometrical features and made of different materials (Susmel and Taylor, 2007). In particular, the above method proved to be capable of estimates falling within the parent material scatter band and this held true independently of the formalisation of the TCD used to predict fatigue lifetime. As an example, Fig. 2.27 shows the experimental, Nf, vs. estimated, Nf,e, number of cycles to failure diagrams obtained by using the LM to estimate medium-cycle fatigue damage in notched specimens of En3B loaded in tension–compression (R = −1), tension–tension (R = 0.1) and three-point bending (R = 0.1). The notch root radii considered ranged between 0.12 mm and 5 mm, resulting in Kt,gross values varying from 16.2 down to 2.9. Figure 2.27 clearly proves that the above reformulation of the TCD is highly accurate, making it suitable for

Fundamentals of fatigue assessment

73

Line Method

Line Method 10 000 000

10 000 000 V-Notched Hole 8 mm Hole 3.5 mm U-Notched

Nf (cycles)

R = 0.1

PS = 95% Nf (cycles)

PS = 95%

Parent material scatter band

100 000

100 000

Parent material scatter band PS = 5%

10 000

10 000

PS = 5%

R = –1 1000 1000

10 000

100 000 1 000 000 10 000 000

1000 1000

10 000

100 000 1 000 000 10 000 000

Nf,e (cycles)

(a)

V-Notched - TT Hole 8 mm Hole 3.5 mm U-Notched - TT Plain - 3PB V-Notched - 3PB U-Notched - 3PB

Nf,e (cycles)

(b)

2.27 LM’s accuracy in estimating fatigue lifetime of notched specimens loaded (a) in fully reversed tension as well as (b) in tension–tension, TT, and in three-point bending, 3PB (data from Susmel and Taylor, 2007).

performing the fatigue assessment of real components in situations of practical interest. In particular, it is worth observing that the estimates summarised in Fig. 2.27b refer also to plain and notched specimens tested under three-point bending, being constants A and B calculated using two calibration curves determined under tension–tension by imposing a load ratio, R, equal to 0.1. Moreover, the samples used to calibrate the LM vs. Nf relationship had thickness equal to 6 mm, whereas in the three-point bending specimens it was 25 mm. The high level of accuracy shown in Fig. 2.27b confirms that the TCD is successful also in accounting for the presence of stress gradients generated by the geometry itself superimposed on the stress gradients generated by the external nominal loading.

2.8

Fatigue assessment under torsional loading

Real mechanical assemblies are often damaged by in-service shear stresses, so that accurate methods are needed to assess components also in the above circumstances. Consider then a plain cylindrical bar loaded in torsion (Fig. 2.28a). Similarly to the uniaxial case, fatigue strength of engineering materials subjected to Mode III loading can be summarised in Wöhler diagrams by simply plotting the amplitude (or range) of the shear stress against the number of cycles to failure, Nf (Fig. 2.28b). As to the correct way of estimating torsional fatigue curves, it has to be pointed out that methodologies similar to those adopted under uniaxial fatigue loading can be used successfully to correctly take into account the statistical dispersion of experimental fatigue results generated under torsion. In other words, when torsional

74

Multiaxial notch fatigue txy,a z y

tA t0

O x

NA (a)

Mt(t)

N0

Nf

(b)

2.28 (a) Plain cylindrical bar loaded in torsion and (b) corresponding Wöhler curve. 600 R = –1 500 t0 t0 (MPa)

s0

400

=1 t0

300

s0

=

1 2

200 t0 100

s0

=

1 3

0 0

100 200 300 400 500 600 700 800 900 s0 (MPa)

2.29 Torsional (t0) vs. uniaxial plain fatigue limit (s0) diagram (data reported in Susmel and Lazzarin, 2002).

loadings are involved, fatigue curves characterised by different values of the probability of survival, PS, can be used to perform the fatigue assessment. Moreover, for a given value of PS, it is always possible to define either the torsional plain fatigue limit, t0, or the torsional endurance limit, tA at NA cycles to failure, by using strategies similar to those already discussed in Section 2.2 and stated by the pertinent standard codes (see Fig. 2.28b for the definition of both t0 and tA). Even if the most accurate way to determine torsional plain fatigue limits is the experimental one, it is worth remembering here that the above material fatigue property can be roughly estimated from the corresponding uniaxial fatigue limit by simply using Von Mises’ criterion. Figure 2.29 shows the correlation existing between t0 and s0. In particular, this

Fundamentals of fatigue assessment

75

diagram clearly proves that the t0 to s0 ratio ranges between 0.5 and 1. According to the investigations of Fukuda and Nisitani (2003), the variation of the actual value of the above ratio is to be ascribed to the material microstructure’s response to fatigue loading. In more detail, they observed that in work-hardening metals the value of t0/s0 is equal to about 0.5. On the contrary, when the material to be assessed is characterised by a banded microstructure, the value of the fatigue limit ratio ranges between 0.55 and 0.6. Finally, in the presence of fully isotropic morphologies such a ratio varies from 0.6 up to 0.75. The above experimental outcomes clearly suggest that assuming a reference value of 1 3 for the t0 to s0 ratio allows fully reversed torsional fatigue limits to be estimated always with an adequate degree of safety. Similarly to the uniaxial case, also under torsional loading the fatigue behaviour of engineering materials is seen to be highly sensitive to many variables, which include geometrical features, size effect, surface treatments, roughness, metallurgical parameters, etc. It is evident that such aspects should always be taken into account when designing real components so as not to overestimate the actual fatigue strength of the material to be assessed. Unfortunately, even if several attempts have been made, as far as the writer is aware there exist no exhaustive investigations systematically considering the effects on the torsional fatigue strength of all the variables commonly considered when performing the uniaxial fatigue assessment. In this scenario, the only exception is represented probably by the detrimental effect of notches, which have been studied since the beginning of the last century – see, for instance, Gough (1949) and Frith (1956). However, although several investigations addressing the notch torsional fatigue problem have been published, there exist only a few experimental outcomes allowing the detrimental effect of notches loaded in torsion to be taken into account efficiently. This is the reason why, even though it is not strictly correct, it is common practice to transfer pieces of quantitative information from the uniaxial directly to the torsional case. The problem of performing the fatigue assessment of notched components loaded in torsion will be addressed in the next subsection; meanwhile it is worth concluding the present section by briefly discussing the aspects related to the mean stress effect in torsional fatigue. In order to explicitly investigate the influence of superimposed torsional static stresses on the overall fatigue strength under cyclic shear stress, Fig. 2.30 plots the ratio between the torsional fatigue limit, t0, and the corresponding experimental value generated under fully reversed loading, t0,R=−1, against the ratio between the maximum shear stress, tmax, experienced during the load cycle by every considered material, and the torsional yield stress, ty. Such a diagram clearly proves that the presence of superimposed static torsional stresses can be neglected as long as the tmax to ty ratio is

76

Multiaxial notch fatigue 1.4 1.2 t0/t0,R = –1 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2 1.4 tmax/ty

2.30 Influence of superimposed static torsional stresses on the highcycle shear fatigue strength (data from Sines, 1959).

lower than unity (Sines, 1959; Davoli et al., 2003): this experimental evidence results in a great simplification of the fatigue assessment problem, especially when fatigue damage in real mechanical components subjected to multiaxial fatigue loading is estimated by using the so-called critical plane approach (see Chapter 3). In any case, it has to be said that many other investigations show that, for certain materials, the presence of nonzero mean shear stresses results in an increase of fatigue damage. For instance, Gough (1949) observed that in cylindrical specimens of S65A a superimposed mean shear stress equal to 266 MPa reduced the torsional fatigue limit from 583 MPa to 553 MPa. Similarly, Wang and Miller (1991) proved that in specimens of 1.99% NiCrMo steel subjected to a torsional stress amplitude equal to about 280 MPa, the number of cycles to failure decreased from 106 to 105 as the mean shear stress increased from 0 to 115 MPa. They ascribed such an increment of fatigue damage to the fact that the presence of non-zero shear stresses strongly influences the crack initiation phenomenon (Stage I) as well as the subsequent propagation of Stage II cracks. In any case, in spite of the above experimental evidence, as said above, when performing the multiaxial fatigue assessment of real components it is common practice to neglect the influence of non-zero mean torsional stresses, because in any case their contribution to the overall fatigue damage is in general very slight.

2.8.1 Notch torsional fatigue As for the uniaxial case, also under torsional fatigue loading the influence of stress concentration phenomena deserves to be considered in detail due to the fact that such phenomena strongly affect the overall fatigue strength

Fundamentals of fatigue assessment

77

1000 Plain Notch root radius = 0.4 mm Notch root radius = 0.2 mm

Dtxy (MPa)

PS = 90%

PS = 10% ΔtAn = 310.5 MPa Kft = 1.2 ΔtAn = 249.5 MPa Kft = 1.5 R = –1 100 10 000

100 000

1 000 000

10 000 000 Nf (cycles)

2.31 Fatigue results generated by testing plain and V-notched cylindrical specimens of low-carbon steel (sUTS = 500 MPa) under fully reversed torsional loading (data from Qilafku et al., 2001).

of engineering materials. As an initial example, the Wöhler diagram sketched in Fig. 2.31 clearly shows the detrimental effect of circumferential V-notches in cylindrical bars of low-carbon steel (in the figure notch data are plotted in terms of range of the nominal shear stress calculated with respect to the net section). In particular, these results were generated by testing under fully reversed torsion V-notches having root radii equal to 0.4 mm and 0.2 mm, resulting in a torsional stress concentration factor, Ktt,net, equal to 1.9 and 2.4, respectively (Qilafku et al., 2001). The state of the art shows that different strategies have been devised to estimate the damaging effect of notches subjected to torsional loading, but, before briefly reviewing the most interesting methods suitable for addressing such a problem, it is worth observing that, when dealing with cylindrical components, the notch torsional fatigue limit can be roughly estimated from the corresponding notch uniaxial fatigue limit by simply using Von Mises’ criterion: Fig. 2.32 clearly shows that such a hypothesis allows notch torsional fatigue limits to be estimated with an adequate degree of safety. In order to correctly address the notch torsional problem, it can be recalled here that, as under uniaxial loading, Peterson (1974) suggested accounting for the detrimental effect of notches in mechanical components loaded in torsion through the fatigue notch factor, Kft, defined as:

78

Multiaxial notch fatigue 600 t0n (MPa)

R = –1 500 t0n = 0.57 ⋅ σ0n

400 300

Von Mises

200

t0n = 100

s0n 3

0 0

100 200 300 400 500 600 700 800 900 s0n (MPa)

2.32 Torsional (t0n) vs. uniaxial notch fatigue limit (s0n) diagram (data from Gough, 1949; Frith, 1956).

K ft =

τ0 τ 0n

2.43

In the above definition t0 is the plain torsional fatigue limit, whereas t0n is the notch torsional fatigue limit. It is evident that the most accurate way to determine Kft is by running appropriate experiments, but when this is not possible, according to Peterson, its value can be estimated by using the following formula (Peterson, 1974): K ft = q ( Ktt − 1) + 1

2.44

where Ktt is the linear-elastic stress concentration factor calculated under shear stress and q is the so-called notch sensitivity factor. Again q is equal to zero in the presence of non-damaging notches, whereas it is equal to unity when the assessed material is fully sensitive to the considered notch. As far as the writer is aware, there exist no universally accepted empirical formulas suitable for estimating the notch sensitivity, q, under torsional loading, so that it is common practice to calculate q by simply using the methods proposed and validated by considering notched specimens subjected to uniaxial fatigue loading. Even if the results obtained by adopting the above stratagem can sometimes be accurate enough, it has to be said that, from a theoretical point of view, this simplification hides some inconsistencies. In particular, it can be observed that, for a given geometrical feature, the linear-elastic stress distribution in the vicinity of the notch tip under uniaxial loading is different from the corresponding one under torsional loading. The fact that the formulas suitable for calculating Kf under uniaxial loading

Fundamentals of fatigue assessment

79

proposed by both Neuber and Peterson are based on the definition of a critical distance constant should make it evident that extending them to torsional situations may result in inaccurate estimates due to the evident differences in the stress field distributions. In spite of the aforementioned difficulties, also the TCD can be used to estimate notch torsional fatigue limits, provided that it is reformulated in order to correctly address such a problem (Susmel and Taylor, 2006). Consider then the notch sketched in Fig. 2.33 and subjected to torsional cyclic loading. Initially, it is easy to observe (see Fig. 2.33) that along the notch bisector (q = 0°) the following condition is always assured: Δτ θ y = Δτ xy = Δσ 1

2.45

According to the above identity, the PM, LM and AM can then be rewritten for the case of torsion as: L L 2.46 Δτ eff = Δτ xy ⎛ θ = 0, r = T ⎞ = Δσ 1 ⎛ θ = 0, r = T ⎞ = Δτ 0 ⎝ ⎝ 2 ⎠ 2 ⎠ Δτ eff = Δτ eff =

1 2 LT 4 πL2T

4 = π L2T

2 LT

2 LT

1

∫ Δτ xy (θ = 0, r ) dr = 2L ∫ Δσ 1 (θ = 0, r ) dr = Δτ 0 T

0

π 2 LT

∫ ∫ Δτ θ y (θ , r ) ⋅ r ⋅ dr ⋅ dθ 0

0

2.48

π 2 LT

∫ ∫ Δσ 1 (θ , r ) ⋅ r ⋅ dr ⋅ dθ = Δτ 0 0

0

x y

z

2.47

0

ΔMt Δtqy r

q

Δtry Δt Δtqy = Δtxy = Δs1

Δs3

Δs1 Δs

Along the notch bisector (q = 0°)

2.33 Notch subjected to cyclic torsion.

ΔMt

80

Multiaxial notch fatigue

where LT is the critical distance value under anti-plane torsion, that is, a material property to be determined by running appropriate experiments. According to the TCD’s philosophy a notched component subjected to torsional loading is then in the fatigue limit condition when the range of the effective torsional stress, Δteff, calculated according to one of the above formulations, equals the material torsional plain fatigue limit, Δt0 (Susmel and Taylor, 2006). Strictly speaking, and similarly to definition (2.23), the critical distance value under torsion can be determined as follows: LT =

1 ⎛ ΔKIII,th ⎞ ⎜ ⎟ π ⎝ Δτ 0 ⎠

2

2.49

where ΔKIII,th is the range of the threshold value of the stress intensity factor under Mode III loading and, as always, Δt0 is the torsional plain fatigue limit. Unfortunately, due to the well-known experimental difficulties in determining ΔKIII,th, a rigorous determination of LT is never easy in practice. As to the determination of LT, initially it is interesting to observe that, in general, the critical distance value under tension is seen to be different from the corresponding value under torsion. For instance, Murakami and coworkers (Endo and Murakami, 1987; Murakami and Takahashi, 1998), by testing cylindrical specimens of 0.46% C steel weakened by superficial artificial defects (where the depth of the drilled hole was equal to its diameter), have observed that the diameter of the defect which does not reduce the specimen fatigue strength under torsion is much larger than that under rotating bending. In more detail, the Kitagawa–Takahashi diagram sketched in Fig. 2.34 shows that, for the material tested by Murakami and co-workers, the critical value of the diameter was equal to about 35 μm under rotating 1000

Stress (MPa)

Ds µ d–1/6 Δ Ds0

480 MPa

Dt0

284 MPa

Bending Torsion

Dt µ d–1/6 ª35 mm

ª130 mm

100 10

100

1000

Diameter of hole, d (mm)

2.34 Critical distance values under both bending and torsion (data from Endo and Murakami, 1987).

Fundamentals of fatigue assessment

81

bending and to about 130 μm under torsion. It can be observed also that in Fig. 2.34 the slope of the two LEFM straight lines is equal to about −1/6 and not −1/2 as expected. This anomalous behaviour can be ascribed to the fact that Murakami and co-workers tested specimens containing holes and not cracks, as well as to the fact that, even if they were cracks, they would not be large enough to behave like long cracks. By using the PM argument, we calculated the critical distance value under bending as well as under torsion by considering several experimental results taken from the literature and generated by testing V-notched cylindrical samples made of different steels (Susmel and Taylor, 2006). The results obtained are summarised in Table 2.4, while the numerical procedure followed to determine both L and LT is summarised in Fig. 2.35. The table Table 2.4 Critical distance values under uniaxial fatigue loading, L, and under torsional fatigue loading, LT, calculated according to the procedure summarised in Fig. 2.35 Material

L (mm)

LT (mm)

L/LT

Reference

0.4% C steel (normalised) 3% Ni steel Cr-Va steel 3.5% NiCr steel (normal impact) 3.5% NiCr steel (low impact) NiCrMo steel (75–80 tons) Low-carbon steel

0.111

0.305

0.364

Gough, 1949

0.145 0.100 0.103

0.215 0.148 0.183

0.674 0.676 0.563

Gough, 1949 Gough, 1949 Gough, 1949

0.098

0.12

0.817

Gough, 1949

0.075

0.208

0.361

Gough, 1949

0.143

0.339

0.423

Qilafku et al., 2001

Torsional fatigue loading

Uniaxial fatigue loading Δsx

Δtxy

Δs0 Δt0 L/2

LT/2

r Δs0n

Δt0n sx

txy

r Δs0n

r

Δt0n

2.35 Procedure used to calculate the critical distance values summarised in Table 2.4.

r

82

Multiaxial notch fatigue

clearly shows that the critical distance value under torsion is larger than the corresponding value determined under uniaxial fatigue loading. Owing to the complexity of the material cracking behaviour under torsional loading, it is always advisable to determine the critical distance under shear stress by running appropriate tests in order to correctly quantify the defect sensitivity of the material to be assessed. Unfortunately, this way of calculating LT is sometimes not compatible with the needs of industrial reality, so that the TCD can in any case be used to estimate notch fatigue limits under torsion by simply assuming that the critical distance value under Mode III torsion is equal to its value determined under uniaxial fatigue loading (Susmel and Taylor, 2006): LT = L =

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

2.50

Even if, as proved by Table 2.4, such an assumption is not correct from a scientific point of view, it was seen to be acceptable from an engineering point of view, allowing the TCD to be applied by simply using pieces of experimental information obtained under simple uniaxial fatigue loading (Susmel and Taylor, 2006). To conclude, the correctness of such a simplification is clearly proved by the error diagrams reported in Fig. 2.36, where the considered results were generated by testing, under fully reversed torsion, V-notched samples as well as cylindrical specimens with fillet: such charts show that the PM, LM and AM applied by assuming LT = L are capable of estimates falling within the same error interval as the typical one shown by the TCD when used to estimate notch fatigue limits under uniaxial cyclic loading.

2.9

Fatigue damage under multiaxial fatigue loading

In many different applications, mechanical assemblies are loaded by complex systems of cyclic forces resulting in multiaxial stress states damaging components’ hot spots. Estimating fatigue strength of engineering materials subjected to multiaxial cyclic loading is a complex task, so that engineers dealing with problems of practical interest need accurate and reliable methodologies to correctly perform the fatigue assessment in the presence of such particular loading paths. Due to its relevance, since the pioneering work done by Gough (1949) major efforts have been made by several researchers both to understand the cracking behaviour of materials damaged by bi/tridimensional cyclic stress/strain states and to devise safe engineering procedures suitable for designing components against multiaxial fatigue. As an initial example, Fig. 2.37 clearly shows the detrimental effect of a cyclic torsional shear stress component when superimposed on a

Fundamentals of fatigue assessment Point Method

1000

1000 Dt0 (MPa)

E = +20%

Line Method

Dt0 (MPa)

Conservative

E = –20%

Conservative

E = +20%

Non-conservative 100 100

1000

Dteff (MPa) 1000

83

E = –20% Non-conservative

100 100

Dteff (MPa) 1000

Area Method

Dt0 (MPa)

Conservative

E = +20%

E = –20% Non-conservative

100 100

Dteff (MPa) 1000

2.36 TCD’s accuracy in estimating torsional notch fatigue limits by assuming L = LT (Susmel and Taylor, 2006).

uniaxial cyclic stress due to bending. In particular, the data reported in the diagram of Fig. 2.37 were generated by testing tube-to-plate welded joints (Fig. 2.37a) subjected to combined fully reversed bending and torsion. The two sets of data plotted in this chart were obtained by testing welded details having the same absolute dimensions but made of structural steel and aluminium, respectively. It is important to highlight that such data have been summarised in Fig. 2.37b in terms of nominal stresses calculated, at the weld toe, with respect to the tube circular section. Such a Wöhler diagram also shows that for the steel specimens the presence of non-zero out-of-phase angles resulted in a increase of fatigue damage with respect to the corresponding biaxial results generated under in-phase loading. On the contrary, the nonproportionality of the applied loading did not affect the biaxial fatigue behaviour of the considered aluminium welded details: this is the reason why for the aluminium samples in-phase and out-of-phase data have been reanalysed together by plotting a unique PS = 50% fatigue curve.

84

Multiaxial notch fatigue 1000

z

PS = 50%

y

Dsx (MPa)

PS = 50%

x

For the biaxial tests:

PS = 50%

PS = 50%

Δsx = 3 Δtxy Steel

100 PS = 50%

Mt(t) Mf(t)

R = –1 10 10 000

Aluminium

Bending In-phase bending/torsion 90° Out-of-phase bending/torsion

100 000

1 000 000 10 000 000 Nf (cycles)

(a) (b)

2.37 Multiaxial fatigue behaviour of tube-to-plate welded joints subjected to in-phase and 90° out-of-phase bending and torsion (data from Sonsino, 1995; Sonsino and Kueppers, 2001; Kueppers and Sonsino, 2003).

As to the response of engineering materials to non-proportional loading, Sonsino (1995) has observed that the presence of non-zero out-of-phase angles does not always result in a increase of fatigue damage as commonly presumed. In particular, when out-of-phase loadings are involved, the principal stress directions may rotate during the load cycle, resulting in a difficult evaluation of fatigue strength, because in such circumstances fatigue damage strongly depends on the mutual interaction between the intrinsic material ductility and the cyclic change of the maximum principal stress direction (Sonsino, 1995; Sonsino amd Kueppers, 2001). This means that, for particular engineering materials, the presence of non-proportional loading can have either a beneficial effect or even no effect at all on the overall fatigue strength of the mechanical component to be assessed. The above remark should make it evident that only by running appropriate experiments can the actual material response to non-proportional loading be correctly evaluated. When engineering materials are damaged in the high-cycle fatigue regime by in-phase bending and torsion, Gough (1949) observed that their highcycle fatigue strength can be summarised in txy,a vs. sx,a diagrams. Figure 2.38 shows two examples of this way of representing biaxial fatigue results. In particular, by performing an accurate and extensive experimental investigation, Gough classified engineering materials into two groups according to the following rule: Ductile materials:

σ0 ≥ 3 τ0

2.51

Fundamentals of fatigue assessment 500 450 txy,a 400 (MPa) 350 300 250 200 150 100 50 0

NiCr steel s0 = 810.0 MPa t0 = 452.3 MPa (Ductile) Failure

Eq. (2.53)

250

85

0.4% C steel (normalised) s0 = 331.9 MPa t0 = 206.9 MPa (Brittle)

txy,a 200 (MPa) 150

Failure Eq. (2.54)

100 50 R = –1

R = –1 0

0

200

400

600 800 1000 sx,a (MPa)

(a)

0

100

200

300 400 sx,a (MPa)

(b)

2.38 (a) Ductile and (b) brittle multiaxial fatigue behaviour according to Gough (data from Gough, 1949).

Brittle materials: 1.2 <

σ0 ≤ 3 τ0

2.52

Gough’s experimental outcomes show that ductile materials under in-phase bending and torsion are at the fatigue limit as long as the condition expressed by the following equation describing a quarter of an ellipse is assured: 2 ⎛ σ x ,a ⎞ + ⎛ τ xy,a ⎞ ≤ 1 ⎜⎝ ⎟ ⎜ ⎟ σ0 ⎠ ⎝ τ0 ⎠ 2

2.53

On the contrary, an arc of an ellipse is suggested to be used to perform the biaxial fatigue assessment of brittle materials: ⎛ τ xy,a ⎞ + ⎛ σ x ,a ⎞ ⎛ σ 0 − 1⎞ + ⎛ σ x ,a ⎞ ⎛ 2 − σ 0 ⎞ ≤ 1 ⎜⎝ ⎟ ⎜ ⎟ ⎜ ⎟⎠ ⎜⎝ ⎟⎜ ⎟ τ0 ⎠ ⎝ σ0 ⎠ ⎝ τ0 σ0 ⎠ ⎝ τ0 ⎠ 2

2

2.54

Gough’s diagrams reported in Fig. 2.38 clearly prove the accuracy of Eqs (2.53) and (2.54) in describing the high-cycle fatigue behaviour of both a ductile and a brittle material subjected to in-phase bending and torsion. Figure 2.39 instead shows the influence of the degree of nonproportionality of the applied loading on the high-cycle fatigue strength of a conventional low-carbon steel loaded in bending and torsion: for such a material the presence of non-zero out-of-phase angles results in an evident decrease of fatigue damage. As briefly said above, the behaviour of engineering materials under non-proportional loading depends on the microstructure response to the cyclic rotation of the principal directions. This experimental evidence can be better understood by observing the example reported in Figure 2.40 (Zenner et al., 2000). In more detail, specimens of 34Cr4 were tested under two different loading paths resulting in the same values of the maximum and minimum principal stresses at any

86

Multiaxial notch fatigue 160

Soft steel 0.1% C

txy,a 140 (MPa) 120

In-phase

R = –1

60° Out-of-phase 90° Out-of-phase

Eq. (2.53)

100 80 60

s0 = 235.4 MPa t0 = 137.3 MPa (Ductile)

40 20 0 0

50

100

150

200 250 sx,a (MPa)

2.39 Out-of-phase angle effect on the high-cycle fatigue strength of plain specimens of 0.1%C steel loaded in combined bending and torsion (data from Nishihara and Kawamoto, 1945).

90°

Case (a) sx(t) = sx,asin(wt) txy(t) = 0.5sx,asin(wt – 90°)

y sy(t)

s(t)

txy(t)

z(t) 0

p

s1(t)

2p

s1(t)

–90° sx(t)

wt

z(t)

t

x s3(t)

90°

x

s2(t) = 0 z(t) Case (b) sx(t) = sx,a + sx,asin(wt) sy(t) = –sx,a + sx,asin(wt)

0

π

2π

wt

–90°

2.40 Influence of the orientation of the principal directions on the high-cycle fatigue strength of specimens made of 34Cr4 (Zenner et al., 2000).

instant of the load history (Fig. 2.40). As clearly shown by the z vs. w t diagrams, the only difference was that in case (a) principal direction 1 rotated during the load cycle, whereas in case (b) it was invariably coincident with the x-axis. The ratio between the fatigue limit for case (a) and the fatigue limit for case (b), both expressed in terms of amplitude of the maximum principal stress, was seen to be equal to about 1.3: this example should better clarify the fact that the rotation of the principal directions during the loading cycle strongly affects the overall fatigue strength of engineering materials, so that any efficient multiaxial fatigue criterion should be capable of efficiently accounting also for the above aspect.

Fundamentals of fatigue assessment 400

87

S65A s0 = 583.6 MPa t0 = 370.6 MPa (Brittle)

2

txy,a 350 (MPa) 300 250 200

0.7

150

txy,a sx,a = 3.0

100 50

(Rs , Rt) –1, –1 0.11, 0.48 –0.30, 0.09 0.26, 0.24 –0.17, –0.19 0.62, 0.15 0.25, –0.31

Eq. (2.54)

0 0

200

400

600 800 sx,a (MPa)

2.41 Mean stress effect under in-phase bending and torsion (data from Gough, 1949).

The multiaxial fatigue behaviour of engineering materials is influenced by the presence of non-zero mean stresses as well. As an example, Fig. 2.41 shows the detrimental effect of uniaxial (Rs = sx,min/sx,max) and torsional (Rt = txy,min/txy,max) load ratios larger than −1 on the biaxial fatigue strength of a steel showing, according to Gough’s classification, brittle behaviour: this chart makes it evident that the presence of superimposed static stresses must always be taken into account when performing the fatigue assessment of real mechanical assemblies so as not to overestimate the actual multiaxial fatigue strength of the employed material. Finally, Gough’s diagram sketched in Fig. 2.42 clearly shows the detrimental effect of V-notches on the high-cycle fatigue strength of cylindrical specimens made of NiCrMo steel and subjected to in-phase bending and torsion (according to continuum mechanics, in this diagram notch data are plotted in terms of amplitude of the nominal stresses calculated with respect to the net area). Figure 2.42 allows the reader to observe also that the high-cycle fatigue strength of the considered notched samples is accurately described by Eq. (2.54) even if the behaviour of the plain material follows the quarter of an ellipse relationship: according to Gough’s experimental findings, notch fatigue limits under in-phase bending and torsion are always described by an arc of an ellipse and this holds true independently of the behaviour of the parent material (Gough, 1949). The considerations reported in the previous paragraphs should make it evident that performing the fatigue assessment of real components subjected to in-service multiaxial fatigue loading is a complex process which involves many different variables. Owing to the complexity of such a problem, since the beginning of the last century many researchers have attempted to devise sound engineering tools suitable for estimating fatigue

88

Multiaxial notch fatigue 450

NiCrMo steel (75–80 tons) s0 = 660.7 MPa

R = –1

txy,a 400 (MPa) 350

t0 = 342.7 MPa

Plain data

(Ductile)

300 250

Eq. (2.54) Eq. (2.53)

200 150 100

Notch data

50 0 0

200

400

600

800

sx,a (MPa)

2.42 Detrimental effect of V-notches in cylindrical specimens of NiCrMo steel subjected to in-phase bending and torsion (data from Gough, 1949).

damage in real mechanical components experiencing biaxial/triaxial states of stress and strain. In particular, according to the state of the art (Garud, 1981a; You and Lee, 1996; Socie and Marquis, 2000), the multiaxial fatigue assessment can be performed by following two different philosophies: in the low-cycle fatigue regime (that is, when cyclic plasticity plays a fundamental role in the fatigue crack initiation phenomenon), strain-based methods are always suggested to be used; on the contrary, in the medium/ high-cycle fatigue field, stress-based approaches are in general preferable. Among those criteria suitable for estimating fatigue lifetime in the lowcycle fatigue regime, certainly the methodology devised by Brown and Miller (1973) deserves to be mentioned first. In particular, they suggested predicting fatigue lifetime by simply considering the strain components perpendicular and parallel to the crack initiation plane (Stage I plane). Moreover, they argued that fatigue damage under multiaxial cyclic loading depends also on the crack propagation direction: those criteria suitable for estimating fatigue lifetime in the presence of cracks growing on components’ surfaces cannot be used when cracks propagate within the material itself. From a practical point of view, when the crack propagation phenomenon is confined to the surface of the component to be assessed, Brown and Miller suggested using the critical plane approach (where the critical plane is the one experiencing the maximum shear strain amplitude) applied in conjunction with the Manson–Coffin curve. Some years later, Wang and

Fundamentals of fatigue assessment

89

Brown (1993) reformulated the above criterion in order to correctly take into account also the influence of the mean stress normal to the critical plane itself. By taking as a starting point the considerations summarised in the previous paragraph, Socie (Socie, 1987; Fatemi and Socie, 1988) has argued that fatigue damage under multiaxial fatigue loading could be estimated more accurately by using the maximum stress perpendicular to the critical plane instead of the normal strain, because such a stress parameter is more closely related to the micro/meso crack growth rate. Starting from such an assumption, he proposed performing the fatigue assessment according to the observed material cracking behaviour: when the crack initiation phenomenon is mainly Mode I dominated, then the critical plane becomes the one on which the normal stress/strain components reach their maximum value, and fatigue lifetime has to be estimated by means of the Smith–Watson– Topper parameter (Socie, 1987); on the contrary, when the early stage of crack propagation is mainly Mode II governed, the critical plane is the one experiencing the maximum shear strain amplitude, and fatigue lifetime can be predicted by directly using the torsional Manson–Coffin curve (Fatemi and Socie, 1988). Another interesting category of multiaxial fatigue criteria is the one taking as a starting point the assumption that energy, calculated at the crack initiation point, gives an accurate measure of fatigue damage. The main advantage of such a philosophy over the other existing techniques is that energy is a scalar quantity independent of the complexity of the stress/strain state damaging the point assumed to be the critical one in the component to be assessed: this implies that a simple uniaxial fatigue curve should be enough to correctly calibrate all the energy-based criteria. By following the above idea, Garud (1981b) proposed the estimation of multiaxial fatigue lifetime by simply using that portion of energy related to the cyclic plastic deformation: unfortunately, even if very appealing from a philosophical point of view, such a method hides some evident inconsistencies. To overcome such problems, Ellyin (Ellyin, 1989; Ellyin et al., 1991) subsequently suggested that accurate predictions could be made by considering not only the plastic but also the positive elastic energy. This assumption took as its starting point the experimental evidence that fatigue damage depends on the elastic energy due to the tensile stress components, because failures can occur also in the high-cycle fatigue regime where the plasticity contribution to the overall fatigue damage is in general negligible (at least in conventional metallic materials). In the high-cycle fatigue field, instead, real mechanical components are usually assessed by adopting stress-based criteria. In this scenario, certainly, the two equations devised by Gough (1949) and considered above form the basis for the most modern multiaxial high-cycle fatigue

90

Multiaxial notch fatigue

assessment methodologies. After the pioneering work done by Gough, several other techniques have been devised and validated, including criteria based on the use of stress invariants, critical plane methods and mesoscopic approaches. Those multiaxial fatigue criteria based on the calculation of stress invariants assume that fatigue damage depends both on the square root of the second invariant of the stress deviator and on the hydrostatic stress. Among such criteria, certainly the formulas proposed by both Crossland (1956) and Sines (1959) have to be mentioned. Even if very efficient in estimating multiaxial high-cycle fatigue strength from a computational point of view, unfortunately such criteria proved to be inaccurate in predicting the damaging effect of non-proportional loading (Papadopoulos et al., 1997). Contrary to the above methods, the critical plane approaches instead were seen to be highly accurate in estimating multiaxial fatigue limits also in the presence of out-of-phase loading. Among such methods, the criteria proposed by Findley (1959), Matake (1977) and McDiarmid (1991, 1994), deserve to be mentioned. The first method assumes that the critical plane is the one on which a linear combination of the shear and the normal stress reaches its maximum value. The other two criteria instead take as their starting point the idea that the critical plane is the one experiencing the maximum shear stress amplitude and multiaxial fatigue damage depends also on the maximum stress perpendicular to such a plane. Following the critical plane philosophy, Carpinteri and Spagnoli (Carpinteri et al., 2000; Carpinteri and Spagnoli, 2001) have devised an original and efficient method suitable for estimating fatigue limits in plain engineering materials subjected to multiaxial fatigue loading. In more detail, such a criterion correlates the orientation of the critical plane with the weighted mean principal stress directions, and fatigue damage is estimated by using a non-linear combination of the maximum normal stress and the shear stress amplitude relative to the plane where the micro/meso crack is assumed to initiate. By taking full advantage of the so-called mesoscopic approach, Dang Van (Dang Van et al., 1989; Dang Van, 1993) devised an interesting method based on a different philosophy from the methodologies briefly reviewed in the previous paragraphs. In particular, the simplest version of this method estimates fatigue damage by linearly combining the maximum shear stress and the hydrostatic pressure. This approach postulates that fatigue failures are avoided as long as, at every instant of the load history, the fatigue damage parameter calculated as above is lower than a reference value which depends on two fatigue limits experimentally determined under simple loading paths (typically, such a criterion is calibrated by using the uniaxial and torsional fully reversed fatigue limits). By following a strategy similar to the one adopted by Dang Van, Papadopoulos (1987, 1995) formulated a very efficient high-cycle multiaxial

Fundamentals of fatigue assessment

91

fatigue assessment methodology by taking as a starting point the fundamentals of the mesoscopic approach applied in conjunction with the shakedown concept. In particular, he proposed performing the multiaxial fatigue assessment by using two different formulas suitable for predicting multiaxial fatigue damage in brittle and in ductile materials, respectively. Such a methodology was seen to be capable of estimates falling within an error interval equal to ±10% when applied to plain components subjected to both in-phase and out-of-phase complex loading. It is also worth noting that Papadopoulos has recently extended the use of his criterion down to the medium-cycle fatigue regime (Papadopoulos, 2001): such a new reformulation of this well-known method proved to be highly accurate and this held true independently of the degree of multiaxiality of the considered loading path. To conclude this section, it can be pointed out that all the approaches briefly reviewed above were specifically devised to perform the multiaxial fatigue assessment of mechanical components made of homogeneous and isotropic materials, so that they cannot be used in the presence of any type of anisotropy (for instance, composite materials): to perform the fatigue assessment of this kind of engineering materials different theories have been proposed and validated and such methods will be reviewed in Chapter 8.

2.10

References

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Atzori, B., Lazzarin, P., Meneghetti, G. (2005) A unified treatment of the mode I fatigue limit of components containing notches or defects. International Journal of Fracture 133, 61–87. DOI: 10.1007/s10704-005-2183-0. Atzori, B., Berto, F., Lazzarin, P., Quaresimin, M. (2006a) Multi-axial fatigue behaviour of a severely notched carbon steel. International Journal of Fatigue 28, 485–493. DOI: 10.1016/j.ijfatigue.2005.05.010. Atzori, B., Lazzarin P., Meneghetti, G. (2006b) Estimation of fatigue limits of sharply notched components. In: Proceedings of Fatigue 2006, Atlanta, GA. Bellett, D., Taylor, D. (2006) The effect of crack shape on the fatigue limit of threedimensional stress concentrations. International Journal of Fatigue 28, 114–123. DOI: 10.1016/j.ijfatigue.2005.04.010. Bellett, D., Taylor, D., Marco, S., Mazzeo, E., Guillois, J., Pircher, T. (2005) The fatigue behaviour of three-dimensional stress concentrations. International Journal of Fatigue 27, 207–221. DOI: 10.1016/j.ijfatigue.2004.07.006. Brown, M. W., Miller, K. J. (1973) A theory for fatigue under multiaxial stress– strain conditions. Proceedings of the Institution of Mechanical Engineers 187, 745–755. Carpinteri, A., Spagnoli, A. (2001) Multiaxial high-cycle fatigue criterion for hard metals. International Journal of Fatigue 23, 135–145. DOI: 10.1016/S0142-1123 (00)00075-X. Carpinteri, A., Brighenti, R., Spagnoli, A. (2000) A fracture plane approach in multiaxial high-cycle fatigue of metals. Fatigue and Fracture of Engineering Materials and Structures 23, 355–364. DOI: 10.1046/j.14602695.2000.00265.x. Ciavarella, M., Meneghetti G. (2004) On fatigue limit in the presence of notches: classical vs. recent unified formulations. International Journal of Fatigue 26, 289– 298. DOI: 10.1016/S0142-1123(03)00106-3. Crossland, B. (1956) Effect of large hydrostatic pressure on the torsional fatigue strength of an alloy steel. In: Proceedings of the International Conference on Fatigue of Metals, Institution of Mechanical Engineers, London, 138–149. Crupi, G., Crupi, V., Guglielmino, E., Taylor, D. (2005) Fatigue assessment of welded joints using critical distance and other methods. Engineering Failure Analysis 12, 129–142. DOI: 10.1016/j.engfailanal.2004.03.005. Dang Van, K. (1993) Macro-Micro Approach in High-Cycle Multiaxial Fatigue. ASTM STP 1191, American Society for Testing and Materials, Philadelphia, PA, 120–130. Dang Van, K., Griveau, B., Messagge, O. (1989) On a new multiaxial fatigue limit criterion: theory and application. In: Biaxial and Multiaxial Fatigue, EGF 3, edited by M. W. Brown and K. J. Miller, Mechanical Engineering Publications, London, 479–496. Davoli, P., Bernasconi A., Filippini M., Foletti S., Papadopoulos, I. V. (2003) Independence of the torsional fatigue limit upon a mean shear stress. International Journal of Fatigue 25, 471–480. DOI: 10.1016/S0142-1123(02)00174-3. Dowling, N. E. (1998) Mechanical Behaviour of Materials, 2nd edition. Prentice-Hall, Englewood Cliffs, NJ. DuQuesnay, D. L., Yu, M. T., Topper, T. H. (1988) An analysis of notch size effect on the fatigue limit. Journal of Testing and Evaluation 4, 375–385. El Haddad, M. H. (1978) A study of the growth of short fatigue cracks based on fracture mechanics. PhD thesis, University of Waterloo, Waterloo, Ontario.

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Kihara, S., Yoshii, A. (1991) A strength evaluation method of a sharply notched structure by a new parameter, the equivalent stress intensity factor. JSME International Journal 34, 70–75. Kitagawa, H., Takahashi, S. (1976) Applicability of fracture mechanics to very small cracks or the cracks in the early stage. In: Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Boston, MA, 627–631. Kueppers, M., Sonsino, C. M. (2003) Critical plane approach for the assessment of the fatigue behaviour of welded aluminium under multiaxial loading. Fatigue and Fracture of Engineering Materials and Structures 26, 507–513. DOI: 10.1046/j.14602695.2003.00674.x. Kuhn, P., Hardraht, H. F. (1952) An Engineering Method for Estimating the Notchsize Effect in Fatigue Tests on Steel. NACA TN2805, Langley Aeronautical Laboratory, Washington, DC. Lazzarin, P., Tovo, R. (1996) A unified approach to the evaluation of linear elastic stress fields in the neighbourhood of cracks and notches. International Journal of Fracture 78, 3–19. DOI: 10.1007/BF00018497. Lazzarin P., Tovo R., Meneghetti G. (1997) Fatigue crack initiation and propagation phases near notches in metals with low notch sensitivity. International Journal of Fatigue 19, 647–657. DOI: 10.1016/S0142-1123(97)00091-1. Lin, S. K., Lee, Y. L., Lu, M. W. (2001) Evaluation of the staircase and the accelerated test methods for fatigue limit distributions. International Journal of Fatigue 23, 75–83. DOI: 10.1016/S0142-1123(00)00039-6. Luise, M. (2001) Caratterizzazione a fatica di leghe leggere in presenza di intagli. Degree thesis, University of Padova, Italy (in Italian). Lukas, P., Klesnil, M. (1978) Fatigue limit of notched bodies. Material Science and Engineering 34, 61–66. Lukas, P., Kunz, L. (1992) Effect of mean stress on short crack threshold. In: Short Cracks, ESIS 13, edited by K. J. Miller and E. R. de los Rios. Mechanical Engineering Publications, London, 265–275. Lukas, P., Kunz, L., Weiss, B., Stickler, R. (1986) Non-damaging notches in fatigue. Fatigue and Fracture of Engineering Materials and Structures 9, 195–204. DOI: 10.1111/j.1460-2695.1986.tb00446.x. Marin, J. (1956) Interpretation of fatigue strength for combined stresses. In: Proceedings of International Conference on Fatigue of Metals, London, 184–192. Matake, T. (1977) An explanation on fatigue limit under combined stress. Bulletin of the JSME 20 141, 257–263. McDiarmid, D. L. (1991) A general criterion for high cycle multiaxial fatigue failure. Fatigue and Fracture of Engineering Materials and Structures 14, 429–453. DOI: 10.1111/j.1460-2695.1991.tb00673.x. McDiarmid, D. L. (1994) A shear-stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction. Fatigue and Fracture of Engineering Materials and Structures 17, 1475–1484. DOI: 10.1111/j.1460-2695. 1994.tb00789.x. Miannay, D. P. (1998) Fracture Mechanics. Springer, New York. Miller, K. J. (1982) The short crack problem. Fatigue and Fracture of Engineering Materials and Structures 5, 223–232. DOI: 10.1111/j.1460-2695.1982.tb01250.x. Miller, K. J. (1993) The two thresholds of fatigue behaviour. Fatigue and Fracture of Engineering Materials and Structures 16, 931–939. DOI: 10.1111/j.1460-2695.1993. tb00129.x.

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Miller, K. J., O’Donnell, W. J. (1999) The fatigue limit and its elimination. Fatigue and Fracture of Engineering Materials and Structures 22, 545–557. DOI: 10.1046/ j.1460-2695.1999.00204.x. Murakami, Y., Takahashi, K. (1998) Torsional fatigue of a medium carbon steel containing an initial small surface crack introduced by tension–compression fatigue: crack branching, non-propagation and fatigue limit. Fatigue and Fracture of Engineering Materials and Structures 21, 1473–1484. DOI: Neuber, H. (1936) Zur Theorie der technischen Formzahl. Forschg Ing-Wes 7, 271–281. Neuber, H. (1958) Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural Form and Material, 2nd edition. Springer Verlag, Berlin. Nishihara, T., Kawamoto, M. (1945) The strength of metals under combined alternating bending and torsion with phase difference. Memoirs of the College of Engineering, Kyoto Imperial University 11, 85–112. Nisitani, H., Endo, M. (1988) Unified treatment of deep and shallow notches in rotating bending fatigue. In: Basic Questions in Fatigue, Vol. I, ASTM STP 924, 136–153. Papadopoulos, I. V. (1987) Fatigue polycyclique des métaux: une nouvelle approche. Thèse de Doctorat, Ecole Nationale des Ponts et Chaussées, Paris. Papadopoulos, I. V. (1995) A high-cycle fatigue criterion applied in biaxial and triaxial out-of-phase stress conditions. Fatigue and Fracture of Engineering Materials and Structures 18, 79–91. DOI: 10.1111/j.1460-2695.1995.tb00143.x. Papadopoulos, I. V. (2001) Long life fatigue under multiaxial loading. International Journal of Fatigue 23, 839–849. DOI: 10.1016/S0142-1123(01)00059-7. Papadopoulos, I. V., Davoli, P., Gorla, C., Filippini, M., Bernasconi, A. (1997) A comparative study of multiaxial high-cycle fatigue criteria for metals. International Journal of Fatigue 19, 219–235. DOI: 10.1016/S0142-1123(96)00064-3. Peterson, R. E. (1959) Notch sensitivity. In: Metal fatigue, edited by G. Sines and J. L. Waisman, McGraw-Hill, New York, 293–306. Peterson, R. E. (1974) Stress Concentration Factors. Wiley, New York. Pook, L. P. (2000) Linear Elastic Fracture Mechanics for Engineers: Theory and Applications. WIT Press, Southampton, UK. Qilafku, G., Kadi, N., Dobranski, J., Azari, Z., Gjonaj, M., Pluvinage, G. (2001) Fatigue of specimens subjected to combined loading. Role of hydrostatic pressure. International Journal of Fatigue 23, 689–701. DOI: 10.1016/S0142-1123(01) 00030-5. Sedeckyj, G. P. (2001) Constant life diagrams – a historical review. International Journal of Fatigue 23, 347–353. DOI: 10.1016/S0142-1123(00)00077-3. Sheppard, S. D. (1991) Field effects in fatigue crack initiation: long life fatigue strength. Transactions of ASME, Journal of Mechanical Design 113, 188–194. DOI: 10.1115/1.2912768. Sines, G. (1959) Behaviour of metals under complex static and alternating stresses. In: Metal Fatigue, edited by G. Sines and J. L. Waisman. McGraw-Hill, New York, 145–169. Smith, R. A., Miller K. J. (1978) Prediction of fatigue regimes in notched components. International Journal of Mechanical Sciences 20, 201–206. DOI: 10.1016/0020-7403 (78)90082-6. Socie, D. F. (1987) Multiaxial fatigue damage models. Transactions of the ASME, Journal of Engineering Materials and Technology 109, 293–298.

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3 The Modified Wöhler Curve Method in fatigue assessment

Abstract: This chapter aims to investigate the main features of the so-called Modified Wöhler Curve Method (MWCM). Key words: critical plane, Modified Wöhler Curve Method.

3.1

Introduction

As briefly said at the end of the previous chapter, since the pioneering work done by Gough, several multiaxial fatigue criteria have been formalised and validated in order to provide structural engineers with safe techniques suitable for performing the fatigue assessment of mechanical components subjected to complex in-field loading paths. Amongst the different strategies which have been explored in order to make reliable and accurate predictions, it is the writer’s opinion that the most interesting and promising methodologies are those based on the critical plane concept. According to this idea, over the last decade we have made a big effort in order to devise a novel approach suitable not only for predicting fatigue damage in plain materials but also for estimating fatigue strength of notched components subjected to multiaxial fatigue loading. This chapter then aims to investigate the main features of the so-called Modified Wöhler Curve Method, MWCM (Susmel, 2000a, 2000b).

3.2

Fatigue damage model

The MWCM takes as its starting point the idea that fatigue damage in homogeneous and isotropic materials subjected to fatigue loading can successfully be predicted by modelling the initiation as well as the initial propagation of micro/meso cracks. It is worth remembering here that, according to the definition proposed by Bolotin (1999), engineering material cracking behaviour under cyclic loading can be investigated at a micro-, a meso- or a macro-level. To be precise, a micro-crack is located within a single grain, a meso-crack covers several grains, and finally a crack that includes a high number of broken grains is called a macrocrack. According to the idea of Miller (1982, 1983), the only way to correctly model the micro/meso crack behaviour is by considering the real 98

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material morphology as well as the elasto-plastic behaviour of the grains. Unfortunately, estimating fatigue strength by rigorously considering all the aspects mentioned above would result in a fatigue assessment methodology too cumbersome to be used in situations of practical interest: such a model would require too many material constants to be determined by running appropriate experiments, unacceptably increasing the time and costs of the design process. This is the reason why inevitable approximations have to be introduced by forming the hypothesis that conventional metallic materials are linear-elastic, homogeneous and isotropic. In spite of the limitations mentioned above, the fatigue damage model on which the MWCM is based attempts to describe the formation and the initial propagation of micro/meso cracks by simply using macroscopic stresses calculated according to continuum mechanics. Consider then a plain material subjected to a complex system of cyclic forces resulting in a multiaxial stress state damaging the fatigue process zone (Fig. 3.1a). The MWCM assumes that, in the medium/high-cycle fatigue regime, both initiation and initial growth of micro/meso cracks depend on the shear stress damaging the grains within the fatigue process zone (Brown and Miller, 1973; Socie and Bannantine, 1988). The validity of such an assumption is supported by the experimental evidence that the formation of the persistent slip bands (PSBs) as well as the initiation and initial propagation of micro/ meso cracks (Fig. 3.1) are related to the cyclic variation of the shearing forces. It is worth remembering here that the presence of the PSBs in grains is to be ascribed to the fact that, during the loading cycle, certain atomic planes can slide along preferential directions (the so-called easy glide directions), resulting in the formation of intrusions and extrusions (Fig. 3.1b). After a certain number of cycles, micro-cracks initiate in the grains either

y

Fatigue process zone

Micro/meso crack

sn,max

sn,max

x ta

ta

O ta ta Critical plane (a)

PSBs sn,max

Critical plane

(b)

3.1 The adopted fatigue damage model.

sn,max (c)

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due to micro-stress concentration phenomena resulting from the presence of deep intrusions or due to the separation occurring within the slip bands. The driving force of the above phenomena is the micro-plastic shear strain acting on the easy glide planes (Suresh, 1991) and, by using sophisticated reasoning, it is possible to demonstrate that such a strain quantity is proportional to the macroscopic linear-elastic shear stress (Papadopoulos, 1997). After initiating, micro/meso cracks keep propagating due to the shearing forces which push their tips (Kaufman and Topper, 2003). Finally, when the above cracks reach a length of the order of a few grains, the so-called Stage I ends and cracks tend to orient themselves in order to experience the maximum Mode I loading (Stage II). The considerations briefly summarised above should make it evident that the shear stress is a quantity which is closely related to the initiation and initial propagation of fatigue cracks, therefore it is possible to assume that such a stress component can successfully be used to estimate fatigue damage. Moreover, by remembering that, under pure torsion, the presence of superimposed static torsional stresses can be neglected (see Section 2.8), it is logical to presume that the initiation of fatigue cracks and their initial propagation depend mainly on the amplitude of the applied shear stress. This implies that, under multiaxial fatigue loading, amongst the infinite planes passing through the material hot spot, the plane on which fatigue damage reaches its maximum value is the one experiencing the maximum shear stress amplitude, ta. It is worth noting also that initiation and propagation phenomena are also influenced by the stress component perpendicular to the assumed crack initiation plane (Socie, 1987). In particular, a normal tensile stress opens the micro/meso-cracks by favouring their growth, whereas a compressive normal stress reduces the growth rate due to the friction between the crack surfaces (Fig. 3.1c). Similarly, the stress perpendicular to the initiation plane influences the formation of the PSBs (Fig. 3.1b); in fact, a compressive stress inhibits the PSB laminar flow, whereas a tensile stress favours their flow (Susmel and Lazzarin, 2002). According to the above fatigue damage model and observing that the presence of superimposed tensile static stresses has a detrimental effect on the overall fatigue strength of conventional engineering materials subjected to uniaxial fatigue loading (see Section 2.3), it is possible to assume that the mean stress effect can efficiently be taken into account through the maximum stress perpendicular to the critical plane, sn,max (Findley, 1959; Matake, 1977). The considerations reported above should make it evident that the classical critical plane approaches formalised in terms of ta and sn,max are somehow related to the observed physical phenomena taking place within the fatigue process zone, even though, to make such criteria usable in situations of practical interest, a number of simplifying hypotheses have to be formed. In Chapter 6, the existing links between MWCM and observed experimental reality will be considered in great detail, trying to better justify, from a physi-

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cal point of view, the reasons why our method is seen to be so accurate and reliable in estimating fatigue damage under multiaxial fatigue loading. To conclude, it is worth noting that, according to the assumptions discussed above, all the critical plane approaches should be applied by defining Nf as the number of cycles needed to exhaust the Stage I process. Unfortunately, due to the evident experimental difficulties resulting from the above definition, in general also the critical plane approaches are applied by defining fatigue failures through the conventional strategies already discussed in Section 2.2.

3.3

Degree of multiaxiality of the stress field damaging the fatigue process zone according to the MWCM

As briefly said above, to perform the fatigue assessment under cyclic loading the classical critical plane approaches make use of both the maximum shear stress amplitude, ta, and the maximum stress, sn,max, perpendicular to the critical plane (Findley, 1959; Matake, 1977; McDiarmid, 1991, 1994). As to the determination of the critical plane, it is worth noting here that, independently of the complexity of the stress state damaging the material point at which the stress analysis is carried out, there always exist two or more material planes experiencing the maximum shear stress amplitude. Amongst all the potential critical planes, the one which has to be used to perform the fatigue assessment is the plane characterised by the highest value of the maximum normal stress: according to the critical plane approach, such a plane is that on which fatigue damage reaches its maximum value. As to devising the MWCM (Susmel, 2000a; Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003), we argued that, in order to correctly estimate fatigue strength, the degree of multiaxiality of the stress field damaging the fatigue process zone of the material to be assessed has to be measured by using the following stress ratio relative to the critical plane: σ 3.1 ρ = n,max τa The main feature of the above stress parameter is that it is sensitive to superimposed static stresses as well as to the presence of nonproportional (that is, out-of-phase) loadings. In order to better understand the peculiarities of the above critical plane stress ratio, consider the specimen sketched in Fig. 3.2. For the sake of simplicity, point O, which belongs to the surface of the sample, is assumed here to experience the following cyclic stress state: ⎡ σ x ( t ) τ xy( t ) 0 ⎤ 0 0⎥ [σ ( t )] = ⎢τ xy( t ) ⎢ ⎢⎣ 0

0

⎥ 0 ⎥⎦

3.2

102

Multiaxial notch fatigue Mt(t) F(t) z y O

x Mt(t)

F(t)

3.2 Cylindrical specimen subjected to combined tension and torsion.

where, if dxy,x is the out-of-phase angle, stress components sx(t) and txy(t) vary during the loading cycle as follows:

σ x( t ) = σ x,m + σ x,a sin (ω t ) τ xy( t ) = τ xy,m + τ xy,a sin (ω t − δ xy, x )

3.3

Consider initially the uniaxial sub-case, that is, assume that, during the loading cycle, txy(t) is invariably equal to zero. Under the above loading path, it is trivial to demonstrate that the relevant stress components relative to the critical plane are as follows:

τa =

σ x, max σ (1 − R) = x,a 4 2

σ n, max = σ n,m =

σ x, max σ x,m + σ x,a = ⇒ σ x,m = 2 (σ n, max − τ a ) 2 2

σ x,m σ x,a ; σ n,a = 2 2

3.4

3.5

3.6

By using the above relationships, it is straightforward to calculate the r ratio under uniaxial fatigue loading, obtaining:

ρ=

σ n, max 2 = τa 1− R

3.7

where, as usual, R is the load ratio (R = sx,min/sx,max). Equation (3.7) makes it evident that, under uniaxial fatigue loading, the stress ratio relative to the critical plane increases as the applied load ratio, R, increases. In particular, under fully-reversed loading r is equal to 1, whereas when R approaches unity, r tends to infinity (see the txy,a/sx,a = 0 case in Fig. 3.3). According to the fact that in the high-cycle fatigue regime the presence of superimposed torsional static stresses can be neglected as long as the

The Modified Wöhler Curve Method in fatigue assessment 8 7

txy,a =0 sx,a

dxy,x = 0°

txy,a sx,a = 0.5

6 5 r

103

txy,a sx,a = 1

4

txy,a sx,a = 2

3

txy,a sx,a = ∞

2 1 0 –1.0

–0.5

0.0

0.5

1.0

R = sx,min/sx,max = txy,min/txy,max

3.3 r vs. R diagram for different txy,a to sx,a ratios.

maximum shear stress is lower than the torsional yield stress (Sines, 1959), ratio r is insensitive to non-zero mean shear stresses: under torsional fatigue loading, ta is invariably equal to the amplitude of the applied torsional stress, txy,a, and the maximum stress perpendicular to the critical plane is equal to zero. As shown in Fig. 3.3, this results in the fact that, under torsional loading, r is equal to zero, independently of the considered load ratio. Fig. 3.3 reports also some other values of r, plotted against the load ratio, R, calculated by imposing an out-of-phase angle, dxy,x, equal to zero: this chart should make it evident that ratio r is highly sensitive to the presence of superimposed tensile static stresses. To conclude, it can be said that, even if, for the sake of simplicity, only the tension (or bending) and torsion case was considered above, a similar sensitivity of parameter r can be seen also when more complex tridimensional stress states are investigated (Susmel and Lazzarin, 2002). The second important feature of ratio r is that its value depends also on the phase shifts amongst the different stress components damaging the mechanical assembly to be assessed. This aspect is very important because, as already discussed in the previous chapter, the degree of nonproportionality of the applied loading is a variable which must always be taken into account when performing the multiaxial fatigue assessment. In order to show the sensitivity of ratio r to non-zero out-of-phase angles, the r vs. dxy,x diagram obtained by considering three different ratios between txy,a and sx,a is reported in Fig. 3.4: this chart confirms that r is sensitive not only to the presence of superimposed tensile static stresses but also to the degree of non-proportionality of the applied loading.

104

Multiaxial notch fatigue 3

txy,a sx,a = 0.5

sx,m = txy,a = 0 Mpa

txy,a sx,a = 1

2

txy,a sx,a = 2

r (ra) 1

0 0

15

30

45

60

75 90 dxy,x (degrees)

3.4 r vs. dx,xy diagram for different txy,a to sx,a ratios.

As already discussed in Section 2.3 (Fig. 2.6), different materials subjected to uniaxial fatigue loading are characterised by different sensitivities to the presence of non-zero mean stresses. In order to better investigate the contribution of the mean stress perpendicular to the critical plane to the overall fatigue damage, it is useful to rewrite Eq. (3.1) as follows (Susmel, 2008):

ρ = ρm + ρa =

σ n,m σ n,a + τa τa

3.8

where sn,m and sn,a are the mean value and the amplitude of the stress normal to the critical plane, respectively. Splitting ratio r into terms rm and ra makes it possible to separately consider the effect of non-zero mean stresses and the influence of the degree of non-proportionality of the applied loading. In more detail, the diagram sketched in Fig. 3.5 shows that, under in-phase tension/torsion, rm increases as the load ratio, R, varies from −1 up to 1. On the contrary, under pure torsional loading (txy,a/sx,a = ∞), rm is equal to zero independently of the considered load ratio, R: Fig. 3.5 confirms that ratio rm is sensitive only to the mean normal stress perpendicular to the critical plane. Observing now that under fully reversed loading ra is equal to r, Fig. 3.4 instead proves that stress ratio ra alone is an efficient parameter suitable for measuring the degree of non-proportionality of the applied loading: for a given ratio between txy,a and sx,a, ra increases as the out-of-phase angle, dxy,x, increases. In order to more accurately evaluate the influence of non-zero mean stresses perpendicular to the critical planes, it is important to remember

The Modified Wöhler Curve Method in fatigue assessment 8

txy,a sx,a = 0

dxy,x = 0°

7

105

txy,a sx,a = 0.5

6 5 rm 4

txy,a sx,a = 1 txy,a sx,a = 2

3

txy,a sx,a = ∞

2 1 0 –1.0

–0.5

0.0

0.5

1.0

R = sx,min/sx,max = txy,min/txy,max

3.5 Sensitivity of rm to superimposed static stresses.

here that Kaufman and Topper (2003) have argued that when the mean normal stress, sn,m, becomes larger than a certain threshold value (which it is logical to suppose to be different for different materials), a further increase of sn,m does not result in any reduction of the corresponding fatigue strength. This experimental evidence was ascribed to the fact that, when the mean normal stress relative to the critical plane is lower than the above material threshold value, the magnitude of the shearing forces pushing the crack tips is reduced due to the interactions amongst the asperities present on the two faces of the micro/meso cracks: it is logical to presume that this phenomenon results in a decrease of the crack growth rate. On the contrary, when micro/meso cracks are open, the shearing forces are fully transmitted to the crack tips causing the Mode II propagation: this is the reason why a further increase of the normal mean stress relative to the critical plane does not affect the crack propagation process any more. According to the above considerations, which are based on an extensive and accurate experimental investigation, an alternative definition of r can be adopted to more efficiently take into account the detrimental effect of the mean stresses normal to the critical planes. In particular, the effective value of the critical plane stress ratio, reff, can be defined as follows (Susmel, 2008):

ρeff =

mσ n,m σ n,a + τa τa

3.9

In the above definition, m is the mean stress sensitivity index, which is assumed to be a material property suitable for determining the portion of

106

Multiaxial notch fatigue

y

Critical plane

Fatigue process zone x

m·sn,m

sn,a

ta O

ta Critical plane

m·sn,m

sn,a

3.6 Improved fatigue damage model adopted to formalise the MWCM.

the mean normal stress relative to the critical plane which effectively opens the micro/meso cracks by favouring the propagation phenomenon. Thanks to definition (3.9), fatigue damage is modelled by the MWCM through three different stress quantities (Fig. 3.6): (i) the initiation phenomenon and the initial propagation of the micro/meso cracks depends on the maximum shear stress amplitude, ta; (ii) the contribution to the overall fatigue damage of the cyclic stress perpendicular to the micro/meso cracks is a function of the amplitude of the stress normal to the critical plane, sn,a; and (iii) the portion of the mean stress perpendicular to the critical plane which effectively contributes to the initiation and propagation phenomenon is equal to msn,m, where m is assumed to be a material constant to be determined experimentally. The fatigue damage model proposed above suggests that m has to range between 0 and 1, in fact, when m is equal to unity, the material to be assessed is fully sensitive to the mean stress perpendicular to the critical plane; on the contrary, an m value equal to zero means that the considered material is not sensitive to the presence of superimposed static tensile stresses at all. To conclude, in the next section the MWCM will initially be formalised in its general form by considering the effective value of the critical plane stress ratio, reff. Subsequently, such an engineering criterion will be rewritten also in terms of ratio r, showing the differences existing between these two ways of formalising the same idea.

3.4

The Modified Wöhler Curve Method

For the sake of simplicity, consider a cylindrical specimen (Fig. 3.2) subjected to combined tension and torsion, so that material point O is damaged by a biaxial cyclic stress state. As usual, such a point is assumed to be also the centre of the absolute frame of reference, Oxyz, whose orientation is chosen as sketched in Fig. 3.2. By post-processing the stress state at point O, the shear stress amplitude, ta, and the normal stress components, sn,m and

The Modified Wöhler Curve Method in fatigue assessment

107

sn,a, relative to the critical plane can be determined by taking full advantage of one of the methodologies reviewed in Chapter 1. The calculated stress quantities can then be used to determine the corresponding effective critical plane stress ratio, reff, provided that the mean stress sensitivity index, m, is known from the experiments (the determination of m will be explained in the next section; in what follows instead such a material constant is assumed to be known). The MWCM takes as its starting point the idea that fatigue strength under cyclic loading can successfully be predicted in terms of shear stress amplitude relative to the critical plane, ta, provided that the fatigue curves used to estimate lifetime are determined by correctly considering, through ratio reff, the degree of multiaxiality of the stress field damaging the fatigue process zone. In order to explain how the above fatigue curves have to be estimated according to the MWCM, assume that the specimen of Fig. 3.2 is initially subjected to fully reversed uniaxial fatigue loading (R = −1). Mohr’s circles describing the resulting stress states at point O are sketched in Fig. 3.7a, where circle A depicts the stress state under the maximum applied loading, sx,max = sx,a, whereas circle B the stress state under the minimum loading, sx,min = −sx,a. The figure suggests that, according to Eqs (3.4), (3.5) and (3.6), the shear stress amplitude and the normal stress components relative to the critical plane turn out to be: τ a = σ x,a 2 3.10 σ n,a = σ x,a 2

σ n,m = 0 making it evident again that, under fully reversed uniaxial loading, ratio reff is invariably equal to unity. Contrary to the conventional representation (see Section 2.2), the uniaxial fatigue strength of the material from which the specimen sketched in Fig. 3.6 is made can be summarised now in a modified log-log Wöhler diagram plotting the shear stress amplitude relative to the critical plane, ta, t

R = –1

tmax

B −sx,a

R = –1 t txy,a A

90° sx,a sn,max

90°

σ

s −txy,a

tmin

(a)

(b)

3.7 Mohr’s circles (a) under fully reversed tension–compression, and (b) under fully reversed torsion.

108

Multiaxial notch fatigue log ta

Torsional curve 1 kt(reff = 0)

tA,Ref(reff = 0)

1 kt(reff = 1)

tA,Ref(reff = 1)

Uniaxial curve N0

log Nf

3.8 Modified Wöhler curves under fully reversed uniaxial (reff = 1) and torsional (reff = 0) fatigue loading.

against the number of cycles to failure, Nf (Fig. 3.8). As usual, by using one of the appropriate statistical procedures, the experimental data can be reanalysed to obtain the corresponding modified Wöhler curve, which is unambiguously described by its inverse slope kt (reff = 1) and by its fatigue limit expressed in terms of shear stress amplitude, tA,Ref (reff = 1), where:

τ A,Ref ( ρeff = 1) =

σ0 2

3.11

By using a similar procedure, a second fatigue curve can be plotted in the above modified Wöhler diagram by running a series of tests under fully reversed torsional loading (R = −1). In such a situation, as shown in Fig. 3.7b, Mohr’s circles under the maximum and minimum applied torque are coincident and the stress components relative to the critical plane turn out to be: τ a = τ xy,a 3.12 σ =σ =0 n,a

n,m

resulting in a reff ratio equal to zero. The torsional straight line plotted in the modified Wöhler diagram of Fig. 3.8 is fully described by the corresponding values of the slope, kt (reff = 0), and of the reference shear stress, tA,Ref (reff = 0), where:

τ A,Ref ( ρeff = 0) = τ 0

3.13

It is worth noting that the modified Wöhler diagram sketched in Fig. 3.8 is built by assuming that fatigue limits are always at N0 cycles to failure, independently of the complexity of the considered loading path. It has to be said that, even though in general such an assumption is not true (see Section 2.2), this problem can easily be overcome by using as reference quantities endurance limits, sA and tA, determined at an appropriate number of cycles to failure, NA, instead of s0 and t0.

The Modified Wöhler Curve Method in fatigue assessment

109

In order to understand the reason why in Fig. 3.8 the torsional fatigue curve is plotted above the uniaxial one, it can be remembered here that, according to the diagram reported in Fig. 2.29, for a given material the torsional fatigue limit can be estimated from the corresponding uniaxial one by simply using Von Mises’ formula. If such a criterion is rewritten in terms of reference shear stresses determined under fully reversed torsional and uniaxial fatigue loading, respectively, it is straightforward to obtain (Susmel and Lazzarin, 2002): τ A,Ref ( ρeff = 0 ) 2 = ≅ 1.155 3.14 τ A,Ref ( ρeff = 1) 3 The above ratio suggests that, in conventional engineering materials, the reference shear stress under fully reversed uniaxial fatigue loading, tA,Ref (reff = 1), is lower than the corresponding reference shear stress determined under torsion, tA,Ref (reff = 0). According to such an experimental evidence, the hypothesis can be formed that, for a given material, tA,Ref (reff) decreases with increasing reff, resulting in the schematisation shown by the modified Wöhler diagram sketched in Fig. 3.9. Such a chart should make it evident that fatigue lifetime can be estimated according to the degree of multiaxiality of the stress field damaging the fatigue process zone measured in terms of reff, provided that the kt vs. reff and tA,Ref vs. reff relationships are properly defined and correctly calibrated by running appropriate experiments. In particular, by an extensive systematic investigation it was proved that the above calibration functions can be expressed by using simple linear laws (Susmel, 2000a; Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003; Susmel and Petrone, 2003), that is: 3.15 kτ ( ρeff ) = a ⋅ ρeff + b τ A,Ref ( ρeff ) = α ⋅ ρeff + β 3.16 log ta 1 kt(reff = 0) 1

tA,Ref(reff = 0)

kt(0 < reff < 1)

tA,Ref(0 < reff < 1)

kt(reff = 1)

tA,Ref(reff = 1)

1 1 kt(reff > 1)

tA,Ref(reff > 1)

N0

3.9 Modified Wöhler diagram.

log Nf

110

Multiaxial notch fatigue

Fj(t)

Fk(t) z y O

sx(t) [s(t)] = txy(t) txz(t)

x Fi(t)

txy(t) sy(t) tyz(t)

txz(t) tyz(t) sz(t)

ta, sn,m, sn,a, reff

tA,Ref(reff), kt(reff)

ta

ta 1 kt(reff)

Nf,e

tA,Ref(reff)

N0

Nf

3.10 In-field use of the MWCM.

where a, b, a and b are material constants to be determined experimentally. Even if the above constants can be calculated by using two different fatigue curves generated under two different values of reff, it is evident that the accuracy of the MWCM in estimating fatigue lifetime is expected to increase as the number of fatigue curves used to calibrate the method itself increases. To conclude, Fig. 3.10 summarises the use of the above ideas to estimate fatigue lifetime. In more detail, from the stress state at point O, the maximum shear stress amplitude, ta, and the effective critical plane stress ratio, reff, can be determined by taking full advantage of the methods summarised in Chapter 1. Subsequently, according to the calculated value of reff, the modified Wöhler curve corresponding to the degree of multiaxiality of the considered stress field acting on the fatigue process zone can be estimated directly from Eqs (3.15) and (3.16). Finally, fatigue lifetime under the investigated loading path can be predicted by using the following trivial relationship:

(ρ ) τ N f,e = N 0 ⎡⎢ A,ref eff ⎤⎥ τa ⎦ ⎣

kt( ρeff )

3.17

The Modified Wöhler Curve Method in fatigue assessment

3.5

111

Use of the MWCM to estimate high-cycle multiaxial fatigue strength

In order to explicitly formalise the MWCM to make it suitable for performing the high-cycle fatigue assessment of plain engineering materials subjected to multiaxial fatigue loading, it is initially useful to introduce the load ratio relative to the critical plane, RCP, which is defined as follows: RCP =

σ n,min σ n,max

3.18

In the above identity sn,min and sn,max are the minimum and the maximum stress perpendicular to the plane of maximum shear stress amplitude, respectively. Owing to the fact that the mean stress sensitivity index must be known to properly calculate the effective value of the critical plane stress ratio, reff, the only way to determine constants a and b is by considering two experimental fatigue limits generated, under two different values of reff, by keeping RCP equal to −1: under the above loading paths, sn,m is invariably equal to zero, making it possible for the MWCM to be calibrated. Obviously, the two simplest pieces of experimental information which can be used to calculate constants a and b in Eq. (3.16) are the fully reversed uniaxial and torsional fatigue curves. In particular, if s0 and t0 are the corresponding fatigue limits, a and b take on the following values:

σ0 − τ0 2 β = τ0

α=

3.19 3.20

By carefully observing now the modified Wöhler diagram sketched in Fig. 3.9, it is easy to understand that, according to such a schematisation, a plain component is at its fatigue limit as long as the following condition is assured (Susmel and Lazzarin, 2002):

τ a ≤ τ A,Ref ( ρeff )

3.21

In particular, by calculating the values of constants a and b through Eqs (3.19) and (3.20), the above condition can be rewritten in explicit form, obtaining the following relationship (Susmel, 2008):

(

τa + τ0 −

)

σ0 ⋅ ρeff ≤ τ 0 2

3.22

As constants a and b are known, the mean stress sensitivity index, m, can then be estimated by using a third fatigue limit generated under a value of RCP larger than −1. In particular, if t*a , s*n,m and s*n,a are the critical plane stress components referred to the above fatigue limit condition, m takes on the following value:

112

Multiaxial notch fatigue m=

τ a* ⎛ τ 0 − τ a* σ *n,a ⎞ − ⎜2 ⎟ * ⎝ 2τ 0 − σ 0 τ a* ⎠ σ n,m

3.23

In practice, the material constant m can easily be determined by simply using a uniaxial fatigue limit generated under a load ratio, R, larger than −1, where the relevant stress components relative to the critical plane can be calculated directly according to Eqs (3.4) and (3.6). On the contrary, if an experimental fatigue limit suitable for estimating index m is not available, the hypothesis can be formed that the material to be assessed is fully sensitive to the presence of non-zero mean stresses perpendicular to the critical planes (i.e., m = 1), so that Eq. (3.21) can be rewritten as (Susmel and Lazzarin, 2002):

τ a ⭐ τ A,Ref ( ρ ) ⇒

3.24

τa − α ⋅ ρ ⭐ β

3.25

The main advantage of Eq. (3.25) over Eq. (3.21) is that only two fatigue limits generated under two different values of ratio r are needed to correctly calculate constants a and b. In particular, if the MWCM is calibrated by using the fully reversed uniaxial and torsional fatigue limit, it is trivial to rewrite our criterion in the following form (Susmel and Lazzarin, 2002):

(

τa + τ0 −

)

σ0 ⋅ ρ ⭐ τ0 2

3.26

After formalising the MWCM, it is useful to better investigate the way it works when used to perform the fatigue assessment of plain materials subjected to multiaxial fatigue loading. In particular, in the presence of superimposed static stresses, it can be remembered here that both reff and r increase as the mean stress normal to the critical plane, sn,m, increases (see Figs 3.3 and 3.5): according to the experimental evidence, the MWCM postulates that an increment of either reff or r results in a decrease of the corresponding fatigue strength. Moreover, in light of the fact that in conventional engineering materials the mean stress sensitivity index, m, ranges between 0 and 1, assuming m equal to unity for those materials characterised by a value of m lower than 1 results in conservative estimates. This is the reason why, when such a material index cannot be determined by running appropriate experiments, the use of the MWCM in the form given by Eq. (3.26) is always advisable. Another feature of the MWCM which has to be considered in great detail is its sensitivity to the presence of non-proportional loading. In particular, as already mentioned in Section 2.9, much experimental evidence clearly shows that the presence of non-zero shift phases can have either a beneficial effect, a detrimental effect or no effect at all on the overall fatigue strength

The Modified Wöhler Curve Method in fatigue assessment

113

of engineering materials (Sonsino, 1995). According to such experimental evidence, the MWCM assumes that the fatigue damage extent under outof-phase loading depends on the mutual interaction between degree of multiaxiality and non-proportionality of the stress field acting on the process zone and material fatigue properties. In order to better understand the way the MWCM works in the presence of out-of-phase loading, for the sake of simplicity consider a smooth cylindrical sample loaded in fully reversed tension (or bending) and torsion (Fig. 3.2). Initially, it is trivial to observe that, in general, under constant values of the ratio between the amplitudes of the applied tensile and torsional nominal stresses, both the maximum shear stress amplitude, ta, and the corresponding amplitude of the reference shear stress, tA,Ref(reff), decreases as the out-of-phase angle, dxy,x, increases. In terms of cyclic strength instead, the MWCM postulates that, when dxy,x ranges from 0° up to 90°, the fatigue damage extent varies as both the ratio between txy,a and sx,a and the ratio between t0 and s0 change (Fig. 3.11). In more detail, the diagrams of Fig. 3.11 plot the sx,a to s0 ratio against angle dxy,x, where sx,a was estimated, for the two considered ratios between t0 and s0, by investigating four different values of txy,a/sx,a, i.e., 0.25, 0.5, 1 and 2. The choice of a t0 to s0 ratio equal to 1 3 (Von Mises’ hypothesis) as well as equal to unity is due to the fact that in conventional engineering materials the above ratio is seen to range from about 0.5 up to unity (see Section 2.8). Fig. 3.11 clearly shows that, in the presence of non-proportional loading, the fatigue damage extent calculated according to our multiaxial fatigue criterion depends on three different aspects: (i) degree of multiaxiality and (ii) non-proportionality of the stress field acting on the fatigue process zone, and (iii) material fatigue behaviour expressed in terms of t0 to s0 ratio. As to the influence of the latter ratio, it is important to highlight that, as clearly proved by Fukuda and Nisitani (2003), the value of the t0 to s0 ratio depends on the material microstructure, which in turns affects the overall response of engineering materials to fatigue. In other words, the MWCM estimates fatigue strength under cyclic loading by accounting for the material morphology in an indirect way, i.e., through the ratio between t0 and s0. It is important to point out here that the validity of the above way of taking into account the presence of non-zero out-of-phase angles is fully supported by the remarkable accuracy shown by the MWCM when used to estimate high-cycle fatigue strength under non-proportional loading paths (Susmel and Lazzarin, 2002; Susmel, 2008). Another aspect which deserves to be considered in detail in the present section is the range of validity of the MWCM as formalised by Eqs (3.21) and (3.24). From a theoretical point of view, it is logical to believe that in the presence of very high values of ratios r and reff (that is, when the stress component perpendicular to the critical plane is much larger than the

114

Multiaxial notch fatigue 1.2

t0 1 = s0 3

sx,a/s0 1

txy,a sx,a = 0.25 txy,a sx,a = 0.5

0.8 0.6

txy,a sx,a = 1

0.4

txy,a sx,a = 2

0.2 0

0

15

30

45

60

75

90

dxy,x (degrees)

1.2

txy,a sx,a = 0.25

sx,a/s0 1

txy,a sx,a = 0.5

0.8 0.6

txy,a sx,a = 1

0.4

0

txy,a sx,a = 2

t0 s0 = 1

0.2 0

15

30

45

60

75

90

dxy,x (degrees)

3.11 High-cycle fatigue damage under out-of-phase fully reversed tension and torsion according to the MWCM.

maximum shear stress amplitude), fatigue damage is no longer only Mode II governed. In other words, the hypothesis can be formed that, when r and reff become larger than a certain material threshold value, rlim, the use of the critical plane approach as it stands is no longer justified due to a change in the mechanisms leading to final breakage (Susmel et al., 2005; Karadag and Stephens, 2003). In order to estimate a reference value for rlim, it is useful to rewrite Eq. (3.26) as follows: σ 3.27 τ a2 − τ 0τ a + τ 0 − 0 σ n,max ⭐ 0 2

(

)

The above equation can easily be plotted in a ta vs. sn,max diagram, obtaining a limit curve having the profile sketched in Fig. 3.12a. Such a schematic diagram should make it evident that the use of the MWCM is justified as long as r is lower than rlim, where (Lazzarin and Susmel, 2003; Susmel et al., 2005): τ0 ρlim = 3.28 2τ 0 − σ 0

The Modified Wöhler Curve Method in fatigue assessment

115

According to Fig. 3.12a, the hypothesis can be formed that the intrinsic mathematical limit characterising Eq. (3.26) corresponds to the material threshold value above which the use of the conventional critical plane approach can correctly be extended only if other assumptions are introduced in order to efficiently model the change in the physical mechanisms resulting in the formation of fatigue cracks. In order to extend the use of the MWCM to those situations in which critical planes experience very high values of either r or reff, we have argued that accurate estimates can be obtained by simply assuming that a plain material is at its fatigue limit if conditions (3.21) and (3.24) are assured by imposing r = reff = rlim when either r or reff is larger than rlim (Susmel, 2008). The diagram sketched in Fig. 3.12b shows how the profile of the tA,Ref vs. reff relationship is modified according to the above assumption. It is also worth noting that when loading paths resulting in RCP values equal to −1 are involved, reff is always equal to r: this consideration should clearly explain the reason way constants a and b in Eqs (3.22) and (3.26) are identical. In other words, the limit curve shown in Fig. 3.12b does not change its profile even when ta is plotted against r and no longer against reff: this should make it evident that the effective value of the critical plane stress ratio affects the prediction only under RCP ratios larger than −1 by supplying a reference shear stress, tA,Ref, higher than the one which would be obtained by assuming m = 1. To conclude, it is important to highlight here that the schematisation adopted to calculate rlim (Fig. 3.12a) does not have to be misinterpreted. In particular, the MWCM postulates that fatigue damage depends on ta and reff (or r) and not on ta and sn,max, so that the correct way of interpreting the MWCM is the one depicted in Fig. 3.12b, whereas the diagram sketched in Fig. 3.12a was used only to directly determine the intrinsic mathematical limit hidden behind Eqs (3.22) and (3.26). In order to show the accuracy and reliability of the MWCM in estimating high-cycle fatigue strength under multiaxial fatigue loading, the diagram tA,Ref t0

ta r=0 t0 s0/2

r=1

s0/2 rlim

r=∞ 0

s0/2

sn,max (a)

0

rlim

1

reff

(b)

3.12 (a) Definition of rlim and (b) proposed correction for the tA,Ref vs. reff (or r) relationship.

116

Multiaxial notch fatigue

reported in Fig. 3.13 proves that, when used to predict fatigue limits under in-phase fully-reversed bending and torsion, our method is capable of estimates falling within an error interval equal to ±10%. It is interesting to highlight that the figure summarises 162 experimental results generated by testing 29 materials under different values of the ratio between sx,a and txy,a (see Fig. 3.2 for the adopted frame of reference): Fig. 3.13 confirms that the MWCM is successful in estimating multiaxial fatigue limits under the above loading paths, with the advantage over other existing methods that there is no need for distinguishing between ductile and brittle behaviour as was done, for instance, by Gough (1949) and Papadopoulos (1987). In order to clearly show how the mean stress sensitivity index, m, affects the MWCM’s accuracy in estimating multiaxial fatigue limits, Fig. 3.14 compares 1000

Gough's data

E = –10%

Frith's data

Non-conservative t0 (MPa) Conservative R = –1

E = +10% 100 100

1000

ta-a·reff (MPa)

3.13 MWCM’s accuracy in estimating multiaxial fatigue limits under in-phase fully reversed bending and torsion (data from Gough, 1949; Frith, 1956). 500 m = 0.41

ta (MPa)

400

E = 10%

m=1

300 E = –10%

200 100 0 0.0

rlim = 2.35 0.5

1.0

1.5

2.0

2.5 r 3.0 eff

3.14 MWCM’s accuracy in estimating fatigue limits generated by testing specimens of S65A under in-phase bending and torsion with superimposed tensile and torsional static stresses (data from Gough, 1949).

The Modified Wöhler Curve Method in fatigue assessment

117

the predictions made by taking into account the experimental value of m to those obtained by assuming full mean stress sensitivity (m = 1). It is worth noting that the above experimental results were generated by testing solid cylindrical samples of S65A under in-phase bending and torsion with superimposed static stresses (Gough, 1949).To build this chart, the actual value of m was estimated from a uniaxial fatigue limit experimentally determined under a load ratio, R, equal to −0.35. The diagram of Fig. 3.14 clearly shows that when the actual value of m is used, the MWCM is highly accurate, giving estimates falling within an error interval of ±10%; on the contrary, and as expected, assuming a mean stress sensitivity index equal to 1 for those materials characterised by an m value lower than unity results in conservative estimates. The diagrams of Figs 3.15 and 3.16 instead show that, when cylindrical specimens subjected to uniaxial and torsional fatigue loading are consid250 0°

ta (MPa)

200

30°

E = 10%

60°

150

90° E = –10%

100 50

rlim = 2.5

RCP = –1

0 0.0

0.5

1.0

1.5

2.0

2.5 r 3.0 eff

(a)

ta (MPa)

200 175

0°

150

60°

125

90°

E = 10%

100 75

E = –10%

50 25

rlim = 3.5

RCP = –1

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

reff

4.0

(b)

3.15 MWCM’s accuracy in estimating fatigue limits generated by testing specimens of (a) hard and (b) mild steel under fully reversed out-of-phase bending and torsion (data from Nishihara and Kawamoto, 1945).

118

Multiaxial notch fatigue

ta (MPa)

450 400

dx,xy = 0°, RCP = –1

dx,xy = 0°, RCP = –0.2

350

dx,xy = 0°, RCP = 0.3

dx,xy = 90°, RCP = –1

300

dx,xy = 90°, RCP = –0.2

dx,xy = 180°, RCP = –0.2

250 200 150

E = 10% E = –10%

100 50

m = 0.35

rlim = 2.13

0 0.0

0.5

1.0

1.5

2.0

2.5 r 3.0 eff

(a)

ta (MPa)

450 400

0°, –1

0°, 0.3

0°, –0.1

350

60°, –1

60°, –0.1

90°, –1

90°, 0

120°, –1

0°, –0.2

300

E = 10%

250 200 150

E = –10%

100 50

m = 0.36

rlim = 2.51

0 0.0

0.5

1.0

1.5

2.0

2.5 r 3.0 eff

(b)

3.16 MWCM’s accuracy in estimating fatigue limits generated by testing specimens of (a) 42CrMo4 and (b) 34Cr4 under in-phase and out-of-phase biaxial loading with superimposed static stresses (data from Zenner et al., 1985).

ered, our multiaxial fatigue criterion is successful in estimating high-cycle fatigue strength also in the presence of non-zero out-of-phase angles. In more detail, these charts prove that the MWCM is capable of correctly taking into account the degree of non-proportionality of the applied loading path not only under fully reversed loading (Fig. 3.15), but also, through the actual value of the mean stress sensitivity index, in the presence of superimposed static stresses (Fig. 3.16). To conclude, Fig. 3.17 fully confirms that the MWCM is highly accurate also when such a criterion is used to estimate multiaxial fatigue limits in the presence of complex biaxial non-proportional loading paths. In particular, the results summarised in these charts were generated by testing tubular samples subjected to the following stress components (see Fig. 3.2 for the definition of the adopted system of coordinates):

The Modified Wöhler Curve Method in fatigue assessment

119

500 0°, 0°, 0.1 90°, 0°, 0.2 0°, 0°, –1 0°, 90°, –1 0°, 90°, 0.2 0°, 90°, 0.4

ta (MPa)

400 300

E = 10%

60°, 0°, 0 180°, 0°, 1 0°, 60°, –1 0°, 60°, 0.3 0°, 60°, 0.4 (dy,x, dxy,x, RCP)

200 E = –10%

100 0

m = 0.43 0.0

0.5

rlim = 2.32 1.0

1.5

2.0

2.5

3.0 r 3.5 eff

(a) 400 0°, 0°, –0.1 0°, 0°, 0.1 60°, 90°, 0.2 0°, 0°, 0.1 180°, 0°, –1

ta (MPa)

300 200

180°, 0°, 1 0°, 90°, 0.2 180°, 90°, 0.6 60°, 0°, 0.2 (dy,x, dxy,x, RCP)

E = 10%

E = –10% 100

rlim = 3.14

m=1 0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0 4.5 reff

(b)

3.17 MWCM’s accuracy in estimating fatigue limits generated by testing specimens of (a) 25CrMo4 and (b) 34Cr4 under complex multiaxial loading paths with superimposed static stresses (data from Zenner et al., 1985).

σ x( t ) = σ x,m + σ x,a sin (ω t ) σ y( t ) = σ y,m + σ y,a sin (ω t − δ y,x ) τ xy( t ) = τ xy,m + τ xy,a sin (ω t − δ xy,x ) and the estimates summarised in Fig. 3.17 were obtained by determining, for every material considered, the actual value of the mean stress sensitivity index.

3.6

Use of the MWCM to estimate finite life under multiaxial fatigue loading

Thanks to its nature, the MWCM can be employed also to predict lifetime in the medium-cycle fatigue regime, provided that the modified Wöhler curves used to evaluate fatigue damage are correctly estimated through Eqs

120

Multiaxial notch fatigue

(3.15) and (3.16). In particular, when the degree of multiaxiality of the stress field relative to the fatigue process zone is evaluated in terms of reff, the constants of functions kt(reff) and tA,Ref(reff) have to be determined by using two different reference curves generated under RCP equal to −1. As usual, if the fully reversed uniaxial and torsional fatigue curves are known from the experiments, Eqs (3.15) and (3.16) can easily be rewritten as follows: kτ ( ρeff ) = [ kτ ( ρeff = 1) − kτ ( ρeff = 0)] ρeff + kτ ( ρeff = 0)

3.29

σ τ A,Ref ( ρeff ) = 0 − τ 0 ρeff + τ 0 2

3.30

(

)

where, in Eq. (3.29), kt(reff = 1) and kt(reff = 0) are the negative inverse slope of the uniaxial and torsional fatigue curve, respectively. It is worth noting here that the value of the mean stress sensitivity index, m, needed to correctly calculate reff can be determined by using the same strategies as those already discussed in the previous section. On the contrary, if there is no experimental information suitable for calculating index m, then the MWCM can simply be applied in terms of r. As already said above, the main advantage of this way of formalising the MWCM is that only two experimental fatigue curves generated under two different values of ratio r are needed to efficiently calibrate the method itself. In particular, if uniaxial and torsional fully reversed experimental results are employed to calibrate Eqs (3.15) and (3.16), rewritten in terms of r, the constants of such relationships take on the following values: kτ ( ρ ) = [ kτ ( ρ = 1) − kτ ( ρ = 0)] ρ + kτ ( ρ = 0) σ τ A,Ref ( ρ ) = 0 − τ 0 ρ + τ 0 2

(

)

3.31 3.32

It is important to point out here that, similarly to what was done in the highcycle fatigue regime, also the above calibration functions have to be corrected in order to correctly take into account the presence of large values of either reff or r. In particular, if the limit value of the above stress ratios, rlim, is again calculated through Eq. (3.28), the hypothesis can be formed that, when either reff or r is larger than rlim, Eqs (3.29) and (3.30), or Eqs (3.31) and (3.32), have to be applied by imposing reff = r = rlim. According to the above assumption, the profiles of the calibration curves are corrected as sketched in Fig. 3.18a: this results in the fact that the multiaxial fatigue strength of engineering materials cannot be lower than the one predicted by the modified Wöhler curve calculated by assuming reff = r = rlim (Fig. 3.18b). In order to show the accuracy of the MWCM when used to estimate lifetime in the medium-cycle multiaxial fatigue regime, three different experimental (Nf) vs. estimated (Nf,e) number of cycles to failure diagrams are reported in Figs 3.19, 3.20 and 3.21. The data summarised in these charts were generated by testing cylindrical samples under in-phase and outof-phase fully reversed bending and torsion. It is important to highlight that

The Modified Wöhler Curve Method in fatigue assessment kt(reff) = a·reff + b

tA,Ref

tA,Ref(reff) = a·reff + b

kt kt(reff = 0)

ta

121

Increasing r

kt(reff = 1)

t0 s0/2

tA,Ref(rlim)

0

rlim

1

reff

N0

(a)

Nf

(b)

3.18 (a) Proposed correction for the kt vs. reff (or r) relationship, and (b) resulting lowest modified Wöhler curve. 10 000 000 Nf (cycles) 1 000 000

Bending Torsion In-phase 90° Out-of-phase

PS = 95%

PS = 5% 100 000

Conservative

Uniaxial scatter band 10 000

Non-conservative Torsional scatter band

1000 1000

10 000

100 000

1 000 000

10 000 000

Nf,e (cycles)

3.19 MWCM’s accuracy in estimating fatigue lifetime of specimens made of SM45C and subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Lee, 1985).

in such charts not only the experimental data but also the scatter bands of the calibration fatigue curves are plotted. In particular, the dashed lines delimit the torsional scatter bands, whereas the continuous straight lines delimit the scatter bands relative to the fatigue results generated under bending. These scatter bands refer to a probability of survival, PS, equal to 5% as well as to 95% and were calculated under the hypothesis of a lognormal distribution of the number of cycles to failure for each stress level with a confidence value equal to 95%. The diagrams of Figs 3.19, 3.20 and 3.21 clearly prove that the MWCM is highly accurate when used to predict multiaxial finite life, allowing the estimates to fall within the widest scatter band between the uniaxial and the torsional one. This result is very interesting, especially in light of the fact that it is unrealistic to believe that it is pos-

122

Multiaxial notch fatigue 1 000 000 Nf (cycles) 100 000

Bending Torsion In-phase 90° Out-of-phase

PS = 95%

Conservative 10 000

PS = 5% Uniaxial scatter band

1000

Torsional scatter band Non-conservative

100 100

1000

10 000

100 000 1 000 000 Nf,e (cycles)

3.20 MWCM’s accuracy in estimating fatigue lifetime of specimens made of SAE 1045 and subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Kurath et al., 1989). 10 000 000 Nf (cycles) 1 000 000

100 000

Bending Torsion In-phase Out-of-phase

PS = 95%

Conservative

Uniaxial scatter band 10 000

1000 1000

Non-conservative PS = 5% Torsional scatter band 10 000

100 000

1 000 000

10 000 000

Nf,e (cycles)

3.21 MWCM’s accuracy in estimating fatigue lifetime of specimens made of 6082-T6 and subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Susmel and Petrone, 2003).

sible to devise a predictive method which is, from a statistical point of view, more accurate than the experimental information used to calibrate the criterion itself. To conclude, it is worth noting that, in order to clearly show our method’s accuracy in estimating fatigue strength under multiaxial fatigue loading, the

The Modified Wöhler Curve Method in fatigue assessment

123

validation exercises summarised in both the present and the previous section were done by calculating constants a, b, a and b by means of fatigue curves characterised by a probability of survival equal to 50%. On the contrary, and in accordance with the available standard codes, when assessing real mechanical components, fatigue curves having a higher value of the probability of survival are always suggested to be adopted: by so doing, the way of using the MWCM is the same as that discussed above, but estimates are obviously characterised by a higher degree of conservatism.

3.7

References

Bolotin, V. V. (1999) Mechanics of Fatigue. CRC Press, Boca Raton, FL. Brown, M. W., Miller, K. J. (1973) A theory for fatigue under multiaxial stress–strain conditions. Proceedings of the Institution of Mechanical Engineers 187, 745–755. Findley, W. N. (1959) A theory for the effect of mean stress on fatigue under combined torsion and axial load or bending. Transactions of the ASME 81, Ser. B, 301–306. Frith, P. H. (1956) Fatigue of wrought high-tensile alloy steel. Proceedings of the Institution of Mechanical Engineers 462–499. Fukuda, T., Nisitani, H. (2003) The background of fatigue limit ratio of torsional fatigue to rotating bending fatigue in isotropic materials and materials with clearbanded structures. In: Biaxial and Multiaxial Fatigue and Fracture, edited by A. Carpinteri, M. de Freitas and A. Spagnoli. Elsevier and ESIS, 285–302. DOI: 10.1016/S1566-1369(03)80016-X. Gough, H. J. (1949) Engineering steels under combined cyclic and static stresses. Proceedings of the Institution of Mechanical Engineers 160, 417–440. Karadag, M., Stephens, R. I. (2003) The influence of high R ratio on unnotched fatigue behaviour of 1045 steel with three different heat treatments. International Journal of Fatigue 25, 191–200. DOI: 10.1016/S0142-1123(02)00116-0. Kaufman, R. P., Topper T. (2003) The influence of static mean stresses applied normal to the maximum shear planes in multiaxial fatigue. In: Biaxial and Multiaxial Fatigue and Fracture, edited by A. Carpinteri, M. de Freitas and A. Spagnoli. Elsevier and ESIS, 123–143. DOI: 10.1016/S1566-1369(03)80008-0. Kurath, P., Downing, S. D., Galliart, D. R. (1989) Summary of non-hardened notched shaft – round robin program. In: Multiaxial Fatigue – Analysis and Experiments, edited by G. E. Leese and D. F. Socie. SAE AE-14, Society of Automotive Engineers, Warrendale, PA, 13–32. Lazzarin P., Susmel L. (2003) A stress-based method to predict lifetime under multiaxial fatigue loadings. Fatigue and Fracture of Engineering Materials and Structures 26, 1171–1187. DOI: 10.1046/j.1460-2695.2003.00723.x. Lee, S. B. (1985) A criterion for fully reversed out-of-phase torsion and bending. In: Multiaxial Fatigue, edited by K. J. Miller and M. W. Brown. ASTM STP, 853, 553–568. Matake, T. (1977) An explanation on fatigue limit under combined stress. Bulletin of the JSME 20 (141), 257–263. McDiarmid, D. L. (1991) A general criterion for high cycle multiaxial fatigue failure. Fatigue and Fracture of Engineering Materials and Structures 14, 429–453. DOI: 10.1111/j.1460-2695.1991.tb00673.x.

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McDiarmid, D. L. (1994) A shear-stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction. Fatigue and Fracture of Engineering Materials and Structures 17, 1475–1484. DOI: 10.1111/j.14602695.1994.tb00789.x. Miller, K. J. (1982) The short crack problem. Fatigue and Fracture of Engineering Materials and Structures 5, 223–232. DOI: 10.1111/j.1460-2695.1982.tb01250.x. Miller, K. J. (1993) The two thresholds of fatigue behaviour. Fatigue and Fracture of Engineering Materials and Structures 16, 931–939. DOI: 10.1111/j.1460-2695.1993. tb00129.x. Nishihara, T., Kawamoto, M. (1945) The strength of metals under combined alternating bending and torsion with phase difference. Memoirs of the College of Engineering, Kyoto Imperial University 11, 85–112. Papadopoulos, I. V. (1987) Fatigue polycyclique des métaux: une nouvelle approche. Thèse de Doctorat, Ecole Nationale des Ponts et Chaussées, Paris, France. Papadopoulos, I. V. (1997) Exploring the high-cycle fatigue behaviour of metals from the mesoscopic scale. Notes of the CISM Seminar, Udine, Italy. Sines, G. (1959) Behaviour of metals under complex static and alternating stresses. In: Metal Fatigue, edited by G. Sines and J. L. Waisman, McGraw-Hill, New York, 145–169. Socie, D. F. (1987) Multiaxial fatigue damage models. Transactions of the ASME, Journal of Engineering Materials and Technology 109, 293–298. Socie, D. F, Bannantine J. (1988) Bulk deformation fatigue damage models. Material Science and Engineering A 103, 3–13. DOI: 10.1016/0025-5416(88)90546-0. Sonsino, C. M. (1995) Multiaxial fatigue of welded joints under in-phase and out-of-phase local strains and stresses. International Journal of Fatigue 17, 55–70. DOI: 10.1016/0142-1123(95)93051-3. Suresh, S. (1991) Fatigue of Materials. Cambridge University Press, Cambridge. Susmel, L. (2000a) Validazione di un criterio di resistenza a fatica multiassiale fondato sull’individuazione di un piano di verifica. In: Proceedings of IGF National Conference, Bari, Italy, 167–174 (in Italian). Susmel, L. (2000b) Una metodologia di verifica a fatica multiassiale per carichi in fase e fuori fase. In: Proceedings of XXIX AIAS National Conference, Lucca, Italy, 435–446 (in Italian). Susmel, L. (2008) Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes. Fatigue and Fracture of Engineering Materials and Structures 31, 295–309. DOI: 10.1111/j.1460-2695.2008.01228.x. Susmel, L., Lazzarin, P. (2002) A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures 25, 63–78. DOI: 10.1046/j.1460-2695.2002.00462.x. Susmel, L., Petrone, N. (2003) Multiaxial fatigue life estimations for 6082-T6 cylindrical specimens under in-phase and out-of-phase biaxial loadings. In: Biaxial and Multiaxial Fatigue and Fracture, edited by A. Carpinteri, M. de Freitas and A. Spagnoli, Elsevier and ESIS, 83–104. DOI: 10.1016/S1566-1369(03)80006-7. Susmel, L., Tovo, R., Lazzarin, P. (2005) The mean stress effect on the high-cycle fatigue strength from a multiaxial fatigue point of view. International Journal of Fatigue 27, 928–943. DOI: 10.1016/j.ijfatigue.2004.11.012. Zenner, H., Heidenreich, R., Richter, I. (1985) Dauerschwingfestigkeit bei nichtsynchroner mehrachsiger Beanspruchung. Zeitschrift für Werkstofftechnik 16, 101–112.

4 Fatigue assessment of notched components according to the Modified Wöhler Curve Method

Abstract: In the present chapter the different ways of using the Modified Wöhler Curve Method (MWCM) to estimate both high-cycle fatigue strength and finite life in the presence of stress concentration phenomena are reviewed. Key words: Modified Wöhler Curve Method, notches, nominal stresses, local stresses, Theory of Critical Distances.

4.1

Introduction

Owing to their well-known detrimental effect on the overall fatigue strength of engineering materials, stress concentration phenomena are always a matter of concern to engineers engaged in performing the fatigue assessment of mechanical assemblies. As to the above problem, the state of the art shows that, when uniaxial fatigue loadings are involved, many different criteria suitable for estimating fatigue damage in notched components have been formalised and validated (see Chapter 2). On the contrary, only few attempts have been made so far to devise sound methodologies capable of accurately predicting fatigue strength when stress raisers are subjected to multiaxial fatigue loading. In this scenario, it can be observed that, thanks to its peculiar features, the MWCM can efficiently be used to perform the fatigue assessment of notches, provided that the stress and strength analyses are coherently performed. In more detail, our method can be applied either in terms of nominal stresses or by directly post-processing the linear-elastic stress fields acting on the fatigue process zone. In other words, the MWCM proved to be highly accurate both when the plain modified Wöhler curves are corrected through suitable fatigue strength reduction factors and when the above critical plane approach is applied in terms of the Theory of Critical Distances (TCD). In the present chapter, then, the different ways of using the MWCM to estimate both high-cycle fatigue strength and finite life in the presence of stress concentration phenomena will be reviewed in detail. 125

126

Multiaxial notch fatigue

4.2

Inherent and external multiaxiality

Among the different methods specifically devised to assess notched components subjected to cyclic loading, it is recognised that the simplest way to take into account the presence of stress concentration phenomena is by using nominal quantities. In particular, as briefly recalled in Chapter 2, the nominal stress based approach suggests estimating fatigue damage in components containing geometrical features by correcting the parent material fatigue curve through the classical fatigue strength reduction factor, Kf (Peterson, 1959). Since the empirical approach devised by Gough (1949), the above strategy has been attempted to be extended also to those situations involving multiaxial fatigue loadings (Tipton and Nelson, 1997; Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003). Unfortunately, it has to be said that, when dealing with components having complex geometries, the definition of nominal quantities is never straightforward, making it difficult for the nominal stress based approach to be used in situations of practical interest. Moreover, when mechanical assemblies are subjected to non-proportional loading, the fact that the principal stress directions may rotate during the load cycle results in a more difficult evaluation of the fatigue damage extent due to the fact that, under the above loading paths, fatigue strength depends on the mutual interaction between material microstructure and cyclic change of the maximum principal stress direction (Sonsino, 1995): in the presence of stress concentration phenomena the material response to the degree of non-proportionality of the applied loading becomes more difficult to model due to the fact that the stress fields damaging the fatigue process zone are not only multiaxial but also characterised by tridimensional stress gradients. The considerations briefly reported above make it evident that, even if effective under certain circumstances, nominal stresses are in any case poorly related to the mechanisms taking place within the fatigue process zone and leading to final breakage. Another interesting aspect which deserves to be considered here is also the fact that the uniaxial and multiaxial fatigue problem are sometimes considered to be somehow different and, for this reason, they are too often addressed by adopting different strategies. It is the writer’s opinion that this approach to the notch fatigue issue is in a way surprising and, above all, not justified from a philosophical point of view. In particular, it is well known that the physical mechanisms leading to the formation of fatigue cracks are governed by the entire stress field damaging the material in the vicinity of crack initiation sites. The above remark should make it evident that performing the fatigue assessment of notched components is always a multiaxial problem; in fact, when stress concentrators are involved, the stress fields acting on the fatigue process zone are always at least biaxial and this holds true also under nominal uniaxial cyclic loading. The most important

Fatigue assessment of notched components

127

implication of the above fact is that real mechanical components can be damaged either by an external or by an inherent multiaxiality, where the latter type of multiaxiality is due to the geometrical feature contained by the component itself, whereas the first one is due to the complexity of the applied loading path. Moreover, even if it is often possible to distinguish between the above two types of multiaxiality, it is evident that, for a given cyclic stress field damaging the process zone, fatigue strength has to be the same independently of the source from which the multiaxiality of the stress field itself arises. It is worth concluding the present section by observing that the distinction drawn above is only valid when the fatigue assessment is performed in terms of local quantities. On the contrary, if the nominal stress based approach is attempted to be used to estimate fatigue damage, the only source of multiaxiality is the complexity of the applied cyclic loading.

4.3.

The MWCM applied in terms of nominal stresses

According to the two different fatigue assessment philosophies mentioned above, in the present section and in the following one the problem of using the MWCM to estimate high-cycle fatigue strength as well as finite life of notched components will be addressed in terms of both nominal and local quantities. As to the way of applying our method in terms of nominal stresses, initially it is worth noting that, as postulated by continuum mechanics, the reference section which should be used to estimate fatigue damage is the net one at the hot-spots of the component to be assessed. In other words, all the considered stresses, that is, both those applied to the component and those relative to the critical plane, should be calculated with respect to the net cross-sectional area. Before considering in detail how to use the TCD in terms of nominal stresses, it is useful to remember here that, according to Peterson (1959, 1974), when mechanical components contain notches, plain fatigue limits (or plain endurance limits) have to be corrected through the fatigue strength reduction factor, Kf, and such a strategy can be used under both uniaxial and torsional fatigue loading (see Sections 2.4 and 2.8.1). A similar strategy can be followed also when fatigue strength is summarised in terms of modified Wöhler curves. In particular, the modified Wöhler diagram sketched in Fig. 4.1 shows how Kf and Kft correct the uniaxial and the torsional plain fatigue curve, respectively, to account for the detrimental effect of the assessed geometrical feature. As clearly suggested by such a sketch, the above fatigue strength reduction factors can easily be rewritten as follows (Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003):

128

Multiaxial notch fatigue reff = 0 (torsional curves)

ta

Kft

Kf

reff = 1

(uniaxial curves) N0

Nf

4.1 Uniaxial and torsional plain material modified Wöhler curves corrected by means of the corresponding fatigue strength reduction factors.

Kf = Kf ( ρeff = 1) =

τ A ,Ref ( ρeff = 1) Plain σ = 0 τ A ,Ref ( ρeff = 1) Notch σ 0 n

4.1

Kft = Kf ( ρeff = 0 ) =

τ A ,Ref ( ρeff = 0 ) Plain τ = 0 τ A ,Ref ( ρeff = 0 ) Notch τ 0 n

4.2

Such identities should make it clear that the MWCM can be used directly to assess notched components by taking full advantage of the classical nominal stress based approach as formalised by Peterson. According to the above remark, the functions defining the slope and the position of any notch modified Wöhler curve can then be expressed as: kτ ( ρeff ) Notch = an ⋅ ρeff + bn

4.3

τ A ,Ref ( ρeff ) Notch = α n ⋅ ρeff + βn

4.4

where the values of calibration constants an, bn, an and bn depend not only on the fatigue properties of the considered material but also on the geometrical feature to be assessed. It is worth noting here that in relationships (4.3) and (4.4) reff is always the effective value of the critical plane stress ratio, which is defined as:

ρeff =

mσ n,m σ n,a + τa τa

4.5

In the above identity the stress components relative to the critical plane have to be calculated by post-processing the stress state at the hot-spot expressed in terms of nominal net stresses.

Fatigue assessment of notched components

129

If relationships (4.3) and (4.4) are calibrated by using the uniaxial and torsional fatigue curves, it is trivial to obtain the following values for an, bn, an and bn: an = kτ ( ρeff = 1) n − kτ ( ρeff = 0 ) n; bn = kτ ( ρeff = 0 ) n

αn =

σ 0n − τ 0 n; βn = τ 0 n 2

4.6 4.7

where subscript n denotes that the needed fatigue constants have to be determined from the corresponding uniaxial and torsional notch fatigue curves. If the above values of constants an and bn are introduced now in Eq. (4.4), the MWCM formalised to address the high-cycle fatigue problem can be rewritten directly as follows (Susmel and Lazzarin, 2002):

(

τa + τ 0n −

)

σ 0n ⋅ ρeff ≤ τ 0 n 2

4.8

The correct determination of the mean stress sensitivity index instead is much more difficult and tricky: owing to the fact that the stress components relative to the critical plane are calculated from the applied nominal net stresses, m depends not only on the material response to superimposed static stresses but also on the assessed geometrical feature. In any case, strictly speaking, m can be estimated, according to Eq. (3.23), though an endurance limit experimentally determined by testing the notch to be assessed under a nominal critical plane load ratio, RCP, larger than −1. Finally, similarly to the use of the MWCM to estimate multiaxial fatigue strength of plain engineering materials, the limit value of the critical plane stress ratio is calculated as follows:

ρlim =

τ 0n 2τ 0 n − σ 0 n

4.9

It has to be said here that, due to the way they are defined, nominal stresses are, in general, poorly related to the physical processes taking place within the fatigue process zone, so that the correct evaluation of both m and rlim is always very difficult. Moreover, even when the values of the above constants are properly calculated, sometimes they do not allow the overall accuracy of the estimates to be increased significantly. This is the reason why, when applying the MWCM in terms of nominal stresses, m is suggested to be kept equal to unity and rlim equal to infinity (Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003): such an assumption allows a higher margin of safety to be reached with the advantage that a reduced number of notch fatigue curves are needed to efficiently calibrate Eqs (4.3) and (4.4).

130

Multiaxial notch fatigue

Apart from the aforementioned problems which may arise when performing the fatigue assessment of notched components characterised by complex geometries, it has to be said that the MWCM applied in terms of nominal net stresses was seen to be capable of accurately estimating highcycle notch fatigue strength (Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003). In more detail, as an initial example, the diagram reported in Fig. 4.2 shows the accuracy of our method in estimating multiaxial fatigue limits in the presence of standard notches subjected to in-phase fully reversed bending and torsion. This diagram summarises the results generated by testing 19 different materials weakened by notches characterised by a Kf value ranging from 1.05 to 2.5 and a Kft value from 1.2 to 2. The charts of Figs 4.3 and 4.4 instead show that our method was seen to be highly accurate and reliable also when used to estimate high-cycle notch fatigue strength under non-proportional loading as well as in the presence of superimposed static stresses. In more detail, these diagrams summarise the results obtained by testing specimens with fillet of SAE 1045 and S65A, respectively, where the notch root radius of the latter samples was equal to about 0.9 mm (Kt = 1.7 and Ktt = 1.2), whereas it was equal to 5 mm (Kt = 1.6 and Ktt = 1.3) in those of SAE 1045. As to the overall accuracy of the MWCM, it is worth noting here that, similarly to the most accurate uniaxial notch fatigue criteria (Chapter 2), the systematic use of our method was seen to result in estimates always falling within an error interval equal to about ±20% (see Figs 4.2, 4.3 and 4.4): this strongly supports the idea that, if properly calibrated, the MWCM can be used successfully to assess real mechanical assemblies in situations of practical interest. 1000 Gough's data

E = –20%

Frith's data

Non-conservative t0n (MPa) Conservative E = +20%

R = –1

100 100

ta – an·reff (MPa)

1000

4.2 Accuracy of the MWCM applied in terms of nominal stresses in estimating multiaxial fatigue limits under in-phase fully reversed bending and torsion (data from Gough, 1949; Frith, 1956).

Fatigue assessment of notched components

131

300 In-phase

ta (MPa)

250

90° Out-of-phase

200

E = 20%

150 100

E = –20%

50

rlim = 1.33

R = –1

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8 2.0 reff

4.3 Accuracy of the MWCM applied in terms of nominal stresses in estimating high-cycle multiaxial fatigue damage in notched specimens of SAE 1045 subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Kurath et al., 1989).

400 RCP = –1

350 E = 20%

ta (MPa)

300

RCP > –1 RCP = 1 (or 0/0)

250 200 E = –20%

150 100 50

rlim = 1.4

m = 0.24, in-phase

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8 2.0 reff

4.4 Accuracy of the MWCM applied in terms of nominal stresses in estimating high-cycle multiaxial fatigue damage in notched specimens of S65A subjected to in-phase bending and torsion with superimposed static stresses (data from Gough, 1949).

The fact that such a high accuracy level can be reached by simply using a linear relationship between ta and reff suggests that a linear law can be successfully employed also to estimate the variation of the multiaxial strength reduction factor as reff increases. In particular, according to identities (4.1) and (4.2), such a linear function can be easily calibrated through the corresponding values of Kf determined under uniaxial and under torsional fatigue loading, respectively. The validity of the above assumption is clearly proved by the diagrams reported in Fig. 4.5, which

132

Multiaxial notch fatigue

2.5 Kf(reff) 2.0

0.4% C steel (normalised)

2.5 Kf(reff) 2.0

Kf(reff) = 0.67·reff + 1.19

1.5

3% Ni steel Kf(reff) = 0.23·reff + 1.38

1.5

1.0 0.0

0.2

0.4

0.6

reff

1.0

1.0 0.0

0.2

0.4

0.6

reff

1.0

4.5 Kf vs. reff relationships for two different V-notched steels tested under in-phase fully reversed bending and torsion (data from Gough, 1949).

plot the multiaxial fatigue strength reduction factor, experimentally determined by testing V-notched specimens under in-phase bending and torsion, against reff: such charts strongly support the idea that, at least in a reff interval ranging between 0 and 1, the Kf vs. reff relationship can be assumed to be linear. In other words, this remark makes it evident that when the MWCM is applied in terms of nominal stresses, the fatigue assessment under multiaxial fatigue loading can be performed by following a strategy similar to the classical one proposed and validated by Peterson (1959). As clearly suggested by Eq. (4.3), the MWCM can be applied in terms of nominal stresses to estimate not only high-cycle fatigue strength but also finite life. It is evident that the method is applied in the same way as already discussed in Section 3.6: the only difference is that now stresses have to be calculated, at the component hot-spot, with respect to the net section. It is important to highlight here that, for the same reasons as above, also when attempting to estimate the lifetime of notched components in terms of nominal stresses m is suggested to be kept equal to unity and rlim to infinity, unless all the pieces of experimental information needed to properly determine such quantities are known from the experiments. The experimental (Nf) vs. estimated (Nf,e) fatigue life diagrams reported in Figs 4.6 and 4.7 summarise the accuracy of the MWCM applied in terms of nominal stresses, when it is used to estimate the finite life of samples containing standard notches. In particular, the data plotted in the first chart were generated by testing, under bending and torsion, specimens with fillet of SAE 1045, where in such samples the notch root radius was equal to 5 mm. The latter data instead were generated by testing, under in-phase and 90° out-of-phase tension and torsion, V-notched samples of C40 steel having root radius equal to 0.5 mm. Such diagrams clearly prove that our method is highly accurate in predicting finite life in the presence of stress concentration phenomena, resulting in estimates falling within the widest scatter band between the two used to calibrate the method itself.

Fatigue assessment of notched components

133

10 000 000 Bending, R = –1 Torsion, R = –1 In-phase, R = –1 Out-of-phase, R = –1

Nf (cycles) 1 000 000

PS = 95%

Conservative

100 000

Uniaxial scatter band 10 000

Non-conservative PS = 5% Torsional scatter band

1000 1000

10 000

100 000

1 000 000 10 000 000 Nf,e (cycles)

4.6 Accuracy of the MWCM applied in terms of nominal stresses in estimating fatigue lifetime of notched specimens made of SAE 1045 and subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Kurath et al., 1989).

10 000 000

Tension, R = –1 Torsion, R = –1 In-phase, R = –1 Out-of-phase, R = –1 In-phase, R = 0 Out-of-phase, R = 0

Nf (cycles) 1 000 000

Conservative

100 000

Uniaxial scatter band

PS = 95%

Non-conservative

10 000 Torsional scatter band

1000 1000

m=1

PS = 5% 10 000

100 000

1 000 000 10 000 000 Nf,e (cycles)

4.7 Accuracy of the MWCM applied in terms of nominal stresses in estimating fatigue lifetime of notched specimens made of C40 and subjected to in-phase and out-of-phase tension and torsion (data from Atzori et al., 2006).

134

Multiaxial notch fatigue

4.4

The MWCM applied along with the TCD to estimate notch fatigue strength

Working in collaboration with our colleague Professor David Taylor (Trinity College, Dublin, Ireland), over the last decade we have made a big effort in order to devise a reliable fatigue assessment procedure capable of overcoming the problems usually encountered when attempting to use the nominal stress based approach to assess real components. In more detail, the engineering method we have proposed takes as its starting point the idea that fatigue damage has to be estimated by considering the degree of multiaxiality of the entire linear-elastic stress field damaging the material in the vicinity of crack initiation sites (Susmel and Taylor, 2003a, 2006, 2007a; Susmel, 2004, 2006; Meneghetti et al., 2007). By taking full advantage of the above hypothesis, the MWCM was then reformulated in terms of the TCD to make it suitable for efficiently accounting for the detrimental effect of stress concentration phenomena (Susmel and Taylor, 2003a). In other words, the scale and the stress gradient effect in notched components subjected to fatigue loading were assumed to be taken into account efficiently by the TCD (Tanaka, 1983; Taylor, 1999, 2001), whereas, thanks to its features, the MWCM was used to estimate the influence on the overall fatigue strength of the degree of multiaxiality and non-proportionality of the stress field acting on the fatigue process zone (Susmel and Lazzarin, 2002; Lazzarin and Susmel, 2003; Susmel et al., 2005; Susmel, 2008a). It is worth noting here also that, among the different formalisations of the TCD, the MWCM was proposed to be applied in terms of the PM simply because complex non-proportional multiaxial load histories are much easier to handle when the stress state needed to perform the fatigue assessment is determined at one single point (Susmel and Taylor, 2003a; Susmel, 2006). As to the material characteristic length to be used to estimate fatigue damage according to the above strategy, it has to be pointed out here that, as postulated by the TCD, the value of the critical distance was assumed not to change as the type of geometrical feature contained by the considered component changes, but, for a given material, the hypothesis was formed that its value increases as the number of cycles to failure decreases (Susmel and Taylor, 2007b, 2008). In the following sections the use of the MWCM in conjunction with the PM to estimate both high-cycle fatigue strength and finite life of mechanical components experiencing, under cyclic loading, multiaxial stress concentration phenomena will be explained in detail.

4.4.1 Estimating high-cycle notch fatigue strength In order to correctly use the MWCM in conjunction with the PM, our multiaxial fatigue criterion has to be calibrated and applied as explained in

Fatigue assessment of notched components

135

Chapter 3. In more detail, the material fatigue properties needed to determine constants a, b, m and rlim in Eqs (3.16), (3.23) and (3.28) have to be determined by using experimental fatigue results generated by testing plain specimens of the same material from which the notched component to be assessed is made (Susmel and Taylor, 2003a; Susmel, 2004). As postulated by the TCD (Taylor, 1999), the characteristic length instead has to be calculated according to the following relationship (see Section 2.7): L=

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

4.10

where, as usual, ΔKth is the range of the threshold value of the stress intensity factor, while Δs0 is the range of the uniaxial plain fatigue limit. It is important to highlight here that to correctly apply the MWCM in terms of the PM, the material characteristic length, L, has to be calculated always by using material fatigue properties determined under fully reversed fatigue loading, because the presence of superimposed static stresses is assumed to be taken into account directly by the MWCM itself. It is worth remembering here that the material characteristic length can be estimated also by using the fully reversed plain fatigue limit, Δs0, and a fully reversed notch fatigue limit, Δs0n, generated by testing samples containing a known geometrical feature (Taylor, 2007; Susmel and Taylor, 2007a). In more detail, initially the stress field distribution along the bisector of the considered notch has to be determined either through a linear-elastic FE model or by using a suitable analytical method (Fig. 4.8). Subsequently,

Δs0n

Δs1 Δs0 L 2

r

Δs0n

4.8 The PM as an alternative strategy to experimentally determine the material characteristic length L.

136

Multiaxial notch fatigue

the material characteristic length is directly estimated from the above stress–distance curve; in fact, as postulated by the PM, at the fatigue limit, Δs0n, the critical distance, L/2, is the distance from the stress concentrator apex at which the linear-elastic maximum principal stress equals the plain material fatigue limit, Δs0 (Fig. 4.8). Finally, it is worth noting also that, in order to accurately determine the material characteristic length according to the procedure briefly summarised above, the use of notches as sharp as possible is always advisable. Figure 4.9 explains the use of the MWCM in conjunction with the PM to estimate high-cycle fatigue strength in the presence of stress concentration phenomena. In more detail, assume that the component sketched in this figure is subjected to a complex system of cyclic forces resulting in a multiaxial stress state damaging the material in the vicinity of the assumed crack initiation site (point A in Figure 4.9). The focus path needed to locate the position of the point whose stress state has to be used to estimate fatigue strength is a straight line emanating from A and perpendicular, at the hotspot itself, to the component surface. It is worth remembering here also that the focus path has to be coincident with the notch bisector when the root radius is imposed to be equal to zero (Susmel, 2004). Finally, the critical point, O, giving all the engineering information suitable for estimating

Fi(t)

Fj(t) A L/2

sx(t) [ε(t)] =

O

txy(t) txz(t)

txy(t) sy(t) tyz(t) txz(t) tyz(t) sz(t)

Focus path

r

Fk(t) ta, reff

ta ≤ tA,Ref (reff) = a⋅reff + a

4.9 In-field use of the MWCM applied along with the PM to estimate notch fatigue limits.

Fatigue assessment of notched components

137

fatigue damage is positioned, along the focus path, at a distance from A equal to L/2 (Fig. 4.9). If a suitable frame of reference, Oxyz, is introduced, in a generic instant t of the considered cyclic load history, the linear elastic stress state at point O can be expressed as follows (Fig. 4.9): ⎡ σ x(t ) τ xy(t ) τ xz(t )⎤ [σ (t )] = ⎢⎢τ xy(t ) σ y(t ) τ yz(t )⎥⎥ ⎢⎣τ xz(t ) τ yz (t ) σ z(t ) ⎥⎦

4.11

Form the above stress tensor, the maximum shear stress amplitude, ta, and the stress components, sn,m and sn,a, perpendicular to the critical plane can then be calculated by taking full advantage of the methods summarised in Chapter 1. The values obtained for ta, sn,m and sn,a have to be used now to determine the effective value of the critical plane stress ratio, reff. Finally, the assessed mechanical component is assumed to be at its fatigue limit as long as the following condition is assured (Susmel, 2004):

τ a ≤ τ A ,Ref ( ρeff ) = aρeff + b

4.12

It is important to point out here that, thanks to the fact that the proposed approach post-processes the linear-elastic stress fields acting on the fatigue process zone, the damaging effect of every applied force can be computed separately, when the component to be assessed is subjected to a complex system of cyclic loadings. The resulting stress tensor (4.11) can then be calculated by superposing the contribution of every considered force, paying attention to keeping unchanged the synchronism among them: the above procedure allows the presence of both superimposed static stresses and non-zero out-of-phase angles to be taken easily into account during the design process. The accuracy and reliability of the above procedure was systematically checked by using several experimental results taken from the literature as well as generated in our laboratories (both in Trinity College, Dublin, Ireland and at the University of Ferrara, Ferrara, Italy). In more detail, such a method was initially attempted to be used to estimate high-cycle fatigue strength of notched samples subjected to uniaxial fatigue loading. As an example, Fig. 4.10 shows the accuracy of the MWCM applied in conjunction with the PM in estimating the classical fatigue results due to Frost generated by testing, under fully reversed tension–compression, V-notched cylindrical samples of mild steel (Frost, 1959). In this figure both experimental values and estimates are plotted in a Frost, Smith and Miller diagram: such a chart makes it evident that our approach is capable of correctly predicting the transition from the blunt- to the sharp-notch regime. Moreover, it can be pointed out that such a high accuracy level was obtained even if the considered V-notched samples were cylindrical: in light of the fact that the material

138

Multiaxial notch fatigue 160 Dt0n 140 (MPa)

Predicted Experimental

120 100

ΔKth

80

F p⋅a

60 40 20

Δs0 Kt,gross

R = –1

0 0

5

10

15

Kt,gross

25

4.10 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of V-notched cylindrical specimens made of mild steel (0.15% C) and subjected to fully reversed tension– compression (data from Frost, 1959).

in the vicinity of the stress concentrator apices experienced in-phase triaxial stress states even if the applied nominal loading was uniaxial, Fig. 4.10 clearly proves that our approach is capable of correctly taking into account the degree of multiaxiality of the stress field acting on the process zone. The overall accuracy of the MWCM applied along with the PM in estimating notch high-cycle fatigue strength under nominal uniaxial fatigue loading is shown instead in the diagram reported in Fig. 4.11. In more detail, such a chart summarises the estimates obtained considering experimental fatigue results generated by testing both flat and cylindrical specimens made of nine different materials and containing a variety of geometrical features. Similarly to the uniaxial TCD (Taylor and Wang, 2000; Susmel and Taylor, 2003b), the multiaxial fatigue approach reviewed in the present section was seen to be capable of estimates falling within an error interval of about ±20% (Susmel, 2004). Subsequently, the accuracy of the MWCM applied along with the PM was checked by using the data we generated by testing V-notched flat samples under in-phase Mode I and II loading (Susmel and Taylor, 2003a). In more detail, as shown in Fig. 4.12a, the considered notches had opening angle equal to 60° and root radius equal to about 0.08 mm. Different ratios between Mode I and II stress components were obtained by machining specimens having angle g (see Fig. 4.12a) equal to 45°, 60° and 90°, respectively. The diagram of Fig. 4.12b clearly proves that also in this case our method was seen to be highly accurate, giving estimates falling within an error interval equal to ±15%. In order to further investigate its accuracy in estimating in-phase fully reversed notch fatigue limits, the MWCM used in conjunction with the PM

Fatigue assessment of notched components 1000

G40.11

E = –20% t0 (MPa)

139

Mild steel (1) AL-2024-T351

Non-conservative

SAE 1045 SM41B

100

AA356-T6 Ni-Cr steel

Conservative

Mild steel (2)

E = 20%

Al-Alloy BS L65 R = –1

10

t0 - a◊reff (MPa)

10

Steel 15313

1000

4.11 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of notched specimens subjected to uniaxial fatigue loading (data reported in Susmel, 2004).

ΔF 350 D = 5 mm D = 3 mm

300

D

60°

ta (MPa)

250 200

R0.08

g 2D

100 50 0 0.0

ΔF

E = 20%

150 E = –20% RCP = –1 0.5

1.0

rlim = 2.48 1.5

2.0

2.5 reff 3.0

(b)

(a)

4.12 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of notched specimens made of En3B and subjected to in-phase Mode I and II loading (data from Susmel and Taylor, 2003a, b).

was applied also to the classical data due to Gough (1949). Again our approach proved to be very successful, accurately predicting the above results generated by testing V-notched samples of different metallic materials under bending and torsion (Fig. 4.13). A similar accuracy level was also obtained when the MWCM was attempted to be applied in terms of the PM to estimate notch high-cycle multiaxial fatigue strength of specimens with fillet tested under in-phase and out-of-phase bending and torsion (Fig. 4.14) with superimposed static stresses (Fig. 4.15).

140

Multiaxial notch fatigue 1000 0.4% C

E = –20% t0 (MPa)

3% Ni Non-conservative

3/3.5% NiCr CrVa 3.5% NiCr (n. i.)

Conservative

3.5% NiCr (l. i.) E = 20%

RCP = –1

NiCrMo

100 100

1000

ta - a◊reff (MPa)

4.13 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of V-notched specimens made of different steels and subjected to in-phase fully reversed bending and torsion (data from Gough, 1949).

350 In-phase

300

90° Out-of-phase

ta (MPa)

250 200 E = 20% 150 100 E = –20%

50

rlim = 3.73

RCP = –1

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

reff

5.0

4.14 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of specimens with fillet made of SAE 1045 (Ck45) and subjected to in-phase and out-of-phase fully reversed bending and torsion (data from Kurath et al., 1989; Sonsino, 1995).

Finally, the diagram reported in Fig. 4.16 summarises the accuracy of our method when applied to the data we generated by testing notched samples of En3B under in-phase and 90° out of-phase tension and torsion, investigating also the detrimental effect of superimposed tensile and torsional static stresses (Susmel and Taylor, 2008). In more detail, the experimental results reported in this diagram were generated by testing V-notched cylindrical specimens having a gross diameter of 8 mm and a net diameter of 5 mm. The opening angle was equal to 60° and three different values of the notch

Fatigue assessment of notched components

141

500 m = 0.41 E = 20%

ta (MPa)

400 300 200 100 0 0.0

E = –20% RCP = –1

RCP = –0.2

RCP = –0.1

RCP = 0

RCP = 0.9

RCP = 1

0.5

1.0

1.5

rlim = 2.35 2.0

2.5 reff 3.0

4.15 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of specimens with fillet made of S65A and subjected to in-phase bending and torsion with superimposed static stresses (data from Gough, 1949). 500 rn = 0.2 mm

ta (MPa)

400 300

rn = 1.25 mm rn = 4 mm

E = 20%

200 E = –20% 100 m = 0.22 0 0.0

0.5

rlim = 1.41 1.0

1.5

2.0

2.5 reff 3.0

4.16 Accuracy of the MWCM applied along with the PM in estimating high-cycle fatigue strength of notched specimens made of En3B and subjected to in-phase and 90° out-of-phase tension and torsion with superimposed static stresses (data from Susmel and Taylor, 2008).

root radius were considered, i.e. 0.2 mm, 1.25 mm and 4 mm. The chart of Fig. 4.16 definitely proves that our approach is highly accurate, correctly estimating high-cycle notch fatigue strength independently of stress raiser sharpness and complexity of the applied loading path. To conclude the present section, it is worth noting that, as shown in Fig. 4.16, the validity of the hypothesis formed to correctly handle those situations characterised by large values of reff is fully supported by the experimental evidence.

4.4.2 Estimating high-cycle fretting fatigue strength Under fretting conditions the material in the vicinity of the contact surface experiences cyclic stress gradients. Taking this fact as a starting point, many

142

Multiaxial notch fatigue

different investigations have shown that fretting fatigue damage can be estimated efficiently by directly using those methods explicitly devised to assess the classical notch fatigue problem (Giannakopoulos et al., 2000; Fouvry et al., 2002; Ciavarella, 2003; Naboulsi and Mall, 2003; Vallelano et al., 2003, 2004). As to the material cracking behaviour in the vicinity of the contact region, it is worth remembering here that the initiation phenomenon and the initial growth are seen to occur under mixed mode loading (Lindley, 1997). For instance, by testing Al 7075-T6 sphere-on-plane contacts, Vallelano et al. (2003, 2004) observed that cracks initially propagated along straight lines at small angles to the surface and subsequently they changed their direction to grow along rectilinear paths which were almost perpendicular to the contact zone surface. The considerations reported above make it evident that, under fretting fatigue loading, the crack initiation phenomenon and the subsequent initial propagation are governed by the cyclic multiaxial stress gradients acting on the process zone: according to the above experimental finding, attempts were then made to use the MWCM reinterpreted in terms of the PM also to estimate high-cycle fretting fatigue damage (Araújo et al., 2007, 2008; Martins et al., 2007). The procedure for the in-field use of our method is briefly summarised in Fig. 4.17. In particular, for the sake of simplicity consider a flat specimen loaded by an axial cyclic force, ΔF. The fretting pads are both pushed against the main plate by a constant force, P, and subjected to an oscillatory tangential force, ΔQ. The focus path emanates from the trailing edge of the contact zone and is perpendicular to the contact surface itself (Fig. 4.17). The MWCM applied along with the PM postulates that the mechanical assembly to be assessed is at its fatigue limit as long as the condition given

ta

ΔF

tA,Ref(reff) L/2

ΔQ

P

r

ΔQ

P

4.17 In-field use of the MWCM applied along with the PM to estimate high-cycle fretting fatigue strength.

Fatigue assessment of notched components

143

150 Failure

125

Run out ta (MPa)

100

E = 20%

75 50

E = –20%

25

rlim = 2.13

m=1 0 0.0

0.5

1.0

1.5

2.0

reff

2.5

4.18 Accuracy of the MWCM in estimating high-cycle fretting fatigue strength (data from Nowell, 1988).

by Eq. (4.12) is assured at a distance from the assumed crack initiation site equal to L/2. As usual, the material characteristic length, L, has to be calculated according to definition (4.9) by using material fatigue properties, i.e. ΔKth and Δs0, experimentally determined under fully reversed loading. The diagram reported in Fig. 4.18 clearly shows that our method proved to be capable of predicting run-out fretting tests with its usual level of accuracy. In more detail, the data reported in this chart were generated by testing cylinder-on-plane contact configurations (Nowell, 1988). Both the fretting pads and the main plates were made of Al 4%Cu, a commercial aluminium alloy. Different loading configurations were considered by varying the pad radius from 12.5 mm up to 150 mm. Figure 4.17 makes it evident that the MWCM applied along with the PM is highly accurate in estimating high-cycle fretting fatigue strength, giving estimates falling within an error interval equal to about ±20%.

4.4.3 Estimating finite life When mechanical components subjected to cyclic loading are weakened by notches and defects, the MWCM can be used in conjunction with the PM to predict not only high-cycle fatigue strength but also finite life. In more detail, to correctly estimate fatigue damage in the medium-cycle fatigue regime, initially the MWCM has to be calibrated by using pieces of experimental information generated by testing plain samples. In other words, constants a, b, a, b, m and rlim have to be determined by following the procedures already discussed in Chapter 3 when reviewing the fundamentals of our multiaxial fatigue criterion. The use of the MWCM applied along with the PM takes as its starting point the idea that reliable estimates can be obtained by adopting a material

144

Multiaxial notch fatigue

characteristic length, LM, whose value depends on the number of cycles to failure, Nf, so that the MWCM is applied by considering the linear-elastic stress state at a material point whose distance from the stress raiser apex increases as Nf decreases. Moreover, according to the TCD’s philosophy, the hypothesis is formed that the LM vs. Nf relationship to be used to estimate finite life is a material property. Hence, for a given material, the critical distance value does not depend on the geometry of the stress concentrator to be assessed (Susmel and Taylor, 2007b, 2008). Similarly to the uniaxial case (see Section 2.7.1), the LM vs. Nf relationship is suggested to be formalised by using the following power law (Susmel and Taylor, 2007b): LM( N f ) = A ⋅ N fB

4.13

where A and B are again two constants depending on the fatigue properties of the assessed material. Strictly speaking, the above relationship could be calibrated by using the characteristic lengths valid for high-cycle fatigue, L, and static, LS, problems, i.e. Eqs (4.10) and (2.40), respectively. The above remark should make it evident that, being both L and LS material properties, LM is in turn a material fatigue property. Unfortunately, as already discussed in Section 2.7.2, constants A and B are very difficult to estimate by using L and LS due to the well-known inadequacy of the stress based approach at describing the fatigue behaviour of engineering materials in the low-cycle fatigue regime, as well as because the number of cycles to failure locating the position of the fatigue curve knee points varies as both the geometry of the tested geometrical feature and the degree of multiaxiality of the considered loading path change. Owing to the difficulties mentioned above, constants A and B in relationship (4.13) are then suggested to be estimated by adopting the alternative procedure already explained in Section 2.7.2, based on the use of two calibration fatigue curves (Susmel and Taylor, 2007b, 2008). It is important to point out here also that constants A and B in the LM vs. Nf relationship have to be determined by using experimental fatigue results that are always generated under fully reversed loading, because the presence of superimposed static stresses is assumed to be taken into account directly by the MWCM itself. Finally, it has to be said that the values of the above constants are assumed to be independent of the degree of multiaxiality of the stress field acting on the fatigue process zone (Susmel and Taylor, 2008). The way of using the MWCM applied along with the PM to estimate lifetime under cyclic loading is shown by the schematic sketch reported in Fig. 4.19. In more detail, initially the linear-elastic stress distribution has to be calculated along the focus path, where such a path is a straight line emanating from the assumed crack initiation point, A, and perpendicular

Fatigue assessment of notched components

ta

145

reff = rlim

Fi(t)

B

A · Nf −r 2 reff

Fj(t)

B

A · Nf,e 2

A

A

Focus path r

=r r

LM(Nf,e) 2

Fk(t) (b) (a)

Nf,e

4.19 In-field use of the MWCM reinterpreted in terms of the PM to predict finite life of notched components subjected to fatigue loading.

to the surface at the hot-spot itself (Fig. 4.19a). It is important to highlight here that, as in the high-cycle fatigue regime, it is suggested that the focus path is taken to be coincident with the notch bisector when the notch root radius of the stress raiser to be assessed is assumed to be equal to zero (Susmel, 2008b). Subsequently, the maximum shear stress amplitude, ta, and the corresponding value of ratio reff have to be determined along the above focus path, obtaining the ta vs. r and reff vs. r curves schematically sketched in Fig. 4.19b. At any distance r from the stress concentrator apex, and according to the calculated values for ta and reff, the corresponding modified Wöhler curve can now be estimated by using the kt vs. reff and tA,Ref vs. reff relationships previously calibrated through the parent material fatigue properties. The number of cycles to failure, Nf, can then be calculated directly at any point belonging to the focus path. Subsequently, by using the estimated values for Nf, the material characteristic length LM has to be determined by means of Eq. (4.13) for any value of distance r. Finally, our method postulates that the assessed component fails at the number of cycles to failure, Nf,e, assuring the following condition (Fig. 4.19b): LM A ⋅ N fB,e =r⇒ =r 2 2

4.14

The error diagrams reported in Figs 4.20, 4.21 and 4.22 show the accuracy of the MWCM used in conjunction with the PM in estimating fatigue lifetime of notched specimens failing in the medium-cycle fatigue regime (the values of the fatigue constants of the three considered materials are listed in Table 4.1).

146

Multiaxial notch fatigue 10 000 000 Nf (cycles) 1 000 000

100 000

Hole 8mm, TC, R = –1 Hole 8mm, TT, R = 0.1 Hole 3.5mm, TC, R = –1 Hole 3.5mm, TT, R = 0.1 U-Notch, TC, R = –1 U-Notch, TT, R = 0.1 V-Notched, TC, R = –1 V-Notched, TT, R = 0.1 U-Notched, 3PB, R = 0.1 V-Notched, 3PB, R = 0.1

PS = 5%

Conservative PS = 95%

Non-conservative

10 000 Fully-reversed plain Uniaxial scatter band 1000 10 000

100 000

1 000 000

10 000 000 100 000 000 Nf,e (cycles)

4.20 Accuracy of the MWCM applied along with the PM in estimating fatigue lifetime of notched specimens made of En3B (Batch 1) and loaded in tension–compression (TC), tension–tension (TT) and threepoint bending (3PB) (data from Susmel and Taylor, 2007b).

100 000 000 Nf (cycles)

Torsional scatter band Uniaxial scatter band

100 000

0.2mm, In-Phase, R = –1 0.2mm, Out-of-Phase, R = –1

Conservative

1 000 000

0.2mm, In-Phase, R = 0 0.2mm, Out-of-Phase, R = 0 1.25mm, In-Phase, R = –1 1.25mm, Out-of-Phase, R = –1

PS = 95% PS = 5%

10 000 Non-conservative 1000 1000

10 000

1.25mm, In-Phase, R = 0 1.25mm, Out-of-Phase, R = 0 4mm, In-Phase, R = –1 4mm, Out-of-Phase, R = –1 4mm, In-Phase, R = 0 4mm, Out-of-Phase, R = 0

100 000 1 000 000 10 000 000 100 000 000 Nf,e (cycles)

4.21 Accuracy of the MWCM applied along with the PM in estimating fatigue lifetime of notched specimens made of En3B (Batch 2) and subjected to in-phase and 90° out-of-phase tension and torsion with superimposed static stresses (data from Susmel and Taylor, 2008).

Fatigue assessment of notched components

147

100 000 000 Uniaxial scatter band Torsional scatter band

Nf (cycles)

Conservative PS = 5%

1 000 000

Torsion, R = –1

100 000

PS = 95%

In-Phase, R = –1 90° Out-of-Phase, R = –1 In-Phase, R = 0

10 000

1000 1000

Nonconservative 10 000

90° Out-of-Phase, R = 0 Fillet (Torsion), R = –1

100 000 1 000 000 10 000 000 100 000 000 Nf,e (cycles)

4.22 Accuracy of the MWCM applied along with the PM in estimating fatigue lifetime of notched specimens made of C40 steel and subjected to in-phase and 90° out-of-phase tension and torsion with superimposed static stresses (data from Atzori et al., 2006).

Table 4.1 Fatigue constants of the considered materials Material

B

A (mm/cycleB)

a

b

a (MPa)

b (MPa)

m

rlim

En3B (Batch 1) En3B (Batch 2) C40 Steel

−0.342 −0.565 −0.345

67.4 118.9 48.7

−5.9 1.0 −1.2

15.3 18.7 17.5

−42.2 −95.3 −63.3

141.0 268.3 194.3

0.52 0.22 1.00

1.669 1.407 1.534

In more detail, the chart of Fig. 4.20 reports the results we generated by testing notched samples of En3B (batch 1) loaded in tension–compression (R = −1), in tension–tension (R = 0.1) and in three-point bending (R = 0.1). The specimens tested under uniaxial cyclic loading had thickness equal to 6 mm whereas it was equal to 25 mm in the three-point bending samples (Susmel and Taylor, 2007b). It is worth noting here also that the LM vs. Nf relationship of the above material was calibrated by using plain and sharply V-notched samples having thickness equal to 6 mm and tested under fully reversed uniaxial fatigue loading. Figure 4.20 clearly proves that our method was seen to be highly accurate in predicting finite life, correctly taking into account not only the presence of stress concentration phenomena arising from notches but also the damaging effect of stress gradients due to the applied nominal loading.

148

Multiaxial notch fatigue

Figure 4.21 instead summarises the experimental results we generated by testing V-notched cylindrical samples of En3B (batch 2) under in-phase and 90° out-of-phase tension and torsion, considering also the effect of superimposed static stresses (Susmel and Taylor, 2008). The tested samples had gross and net diameter equal to 8 mm and 5 mm, respectively, whereas the investigated V-notches had opening angle equal to 60°. Three different values of the notch root radius were considered, i.e. 0.2 mm, 1.25 mm and 4 mm. This chart clearly proves that our method is successful in predicting fatigue strength in the medium-cycle fatigue regime, allowing the estimates to fall within the widest scatter band between the two used to calibrate the MWCM itself. Finally, the accuracy and reliability of our fatigue life estimation technique is fully confirmed also by the diagram reported in Fig. 4.22 summarising the data generated by Atzori et al. (2006) by testing V-notched samples and specimens with fillet (both made of C40 steel) under in-phase and 90° out-of-phase tension and torsion. The tests were run investigating two different values of the load ratio, R, i.e., −1 and 0, and the two considered geometrical features had root radius equal to 0.5 mm. Again estimates are seen to fall within the widest scatter band between the two used to calibrate the MWCM’s governing equations.

4.4.4 Mesh size in finite element models to apply the MWCM Owing to its nature, our method can be used successfully to estimate fatigue damage in real components by directly post-processing simple linear-elastic FE models, provided that both stress raisers’ shape and size of the mesh used to determine the stress fields acting on the fatigue process zones are properly defined. As to the above aspects, the main outcomes of the investigation carried out by Chaves and Taylor (2002) by considering the uniaxial sub-case can be extended directly to those situations involving multiaxial fatigue loading. In more detail, Chaves and Taylor addressed the above problem by taking as a starting point the following two aspects: •

•

Component geometry often has to be poorly modelled due to the lack of information. In particular, it is common practice to replace notches having finite root radius (rn ≠ 0) with zero-radius geometrical features (rn = 0). When the mechanical components to be assessed are very large, the stress fields in the vicinity of crack initiation sites are often determined by using a coarse mesh.

Fatigue assessment of notched components

149

According to the above schematisation, Chaves and Taylor then investigated two different categories of FE models, that is, ‘zero radius’ and ‘coarse mesh’ models, coming to the following conclusions: •

•

Using notch tip radii equal to zero is seen to result in conservative estimates, with an error level always lower than 30%, only when the rn to L ratio is less than 5. On the contrary, when the above ratio becomes larger than 5, simplified models give too-conservative predictions. When complex geometries are modelled by using coarse mesh, predictions are accurate and conservative only when the ratios between mesh size and rn and between mesh size and L are lower than unity. On the contrary, when the above two ratios become larger than 1, estimates are non-conservative and inaccurate.

According to our experience, the above simple practical rules represent very useful guidelines, which should always be borne in mind when the linear-elastic stress fields needed to perform the fatigue assessment according to the engineering method reviewed in the present chapter are calculated by means of linear-elastic FE models.

4.5

References

Araújo, J. A., Susmel, L., Taylor, D., Ferro, J. C. T., Mamiya, E. N. (2007) On the use of the theory of critical distances and the modified Wöhler curve method to estimate fretting fatigue strength of cylindrical contacts. International Journal of Fatigue 29, 95–107. DOI: 10.1016/j.ijfatigue.2006.02.041. Araújo, J. A., Susmel, L., Taylor, D., Ferro, J. C. T., Ferreira, J. L. A. (2008) On the prediction of high-cycle fretting fatigue strength: theory of critical distances vs. hot spot approach. Engineering Fracture Mechanics 75 (7), 1763–1778. DOI: 10.1016/j.engfracmech.2007.03.026. Atzori, B., Berto, F., Lazzarin, P., Quaresimin, M. (2006) Multi-axial fatigue behaviour of a severely notched carbon steel. International Journal of Fatigue 28, 485–493. DOI: 10.1016/j.ijfatigue.2005.05.010. Chaves, V., Taylor, D. (2002) Use of simplified models in fatigue predictions of real components. In: Proceedings of Fatigue 2002, Vol. 5, edited by A. F. Blom, Engineering Materials Advisory Services, 2799–2807. Ciavarella, M. (2003) A ‘crack-like’ notch analogue for a safe-life fretting fatigue design methodology. Fatigue and Fracture of Engineering Materials and Structures 26, 1159–1170. DOI: 10.1046/j.1460-2695.2003.00721.x. Fouvry, S., Elleuch, K., Simeor, G. (2002), Prediction of crack nucleation under partial slip fretting conditions. Journal of Strain Analysis for Engineering Components 37 (6), 549–564. DOI: 10.1243/030932402320950152. Frith, P. H. (1956) Fatigue of wrought high-tensile alloy steel. Proceedings of the Institution of Mechanical Engineers, 462–499. Frost, N. E. (1959) Non-propagating cracks in V-notched specimens subjected to fatigue loading. Aeronautical Quarterly VIII, 1–20.

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Giannakopoulos, A. E., Suresh, S., Lindley, T. C., Chenut, C. (2000) Similarities of stress concentrations in contact at round punches and fatigue at notches: implications to fretting fatigue crack initiation. Fatigue and Fracture of Engineering Materials and Structures 23, 561–571. DOI: 10.1046/j.1460-2695.2000.00306.x. Gough, H. J. (1949) Engineering steels under combined cyclic and static stresses. Proceedings of the Institution of Mechanical Engineers 160, 417–440. Kurath, P., Downing, S. D., Galliart, D. R. (1989) Summary of non-hardened notched shaft – round robin program. In: Multiaxial Fatigue – Analysis and Experiments, edited by G. E. Leese and D. F. Socie, SAE AE-14, Society of Automotive Engineers Warrendale, PA, 13–32. Lazzarin, P., Susmel, L. (2003) A stress-based method to predict lifetime under multiaxial fatigue loadings. Fatigue and Fracture of Engineering Materials and Structures 26, 1171–1187. DOI: 10.1046/j.1460-2695.2003.00723.x. Lindley, T. C. (1997) Fretting fatigue in engineering alloys. International Journal of Fatigue 19, 39–49. DOI: 10.1016/S0142-1123(97)00039-X. Martins, L. H. L., Ferro, J. C. T., Ferreira, J. L. A., Araújo, J. A., Susmel, L. (2007) A notch methodology to estimate fretting fatigue strength. Journal of the Brazilian Society of Mechanical Sciences and Engineering XXIX, 76–84. Meneghetti, G., Susmel, L., Tovo, R. (2007) High-cycle fatigue crack paths in specimens having different stress concentration features. Engineering Failure Analyses 14, 656–672. DOI: 10.1016/j.engfailanal.2006.02.004. Naboulsi, S., Mall, S. (2003) Fretting fatigue crack initiation behaviour using process volume approach and finite element analysis. Tribology International 36, 121–131. DOI: 10.1016/S0301-679X(02)00139-1. Nowell, D. (1988) An analysis of fretting fatigue, D.Phil Thesis, Oxford University. Peterson, R. E. (1959) Notch sensitivity. In: Metal Fatigue, edited by G. Sines and J. L. Waisman, McGraw-Hill, New York, 293–306. Peterson, R. E. (1974) Stress Concentration Factors. John Wiley & Sons, New York, USA. Sonsino, C. M. (1995) Multiaxial fatigue of welded joints under in-phase and out-of-phase local strain and stresses. International Journal of Fatigue 17, 55–70. DOI: 10.1016/0142-1123(95)93051-3. Susmel, L. (2004) A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings. Fatigue and Fracture of Engineering Materials and Structures 27, 391–411. DOI: 10.1111/j.1460-2695.2004.00759.x. Susmel, L. (2006) On the use of the modified Wöhler curve method to estimate notch fatigue limits. Materialprüfung 48 (1–2), 27–35. Susmel, L. (2008a) Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes. Fatigue and Fracture of Engineering Materials and Structures 31, 295–309. DOI: 10.1111/j.1460-2695.2008.01228.x. Susmel, L. (2008b) Modified Wöhler curve method, Theory of Critical Distances and EUROCODE 3: a novel engineering procedure to predict the lifetime of steel welded joints subjected to both uniaxial and multiaxial fatigue loading. International Journal of Fatigue 30, 888–907. DOI: 10.1016/j.ijfatigue.2007.06.005. Susmel, L., Lazzarin, P. (2002) A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures 25, 63–78. DOI: 10.1046/j.1460-2695.2002.00462.x.

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Susmel, L., Taylor, D. (2003a) Two methods for predicting the multiaxial fatigue limits of sharp notches. Fatigue and Fracture of Engineering Materials and Structures 26, 821–833. DOI: 10.1046/j.1460-2695.2003.00683.x. Susmel, L., Taylor, D. (2003b) Fatigue design in the presence of stress concentrations. International Journal of Strain Analysis for Engineering Components 38 (5), 443–452. DOI: 10.1243/03093240360713496. Susmel, L., Taylor, D. (2006) Can the conventional high-cycle multiaxial fatigue criteria be re-interpreted in terms of the Theory of Critical Distances? Structural Durability and Health Monitoring 2 (2), 91–108. Susmel, L., Taylor, D. (2007a) Non-propagating cracks and high-cycle fatigue failures in sharply notched specimens under in-phase Mode I and II loading. Engineering Failure Analyses 14, 861–876. DOI: 10.1016/j.engfailanal.2006.11.038. Susmel, L., Taylor, D. (2007b) A novel formulation of the Theory of Critical Distances to estimate lifetime of notched components in the medium-cycle fatigue regime. Fatigue and Fracture of Engineering Materials and Structures 30, 567–581. DOI: 10.1111/j.1460-2695.2007.01122.x. Susmel, L., Taylor, D. (2008) The Modified Wöhler Curve Method applied along with the Theory of Critical Distances to estimate finite life of notched components subjected to complex multiaxial loading paths. Fatigue and Fracture of Engineering Materials and Structures 31 (12), 1047–1064. DOI: 10.1111/j.1460-2695.2008. 01296.x. Susmel, L., Tovo, R., Lazzarin, P. (2005) The mean stress effect on the high-cycle fatigue strength from a multiaxial fatigue point of view. International Journal of Fatigue 27, 928–943. DOI: 10.1016/j.ijfatigue.2004.11.012. Tanaka, K. (1983) Engineering formulae for fatigue strength reduction due to crack-like notches. International Journal of Fracture 22, R39–R45. DOI: 10.1007/ BF00942722. Taylor, D. (1999) Geometrical effects in fatigue: a unifying theoretical model. International Journal of Fatigue 21, 413–420. DOI: 10.1016/S0142-1123(99) 00007-9. Taylor, D. (2001) A mechanistic approach to critical-distance methods in notch fatigue. Fatigue and Fracture of Engineering Materials and Structures 24, 215–224. DOI: 10.1046/j.1460-2695.2001.00401.x. Taylor, D. (2007) The Theory of Critical Distances: A New Perspective in Fracture Mechanics. Elsevier Science. Taylor, D., Wang, G. (2000) The validation of some methods of notch fatigue analysis. Fatigue and Fracture of Engineering Materials and Structures 23, 387–394. DOI: 10.1046/j.1460-2695.2000.00302.x. Tipton, S. M., Nelson, D. V. (1997) Advances in multiaxial life prediction for components with stress concentrator. International Journal of Fatigue 19, 503–515. DOI: 10.1016/S0142-1123(96)00070-9. Vallelano, C., Dominguez, J., Navarro, A. (2003) On the estimation of fatigue failure under fretting conditions using notch methodologies. Fatigue and Fracture of Engineering Materials and Structures 26, 469–478. DOI: 10.1046/j.1460-2695. 2003.00649.x. Vallelano, C., Dominguez, J., Navarro, A. (2004) Predicting the fretting fatigue limit for spherical contact. Engineering Failure Analysis 11, 727–736. DOI: 10.1016/j. engfailanal.2003.09.002.

5 Multiaxial fatigue assessment of welded structures

Abstract: The present chapter reviews the different ways of using the Modified Wöhler Curve Method (MWCM), applied in terms of nominal, hot-spot and local stresses, to estimate fatigue damage in welded details made of both steel and aluminium and subjected to constant amplitude fatigue loading. Key words: Modified Wöhler Curve Method, weldments, nominal stresses, hot-spot stresses, local stresses.

5.1

Introduction

Engineers engaged in assessing real structures can take full advantage of many approaches which have specifically been devised to estimate fatigue damage in welded details subjected to fatigue loading – see, for instance, Hobbacher (2007) and Radaj et al. (2007) for an up-to-date review of the state of the art. Among them, it is recognised that the simplest way to perform the fatigue assessment of weldments is by using nominal stresses, where fatigue strength is directly estimated from the specific S–N curve supplied, for the type of considered welded detail, by the available standard codes. On the contrary, when either nominal stresses cannot directly be calculated or the standard fatigue curve for the geometry of the welded detail to be assessed is not available, then either a structural or a local stress based approach are always recommended to be used (Niemi, 1995; Radaj, 1996; Hobbacher, 1996). In other words, the first problem to be addressed when designing welded connections against fatigue is the choice of the most suitable stress analysis to be performed to calculate a stress quantity which can be used to estimate fatigue strength efficiently. The state of the art shows that the most modern methodologies specifically devised to perform the fatigue assessment of welded connections are based on the use of local parameters. In particular, methods as the N-SIF approach (Lazzarin and Tovo, 1998; Lazzarin and Livieri, 2001; Livieri and Lazzarin, 2005), the strain energy density (SED) parameter (Lazzarin and Zambardi, 2001; Livieri and Lazzarin, 2005), the fictitious notch radius approach (Morgenstern et al., 2006; Radaj et al., 2007), the Theory of Critical Distances (Taylor et al., 2002; Crupi et al., 2005) and the criteria based on the 152

Multiaxial fatigue assessment of welded structures

153

LEFM concepts (Maddox, 1987) proved to be highly accurate, allowing also the scale effect in weldment fatigue to be taken into account correctly. The accuracy of the methods mentioned above was checked considering mainly uniaxial fatigue situations. On the contrary, only a few attempts have also been made so far to extend the use of such strategies to those situations involving complex fatigue loading paths. As to the multiaxial fatigue behaviour of welded connections, it is useful to remember here that under proportional loading all the classical approaches (like, for instance, Von Mises’ criterion, the maximum principal stress criterion, the maximum shear stress criterion, etc.) were seen to be capable of estimating fatigue damage always with an adequate accuracy level (Sonsino and Kueppers, 2001; Bäckström and Marquis, 2001). On the contrary, in the presence of non-proportional loading, the classical stress quantities were seen to result in non-conservative estimates, especially when steel welded details are considered. In order to correctly take into account the degree of non-proportionality of the applied loading, a few methods have been formalised and validated and, apart from those stated by some design codes and based on the use of nominal stresses, the most successful multiaxial fatigue criteria were seen to be those post-processing either local or structural quantities. For instance, Sonsino (Sonsino, 1995; Sonsino and Kueppers, 2001) suggested using the effective shear stress applied along with the fictitious radius idea. Bäckström and Marquis instead proposed to use Findley’s criterion together with the hot-spot concept to calculate the relevant stresses relative to the critical plane (Bäckström and Marquis, 2001). Finally, the theoretical work done by Lazzarin and co-workers deserves to be mentioned: by reanalysing a large amount of experimental results generated by testing both aluminium and steel welded details they showed that accurate predictions can be made by applying both the N-SIF approach (Lazzarin et al., 2004) and the SED parameter (Lazzarin et al., 2008). Owing to the important role played by weldments in practical applications, in recent years we have made a big effort in order to check the accuracy of the MWCM when used to assess welded connections subjected to complex loading paths. The present chapter then reviews the different ways of applying our multiaxial fatigue criterion to estimate fatigue damage in weldments made of both steel and aluminium and subjected to constant amplitude fatigue loading.

5.2

Stress quantities used to assess welded structures

Owing to its nature, the MWCM can be used to estimate fatigue strength of welded components subjected to multiaxial cyclic loading by applying it

154

Multiaxial notch fatigue

in terms of nominal, hot-spot and local stresses (Susmel and Tovo, 2004, 2006; Susmel, 2007, 2008, 2009). Before considering in depth the different strategies which can be followed to use our multiaxial fatigue criterion to perform the fatigue assessment of welded components, it is useful to review here the different ways of determining the stress quantities commonly employed to estimate fatigue damage in weldments. In more detail, nominal stresses have to be calculated by using the classical continuum mechanics theory, without taking into account the stress concentration phenomena due to the weld toe. All the calculations have to be done by assuming an overall elastic behaviour of the material to be assessed, and the effects of macro-geometrical features as well as the presence of concentrated loadings must always be taken into account. Moreover, it is worth remembering here also that the presence of angular misalignments, when larger than the maximum amount which is considered to be acceptable according to the followed standard code, have to be included in the stress analysis. When conventional structural details are assessed, all the available standard codes clearly indicate the nominal reference sections to be adopted for the calculations. On the contrary, in the presence of hyperstatic structures or when no analytical solutions are available for determining nominal stresses in welded components containing non-conventional geometrical features, the stress analysis can be performed by using simple ‘coarse mesh’ FE models (Hobbacher, 2007). Same examples showing the position of the reference sections to be used to calculate nominal stresses are sketched in Fig. 5.1. The use of the hot-spot approach instead is recommended either when the geometry of the welded detail to be assessed is not stated by the available standard codes or when a nominal section cannot be unequivocally defined due to the shape of the connection itself. By definition, structural

Nominal sections

5.1 Nominal sections in standard welded details.

Multiaxial fatigue assessment of welded structures

155

Stress Geometrical stress sHS

Stress distribution on the surface Hot-spot

Reference points

5.2 Definition of geometrical hot-spot stress.

stresses take into account all the stress concentration phenomena acting on the fatigue process zone excluding the contribution due to the local weld profile (Hobbacher, 2007). In general, the use of the hot-spot approach is not recommended for those situations in which cracks initiate at weld roots, unless other hypotheses are formed in order to correctly address the above problem (Hobbacher, 2007). Geometrical stresses can be determined either by post-processing linearelastic FE models or by directly measuring local strains through strain gauges. Figure 5.2 summarises the basic idea on which the calculation of structural stresses is based: the linear-elastic stress has to be determined on the component surface at either two or three reference points and the structural stress, sHS, is extrapolated to the weld toe at the hot spot. The positions of the reference points depend on the main plate thickness, t, whereas, if the stress distribution is independent of t, then the reference points are taken at given distances from the weld toe itself (Hobbacher, 2007). Correctly calculating structural hot-spot stresses is a tricky problem which has to be addressed properly, so that the reader is referred to those books explaining in detail the proper way to determine the above stress quantities – see, for instance, Niemi (1995), Hobbacher (2007) and Radaj et al. (2007). Another efficient stress-based method suitable for assessing welded connections subjected to fatigue loading which deserves to briefly be reviewed here is the so-called N-SIF approach (Lazzarin and Tovo, 1998). Unfortunately, such a method is not yet included among those suggested by the available standard codes, but because of its well-known accuracy it will be used in Section 5.6 to calculate a critical distance value suitable for

156

Multiaxial notch fatigue

Opening angle

Weld bead

sq t rq

r

q

sr Notch bisector L z Δsnom

Δsnom t

5.3 Cruciform welded joint subjected to uniaxial fatigue loading and definition of the local frame of reference.

using the MWCM in conjunction with the Point Method (PM) to estimate fatigue strength of welded details subjected to both uniaxial and multiaxial fatigue loading. In order to understand the main features of the N-SIF approach, consider the cruciform welded joint sketched in Fig. 5.3, where the weld toe radius is assumed to be equal to zero (Lazzarin and Tovo, 1998). According to the frame of reference defined in this figure, under nominal uniaxial fatigue loading the linear-elastic stress field in the vicinity of the weld toe can be efficiently described through Eqs (1.71) and (1.72) rewritten in terms of ranges (Lazzarin and Tovo, 1996): ⎧Δσ θ ⎫ ⎪ ⎪ ⎨Δσ r ⎬ = ⎪⎩Δτ ⎪⎭ rθ

r λ1 − 1 ΔK1 ⋅ 2 π (1 + λ1 ) + χ 1 (1 − λ1 ) 1

⎡ ⎧( 1 + λ1) cos ( 1 − λ1 ) θ ⎫ ⎧cos (1 + λ1 ) θ ⎫⎤ ⎢ ⎪⎨( 3 − λ1 ) cos ( 1 − λ1) θ ⎪⎬ + χ 1 (1 − λ1 ) ⎪⎨− cos (1 + λ1 ) θ ⎪⎬⎥ ⎥ ⎢⎪ ⎪⎩sin (1 + λ ) θ ⎪⎭⎥⎦ ⎢⎣ ⎩( 1 − λ ) sin ( 1 − λ ) θ ⎪⎭ 1

1

1

5.1

Multiaxial fatigue assessment of welded structures ⎧Δσ θ ⎫ ⎪ ⎪ ⎨Δσ r ⎬ = ⎪⎩Δτ ⎪⎭ rθ

157

1 r λ2 −1ΔK 2 ⋅ 2 π (1 − λ 2 ) + χ 2 (1 + λ 2 ) 5.2

⎡ ⎧− ( 1 + λ 2) sin ( 1 − λ 2 )θ ⎫ ⎧− sin (1 + λ 2 )θ ⎫⎤ ⎢ ⎪⎨− ( 3 − λ 2 ) sin ( 1 − λ 2)θ ⎪⎬ + χ 2 (1 + λ 2 ) ⎪⎨sin (1 + λ 2 )θ ⎪⎬⎥ ⎥ ⎢⎪ ⎪⎩cos (1 + λ )θ ⎪⎭⎥⎦ ⎢⎣ ⎩( 1 − λ 2 ) cos ( 1 − λ 2 )θ ⎪⎭ 2

By taking as a starting point the fact that Mode II stress components are no longer singular when notch opening angles become larger than about 100°, Lazzarin and Tovo (1998) argued that fatigue strength in the presence of weld bead opening angles equal to 135° can be estimated efficiently in terms of Mode I N-SIF range, ΔKI – see Eqs (1.74) for the definition of KI – because such a stress parameter is capable of describing in a very concise and effective way the entire singular stress field acting on the process zone. According to the above intuition, they suggested performing the fatigue assessment of conventional welded joints by directly using the two unifying reference fatigue curves reported in Fig. 5.4 (Lazzarin and Tovo, 1996; Lazzarin and Livieri, 2001; Livieri and Lazzarin, 2005). To conclude, it is worth noting also that the N-SIF approach proved to be highly accurate in accounting for the scale effect in weldment fatigue, confirming that simple linear-elastic stress quantities can be used successfully to estimate fatigue damage in welded connections, with the advantage over other existing methods that complex and time-consuming elasto-plastic analyses do not have to be carried out. 10 000 DΚI (MPa mm0.326)

R≈0

100

PS = 97.7%

Steel

PS = 2.3% PS = 97.7%

PS = 2.3%

Aluminium From the statistical reanalysis: Steel: k = 3.0; ΔΚI,50% = 211 MPa mm0.326 (at 5·106 cycles to failure) Aluminium: k = 4.0; ΔKI,50% = 99 MPa mm0.326 (at 5·106 cycles to failure)

10 10 000

100 000

1 000 000

Nf (cycles)

10 000 000

5.4 Reference fatigue curves for steel and aluminium cruciform joints determined according to the N-SIF approach (data from Livieri and Lazzarin, 2005).

158

Multiaxial notch fatigue

5.3

Preliminary assumptions

In order to correctly use the MWCM to perform the fatigue assessment of weldments, our criterion has to be reformulated to fully comply with the recommendations of the pertinent standard codes (Anon, 1988, 1999; Hobbacher, 2007). In particular, the most tricky problem is correctly incorporating in the method itself the influence of non-zero mean stresses. As to this aspect, many experimental investigations suggest that when as-welded connections are subjected to uniaxial cyclic loading, the presence of superimposed static stresses plays a secondary role in the overall fatigue strength of welded details. This experimental evidence is to be ascribed to the effect of the residual stresses arising from the welding process: such stress components change the actual value of the load ratio in the vicinity of crack initiation sites, so that, under high tensile residual stresses, the local value of R can become larger than zero also when the nominal load ratio approaches −1. This is the reason why joints in as-welded condition can be assessed efficiently by simply using reference fatigue curves generated under R ratios larger than zero, and this holds true independently of the magnitude of the superimposed static stresses applied to the connection itself. The detrimental effect of residual stresses can be efficiently reduced by simply relieving the material within the fatigue process zone by means of appropriate technological processes. In so doing, the fatigue strength of welded joints increases, but at the same time they become much more sensitive to the presence of superimposed static stresses: this is the reason why, to efficiently take into account the presence of non-zero mean stresses in stress-relieved welded joints, standard codes state some procedures suitable for correcting the supplied reference fatigue curves. However, as to the effect of load ratios lower than zero, it has to be said that, in practice, fatigue endurance of welded specimens tested in the laboratory is seen to be influenced much more by the presence of superimposed static stresses than real structures are (Atzori, 1983); therefore it is reasonable to assume that the mean stress effect can be disregarded when assessing mechanical components in situations of practical interest (at least, during the pre-design process). In other words, a conservative fatigue assessment can be performed without taking into account the influence of the load ratio, provided that fatigue damage is estimated by using design curves which already account for the detrimental effect of high-tensile mean or high-tensile residual stresses. At the same time, it has to be pointed out that, according to one’s own in-field experience, one can follow the recommendations of the pertinent standard codes in order to estimate fatigue damage more efficiently by taking into account also the effect, in stress-relieved weldments, of R ratios lower than zero.

Multiaxial fatigue assessment of welded structures

159

In the present chapter, then, the MWCM will be formalised by disregarding the influence of superimposed static stresses and will be calibrated through fatigue curves suitable for estimating fatigue damage in as-welded connections, simply because this represents the safest way of using our approach to assess real welded components: it is clear that other assumptions can be made in order to estimate more accurately fatigue damage in weldments subjected to cyclic loading, but this is left to the in-field experience of the reader. According to the above considerations, welded materials can then be assumed to be characterised by a mean stress sensitivity index, m, equal to zero, so that the stress ratio relative to the critical plane can be rewritten as follows (Susmel and Tovo, 2004):

ρw =

Δσ n Δτ

5.3

where Δt = 2ta is the maximum shear stress range and Δsn is the range of the stress perpendicular to the critical plane. To conclude, it can be noted that, as clearly shown by identity (5.3) and in accordance with the symbolism adopted by the available standard codes, in the next sections the problem of performing the multiaxial fatigue assessment of welded structures will be addressed in terms of ranges and not in terms of amplitudes as was done in the previous chapter when considering conventional notches.

5.4

MWCM and nominal stresses

In order to explain how to apply the MWCM in terms of nominal stresses to estimate multiaxial fatigue damage in welded connections, consider a tube-to-plate joint loaded in combined tension and torsion (Fig. 5.5a). According to the geometry and the absolute dimensions of the connection to be assessed, initially the appropriate uniaxial and torsional reference fatigue curves have to be selected among those stated by the pertinent standard codes. Such curves can then be used to directly calculate constants an, bn, an and bn in Eqs (4.3) and (4.4), obtaining: kτ ( ρw ) = [ kτ ( ρw = 1) − kτ ( ρw = 0)] ρw + kτ ( ρw = 0)

5.4

Δσ A Δτ A,Ref ( ρw ) = ⎛ − Δτ A ⎞ ρw + Δτ A ⎝ 2 ⎠

5.5

In the above relationships kt (rw = 1) and kt (rw = 0) are the negative inverse slopes of the uniaxial and torsional fatigue curve, respectively, whereas ΔsA and ΔtA are the ranges of the corresponding reference stresses extrapolated

160

Multiaxial notch fatigue txy(t) 0

sx(t) [s(t)] = txy(t) 0

0

0

0

0

Δt, rw Mt(t) (a) F(t) Nominal section

ΔtA,Ref(rw), kt(rw)

Δt

Δt 1 kt(rw)

ΔtA,Ref(rw)

(b) Nf

NA Cycles to failure

5.5 In-field use of the MWCM applied in terms of nominal stresses to estimate fatigue strength of welded connections subjected to multiaxial fatigue loading.

at NA cycles to failure. Subsequently, according to the calculated value for rw, the corresponding modified Wöhler curve has to be determined through Eqs (5.4) and (5.5). Finally, the number of cycles to failure can be estimated directly by using the following well-known relationship (Fig. 5.5b): N f,e = N A ⎡ ⎢⎣

Δτ A,ref ( ρ w ) ⎤ ⎥⎦ Δτ

kt ( ρ w )

5.6

It is useful to remember here that, as already explained in Section 4.3, all the stress quantities relative to the critical plane needed to perform the fatigue assessment must always be determined by directly post-processing the stresses calculated with respect to the nominal section. Similarly to what was done when using the MWCM to estimate fatigue strength of conventional notched components, Eqs (5.4) and (5.5) can also be corrected in order to manage more efficiently those situations characterised by large values of ratio rw. Unfortunately, because nominal stresses are poorly related to the physical processes leading to the crack initiation phenomenon, a reliable value for rlim is never easy to be estimated. In any case, as a general guideline and according to our in-field experience, we

Multiaxial fatigue assessment of welded structures

161

(b) (a)

(c)

(d)

(e)

5.6 Geometries of the considered welded details.

suggest using a rlim value equal to 1.4–1.5 to avoid an excessive degree of conservatism in the presence of large values of the Δsn to Δt ratio. The MWCM applied in terms of nominal stresses was seen to be successful in assessing welded connections failing in the medium to high-cycle fatigue regime. In particular, the estimated (Nf,e) vs. experimental (Nf) fatigue life diagrams reported in Figs 5.7 and 5.8, respectively, clearly show that our approach is highly accurate in predicting fatigue damage in standard welded details (see Fig. 5.6 for the geometries of the considered joints), giving estimates falling mainly within the widest scatter band between the two used to calibrate the MWCM itself. It is worth pointing out here that the multiaxial fatigue strength of the above welded connections was predicted by calculating the constants of Eqs (5.4) and (5.5) through the design curves supplied by the IIW (Hobbacher, 2007) and summarised in Table 5.1 (the values of the listed reference stresses, extrapolated at NA = 2 × 106 cycles to failure, refer to a probability of survival, PS, of 97.7% and the negative inverse slopes of the uniaxial and torsional fatigue curves are equal to 3 and 5, respectively). To conclude, it can be highlighted also that, in order to better show the accuracy and reliability of our method in estimating fatigue lifetime of weldments, the diagrams reported in Figs 5.7 and 5.8 were built by calibrating Eq. (5.5) through the standard reference stresses recalculated for PS = 50%: these charts strongly support the idea that our approach can be used successfully to perform the fatigue assessment of real welded components subjected to multiaxial cyclic loading, especially in light of the fact that, when applied in accordance with the recommendations of the pertinent standard codes, the MWCM always allows an adequate degree of safety to be reached.

162

Multiaxial notch fatigue 100 000 000

Uniaxial Torsion In-phase Out-of-phase

Nf (cycles) 1 000 000 100 000

PS = 97.7%

Conservative

Uniaxial scatter band

10 000

PS = 2.3%

1000

Non-conservative

Torsional scatter band 100 100

1000

10 000 100 000 1 000 000 10 000 000 100 000 000

Nf,e (cycles)

5.7 Accuracy of the MWCM applied in terms of nominal stresses in estimating fatigue lifetime of steel welded joints subjected to biaxial cyclic loading.

Table 5.1 Summary of the considered experimental results generated by testing steel and aluminium welded joints under biaxial fatigue loading Material

R

ΔsA (MPa)

ΔtA (MPa)

Geometry

Reference

StE 460* StE 460* StE 460* A519

−1 −1, 0 −1, 0 −1

90 71 71 71

100 100 100 80

Fig. Fig. Fig. Fig.

5.6a 5.6a 5.6a 5.6b

A519-A36* BS4360 Gr. 50E Fe 52 steel BS4360 6082-T6

−1, 0 0 −1 0 −1

80 71 45 80 32

100 80 100 80 36

Fig. Fig. Fig. Fig. Fig.

5.6b 5.6b 5.6c 5.6d 5.6a

6060-T6*

−1, 0.05

32

36

Sonsino, 1995 Yousefi et al., 2001 Amstutz et al., 2001 Young and Lawrence, 1986 Siljander et al., 1992 Razmjoo, 1996 Bäckström et al., 1997 Archer, 1987 Kueppers and Sonsino, 2003 Costa et al., 2005

Fig. 5.6e

* Stress relieved.

5.5

MWCM and hot-spot stresses

Owing to the advantages outlined above, the MWCM can also be used to estimate the fatigue lifetime of welded connections by taking full advantage of the hot-spot concept (Susmel and Tovo, 2006). Before reviewing and validating the procedure for applying our criterion in terms of structural quantities, it is worth remembering here that when weld beads experience

Multiaxial fatigue assessment of welded structures

163

100 000 000

Uniaxial Torsion In-phase Out-of-phase

Nf (cycles) 1 000 000

PS = 97.7%

Conservative 100 000

Uniaxial scatter band

10 000

PS = 2.3% Non-conservative 1000

Torsional scatter band 100 100

1000

10 000 100 000 1 000 000 10 000 000 100 000 000

Nf,e (cycles)

5.8 Accuracy of the MWCM applied in terms of nominal stresses in estimating fatigue lifetime of aluminium welded joints subjected to biaxial cyclic loading. sx

Weld bead sHS

Linear elastic stress distributions

Surface

tHS txy Reference points

z

x

y x

5.9 Definition of normal and shear hot-spot stresses.

complex loading paths, the conventional hot-spot approach postulates that the extent of fatigue damage depends on the maximum principal geometrical stress calculated at the assumed crack initiation point (Anon, 1988; Hobbacher, 2007). Instead, it is recommended that the MWCM is applied by post-processing the structural stress components perpendicular and parallel to the weld bead (Fig. 5.9).

164

Multiaxial notch fatigue

The above strategy takes as its starting point the idea that, in the presence of geometrical features characterised by opening angles larger than about 100°, fatigue strength depends mainly on the Mode I and III stress components (Lazzarin et al., 2004), whereas as a first approximation the contribution due to Mode II loading can be disregarded because the resulting stresses are not singular (Lazzarin and Tovo, 1998). By observing now that, according to how they are calculated, hot-spot quantities are closely related to the actual distribution of the linear-elastic stress field acting on the fatigue process zone (Tovo and Lazzarin, 1999), it is logical to presume that the structural stresses perpendicular to the weld bead are somehow related to the corresponding Mode I stress components, whereas the structural shear stresses depend mainly on the contribution of the anti-plane stress to the overall distribution of the local stress field (Susmel and Tovo, 2006). The procedure for the in-field application of the MWCM is the same as that explained in the previous section (Fig. 5.5). The only difference is that now the stress tensor at the assumed crack initiation point has to be expressed in terms of geometrical quantities. In other words, according to the hot-spot concept, the stress components relative to the critical plane that are needed to estimate fatigue damage have to be calculated by starting from the ranges of the structural stresses, determined at the joint hot-spot, perpendicular and parallel to the weld bead. In order to show the accuracy of the MWCM in estimating fatigue strength of welded connections when applied in terms of structural quantities, our criterion was initially applied to tubular joints subjected to bending (or tension) and torsion. The considered experimental results are summarised in Table 5.2. Figure 5.10 shows the strategy we adopted to determine Table 5.2 Experimental results used to show the accuracy of the MWCM applied in terms of hot-spot stresses in estimating fatigue strength of steel and aluminium welded joints Material

R

ΔsA (MPa)

ΔtA (MPa)

Geometry

Reference

StE 460* StE 460* StE 460* A519

−1 −1, 0 −1, 0 −1

90 90 90 90

100 100 100 100

Fig. Fig. Fig. Fig.

A519-A36* BS4360 Gr. 50E 6082-T6

−1, 0 0 −1

90 90 36

100 100 36

Fig. 5.6b Fig. 5.6b Fig. 5.6a

6060-T6*

−1, 0.05

36

36

Fig. 5.6e

Sonsino, 1995 Yousefi et al., 2001 Amstutz et al., 2001 Young and Lawrence, 1986 Siljander et al., 1992 Razmjoo, 1996 Kueppers and Sonsino, 2003 Costa et al., 2005

* Stress relieved.

5.6a 5.6a 5.6a 5.6b

Multiaxial fatigue assessment of welded structures ΔsHS

165

Δsx

ΔtHS z y Δtxy

0.4t

x t x

5.10 Adopted procedure to calculate the hot-spot stresses perpendicular and parallel to the weld bead in the considered tubular welded joints.

the normal and shear hot-spot stresses. To be precise, the above stress quantities were extrapolated to the weld toe by using two reference points at distances from the assumed hot-spot of 0.4t and t, respectively (Hobbacher, 2007), where the thickness, t, of the main tubes was used as reference geometrical information to determine the positions of the above two points. Finally, the mesh density on the surface of every considered welded detail was set in accordance with Niemi’s recommendations (Niemi, 1995). The experimental (Nf) vs. estimated (Nf,e) fatigue life diagrams reported in Figs 5.11 and 5.12 clearly show that our criterion is successful in estimating fatigue damage in both steel and aluminium welded details also when applied in terms of geometrical stresses. In particular, these charts confirm that the MWCM predicts the finite life of welded joints subjected to multiaxial fatigue loading always with an adequate degree of safety, even though the estimates plotted in Figs 5.11 and 5.12 were obtained by calibrating the criterion through standard fatigue curves recalculated for a probability of survival equal to 50% (where the FAT 90 uniaxial and the FAT 100 torsional curves were used as reference information for estimating fatigue lifetime of steel weldments, whereas the FAT 36 uniaxial and torsional standard curves were used to calibrate the MWCM for predicting fatigue strength of aluminium welded joints). Finally, it is worth noting that the above estimates were obtained by taking rlim equal to 1.4. In order to better explain how the MWCM works when used in terms of hot-spot stresses, it is useful to briefly review here the main outcomes of an experimental investigation we carried out, working in collaboration with

166

Multiaxial notch fatigue 100 000 000

1 000 000 100 000

PS = 97.7%

Uniaxial Torsion In-phase Out-of-phase

Nf (cycles)

Conservative Uniaxial scatter band

10 000

PS = 2.3%

1000

Non-conservative

Torsional scatter band 100 100

1000

10 000 100 000 1 000 000 10 000 000 100 000 000 Nf,e (cycles)

5.11 Accuracy of the MWCM applied in terms of geometrical stresses in estimating fatigue lifetime of steel welded joints subjected to biaxial cyclic loading. 100 000 000

Nf (cycles) 1 000 000

Uniaxial Torsion In-phase Out-of-phase Conservative

PS = 97.7%

PS = 2.3%

100 000

Uniaxial scatter band

10 000

Non-conservative 1000

Torsional scatter band 100 100

1000

10 000 100 000 1 000 000 10 000 000 100 000 000

Nf,e (cycles)

5.12 Accuracy of the MWCM applied in terms of geometrical stresses in estimating fatigue lifetime of aluminium welded joints subjected to biaxial cyclic loading.

our colleague Roberto Tovo, by testing the welded connections sketched in Fig. 5.13 (Susmel and Tovo, 2006). The material employed was S355, a conventional structural steel, and the welding process adopted was the same for the two types of specimens. The above welded joints were tested under nominal uniaxial fatigue loading, by imposing a load ratio, R, equal to 0.1. Even if the fatigue lifetime of both types of joints could be estimated

Multiaxial fatigue assessment of welded structures

Δsnom

Δsnom

Δsnom

Δsnom

f

A-type joint

167

f

B-type joint

5.13 Geometries of the welded connections used to check the accuracy of the MWCM, applied in terms of geometrical stress, in locating the position of crack initiation points (Susmel and Tovo, 2006).

successfully simply in terms of nominal stresses by using the corresponding design curve supplied by Eurocode 3 (Anon, 1988), the direct inspection of the broken samples revealed an unexpected material cracking behaviour. In more detail, fatigue cracks were seen to emanate always from the weld toe, but in A-type joints they initiated at the intersection point between the weld toe circumference and the axial direction (i.e., f = 0° in Fig. 5.13), whereas in B-type joints the crack initiation points were seen to move as the range of the applied loading increased. In more detail, in the mediumcycle fatigue regime cracks initiated at f angles equal to about 45°, whereas in the high-cycle fatigue field the position of the crack initiation points was seen to move back to f = 0° (Fig. 5.13). When trying to interpret the experimental results, Roberto Tovo argued that such a particular cracking behaviour could be ascribed to the inherent multiaxiality resulting from the particular shapes of the investigated welded details. By taking this idea as a starting point, we calculated the tangential and normal hot-spot stresses along the weld bead, obtaining the two diagrams reported in Fig. 5.14. Subsequently, by using such structural quantities, the MWCM was used to calculate the resulting fatigue damage along the weld toe circumference. Finally, the fatigue curves estimated according to the MWCM for several values of angle f were recalculated in terms of ranges of the nominal stress, building the two diagrams sketched in Fig. 5.15: according to the observed material cracking behaviour, our method correctly predicted that in A-type joints the point at which fatigue damage reached its maximum value was always at f = 0°; on the contrary, our multiaxial fatigue criterion applied in terms of geometrical stresses predicted that in B-type joints the initiation phenomenon had to occur at f = 45° in the

168

Multiaxial notch fatigue

sHS/snom 1.4 tHS/snom 1.2 1 0.8 0.6 0.4 0.2 0 –0.2

A-type joint

sHS/snom 1.4 tHS/snom 1.2 1 0.8 0.6 0.4 0.2 0 –0.2

sHS tHS

0

15

30

B-type joint

45

60

75

90

sHS tHS

0

15

30

45

f (degrees)

60

75

90

f (degrees)

5.14 Hot-spot stress distributions along the weld toe circumferences of the welded details sketched in Fig. 5.13.

A-type joint 1000 Dsnom (MPa)

Results f = 0° f = 22.5° f = 45° f = 67.5°

100

10 10 000

100 000

1 000 000

10 000 000 Nf (cycles)

1 000 000

10 000 000 Nf (cycles)

B-type joint 1000 Dsnom (MPa) 100

Results f = 0° f = 22.5° f = 45° f = 67.5°

10 10 000

100 000

5.15 Experimental results and fatigue curves, determined according to the MWCM, at different values of angle f.

Multiaxial fatigue assessment of welded structures

169

medium-cycle fatigue regime and at f = 0° in the high-cycle fatigue field. Such a validation exercise allowed us to further confirm that the MWCM applied in terms of structural stresses can be used successfully to perform the fatigue assessment of welded joints by taking full advantage of the extensive work done to provide engineers engaged in assessing real components with safe and reliable rules to calculate geometrical stresses – see, for instance, Hobbacher (2007) and references reported therein.

5.6

The MWCM applied in terms of the PM to perform the fatigue assessment of weldments

In recent years, we have made a big effort to formalise a sound and reliable procedure (based on the use of the MWCM applied along with the PM) suitable for estimating fatigue damage in steel and aluminium welded details by directly post-processing linear-elastic FE models (Susmel, 2007, 2008, 2009). Basically, the strategy to be followed to employ our method to predict fatigue strength of weldments subjected to cyclic loading is the same as the one already explained in Section 4.4.3. The only difference is that, when welded connections are involved and the MWCM is calibrated by using appropriate standard fatigue curves, the multiaxial critical distance value, M-DV, is seen to be independent of the number of cycles to failure, resulting in an evident simplification of the addressed problem. The reasons why the critical distance can be assumed to be constant in both the mediumand high-cycle fatigue regimes will be explained in detail in the next section. In what follows instead, the in-field use of the MWCM applied along with the PM is briefly reviewed. In particular, our multiaxial fatigue life estimation technique takes as a starting point the following assumptions (Susmel, 2008, 2009): • Fatigue strength is estimated assuming a linear-elastic behaviour of the parent material. • In welded details the radius at the weld toe (or at the weld root) is assumed to be equal to zero. • The focus path to be used to estimate fatigue strength is coincident with the notch bisector, that is, with that direction along which the stress components due to the three fundamental modes are uncoupled. By following a fairly articulated reasoning, it was proved that, when calculated for a probability of survival, PS, equal to 50%, the multiaxial critical distance value, M-DV, is equal to 0.5 mm and 0.075 mm in steel and aluminium weldments, respectively (Susmel, 2008, Susmel, 2009). The ΔtA,Ref vs. rw and kt vs. rw relationships instead have the profiles sketched in Fig. 5.16, where the values of the reference shear stress, determined for

170

Multiaxial notch fatigue 200 DtA,Ref 180 (MPa) 160

6 5 k = –2rw + 5

140

4

120

k=3

FAT 80

100 80

3

FAT 125 ΔtRef = –32rw + 96 MPa (PS = 50%)

60 40

0

1

2

3

rw

2 1

ΔtRef = 32 MPa

N-SIF approach

20 0

k

Steel, as-welded

4

0

(a)

6

80 DtA,Ref 70 (MPa)

k

Aluminium, as-welded

5

k = –0.5rw + 5

60

4

50 k=3

40

FAT 28

30

N-SIF approach

3

ΔtRef = 28.4 MPa

55-4.5

2

20 1

ΔtRef = –1.3rw + 33.6 MPa (PS = 50%)

10 0

0 0

1

2

3

4

5

rw

6

(b)

5.16 ΔtA,Ref vs. rw and kt vs. rw relationships to be used to assess (a) steel and (b) aluminium welded connections.

PS equal to 50%, were extrapolated at NA = 5 × 106 cycles to failure. Moreover, it has to be highlighted that, in accordance with Eurocode 3, Eurocode 9 and the IIW recommendations (Anon, 1988, 1999), the above ΔtA,Ref vs. rw relationships as they stand are strictly valid only for assessing as-welded connections: as already said in Section 5.3, the corrections suitable for taking into account the beneficial effects due to heat treatments and other technological processes are left to the reader’s experience. The procedure to be followed to estimate fatigue damage in welded mechanical assemblies according to our approach is summarised in Fig. 5.17. In particular, consider a welded connection experiencing a complex system of cyclic forces. The stress state to be used to determine the quantities relative to the critical plane has to be determined along the weld toe bisector

Multiaxial fatigue assessment of welded structures

171

Fk(t) Weld bead

M-DV

sx(t)

txy(t) tyz(t)

[s(t)] = txy(t) txz(t)

sy(t) tyz(t) tyz(t) sz(t)

Fj(t)

Fi(t)

Δt,Δsn,rw

ΔtRef(rw),k(rw)

Δt (MPa) Δt 1 kt(ρw)

ΔtA,Ref(rw)

Nf,e

NA Nf (cycles)

5.17 In-field use of the MWCM applied along with the PM to estimate fatigue damage in welded joints.

(or along the weld root bisector) at a distance from the weld toe apex equal to M-DV. It is worth highlighting here that, as a first approximation, the root radius in the model can be taken to be equal to zero. It can be remembered here also that, as already explained in the previous chapter, because the stress tensor needed to apply the MWCM is calculated by assuming a linearelastic behaviour of the parent material, the contribution of every considered force can be computed separately, determining the resulting stress tensor by superposing the contribution of every applied force. Another aspect which deserves to be discussed in detail here is the fact that the stress tensor at a distance equal to M-DV from the assumed crack initiation point has to be determined by considering the stress components due to Mode I, II and III loading. As highlighted above, we believe that fatigue damage in conventional welded connections depends mainly on Mode I and III stress components, whereas the contribution due to Mode II stresses can be neglected because they are not singular in the presence of opening angles

172

Multiaxial notch fatigue

larger than about 100° (Susmel and Tovo, 2006). In spite of the above fact, we suggested using the MWCM in conjunction with the PM by considering the entire stress field at the reference point in order to formalise a procedure suitable for application by directly post-processing linear elastic FE models (i.e., without the need for distinguishing a priori among the contributions of the three fundamental modes). From the stress tensor determined at a distance from the assumed crack initiation point equal to M-DV, the range of the maximum shear stress, Δt, as well as the range of the stress, Δsn, perpendicular to the critical plane can be determined by taking full advantage of the procedures reviewed in Chapter 1. By using the resulting value of ratio rw, the inverse slope, kt, and the reference shear stress range, ΔtA,Ref, can then be determined from the calibration functions plotted in Fig. 5.16. Subsequently, through the calculated values for both kt and ΔtA,Ref, the corresponding modified Wöhler curve can be estimated (Fig. 5.17). Finally, the shear stress range, Δt, relative to the critical plane and the above design curve can be used directly to predict, through Eq. (5.6), the number of cycles to failure, Nf,e (see Fig. 5.17).

5.6.1 Calibration of the MWCM’s governing equations to be used to assess steel weldments In the present section as well as in the next one, the strategy we adopted to determine the calibration functions plotted in the diagrams of Fig. 5.16 will be explained in detail. Before doing that, it is worth highlighting that the ΔtA,Ref vs. rw relationships suggested here to assess both steel and aluminium welded details were calibrated by using standard curves suitable for estimating fatigue damage in as-welded joints by neglecting the effect of superimposed static stresses. As said briefly in the previous sections, it is evident that the same strategy as that which follows could be adopted by using different reference curves in order to take into account, as suggested by the pertinent standard codes, the other design parameters affecting weldment fatigue strength (like, for instance, the beneficial effects of heat treatments): according to the philosophy we decided to embrace when writing the present chapter, such corrections are left to the in-field experience of the reader. In other words, in accordance with the recommendations of the available standard procedures (i.e., Eurocode 3, Eurocode 9 and IIW’s documents), the calibration functions proposed here for performing the fatigue assessment of welded connections should always allow an adequate margin of safety to be reached. The first problem to be addressed, then, is the choice of adequate design curves suitable for calculating constants a, b, a and b in the MWCM’s governing equations, where:

Multiaxial fatigue assessment of welded structures

173

kτ ( ρ w ) = aρ w + b

5.7

Δτ A,Ref ( ρ w ) = αρ w + β

5.8

Eurocode 3 suggests using the FAT 125 curve to estimate lifetime of ground butt welds in as-welded condition subjected to uniaxial fatigue loading, whereas the most conservative curve to be adopted to predict fatigue damage in weldments loaded in torsion is the FAT 80 curve. From an experimental point of view, the accuracy of such curves in addressing the above situations is fully confirmed by the diagrams reported in Fig. 5.18, where the fatigue strength of both ground butt welds subjected to uniaxial fatigue loading (Fig. 5.18a) and tube-to-plate welded joints loaded in torsion (Fig. 5.18b) is summarised in terms of nominal stress range. As to the torsional results, it is worth highlighting the evident beneficial effect of heat treatments on the fatigue behaviour of the considered welded joints. By using the FAT 125 and FAT 80 standard curves to calibrate the MWCM, Eqs (5.7) and (5.8) can easily be rewritten as: kτ ( ρw ) = −2 ρw + 5

5.9

Δτ A,Ref ( ρw ) = −29.6 ρw + 96 ( MPa )

5.10

where the latter relationship estimates, for a probability of survival equal to 50%, the reference shear stress range, ΔtA,Ref, at 5 × 106 cycles to failure. The above calibration functions now have to be adjusted in order to fully comply with Eurocode 3’s recommendations, so that an adequate degree of safety can always be guaranteed when using our method to assess components in situations of practical interest. In more detail, initially it can be noted that under uniaxial fatigue loading (rw = 1) relationship (5.9) gives a kt value equal to 3, whereas under torsional cyclic loading (rw = 0)

10 000

Ground butt welds

Dsnom (MPa)

10 000

Tube-to-plate welded joints

Dtnom (MPa)

PS = 2.3%

1000

1000

Stress relieved As-welded

PS = 2.3%

PS = 97.7% 100

10 100

FAT 125, k = 3

1000

10 000

(a)

100 000 1 000 000 10 000 000

100

10 100

PS = 97.7% FAT 80, k = 5

1000

10 000

100 000 1 000 000 10 000 000

Nf (cycles)

Nf (cycles)

(b)

5.18 Standard curves and experimental results (data reported in Susmel, 2008).

174

Multiaxial notch fatigue

it estimates a negative inverse slope equal to 5. If conventional joints having a weld bead opening angle equal to 135° and subjected to uniaxial fatigue loading are considered, by directly rearranging Eqs (5.1) and (5.2) it is straightforward to see that the stress fields acting on the fatigue process zone are always characterised by rw values larger than unity. All the standard curves supplied by Eurocode 3 and suitable for performing the uniaxial fatigue assessment of welded connections have inverse slope equal to 3. This suggests that the kt vs. rw relationship suitable for using the MWCM reinterpreted in terms of the PM in accordance with Eurocode 3’s philosophy can simply be rewritten as follows: kτ ( ρw ) = −2 ρw + 5

for ρw ≤ 1

5.11

kτ ( ρw ) = 3

for ρw > 1

5.12

The N-SIF approach can now be employed to estimate the multiaxial critical distance value, M-DV, to be used in conjunction with the MWCM to evaluate fatigue damage in welded steel connections. In particular, as to how the N-SIF approach works, it is initially important to remember here that such a method is successful in accounting for the scale effect in fatigue, because, according to Eqs (5.1) and (5.2), for a given value of both ΔKI and ΔKII, the stress field distribution close to the root of the weld bead does not depend on the absolute dimensions of the considered connection. In more detail, as far as cruciform joints loaded either in tension or in bending are concerned, the N-SIFs due to Mode I and II loading can be estimated, respectively, as (Lazzarin and Tovo, 1998): ΔKI = kI ⋅ Δσ nom ⋅ t 1− λ1

5.13

ΔKII = kII ⋅ Δσ nom ⋅ t 1− λ2

5.14

In the above identities ki (i = I, II) are non-dimensional quantities which depend on the geometry of the considered welded joint, Δsnom is the range of the nominal stress, and t1−li (i = 1, 2) accounts for the scale effect through the thickness, t, of the main plate. The N-SIF approach postulates that in cruciform joints characterised by different absolute dimensions the fatigue damage extent is the same as long as the ΔKI value characterising the linear-elastic stress field acting on the material in the vicinity of stress concentrator apices is the same (Lazzarin and Tovo, 1998). According to Eq. (5.13), both kI and t1−l1 vary as the absolute dimensions of the considered welded joint change, so that ΔKI can remain constant only if the range of the applied nominal stress, Δsnom, is varied. This implies that the distribution of the stresses due to Mode II loading, Eq. (5.14), changes

Multiaxial fatigue assessment of welded structures

175

as the dimensions of the considered geometry are scaled, because in general kI ≠ kII and t1−l1 ≠ t1−l2. In other words, for a given joint shape, the profile of the entire linear elastic stress field acting on the process zone varies as the absolute dimensions of the considered geometry increase, even though both the range of the applied nominal loading and the size of the joint are varied by keeping ΔKI constant. This results in further complications, because, if fatigue damage is attempted to be estimated by post-processing the entire stress field in the vicinity of crack initiation sites, the N-SIF reference curves (Fig. 5.4) cannot be used as they stand and some other simplifying hypothesis has to be formed in order to take full advantage of such an approach. It is worth remembering here that we believe that fatigue damage in welded connections depends mainly on Mode I and III stress components, but, as said above, we decided to consider the entire stress field acting on the fatigue process zone simply because we were interested in devising a predictive method suitable for performing the fatigue assessment of welded components by directly post-processing linear-elastic FE results. Bearing in mind the considerations reported above, the N-SIF unified scatter band plotted in terms of ΔKI can then be used to determine the profiles of the stress field acting on the fatigue process zone which result in fatigue breakages at different numbers of cycles to failure. In particular, for welded steel connections, ΔKI is equal to 1675 MPa ⋅ mm0.326 at 104 cycles to failure, whereas it is equal to 211 MPa ⋅ mm0.326 at 5 × 106 cycles to failure (Fig. 5.4). Moreover, as far as conventional cruciform welded joints are concerned, it is possible to observe that, by taking into account also the contribution due to Mode II loading, rw is roughly constant along the bisector and equal to about 2. Using the above values for ΔKI, it is possible to plot, in terms of maximum shear stress range, Δt, the stress distribution along the weld toe bisector. To be precise, Fig. 5.19 shows the shear stress– distance curves calculated by considering the entire stress field in the vicinity of the weld bead root and determined at Nf equal to both 5 × 106 and 104 cycles to failure. Moreover, it is important to highlight that because, for the reasons discussed above, the profile of the linear elastic stress field depends also on the absolute dimensions of the joint when Mode II stress components are included in the calculation, a reference geometry having the following characteristic lengths was assumed: t = L = 25 mm, z = 8 mm (see Fig. 5.3 for the adopted symbolism). These values for t, L and z were chosen simply because, according to Eurocode 3, the supplied standard curves have to be corrected in order to properly take into account the scale effect in weldment fatigue only when the thickness of the main plate becomes larger than 25 mm. In any case, because Mode II stress components are not singular, the absolute dimensions of the considered welded joint play a secondary role in determining the overall profile of the stress–distance curves plotted in Fig. 5.19.

176

Multiaxial notch fatigue 350 Dt (MPa) 300

Δt ≅ 294 MPa

250

Nf = 104 cycles to failure

200 150 M-DV = 0.5mm

100 50 0 0.0

Nf = 5·106 cycles to failure ΔtΑ,Ref ≅ 37 MPa 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 1.0 r (mm)

5.19 Linear-elastic stress field distributions, in terms of Δt, along the focus paths in steel cruciform welded joints subjected to uniaxial cyclic loading.

Such curves were determined by directly using Eqs (5.1) and (5.2) directly to calculate sq, sr and trq, whereas sz was estimated as sz = ν(sq + sr): a plane strain hypothesis (Lazzarin and Zambardi, 2001) was considered to be the most appropriate for describing the required stress fields because in conventional cruciform specimens subjected to uniaxial fatigue loading the crack initiation phenomenon is seen to occur mainly at the mid-section of the main plate, that is, at that joint section experiencing the stress state characterised by the highest degree of triaxiality. Finally, a convenient value for the multiaxial critical distance, M-DV, was calculated by using the shear stress ranges, estimated from Eqs (5.10)– (5.12) for a rw value equal to 2, breaking ground butt welds at Nf equal to 5 × 106 and 104 cycles to failure, respectively: the two straight horizontal lines plotted in Fig. 5.19 allow an M-DV value of about 0.5 mm to be extrapolated. The most relevant result of the procedure followed to estimate the multiaxial critical distance value and summarised in Fig. 5.19 is that, contrary to what was shown in Section 4.4.3, M-DV is independent of the number of cycles to failure: the critical distance value is constant because both the reference curve calculated according to Eqs (5.10)–(5.12) and the N-SIF fatigue curve have inverse slope equal to 3. The ΔtRef vs. rw relationship can now be calibrated more accurately by following a procedure similar to the one summarised above, but imposing that the multiaxial critical distance value is known a priori and equal to 0.5 mm. To be precise, by considering the absolute dimensions of the cruciform joints used by Lazzarin, Tovo and Livieri to calculate the N-SIF unified scatter band (Lazzarin and Tovo, 1998; Lazzarin and Livieri, 2001), it is possible to observe that, under a plane strain hypothesis, the

Multiaxial fatigue assessment of welded structures

177

rw value at M-DV = 0.5 mm ranges between 1.7 and 1.95, with an average value equal to 1.85. The corresponding reference shear stress range, ΔtA,Ref, determined, for a probability of survival equal to 50%, at 5 × 106 cycles to failure results in an average value equal to about 37 MPa (Susmel, 2008). If the FAT 80 torsional fatigue curve (rw = 0), the FAT 125 uniaxial fatigue curve (rw = 1) and the above ΔtA,Ref value (rw = 1.85) are reconsidered all together, constants a and b in Eq. (5.8) can be recalculated, again for a probability of survival equal to 50%, in a more accurate way, obtaining (Susmel, 2008): Δτ A,Ref ( ρw ) = −32 ρw + 96 ( MPa )

5.15

The above ΔtA,Ref vs. rw as well as the corresponding kt vs. rw relationship – the latter defined according to Eqs (5.11) and (5.12) – are plotted together in Fig. 5.16a. Finally, in order to properly manage also those situations characterised by large values of rw, according to what has already been discussed in Chapter 3, the limit value for the critical plane stress ratio was determined by using the following relationship:

ρw,lim =

Δτ A 2 Δτ A − Δσ A

5.16

where ΔsA and ΔtA are the endurance limits calculated, for a probability of survival equal to 50%, from the FAT 125 and FAT 80 standard curve, respectively, at 5 × 106 cycles to failure. Even if, strictly speaking, Eq. (5.16) supplies a limit value for rw equal to about 1.7, according to the ΔtA,Ref values plotted in Fig. 5.16a and considering the variability shown by rw when calculated for cruciform joints by taking into account also the influence of the Mode II stress components, a limit value for rw equal to 2 was then assumed (Susmel, 2008). To conclude, it is worth noting that the above procedure can be used to calibrate Eq. (5.8) also for a probability of survival higher than 50% (for instance, 97.7%): in general, not only the values of the constants in the MWCM’s governing equations change, but also the value of the multiaxial critical distance, M-DV, varies.

5.6.2 Calibration of the MWCM’s governing equations to be used to assess aluminium weldments The procedure suitable for calculating the relevant fatigue constants needed to apply our method to perform the fatigue assessment of aluminium welded joints is similar to the one reviewed in the previous section. In particular, the first problem to be addressed is the choice of the correct uniaxial and

178

Multiaxial notch fatigue Ground butt welds 1000 Dsnom (MPa)

100

PS = 2.3%

PS = 97.7% sA = 55 MPa, k = 4.5

10 10 000

100 000

1 000 000

10 000 000

Nf (cycles)

5.20 Eurocode 9’s uniaxial standard curve and experimental results (data from Person, 1971).

torsional design curves to be used to calibrate the MWCM’s governing equations. Figure 5.20 compares the experimental fatigue strength of ground butt welds made of different aluminium alloys to the corresponding standard curve supplied by Eurocode 9 (the reference stress, ΔsA, at 2 × 106 cycles to failure of such a curve is equal to 55 MPa, whereas its negative inverse slope, k, is equal to 4.5). The chart of Fig. 5.20 clearly shows that the predictions made by the above design curve are definitely conservative. As to the choice of a reference curve to be used to perform the torsional fatigue assessment, the IIW suggests using the FAT 28 curve to estimate fatigue damage in as-welded joints loaded in torsion, that is, a design curve having negative inverse slope, k, equal to 5 and reference shear stress, ΔtA, at 2 × 106 cycles to failure equal to 28 MPa. Unfortunately, we did not find enough experimental results in the literature to allow us to systematically check the accuracy of such a design curve as was done in Fig. 5.20 for the uniaxial case. However, accepting the IIW’s recommendations, the FAT 28 standard curve will be used in what follows as second calibration information. If the two design curves mentioned above are recalculated for a probability of survival, PS, equal to 50% and the corresponding reference shear stresses, ΔtA,Ref, are extrapolated at 5 × 106 cycles to failure, the constants of Eqs (5.7) and (5.8) take on the following values (Susmel, 2009): kτ ( ρw ) = −0.5ρw + 5

5.17

Δτ A,Ref ( ρw ) = −1.3ρw + 33.6 ( MPa )

5.18

Multiaxial fatigue assessment of welded structures

179

The N-SIF approach together with the above relationships can now be employed to estimate the multiaxial critical distance, M-DV, to be used to assess aluminium welded joints. In particular, by extrapolating from the master curve plotted in Fig. 5.4 the values of ΔKI at both 104 and 5 × 106 cycles to failure, the corresponding stress fields can be calculated directly through Eqs (5.1) and (5.2). The profiles of the shear stress–distance curves reported in Fig. 5.21 were then determined by assuming as reference geometry a cruciform joint having t = L = z = 12 mm (see Fig. 5.3 for the definition of the adopted symbolism) and the linear elastic stress field acting on the process zone was determined under the plane strain hypothesis (Livieri and Lazzarin, 2005).The above reference dimensions were chosen simply because Livieri and Lazzarin’s master curve was determined by testing aluminium cruciform joints having t, L and z ranging in the interval from 6 to 24 mm. According to the assumed lengths for t, L and z, the corresponding values of ΔKI and ΔKII were calculated from Eqs (5.13) and (5.14) to estimate rw through Eqs (5.1) and (5.2). Subsequently, by assuming that, for the considered welded geometry, rw could be kept constant along the notch bisector and equal to about 1.95, the maximum shear stress range, Δt, was determined from Eqs (5.17) and (5.18) at both 5 × 106 and 104 cycles to failure, obtaining Δt = 31.1 MPa and Δt = 146.9 MPa, respectively. Finally, the procedure summarised by the diagram of Fig. 5.21 allowed a multiaxial critical distance value, M-DV, equal to 0.075 mm to be estimated for PS = 50%. It is worth noting here that the value obtained for M-DV is again independent of the number of cycles to failure simply because, for rw equal to 1.95, the negative inverse slope of the modified Wöhler curve given by Eq. (5.17) is practically

400 Dt (MPa) 350 300

Nf = 104 cycles to failure

250 M-DV = 0.075 mm

200

Δt = 146.9 MPa

150 100

Nf = 5·106 cycles to failure

ΔtA,Ref = 31.1 MPa

50 0 0.00

0.05

0.10

0.15

0.20

0.25 r (mm)

5.21 Linear-elastic stress field distributions, in terms of Δt, along the focus paths in aluminium cruciform welded joints subjected to uniaxial cyclic loading.

180

Multiaxial notch fatigue

the same as that of the N-SIF curve suggested to be used to assess a cruciform welded joint made of aluminium alloy (Fig. 5.4). The diagram reported in Fig. 5.16b summarises the profiles of the ΔtA,Ref vs. rw and kt vs. rw relationships calculated according to the above procedure. To conclude, it is worth noting that, as sketched in this chart, the limit value for rw was taken to be equal to 4 simply because, according to the IIW’s recommendations, the negative inverse slope of the uniaxial standard curves cannot be lower than 3.

5.6.3 Accuracy of the MWCM applied in terms of the PM in estimating finite life of weldments In order to briefly show the accuracy of the MWCM, applied along with the PM, in estimating fatigue lifetime of welded joints subjected to both uniaxial and multiaxial nominal fatigue loading, several experimental results were selected from the literature. Initially, to check whether the systematic use of our method results in estimates which comply with Eurocode 3’s recommendations, about 800 experimental data generated by testing a variety of steel welded joints subjected to different loading paths were considered (Susmel, 2008). In more detail, we investigated the following geometry/loading configurations: both as-welded and stress relieved cruciform specimens tested under either tension or bending, welded joints having complex geometries (like lap joints, T-joints, tube-to-plate joints, etc.) and tested, in as-welded condition, under uniaxial fatigue loading, cruciform joints loaded in tension where fatigue cracks initiated at the weld roots and, finally, both as-welded and stress-relieved tubular joints tested under in-phase and out-of-phase bending (or tension) and torsion. The diagram reported in Fig. 5.22 shows the accuracy of the MWCM, calibrated by using the relationships plotted in Fig. 5.16a and applied assuming a multiaxial critical distance value, M-DV, equal to 0.5 mm, in estimating fatigue lifetime of as-welded steel joints: such a chart clearly proves that our predictive method is highly accurate, giving estimates falling within the widest scatter band between the two used to calibrate the MWCM’s governing equations. As clearly explained in Section 5.6.1, the calibration functions plotted in Figure 5.16a have been calculated by using two reference standard curves suitable for estimating fatigue strength of as-welded steel joints. This makes it evident that, if the same relationships are used to estimate fatigue damage also in stress-relieved weldments, predictions are expected to be characterised by a higher degree of conservatism. In order to increase the accuracy of our method in addressing the above situations, many different strategies could be followed. For instance, similarly to what is stated by Eurocode 3, an effective range of the shear stress, Δt*, suitable for taking into account the beneficial effect of heat treatments could be calculated by adding the

Multiaxial fatigue assessment of welded structures 100 000 000 Nf (cycles) 1 000 000

Cruciform (failure at weld toe) Cruciform (failure at weld root) Uniaxial loading In-phase multiaxial loading Out-of-phase multiaxial loading

PS = 2.3%

Conservative

100 000 PS = 97.7%

Standard uniaxial scatter band Standard torsional scatter band

10 000

Non-conservative 1000 1000

181

10 000

PS = 50%

100 000 1 000 000 10 000 000 100 000 000 Nf,e (cycles)

5.22 Accuracy of the MWCM applied in terms of the PM in estimating fatigue lifetime of as-welded steel joints (Susmel, 2008).

tensile part to 60% of the compressive portion of the shear stress range (Fig. 5.23a): Δτ * = τ max + 0.6 τ min

5.19

Such an effective shear stress range can then be used together with the modified Wöhler curve estimated through the calibration relationships plotted in Fig. 5.16a to predict the fatigue lifetime of stress-relieved welded connections (Susmel, 2008). According to Eurocode 3’s recommendations, the above correction is not mandatory. However, if Δt* defined as suggested by Eq. (5.19) is used to estimate the fatigue lifetime of stress-relieved weldments, estimates are seen to be highly accurate, even though still characterised by a little degree of conservatism (Fig. 5.23b). Similarly to the procedure described above, the accuracy and reliability of our approach in estimating fatigue strength of weldments was systematically checked also by considering aluminium joints. The results obtained are summarised in the experimental (Nf) vs. estimated (Nf,e) fatigue life diagram reported in Fig. 5.24 (Susmel, 2009). This shows the accuracy obtained by investigating the following geometry/loading configurations: cruciform joints loaded both in tension and in bending, butt welds with complete and incomplete penetration subjected to uniaxial cyclic loading, single and double lap joints, specimens with stiffeners characterised by different geometries, rectangular hollow section welded joints loaded in four-point bending and, finally, tubular connections subjected to in-phase and outof-phase bending and torsion – a detailed description of the investigated geometries can be found in Susmel (2009). Finally, it is worth pointing out

182

Multiaxial notch fatigue t(t) (MPa) tmax Δs* = tmax + 6.0 tmin Time tmin (a)

100 000 000 Cruciform (failure at weld toe) Uniaxial loading

Nf (cycles)

In-phase multiaxial loading Out-of-phase multiaxial loading

Conservative

1 000 000

100 000

PS = 2.3%

PS = 97.7%

Standard uniaxial scatter band Standard torsional scatter band

10 000

Non-conservative 1000 1000

10 000

PS = 50%

100 000 1 000 000 10 000 000 100 000 000 Nf,e (cycles) (b)

5.23 (a) Correction adopted to take into account the beneficial effect of heat treatments, and (b) accuracy of the MWCM in estimating fatigue lifetime of stress-relieved steel welded joints (Susmel, 2008).

also that the considered aluminium alloys belonged to the 5000, 6000 and 7000 series. In order to correctly interpret the diagram of Fig. 5.24, initially it is important to remember that, according to the strategy adopted by both Eurocode 9 (Anon, 1999) and the IIW, the fatigue strength of aluminium welded joints is assumed to be independent of the mechanical properties of the material to be assessed. In other words, only technological and geometrical aspects (including the detrimental effect of defects) are considered, so that aluminium alloys are assumed to have the same fatigue strength independently of their actual chemical composition: this implies that, for evident reasons, estimates can sometimes be too conservative, especially when high-strength aluminium alloys are assessed. Moreover, the curves

Multiaxial fatigue assessment of welded structures Nf (cycles) 100 000 000

Uniaxial, R = –1 Uniaxial, R ≈ 0 Torsion, R = –1 In-phase Out-of-phase

183

Conservative PS = 97.7%

1 000 000

Uniaxial scatter band 10 000

PS = 2.3% Non-conservative Torsional scatter band

100 100

10 000

1 000 000

100 000 000 Nf,e (cycles)

5.24 Accuracy of the MWCM, applied in terms of the PM, in estimating fatigue lifetime of aluminium welded joints (Susmel, 2009).

stated by the available standard codes are suggested to be used as they stand to estimate fatigue damage in as-welded connections, whereas some tricky corrections are proposed to take into account the beneficial effect of post-welding treatments. In particular, if the welded component to be assessed is stress relieved, the influence of negative load ratios, R, can be accounted for by directly correcting the standard curves valid for as-welded components. As to the mean stress effect in aluminium weldment fatigue, it is important to mention here also the fact that the influence of negative load ratios can be very pronounced not only in stress-relieved but also in as-welded connections – see, for instance, Morgenstern et al. (2006). This results in the fact that, in many cases, under R ratios lower than zero, estimates made in accordance with the available standard codes are conservative also in the presence of high tensile residual stresses. In light of the above considerations, the trend shown by the diagram of Fig. 5.24 is fully justified by the fact that, as explained in Section 5.6.2, we deliberately decided to ignore in the calibration process those corrections suitable for taking into account both the beneficial effects of the technological processes suitable for relieving residual stresses and the presence of superimposed static stresses. In more detail, the chart of Fig. 5.24 suggests that, by calibrating the MWCM as explained in Section 5.6.2, predictions are seen to fall mainly within the calibration scatter bands when as-welded joints are considered, and it holds true independently of

184

Multiaxial notch fatigue

geometry and applied loading path. On the contrary, and as expected, the points failing in the conservative zone refer mainly to experimental results generated by testing either stress-relieved connections or welded joints subjected to load ratios, R, lower than zero. It is possible to conclude by saying that, in spite of the well-known practical difficulties always encountered when assessing aluminium welded connections, the errors made by our method are seen to be always on the conservative side (Fig. 5.24): in the light of such experimental evidence, our approach can be considered as an interesting and efficient alternative to other existing fatigue life estimation techniques.

5.6.4 Concluding remarks The above sections attempt to summarise the procedures we devised to use the MWCM, applied in conjunction with the PM, to estimate fatigue damage in welded joints made of both steel and aluminium alloy. In particular, as shown in Sections 5.6.1 and 5.6.2, the calibration of our method was based on a fairly articulated reasoning involving several simplifying hypotheses. It is important to point out that the complexity of the procedure followed to determine the needed ΔtA,Ref vs. rw and kt vs. rw relationships was primarily a consequence of the fact that we attempted to calibrate the MWCM’s governing equations by taking full advantage of the recommendations of the pertinent standard codes. Such a choice resulted in the necessity of making several assumptions in order to coherently use the available design fatigue curves (which necessarily represent a compromise between the need for having information of general validity and the need for performing the fatigue assessment always with an adequate margin of safety). This results in inevitable approximations because it is well known that the overall fatigue strength of welded connections depends on a variety of parameters, whose influence can fully be taken into account only by running appropriate experiments. Owing to its nature, the MWCM applied along with the TCD is highly sensitive to the actual fatigue properties of the material to be assessed, therefore it is not surprising that, when calibrated by using standard curves, its systematic use results in a certain degree of inaccuracy. In any case, thanks to the hypotheses formed, when predictions do not fall within the reference calibration scatter bands, the errors made by our approach are, in general, on the conservative side. This strongly supports the idea that the MWCM, applied in terms of the PM, can be used successfully to assess welded components in situations of practical interest, always allowing an adequate margin of safety to be reached (see Figs 5.22, 5.23 and 5.24). At the same time, it can be pointed out that welded joints can be treated directly as conventional notched components (Lazzarin and Tovo, 1998), so

Multiaxial fatigue assessment of welded structures

185

that if our approach were calibrated by running appropriate experiments (i.e., if it were used as explained in the previous chapter), its accuracy in estimating fatigue strength of welded joints would increase greatly. Unfortunately, this is not always possible in practice and, in any case, it would definitely increase also the costs and time of the design process: this explains why we have made such a big effort to formalise a calibration procedure fully complying with the recommendations of the available standard codes. Moreover, if our method were calibrated by using appropriate experimental results, the influence of other design parameters could easily be incorporated in the calibration process itself. For instance, the presence of non-zero mean stresses could be taken into account directly by including in the calculation their effects through the mean stress sensitivity index, m. Another aspect of the problem which deserves to be considered in great detail is the assumption of using a root radius equal to zero to model the weld bead (or the weld root). As far as conventional welding techniques are concerned, the above assumption is considered to be acceptable: this is the reason why many other fatigue life estimation methods starting from the same assumption proved to be so accurate in estimating fatigue damage in conventional welded details – see, for instance, Lazzarin and Tovo (1998), Taylor et al. (2002), Crupi et al. (2005) and Livieri and Lazzarin (2005). Owing to its nature, the MWCM, when reinterpreted in terms of the PM, can also be successfully applied by explicitly modelling the notch root radius. Of course, considering the actual value of the weld toe radius increases the time needed to do the corresponding FE model, but if the MWCM is calibrated by using pieces of information coming from appropriate experiments, the overall accuracy of our method is expected to increase considerably. In other words, the problem of assessing welded components subjected to fatigue loading can be treated by rigorously modelling the geometries of the stress raisers and, in this case, the MWCM has to be applied in terms of the PM as reviewed in the previous chapter. In any case, it is obvious that imposing a root radius equal to zero results in a higher margin of safety, so that this way of addressing the problem is always advisable when the geometry of the stress raisers to be assessed cannot be modelled accurately. In this regard, owing to the high number of variables affecting the overall fatigue strength of welded connections, it is seen from the experiments that crack initiation does not always occur at those material points experiencing the highest values of the stress concentration factor (Huo, 2007): this experimental evidence strongly supports the idea that accurate and conservative estimates can be obtained by simply keeping the notch root radius equal to zero. At the same time, it has to be admitted that in the presence of very smooth transitions from the parent to the filler material, accurate prediction can be made only by explicitly introducing

186

Multiaxial notch fatigue

in the FE model the actual value of the root radius: in the above situations assuming a root radius equal to zero would result in an unacceptable degree of conservatism. In this scenario, it is worth remembering also that, in accordance with the necessary simplifying assumptions made to do the FE model of the component to be assessed, the mesh density in the vicinity of the assumed crack initiation sites can be set by directly using the practical rules already mentioned in Section 4.4.4. In conclusion, the considerations reported above should better justify the strategy we decided to adopt when writing the present chapter. In particular, it is well known that assessing welded structures is a tricky problem which involves several design parameters: owing to its nature the MWCM is a flexible engineering tool which can be calibrated efficiently by incorporating in it many different aspects of the addressed problem, so that it is desirable that engineers engaged in assessing real components calculate the constants of the MWCM’s governing equations by taking full advantage of their in-field experience.

5.7

References

Amstutz, H., Storzel, K., Seeger, T. (2001) Fatigue crack growth of a welded tubeflange connection under bending and torsional loading. Fatigue and Fracture of Engineering Materials and Structures 24, 357–368. DOI: 10.1046/j.1460-2695. 2001.00408.x. Anon. (1988) Design of steel structures. ENV 1993-1-1, Eurocode 3. Anon. (1999) Design of aluminium structures – Part 2: Structures susceptible to fatigue. prENV 1999, Eurocode 9. Archer, R. (1987) Fatigue of a welded steel attachment under combined direct stress and shear stress. In: Proceedings of Fatigue of Welded Constructions, edited by S. J. Maddox, Brighton, UK, pp. 63–72. Atzori, B. (1983). Trattamenti termici e resistenza a fatica delle strutture saldate. Rivista Italiana della Saldatura, edited by A.T.A – Genova (Italy), N. 1, 3–16. Bäckström, M., Marquis, G. (2001) A review of multiaxial fatigue of weldments: experimental results, design code and critical plane approaches. Fatigue and Fracture of Engineering Materials and Structures 24, 279–291. DOI: 10.1046/j.14602695.2001.00284.x. Bäckström, M., Siljander, A., Kuitunen, R., Ilvonen, R. (1997) Multiaxial fatigue experiments of square hollow section tube-to-plate welded joints. In: Welded High-Strength Steel Structures, edited by A. F. Blom, EMAS, London, 163–177. Costa, J. D. M., Abreu, L. M. P., Pinho, A. C. M., Ferreira, J. A. M. (2005) Fatigue behaviour of tubular AlMgSi welded specimens subjected to bending–torsion loading. Fatigue and Fracture of Engineering Materials and Structures 28, 399–407. DOI: 10.1111/j.1460-2695.2005.00875.x. Crupi, G., Crupi, V., Guglielmino, E., Taylor, D. (2005) Fatigue assessment of welded joints using critical distance and other methods. Engineering Failure Analysis 12, 129–142. DOI: 10.1016/j.engfailanal.2004.03.005.

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Hobbacher, A. (1996) Fatigue Design of Welded Joints and Components. Woodhead, Cambridge, UK. Hobbacher, A. (2007) Recommendations for fatigue design of welded joints and components. IIW Document XIII-2151-07/XV-1254-07, May 2007. Huo, C.-Y. (2007) Fatigue analysis of welded joints with the aid of real threedimensional weld toe geometry. International Journal of Fatigue 29, 772–785. DOI: 10.1016/j.ijfatigue.2006.06.007. Kueppers, M., Sonsino, C.M. (2003) Critical plane approach for the assessment of the fatigue behaviour of welded aluminium under multiaxial loading. Fatigue and Fracture of Engineering Materials and Structures 26, 507–513. DOI: 10.1046/j.14602695.2003.00674.x. Lazzarin, P., Livieri, P. (2001) Notch stress intensity factors and fatigue strength of aluminium and steel welded joints. International Journal of Fatigue 23, 225–232. DOI: 10.1016/S0142-1123(00)00086-4. Lazzarin, P., Tovo, R. (1996) A unified approach to the evaluation of linear elastic stress fields in the neighbourhood of cracks and notches. International Journal of Fracture 78, 3–19. DOI: 10.1007/BF00018497. Lazzarin, P., Tovo, R. (1998) A notch stress intensity factor approach to the stress analysis of welds. Fatigue and Fracture of Engineering Materials and Structures 21, 1089–1104. DOI: 10.1046/j.1460-2695.1998.00097.x. Lazzarin, P., Zambardi, R. (2001) A finite-volume-energy based approach to predict the static and fatigue behaviour of components with sharp V-shaped notches. International Journal of Fracture 112, 275–298. DOI: 10.1023/A:1013595 930617. Lazzarin, P., Sonsino, C. M., Zambardi, R. (2004) A notch stress intensity approach to assess the multiaxial fatigue strength of welded tube-to-flange joints subjected to combined loadings. Fatigue and Fracture of Engineering Materials and Structures 27, 127–140. DOI: 10.1111/j.1460-2695.2004.00733.x. Lazzarin, P., Livieri, P., Berto, F., Zappalorto, M. (2008) Local strain energy density and fatigue strength of welded joints under uniaxial and multiaxial loading. Engineering Fracture Mechanics 75(7), 1875–1889. DOI: 10.1016/j.engfracmech. 2006.10.019. Livieri, P., Lazzarin, P. (2005) Fatigue strength of steel and aluminium welded joints based on generalised stress intensity factors and local stain energy. International Journal of Fracture 133, 247–276. DOI: 10.1007/s10704-005-4043-3. Maddox, S. J. (1987). The Effect of Plate Thickness on the Fatigue Strength of Fillet Welded Joints. Abington Publishing, Cambridge, UK. Morgenstern, C., Sonsino, C. M., Hobbacher, A., Sorbo, F. (2006) Fatigue design of aluminium welded joints by the local stress concept with the fictitious notch radius of rf = 1 mm. International Journal of Fatigue 28, 881–890. DOI: 10.1016/j. ijfatigue.2005.10.006. Niemi, E. (1995) Stress Determination for Fatigue Analysis of Welded Components. Abington Publishing, Cambridge, UK. Person, N. L. (1971) Fatigue of aluminium alloy welded joints. Welding Research, Supplement 50 2, 77s–87s. Radaj, D. (1996) Review of fatigue strength assessment of nonwelded and welded structures based on local parameters. International Journal of Fatigue 18, 153–170. DOI: 10.1016/0142-1123(95)00117-4. Radaj, D., Sonsino, C. M., Fricke, W. (2007) Fatigue Assessment of Welded Joints by Local Approaches. Woodhead, Cambridge, UK.

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Razmjoo, G. R. (1996). Fatigue of load-carrying fillet welded joints under multiaxial loadings. TWI, Abington, Cambridge, UK, TWI REF. 7309.02/96/909. Siljander, A., Kurath, P., Lowrence, F. V. (1992) Non-proportional fatigue of welded structures. In: Advances in Fatigue Lifetime Predictive Techniques, edited by M. R. Mitchell and R. Landgraf, ASTM STP 1122, Philadelphia, PA, 319–338. Sonsino, C. M. (1995) Multiaxial fatigue of welded joints under in-phase and outof-phase local strains and stresses. International Journal of Fatigue 17, 55–70. DOI: 10.1016/0142-1123(95)93051-3. Sonsino, C. M., Kueppers, M. (2001) Multiaxial fatigue of welded joints under constant and variable amplitude loadings. Fatigue and Fracture of Engineering Materials and Structures 24, 309–327. DOI: 10.1046/j.1460-2695.2001.00393.x. Susmel, L. (2007) Eurocode 3’s standard curves and Theory of Critical Distances to estimate fatigue lifetime of steel weldments. Key Engineering Materials 348–349, 21–24. Susmel, L. (2008) Modified Wöhler Curve Method, Theory of Critical Distances and EUROCODE 3: a novel engineering procedure to predict the lifetime of steel welded joints subjected to both uniaxial and multiaxial fatigue loading. International Journal of Fatigue 30, 888–907. DOI: 10.1016/j.ijfatigue.2007.06.005. Susmel, L. (2009) The Modified Wöhler Curve Method calibrated by using standard fatigue curves and applied in conjunction with the Theory of Critical Distances to estimate fatigue lifetime of aluminium weldments. International Journal of Fatigue 31, 197–212. DOI: 10.1016/j.ijfatigue.2008.04.004. Susmel, L., Tovo, R. (2004) On the use of nominal stresses to predict the fatigue strength of welded joints under biaxial cyclic loadings. Fatigue and Fracture of Engineering Materials and Structures 27, 1005–1024. DOI: 10.1046/j.1460-2695. 2001.00397.x. Susmel, L., Tovo, R. (2006) Local and structural multiaxial stress states in welded joints under fatigue loading. International Journal of Fatigue 28, 564–575. DOI: 10.1016/j.ijfatigue.2005.08.009. Taylor, D., Barrett, N., Lucano, G. (2002) Some new methods for predicting fatigue in welded joints. International Journal of Fatigue 24, 509–518. DOI: 10.1016/S01421123(01)00174-8. Tovo, R., Lazzarin, P. (1999) Relationships between local and structural stress in the evaluation of the weld toe stress distribution. International Journal of Fatigue 21, 1063–1078. DOI: 10.1016/S0142-1123(99)00089-4. Young, J. Y., Lawrence, F. V. (1986) Predicting the fatigue life of welds under combined bending and torsion. Report No. 125, UIL-ENG 86-3602, University of Illinois at Urbana – Champaign, IL. Yousefi, F., Witt, M., Zenner, H. (2001) Fatigue strength of welded joints under multiaxial loading: experiments and calculation. Fatigue and Fracture of Engineering Materials and Structures 24, 339–355. DOI: 10.1046/j.1460-2695. 2001.00397.x.

6 The Modified Wöhler Curve Method and cracking behaviour of metallic materials under fatigue loading

Abstract: In spite of the theoretical barriers which emerge when attempting to create a sound link between experimental reality and the Modified Wöhler Curve Method’s (MWCM’s) ‘modus operandi’, in the present chapter some considerations concerning the simplified way our criterion models the cracking behaviour of metallic materials are briefly reported. The chapter attempts to summarise some possible connections between the material planes used by the MWCM to estimate fatigue strength and the observed crack paths under both uniaxial and multiaxial cyclic loading. Key words: Modified Wöhler Curve Method, material cracking behaviour.

6.1

Introduction

Previous chapters described how to use the Modified Wöhler Curve Method (MWCM) to perform the fatigue assessment of metallic materials, showing its accuracy and reliability when used to estimate fatigue damage in conventional specimens subjected to both uniaxial and multiaxial fatigue loading. Moreover, it was seen that our multiaxial fatigue criterion is successful in predicting the fatigue strength of plain as well as notched components. Concerning the problem of performing fatigue assessment in the presence of stress concentration phenomena, it is worth mentioning here the MWCM’s capability, when such a critical plane approach is applied in conjunction with the Theory of Critical Distances (TCD), of correctly handling not only the degree of multiaxiality of the stress field acting on the process zone but also the detrimental effect of tridimensional stress gradients. In the light of such a high accuracy level, a simple but cruel question arises: ‘Why does it work?’. Unfortunately, it is very difficult to answer this question by explaining, from a physical point of view, why our criterion is seen to be so successful in estimating fatigue damage in metals. In spite of the theoretical barriers which emerge when attempting to create a sound link between experimental reality and the MWCM’s ‘modus operandi’, in the present chapter some considerations of the simplified way our criterion models the cracking behaviour of metallic materials are briefly reported. The following sections attempt then to summarise some possible connec189

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tions between the material planes used by the MWCM to estimate fatigue strength and the observed crack paths under both uniaxial and multiaxial cyclic loading. Even if it is possible to find many points in common between our fatigue life estimation technique and the observed experimental reality, it is evident that more work has to be done in this area to formalise a rigorous theory capable of fully and soundly explaining why the MWCM is so effective in estimating fatigue damage.

6.2

Crack initiation in single crystals

It is well known that metals are made up of many grains (i.e. crystals), which are assemblies of atoms packed together so that the potential energy of every elementary cell is minimised. This way of defining grains represents an ideal and, somehow, simplistic schematisation of a more complex reality. In fact, real metallic crystals contain not only atoms but also different kinds of defects. Among them, certainly the most important role in the overall behaviour of crystals experiencing states of stress/ strain is played by the so-called dislocations. From a continuum viewpoint, a dislocation can be thought of as an extra half-plane of atoms whose presence results in a distortion of the atom distribution (Ashby and Jones, 1996). When grains experience a state of stress/strain, dislocations can move and their motion occurs on the so-called easy glide planes. Moreover, on every easy glide plane it is always possible to locate at least one easy glide direction. In real grains dislocations move due to the shearing forces acting on the easy glide planes, resolved along the corresponding easy glide directions. One of the most important consequences of the dislocation motion is that crystals can deform plastically when dislocations migrate from place to place. It is also worth remembering here that there exist two types of dislocations, i.e. edge and screw dislocations. In particular, both types move due to shearing forces, but edge dislocations follow a path which is perpendicular to the dislocation line, whereas screw dislocations follow a path which is parallel to the above line (Ashby and Jones, 1996). Consider now a single crystal subjected to an external uniaxial cyclic loading (Fig. 6.1). For the sake of simplicity, assume that in this grain only one easy glide plane is active. Under the cyclic shearing forces relative to this plane, dislocations can move, resulting in the formation of thin lamellae, which are called persistent slip bands (PSBs). Owing to the presence of the PSBs the profile of the grain becomes quite irregular, showing an alternation of extrusions and deep intrusions (Fig. 6.1). After the formation of the PSBs, micro-cracks initiate within the fatigued crystal due to micro-stress concentration phenomena occurring either at the deepest intrusions or at the interface between matrix and PSBs – see Ma and Laird (1984), Basinski and Basinski (1985), Mugharabi (1986), Hunsche and Neumann (1988),

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191

ΔF

PSBs Easy glide plane Extrusions Micro-crack initiation sites Single crystal Intrusions ΔF

6.1 PSBs in a single crystal subjected to fatigue loading and possible micro-crack initiation sites.

Suresh (1991) and Ellyin (1997) and references reported therein for a detailed description of the phenomena resulting in the formation of the PSBs as well as for a summary of the possible locations of micro-crack initiation sites. After the formation of the PSBs and the subsequent initiation of a micro-crack, such a crack propagates within the grain due to plastic decohesion occurring within the slip bands (Tomkins, 1968). Many experimental findings suggest that all the phenomena mentioned above are due to the cyclic shearing forces acting on the easy glide planes, resolved along the corresponding easy glide directions, and such driving forces can be expressed in terms of either microscopic shear strains or microscopic shear stresses (Suresh, 1991; Ellyin, 1997). Finally, it is important to highlight that, for the sake of simplicity, all the considerations reported in the previous paragraphs were formulated by assuming that, in the considered crystal, only one easy glide plane was present. Unfortunately, reality is much more complex because real grains contain several easy glide planes as well as several easy glide directions characterised by different orientations. Under the applied microscopic shearing forces, both edge and screw dislocations can move on intersecting easy glide planes, obstructing each other. Because many phenomena take place simultaneously within a real grain subjected to an external cyclic loading, the formation of the PSBs and the consequent micro-crack initiation process are much more difficult to model. In any case, it is reasonable to believe that, even though characterised by a higher degree of complexity, the fundamental mechanisms leading to the formation of micro-cracks in real grains

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Multiaxial notch fatigue

can be brought back to those schematically and briefly discussed above by considering an ideal crystal having only one single easy glide plane.

6.3

Stage I and Stage II in polycrystals subjected to uniaxial fatigue loading

As said at the beginning of the previous section, real metallic materials are made up of many grains which are packed together by meeting with each other at grain boundaries (whose presence results in a distortion of the crystal structure). Grains in polycrystals have different orientations, so that if at a macroscopic level conventional metallic materials can be thought of as homogeneous and isotropic, at a microscopic scale the polycrystal behaviour strongly depends on the actual material morphology as well as on the presence of hard inclusions such as grain boundaries, non-metallic particles, precipitates, defects, etc. Consider a polycrystal subjected to an external system of forces (Fig. 6.2). According to continuum mechanics, and assuming that at a macroscopic scale the considered material is homogeneous and isotropic, it is trivial to determine the resulting macroscopic stress tensor, [s(t)], at any material point and at any instant, t, of the load history (Fig. 6.2a). On the contrary, calculating the microscopic stress/strain state acting on the grains and resulting in the formation of the PSBs is not trivial at all, because the magnitudes of the normal and shear stress/strain components relative to any considered easy glide plane depend on the actual mechanical properties of every single crystal, on the orientation of the grains in the bulk material and on the influence of the grain boundaries (together with the effects due to the presence of hard inclusions). As to the calculation of the microscopic

[s(t)] Single grain msn(t)≡s n(t) (b) Easy glide plane

mt(t)∝t(t)

Component surface (a) Polycrystal

6.2 Polycrystal subjected to an external system of forces and existing relationship between macro- and micro-stress components according to the mesoscopic approach.

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193

stress/strain quantities, it is worth remembering here that the mesoscopic approach, as formalised by Papadopoulos by taking full advantage of the pioneering work done by Dang Van (Dang Van, 1993; Papadopoulos, 1997; Morel, 1998), postulates that, under macroscopic elastic deformations and when only one easy glide plane is active in the considered grain, normal micro-stresses, μs, are invariably equal to the corresponding macroscopic quantities, whereas microscopic shear stresses, μt, are proportional to the corresponding quantities calculated according to the continuum mechanics concepts (Fig. 6.2b). Consider now a polycrystal subjected to an external uniaxial fatigue loading (Fig. 6.3). Due to the applied cyclic force, after a certain number of cycles a micro-crack initiates in the material (initiation phase). Subsequently, such a crack propagates under the applied fatigue loading by gradually lengthening (propagation phase). Finally, when the maximum force applied during the loading cycle results in a local stress higher than the material ultimate tensile strength, the considered polycrystal fails due to a quasistatic fast fracture (final breakage). By performing an accurate experimental investigation, Forsyth has observed that, in general, the micro/meso-crack propagation phenomenon can be subdivided into two different stages (Forsyth, 1961): Stage I cracks grow along the crystallographic planes of maximum shear and their propagation is mainly Mode II governed; Stage II cracks instead take over from Stage I propagation and tend to orient themselves in order to experience the maximum Mode I loading (Fig. 6.3). In more detail, Stage I growth is controlled by the microscopic shear stress/strain relative to those easy glide planes subjected to the maximum shear and the Stage I crack length depends

Δs Stage II

Stage I

Component surface

Δs

6.3 Stage I and II cracking in a polycrystal subjected to uniaxial fatigue loading.

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Multiaxial notch fatigue

on the morphology of the investigated material as well as on the amplitude of the applied loading. In general, the maximum extension of Stage I micro/ meso-cracks is of the order of a few grains (Ellyin, 1997). According to Tomkins’ idea, Stage II propagation occurs due to ‘plastic de-cohesion on the planes of maximum shear strain gradient at the crack tip’; moreover, Tomkins observed that ‘the same mechanism is operative also in Stage I growth, but de-cohesion occurs on only one of the available shear planes’ (Tomkins, 1968). It is worth noting here also that, in general, fatigue cracks initiate at the surface of the fatigued component (Ellyin, 1997), unless there exist localised stress concentration phenomena within the material due to the presence of defects or hard inclusions (Suresh, 1991). Moreover, it is important to highlight that it is generally believed that fatigue is mainly a propagation phenomenon (Tomkins, 1968; Miller, 1993). In other words, the largest part of the fatigue lifetime of conventional metallic specimens is spent in making the cracks propagate rather than in the initiation process. Moreover, at room temperature propagation in metals occurs mainly in a trans-crystalline mode (Ellyin, 1997) and the micro-structural barriers, like grain boundaries, play a fundamental role, especially in the initial growth phase (Miller, 1993). To conclude, it is worth mentioning that the presence of either high stresses or stress concentration phenomena due to macroscopic geometrical features reduces the length of Stage I cracks, which are in any case always present (Hertzberg, 1996). For instance, by testing U-notched specimens subjected to uniaxial fatigue loading and made of a conventional low carbon steel, we observed that Stage I cracks were never parallel to the notch bisector, independently of the sharpness of the tested notch (Meneghetti et al., 2007). In particular, by considering specimens having a stress concentration factor calculated with respect to the gross section ranging between 3.8 and 25, we determined that the average value of the angle between Stage I planes and notch bisector was equal to about 25°. In other words, such an experimental investigation seems to strongly support the idea that Stage I cracks are always present also in sharply notched specimens, but their orientation depends mainly on the actual crystallographic features of the grains in the vicinity of the stress concentrator apices.

6.4

Mesoscopic cracking behaviour of metallic materials under multiaxial fatigue loading

The orientation of the crack paths in plain metals subjected to multiaxial fatigue loading has been extensively investigated over the last 60 years, so that nowadays many experimental outcomes explaining the cracking behaviour of smooth components experiencing multiaxial cyclic stress/

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strain states are available in the technical literature. In general, the above investigations were carried out by testing thin-walled tubular specimens subjected either to tension (or bending) and torsion or to tension with internal/external pressure. The same problem has been addressed also by using solid cylindrical samples, even though the results obtained are, in many cases, much more difficult to interpret due to the influence of the bulk material on the crack propagation phenomenon. Initially, it is worth observing that, as clearly highlighted by Brown and Miller (1973), in the presence of multiaxial stress/strain states micro/mesocracks tend to propagate either on the component surface (Case A) or inwards (Case B), where Case B seems to be much more damaging than Case A (Fig. 6.4). The validity of the above schematisation has been fully confirmed by several experimental investigations, notably the work done by McDiarmid (1985, 1989, 1991). All the experimental investigations published in the technical literature and carried out by testing a variety of metallic materials in the low-, mediumand high-cycle fatigue regimes seem to converge to a common conclusion: at room temperature, fatigue cracks always initiate on Stage I planes independently of the degree of multiaxiality of the stress/strain field acting on the fatigue process zone – see Kanazawa et al. (1977), Matake (1977), Brown and Miller (1979), Hua and Socie (1984), Socie et al. (1985), Socie (1987), Socie and Bannantine (1988), Wang and Miller (1991) and Marquis and Socie (2000) and the references reported therein for a summary of the tested materials as well as for a detailed analysis of the observed mesoscopic crack paths. In more detail, some materials show evident Stage I cracks, whose length covers several grains – as, for instance, in the samples of SAE 1045 tested by Hua and Socie (1984) – whereas, in other cases, Stage I cracks are so short that the overall cracking behaviour at a meso/ macroscopic level is mainly Mode I governed – see, for instance, the results Surface plane

Case A

μτ(t)

Stage I crack Stage I crack

Surface plane

mt(t) Case B

6.4 Case A and Case B cracking behaviour.

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Multiaxial notch fatigue

generated by Socie and co-workers by testing specimens made of AISI 304 (Socie, 1987; Socie and Bannantine, 1988). We believe that the cracking behaviour under multiaxial fatigue loading of conventional metallic materials was very well summarised by Kanazawa et al. (1977) who affirmed: ‘Stage I cracks form on crystallographic planes, being slip planes within individual grains of metal. These are not necessarily the planes of maximum shear in the macroscopic sense, but rather the slip system most closely aligned to these planes. Clearly, the slip systems which experience the greatest amount of deformation are those which align precisely with the maximum shear direction, and therefore most fatigue cracks initiate in these grains. But slip systems with lesser degrees of shear also initiate cracks at a slower rate’. The above considerations, which are based on an extensive experimental investigation, clearly prove that the orientation of Stage I planes can be predicted correctly only by fully accounting for the morphology of the material within the fatigue process zone. Moreover, in light of the fact that initiation and growth of Stage I cracks is mainly a short crack problem, the only way to rigorously model their formation is by taking into account not only the actual orientation of the slip systems in those grains close to initiation sites, but also the elasto-plastic behaviour of the grains themselves, which is different from the one shown by the bulk material (Miller, 1993). Another interesting consequence of the above experimental outcome is that under non-proportional loading, that is, when the principal directions rotate during the loading cycle, several slip systems are activated simultaneously so that microscopic Stage I cracks tend to propagate in several directions. On the contrary, under proportional loading the above microcracks are seen to propagate mainly along a preferential plane, resulting in smaller deviations of the growth directions with respect to the one of maximum shear. An interesting exception is represented by cast irons: under fatigue loading micro-meso cracks initiate from the interface between hard inclusions of graphite and matrix and their propagation is mainly Mode I governed (Marquis and Socie, 2000; Marquis and KarjalainenRoikonen, 2003). In any case, and as said above, in conventional metallic materials the crack initiation phenomenon is, in general, seen to occur on Stage I planes, and this holds true also in the presence of superimposed static stresses (Socie et al., 1985). As to the mean stress influence on the propagation of microscopic shear cracks, by carrying out an accurate and extensive experimental investigation Kaufman and Topper (2003) observed that, when the mean stress perpendicular to the Stage I planes is larger than a certain material threshold value, an increase of this mean stress does not result in a further increase of fatigue damage. This experimental evidence was ascribed to the fact that, in the presence of large values of the mean

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stress perpendicular to the growth direction, micro/meso shear cracks are already fully open, so that the shearing forces are directly transmitted to the crack tips by favouring the Mode II growth. On the contrary, when the mean stress normal to the Stage I planes is lower than the above material threshold value, the effect of the shearing forces pushing the crack tips is reduced due to the interactions among the asperities present on the two faces of the cracks themselves. As already explained in Chapter 3, this effect was incorporated in the MWCM through the mean stress sensitivity index, m. When metals weakened by notches are subjected to multiaxial fatigue loading, the material cracking behaviour in the vicinity of crack initiation sites is seen to be very similar to that briefly described above, even if the problem is complicated by the presence of cyclic multiaxial stress gradients. In particular, by testing V-notched specimens under in-phase bending and torsion, Matake and Imai (1980) initially observed that also in materials containing notches micro-cracks can propagate either on the surface or inwards (Case A and Case B in Fig. 6.4, respectively) and the orientation of the growth directions depends on material morphology, geometrical features and degree of multiaxiality of the stress field acting on the fatigue process zone. Subsequently, by observing that Stage I cracks tend to propagate along those slip systems most closely aligned to the planes of maximum shear, they suggested estimating fatigue damage by using a stress parameter depending on the maximum shear stress amplitude as well as on the stress perpendicular to the plane of maximum shear stress amplitude, both averaged over a line: as far as the writer is aware, Matake and Imai’s criterion represented the first attempt at using the critical plane concept in conjunction with the Line Method to perform multiaxial fatigue assessment of notched components. By testing sharply V-notched specimens made of a commercial low carbon steel under in-phase Mode I and II loading (Susmel and Taylor, 2007), we observed a material cracking behaviour similar to that seen in plain metals under uniaxial fatigue loading: in the medium/high-cycle fatigue regime two stages, characterised by different crack propagation directions, were always evident, independently of the degree of multiaxiality of the stress field acting on the material in the vicinity of the tested notches. Moreover, the Stage I-like crack was seen to increase in length as the contribution of Mode II stress components to the overall fatigue damage increased. To conclude, due to its features, the Stage I process is one of the aspects of the fatigue problem that could be used efficiently in fatigue assessment of uncracked bodies. Unfortunately, even if this idea is very appealing from an engineering point of view, the physical phenomena taking place within the fatigue process zone and resulting in the formation of Stage I cracks are so complex that, inevitably, simplifying assumptions have to be made in

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Multiaxial notch fatigue

order to devise a fatigue life estimation technique which can be used in situations of practical interest by fully meeting industry’s requirements. In the following section we therefore discuss in detail how the MWCM models the Stage I process.

6.5

The MWCM and structural volume

If Stage I is assumed to be the most important stage in trying to estimate fatigue damage in uncracked mechanical components, it is evident that two different philosophies can be followed: either we attempt to estimate the stress/strain components relative to the most unfavourably oriented crystallographic systems, or an ideal and simplified Stage I process is modelled in order to reduce the complexity of the required calculations. For instance, the mesoscopic approach as formalised by Dang Van and Papadopoulos (Dang Van, 1993; Papadopoulos, 1997) represents a sophisticated attempt at devising an analytical method suitable for calculating the stress quantities relative to an easy glide plane. On the contrary, the MWCM estimates fatigue damage making use of the structural volume concept, which is an efficient engineering tool allowing the addressed problem to be greatly simplified without much loss of accuracy (Neuber, 1958; Susmel and Lazzarin, 2002; Susmel, 2004). In more detail, consider a notched component subjected to an external system of cyclic forces resulting in a multiaxial stress field damaging the material in the vicinity of the notch tip (Fig. 6.5). According to the considerations reported in the previous sections, the formation of Stage I cracks depends on the micro-stress components relative to the slip planes most closely aligned to the macroscopic material planes experiencing the maximum shear stress amplitude. Owing to the fact that, Stage I crack

Structural volume

Structural volume centre

ta

Notch tip

m⋅sn,m Macroscopic critical plane

sn,a

Critical distance

6.5 Structural volume and MWCM.

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199

when assessing real mechanical components, the orientation of the above crystallographic planes is never known, by using the Volume Method (VM) argument (Bellett et al., 2005) it can be assumed that a macroscopic stress state representative of the fatigue damage in those grains situated in the vicinity of the notch tip can be calculated by simply averaging the linearelastic stress over the structural volume. Moreover, according to the TCD (Taylor, 1999; Bellett et al., 2005), the reference stress state estimated as above, that is in terms of the VM, is the same as that determined at a given distance from the stress concentrator apex, that is, calculated according to the Point Method (PM) (Fig. 6.5). If the stress state determined at the centre of the structural volume is then assumed to give engineering information representative of the average microscopic stress state acting on the grains close to the crack initiation site, the hypothesis can be formed that the shear and normal macroscopic linearelastic stresses relative to the plane of maximum shear stress amplitude are somehow proportional to the corresponding microscopic quantities acting on the most damaged glide planes (Fig. 6.5) – see Susmel and Lazzarin (2002) and Susmel (2004). In other words, even though based on the calculation of linear-elastic stresses, it is possible to presume that the MWCM, applied in terms of the PM, is seen to be so accurate in estimating fatigue strength because it is somehow capable of capturing, in an engineering way, the essence of those microscopic phenomena leading to the formation of Stage I cracks. According to the above idea, the structural volume can then be thought of as the material portion controlling the overall fatigue strength of the component to be assessed (Susmel, 2004). This implies that, at a given number of cycles to failure, Nf, the size of the above volume is a material property to be determined by running appropriate experiments (Susmel and Taylor, 2007). Even if many different strategies could be followed to determine the material critical distance (Susmel, 2008), in the light of its well-known accuracy and reliability we suggest using Taylor’s TCD (Taylor, 1999; Susmel and Taylor, 2007) to determine the position of the structural volume’s centre (see Chapter 4), that is, to locate the position of that material point whose macroscopic stress state is assumed to give all the engineering information suitable for performing the fatigue assessment of mechanical components. Unfortunately, it has to be said that the above strategy does not allow us to form any reasonable hypothesis concerning the actual shape of the structural volume, so that the circle drawn in Fig. 6.5 is just a schematic and simplistic representation of the structural volume idea (Susmel, 2004). Apart from the above theoretical difficulties, it is important to highlight that, under uniaxial fatigue loading, there exists a perfect correspondence between the estimates obtained by applying the conventional TCD (Taylor, 1999) and those obtained by applying the MWCM in terms of the PM

200

Multiaxial notch fatigue Δs0n R = –1

y sy,a

sx,a

x

t

ta

L/2

sn,a sx,a

sy,a s

Δs0n

6.6 V-notched flat plate subjected to fully reversed uniaxial fatigue loading and Mohr’s circles determined, in terms of stress amplitudes, at the critical point.

(Susmel and Taylor, 2003; Susmel, 2004). In more detail, consider a Vnotched specimen subjected to fully reversed uniaxial fatigue loading (Fig. 6.6). The amplitude of the nominal cyclic force is set so that the tested sample is at its fatigue limit. The opening angle of the considered notch is assumed to be lower than 90° and the root radius to be small compared to the other notch dimensions: according to the above hypotheses the singularity of the stress field in the vicinity of the notch apex is very close to that of a conventional crack. At the fatigue limit, the stress state relative to the point positioned, along the notch bisector, at a distance from the notch tip equal to L/2 – where L has to be estimated according to Eq. (2.23) – can easily be rearranged, allowing the following identities to be written (Fig. 6.6), identities which are valid provided that the stress state in the vicinity of the notch apex is assumed to be biaxial (plane stress):

σ y,a = σ 1,a; σ x,a = σ 2,a; τ a = σ y,a 2 ; σ n,m = 0; σ n,a = σ y,a 2

6.1

By substituting the above identities into Eq. (3.22), it is straightforward to obtain identity (Eq. 2.33) (Susmel and Taylor, 2003):

(

τa + τ0 −

)

(

)

σ y ,a σ σ y ,a 2 σ 0 ⎛ mσ n,m + σ n,a ⎞ = τ0 + τ0 − 0 ⎜ ⎟⎠ = τ 0 ⇒ τa 2 ⎝ 2 2 σ y ,a 2

(

)

σ σ σ σ ⇒ 1,a + τ 0 − 0 = τ 0 ⇒ 1,a − 0 = 0 ⇒ σ 1,a = σ 0 2 2 2 2

6.2

The Modified Wöhler Curve Method and cracking behaviour

201

where, as usual, s0 and t0 are the fully reversed plain uniaxial and torsional fatigue limit, respectively. The above reasoning demonstrates the equivalence, at the fatigue limit, between the conventional PM as formalised by Taylor and the MWCM applied by taking full advantage of the TCD. It is important to highlight that the above procedure can be directly extended also to the finite life regime by simply applying the TCD as explained in Section 2.7.2 and the MWCM as reviewed in Section 4.4.3. Unfortunately, it has to be admitted here also that, strictly speaking, it is difficult to obtain a similar equivalence in the presence of triaxial stress states, i.e. in plane strain (Susmel and Taylor, 2006). In any case, the high accuracy level shown by the MWCM applied in conjunction with the PM seems to strongly support the idea that our method can be used successfully to estimate fatigue damage in notched components independently of the degree of multiaxiality of the stress field acting on the process zone. However, these considerations make it clear that more work has to be done in this area to better explore the existing links between our fatigue life estimation technique and the critical distance value calculated by taking full advantage of the TCD concepts. In a few words, according to the structural volume idea, the MWCM applied in terms of the PM estimates fatigue damage by assuming that the stress quantities relative to the macroscopic material plane experiencing the maximum shear stress amplitude give an engineering measurement of the driving forces resulting in the formation of Stage I cracks (Fig. 6.5). In more detail, our method postulates that the initiation phenomenon and the initial propagation of micro/meso cracks depend on the maximum shear stress amplitude, ta, relative to the macroscopic critical plane (see Figs 3.1 and 3.6). Moreover, the overall fatigue damage is a function also of the stress perpendicular to the critical plane itself, sn,a, because it cyclically opens and closes Stage I cracks. Finally, the portion of the mean stress normal to the critical plane which effectively contributes to the Stage I process is equal to msn,m, where m is the mean stress sensitivity index as defined in Chapter 3 and, according to the idea of Kaufman and Topper (2003), the value of such a material constant is closely related to the morphology of the surfaces of the micro/meso cracks. The considerations reported in the previous paragraphs were formulated by considering a notched component subjected to a generic multiaxial fatigue loading. It is evident that the same reasoning is always valid also in the absence of stress concentration phenomena. In particular, if the size of the structural volume is small compared to the absolute dimensions of the plain component to be assessed (as happens in practice), it is evident that the stress state at the superficial crack initiation point is practically the same as at the centre of the structural volume. This suggests that even when using the stress state on the surface of the component (as shown in Chapter 3),

202

Multiaxial notch fatigue

fatigue damage in plain materials is estimated by using stress quantities which are representative of the entire stress field damaging the fatigue process zone, that is, representative of the microscopic stress components relative to the slip systems which experience the maximum shear. These considerations should also better clarify that, strictly speaking, the fatigue lifetime predicted by the MWCM should be thought of as coincident with the number of cycles required to exhaust the Stage I process. To conclude, it can be said that the plane of maximum shear stress amplitude determined by considering the macroscopic stress state at the centre of the structural volume obviously gives only an approximate estimate of the actual orientation of Stage I cracks. In fact, the initiation and growth of such cracks depend on the morphology of the material within the fatigue process zone as well as on the elasto-plastic behaviour of the grains themselves, whereas our engineering methods simply model an ideal process resulting in the formation of a shear crack in a linear-elastic, homogeneous and isotropic material.

6.6

The problem of estimating fatigue limits

In practical applications, mechanical components are considered to be broken when visible macro-cracks appear on their surfaces. However, engineers learnt by experience that the presence of micro/meso-cracks cannot be dangerous for the structural integrity of the considered mechanical assembly, provided that the stress field in the vicinity of crack initiation sites does not favour the propagation phenomenon. As to the above aspect of the fatigue problem, it is worth remembering here that, from a scientific point of view, the so-called fatigue limit, when it exists, can be defined in terms of the formation of micro/meso non-propagating cracks (NPCs) which initiate at components’ weakest points (Frost, 1957; 1959; Miller, 1993; Akiniwa et al., 2001). In plain materials, or in the presence of very blunt notches, the propagation of such cracks is arrested either by the first grain boundary or by the first micro-structural barrier. On the contrary, in the presence of sharp notches, NPCs are seen to be much longer (Frost, 1957, 1959) and their length depends on the morphology as well as on the fatigue properties of the considered material. In particular, by approximating Kitagawa and Takahashi’s curve to its two straight asymptotic lines (see Section 2.6), Yates and Brown (1987) argued that the maximum of NPCs’ length is simply equal to El Haddad’s short crack constant, that is: a0 =

1 ⎛ Δ Kth ⎞ ⎜ ⎟ π ⎝ F ⋅ Δσ 0 ⎠

2

6.3

where, as usual, Δs0 is the plain fatigue limit, ΔKth is the range of threshold value of the stress intensity factor and F is the geometrical correction factor

The Modified Wöhler Curve Method and cracking behaviour

203

for the LEFM SIF. It is important to point out here that, according to definition (6.3), the maximum length of NPCs depends not only on the material fatigue properties but also, through factor F, on the geometry of the considered notched component. On the contrary, by following a different reasoning, Taylor has observed that the maximum NPC length should be equal to about 2L (Taylor, 2001), where L is the material characteristic length defined as (Taylor, 1999): L=

1 ⎛ Δ Kth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

6.4

In other words, Taylor’s idea is that the maximum length of NPCs depends only on the fatigue properties of the assessed material. It is also worth noting here that, in general, L is equal to about 10 times the average grain size, so that the formation of NPCs emanating from sharp notches is a process taking place at a mesoscopic scale. The considerations reported above suggest that micro/meso cracks are always present within the process zone also at the fatigue limit, and the phenomena leading to their formation involve a number of grains located beside the crystal where the Stage I crack initiates. If the range of the applied stress is larger than the corresponding threshold value, micro-cracks can propagate, eventually becoming macro-cracks, simply because the forces driving the growth phenomenon are large enough to oblige the cracks themselves to break through the micro-structural barriers. As said above, the formation of both micro- and meso-cracks strongly depends on the material morphology as well as on the elasto-plastic behaviour of the grains. Only when cracks become long cracks (i.e., when their length is larger than about 10L) can the considered material be treated, from the crack point of view, as homogeneous and isotropic, so that only in this case are the linear-elastic stress fields acting on the process zone really representative of the stress state forcing the cracks themselves to propagate. This is the reason why the classical LEFM theory can be used successfully only to study the behaviour of long cracks. On the contrary, plasticity plays a primary role in the overall behaviour of short cracks, therefore its contribution, together with the real material morphology, should always be taken into account when attempting to address such a problem (Miller, 1993). Consider now an infinite plate containing a central crack (F = 1) and subjected to fully reversed uniaxial fatigue loading (Fig. 6.7a). The above configuration can be considered to be representative of the ‘pure’ material cracking behaviour, because, under the above circumstances, geometry and nominal stress gradients do not affect the crack propagation phenomenon at all. At the threshold condition, the fatigue strength of the considered material is fully described by the corresponding Kitagawa–Takahashi curve as

204

Multiaxial notch fatigue Δsgross

log Δs0n

Δs0

Δs0n =

ΔKth p⋅a

L Short-crack region

2a

Long-crack region

Structural volume

L

≈10L

log a (b)

(a) Δsgross

6.7 (a) Infinite plate with a central crack; (b) corresponding Kitagawa and Takahashi diagram.

the crack semi-length, a, increases (Fig. 6.7b). If the above curve is approximated to its two straight asymptotic lines, it can be said that cracks can propagate up to a half-length equal to L, without an evident reduction of the nominal fatigue limit. By taking this simplistic idea as a starting point, the hypothesis can be formed that the size of the structural volume at the fatigue limit is closely related to the material characteristic length defined according to Eq. (6.4). In fact, as long as all the cracking processes are confined within the structural volume, the value of the nominal fatigue limit does not drop significantly. According to the above fatigue damage model, and bearing in mind the considerations reported in the previous section about the theoretical difficulties encountered when attempting to rigorously define the shape of the structural volume, it is possible to assume that in two-dimensional problems the structural volume has a circular shape, whereas in three-dimensional problems it becomes almost spherical and the radius is always equal to about L/2 (Susmel, 2004). Observing now that in plain materials at their fatigue limit microstructural NPCs have a length equal to about the average value of the grain size (Stage I crack), whereas, according to Eq. (6.3), mesoscopic NPCs have length lower than L, shape factor F being larger than unity in real sharply notched components, it is logical to presume that NPCs are always within the structural volume even when they reach their maximum length: this seems to strongly support the idea that fatigue limits can be estimated by directly focusing attention on the physical processes taking place within the fatigue process zone defined as above.

The Modified Wöhler Curve Method and cracking behaviour

205

The orientation of the micro/meso-crack paths within the structural volume depends on the material morphology, the degree of multiaxiality of the stress field acting on the process zone, and the sharpness of the stress raiser. In particular, in plain materials Stage I fatigue cracks tend to initiate in those crystallographic systems closely aligned to the planes experiencing the maximum shear and their growth is arrested by the first micro-structural barrier. In the presence of sharp notches instead the material cracking behaviour is seen to be much more complex. In more detail, as demonstrated by Frost (1957, 1959) by performing an accurate experimental investigation, in general mesoscopic NPCs tend to orient themselves in order to experience the maximum Mode I loading, but in same cases a Stage II crack does not take over from the initial Stage I propagation, so that mesoscopic NPCs are seen to be at 45° to the applied loading. Moreover, the formation of the above cracks is strongly influenced also by other factors, including initiations that do not occur at the stress raiser apices, zig-zag paths, multiple cracks, branching, etc. As to the NPC crack paths, it is worth noting that by investigating the cracking behaviour of both blunt and sharp notches subjected to Mode I and to mixed Mode I and II loading (Meneghetti et al., 2007; Susmel and Taylor, 2007), we observed that, in a near-threshold condition, Stage II cracks always took over from Stage I propagation. Moreover, it was seen that the difference, expressed in terms of number of cycles to failure, between the time required to exhaust the Stage I process and the time needed to make the NPCs reach their maximum length was never very pronounced. Further, direct inspection of the broken samples revealed that, in a near-threshold condition, the transition from Stage I to Stage II propagation occurred at a distance from the notch tip equal to about L/2 (Susmel et al., 2003; Meneghetti et al., 2007; Susmel and Taylor, 2007). The above considerations seem to strongly support the idea that one can attempt to estimate fatigue limits by directly modelling the Stage I process also in the presence of sharp notches. The experimental investigations we carried out in our laboratories (Trinity College, Dublin, Ireland, and the University of Ferrara, Italy) showed that, in general, there always exists an acceptable correspondence between the observed Stage I directions and those predicted by the MWCM, even though, according to the philosophy of the TCD, such orientations are determined by directly post-processing linear-elastic FE results obtained without taking into account size and crystallographic features of the grains in the fatigue process zone. The fact that such estimates are not very accurate is not surprising. In fact, as clearly said above, predicting the orientation of the Stage I planes in uncracked bodies is mainly a short-crack problem, so that micro/meso-crack paths could be rigorously estimated only by taking into account the elasto-plastic behaviour of the grains as well as the actual

206

Multiaxial notch fatigue

orientation of their easy glide planes. On the contrary, the MWCM applied in conjunction with the PM maximises the fatigue damage due to an ideal Stage I process taking place in homogeneous and isotropic linear-elastic materials. In our opinion, such simplifications explain why there always exists a certain level of inaccuracy in predicting the crack path orientation within the structural volume. To conclude, it can be said that, at the fatigue limit, the structural volume can be thought of as the minimum amount of material which can still be treated as homogeneous and isotropic and, according to the PM, the linear-elastic stress state calculated at the centre of the structural volume supplies a reference value representative of the entire stress field acting on the fatigue process zone, that is, a reference stress representative of the microscopic stress state damaging the grains in the vicinity of crack initiation sites.

6.7

Concluding remarks

The previous sections summarise an attempt to form some reasonable hypotheses capable of explaining why the MWCM is seen to be so successful in estimating fatigue damage in both plain and notched metallic materials. In short, we believe that our methodology works simply because it represents a pragmatic compromise between the need for rigorous modelling of the physical processes taking place within the fatigue process zone and the need for a simple engineering tool suitable for use in situations of practical interest. The simplifying assumptions that need to be made in order to fully meet the above two requirements explain why there always exists a certain level of discrepancy between predicted and actual Stage I plane orientations. However, according to the reasoning summarised in the present chapter, it is possible to say that if crack initiation is assumed to be the most important stage to be considered when attempting to estimate fatigue damage, then the critical plane approach as formalised by the MWCM can be used in conjunction with the TCD to describe physical reality in a very simplified but effective way. It is the writer’s opinion that such a strategy is successful because it allows the most important phenomena damaging the material within the structural volume to be modelled. In other words, even though based on the calculation of linear-elastic stresses, the MWCM reinterpreted in terms of the TCD is capable of capturing the essence of the most important phenomena resulting in the formation of micro/meso cracks within the structural volume, which is the reference material portion controlling fatigue strength. In conclusion, it has to be admitted that, even though many attempts have already been made to understand why our multiaxial fatigue method, and the TCD in general, works so well, much work still needs to be done in this

The Modified Wöhler Curve Method and cracking behaviour

207

area in order to correctly model the existing links between such an approach and the real processes taking place within the fatigue process zone and leading to final breakage.

6.8

References

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Kaufman, R. P., Topper, T. (2003) The influence of static mean stresses applied normal to the maximum shear planes in multiaxial fatigue. In: Biaxial and Multiaxial Fatigue and Fracture, edited by A. Carpinteri, M. de Freitas and A. Spagnoli, Elsevier and ESIS, 123–143. DOI: 10.1016/S1566-1369(03)80008-0. Ma, B. T., Laird, C. (1984) Overview of fatigue behaviour in copper single crystals. I. Surface morphology and stage I crack initiation sites for tests at constant strain amplitude. Acta Metallurgica 37, 325–336. Marquis, G., Socie, D. (2000) Long-life torsion fatigue with normal mean stresses. Fatigue and Fracture of Engineering Materials and Structures 23, 293–300. DOI: 10.1046/j.1460-2695.2000.00291.x. Marquis, G. B., Karjalainen-Roikonen, P. (2003) Long-life multiaxial fatigue of a nodular graphite cast iron. In: Biaxial and Multiaxial Fatigue and Fracture, edited by A. Carpinteri, M. de Freitas and A. Spagnoli, Elsevier and ESIS, 105–122. DOI: 10.1016/S1566-1369(03)80007-9. Matake, T. (1977) An explanation on fatigue limit under combined stress. Bulletin of JSME 20 (141), 257–263. Matake, T., Imai, Y. (1980) Fatigue strength of notched specimen under combined stress. Bulletin of JSME 23 (179), 623–629. McDiarmid, D. L. (1985) Fatigue under out-of-phase biaxial stresses of different frequencies. In: Multiaxial Fatigue, edited by K. J. Miller and M. W. Brown, ASTM STP 853, American Society for Testing and Materials, Philadelphia, PA, 606–621. McDiarmid, D. L. (1989) Crack systems in multiaxial fatigue. In: Advances in Fracture Research, Proceedings of 7th International Conference on Fracture, edited by K. Salama, K. Ravi-Chandor, D. M. R. Taplin and P. Rama Rao, Houston, TX. Pergamon Press, Oxford, 1265–1277. McDiarmid, D. L. (1991) Mean stress effects in biaxial fatigue where the stresses are out-of-phase and at different frequencies. In: Fatigue Under Biaxial and Multiaxial Loading, ESIS 10, edited by K. Kussmaul, D. McDiarmid and D. Socie, Mechanical Engineering Publications, London, 321–335. Meneghetti, G., Susmel, L., Tovo, R. (2007) High-cycle fatigue crack paths in specimens having different stress concentration features. Engineering Failure Analyses 14, 656–672. DOI: 10.1016/j.engfailanal.2006.02.004. Miller, K. J. (1993) The two thresholds of fatigue behaviour. Fatigue and Fracture of Engineering Materials and Structures 16, 931–939. DOI: 10.1111/j.1460-2695.1993. tb00129.x. Morel, F. (1998) A fatigue life prediction method based on a mesoscopic approach in constant amplitude multiaxial loading. Fatigue and Fracture of Engineering Materials and Structures 21, 241–256. DOI: 10.1046/j.1460-2695.1998.00452.x. Mugharabi, H. (1986) Cyclic deformation and fatigue: some current problems. In: Strength of Metals and Alloys (Proc. ICSMA 7), Vol. 3, edited by H. J. McQeen et al., Pergamon Press, Oxford, 1917–1942. Neuber, H. (1958) Theory of Notch Stresses: Principles for Exact Calculation of Strength with Reference to Structural Form and Material, 2nd edition. Springer Verlag, Berlin. Papadopoulos, I. V. (1997) Exploring the high-cycle fatigue behaviour of metals from the mesoscopic scale. Notes of the CISM Seminar, Udine, Italy. Socie, D. F. (1987) Multiaxial fatigue damage models. Transactions of the ASME, Journal of Engineering Materials and Technology 109, 293–298.

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Socie, D. F., Bannantine, J. (1988) Bulk deformation fatigue damage models. Materials Science and Engineering A103, 3–14. Socie, D. F., Waill, L. A., Dittmer, D. F. (1985) Biaxial fatigue of Inconel 718 including mean stress effects. In: Multiaxial Fatigue, edited by K. J. Miller and M. W. Brown, ASTM STP 853, American Society for Testing and Materials, Philadelphia, PA, 463–481. Suresh, S. (1991) Fatigue of Materials. Cambridge University Press, Cambridge. Susmel, L. (2004) A unifying approach to estimate the high-cycle fatigue strength of notched components subjected to both uniaxial and multiaxial cyclic loadings. Fatigue and Fracture of Engineering Materials and Structures 27, 391–411. DOI: 10.1111/j.1460-2695.2004.00759.x. Susmel, L. (2008) The Theory of Critical Distances: a review of its applications in fatigue. Engineering Fracture Mechanics 75, 1706–1724. DOI: 10.1016/ j.engfracmech.2006.12.004. Susmel, L., Lazzarin, P. (2002) A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures 25, 63–78. DOI: 10.1046/j.1460-2695.2002.00462.x. Susmel, L., Taylor, D. (2003) Two methods for predicting the multiaxial fatigue limits of sharp notches. Fatigue and Fracture of Engineering Materials and Structures 26, 821–833. DOI: 10.1046/j.1460-2695.2003.00683.x. Susmel, L., Taylor, D. (2006) Can the conventional high-cycle multiaxial fatigue criteria be re-interpreted in terms of the theory of critical distances? Structural Durability and Health Monitoring 2 (2), 91–108. Susmel, L., Taylor, D. (2007) Non-propagating cracks and high-cycle fatigue failures in sharply notched specimens under in-phase Mode I and II loading. Engineering Failure Analyses 14, 861–876. DOI: 10.1016/j.engfailanal.2006.11.038. Susmel, L., Taylor, D., Tovo, R. (2003) Crack paths in sharply notched specimens under in-phase biaxial loadings. In: Proceedings of International Conference on Fatigue Crack Paths, edited by A. Carpinteri and L. Pook, Parma, Italy, September 2003. Taylor, D. (1999) Geometrical effects in fatigue: a unifying theoretical model. International Journal of Fatigue 21, 413–420. DOI: 10.1016/S0142-1123(99) 00007-9. Taylor, D. (2001) A mechanistic approach to critical-distance methods in notch fatigue. Fatigue and Fracture of Engineering Materials and Structures 24, 215–224. DOI: 10.1046/j.1460-2695.2001.00401.x. Tomkins, B. (1968) Fatigue crack propagation – an analysis. Philosophical Magazine 18, 1041–1066. Wang, C. H., Miller, K. J. (1991) The effect of mean shear stress on torsional fatigue behaviour. Fatigue and Fracture of Engineering Materials and Structures 14, 293–307. DOI: 10.1111/j.1460-2695.1991.tb00659.x. Yates, J. R., Brown, M. W. (1987) Prediction of the length of non-propagating fatigue cracks. Fatigue and Fracture of Engineering Materials and Structures 10, 187–201. DOI: 10.1111/j.1460-2695.1987.tb00477.x.

7 The Modified Manson–Coffin Curve Method in fatigue assessment

Abstract: The present chapter explains the use of the Modified Manson–Coffin Curve Method to perform multiaxial fatigue assessment of plain and notched mechanical components failing in the low/ medium-cycle fatigue regime. Key words: Modified Manson–Coffin Curve Method, low-cycle fatigue, notches.

7.1

Introduction

It is well known that stress-based approaches are not adequate at describing the fatigue behaviour of metals in the low-cycle fatigue regime, so the MWCM as formalised and validated in the previous chapters cannot be used directly to address such a problem. In order to propose a fatigue life estimation technique suitable for predicting lifetime in the low-cycle fatigue field, we attempted, working in collaboration with our colleagues Prof. Bruno Atzori and Dr Giovanni Meneghetti (University of Padova, Italy), to reformulate in terms of strains the same basic idea as that on which the MWCM is based (Atzori et al., 2004, 2005, 2007). In particular, the critical plane is assumed to be the material plane experiencing the maximum shear strain amplitude, and the degree of multiaxiality of the stress state at crack initiation sites is suggested to be evaluated in terms of ratio r (where r = sn,max/ta). Taking the above idea as a starting point, finite life was then estimated by using non-conventional bi-parametric Manson–Coffin curves. The systematic validation exercise we performed, using a large number of experimental results, proved that our low-cycle fatigue criterion is highly accurate, allowing fatigue lifetime always to be estimated with an adequate degree of safety. The present chapter attempts to explain how to use such a multiaxial fatigue criterion to perform the fatigue assessment of mechanical components failing in the low/medium-cycle fatigue regime. To conclude, only the fundamental concepts on which the strain-based approach is based will be recalled; therefore those readers interested in studying the above concepts in depth are referred to textbooks that address such a problem in detail – see, for instance, Dowling (1993), Zahavi (1996) and Ellyin (1997). 210

The Modified Manson–Coffin Curve Method

7.2

211

Strain quantities used in low-cycle fatigue problems

Consider a body subjected to an external system of forces resulting in a triaxial strain state at point O (Fig. 7.1). Such a point is also assumed to be the centre of the absolute system of coordinates, Oxyz. The state of strain at the above material point is fully described by the following tensor: ⎡ εx [ε ] = ⎢⎢ 12 γ xy ⎢⎣ 12 γ xz

γ xy εy 1 2 γ yz

1 2

1 2 1 2

γ xz ⎤ γ yz ⎥ ⎥ ε z ⎥⎦

7.1

where ex, ey and ez are the three normal strains, and gxy, gxz and gyz are the total shear strains. The three principal strains and the corresponding principal directions can be determined directly by calculating the eigenvalues and the eigenvectors, respectively, of tensor (7.1). If a new frame of reference having axes coincident with the principal directions is introduced, the state of strain at point O can easily be rewritten as follows: ⎡ε 1 0 [ε ] = ⎢ 0 ε 2 ⎢ ⎢⎣ 0 0

0⎤ 0⎥ ⎥ ε 3 ⎥⎦

where ε 1 ≥ ε 2 ≥ ε 3

7.2

In other words, the shear strains relative to the planes perpendicular to the principal directions are invariably equal to zero. Consider now a superficial point O experiencing the following strain state: ⎡ εx ε = [ ] ⎢⎢ 12 γ xy ⎢⎣ 0

1 2

γ xy 0 ⎤ εy 0⎥ ⎥ ε z ⎥⎦ 0

7.3

Fj

Fk z

[e] =

ex

1g 2 xy

1g 2 xz

1g 2 xy

ey

1g 2 yz

1g 2 xz

1g 2 yz

ez

y O x Fi

7.1 Body subjected to an external system of forces and strain tensor at point O.

212

Multiaxial notch fatigue z Component surface y

q = 90°

O

n f

x

7.2 Material plane perpendicular to the component surface and having normal unit vector n at angle f to the x-axis.

The normal and shear strains relative to a generic plane perpendicular to the component surface and having normal unit vector n at angle f to axis x (see Fig. 7.2) take on the following values (Socie and Marquis, 2000):

ε n (φ ) =

εx + εy εx − εy γ xy + cos ( 2φ ) + sin ( 2φ ) 2 2 2

ε −ε γ γn (φ ) = x y sin ( 2φ ) − xy cos ( 2φ ) 2 2 2

7.4 7.5

By carefully comparing the above identities to Eqs (1.11) and (1.12) it is easy to see the correspondence between the two sets of formulas. In particular, all the equations reviewed in Chapter 1 and obtained by considering a generic triaxial stress state at a given material point can be rewritten directly in terms of strains by carefully replacing sx, sy and sz with ex, ey and ez as well as by replacing txy, txz and tyz with one-half of gxy, gxz and gyz (Socie and Marquis, 2000). The first important implication of the above substitution is that strain analyses done according to Mohr’s theory have to be carried out by plotting the required circles in g/2 vs. e diagrams. Moreover, it has to be borne in mind that the degree of multiaxiality of the applied stress state is, in general, different from that of the corresponding strain state. For instance, in a material obeying a linear-elastic constitutive law, a uniaxial stress state results in a triaxial strain state; in fact, according to Hooke’s relationships, it is trivial to obtain: e1 = s1/E and e2 = e3 = −nee1, where E is Young’s modulus and ne is Poisson’s ratio. It is worth concluding the present section by observing that strains have to be expressed in terms of amplitudes, ranges, mean values, etc., when they

The Modified Manson–Coffin Curve Method

213

are attempted to be used to perform the fatigue assessment. Again the quantities needed can be calculated directly by taking full advantage of the methods summarised in Chapter 1 where, as suggested above, the normal strains have to be substituted for the corresponding normal stresses and one-half of the shear strains for the corresponding shear stresses.

7.3

Stress–strain behaviour of metallic materials

In order to correctly use the strain-based approach to perform the fatigue assessment of mechanical components, it is crucial to understand the stress– strain response of metals under both monotonic and cyclic loading. This section then summarises the fundamental concepts which should always be kept in mind when addressing such a problem.

7.3.1 Stress–strain behaviour under quasi-static uniaxial loading Consider a plain metallic specimen having gauge length equal to l0 (Fig. 7.3) and subjected to an external axial loading, F. Due to the applied force the reference length of the sample increases from its initial value, l0, up to l and the resulting uniaxial engineering strain is equal to:

ε eng =

l − l0 l0

7.6

The corresponding engineering stress turns out to be:

σ eng =

F A0

7.7

where A0 is the initial value of the sample cross-sectional area (Fig. 7.3). If the applied force, F, is gradually increased up to complete breakage, the experimental stress–strain curve plotted in terms of engineering quantities has a profile similar to that sketched in Fig. 7.4a (the shape of such a A0

O

z y

l0

x

7.3 Plain cylindrical specimen and its gauge length.

214

Multiaxial notch fatigue strue

seng

Monotonic curve A

sUTS

Final breakage sY

D

sY

E

E eeng

sf, e f

B e true

C (b)

(a)

7.4 Uniaxial stress–strain curve plotted in terms of (a) engineering and (b) true quantities.

curve is typical of conventional carbon steels). As long as the applied stress is lower than the material yield stress, sY, the tested specimen deforms elastically and all the physical processes taking place within the material are reversible at a macroscopic level. On the contrary, when the applied stress becomes larger than sY, the material undergoes plastic (i.e., irreversible) deformation. The maximum value of the stress in the engineering stress–strain curve, i.e. the stress at onset of necking (Ashby and Jones, 2002), is called the ultimate tensile strength, sUTS (Fig. 7.4a). According to the way engineering quantities are defined, this curve can be used to describe the elasto-plastic behaviour of metallic materials as long as deformations are relatively small, that is, lower than about 4–5% strain (Ellyin, 1997). On the contrary, when materials undergo large-scale plastic deformation, definitions (7.6) and (7.7) are no longer suitable for describing the investigated phenomenon and the monotonic stress–strain behaviour has to be modelled in terms of true quantities. In more detail, if l is the reference length under the applied force, F, and A is the corresponding value of the cross-sectional area, then the resulting true strain and true stress are defined as follows, respectively (Atzori, 2000):

ε true = σ true =

l

dl

l

∫ = ln l = ln (1 + ε eng ) 0 l0 l F = σ ing(1 + ε eng ) A

7.8

7.9

If the stress–strain response of conventional metallic material is described by using definitions (7.8) and (7.9), the corresponding curve has a monotonic profile similar to the one sketched in Fig. 7.4b. We reported the above definitions for the sake of completeness, but from a practical point of view the magnitude of the strains involved in fatigue problems is in general not

The Modified Manson–Coffin Curve Method

215

very high, therefore accurate estimates can be obtained by simply using engineering quantities (Ellyin, 1997). It is common practice to describe the monotonic stress–strain curve through the well-known Ramberg–Osgood relationship, which calculates the total strain by separately taking into account the elastic and the plastic contributions, i.e.:

( )

σ σ ε tot = ε e + ε p = + E K

1 n

7.10

where E is Young’s modulus, K is the strength coefficient and n is the strain hardening exponent. Consider now the monotonic curve sketched in Fig. 7.4b. As mentioned above, when the applied stress is larger than the corresponding elastic threshold, the material begins to deform plastically and the phenomena taking place within the process zone become irreversible also at a macroscopic scale. It is worth remembering here that the plastic behaviour of metallic materials depends on the dislocation motion, where dislocations act like carriers of deformation (Ashby and Jones, 2002). If the applied loading is gradually reduced from point A down to zero (Fig. 7.4b), the unloading curve is seen to be initially linear and almost parallel to the elastic portion of the monotonic curve. At a given stress level (point B in Fig. 7.4b), whose value depends on the elasto-plastic properties of the considered material, such a curve bends, deviating from the linear behaviour. The subsequent loading curve is again characterised by an initial linearelastic portion followed by an elasto-plastic response and the new material yield point, D, is seen to be at a stress level different from sY (Fig. 7.4b). The considerations reported above suggest that fatigue damage can be estimated in terms of strains only if the cyclic elasto-plastic behaviour of the material to be assessed is correctly modelled. The fundamental concepts used in practice to address such a problem will briefly be reviewed in the next subsections.

7.3.2 Stress–strain behaviour under uniaxial cyclic deformation or loading Consider a plain specimen subjected to a uniaxial cyclic force (Fig. 7.3). The applied loading induces, at any instant of the load history and at any material point belonging to the specimen’s surface, a state of stress as well as a state of strain, where the cyclic stress state is uniaxial, whereas the cyclic strain state is triaxial. Such a test can be run by controlling either the applied loading (stresscontrolled test) or the axial strain (deformation-controlled test). This

216 e

Multiaxial notch fatigue e

e max Re = –1

εmax

Rε = –1

e max

e

e min

t

t εmin

e min

s

s

t

s

t

t

Strain hardening

Re > 0

Strain softening

t

Mean stress relaxation

7.5 Material transient behaviours under cyclic deformation.

distinction is very important because, depending on the variable used to control the test itself, the material shows different transient behaviours. In particular, if the test is run by taking the amplitude of the axial deformation constant, two different responses can be observed (Fig. 7.5): either the stress gradually increases cycle by cycle (strain hardening) or it gradually decreases (strain softening) until a stable regime is reached. Moreover, in the presence of superimposed static axial strain (Fig. 7.5), the resulting tensile mean stress is seen to decrease as the number of cycles increases (mean stress relaxation): under certain circumstances such a phenomenon can be so pronounced that, in the stabilised configuration, the actual load ratio, R = sx,min/sx,max, approaches −1 even if the applied strain ratio, Re = ex,min/ex,max, is larger than zero. The stress–strain response of engineering materials is also rather complex when they are tested by controlling the amplitude of the applied cyclic force. In particular, either strain hardening or strain softening is seen to occur (Fig. 7.6). Moreover, in the presence of superimposed static tension (Fig. 7.6) or compression, the corresponding mean axial strain increases or decreases cycle by cycle (cyclic creep): unfortunately, the physical mechanisms leading to the increment or decrement of the mean axial strain are very complex, so that sophisticated methodologies are required to model them correctly (Ellyin, 1997). When metals are subjected to uniaxial strain-controlled tests, the transient phenomena briefly mentioned above tend to exhaust themselves, so that the material stress–strain response eventually reaches a stable regime. It is worth remembering here that, in general, from 30% to 50% of the total fatigue life is spent in obtaining a stable response from the tested material.

The Modified Manson–Coffin Curve Method s

s max

R = –1

s

R = –1

s max

s max

s

t

t s min

s min

R > –1

e

e

t

s min

e

t

Strain hardening

217

t

t

Cyclic creep

Strain softening

7.6 Material transient behaviours under uniaxial cyclic stress. Cyclic stress–strain curve sx

s x,a

Increasing e x,a ex

(a)

e x,a =

s x,a

E

+

⎛ ⎜ ⎝

s x,a ⎞

K′

⎟ ⎠

1 n′

Re = –1 E e px,a

e ex,a

e x,a

(b)

7.7 Cyclic stress–strain curve.

Modelling the entire process resulting in the stabilised behaviour is very complicated; therefore it is common practice to summarise the material’s stable response under fully reversed uniaxial cyclic strain by using the socalled cyclic stress–strain curve. Such a curve is determined by interpolating the apices of the stable hysteresis loops experimentally generated by testing the investigated material at different strain amplitudes (Fig. 7.7a). Because that the above experimental procedure is extremely timeconsuming, many other methodologies have been devised and validated in order to reduce the number of samples needed to accurately define cyclic stress–strain curves – see, for instance, Landgraf et al. (1969). To conclude the present section, it is important to highlight that cyclic stress–strain curves are usually described by using a Ramberg–Osgood type equation (Fig. 7.7b), that is:

218

Multiaxial notch fatigue

ε x ,a =

( )

σ x ,a σ x ,a + E K′

1 n′

7.11

where E is Young’s modulus, K′ is the cyclic strength coefficient and n′ is the cyclic strain hardening exponent. The above relationship represents one of the governing equations of the so-called strain-based approach, whose fundamental features will be reviewed in Section 7.4.

7.3.3 Multiaxial stress–strain relationships The concepts summarised in the previous sections were reviewed considering nominal uniaxial situations. On the contrary, real mechanical components experience, at their weakest points, complex states of stress and strain; therefore sophisticated constitutive models are needed to correctly take into account the degree of multiaxiality of the stress/strain field acting on the process zone. Owing to the complexity of this problem, only the fundamental concepts will be recalled in what follows; the reader is referred to other textbooks for their mathematical formalisation – see, for instance, Dowling (1993), Ellyin (1997) and Socie and Marquis (2000) and references reported therein. The first step in devising a constitutive model is the definition of an appropriate yield surface. Amongst the available methods, the most commonly adopted yield condition is the one due to Von Mises (Iurzolla, 1991). This postulates that the initial yield function is defined by the following relationship:

σ=

1 (σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2 = σ Y 2

7.12

which can also be written as follows:

σ=

1 (σ x − σ y )2 + (σ y − σ z )2 + (σ z − σ x )2 + 6 (τ xy2 + τ xz2 + τ yz2 ) = σ Y 2

7.13 According to definition (7.13), under combined tension and torsion, the initial yield surface can be plotted directly in 3τ xy vs. sx diagrams, obtaining a circular elastic domain (Fig. 7.8a). After choosing an appropriate yield criterion, the second problem to be addressed is the definition of the hardening rule to be used to describe how the yield surface changes when the material undergoes plastic deformation. The state of the art shows that engineers engaged in assessing real mechanical components can take full advantage of many different rules specifically formulated to address such a problem. In particular, according to the isotropic hardening rule, the initial yield surface is supposed to expand

The Modified Manson–Coffin Curve Method

219

Initial yield surface

3txy

3txy

sx

(a)

3txy

Kinematic hardening

sx

Isotropic hardening

sx

(c) (b)

Yield surface after plastic deformation

7.8 Initial yield surface according to (a) Von Mises, (b) isotropic and (c) kinematic hardening.

symmetrically in all directions without changing the position of its centre (Fig. 7.8b). On the contrary, the kinematic hardening rule postulates that when materials deform plastically, the yield surface changes neither its shape nor its size, but the position of the centre is translated (Fig. 7.8c). These two rules can be coupled in order to describe more accurately the change of the yield surface due to plastic strain (Ellyin, 1997). Certainly, isotropic and kinematic hardening rules represent the most adopted hypotheses used to describe the stress–strain response of engineering materials to cyclic stress/strain states, but other techniques have been proposed and validated, including those devised by Mróz (1967a), Iwan (1967), Dafalias and Popov (1975) and Krieg (1975). It is important to remember here also that the mathematical formalisation of the existing links between stresses and plastic strain increments is in general based on the assumption that the vector describing the plastic strain increment is always perpendicular to the yield surface (Drucker, 1952; Ellyin, 1997). When metallic materials are subjected to multiaxial cyclic loading, the presence of non-zero out-of-phase angles can result in an additional cyclic hardening (Brown and Miller, 1979; McDowell, 1985; Socie, 1987; Fatemi and Socie, 1988; Ellyin, 1997), which should always be taken into account in order to correctly model the material stress–strain response under complex time-variable states of stress and strain. Figure 7.9 schematically compares, in terms of equivalent stress and equivalent strain calculated according to Von Mises, the stabilised behaviour under proportional loading to the corresponding one under 90° out-of-phase loading. As suggested by Socie and Marquis, non-proportional hardening can be incorporated efficiently in stress–strain constitutive laws through the non-proportional hardening coefficient, which is defined as the ratio, calculated at high plastic strains, between the amplitude of the equivalent stress under 90° outof-phase loading and the amplitude of the equivalent stress determined

220

Multiaxial notch fatigue sa

90° out-of-phase Non-proportional hardening In-phase

ea

7.9 Definition of non-proportional hardening.

under proportional loading (Socie and Marquis, 2000). As to Fig. 7.9, it is worth remembering here that the amplitude of Von Mises’ equivalent stress is calculated as: 1 2 σa = (σ x,a − σ y,a )2 + (σ y,a − σ z,a )2 + (σ z,a − σ x,a )2 + 6 (τ xy2 ,a + τ xz2 ,a + τ yz,a ) 2 7.14 whereas the amplitude of the equivalent strain has to be determined by adding the elastic contribution, ε¯ ea, to the plastic one, ε¯ pa, i.e.:

ε a = ε ae + ε ap

7.15

where:

ε ae =

(ε

1 (1 + ν e ) 2

e x ,a

−ε

) + (ε

2 e y ,a

e y ,a

−ε

) + (ε

2 e z ,a

e z, a

−ε

)

2 e x ,a

2 e e e ⎡⎛ γ xy ⎛ γ ⎞ ⎛ γ yz,a ⎞ ⎤ ,a ⎞ + ⎜ xz,a ⎟ + ⎜ + 6 ⎢⎜ ⎟ ⎟ ⎥ ⎣⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ 2

2

7.16

1 ε = (1 + ν p ) 2 p a

(ε xp,a − ε yp,a ) + (ε yp,a − ε zp,a ) + (ε zp,a − ε xp,a ) 2

2

2 p p p ⎡⎛ γ xy ⎛ γ xz,a ⎞ ⎛ γ yz,a ⎞ ⎤ ,a ⎞ + + + 6 ⎢⎜ ⎟ ⎜ ⎟ ⎥ ⎟ ⎜ ⎣⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ 2

2

2

7.17 In Eqs (7.16) and (7.17) ne and np are Poisson’s ratio for elastic and plastic strain, respectively. If the elasto-plastic behaviour of the considered material is assumed to be rate-independent, the above concepts can be combined directly to formalise efficient constitutive models for transient proportional and

The Modified Manson–Coffin Curve Method

221

non-proportional plasticity – see, for instance, Mróz (1967b), Eisenberg (1976), Drucker and Plagen (1981), Ohno (1982), McDowell (1985) and Ellyin and Xia (1989). On the contrary, when the effects due to ratcheting and mean stress relaxation cannot be disregarded, more sophisticated methods have to be used in order to correctly describe the material response under cyclic states of stress and strain. Among those available in the technical literature and specifically devised to address the above situations, certainly the two approaches proposed by Jiang and Sehitoglu (Jiang and Sehitoglu, 1996; Socie and Marquis, 2000) and by Ellyin and Xia (Ellyin and Xia, 1991a, 1991b; Ellyin, 1997), respectively, deserve to be mentioned. To conclude, it can be pointed out that all the phenomena briefly reviewed in the previous paragraphs should always be taken into account very carefully when attempting to model the existing relationships between stresses and strains under multiaxial loading paths: owing to the complexity of such a problem the reader is referred to the papers and books quoted in the present chapter, where the different aspects of the problem are rigorously formalised by giving the necessary equations in explicit form.

7.4

Uniaxial and torsional fatigue assessment according to Manson and Coffin’s idea

The strain-based approach postulates that fatigue lifetime can be estimated accurately by simultaneously considering the contributions to fatigue damage of both the elastic and plastic parts of the total strain amplitude (Coffin, 1954; Manson, 1954; Morrow, 1965). Consider the plain specimen sketched in Fig. 7.3 and assume that it is subjected to fully reversed uniaxial fatigue loading. According to Basquin’s idea (Basquin, 1910) the relationship between the stress amplitude, sx,a, and the number of reversals to failure, 2Nf, can be expressed as follows (Morrow, 1965):

σ x,a = σ f′( 2 N f )b

7.18

where the fatigue strength coefficient, s′f, and the fatigue strength exponent, b, are material constants to be determined by running appropriate experiments. By taking full advantage of Eq. (7.18), it is trivial to calculate the amplitude of the elastic part of the total strain, eex,a, as the number of reversals to failure increases, that is:

ε xe,a =

σ x,a σ f′ ( 2 N f )b = E E

7.19

222

Multiaxial notch fatigue

Similarly, the relationship between the plastic strain amplitude, epx,a, and 2Nf can be expressed as follows (Morrow, 1965):

ε xp,a = ε f′( 2 N f )c

7.20

where the fatigue ductility coefficient, e′f, and the fatigue ductility exponent, c, are again material properties to be determined experimentally. Finally, if the elastic contribution to the overall fatigue damage is added to the corresponding plastic contribution, the relationship between the total strain amplitude, ex,a, and the number of reversals to failure can be written directly in the following explicit form:

ε x,a = ε xe,a + ε xp,a =

σ f′ ( 2 N f )b + ε f′( 2 N f )c E

7.21

This relationship is called the Manson–Coffin equation. According to the way they are defined, Eqs (7.19), (7.20) and (7.21) can be plotted in the same ex,a vs. 2Nf diagram: Fig. 7.10 schematically shows the profile of the Manson–Coffin curve. Moreover, in this figure the meaning of the constants s′f, e′f, b and c is clearly explained. As a practical example, Fig. 7.11 reports the experimental cyclic stress– strain curve and the Manson–Coffin diagram we obtained by testing, under fully reversed axial strain, plain cylindrical specimens of En3B, a commercial low-carbon steel (Atzori et al., 2007). In particular, the cyclic curve described according to Eq. (7.11) was determined by interpolating the tips of the stabilised hysteresis loops (see Fig. 7.7a) generated at seven different strain amplitudes. The constants of the Manson–Coffin curve were calculated by separately determining the relationships between the elastic strain amplitude and 2Nf, Eq. (7.19), and between the plastic strain amplitude and 2Nf, Eq. (7.20), where the elastic and plastic contributions to the total strain were estimated as shown in Fig. 7.7b.

log ex,a e′f

Re = –1 c

s ′f E

1

Eq. (7.19) b

Eq. (7.21) 1 Eq. (7.20)

1

7.10 Uniaxial Manson–Coffin curve.

log 2Nf

The Modified Manson–Coffin Curve Method

223

600 sx,a (MPa) 400 300 Cyclic curve 200 ex,a =

100

sx,a

208500

+

⎛ ⎜ ⎝

sx,a

1

⎞ 0.197 ⎟ 1073 ⎠

Re=–1 0 0

0.005

0.010

0.015

0.020

0.025

ex,a (mm/mm) (a)

ex,a (mm/mm)

0.1 ex,a =

688 208500

(2Nf)−0.081 + 0.212⋅(2Nf)−0.486

0.01

Elastic part

0.001

Plastic part

Re=–1 0.0001 1000

10 000

100 000 2Nf (cycles)

(b)

7.11 (a) Cyclic stress–strain curve and (b) Manson–Coffin diagram determined by testing plain cylindrical samples of En3B (data from Atzori et al., 2007).

The Manson–Coffin equation can also be adapted to describe the low/ medium-cycle fatigue behaviour of engineering materials loaded in cyclic torsion. Under this loading path Eq. (7.21) can be rewritten directly as follows: e p γ xy,a = γ xy ,a + γ xy ,a =

τ f′ ( 2 N f )b0 + γ f′( 2 N f )c0 G

7.22

where t′f, g′f, b0 and c0 are material constants to be determined by running appropriate experiments. When these quantities are not known,

224

Multiaxial notch fatigue

their values can be estimated by using the following practical rules (Kim et al., 2002):

τ f′ =

σ f′ ; γ f′ = 3ε f′; b0 = b, c0 = c 3

if Von Mises’s criterion is adopted, or

τ f′ =

3 σ f′ ; γ f′ = ε f′; b0 = b, c0 = c 2 2

when the above constants are estimated through the maximum shear strain criterion. Also, when not available, the cyclic shear stress–shear strain curve can be estimated directly from the uniaxial cyclic curve rewritten in terms of the Von Mises equivalent stress and strain, i.e.:

( )

σ σ εa = a + a E K′

1 n′

7.23

The above assumption is usually adopted to model the stable material response not only under torsional fatigue loading but also in the presence of multiaxial cyclic stress/strain states.

7.5

The Modified Manson–Coffin Curve Method

The Modified Manson–Coffin Curve Method (MMCCM) takes as a starting point the idea that low/medium-cycle fatigue strength of metallic materials depends on the shear strain amplitude, ga, relative to the plane of maximum shear strain amplitude (critical plane). The degree of multiaxiality of the stress state damaging crack initiation locations is measured instead in terms of ratio r, which, as already discussed in Chapter 3, is defined as (Susmel et al., 2009a):

ρ=

σ n,max τa

7.24

In the above identity ta and sn,max are the shear stress amplitude and the maximum normal stress relative to the plane experiencing the maximum shear strain amplitude. In order to correctly evaluate the degree of multiaxiality of the stress state at crack initiation sites, ratio r has to be calculated by post-processing the stabilised stress state at the assumed hot-spot. Moreover, amongst all the potential critical planes, the one to be used to estimate multiaxial fatigue damage is the material plane experiencing the largest value of ratio r. According to the considerations reported above, the MMCCM postulates that fatigue damage can be estimated successfully in terms of ga and r by

The Modified Manson–Coffin Curve Method

225

using non-conventional bi-parametric Manson–Coffin curves: the fundamental ingredients of such an approach will be summarised in Section 7.5.1, while its governing equations will be discussed in Section 7.5.2.

7.5.1 The modified Manson–Coffin diagram Consider a plain metallic sample subjected to an axial cyclic force resulting in a strain ratio, Re, equal to −1 (Fig. 7.3). As postulated by the method devised by Manson and Coffin, the fatigue behaviour of such a material is fully described by Eq. (7.21). Such a relationship can be rewritten in terms of maximum shear strain amplitude, ga, obtaining (Zahavi, 1996):

γ a = (1 + ν e )

σ f′ ( 2 N f )b + (1 + ν p ) ε f′( 2 N f )c E

7.25

where np is Poisson’s ratio for plastic strain. It is worth noting that under uniaxial fatigue loading the resulting stress state at the assumed crack initiation point is uniaxial, whereas the corresponding strain state is triaxial, where:

ε y,a = ε z,a = −ν eff ε x,a = −ν eε xe,a − ν pε xp,a

7.26

The resulting Mohr circles under uniaxial fatigue loading are sketched in Fig 7.12a and 7.12b in terms of stress amplitude and strain amplitude, respectively: these figures make it evident that in this case there exist two critical planes which are at f = ±45° to the x-axis (see Figs 7.2 and 7.3 for the symbolism adopted). Moreover, as clearly shown in Fig. 7.12a, the

Uniaxial fatigue loading t

g /2

Critical planes sx,a s

ex,a

−neffex,a

e

(a) (b) Torsional fatigue loading t

ta

Critical planes

g /2

ga/2

s

(c)

e

(d)

7.12 Mohr’s circles under uniaxial and torsional fatigue loading.

226

Multiaxial notch fatigue ga g ′f

Torsional fatigue curve, r = 0

t ′f G

(1 + n p) e ′f s′ (1 + n e) f E

b0

Eq. (7.22) 1 c0 Eq. (7.25)

b

1

1

Uniaxial fatigue curve, r = 1 c

1 2Nf

7.13 Modified Manson–Coffin diagram and uniaxial and torsional fatigue curves.

r ratio relative to such planes is invariably equal to unity. Equation (7.25) can now be plotted in a ga vs. 2Nf diagram, obtaining a Manson–Coffin curve characterised by r = 1 (Fig. 7.13). Assume now that the considered specimen is subjected to cyclic torque. The corresponding stress and strain states at the crack initiation point are shown in Figs 7.12c and 7.12d, respectively. In particular, Mohr’s circles reported in these figures suggest that under fully reversed shear strain, the two critical planes are at 0° and 90° to the x-axis and the r ratio relative to such planes is invariably equal to zero. The torsional fatigue behaviour of the considered material can then be summarised in the ga vs. 2Nf diagram of Fig. 7.13 as well, obtaining a new fatigue curve (r = 0) whose profile is fully described by Eq. (7.22), gxy,a being equal to ga. According to the considerations reported above and similarly to what was done to formalise the MWCM, the hypothesis can now be formed that modified Manson–Coffin curves move downward in ga vs. 2Nf diagrams as the ratio r increases (Fig. 7.14). The fatigue damage model on which the above assumption is based is shown in Fig. 7.15. The initiation phenomenon and the initial propagation of micro/meso cracks is assumed to depend on the maximum shear strain amplitude, ga, whereas the contribution to the overall fatigue damage of the forces which open micro/meso cracks is assumed to be a function of the maximum normal stress, sn,max, weighed through the shear stress amplitude relative to the critical plane itself, ga. Moreover, the fact that

σ n,max = σ n,m + σ n,a makes it clear that, according to Socie’s multiaxial fatigue damage model (Socie, 1987), the use of sn,max allows also the mean stress effect to be taken into account.

The Modified Manson–Coffin Curve Method

227

ga 0 < ri < 1 < ri

r=0 ri r=1 rj Increasing r 2Nf

7.14 Modified Manson–Coffin curves.

y

Critical plane

Fatigue process zone

sn,max

x

ta ga O

Critical plane

ga ta sn,max

7.15 Adopted fatigue damage model.

The idea briefly explained above suggests that, for a given value of ratio r, the corresponding modified Manson–Coffin curve can be expressed as follows:

γa =

τ f′( ρ ) ( 2 N f )b( ρ ) + γ f′( ρ ) ⋅ ( 2 N f )c( ρ ) G

7.27

where t′f(r), g′f(r), b(r) and c(r) are suitable material functions of ratio r. The procedure we have proposed should be adopted to determine and calibrate such relationships will be discussed in the next subsection. To conclude the present section, Fig. 7.16 summarises how to use the MMCCM to estimate the lifetime of mechanical components failing in the low/medium cycle fatigue regime. From the strain state at the assumed crack initiation point, O, the maximum shear strain amplitude,

228

Multiaxial notch fatigue

Fj(t)

Fk(t) z

y O [e(t)] = x Fi(t)

[s(t)] =

sx(t)

txy(t)

txz(t)

txy(t)

sy(t)

tyz(t)

txz(t)

tyz(t)

sz(t)

ex(t)

1 g (t) 2 xy

1 g (t) 2 xz

1 g (t) 2 xy

ey(t)

1 g (t) 2 yz

1 g (t) 2 xz

1 g (t) 2 yz

ez(t)

g a, f, q

ga g ′f(r)

t a, sn,max, r

γa t′f(r) G

Eq. (7.27) b(r) 1 t′f(r), g ′f(r), b(r), c(r)

c(r) 1 2Nf,e

2Nf

7.16 In-field use of the MMCCM.

ga, can be determined by taking full advantage of one of the methods reviewed in Chapter 1. After determining the orientation of the critical plane, the shear stress amplitude and the maximum normal stress relative to the plane of maximum shear strain amplitude have to be determined by post-processing the stabilised stress tensor at point O. Subsequently, according to the calculated value of r, the corresponding modified Manson–Coffin curve can be estimated directly from Eq. (7.27). Finally, as shown by the modified Manson–Coffin diagram sketched in Fig. 7.16, such a curve allows the number of reversals to failure, 2Nf,e, to be predicted directly.

The Modified Manson–Coffin Curve Method

229

7.5.2 Governing equations and accuracy of the MMCCM in estimating fatigue lifetime The above section should make it evident that there exist several analogies between the MWCM and the MMCCM, so that in determining the calibration relationships needed to apply the MMCCM we will impose the requirement that, when materials undergo elastic cyclic deformations, the use of the above two criteria has to result in the same (or, at least, in comparable) predictions. In particular, as already discussed in Chapter 3, the MWCM can be calibrated successfully by using the fully reversed uniaxial and torsional fatigue curves. Figure 7.17 shows these curves plotted in a modified Wöhler diagram (Susmel and Lazzarin, 2002). For a given value of r, the corresponding multiaxial design curve can be estimated through the following relationships (Lazzarin and Susmel, 2003): kτ ( ρ ) = ( k − k0 ) ⋅ ρ + k0

τ A ,Ref ( ρ ) =

(σ

7.28

)

− τA ⋅ ρ + τA

A

2

7.29

where the meaning of the adopted symbols is explained in Fig. 7.17. According to Wöhler’s schematisation, the uniaxial and torsional fatigue curves can be expressed, respectively, as: 1

N k σ x,a = σ A ⎛ A ⎞ ⎝ Nf ⎠

7.30

1

N k0 τ xy,a = τ A ⎛ A ⎞ ⎝ Nf ⎠

7.31

log ta

Torsional curve 1 k0

tA,Ref(r = 0) = tA

1 k

tA,Ref(r = 1) = sA/2

Uniaxial curve NA

log Nf

7.17 Modified Wöhler curves under fully reversed uniaxial (r = 1) and torsional (r = 0) fatigue loading.

230

Multiaxial notch fatigue

By assuming now that the tested plain material undergoes only elastic deformations during the loading cycle, Basquin’s equations for the uniaxial and torsional case can be rewritten as follows:

σ x,a = σ f′( 2 N f )b

7.32

τ xy,a = τ f′( 2 N f )b0

7.33

Comparison between Figs 7.13 and 7.17 clearly proves that the k vs. b and k0 vs. b0 relationships turn out to be: b=−

1 1 ; b0 = − k k0

7.34

By substituting the above identities into Eq. (7.28), function b(r) can be calculated directly as follows: k (ρ) = −

1 1 1 1 b ⋅ b0 = ⎛⎜ − + ⎞⎟ ρ − ⇒ b ( ρ ) = (b0 − b) ρ + b b ( ρ ) ⎝ b b0 ⎠ b0

7.35

By taking full advantage of Eqs (7.32), (7.33) and (7.34), the constants of Wöhler’s uniaxial and torsional fatigue curves can be recalculated at a number of reversals to failure, 2Nf, equal to unity, obtaining:

σA =

σ f′ b = σ f′( 2 N A ) ( 2 N A )1 k

7.36

τA =

τ f′ b = τ f′( 2 N A ) 0 1 k0 ( 2 NA )

7.37

By substituting the above identities into Eq. (7.29), it is straightforward to rewrite the tA,Ref vs. r relationship in terms of s′f and t′f, that is:

σ′ τ A ,Ref ( ρ ) = ⎡ f ( 2 N A )b − τ f′( 2 N A )b0 ⎤ ρ + τ f′( 2 N A )b0 ⎣⎢ 2 ⎦⎥ σ′ b b ⇒ τ A ,Ref ( ρ ) = ρ f ( 2 N A ) + (1 − ρ ) τ f′( 2 N A ) 0 2

7.38

If the above reference shear stress, which is extrapolated at NA cycles to failure, is recalculated at a number of reversals to failure equal to unity, it is trivial to obtain the following relationship:

τ f′( ρ ) = τ A ,Ref ( ρ ) ⋅ ( 2 N A )1 k( ρ ) = τ A ,Ref ( ρ ) ⋅ ( 2 N A )− b( ρ ) ⇒ τ f′( ρ ) =

τ A ,Ref ( ρ ) ( 2N A )b( ρ )

7.39

By substituting Eq. (7.38) into Eq. (7.39), function t′f(r) can then be expressed in explicit form, that is:

The Modified Manson–Coffin Curve Method

σ f′ + f2(1 − ρ ) τ f′ 2

τ f′( ρ ) = f1 ρ

231 7.40

where: b − b( ρ )

f1 = ( 2 N A )

b − b( ρ )

; f2 = ( 2 N A ) 0

7.41

Finally, if the left- and right-hand sides of Eq. (7.40) are divided by the torsional elastic modulus, G, function t′f(r)/G can be written as:

τ f′( ρ ) σ′ τ′ = f1 ρ (1 + ν e ) f + f2(1 − ρ ) f G E G

7.42

where G = E/[2(1 + n)]. According to the formed hypotheses, the above identity, together with Eq. (7.35), makes it evident that, given NA, the use of the MMCCM and the MWCM results in the same estimates when the material to be assessed undergoes cyclic elastic deformation (i.e., in the medium/high-cycle fatigue regime). In any case, the most important implication of the reasoning summarised above and leading to the determination of Eq. (7.42) is that assuming a linear relationship between sA/2 and tA at NA cycles to failure results in a linear relationship, weighed through functions f1 and f2, between (1 + ne)s′f /E and t′f /G. Moreover, functions f1 and f2 are invariably equal to unity when either 2NA = 1 or b = b0. Unfortunately, determining a similar link between the MWCM and the MMCCM when metals undergo plastic cyclic deformation is not trivial at all: in such circumstances the stress to strain relationship becomes nonlinear, making it difficult for the contribution of the plastic part of the total deformation to be correctly computed when fatigue lifetime is estimated by using the stress based approach. According to the above considerations, and observing that fatigue constants in the strain based approach are in any case extrapolated at 2NA = 1, the calibration functions of the linear-elastic part of the modified Manson– Coffin curves are suggested here to be defined simply as (Susmel et al., 2009a):

τ f′( ρ ) σ′ τ′ = ρ (1 + ν e ) f + (1 − ρ ) f G E G b(ρ) =

b ⋅ b0 (b0 − b) ρ + b

7.42a

7.43

In addition, similarly to the procedure followed for the uniaxial and torsional elastic curves, the hypothesis can be formed that the calibration functions suitable for estimating the plastic part of modified Manson–Coffin

232

Multiaxial notch fatigue

curves can be determined directly by interpolating (1 + np)e′f and g′f as well as c and c0, that is:

γ f′( ρ ) = ρ (1 + ν p ) ε f′ + (1 − ρ ) γ f′ c (ρ) =

7.44

c ⋅ c0 (c0 − c ) ρ + c

7.45

In other words, according to the hypotheses formed, the use of the MWCM and the MMCCM results in similar (but not identical!) estimates when the above two different formalisations of the same idea are used to predict fatigue damage in the high-cycle fatigue regime. On the contrary, nothing definitive can be said about analogies and differences between the two methods when used to predict fatigue lifetime in the presence of cyclic plastic deformations. In any case, it is logical to believe that, thanks to its nature, the strain based approach, generalised to multiaxial fatigue situations in terms of the MMCCM, is more appropriate to address the low-cycle fatigue problem. To support the validity of the above conviction, Fig. 7.18 shows the accuracy of our criterion in describing the low/medium-cycle multiaxial fatigue behaviour of tubular plain specimens subjected to combined axial and torsional cyclic loading and made of the materials listed in Table 7.1. In the same table also the values of the static and fatigue constants used to calibrate the MMCCM’s governing equations are reported. It is worth noting that not only conventional in-phase and out-of-phase data were considered but also experimental results generated under the loading paths sketched in Fig. 7.19. Moreover, the capability of the MMCCM in estimating fatigue 10 000 000 Nf (cycles) 100 000

In-phase Out-of-phase Path E-A Path E-B Path E-C Path E-D Path E-E Path E-F Non-zero mean strain

Conservative

10 000 1000 Non-conservative 100 10 10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

7.18 Accuracy of the MMCCM in estimating low-cycle multiaxial fatigue lifetime.

The Modified Manson–Coffin Curve Method

233

Table 7.1 Summary of the investigated materials and their static and fatigue properties Material SAE 1045 AISI 304 S45C Reference Kurath et al., Socie, 1987 Kim et al., 1989 1999 E (MPa) G (MPa) e′f s′f (MPa) b c g′f t′f (MPa) b0 c0 K′ (MPa) n′

204000 80300 0.298 930 −0.106 −0.49 0.413 505 −0.097 −0.445 1258 0.208

183000 82800 0.171 1000 −0.114 −0.402 0.279 640 −0.124 −0.339 1660 0.287

g 3

186000 193000 208500 70600 74300 80200 0.359 0.8072 0.1572 923 1124 834 −0.099 −0.0905 −0.0793 −0.519 −0.6652 −0.4927 0.198 0.8118 0.213 685 644 529 −0.12 −0.0876 −0.0955 −0.36 −0.5334 −0.418 1215 1115 1115 0.217 0.1304 0.161

e

e

e

Path E-B g 3

e

Path E-D

g 3

g 3

Path E-A g 3

1Cr-18Ni-9Ti S460N Chen et al., Jiang et al., 2004 2007

Path E-C g 3

e

e

Path E-E

Path E-F

7.19 Investigated multiaxial loading paths.

damage due to superimposed axial and torsional static strains was also investigated. Another important aspect is that, for the selected experimental data, not only the controlled cyclic strain states but also the resulting stabilised stresses were available in the original sources. Figure 7.18 clearly proves that the MMCCM calibrated through Eqs (7.42a)–(7.45) is highly accurate, allowing predictions within a factor of about 3 to be made. As to the data scattering, it is important to point out that almost all the above estimates are seen to fall within an error band as wide as the one obtained by directly using Eqs (7.21) and (7.22) to predict fatigue lifetime under uniaxial and torsional fatigue loading, respectively

234

Multiaxial notch fatigue 10 000 000 Nf (cycles) 100 000

SAE 1045, Uniaxial SAE 1045, Torsion AISI 304, Uniaxial AISI 304, Torsion S45C, Uniaxial S45C, Torsion 1Cr-18Ni-9Ti, Uniaxial 1Cr-18Ni-9Ti, Torsion S460N, Uniaxial S460N, Torsion

Conservative

10 000 1000 Non-conservative 100 Re = Rg = –1 10 10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

7.20 Accuracy of Manson–Coffin approach in predicting lifetime under both uniaxial and torsional fully reversed fatigue loading.

(see Fig. 7.20): this seems to strongly support the idea that the MMCCM can be considered as a powerful engineering tool allowing low/mediumcycle multiaxial fatigue damage to be estimated always with an adequate margin of safety. It is possible to conclude the present section by saying that the accuracy level shown by the error diagram of Fig. 7.18 is satisfactory, especially in the light of the fact that we cannot ask a predictive method to be more accurate than the experimental information that is used to calibrate the method itself (Fig. 7.20).

7.6

Low-cycle fatigue assessment of notched components

Performing the low-cycle fatigue assessment of real mechanical components containing geometrical features is a complex problem, because sophisticated methods are always required to accurately estimate the actual stress/strain states damaging materials in the vicinity of the notch apices (Fash et al., 1985; Tipton and Nelson, 1989). In this regard, the strainbased approach postulates that fatigue damage can be evaluated efficiently through the cyclic state of strain experienced by the material to be assessed and the same idea can also be extended directly to notches (Fig. 7.21). The state of the art shows that many different analytical approaches suitable for determining cyclic stress/strain states at the tip of notches subjected to nominal multiaxial fatigue loading have been devised and validated. Among the available tools, the methods proposed by Hoffmann and Seeger (1985a, 1985b, 1989), Barkey et al. (1994) and Köttingen et al. (1995)

The Modified Manson–Coffin Curve Method

sa

235

Plain material Manson–Coffin curve

2Nf

Fatigue process zone

7.21 The strain-based approach to predict fatigue lifetime of notched components.

deserve to be mentioned. As far as standard cylindrical shafts containing different geometrical features are concerned, such techniques proved to be highly accurate in determining notch stresses and strains under multiaxial loading paths. On the contrary, their use in the presence of complex geometries is not so straightforward, so that many attempts have also been made to estimate the required stress/strain quantities by post-processing elasto-plastic FE models (Yip and Jen, 1996, 1997). In this regard, nowadays engineers engaged in assessing real mechanical components can take full advantage of many software packages which allow notch stresses and strains to be determined accurately in a relatively simple way. In the light of this fact, we decided not to review the available analytical tools in the present section, so that those readers who are interested in studying such methods in depth are referred to the papers quoted above. As to the accuracy of the FE method, it has to be said that the use of commercial software packages is very tricky and, in particular circumstances, the calculated quantities can lead to non-conservative estimates: in general, elasto-plastic stress/strain analyses can be carried out under different hypotheses obtaining different solutions, so that sound in-field experience is required to handle such numerical methods correctly. Concerning to the use of the MMCCM to estimate fatigue damage in the presence of notches (Susmel et al., 2009b), the error diagram reported in Fig. 7.22 shows its accuracy in estimating the lifetime of cylindrical shafts with a fillet made of SAE 1045 and subjected to in-phase and 90° outof-phase fully reversed bending and torsion (Kurath et al., 1989). The root radius of the investigated samples was equal to 5 mm, resulting in a value of the linear-elastic stress concentration factor, Kt, equal to 1.42 in bending and to 1.23 in torsion (Tipton and Nelson, 1985). Figure 7.22 clearly proves

236

Multiaxial notch fatigue 10 000 000 Nf (cycles) 100 000 10 000

Plain Notched, bending Notched, torsion Notched, in-phase Notched, 90° out-of-phase

Conservative

1000

Non-conservative

100 R = –1 10 10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

7.22 Accuracy of the MMCCM in estimating multiaxial fatigue lifetime of cylindrical bars with fillet made of SAE 1045 and subjected to inphase and 90° out-of-phase bending and torsional loading (data from Kurath et al., 1989).

that our multiaxial fatigue criterion is successful also in predicting the low/ medium-fatigue lifetime of the above specimens, allowing the estimates to fall in an error band as wide as the one obtained when the MMCCM is used to predict multiaxial fatigue damage in plain specimens made of the same carbon steel. It is worth noting here also that our method was applied to the above data by estimating notch stresses and strains through the analytical method devised by Köttingen et al. (1995), used in conjunction with the rule proposed by Jiang and Sehitoglu (1996). To conclude, even though the accuracy shown by the chart of Fig. 7.22 is very encouraging, it has to be admitted that more work needs to be done in this area to better explore all the peculiarities of the MMCCM when used to estimate low-cycle fatigue damage in components containing sharp notches.

7.7

References

Ashby, M. F., Jones, D. R. H. (2002) Engineering Materials 1. Butterworth-Heinemann, Oxford. Atzori, B. (2000) Appunti di Costruzione di Macchine. Edizioni Libreria Cortina, Padova, Italy (in Italian). Atzori, B., Meneghetti, G., Susmel, L. (2004) Multiaxial fatigue life predictions by using modified Manson–Coffin curves. In: Proceedings of the 7th International Conference on Biaxial/Multiaxial Fatigue and Fracture, Berlin, 28 June–1 July 2004, 129–134.

The Modified Manson–Coffin Curve Method

237

Atzori, B., Meneghetti, G., Susmel, L. (2005) On the use of the modified Manson– Coffin curves to predict fatigue lifetime in the low-cycle fatigue regime. In: Proceedings of IGF Workshop on Multiaxial Fatigue, edited by L. Susmel and R. Tovo, Ferrara, Italy, 5–6 June, 97–106. Atzori, B., Meneghetti, G., Susmel, L., Taylor, D. (2007) The modified Manson–Coffin method to estimate low-cycle fatigue damage in notched cylindrical bars. In: Proceedings of the 8th International Conference on Multiaxial Fatigue and Fracture, edited by U. S. Fernando, Sheffield, UK, July 2007. Barkey, M. E., Socie, D. F., Hsia, K. J. (1994) A yield surface approach to the estimation of notch strains for proportional and non-proportional cyclic loading. Transactions of the ASME, Journal of Engineering Materials and Technology 116, 104–112. DOI: 10.1115/1.2904269. Basquin, O. H. (1910) The exponential law of endurance tests. Proceedings of American Society for Testing and Materials 10, 625–630. Brown, M. W., Miller, K. J. (1979) Biaxial cyclic deformation behaviour of steels. Fatigue and Fracture of Engineering Materials and Structures 1, 93–106. DOI: 10.1111/j.1460-2695.1979.tb00369.x. Chen, X., An, K., Kim, K. S. (2004) Low-cycle fatigue of 1Cr-18Ni-9Ti stainless steel and related weld metal under axial, torsional and 90° out-of-phase-loading. Fatigue and Fracture of Engineering Materials and Structures 27, 439–448. DOI: 10.1111/ j.1460-2695.2004.00740.x. Coffin, L. F. (1954) A study of the effects of cyclic thermal stresses on a ductile metal. Transactions of the ASME 76, 931–950. Dafalias, Y., Popov, E. P. (1975) A model for non-linear hardening materials for complex loading. Acta Mechanica 21, 173–192. Dowling, N. E. (1993) Mechanical Behaviour of Materials. Prentice-Hall, Englewood Cliffs, NJ. Drucker, D. C. (1952) A more fundamental approach to plastic stress-strain relations. In: Proceedings of the 1st US Congress of Applied Mechanics, American Society of Mechanical Engineers, 487–491. Drucker, D. C., Plagen, L. (1981) On stress-strain relations suitable for cyclic and other loading. Transaction of the ASME, Journal of Applied Mechanics 48, 479–485. Eisenberg, M. A. (1976) A generalisation of plastic flow theory with application to cyclic hardening and softening phenomena. Transactions of the ASME, Journal of Engineering Materials and Technology 98, 221–228. Ellyin, F. (1997) Fatigue Damage, Crack Growth and Life Prediction. Chapman & Hall, London. Ellyin, F., Xia, Z. (1989) A rate-independent constitutive model for transient nonproportional loading. Journal of the Mechanics and Physics of Solids 37, 71–91. DOI: 10.1016/0022-5096(87)90005-6. Ellyin, F, Xia, Z. (1991a) A rate-dependent inelastic constitutive model – Part I: Creep deformation including prior plastic strain effects. Transactions of the ASME, Journal of Engineering Materials and Technology 13, 314–323. DOI: 10.1115/ 1.2903412. Ellyin, F, Xia, Z. (1991b) A rate-dependent inelastic constitutive model – Part II: Elastic-plastic flow. Transactions of the ASME, Journal of Engineering Materials and Technology 13, 344–328. DOI: 10.1115/1.2903413.

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Fash J. W., Socie, D. F., McDowell, D. L. (1985) Fatigue life estimates for a simple notched component under biaxial loading. In: Multiaxial Fatigue, edited by K. J. Miller and M. W. Brown, ASTM STP 853, American Socioty for Testing and Materials, Philadelphia, PA, 497–513. Fatemi, A., Socie, D. F. (1988) A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue and Fracture of Engineering Materials and Structures 11, 149–166. DOI: 10.1111/j.1460-2695.1988.tb01169.x. Hoffmann, M., Seeger, T. (1985a) A generalised method for estimating multiaxial elastic–plastic notch stresses and strains. Part 1: Theory. Transactions of the ASME, Journal of Engineering Materials and Technology 107, 250–254. Hoffmann, M., Seeger, T. (1985b) A generalised method for estimating multiaxial elastic–plastic notch stresses and strains. Part 2: Application and general discussion. Transactions of the ASME, Journal of Engineering Materials and Technology 107, 255–260. Hoffmann, M., Seeger, T. (1989) Stress-strain analysis and life prediction of a notched shaft under multiaxial loading. In: Multiaxial Fatigue – Analysis and Experiments, edited by G. E. Leese and D. F. Socie, SAE AE-14, Society of Automotive Engineers, Warrendale, PA, 81–100. Iurzolla, E. (1991) I Criteri di Resistenza. Edizioni Libreria Cortina, Padova, Italy (in Italian). Iwan, W. D. (1967) On a class of models for the yielding behaviour of continuous and composite systems. Transactions of the ASME, Journal of Applied Mechanics 34, 612–617. Jiang, Y., Sehitoglu, H. (1996) Modelling of cyclic ratcheting plasticity, Part I: development of constitutive relations. Transactions of the ASME, Journal of Applied Mechanics 63, 720–725. DOI: 10.1115/1.2823355. Jiang, Y., Hertel, O., Vormwald, M. (2007) An experimental evaluation of three critical plane multiaxial fatigue criteria. International Journal of Fatigue 29, 1490–1502. DOI: 10.1016/j.ijfatigue.2006.10.028. Kim, K. S., Park, J. C., Lee, J. W. (1999) Multiaxial fatigue under variable amplitude loads. Transactions of the ASME, Journal of Engineering Materials and Technology 121, 286–293. DOI: 10.1115/1.2812377. Kim, K. S., Chen, X., Han, C., Lee, H. W. (2002) Estimations methods for fatigue properties of steels under axial and torsional loading. International Journal of Fatigue 24, 783–793. DOI: 10.1016/S0142-1123(01)00190-6. Köttingen, V. B., Barkey, M. E., Socie, D. F. (1995) Pseudo stress and pseudo strain based approaches to multiaxial notch analysis. Fatigue and Fracture of Engineering Materials and Structures 18, 981–1006. DOI: 10.1111/j.1460-2695.1995.tb00922.x. Krieg, R. D. (1975) A practical two surface plasticity theory. Transactions of ASME, Journal of Applied Mechanics 42, 641–646. Kurath, P., Downing, S. D., Galliart, D. R. (1989) Summary of non-hardened notched shaft – round robin program. In: Multiaxial Fatigue – Analysis and Experiments, edited by G. E. Leese and D. F. Socie, SAE AE-14, Society of Automotive Engineers, Warrendale, PA, 13–32. Landgraf, R. W., Morrow, J. D., Endo, T. (1969) Determination of the cyclic stress– strain curve. Journal of Materials 4, 176–188. Lazzarin P., Susmel L. (2003) A stress-based method to predict lifetime under multiaxial fatigue loadings. Fatigue and Fracture of Engineering Materials and Structures 26, 1171–1187. DOI: 10.1046/j.1460-2695.2003.00723.x.

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Manson, S. S. (1954) Behaviour of materials under conditions of thermal stress. NACA TN-2933, National Advisory Committee for Aeronautics. McDowell, D. L. (1985) A two surface model for transient nonproportional cyclic plasticity, Parts 1 and 2. Transactions of the ASME, Journal of Applied Mechanics 52, 298–308. Morrow, J. D. (1965) Cyclic plastic strain energy and fatigue of metals. In: Internal Friction, Damping and Cyclic Plasticity, ASTM STP 378, American Society for Testing and Materials, Philadelphia, PA, 45–84. Mróz, Z. (1967a) On the description of anisotropic work hardening. Journal of Mechanics and Physics of Solids 15, 163–175. Mróz, Z. (1967b) An attempt to describe the behaviour of metals under cyclic loading using a more general work hardening model. Acta Mechanica 7, 199–212. Ohno, N. (1982) A constitutive model of cyclic plasticity with a non-hardening strain region. Transactions of the ASME, Journal of Applied Mechanics 49, 721–727. Socie, D. F. (1987) Multiaxial fatigue damage models. Transactions of the ASME, Journal of Engineering Materials and Technology 109, 293–298. Socie, D. F. , Marquis, G. B. (2000) Multiaxial Fatigue, Society of Automotive Engineers, Warrendale, PA. Susmel, L., Lazzarin, P. (2002) A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures 25, 63–78. DOI: 10.1046/j.1460-2695.2002.00462.x. Susmel, L., Meneghetti, G., Atzori, B. (2009a) A simple and efficient reformulation of the classical Manson–Coffin curve to predict lifetime under multiaxial fatigue loading. Part I: plain materials. Transactions of the ASME, Journal of Engineering Materials and Technology (in press). Susmel, L., Meneghetti, G., Atzori, B. (2009b) A simple and efficient reformulation of the classical Manson–Coffin curve to predict lifetime under multiaxial fatigue loading. Part II: notches. Transactions of the ASME, Journal of Engineering Materials and Technology (in press). Tipton, S. M., Nelson, D. V. (1985) Fatigue life predictions for a notched shaft in combined bending and torsion. In: Multiaxial Fatigue, edited by K. J. Miller and M. W. Brown, ASTM STP 853, American Society for Testing and Materials, Philadelphia, PA, 514–550. Tipton, S. M., Nelson, D. V. (1989) Methods for estimating cyclic notch strains in the SAE specimen. In: Multiaxial Fatigue – Analysis and Experiments, edited by G. E. Leese and D. F. Socie, SAE AE-14, Society of Automotive Engineers, Warrandale, PA, 101–106. Yip, M.-C., Jen, Y.-M. (1996) Biaxial fatigue crack initiation life prediction of solid cylindrical specimens with transverse circular hole. International Journal of Fatigue 18, 111–117. DOI: 10.1016/0142-1123(95)00057-7. Yip, M.-C., Jen, Y.-M. (1997) Mean strain effect on crack initiation lives for notched specimens under biaxial nonproportional loading paths. Transactions of the ASME, Journal of Engineering Materials and Technology 119, 104–112. DOI: 10.1115/1.2805962. Zahavi, E. (1996) Fatigue Design. CRC Press, Boca Raton, FL.

8 Multiaxial fatigue of composite materials

Abstract: The present chapter attempts to systematically summarise the most relevant findings in this particular field of the structural integrity discipline to single out those parameters playing a primary role in the overall fatigue behaviour of composites subjected to complex loading paths as well as to check the accuracy of the available multiaxial fatigue criteria. Key words: composites, multiaxial fatigue.

8.1

Introduction

Designing composites against fatigue is a tricky problem which has to be addressed by using predictive methods capable of correctly modelling the complex physical processes taking place within the fatigue process zone. As to the available life estimation techniques, the state of the art shows that, since about the middle of the last century, a lot of work has been done in this area to formalise reliable methods suitable for performing the fatigue assessment of composite materials subjected to nominal uniaxial fatigue loading. On the contrary, only a few attempts have been made so far to devise approaches capable of estimating fatigue strength when composites are damaged by multiaxial cyclic loading. Working in collaboration with our colleague Prof. Marino Quaresimin (University of Padova, Italy), over the last decade we have attempted to systematically summarise the most relevant findings in this particular field of the structural integrity discipline to single out those parameters playing a primary role in the overall fatigue behaviour of composites subjected to complex loading paths as well as to check the accuracy of the available multiaxial fatigue criteria (Susmel and Quaresimin, 2001; Quaresimin and Susmel, 2002; Quaresimin et al., 2009). The present chapter attempts to review those pivotal concepts which should always be borne in mind when designing composites against multiaxial fatigue.

8.2

Stress quantities used to assess composite materials subjected to cyclic loading

Even if they are sometimes poorly related to the physical processes leading to the initiation of fatigue cracks, the most effective fatigue criteria specifically devised to design composite materials against multiaxial fatigue are 240

Multiaxial fatigue of composite materials

241

y 1 q

sy(t)

1

s1(t)

q txy(t)

2 s2(t)

2 txy(t) sx(t)

O

x

s6(t)

s6(t)

O

(a)

(b)

8.1 (a) Geometrical and (b) material principal stresses.

those based on the use of stress quantities. In order to briefly review the correct way of performing the stress analysis under complex loading paths, consider then the portion of material sketched in Fig. 8.1a. This composite is assumed to be subjected to a system of external forces resulting in a cyclic plane stress state at point O and such a point is also taken as the centre of the absolute system of coordinates, Oxyz. In composite materials the degree of multiaxiality of the stress state at crack initiation sites can be concisely summarised in terms of biaxiality ratios, which are defined as follows (Quaresimin and Susmel, 2002):

λC =

σ y ,a σ x ,a

8.1

λT =

τ xy,a σ x ,a

8.2

where sx,a, sy,a and txy,a are the amplitudes of the geometrical stress components (Fig. 8.1a). The degree of multiaxiality of the stress state damaging point O can also be expressed in terms of material principal stresses, i.e. in terms of stress components calculated with respect to system of coordinates 0123 (see Fig. 8.1b for the definition of such a frame of reference). In more detail, if s1,a and s2,a are the amplitudes of the material normal stresses, whereas s6,a is the amplitude of the material shear stress, biaxiality ratios l1 and l2 are defined as follows (Quaresimin and Susmel, 2002):

λ1 =

σ 2 ,a σ 1,a

8.3

λ2 =

σ 6 ,a σ 1,a

8.4

242

Multiaxial notch fatigue

As to the determination of the material principal stresses, it is worth remembering here that, at any instant t of the load history, the values of s1(t), s2(t) and s6(t) can be calculated directly from the corresponding geometrical stresses through the following well-known relationships:

σ 1(t ) = σ x(t ) cos 2 θ + σ y(t ) sin 2 θ + 2τ xy(t ) cos θ sin θ σ 2(t ) = σ x(t ) sin 2 θ + σ y(t ) cos 2 θ − 2τ xy(t ) cos θ sin θ

8.5

σ 6(t ) = −σ x(t ) cos θ sin θ + σ y(t ) cos θ sin θ + τ xy(t ) ( cos θ − sin θ ) 2

2

where q is the off-axis angle as defined in Fig. 8.1. By carefully observing Eqs (8.5) it is possible to notice also that, in terms of material principal stresses, composites can be damaged by two different types of multiaxiality, i.e. external and inherent multiaxiality. The first type of multiaxiality is due to the complexity of the applied loading path, whereas the latter type is due to the lay-up of the material itself. This should make it clear why performing the stress analysis in terms of geometrical stresses can be inappropriate when multiaxial nominal loadings are involved: such stresses can hide the actual degree of multiaxiality of the local stress fields, resulting in possible underestimation of fatigue damage. Finally, as to the use of geometrical and material principal stresses to estimate fatigue damage in composites, it is important to point out here that the polarity of geometrical shear stresses always has to be taken into account when performing the stress analysis (Vasiliev and Morozov, 2001), whereas any possible ambiguity is overcome when the cyclic stress state at the assumed crack initiation site is calculated with respect to the material principal axes. In order to better understand the way external and inherent multiaxiality can interact with each other, assume now that the composite material sketched in Fig. 8.1 is subjected to the following synchronous sinusoidal geometrical stress components:

σ x(t ) = σ x,m + σ x,a sin (ω t ) σ y(t ) = σ y,m + σ y,a sin (ω t − δ y, x )

8.6

τ xy(t ) = τ xy,m + τ xy,a sin (ω t − δ xy, x ) where, as usual, dy,x and dxy,x are the out-of-phase angles and w is the angular velocity. By substituting Eqs (8.6) into Eqs (8.5), the material principal stresses can be rewritten directly as follows:

σ 1(t ) = σ 1,m + σ 1,a sin (ω t ) σ 2(t ) = σ 2,m + σ 2,a sin (ω t − δ 2,1 ) σ 6(t ) = σ 6,m + σ 6,a sin (ω t − δ 6,1 )

8.7

where d2,1 and d6,1 are the phase shifts between s2(t) and s1(t) and between s6(t) and s1(t), respectively.

Multiaxial fatigue of composite materials

243

In order to show the existing relationship between geometrical and material principal stresses when unidirectional composites are subjected to fully reversed sinusoidal uniaxial fatigue loading, Fig. 8.2 compares a uniaxial load history expressed in terms of nominal stresses to the corresponding one plotted in terms of s1(t), s2(t) and s6(t). These two diagrams clearly prove that unidirectional laminates having off-axis angle q larger than zero are always damaged by multiaxial stress states (calculated with respect to the material principal directions) also when subjected to nominal uniaxial fatigue loading. Figures 8.3a and 8.3b, on the other hand, show that in a composite material having q = 45° and subjected to fully reversed in-phase tension and torsion, a lT ratio equal to 0.5 results in a phase angle between s1(t) and s6(t) equal to 180°, whereas s2(t) is invariably equal to zero. On the contrary, when the txy,a to sx,a ratio is larger than 0.5, s1(t), s2(t) and s6(t) all vary during the loading cycle, and out-of-phase angles d2,1 and d6,1 are equal to 180° (Figs 8.3c and 8.3d). Finally, under 90° out-of-phase tension and torsion (Fig. 8.3e) the load history becomes extremely complex when expressed in terms of material principal stresses: if the off-axis angle is equal to 45°, s1(t), s2(t) and s6(t) all vary and shift phases d2,1 and d6,1 are different from zero. To conclude the present section it can be highlighted that, as shown by the examples reported in Figs 8.2 and 8.3, one can attempt to estimate fatigue strength under complex loading paths only by accurately performing the stress analysis; in fact, due to their nature, composites can be damaged by very complex local stress states even when the applied nominal loading path is relatively simple.

8.3

Design parameters affecting multiaxial fatigue strength of composites

The present section briefly investigates the most important design variables influencing the overall fatigue strength of composite materials when multiaxial fatigue loadings are involved (Quaresimin and Susmel, 2002). Before considering such parameters in detail, it is important to point out that, as postulated by continuum mechanics, inherent and external multiaxiality have to result in the same fatigue damage extent as long as the local stress fields are the same in terms of both biaxiality ratios l1 and l2, the load ratio of every applied stress component, and the values of the out-of-phase angles. It is also worth noting here that in what follows, the effect of the most important design parameters will be investigated by considering mainly experimental data generated by testing tubular and cruciform specimens (Fig. 8.4) subjected to sinusoidal loading paths. However, according to the experimental results generated by Kawakami et al. (1996) by testing tubular

Multiaxial notch fatigue

Stress (MPa)

150 100

q = 45° sx,a = 100 MPa

sx(t)

50 0

sy,a = 0 MPa txy,a = 0 MPa

ty(t) = txy(t) = 0

–50

dy,x = 0° dxy,x = 0° R = –1 f = 10 Hz

–100 –150

0

0.02 0.04 0.06 0.08 1 Time (s)

150 Stress (MPa)

244

100

q = 45° s1,a = 50 MPa

s1(t) = s2(t)

50

s2,a = 50 MPa s6,a = 50 MPa

0

d2,1 = 0°

–50

–100 –150

d6,1 = 180° R = –1 f = 10 Hz

s6(t)

0

0.02 0.04 0.06 0.08 1 Time (s)

8.2 Uniaxial nominal load history re-plotted in terms of material principal stresses.

100

sx(t)

50 0

sy(t) = 0

–50

txy(t)

–100 –150

0

q = 45° sx,a = 100 MPa sy,a = 0 MPa txy,a = 50 MPa dy,x = 0° dxy,x = 0° R = –1 f = 10 Hz

0.02 0.04 0.06 0.08 0.1 Time (s)

150 Stress (MPa)

Stress (MPa)

150

50

s2(t) = 0

0 –50

s6(t)

–100 –150

0

0.02 0.04 0.06 0.08 0.1 Time (s)

(a)

(b)

sx(t) = txy,a(t)

50 0

sy(t) = 0

–50

–100 –150

0

q = 45° sx,a = 100 MPa sy,a = 0 MPa txy,a = 100 MPa dy,x = 0° dxy,x = 0° R = –1 f = 10 Hz

0.02 0.04 0.06 0.08 0.1 Time (s)

150 Stress (MPa)

Stress (MPa)

150 100

100

s1(t)

50 0 –50

s2(t) = s6(t)

–100 –150

0

(d)

100 txy,a(t) sy(t) = 0

–50

sx(t)

–100 –150

0

q = 45° sx,a = 100 MPa sy,a = 0 MPa txy,a = 100 MPa dy,x = 0° dxy,x = 90° R = –1 f = 10 Hz

0.02 0.04 0.06 0.08 0.1 Time (s)

(e)

150 Stress (MPa)

Stress (MPa)

150

0

q = 45° s1,a = 150 MPa s2,a = 50 MPa s6,a = 50 MPa d2,1 = 180° d6,1 = 180° R = –1 f = 10 Hz

0.02 0.04 0.06 0.08 0.1 Time (s)

(c)

50

q = 45° s1,a = 100 MPa s2,a = 0 MPa s6,a = 50 MPa d2,1 = 0° d6,1 = 180° R = –1 f = 10 Hz

s1(t)

100

s2(t)

100

s1(t)

50 0 –50

s6(t)

–100 –150

0

q = 45° s1,a ≈ 112 MPa s2,a ≈ 112 MPa s6,a = 50 MPa d2,1 ≈ –127° d6,1 ≈ 116.5° R = –1 f = 10 Hz

0.02 0.04 0.06 0.08 0.1 Time (s)

(f)

8.3 Multiaxial load histories plotted in terms of geometrical stresses and corresponding load histories recalculated in terms of material principal stresses.

Multiaxial fatigue of composite materials

245

z z O

O

y

y x

x

8.4 Geometries of the considered samples.

specimens under tension and torsion, the wave form seems to have a negligible effect on the overall multiaxial fatigue behaviour of composites, so that the considerations reported in the following subsections can be considered to be of general validity. Finally, all the Wöhler curves plotted in the diagrams below were calculated, for a probability of survival equal to 50%, by assuming a log-normal distribution of the number of cycles to failure with a confidence level equal to 95%. They are described in terms of negative inverse slope, k, endurance limit extrapolated at NA = 2 × 106 cycles to failure, sx,A or s1,A, and scatter ratio of the endurance limit for 90% and 10% probabilities of survival, Ts.

8.3.1 Off-axis angle and biaxiality ratios lC and lT It is well known that the overall fatigue behaviour of composites is strongly influenced by the off-axis angle q (i.e., by the material lay-up), not only under uniaxial (Hashin and Rotem, 1973; Sims and Brogdon, 1977; Awerbuch and Hahn, 1981; Philippidis and Vassilopoulos, 1999) but also under multiaxial nominal fatigue loading (Smith and Pascoe, 1989a, 1989b; Aboul Wafa et al., 1997; Qi and Cheng, 2007). Even though in the present section the effect of such a design parameter will be investigated in terms of geometrical stresses through ratios lC and lT, it is worth noting here that, from a multiaxial fatigue point of view, its influence can be more efficiently studied in terms of material principal stresses: in the next section the effect of ratios l1 and l2 on the fatigue strength of composites will be discussed in detail. In order to investigate the influence of the material lay-up on the multiaxial fatigue behaviour of composites, consider initially the diagram reported in Fig. 8.5, which summarises the results generated by Qi and Cheng (2007) by testing glass/epoxy tubular samples subjected to fully reversed in-phase tension and torsion. These experimental results show that the fatigue strength of such a composite varied as the material lay-up changed, even though all the tests were run under lT = 0.5. On the contrary, the results generated by Aboul Wafa et al. (1997) by testing tubular samples of glass/polyester under in-phase fully reversed bending and torsion (see Fig. 8.6) suggest that the influence of the material

246

Multiaxial notch fatigue 100

k = 11.6, sx,A = 31.2 MPa, Ts = 1.355

[±35] [±55] [±75]

sx,a (MPa)

k = 11.9, sx,A = 20.4 MPa, Ts = 1.189 k = 19.0, sx,A = 18.5 MPa, Ts = 1.156

10

10

100

1000

l T = 0.5, R = –1 10 000

100 000

Nf (cycles)

8.5 Influence of the material lay-up on the fatigue strength of glass/ epoxy tubes subjected to fully reversed in-phase tension and torsion (data from Qi and Cheng, 2007). 100 sx,a (MPa)

k = 9.2, sx,A = 27.9 MPa, Ts = 1.223 k = 7.1, sx,A = 16.3 MPa, Ts = 1.430

[±45] l = 2 [0/90] T [±45] [0/90] lT = 1 [±45] [0/90] lT = 0.5

R = –1

k = 5.8, sx,A = 7.9 MPa, Ts = 1.622

10 1000

10 000

100 000

1 000 000 Nf (cycles)

8.6 Influence of the material lay-up on the fatigue strength of glass/ polyester tubes subjected to fully reversed in-phase bending and torsion (data from Aboul Wafa et al., 1997).

lay-up on the overall fatigue strength of such a composite was negligible. According to this experimental outcome, the tests run under the same value of lT were then reanalysed together in the chart of Fig. 8.6, by plotting a unique fatigue curve for any considered value of the biaxiality ratio lT. Figure 8.6 also shows that such specimens were highly sensitive to the degree of multiaxiality of the nominal stress field acting on the process zone, an increase of ratio lT resulting in an evident decrease of fatigue strength. Finally, the Wöhler diagram reported in Fig. 8.7 shows the influence of the off-axis angle on the fatigue behaviour of cruciform specimens made of glass/polyester (Smith and Pascoe, 1989a), suggesting that the role played

Multiaxial fatigue of composite materials 1000 sx,a (MPa)

247

k = 9.8, sx,A = 61.6 MPa, Ts = 1.136 [0/90] l =0 [22.5/112.5] C [0/90] k = 10.1, sx,A = 56.8 MPa, [22.5/112.5] lC = 0.5 Ts = 1.176

100

k = 10.7, sx,A = 37.4 MPa, Ts = 1.462 k = 10.9, sx,A = 45.5 MPa, Ts = 1.107

10 10

100

1000

10 000

R = –1

100 000

1 000 000

Nf (cycles)

8.7 Influence of the material lay-up on the fatigue strength of glass/ polyester cruciform specimens subjected to in-phase tension–tension (data from Smith and Pascoe, 1989a).

by the material lay-up in determining the overall strength of this composite prevailed over the detrimental effect of a lC ratio larger than zero. To conclude the present subsection, it can be pointed out again that if the response of composites to multiaxial fatigue loading is investigated in terms of geometrical stresses, the most accurate way to correctly model the influence of the material lay-up on the fatigue behaviour of the composite to be assessed is by running appropriate experiments. Further, for a given value of the off-axis angle q, the effect of ratio lT is seen to be more pronounced than the effect of the biaxiality ratio lC (Quaresimin and Susmel, 2002; Quaresimin et al., 2009).

8.3.2 Biaxiality ratios l1 and l2 As said at the beginning of the previous subsection, making use of the geometrical stresses to design composites against multiaxial fatigue can result in erroneous estimates because in some circumstances the actual degree of multiaxiality of the stress state at the assumed crack initiation site cannot be evaluated correctly. This is why it is always suggested that multiaxial fatigue strength of composite materials should be estimated by determining the relevant stress states in terms of s1(t), s2(t) and s6(t). In order to show the effectiveness of such a modus operandi, Fig. 8.8 proves that, for a given composite, fatigue damage is the same independently of the material lay-up as long as the stress states at the assumed crack initiation points are characterised by the same values of both l1 and l2. These results were generated by testing glass/polyester tubular samples under fully reversed bending and torsion (Aboul Wafa et al., 1997). According to Figs 8.3a and 8.3b, in [±45] samples subjected to fully

248

Multiaxial notch fatigue 100

R = –1

s1,a (MPa)

k = 9.2, s1,A = 27.9 MPa, Ts = 1.223

10 1000

[0/90] l1 = 0, l2 = 0.5, d6,1 = 0° [±45] l1 = 0, l2 = 0.5, d6,1 = 180° 10 000

100 000

1 000 000 Nf (cycles)

8.8 Accuracy of biaxiality ratios l1 and l2 in modelling fatigue damage in glass/polyester tubes subjected to fully reversed in-phase bending and torsion (data from Aboul Wafa et al., 1997).

reversed in-phase bending and torsion, a lT ratio equal to 0.5 results in a phase angle between s1(t) and s6(t) equal to 180°, whereas s2(t) is invariably equal to zero (i.e., l1 = 0 and l2 = 0.5). By taking full advantage of Eqs (8.5) it is trivial to observe that the same cyclic stress state is obtained also in [0/90] specimens when tested under lT = 0.5: according to this remark, Fig. 8.8 clearly proves that, even if the considered results were generated by testing specimens having different material lay-ups, the fact that the two sets of tests were characterised by the same values of both l1 and l2 allows the generated data to fall within a unique scatter band. This fully confirms that post-processing the stress states expressed in terms of material principal stresses is the most efficient way to estimate the actual degree of multiaxiality of the linear-elastic stress field damaging composite materials in the vicinity of crack initiation sites. In order to explicitly investigate the influence of biaxiality ratio l1 on the overall fatigue strength of composites, consider now Fig. 8.9, in which the plotted data were generated by testing cruciform samples of glass/polyester (Smith and Pascoe, 1989a). This chart clearly shows that the effect of l1 is very slight, even though, as expected, an increase of such a biaxiality ratio resulted in a slight decrease of fatigue strength. On the contrary, Fig. 8.10 shows that, when l1 is kept almost constant, fatigue damage significantly increases with the increase of l2. In other words, the presence of a shear stress component larger than zero, even if small compared to the magnitude of s1,a, has a pronounced detrimental effect on the overall fatigue strength of composites subjected to cyclic loading (Quaresimin et al., 2009). This is fully supported also by the Wöhler diagrams reported in Figs 8.11 and 8.12, respectively, where the

Multiaxial fatigue of composite materials 1000

R = –1

k = 10.1, s1,A = 56.8 MPa, Ts = 1.176

s1,a (MPa)

249

k = 9.8, s1,A = 61.6 MPa, Ts = 1.136

100 k = 10.5, s1,A = 50.1 MPa, Ts = 1.194

10

[0/90] l1 = 0, l2 = 0 [0/90] l1 = 0.5, l2 = 0, d2,1 = 0° [0/90] l1 = 1, l2 = 0, d2,1 = 0° 10

100

1000

10 000

100 000

1 000 000

Nf (cycles)

8.9 Influence of biaxiality ratio l1 on the fatigue strength of glass/ polyester cruciform specimens subjected to in-phase tension–tension (data from Smith and Pascoe, 1989a). 1000 s1,a (MPa)

R = –1 k = 9.8, s1,A = 66.1 MPa, Ts = 1.136 k = 10.9, s1,A = 45.0 MPa, Ts = 1.107

100

[0/90] l1 = 0.5, l2 = 0, d2,1 = 0° [22.5/112.5] l1 = 0.6, l2 = 0.2, d2,1 = 0° 10 100

1000

10 000

100 000 1 000 000 Nf (cycles)

8.10 Influence of biaxiality ratio l2 on the fatigue strength of glass/ polyester cruciform specimens subjected to in-phase tension–tension (data from Smith and Pascoe, 1989a).

plotted experimental results were generated by testing [0/90] glass/ polyester tubes under in-phase combined zero-tension and zero-torsion (Kawakami et al., 1996; Amijima et al., 1991): the fatigue curves plotted in Figs 8.11 and 8.12 clearly prove that an increase of l2 results in evident increase of fatigue damage. The above considerations can be extended also to the uniaxial off-axis case, allowing the detrimental effect of angle q on the overall fatigue strength of unidirectional laminates to be investigated from a different point of view. The diagram of Fig. 8.13 simply shows the well-known fact

250

Multiaxial notch fatigue 100

k = 7.3, s1,A = 29.5 MPa, Ts = 1.482

s1,a (MPa)

k = 7.6, s1,A = 27.9 MPa, Ts = 1.519

k = 8.6, s1,A = 24.3 MPa, Ts = 1.421 k = 10.7, s1,A = 13.7 MPa, Ts = 1.317

10 100

[0/90] l2 = 0 [0/90] l2 = 0.14, d6,1 = 0° [0/90] l2 = 0.33, d6,1 = 0° [0/90] l2 = 1, d6,1 = 0° 1000

10 000

l1 = 0, R = 0 100 000 1 000 000 Nf (cycles)

8.11 Influence of biaxiality ratio l2 on the fatigue strength of glass/ polyester tubular specimens subjected to in-phase tension–torsion (data from Kawakami et al., 1996). 100

k = 6.9, s1,A = 18.7 MPa, Ts = 2.113

s1,a (MPa)

10 k = 8.0, s1,A = 13.3 MPa, Ts = 1.654

1 100

[0/90] l2 = 0 k = 8.8, s1,A = 7.0 MPa, Ts = 1.749 [0/90] l2 = 0.44, d2,1 = 0° [0/90] l2 = 1, d2,1 = 0° l = 0, R = 0 1

1000

10 000

100 000

1 000 000

Nf (cycles)

8.12 Influence of biaxiality ratio l2 on the fatigue strength of glass/ polyester tubular specimens subjected to in-phase tension–torsion (data from Amijima et al., 1991).

that, under uniaxial cyclic loading, an increase of angle q results in a decrease of fatigue strength. Such a trend can then be justified by observing that, according to Eqs (8.5), as q increases both l1 and l2 increase, resulting in an increase of fatigue damage. In other words, under off-axis loading the presence of stress components s2(t) and s6(t), which are due to inherent multiaxiality, lowers the fatigue strength of unidirectional composites. These considerations seem to strongly support the idea that the actual degree of multiaxiality of the stress fields acting on the fatigue zone can be taken into account efficiently by directly considering the material principal stress components.

Multiaxial fatigue of composite materials 1000 s1,a (MPa)

q = 0°, l1 = 0, l2 = 0 q = 5°, l1 = 0.01, l2 = 0.09

k = 10.9, s1,A = 265.2 MPa, Ts = 1.019

q = 10°, l1 = 0.03, l2 = 0.18 q = 15°, l1 = 0.07, l2 = 0.27

100

k = 9.1, s1,A = 58.8 MPa, Ts = 1.066

k = 9.6, s1,A = 29.3 MPa, Ts = 1.054

k = 11.4, s1,A = 20.0 MPa, Ts = 1.085

10 100

251

1000

10 000

R = 0.1 100 000

1 000 000

Nf (cycles)

8.13 Influence of the off-axis angle expressed in terms of biaxiality ratios l1 and l2 on the fatigue strength of undirectional glass/epoxy flat specimens subjected to uniaxial fatigue loading (data from Hashin and Rotem, 1973).

8.3.3 Non-proportional loading As already discussed in the previous chapters, when addressing the problem of performing the fatigue assessment of isotropic materials, the degree of non-proportionality of the applied loading can have a beneficial effect, a detrimental effect or no effect at all on the overall fatigue strength of the component to be assessed. In other words, different materials are characterised by different sensitivities to the presence of non-zero out-of-phase angles and the most accurate way to quantify such a sensitivity is by running appropriate experiments. By systematically reanalysing several experimental results generated by testing different composite materials subjected to non-proportional multiaxial fatigue loading, we observed that the influence of non-zero out-of-phase angles is not very pronounced, so that, in general, when estimating fatigue damage the degree of non-proportionality of the applied loading path can be neglected without much loss of accuracy (Quaresimin and Susmel, 2002; Quaresimin et al., 2009). In order to prove the validity of the above remark, consider the Wöhler diagram reported in Fig. 8.14, where the plotted data are again those generated by Smith and Pascoe (1989a). The fatigue curves in Fig. 8.14 clearly suggest that the influence of out-of-phase angle d2,1 is very slight: for the tests run under l1 = 1 and l2 = 0 a value of d2,1 equal to 180° resulted in a slight increase of fatigue damage, whereas in those tests run by imposing l1 = l2 = 1 a shift phase, d2,1, equal to 180° had a slight beneficial effect. Finally, Fig. 8.15 shows that under fully reversed bending and torsion the presence of an out-of-phase angle, d2,1, equal to 90° had no effect at all on

252

Multiaxial notch fatigue 1000 s1,a (MPa)

l1 = 1, l2 = 0, d2,1 = 0° l1 = 1, l2 = 0, d2,1 = 180° [±45] l1 = 1, l2 = 1, d2,1 = 0°, d6,1 = 180° [22.5/112.5] l1 = 1, l2 = 1, d2,1 = 180°, d6,1 = 180° [0/90] [0/90]

R = –1

k = 10.6, s1,A = 50.1 MPa, Ts = 1.244 k = 9.7, s1,A = 41.5 MPa, Ts = 1.182

100

k = 9.9, s1,A = 12.6 MPa, Ts = 1.546

10 10

k = 10.5, s1,A = 16.6 MPa, Ts = 1.308

100

1000

10 000

100 000 1 000 000 Nf (cycles)

8.14 Influence of out-of-phase angle d2,1 on the fatigue strength of glass/polyester cruciform specimens subjected to tension–tension (data from Smith and Pascoe, 1989a). 100 s1,a (MPa)

0° l = 0.5 90° 2 0° 90° l2 = 1 0° 90° l2 = 2

k = 9.2, s1,A = 28.3 MPa, Ts = 1.223

k = 10.3, s1,A = 20.5 MPa, Ts = 1.238

10 1000

k = 9.6, s1,A = 11.4 MPa, Ts = 1.382

10 000

R = –1 100 000

1 000 000 Nf (cycles)

8.15 Fatigue behaviour of [0/90] glass/polyester tubes subjected to in-phase and 90° out-of-phase bending and torsion (data from Aboul Wafa et al., 1997).

the overall multiaxial fatigue strength of the [0/90] glass/polyester tubular samples tested by Aboul Wafa et al. (1997), and it holds true independently of the value of biaxiality ratio l2.

8.3.4 Notch fatigue strength under multiaxial cyclic loading Also in composite materials stress concentrators have a detrimental effect, so that their presence should always be taken into account when performing

Multiaxial fatigue of composite materials

253

the fatigue assessment of real components to allow an adequate margin of safety to be obtained. Unfortunately, as far as the writer is aware, there exist no methods specifically devised to design notched composites against multiaxial fatigue. Moreover, in a systematic literature search, we found only two papers reporting some experimental results generated by testing, under multiaxial loading paths, specimens containing stress concentrators (Francis et al., 1977; Fujii et al., 1994). This suggests that more theoretical and experimental work needs to be done in this area in order to devise and validate predictive methods capable of correctly taking into account the presence of stress raisers weakening mechanical components made of composite material. However, apart from the obvious fact that the presence of stress raisers results in a decrease of fatigue strength, it is important to highlight that, as clearly shown in Figs 8.16 and 8.17, the response of notched composites to multiaxial cyclic loading strongly depends on biaxiality ratio l2. The results reported in these diagrams were generated by testing, under in-phase tension and torsion, tubular samples with a transverse hole, in which the first series of specimens were made of glass/polyester (Fujii et al., 1994) and the second series of graphite/epoxy (Francis et al., 1977); the data are summarised in Figs 8.16 and 8.17 by plotting them in terms of nominal material principal stresses calculated with respect to the gross section. Figure 8.16 shows that an increase of ratio l2 resulted in an evident decrease of fatigue strength, while Fig. 8.17 seems to strongly support the idea that, similarly to what was seen for plain materials under multiaxial fatigue loading, the effect of biaxiality ratio l1 is very slight.

100 s1,a (MPa)

l2 = 0.14 l2 = 0.33 l2 = 1

l1 = 0, R = 0 k = 10.2, s1,A = 13.6 MPa, Ts = 1.330

k = 9.2, s1,A = 12.4 MPa, Ts = 1.251

k = 9.1, s1,A = 8.6 MPa, Ts = 1.375 10 100 1000 10 000

100 000

1 000 000

Nf (cycles)

8.16 Influence of biaxiality ratio l2 on the fatigue strength of [0/90] glass/polyester tubular specimens with transverse hole subjected to in-phase tension–torsion (data from Fujii et al., 1994).

254

Multiaxial notch fatigue 1000 s1,a (MPa)

k = 9.8, s1,A = 42.6 MPa, Tσ = 1.799 k = 17.4, s1,A = 51.3 MPa, Ts = 1.292

l1 = 1, l2 = 1, d2,1 = 0°, d6,1 = 180° l1 = 1, l2 = 0, d2,1 = 180° l1 = 0, l2 = 0.5, d6,1 = 180° l1 = 0.3, l2 = 0.3, d2,1 = 180°, d6,1 = 180° l1 = 0.6, l2 = 0.2, d2,1 = 180°, d6,1 = 180°

100

k = 14.5, s1,A = 43.9 MPa, Ts = 1.959

k = 15.4, s1,A = 39.1 MPa, Ts = 1.267

k = 15.4, s1,A = 20.7 MPa, Ts = 1.363 R = 0.1

10 1

10

100

1000

10 000 100 000 Nf (cycles)

8.17 Influence of biaxiality ratios l1 and l2 on the fatigue strength of [±45] graphite/epoxy tubular specimens with transverse hole subjected to in-phase tension–torsion (data from Francis et al., 1977).

To conclude, it can be pointed out that, unfortunately, the experimental investigations we found in the technical literature do not allow a definitive verdict to be expressed concerning the response of notched composites to the degree of non-proportionality of the applied loading path.

8.3.5 Concluding remarks The reanalysis summarised in the previous subsections, based on a significant number of experimental results generated by testing different composite materials under multiaxial fatigue loading, seems to strongly support the idea that interpreting the stress states at crack initiation sites in terms of material principal stresses is the most effective way to correctly take into account the actual degree of multiaxiality of the linear-elastic stress field damaging the fatigue process zone. This implies that either using geometrical stresses or considering the material lay-up in terms of off-axis angle q can result in a possible underestimation of fatigue damage when multiaxial fatigue loadings are involved. The main advantage of studying the relevant stress states in terms of s1(t), s2(t) and s6(t) is that, from an engineering point of view, to a first approximation both the effect of biaxiality ratio l1 and the presence of non-zero out-of-phase angles can be neglected without much loss of accuracy. This results in a great simplification of the problem, because we can now attempt to estimate multiaxial fatigue strength of composites subjected to multiaxial loading paths by focusing attention mainly on the effects of l2.

Multiaxial fatigue of composite materials

255

Another interesting outcome of the systematic reanalysis summarised above is that, for a given material, the value of the negative inverse slope is seen not to vary significantly as the degree of multiaxiality of the stress field damaging the fatigue process zone changes. In other words, if the multiaxial fatigue behaviour of the material to be assessed cannot be investigated by running appropriate experiments, the lifetime under complex loading paths can be estimated roughly by using the inverse slope of a fatigue curve generated under uniaxial fatigue loading, i.e. determined by using standard testing equipment. To conclude, it has to be admitted that these considerations apply only to composite materials characterised by relatively simple lay-ups: more work still has to be done in this area to check whether the above simplifying hypotheses remain valid also in the presence of more complex material lay-ups.

8.4

Multiaxial fatigue assessment of composite materials

In the course of a thorough literature search, we found a few criteria specifically devised for estimating fatigue damage in plain composites subjected to multiaxial cyclic loading. In particular, the life estimation techniques proposed by Hashin and Rotem (1973), Kawakami et al. (1996), Fawaz and Ellyin (1994, 1995) and Smith and Pascoe (1989b) and, finally, those methods based on the use of polynomial functions (Labosierre and Neale, 1988) deserve to be mentioned. By systematically checking the accuracy of the above multiaxial fatigue criteria in estimating a large number of experimental results taken from the literature and generated under both external and inherent multiaxiality (Susmel and Quaresimin, 2001; Quaresimin et al., 2009), it was seen that the most accurate methods are that due to Smith and Pascoe and those based on polynomial functions. On the contrary, for different reasons the use of the other criteria was seen to be very tricky, therefore such approaches are not suggested to be employed to address problems of practical interest. In more detail, the criterion proposed by Hashin and Rotem was specifically devised to perform the fatigue assessment under inherent multiaxiality, so that, as it stands, such a method is not easy to use in those situations in which the degree of multiaxiality of the relevant stress fields is due to the complexity of the applied loading path. The criterion formalised by Kawakami, Fujii and Morita instead is based on the stiffness decrease theory, therefore its application in practice is difficult because it is not so easy to find or generate the pieces of experimental information needed to calibrate the criterion itself.

256

Multiaxial notch fatigue

Finally, the reasons why the method proposed by Fawaz and Ellyin will not be considered below deserve to be discussed in detail. According to these authors, their criterion, which estimates fatigue lifetime by postprocessing the relevant stress states expressed in terms of geometrical stresses, should be capable of predicting fatigue damage in the presence of both inherent and external multiaxiality. Moreover, for a given composite, it should take into account also the effects of different lay-ups as well as of different values of the applied load ratio, R. Unfortunately, even though appealing from an engineering point of view, the above method is not recommended for the design of real components against multiaxial fatigue because its accuracy was seen to depend strongly on the fatigue curve used to calibrate the method itself (Degrieck and Van Paepegem, 2001; Philippidis and Vassilopoulos, 2004; Quaresimin et al., 2009). In particular, as the degree of multiaxiality of the stress field damaging the fatigue process zone of the samples used to generate the calibration curve changes, the use of such a criterion to estimate fatigue damage under different loading paths can result in predictions which are either too conservative or too non-conservative, without the possibility of keeping the design process under control. Following from the above considerations, in the next subsections only the criterion devised by Smith and Pascoe and those methods based on the use of polynomial functions will be considered, not only by reviewing their governing equations but also by showing their accuracy in estimating fatigue damage under multiaxial fatigue loading. Finally, for the reasons already discussed in Subsection 8.3.3, none of the criteria reported below take explicitly into account the presence of non-zero out-of-phase angles.

8.4.1 Smith and Pascoe’s criterion The criterion proposed by Smith and Pascoe is based on an extensive experimental investigation carried out by testing, under tenson–tension, glass/polyester cruciform samples characterised by different material layups (Smith and Pascoe, 1989a, 1989b). The main peculiarity of such a criterion is that it attempts to estimate fatigue lifetime by modelling the most important mechanisms leading to the initiation of fatigue cracks, i.e. rectilinear cracking, shear deformation along the fibre plane and combined rectilinear cracking/matrix shear deformation. In more detail, by observing that under l2 = 0 only the first damage mechanism is active, Smith and Pascoe suggested estimating fatigue damage in such circumstances by using the following strain energy density based relationship: U F ,a =

{

}

1

1 σ 12,a ⎛ ν12 ν 21 ⎞ N k σ2 − + σ 1,aσ 2,a + 2,a = U F ,A ⎛ A ⎞ ⎝ Nf ⎠ 2 E1 ⎝ E1 E2 ⎠ E2

8.8

Multiaxial fatigue of composite materials

257

where UF,a is the amplitude of the strain energy density, UF,A is the reference amplitude of the strain energy density extrapolated at NA cycles to failure, and k is the negative inverse slope of the UF,a vs. Nf fatigue curve. Under cyclic shear stress instead, i.e. when fatigue breakage occurs due to shear deformation along the fibre plane, fatigue strength is assumed to depend on s6(t), i.e.: 1

N k6 σ 6 ,a = σ 6 , A ⎛ A ⎞ ⎝ Nf ⎠

8.9

where s6,A and k6 are the reference shear stress amplitude at NA cycles to failure and the negative inverse slope of the fatigue curve describing the torsional behaviour of the considered material, respectively. After uncoupling the effects of the two main damaging mechanisms by separately modelling them through Eqs (8.8) and (8.9), Smith and Pascoe suggested considering their combined effects by means of the following simple relationship: 1 1 1 = + σ 12,a( N f ) σ 12,a( N f ) σ 62,a( N f )

8.10

On the right-hand side of the above identity, s1,a is the amplitude of the stress, to be estimated through Eq. (8.8), which would break the component to be assessed at Nf cycles to failure if only the rectilinear cracking mechanism were active. Similarly, s6,a is the amplitude of the shear stress which would break the material at Nf cycles to failure if fatigue cracks initiated only due to shear deformation along the fibre plane. Since the most efficient way to estimate multiaxial fatigue damage in composites is by expressing the relevant stress states in terms of biaxiality ratios l1 and l2, criterion 8.10 can be rewritten as follows (Susmel and Quaresimin, 2001; Quaresimin et al., 2009):

λ2 ⎧ 1 ⎛ ν12 ν 21 ⎞ 1 1 2⎫ λ1 + 1 − + ⎪ ⎝ E1 E2 ⎠ E2 ⎛ N f ⎞ k ⎡⎢ λ 2 ⎛ N f ⎞ k6 ⎤⎥ ⎪⎪ 2 ⎪ E1 σ 1,a ⎨ + ⎬=1 ⎝ NA ⎠ 2U F ,A ⎢⎣ σ 6,A ⎝ N A ⎠ ⎥⎦ ⎪ ⎪ ⎪⎩ ⎪⎭

8.11

where the first term between the braces gives the contribution of rectilinear cracking to the overall fatigue damage, while the second measures the damage resulting from the shear deformation along the fibre plane. In order to check the accuracy of Smith and Pascoe’s approach in estimating fatigue lifetime under multiaxial fatigue loading, such a criterion was systematically applied to several datasets taken from the literature as reported in Table 8.1. The experimental (Nf) vs. estimated (Nf,e) number of cycles to failure diagram shown in Fig. 8.18 makes it evident that the use of

258

Multiaxial notch fatigue

Table 8.1 Summary of the experimental results used to check the accuracy of the considered multiaxial fatigue criteria Matrix

Fibres

s1,UTS s6,UTS Loading Reference (MPa) (MPa) path*

Lay-up

Polylite Glass [0/90]n FG-284 MG-252 Sirpol 8231 Glass E [±45]

164.5

71.5

Te-T

–

–

B-T

Sirpol 8231 Glass E

[0/90]

–

–

B-T

BP 2785 CV Marglass 266 BP 2785 CV Marglass 266 BP 2785 CV Marglass 266 Polylite Glass FG-284 MG-252

[0/90]13

238

82.5

Te-Te

[22.5/112.5]13 238

82.5

Te-Te

[±45]13

238

82.5

Te-Te

[0/90]

224

73.2

Te-T

Amijima et al., 1991 Aboul Wafa et al., 1997 Aboul Wafa et al., 1997 Smith and Pascoe, 1989a Smith and Pascoe, 1989a Smith and Pascoe, 1989a Kawakami et al., 1996

* Te = tension, T = torsion, B = bending.

10 000 000 [0/90]

Nf (cycles)

[±45] [22.5/112.5]

100 000 Conservative

10 000 1000

Non-conservative 100 10

10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

8.18 Accuracy of Smith and Pascoe’s criterion in estimating fatigue lifetime under multiaxial cyclic loading.

Multiaxial fatigue of composite materials

259

the above criterion resulted in estimates characterised by an acceptable accuracy level, independently of material lay-up and degree of multiaxiality and non-proportionality of the applied loading. In more detail, for all the considered datasets, UF,A and k were estimated by using the experimental results generated under l1 = l2 = 0, whereas s6,A and k6 were estimated by using those tests run under l1 = 0/0 and l2 = ∞. Moreover, the calibration fatigue curves were calculated for a probability of survival equal to 50% by assuming a log-normal distribution of the number of cycles to failure for each stress level with a confidence level equal to 95%. To conclude, even if the influence of load ratio R cannot be taken into account directly, the in-field application of the criterion due to Smith and Pascoe is straightforward and only two fatigue curves are needed to efficiently calibrate it. Unfortunately, such an approach is not suitable for estimating fatigue lifetime of unidirectional laminates when cracks initiate due to inherent multiaxiality, because it can be used successfully only to perform the fatigue assessment of composites which show symmetrical behaviour in terms of strength with respect to principal axes 1 and 2.

8.4.2 Multiaxial fatigue criteria based on the use of polynomial functions The mathematical formalisation of the criteria based on the use of polynomial relationships takes as its starting point the idea that the size of the reference failure surface, whose shape is a function of biaxiality ratios l1 and l2, varies as the number of cycles to failure, Nf, changes (Owen and Griffiths, 1978; Owen and Rice, 1981; Labosierre and Neale, 1988; Fawaz and Neale, 1990). Such an idea is schematically explained by the sketch reported in Fig. 8.19: theoretically speaking, for any value of Nf (Nf,i in Fig. 8.19) it should always be possible to define a failure curve to be used to estimate fatigue damage independently of the degree of multiaxiality of the assessed stress state. Unfortunately, although the above intuition is very simple and appealing from an engineering point of view, its mathematical formalisation is not at all straightforward due to the large number of variables which are simultaneously involved in the design process. Moreover, it was proved through accurate experimental investigations that, for a given material, the shape of the failure curves can change as the number of cycles to failure increases – see, for instance, Owen et al. (1976). However, if the problem is greatly simplified by forming the hypothesis that the profile of the above curves does not vary when moving from the low- to the high-cycle fatigue regime, then the entire failure surface can be fully described by extending to the fatigue field one of those polynomial functions usually adopted to perform the static assessment of composite

260

Multiaxial notch fatigue Nf

Failure curve at Nf,i cycles to failure

s6,a vs. Nf fatigue curve

Nf,i

s6,a

s2,a vs. Nf fatigue curve

s1,a vs. Nf fatigue curve

s2,a Failure surface s1,a

8.19 Failure surface under multiaxial fatigue loading.

materials (Fawaz and Neale, 1990). In more detail, if the following fatigue functions are calibrated by running appropriate experiments: Ki = Ki ( N f ) , where i = 1, 2, . . . , 6

8.12

then the use of the classical static criteria, expressed in their most general form, can be extended to multiaxial fatigue situations by rewriting them as follows (Quaresimin et al., 2009): 2

2

2

σ 1,a σ 2 ,a σ 1,aσ 2,a ⎡ σ 6,a ⎤ ⎡ σ 1,a ⎤ ⎡ σ 2,a ⎤ ⎢⎣ K ( N ) ⎥⎦ + ⎢⎣ K ( N ) ⎥⎦ + K ( N ) + K ( N ) + K 2( N ) + ⎢⎣ K ( N ) ⎥⎦ = 1 1 5 f 6 f 2 f f 3 f 4 f

8.13

It is evident that by making different assumptions concerning the stress quantities which play a fundamental role in the damage evolution, different polynomial functions can be obtained from general equation (8.13). Moreover, it is worth noting also that the in-field use of such criteria requires one experimental fatigue curve for any Ki(Nf) function needed to calibrate the criteria themselves. In other words, the approaches based on polynomial functions require expensive experimental investigations to correctly evaluate the fatigue properties of the material to be assessed. Finally, relationships Ki(Nf) (i = 1, 2, . . . , 6) have to be determined by post-processing experimental results generated under the same load ratio, R, as that characterising the load history applied to the component to be assessed. In the present section, in order to briefly investigate the ability of polynomial functions to estimate fatigue lifetime under external multiaxial cyclic loading, Eq. (8.13) is formalised as suggested by Tsai and Hill (Tsai and Wu, 1971), that is:

Multiaxial fatigue of composite materials 2

261

2

2

⎡ σ 1,a ⎤ ⎡ σ 2,a ⎤ σ 1,aσ 2,a ⎡ σ 6,a ⎤ ⎢⎣ K ( N ) ⎥⎦ + ⎢⎣ K ( N ) ⎥⎦ − K 2( N ) + ⎢⎣ K ( N ) ⎥⎦ = 1 f 1 2 f f 1 f 6

8.14

where calibration functions Ki (i = 1, 2, 6) can simply be expressed as Wöhler curves (Quaresimin et al., 2009), i.e.: 1

N ki K i( N f ) = σ i ,A ⎛ A ⎞ ⎝ Nf ⎠

( i = 1, 2, 6)

8.15

The error diagram reported in Fig. 8.20 shows the accuracy of criterion (8.14) in predicting the experimental results summarised in Table 8.1. This chart was built by determining, for any considered material, functions K1(Nf) and K6(Nf) from the experimental results generated under l1 = l2 = 0 and under l1 = 0/0 and l2 = ∞, respectively. Moreover, the fatigue curves used to calibrate Eqs (8.15) were calculated for a probability of survival equal to 50%, by assuming a log-normal distribution of the number of cycles to failure with a confidence level equal to 95%. Finally, it is worth noting that, in the light of the fact that the datasets summarised in Table 8.1 were generated by testing woven materials, function K2(Nf) was assumed to be always equal to K1(Nf). The experimental (Nf) vs. estimated (Nf,e) number of cycles to failure diagram reported in Fig. 8.20 shows that the extension of Tsai–Hill’s criterion to multiaxial fatigue situations results in estimates as accurate (or as scattered!) as those obtained by applying Smith and Pascoe’s criterion (Fig. 8.18).

10 000 000 Nf (cycles)

[0/90] [±45] [22.5/112.5]

100 000 10 000

Conservative

1000 Non-conservative

100 10 10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

8.20 Accuracy of Tsai–Hill’s criterion in estimating fatigue lifetime under multiaxial cyclic loading.

262

Multiaxial notch fatigue

Table 8.2 Summary of the experimental results used to check the accuracy of Tsai–Hill’s criterion in estimating fatigue damage under inherent multiaxiality Matrix

Fibres*

Off-axis angle, q (degree)

R

s1,Ult (MPa)

Reference

Epoxy

Graphite (U)

0.1

1836

Epoxy

Carbon (W)

0.1

600

Epoxy

Carbon (U)

0.5, 0.1, −1

2472

Epoxy

Carbon (U)

0, 10, 20, 30, 45, 60, 90 0, 15, 30, 45, 90 0, 10, 15, 30, 45, 90 0, 10, 15, 30, 45, 90

0.1

1934

Awerbuch and Hahn, 1981 Kawai and Taniguchi, 2006 Kawai and Suda, 2004 Kawai et al., 2001

* U = unidirectional, W = woven. 10 000 000 Nf (cycles) Conservative 100 000 10 000 Non-conservative 1000 Awerbuch and Hahn, 1981 Kawai and Taniguchi, 2006

100

Kawai and Suda, 2004 Kawai et al., 2001

10 10

100

1000

10 000 100 000 1 000 000 10 000 000 Nf,e (cycles)

8.21 Accuracy of Tsai–Hill’s criterion in estimating fatigue lifetime under inherent multiaxiality.

Another important aspect which deserves to be considered here is that, thanks to its peculiar features, Tsai–Hill’s criterion can be used also to estimate fatigue damage in composites damaged by inherent multiaxiality. In order to show its accuracy in predicting lifetime in such circumstances, criterion (8.14) was then applied also to the off-axis data summarised in Table 8.2, obtaining the error diagram reported in Fig. 8.21 (Quaresimin et al., 2009). This chart clearly proves that Tsai–Hill’s criterion is capable of correctly estimating fatigue damage due to inherent multiaxiality, giving estimates characterised by an evident degree of conservatism.

Multiaxial fatigue of composite materials

263

To conclude, it can be said that the criteria based on polynomial relationships are simple to be used in situations of practical interest, being capable of estimating fatigue strength in the presence of not only external but also inherent multiaxiality. The main drawbacks are that their use requires extensive experimental investigations to correctly calibrate functions Ki(Nf) (i = 1, 2, . . . , 6) and that they do not explicitly take into account the influence of the load ratio, R.

8.5

Concluding remarks

The considerations reported in the previous sections suggest that our understanding of the response of composite materials to multiaxial fatigue loading is still at an embryonic stage, so that more work needs to be done, from both an experimental and a theoretical point of view, in order to devise accurate life estimation techniques which allow composites to be designed accurately against multiaxial fatigue. In particular, the accuracy shown by the two criteria reviewed in the present chapter makes it evident that, unfortunately, an adequate margin of safety can be reached systematically when performing the multiaxial fatigue assessment of real components only by using large values of the safety factor, i.e., much larger than 10 in life. This implies that still the only way to take full advantage of the peculiarities of composites in terms of structural performance is by performing accurate and expensive experimental investigations. To conclude, it can be said that, according to the writer’s understanding of the problem, the low accuracy level in estimating multiaxial fatigue lifetime of the available multiaxial fatigue criteria has to be ascribed mainly to the fact that, in general, they are poorly related to those physical processes taking place within the process zone and leading to the initiation of fatigue cracks. This suggests that the only way of devising accurate lifetime estimation techniques to supply structural engineers with reliable design methodologies is to keep investigating in depth the fundamental phenomena damaging composites at a microscopic level when fatigue loadings are involved.

8.6

References

Aboul Wafa, M. N., Hamdy, A. H., El-Midany, A. A. (1997) Combined bending and torsional fatigue of woven roving (GRP). Transactions of the ASME, Journal of Engineering Materials and Technology 119, 181–185. DOI: 10.1115/1.2805991. Amijima, A., Fujii, T., Hamaguchi, M. (1991) Static and fatigue tests of a woven glass fabric composite under biaxial tension–torsion loading. Composites 22(4), 281–289.

264

Multiaxial notch fatigue

Awerbuch, J., Hahn, H. T. (1981) Off-axis fatigue of graphite/epoxy composite. In: Fatigue of Fibrous Composite Materials, ASTM STP 723, American Society for Testing and Materials, Philadelphia, PA, 243–273. Degrieck, J., Van Paepegem, W. (2001) Fatigue damage modelling of fibre-reinforced composite materials: Review. Transactions of the ASME, Applied Mechanics Reviews 54, 279–300. DOI: 10.1115/1.1381395. Fawaz, Z., Ellyin, F. (1994) Fatigue failure model for fibre-reinforced materials under general loading conditions. Journal of Composite Materials 28(15), 1432–1451. DOI: 10.1177/002199839402801503. Fawaz, Z., Ellyin, F. (1995) A new methodology for the prediction of fatigue failure in multidirectional fiber-reinforced laminates. Composites Science and Technology 53, 45–55. DOI: 10.1016/0266-3538(94)00077-8. Fawaz, Z., Neale, K. W. (1990) A parametric criterion for biaxial fatigue failure of fibre-reinforced composite laminate. Transactions of the Canadian Society for Mechanical Engineering 14(4), 93–99. Francis, P. H., Walrath, D. E., Sims, D. F., Weed, D. N. (1977) Biaxial fatigue loading of notched composites. Journal of Composite Materials 11, 488–501. DOI: 10.1177/002199837701100410. Fujii, T., Shina, T., Okubo, K. (1994) Fatigue notch sensitivity of glass woven fabric composites having a circular hole under tension/torsion biaxial loading. Journal of Composite Materials 28(3), 234–251. DOI: 10.1177/00219983940 2800303. Hashin, Z., Rotem, A. (1973) A fatigue failure criterion for fibre reinforced materials. Journal of Composite Materials 7, 448–464. DOI: 10.1177/002199837300 700404. Kawai, M., Suda, H. (2004) Effects of non-negative mean stress on the off-axis fatigue behaviour of unidirectional carbon/epoxy composites at room temperature. Journal of Composite Materials 38, 833–854. DOI: 10.1177/002199830 4042477. Kawai, M., Taniguchi, T. (2006) Off-axis fatigue behaviour of plain weave carbon/ epoxy fabric laminates at room and high temperatures and its mechanical modelling. Composites (Part A) 37, 243–256. Kawai, M., Yajima, S., Hachinohe, A., Takano, Y. (2001) Off-axis fatigue behaviour of unidirectional carbon fiber-reinforced composites at room and high temperatures. Journal of Composite Materials 35, 545–576. DOI: 10.1177/002199801772 662073. Kawakami, H., Fujii, T. J., Morita, Y. (1996) Fatigue degradation and life prediction of glass fabric polymer composite under tension/torsion biaxial loadings. Journal of Reinforced Plastics and Composites 15, 183–195. DOI: 10.1177/07316844960 1500204. Labosierre, P., Neale, K. W. (1988) A general strength theory for orthotropic fibrereinforced composite laminae. Polymer Composites 9, 306–317. DOI: 10.1002/pc. 750090503. Owen, M. J., Griffiths, J. R. (1978) Evaluation of biaxial failure surfaces for a glass fabric reinforced polyester resin under static and fatigue loading. Journal of Materials Science 13, 1521–1537. DOI: 10.1007/BF00553209. Owen, M. J., Rice, D. J. (1981) Biaxial strength behaviour of glass fabric-reinforced polyester resins. Composites 12(1), 13–25.

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265

Owen, M. J., Griffiths, J. R., Found, M. S. (1976) Biaxial stress fatigue testing of thinwalled GRP cylinders. International Conference on Composite Materials, AIME, New York, 917–941. Philippidis, T. P., Vassilopoulos, A. P. (1999) Fatigue of composite laminates under off-axis loading. International Journal of Fatigue 21, 253–262. DOI: 10.1016/S01421123(98)00073-5. Philippidis, T. P., Vassilopoulos, A. P. (2004) Complex stress state effect on fatigue life of GRP laminates. Part II, Theoretical formulation. International Journal of Fatigue 24, 825–830. DOI: 10.1016/S0142-1123(02)00004-X. Qi, D., Cheng, G. (2007) Fatigue behaviour of filament-wound glass fibre reinforced epoxy composite tubes under tension/torsion biaxial loading. Polymer Composites 28(1), 116–123. DOI: 10.1002/pc.20275. Quaresimin, M., Susmel, L. (2002) Multiaxial fatigue behaviour of composite laminates. Key Engineering Materials 221–222, 71–80. Quaresimin, M., Susmel, L., Talreja, R. (2009) Fatigue behaviour and life assessment of composites under multiaxial loadings. International Journal of Fatigue. (in press) Sims, D. F., Brogdon, V. H. (1977) Fatigue behaviour of composites under different loading modes. In: Fatigue of Filamentary Composite Materials, ASTM STP 636, edited by K. L. Reifsnider and K. L. Lauraitis, American Society for Testing and Materials, Philadelphia, PA, 185–205. Smith, E. W., Pascoe, K. J. (1989a) Biaxial fatigue of a glass-fibre reinforced composite. Part 1: Fatigue and fracture behaviour. In: Biaxial and Multiaxial Fatigue, edited by M. W. Brown and K. J. Miller, EGF 3, Mechanical Engineering Publications, London, 367–396. Smith, E. W., Pascoe, K. J. (1989b) Biaxial fatigue of a glass-fibre reinforced composite. Part 2: Failure criteria for fatigue and fracture. In: Biaxial and Multiaxial Fatigue, edited by M. W. Brown and K. J. Miller, EGF 3, Mechanical Engineering Publications, London, 397–421. Susmel, L., Quaresimin, M. (2001) Multiaxial fatigue life prediction criteria for composite materials. In: Proceedings of the 6th International Conference on Biaxial/Multiaxial Fatigue and Fracture, edited by M. De Freitas, Lisbon, Portugal, 25–28 June 2001, 379–386. Tsai, S. W., Wu, E. M. (1971) A general theory of strength for anisotropic materials. Journal of Composite Material 5, 58–80. DOI: 10.1177/002199837100500106. Vasiliev, V. V., Morozov, E. V. (2001) Mechanics and Analysis of Composite Materials. Elsevier Science, Oxford.

Appendix A Experimental values of the material characteristic length, L

Abstract: This appendix summarises about 100 values of the material characteristic length, L, experimentally determined by testing specimens made of different engineering materials. Key words: material characteristic length.

A.1

Introduction

This appendix summarises about 100 values of the material characteristic length, L, experimentally determined by testing specimens made of different engineering materials. From Section 2.7.1, this fatigue property is defined as follows: L=

1 ⎛ ΔKth ⎞ ⎜ ⎟ π ⎝ Δσ 0 ⎠

2

where Δs0 is the range of the plain fatigue limit and ΔKth is the range of the threshold value of the stress intensity factor. The experimental results reported below are classified into four groups, i.e., steel (Section A.2), cast iron (Section A.3), aluminium alloy (Section A.4) and other materials (Section A.5).

266

Steel

Troshchenko, 1994 Usami, 1987 Lukas et al., 1986 Harkegard, 1981; Ting and Lawrence, 1993 Fujimoto et al., 2001 Fouvry et al., 2008 McEvily, 1996

Nisitani and Endo, 1988; Atzori et al., 2003 Nisitani and Endo, 1988; Atzori et al., 2003 Shang et al., 2002 Murakami, 2002 Frost, 1957; Ting and Lawrence, 1993 El Haddad et al., 1979 Fujimoto et al., 2001

12KhN3A 13Cr steel 2.25Cr–1Mo 304 Stainless steel

C36

G40.11 HT80

C45 C46 EN 26 (NiCr steel)

C45

A533 AISI 1034 AISI 304

References

Material

Table A.1

A.2

794

820

957

823

377

364

623

632

721 350

380 222

530

769 600 685

700

sY (MPa)

950

sUTS (MPa)

ΔKth (MPa m1/2) 6.0 6.1 12.0 12.0 10.5 7.0 6.0 7.6 8.1 10.3 15.0 12.8 13.0 13.0

R −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1

518 800

498 1780 1000

582

446

750 270 720

790 1080 440 720

Δs0 (MPa)

0.200 0.084

0.136 0.023 0.052

0.062

0.092

0.062 0.214 0.022

0.018 0.010 0.237 0.088

L (mm)

Rolled, annealed for 1 h at 845°C Hot rolled Quenched 850°C oil quench, 650°C air cool

Stress relieved for 1 h at 600°C Annealed

Remarks

Experimental values of the material characteristic length, L 267

Tokaji et al., 1986a

Tokaji et al., 1986a

Nakai and Ohjim, 1992 Nakai and Ohjim, 1992 McEvily, 1996

Frost et al., 1974

McEvily, 1996

Usami, 1987

Frost, 1959; Harkegard, 1981 440 McEvily, 1996 925

JIS 10C

JIS 10C

JIS SNCM439 JIS SNCM439 Low-alloy steel

Mild steel

Mild steel

Mild steel

Mild steel – 0.15%C Ni-Cr alloy steel

530

1700 934 835

402

433

655

McEvily, 1996

Inconel

sUTS (MPa)

References

Material

Table A.1 Continued

334

1360 839

233

286

sY (MPa) 6.4 8.1 8.2 3.9 9.5 6.3 6.5 6.4 6.5 12.8 6.4

−1 −1 −1 −1 −1 −1 −1 −1 −1 −1

ΔKth (MPa m1/2)

−1

R

420 1000

364

400

520

438 438 920

380

440

440

Δs0 (MPa)

0.296 0.013

0.102

0.081

0.050

0.025 0.150 0.015

0.148

0.108

0.067

L (mm)

Stress relieved for 1 h at 570°C

Stress relieved for 1 h at 650°C

Stress relieved for 1 h at 600°C Normalised for 1 h at 900°C Annealed for 4 h at 1100°C Tempered at 200°C Tempered at 600°C Stress relieved for 1 h at 570°C

Remarks

268 Multiaxial notch fatigue

Murakami and Endo, 1983; Atzori et al., 2005 Tanaka, 1987 Murakami and Endo, 1983; Atzori et al., 2005 Legris et al., 1981 Yu et al., 1991

Yu et al., 1991

Yu et al., 1991

Yu et al., 1991 Yu et al., 1988 DuQuesnay et al., 1986 Yu et al., 1988

S10C

SAE 1010 SAE 1010 – CR22

SAE 1010 – CR56

SAE 1010 – CR76

SAE SAE SAE SAE

Yu et al., 1988

Yu et al., 1988

Yu et al., 1991

SAE 1045-T600

SAE 1045-T900

SAE 945X - CR30

1010 – HR 1045 1045 1045-T1200

S20C S45C

References

Material

Table A.2

621

1182

1984

326 720 745 798

689

525

364 476

543

353

sUTS (MPa)

1054

1617

390 472 645

227

194 284

206

sY (MPa) 11.0 6.2 10.5 11.6 10.2 8.4 6.4 11.8 9.0 6.9 7.7 7.5 7.0 12.4

−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1

588

1134

1200

320 604 606 760

614

546

234 410

163 480

362

Δs0 ΔKth (MPa m1/2) (MPa)

R

0.142

0.012

0.012

0.433 0.071 0.041 0.033

0.035

0.075

0.782 0.197

0.461 0.152

0.294

L (mm)

Austenised at 843°C, quenched, tempered at 649°C Austenised at 843°C, quenched, tempered at 315°C Austenised at 843°C, quenched, tempered at 482°C Cold rolled, thickness reduction = 30%

Cold rolled, thickness reduction = 22% Cold rolled, thickness reduction = 56% Cold rolled, thickness reduction = 76% Hot rolled As received

Annealed

Annealed

Remarks

Experimental values of the material characteristic length, L 269

445 666 516 803 476 520

Yu et al., 1991 Tokaji et al., 1986b Tokaji et al., 1986b Tanaka and Nakai, 1983

Fujimoto et al., 2001 Fujimoto et al., 2001 Troshchenko, 1994 Troshchenko, 1994 Akiniwa et al., 2002

Akiniwa et al., 2002

SAE 945X - HR SCM43S SCM43S SM41B

SM41B(S) SM58 St 45 St40kn SUF medium grained SUF ultrafine grained

558 965 1011 423

752

Yu et al., 1991

SAE 945X - CR61

sUTS (MPa)

References

Material

Table A.2 Continued

446

280 588 339 683 360

847 900 194

sY (MPa) 12.0 13.4 6.9 7.5 12.4 10.2 12.7 11.6 7.4 8.0 9.8

−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1

570

396 630 440 860 490

500 1000 920 326

630

Δs0 ΔKth (MPa m1/2) (MPa)

R

0.094

0.211 0.129 0.221 0.024 0.085

0.229 0.015 0.021 0.461

0.115

L (mm)

Cold rolled, thickness reduction = 61% Hot rolled Fine grained Coarse grained Annealed at 1200°C for 5 h in vacuum

Remarks

270 Multiaxial notch fatigue

Kitagawa, 1981 Beretta and Matteazzi, 1996 Atzori et al., 2006 Gach et al., 2002 Kihara and Yoshii, 1991 Kihara and Yoshii, 1991 Tanaka and Nakai, 1983

SM50 0.76C

35KHS2N3MF

JIS SM41 08kp

FeP04 High-strength steel HT60 SS41 SM41B

Usami and Shida, 1979 Ostash and Panasyuk, 2001 Ostash and Panasyuk, 2001

Lukas et al., 1986 Glodez et al., 2002 Usami, 1987 Kitagawa and Takahashi, 1976 Usami and Shida, 1979 Yu et al., 1988 Tanaka and Nakai, 1983

2.25Cr-1Mo 42CrMo4 HT60 HT80

JIS SM41 SAE 1045 SM41B

References

Material

Table A.3

1950

483 270

310 1770 598 448 423

1855

1700

251 190

185 1620 510 323 194

377 1540

251 390 194

726

785 483 720 423

380

sY (MPa)

530

sUTS (MPa)

4.6 6.7 3.5

≈0.1

10.0 4.2 6.6 6.4 6.4

4.8 4.5

6.5 13.8 8.4

6.4 8.5 13.0 4.8

ΔKth (MPa m1/2)

0.5 ≈0.1

0.1 0.1 0.1 0.1 0.4

0 0.1

0 0 0

0 0 0 0

R

850

281 190

247 1050 425 231 244

378 450

318 448 274

340 550 580 550

Δs0 (MPa)

0.005

0.085 0.396

0.522 0.005 0.077 0.244 0.219

0.051 0.032

0.133 0.302 0.299

0.113 0.076 0.160 0.024

L (mm)

Fatigue limit extrapolated at 106 cycles to failure Fatigue limit extrapolated at 106 cycles to failure

Annealed at 1200°C for 5 h in vacuum

Wire steel

As received Annealed at 1200°C for 5 h in vacuum

Remarks

Experimental values of the material characteristic length, L 271

Cast iron

References

Wang et al., 2000 Taylor et al., 1996 Taylor, 2007 Personal communication Personal communication Personal communication Usami, 1987 Taylor et al., 1996 Clement et al., 1984

Gasparini and Meneghetti, 2002 Taylor et al., 1996 Taylor et al., 1996 Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Material

Grey cast iron Grey cast iron S. G. cast iron Perlitic Ferritic GH40 Cast iron Grey cast iron Ferritic

EN-GJS-800-8

Grey cast iron Grey cast iron VCH50

VCH90

VCHN10

Table A.4

A.3

450

980 320

850

202 202 310

202 270

249 450

249 249 510

202 202

sY (MPa)

249 249

sUTS (MPa)

8.0 5.2 8.9 8.0 9.1

≈0.1 ≈0.1

8.1

0.1 0.5 0.7 ≈0.1

15.9 15.9 23.5 24.4 24.4 18.8 5.9 11.2 8.0

ΔKth (MPa m1/2)

1 1 1 1 1 1 1 0.1 0.1

R

205

410

68 48 255

440

190 155 590 451 378 450 210 99 410

Δs0 (MPa)

0.627

0.121

4.406 3.736 0.388

0.108

2.240 3.350 0.505 0.932 1.326 0.556 0.251 4.074 0.121

L (mm)

Grade 17 Grade 17 Fatigue limit extrapolated at 106 cycles to failure Fatigue limit extrapolated at 106 cycles to failure Fatigue limit extrapolated at 106 cycles to failure

Grade 17 Ferritic nodular graphite cast iron

Grade 17

Remarks

272 Multiaxial notch fatigue

Aluminium alloy

DuQuesnay et al., 1986 Fujimoto et al., 2001 Wang et al., 1999 Frost, 1957; Ting and Lawrence, 1993 Vallelano et al., 2003 Stanzl-Tschegg and Mayer, 2001 Blom et al., 1986 Blom et al., 1986 Plumtree and Schafer, 1986 Atzori et al., 2004 DuQuesnay et al., 1988; Atzori et al., 2003 Atzori et al., 2005 Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

2024-T351 2024-T3 LM25 AL-B.S.L. 65

D16chAT1

AA 356-T6 D16chAT

AA356-T6 2024-T351

2024 7075 AA356-T6

7075-T6 2024-T351

References

Material

Table A.5

A.4

475

452

260 466

450 523 342

460

466 483 275 486

sUTS (MPa)

442

182 334

192 357

345 467 256

352

425

357 322

sY (MPa)

4.0 4.0

−1 0

2.2

2.2 1.6 1.6

−1 −1 −1

≈0.1

4.0 4.2

−1 −1

5.0 3.2

3.5 6.4 5.9 4.2

−1 −1 −1 −1

0.1 ≈0.1

ΔKth (MPa m1/2)

R

195

140 130

231 172

256 296 125

428 323.6

248 300 77.5 300

Δs0 (MPa)

0.041

0.406 0.193

0.093 0.172

0.022 0.010 0.052

0.028 0.054

0.063 0.147 1.845 0.062

L (mm)

Cast aluminium alloy Fatigue limit extrapolated at 106 cycles to failure Fatigue limit extrapolated at 106 cycles to failure

Al-Si Alloy 505°C water quench age 160–190°C

Remarks

Experimental values of the material characteristic length, L 273

References

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Ostash and Panasyuk, 2001

Material

D16chATN

D16chAT

V95pchT1

V95pchT2

V95pchT3

Amg-6

1420T1

1201T1

Table A.5 Continued

442

431

340

498

510

541

454

533

sUTS (MPa)

340

282

175

432

456

479

330

417

sY (MPa)

ΔKth (MPa m1/2) 2.6 3.3 2.8 2.4 2.1 3.2 4.2 2.5

R ≈0.1 ≈0.1 ≈0.1 ≈0.1 ≈0.1 ≈0.1 ≈0.1 ≈0.1

135

250

150

165

125

125

138

168

Δs0 (MPa)

0.109

0.090

0.145

0.052

0.117

0.160

0.182

0.076

L (mm)

Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure Fatigue limit extrapolated 106 cycles to failure

Remarks

at

at

at

at

at

at

at

at

274 Multiaxial notch fatigue

Other materials

References

Weiss et al., 1979 Weiss et al., 1979 Weiss et al., 1979 Weiss et al., 1979 Torres et al., 2002 Torres et al., 2002 Chuan-Yao and Da-Xing, 1983

Material

Mo Mo-0.8W Mo-1.5W Ti-Zr-Mo 16F 16M Mg-Al-Zn

Table A.6

A.5

473 522 488 559 2742 1813 176

sUTS (MPa)

88

356 406 409 416

sY (MPa)

ΔKth (MPa m1/2) 4.9 5.9 6.6 7.0 5.4 5.7 1.6

R −1 −1 −1 −1 0.1 0.1 0.1 674 750 738 766 1644 1091 132

Δs0 (MPa)

0.017 0.020 0.025 0.027 0.003 0.009 0.049

L (mm)

WC-Co cemented carbides WC-Co cemented carbides Cast magnesium alloy

Remarks

Experimental values of the material characteristic length, L 275

276

Multiaxial notch fatigue

A.6

References

Akiniwa, Y., Tanaka, K., Kimura, H., Ishikawa, T. (2002) Fatigue crack propagation behaviour in ultrafine-grained steel. In: Proceedings of the 8th International Fatigue Congress, edited by A. F. Blom, Vol. 1, Cradley Heath, UK, 473–480. Atzori, B., Lazzarin, P., Meneghetti, G. (2003) Fracture mechanics and notch sensitivity. Fatigue and Fracture of Engineering Materials and Structures 26, 257– 267. DOI: 10.1046/j.1460-2695.2003.00633.x. Atzori, B., Meneghetti, G., Susmel, L. (2004) Fatigue behaviour of AA356-T6 cast aluminium alloy weakened by cracks and notches. Engineering Fracture Mechanics 71, 759–768. DOI: 10.1016/S0013-7944(03)00036-5. Atzori, B., Lazzarin, P., Meneghetti, G. (2005) A unified treatment of the mode I fatigue limit of components containing notches or defects. International Journal of Fracture 133, 61–87. DOI: 10.1007/s10704-005-2183-0. Atzori, B., Lazzarin, P., Meneghetti, G. (2006) Estimation of fatigue limits of sharply notched components. In: Proceedings of Fatigue 2006, Atlanta, GA (oral reference: FT 419). Beretta, S., Matteazzi, S. (1996) Short crack propagation in eutectoid steel wires. International Journal of Fatigue 18, 451–456. DOI: 10.1016/0142-1123(96) 00014-X. Blom, A. F. et al. (1986) Short fatigue crack growth behaviour in Al 2024 and Al 7075. In: The Behaviour of Short Fatigue Cracks, EGF Pub. 1, edited by K. J. Miller and E. R. de los Rios, Mechanical Engineering publications, London, 37–66. Chuan-Yao, C., Da-Xing, G. (1983) Fatigue crack propagation in a cast magnesium alloy. Fatigue and Fracture of Engineering Materials and Structures 6, 167–176. DOI: 10.1111/j.1460-2695.1983.tb00333.x. Clement, P., Angeli, J. P., Pineau, A. (1984) Short crack behaviour in nodular cast iron. Fatigue and Fracture of Engineering Materials and Structures 7, 251–265. DOI: 10.1111/j.1460-2695.1984.tb00194.x. DuQuesnay, D. L., Topper, T. H., Yu, M. T. (1986) The effect of notch radius on the fatigue notch factor and the crack propagation of short cracks. In: The Behaviour of Short Fatigue Cracks, EGF Pub. 1, edited by K. J. Miller and E. R. de los Rios, Mechanical Engineering Publications, London, 323–335. DuQuesnay, D. L., Yu, M. T., Topper, T. H. (1988) An analysis of notch size effect on the fatigue limit. Journal of Testing and Evaluation 4, 375–385. El Haddad, M. H., Topper, T. H., Smith, K. N. (1979) Prediction of non propagating cracks. Engineering Fracture Mechanics 11, 573–584. DOI: 10.1016/0013-7944 (79)90081-x. Fouvry, S., Nowell, D., Kubiak, K., Hills, D. A. (2008) Prediction of fretting crack propagation based on a short crack methodology. Engineering Fracture Mechanics 75, 1605–1622. DOI: 10.1016/j.engfracmech.2007.06.011. Frost, N. E. (1957) Non-propagating cracks in V-notched specimens subjected to fatigue loading. Aeronautical Quarterly VIII, 1–20. Frost, N. E. (1959) A relation between the critical alternating propagation stress and crack length for mild steel. Proceedings of the Institution of Mechanical Engineers 173, 811–834. Frost, N. E., Marsh, K. J., Pook, L. P. (1974) Metal Fatigue. Clarendon Press, Oxford.

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277

Fujimoto, Y., Kunihiro, H., Shintaku, H., Pirker, G. (2001) Inherent damage zone model for strength evaluation of small fatigue cracks. Engineering Fracture Mechanics 68, 455–473. DOI: 10.1016/S0013-7944(00)00116-8. Gach, E., Daxelmüller, M., Ponemayr, H., Pippan, R., Kolednik, O. (2002) Fatigue behaviour of predamaged high-strength steels. In: Proceedings of the 8th International Fatigue Congress, edited by A. F. Blom, Vol. 1, Cradley Heath, UK, 489–497. Gasparini, E., Meneghetti, G. (2002) Una banca dati sul comportamento a fatica delle ghise sferoidali austemperate. La Metallurgia Italiana 3, 29–35 (in Italian). Glodez, S., Sraml, M., Kramberger, J. (2002) A computational model for determination of service life of gears. International Journal of Fatigue 24, 1013–1020. DOI: 10.1016/S0142-1123(02)00024–5. Harkegard, G. (1981) An effective stress intensity factor and the determination of the notched fatigue limit. In: Fatigue Thresholds: Fundamentals and Engineering Application, Vol. II, edited by J. Backlund, A. F. Blom and C. J. Beevers, Chameleon Press, London, 867–879. Kihara, S., Yoshii, A. (1991) A strength evaluation method of a sharply notched structure by a new parameter, the equivalent stress intensity factor. Japan Society of Mechanical Engineers International Journal 34, 70–75. Kitagawa, H. (1981) Limitations in the applications of fatigue threshold ΔKth. In: Fatigue Thresholds: Fundamentals and Engineering Application, Vol. II, edited by J. Backlund, A. F. Blom and C. J. Beevers, Chameleon Press, London, 1051–1068. Kitagawa, H., Takahashi, S. (1976) Applicability of fracture mechanics to very small cracks or the cracks in the early stage. In: Proceedings of 2nd International Conference on Mechanical Behaviour of Materials, Boston, MA, 627–631. Legris, L. et al. (1981) The effect of cold rolling on the fatigue properties of a SAE 1010 steel. In: Proceedings of FATIGUE ’81, edited by F. Sherratt and J. B. Sturegeon, 97–105. Lukas, P., Kunz, L., Weiss, B., Stickler, R. (1986) Non-damaging notches in fatigue. Fatigue and Fracture of Engineering Materials and Structures 9, 195–204. McEvily, A. J. (1996) Fatigue crack thresholds. In: ASTM Handbook – Fatigue and Fracture, Vol. 19, 134–152. Murakami, Y. (2002) Metal Fatigue: Effects of Small Defects and Non-metallic Inclusions. Elsevier, Oxford. Murakami, Y., Endo, M. (1983) Quantitative evaluation of fatigue strength of metals containing various small defects or cracks. Engineering Fracture Mechanics 17, 1–15. DOI: 10.1016/0013-7944(83)90018-8. Nakai, Y., Ohjim, K. (1992) Predictions of growth rate and closure of short fatigue cracks. In: Short Fatigue Cracks, edited by K. J. Miller and E. R. de los Rios, ESIS 13, 169–189. Nisitani, H., Endo, M. (1988) Unified treatment of deep and shallow notches in rotating bending fatigue. In: Basic Questions in Fatigue, Vol. I, ASTM STP 924, 136–153. Ostash, O. P., Panasyuk, V. V. (2001) Fatigue process zone at notches. International Journal of Fatigue 23, 627–636. DOI: 10.1016/S0142-1123(01)00004-4.

278

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Plumtree, A., Schafer, S. (1986) Initiation and short crack behaviour in aluminium alloy casting. In: The Behaviour of Short Fatigue Cracks, EGF Pub. 1, edited by K. J. Miller and E. R. de los Rios, Mechanical Engineering publications, London, 215–227. Shang, D.-G., Yao, W.-X., Wang, D.-J. (2002) A new approach to the determination of fatigue crack initiation size. International Journal of Fatigue 20, 683–687. DOI: 10.1016/S0142-1123(98)00035-8. Stanzl-Tschegg, S. E., Mayer, H. (2001) Fatigue and fatigue crack growth of aluminium alloys at very high numbers of cycles. International Journal of Fatigue 23, S231–S237. DOI: 10.1016/S0142-1123(01)00167-0. Tanaka, K. (1987) Short-crack fracture mechanics in fatigue conditions. In: Current Research on Fatigue Cracks, Vol. I, edited by T. Tanaka, M. Jono and K. Komai, Elsevier Applied Science, London, 119–147. Tanaka, K., Nakai, Y. (1983) Propagation and non-propagation of short fatigue cracks at a sharp notch. Fatigue and Fracture of Engineering Materials and Structures 6, 315–327. DOI: 10.1111/j.1460-2695.1983.tb00347.x. Taylor, D. (2007) The Theory of Critical Distances: A New Perspective in Fracture Mechanics. Elsevier Science, Oxford. Taylor, D., Hughes, M., Allen, D. (1996) Notch fatigue behaviour in cast irons explained using a fracture mechanics approach. International Journal of Fatigue 18, 439–445. DOI: 10.1016/0142-1123(96)00018-7. Ting, J. C., Lawrence, F. V. (1993) A crack closure model for predicting the threshold stresses of notches. Fatigue and Fracture of Engineering Materials and Structures 16, 93–114. DOI: 10.1111/j.1460-2695.1993.tb00073.x. Tokaji, K., Ogawa, T., Harada, Y. (1986a) The growth of small fatigue cracks in a low carbon steel: the effect of microstructure and limitations of linear elastic fracture mechanics. Fatigue and Fracture of Engineering Materials and Structures 9, 205– 217. DOI: 10.1111/j.1460-2695.1986.tb00447.x. Tokaji, K., Ogawa, T., Harada, Y., Ando, Z. (1986b) Limitations of linear elastic fracture mechanics in respect of small fatigue cracks and microstructure. Fatigue and Fracture of Engineering Materials and Structures 9, 1–14. DOI: 10.1111/j. 1460-2695.1986.tb01207.x. Torres, Y., Anglada, M., Llanes, L. (2002) Fatigue limit – fatigue crack growth threshold correlation for hard metals: influence of microstructure. In: Proceedings of the 8th International Fatigue Congress, edited by A. F. Blom, Vol. 1, Cradley Heath, UK, 1171–1178. Troshchenko, V. T. (1994) Stable and unstable fatigue crack propagation in metals. In: Handbook of Fatigue Crack Propagation in Metallic Structures, Vol. 1, edited by A. Carpinteri, Elsevier, Oxford, 581–612. Usami, S. (1987) Short crack fatigue properties and component life estimation. In: Current Research on Fatigue Cracks, Vol. I, edited by T. Tanaka, M. Jono and K. Komai, Elsevier Applied Science, London, 119–147. Usami, S., Shida, S. (1979) Elastic–plastic analysis of the fatigue limit for a material with small flaws. Fatigue and Fracture of Engineering Materials and Structures 1, 471–481. DOI: 10.1111/j.1460-2695.1979.tb01334.x. Vallelano, C., Dominguez, J., Navarro, A. (2003) On the estimation of fatigue failure under fretting conditions using notch methodologies. Fatigue and Fracture of Engineering Materials and Structures 26, 469–478. DOI: 10.1046/j.1460-2695. 2003.00649.x.

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279

Wang, G., Taylor, D., Ciepalowicz, A., Devlukia, J. (1999) Prediction of fatigue failure in cast aluminium alloy components using the crack modelling method. In: Proceedings of Fatigue ’99, Higher Education Press, Beijing, 735–740. Wang, G., Taylor, D., Bouquin, B., Devlukia, J., Ciepalowicz, A. (2000) Prediction of fatigue failure in a camshaft using the crack modelling method. Engineering Failure Analysis 7, 189–197. DOI: 10.1016/S1350-6307(99)00015-1. Weiss, B., Stickler, R., Femböck, J., Pfaffinger, K. (1979) High cycle fatigue and threshold behaviour of powder metallurgical Mo and Mo-alloys. Fatigue and Fracture of Engineering Materials and Structures 2, 73–84. DOI: 10.1111/j.14602695.1979.tb01344.x. Yu, T. M., DuQuesnay, D. L., Topper, T. H. (1988) Notch fatigue behaviour of SAE1045 steel. International Journal of Fatigue 10, 109–116. DOI: 10.1016/01421123(88)90038-2. Yu, T. M., DuQuesnay, D. L., Topper, T. H. (1991) Notched fatigue behaviour of two cold rolled steels. Fatigue and Fracture of Engineering Materials and Structures 14, 89–101. DOI: 10.1111/j.1460-2695.1991.tb00645.x.

Appendix B Experimental results generated under multiaxial fatigue loading

Abstract: This appendix summarises about 4500 experimental results generated by testing plain and notched specimens under both strain and stress controlled multiaxial fatigue loading paths. A number of experimental results generated by testing aluminium and steel welded joints as well as samples of composite material are also reported. Key words: multiaxial fatigue results.

B.1

Introduction

This appendix summarises about 4500 experimental results generated by testing plain and notched specimens under both strain and stress controlled multiaxial fatigue loading paths. A number of experimental results generated by testing aluminium and steel welded joints as well as samples of composite material are also reported. The outline of this appendix is as follows:

B.2 Adopted symbolism and nomenclature

B.3 High-cycle fatigue strength of plain specimens. This section summarises multiaxial plain fatigue limits (or endurance limits) generated under tension/ torsion, bending/torsion and internal/external pressure/tension. The static and fatigue properties of the considered materials are summarised in Section B.3.1, and the experimental fatigue results are listed in Section B.3.2.

B.4 High-cycle fatigue strength of notched specimens. This section reports the experimental fatigue limits (or endurance limits) determined by testing, 280

Experimental results generated under multiaxial fatigue loading

281

under bending and torsion, samples containing different geometrical features. The selected materials together with the corresponding fully reversed uniaxial and torsional notch fatigue limits are summarised in Section B.4.1. The experimental values of the notch multiaxial results are summarised in Section B.4.2.

B.5 Fatigue results generated by testing plain and notched specimens under strain control. The static and fatigue properties of the selected materials are reported in Section B.5.1, and the corresponding experimental results in Section B.5.2.

B.6 Low/medium-cycle fatigue results generated under stress control. The designations of the selected materials together with their static properties are listed in Section B.6.1, and the corresponding experimental results are reported in Section B.6.2.

B.7 Fatigue results generated by testing steel and aluminium welded specimens under multiaxial fatigue loading. The materials from which the considered welded specimens are made are summarised in Section B.7.1, and the experimental results generated under multiaxial cyclic loading are reported in Section B.7.2.

B.8 Multiaxial fatigue strength of composite materials. The considered materials together with their static properties are summarised in Section B.8.1, and the experimental results generated under multiaxial cyclic loading are listed in Section B.8.2.

B.9 Geometries of the notched/welded samples. This section presents the technical drawings of the considered notched and welded specimens (dimensions in millimetres).

B.10 References

282

Multiaxial notch fatigue

z y O

x

B.1 Definition of the frame of reference used to calculate stress and strain components in cylindrical (tubular) specimens.

B.2

Adopted symbolism and nomenclature

All the stress-based data listed in this appendix are summarised in terms of nominal stresses calculated with respect to the nominal net section. Only for those specimens characterised by complex geometries are different assumptions made to calculate nominal stresses correctly and efficiently. In the above situations, the reference sections used to calculate normal and shear nominal stresses are shown in Section B.9. For the data generated by testing cylindrical samples, the adopted frame of reference is defined in Fig. B.1. In the presence of more complex specimen shapes, the defined frames of reference are shown in Section B.9. According to the adopted system of coordinates, the stress components applied to the considered cylindrical/tubular specimens at any instant of the cyclic load histories are calculated as follows:

σ x ( t ) = σ x ,m + σ x ,a sin (ω t ) σ y ( t ) = σ y,m + σ y,a sin (ω t + δ y, x ) τ xy ( t ) = τ xy,m + τ xy,a sin ( ω t + δ xy, x ) Similarly, the strain components during the loading cycle are expressed as:

ε x ( t ) = ε x ,m + ε x ,a sin (ω t ) ε y ( t ) = ε y,m + ε y,a sin (ω t + δ y, x ) γ xy ( t ) = γ xy,m + γ xy,a sin (ω t + δ xy, x ) For those results generated under more complex load histories, the considered loading paths are sketched in Fig. B.2.

Experimental results generated under multiaxial fatigue loading g

g

g

g

3

3

3

3

e

Path E-A g

e

e

Path E-B g

3

Path E-E g

e

e

Path E-O g

e

3

e

e

Path E-P

Path E-Q

s

3

e

s

e

e

Path a

Path E-S

B.2 Loading paths.

Path E-M g

3

e

Path E-R

e

Path E-L g

3

Path b

3

e

Path E-K g

3

Path E-H g

3

e

Path E-N g

e

Path E-G g

3

3

3

e

Path E-F g

Path E-I g

Path E-D g

3

e

3

e

Path E-C g

3

e

283

Path W-A

Path W-B

284

Multiaxial notch fatigue

B.2.1 Nomenclature b c b0 c0 fx fy fxy n n′ n′0 t E G B I/II IP IEP K K′ K′0 Nf OH T Te VN dy,x dxy,x sUTS s0 s0n t0 t0n s′f e′f t′f g ′f w sy sx,a, sy,a sx,m, sy,m

Fatigue strength exponent Fatigue ductility exponent Fatigue strength exponent under shear strain Fatigue ductility exponent under shear strain Frequency of the x-stress/strain component Frequency of the y-stress/strain component Frequency of the xy-stress/strain component Strain hardening exponent Cyclic strain hardening exponent Cyclic shear strain hardening exponent Time Young’s modulus Torsional elastic modulus Bending Mode I and II loading Internal pressure Internal/external pressure Strength coefficient Cyclic strength coefficient Cyclic shear strength coefficient Number of cycles to failure Tubular specimen with radial oil hole Torsion Tension Cylindrical V-notched specimen Phase angle between y- and x-components Phase angle between xy- and x-components Ultimate tensile strength Fully reversed plain uniaxial fatigue limit (or endurance limit) Fully reversed notch uniaxial fatigue limit (or endurance limit) Fully reversed plain torsional fatigue limit (or endurance limit) Fully reversed notch torsional fatigue limit (or endurance limit) Fatigue strength coefficient Fatigue ductility coefficient Fatigue strength coefficient under shear strain Fatigue ductility coefficient under shear strain Angular velocity Yield stress Amplitude of the normal stress components Mean value of the normal stress components

Experimental results generated under multiaxial fatigue loading txy,a txy,m ex,a, ey,a ex,m, ey,m gxy,a gxy,m

Shear stress amplitude Mean value of the shear stress Amplitude of the normal strain components Mean value of the normal strain components Shear strain amplitude Mean value of the shear strain

285

High-cycle fatigue strength of plain specimens

Material

0.1% C steel (normalised) 0.4% C steel (normalised) 0.4% C steel (spheroidised) 0.9% C steel (pearlitic) 3% Ni steel 3/3.5% Ni steel Cr-Va steel 3.5% NiCr steel (n. impact) 3.5% NiCr steel (l. impact) NiCrMo steel (60–70 tons) NiCrMo steel (75–80 tons) NiCr steel Silal cast iron Nicrosilal cast iron S65A steel CrMo steel CrMo steel CrMo steel CrMo steel

Code

HSF1 HSF2 HSF3 HSF4 HSF5 HSF6 HSF7 HSF8 HSF9 HSF10 HSF11 HSF12 HSF13 HSF14 HSF15 HSF16 HSF17 HSF18 HSF19

Table B.1

Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough et al., 1951 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956

Reference

B.3.1 Static and fatigue properties of the selected materials

B.3

151.3 206.9 155.9 240.8 205.3 267.1 257.8 352.0 324.2 384.7 342.7 452.3 219.2 211.5 370.5 425.3 412.8 306.9 367.6

(MPa)

268.6 331.9 274.8 352.0 342.7 352.0 429.1 540.3 509.4 625.6 660.7 810.4 240.8 253.2 583.5 713.2 688.9 509.0 589.6

t0

s0

(MPa)

430.7 648.4 477.0 847.5 526.4 722.5 751.8 895.3 896.9 1000.3 1242.7 1667.2 230.0 219.2 1000.0 946.5 946.5 954.0 944.8

(MPa)

sUTS

B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T

Loading path

286 Multiaxial notch fatigue

HSF20 HSF21 HSF22 HSF23 HSF24 HSF25 HSF26 HSF27 HSF28 HSF29 HSF30 HSF31 HSF32 HSF33 HSF34 HSF35 HSF36 HSF37 HSF38 HSF39

CrMo steel CrMo steel NiCrMo steel S81 NiCrMoVa steel CrMoVa steel DTD551 CrMoVa steel DTD551 CrMoVa steel DTD551 CrMoVa steel DTD551 NiCr steel NiCr steel NiCr steel SAE 4340 Brass High-strength steel Mild steel 0.34% C steel Carbon steel 30NCD16 (Batch 1) 30NCD16 (Batch 2) Cast iron

Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Findley et al., 1956 Matake, 1977 Matake, 1977 Matake, 1977 Nishihara and Kawamoto, 1941 Kitaioka et al., 1986 Froustey, 1986 Froustey and Lasserre, 1989 Ros, 1950

593.3 628.3 589.7 660.7 667.8 659.9 706.1 737.7 666.7 653.2 771.9 462.0 83.0 460.0 196.0 378.0 261.0 586.0 660.0 151.0

350.7 366.6 331.9 342.7 398.3 386.5 412.5 447.4 369.7 339.6 452.3 286.0 74.0 275.0 186.0 218.0 160.0 405.0 410.0 92.0

944.8 954.0 1103.8 1242.7 1397.1 1397.1 1368.7 1368.7 1398.4 1398.4 1667.2 – – – – – – – 1880.0 –

B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T B-T IP-T

Experimental results generated under multiaxial fatigue loading 287

Material

0.35% C steel Mild steel Hard steel 34Cr4 42CrMo4 St35 En24T XC18 34Cr4 25CrMo4 Fe-Cu 25CrMo4 Cast iron

Code

HSF40 HSF41 HSF42 HSF43 HSF44 HSF45 HSF46 HSF47 HSF48 HSF49 HSF50 HSF51 HSF52

Table B.2

Rotvel, 1970 Nishihara and Kawamoto, 1945 Nishihara and Kawamoto, 1945 Zenner et al., 1985 Lempp, 1977 Issler, 1973 McDiarmid, 1985 Froustey et al., 1992 Zenner et al., 1985 Kaniut, 1983 Fogué and Bahuand, 1985 Zenner et al., 1985 Nishihara and Kawamoto, 1945

Reference 220.0 235.4 313.9 410.0 398.0 206.0 405.0 332.0 343.0 340.0 319.0 361.0 96.1

s0 (MPa) 142.0 137.3 196.2 256.0 267.0 123.0 270.0 186.0 204.0 228.0 220.0 228.0 91.2

t0 (MPa)

375.0 680.0 795.0 1025.0 395.0 – 1530.0 710.0 – – 780.0 185.0

–

sUTS (MPa)

IEP-Te B-T B-T B-T B-T IP-Te IP-Te B-T IP-B-T B-T B-T IP-Te-T B-T

Loading path

288 Multiaxial notch fatigue

HSF53 HSF54 HSF55 HSF56 HSF57 HSF58 HSF59 HSF60 HSF61 HSF62 HSF63 HSF64 HSF65

Grey cast iron CK45 FGS 800-2 FeE 460 FGS 700/2 Ti-6Al-4V High-strength steel High-strength steel High-strength steel R7T (axial) R7T (circum.) 42CrMo4 39NiCrMo3

Achtelik et al., 1983 Simbürger, 1975 Palin-Luc and Lasserre, 1998 Sonsino, 2001 Akrache and Lu, 1999 Delahay and Palin-Luc, 2005 Altenbach and Zolochevsky, 1996 Altenbach and Zolochevsky, 1996 Altenbach and Zolochevsky, 1996 Bernasconi et al., 2006 Bernasconi et al., 2006 Froeschl et al., 2007 Bernasconi et al., 2008

164.0 423.0 294.0 272.0 294.0 652.0 630.0 – 320.0 – – 996.0 367.5

124.7 287.0 220.0 174.0 218.0 411.0 364.0 172.0 – 297.0 310.0 525.7 265.0

278.8 – 815.0 670.0 750.0 1090.0 – – – 820–940 820–940 336.3 856

B-T B-T B-T Te-T B-T B-T B-T B-T B-T Te-T Te-T Te-T Te-T

Experimental results generated under multiaxial fatigue loading 289

Material

0.1% C steel (normalised)

0.4% C steel (normalised)

0.4% C steel (spheroidised)

Code

HSF1

HSF2

HSF3

Table B.3

76.9 153.1 207.9 247.8 275.3

101.7 193.3 252.1 297.0 329.0

80.9 147.4 207.3 241.1 263.7 72.6 138.5 202.3 240.2 257.8

sx,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

B.3.2 Plain multiaxial fatigue limits (or endurance limits)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

145.6 131.7 103.7 73.8 40.0

189.7 164.1 126.1 86.3 46.0

147.1 130.5 105.6 66.2 32.6 136.2 123.0 104.0 71.2 33.7

txy,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dy,x

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dxy,x

290 Multiaxial notch fatigue

0.9% C steel (pearlitic)

3% Ni steel

3/3.5% Ni steel

Cr-Va steel

HSF4

HSF5

HSF6

HSF7

134.8 245.8 328.0 377.6 407.4

134.9 247.9 334.8 394.9 425.8

104.8 190.5 266.6 306.0 329.0

113.0 213.5 283.7 323.9 355.8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

239.3 208.4 257.7 100.5 59.3

249.8 211.0 163.0 112.7 52.2

192.4 163.6 134.3 88.9 41.5

208.1 178.0 140.9 91.1 44.2

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 291

Material

3.5% NiCr steel (normal impact)

3.5% NiCr steel (low impact)

NiCrMo steel (60–70 tons)

Code

HSF8

HSF9

HSF10

Table B.4

179.5 344.9 473.9 539.7 566.6

167.0 316.2 399.1 469.9 514.7

178.0 330.4 410.3 490.3 541.2 164.9 297.2 409.7 456.2 501.4

sx,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

328.2 292.9 234.0 153.3 72.6

316.2 274.2 200.5 135.1 72.3

329.7 280.8 200.4 141.3 68.9 306.9 256.7 207.2 128.4 67.6

txy,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dxy,x (°)

292 Multiaxial notch fatigue

NiCrMo steel (75–80 tons)

NiCr steel

Silal cast iron

Nicrosilal cast iron

HSF11

HSF12

HSF13

HSF14

93.2 169.7 189.6 228.3 243.9

87.8 151.1 193.6 208.9 233.7

228.3 429.8 605.9 735.3 788.7

173.1 354.1 506.2 545.6 623.5

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

170.1 147.7 94.3 62.4 31.7

164.6 134.5 95.7 58.2 32.9

411.4 365.4 298.3 208.3 103.6

337.0 303.0 248.9 156.7 84.0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 293

Material

S65A steel

Code

HSF15

Table B.5

583.6 552.8 532.7 0 0 0 549.7 540.4 555.8 555.8 469.4 472.5 0 0 0 0 0 0 546.6 389.1 168.3 496.4 373.6 160.6 428.5 315.0 126.6 386.0 382.9

sx,a (MPa) 0.0 266.3 532.7 0 0 0 0 0 266.3 266.3 532.7 532.7 266.3 532.7 266.3 532.7 266.3 532.7 0 0 0 266.3 266.3 266.3 532.7 532.7 532.7 266.3 0.0

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 370.6 338.9 342.8 0 0 0 0 0 0 311.9 284.1 304.2 281.0 308.8 293.4 155.9 259.4 335.0 141.3 248.6 321.2 121.2 210.0 251.7 256.3 254.0

txy,a (MPa) 0.0 0.0 0.0 0.0 169.8 343.5 169.8 343.5 169.8 343.5 169.8 343.5 0.0 0.0 169.8 169.8 343.5 343.5 0 0 0 169.8 169.8 169.8 343.5 343.5 343.5 0 169.8

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

294 Multiaxial notch fatigue

CrMo steel

CrMo steel

HSF16

HSF17

617.3 606.8 574.9 480.7 373.6 207.6

683.3 669.1 605.9 530.7 401.2 216.6 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 82.7 113.9 165.8 240.4 319.4 387.5

91.5 125.8 174.9 265.2 347.3 404.2 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 295

Material

CrMo steel

CrMo steel

CrMo steel

CrMo steel

Code

HSF18

HSF19

HSF20

HSF21

Table B.6

600.7 591.7 560.7 465.7 355.2 186.0

560.2 549.1 519.9 429.2 322.6 173.8

523.2 445.7 343.2

490.0 478.7 460.0 394.3 284.8 152.2

sx,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sy,m (MPa)

80.4 111.2 161.9 233.0 307.4 347.2

75.0 103.1 150.1 214.6 279.4 324.0

151.0 222.8 297.2

65.6 90.0 132.8 197.0 246.5 283.9

txy,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dxy,x (°)

296 Multiaxial notch fatigue

NiCrMo steel S81

NiCrMoVa steel

CrMoVa steel DTD 551

HSF22

HSF23

HSF24

645.1 617.8 591.6 499.4 375.1 200.1

625.2 546.5 506.3 355.1 183.7

562.5 537.8 499.2 428.7 306.7 167.3

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

86.5 116.1 170.7 249.6 324.7 373.1

83.4 157.5 253.2 307.2 342.7

75.3 101.1 144.2 214.3 265.2 312.3

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 297

Material

CrMoVa steel DTD 551

CrMoVa steel DTD 551

CrMoVa steel DTD 551

NiCr steel

Code

HSF25

HSF26

HSF27

HSF28

Table B.7

649.9 615.8 586.8 484.1 343.0 189.6

726.6 703.9 678.9 575.5 426.8 229.9

626.6 555.7 414.2

636.0 608.7 577.2 504.5 372.4 196.5

sx,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sy,m (MPa)

87.1 115.6 169.4 242.1 296.9 353.7

97.3 132.3 196.1 287.8 369.6 428.7

180.8 277.9 359.1

85.2 114.4 166.6 252.3 322.3 366.8

txy,a (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dxy,x (°)

298 Multiaxial notch fatigue

NiCr steel

NiCr steel

SAE 4340

Brass

High-strength steel

HSF29

HSF30

HSF31

HSF32

HSF33

186.0 296.0

46.0 64.0

354.0 265.0 155.0

790.4 734.8 608.2 429.2 225.4

609.2 579.4 542.5 471.3 336.8 174.4

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

235.0 171.0

54.0 37.0

177.0 230.0 276.0

106.5 211.5 302.6 372.0 418.4

81.5 108.8 156.5 235.4 291.5 324.5

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0

0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 299

Material

Mild steel

0.34% C steel

Carbon steel

30NCD16 (Batch 1)

Code

HSF34

HSF35

HSF36

HSF37

Table B.8

315.0 450.0 435.0 500.0 420.0

242.0 204.0 248.0

110.0 109.0 204.0 207.0 280.0 285.0 331.0 338.0 365.0 374.0

106.0 167.0

sx,a (MPa)

0 0 0 300.0 600.0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

sx,m (MPa)

0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

sy,a (MPa)

0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

sy,m (MPa)

315.0 225.0 215.0 125.0 140.0

134.0 104.0 71.0

201.0 205.0 174.0 179.0 137.0 141.0 96.0 93.0 47.0 48.0

126.0 96.0

txy,a (MPa)

0 0 200.0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

txy,m (MPa)

0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

dy,x (°)

0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0

dxy,x (°)

300 Multiaxial notch fatigue

30NCD16 (Batch 2)

Cast iron

0.35% C steel

HSF38

HSF39

HSF40

221.0 224.0 158.0 120.0

137.5 120.0 128.5 133.0 102.5

485.0 480.0 480.0 480.0 470.0 473.0 590.0 565.0 540.0 211.0

0 0 0 0

137.5 120.0 128.5 133.0 102.5

0 0 300.0 300.0 300.0 300.0 300.0 300.0 300.0 300.0

174.0 169.0 123.0 156.0

68.0 117.5 128.5 62.0 107.5

0 0 0 0 0 0 0 0 0 0

0.0 0.0 0.0 0.0

68.0 117.5 128.5 62.0 107.7

0 0 0 0 0 0 0 0 0 0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

280.0 277.0 277.0 277.0 270.0 273.0 148.0 141.0 135.0 365.0

0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 180 180

0 0 0 0 180

0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0

0 90 0 45 60 90 0 45 90 0

Experimental results generated under multiaxial fatigue loading 301

Material

Mild steel

Hard steel

Code

HSF41

HSF42

Table B.9

131.8 245.3 299.1 140.4 249.7 145.7 252.4 150.2 258.0 304.5

99.9 180.3 213.2 103.6 191.4 108.9 201.1 230.2

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

sy,m (MPa)

167.1 122.7 62.8 169.9 124.9 176.3 126.2 181.7 129.0 63.9

120.9 90.2 44.8 125.4 95.7 131.8 100.6 48.3

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 30 30 60 60 90 90 90

0 0 0 60 60 90 90 90

dxy,x (°)

302 Multiaxial notch fatigue

34Cr4

42CrMo4

HSF43

HSF44

328.0 233.0 266.0 280.0 286.0 213.0 283.0 271.0 330.0

314.0 316.0 279.0 355.0 315.0 314.0 316.0 224.0 380.0 315.0 284.0 212.0 129.0 315.0 0 0 0 280.0 0 0 0 271.0 0

0 0 279.0 0 0 0 0 0 0 0 284.0 212.0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 157.4 223.7 127.7 134.4 137.3 204.5 135.8 130.1 158.4

157.0 158.0 140.0 89.0 158.0 157.0 158.0 224.0 95.0 158.0 142.0 212.0 258.0 158.0 0 0 127.7 0 0 0 135.8 0 158.4

0 158.0 0 178.0 0 157.0 0 0 0 158.0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 90 90 90 90 180

0 0 0 0 60 60 90 90 90 90 90 90 90 120

Experimental results generated under multiaxial fatigue loading 303

Material

St35

En24T

XC18

Code

HSF45

HSF46

HSF47

Table B.10

0 0 0

0 0 0 0 0 0

270.0a 355.0a 260.0b 250.0b 260.0c 230.0c

246.0 246.0 264.0

153.2 169.4 172.6 161.8 164.0 142.4 156.4 159.7 120.8 110.0 134.8 129.5

sx,m (MPa)

139.0 154.0 156.9 147.1 149.1 129.5 142.2 145.1 109.8 100.0 122.6 117.7

sx,a (MPa)

0 0 0

270.0 355.0 260.0 250.0 260.0 230.0

139.0 154.0 156.9 147.1 149.1 129.5 142.2 145.1 109.8 100.0 122.6 117.7

sy,a (MPa)

0 0 0

0 0 0 0 0 0

153.2 169.4 172.6 161.8 164.0 142.4 156.4 159.7 120.8 110.0 134.8 129.5

sy,m (MPa)

138.0 138.0 148.0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa)

0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0

180 0 90 0 180 0

0 0 0 60 60 90 90 90 120 180 180 180

dy,x (°)

0 45 90

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

304 Multiaxial notch fatigue

Fe-Cu

HSF50

d

0 0 202.5 213.0

0 0 0 0

210.0d 220.0e 242.0e 196.0f

242.4 255.4 202.5 213.0

250.0 214.0 221.0 238.0 234.0 220.0 275.0 275.0 232.0

205.0 175.0 181.0 195.0 192.0 180.0 225.0 225.0 190.0

fy/fx = 1; b fy/fx = 2; c fy/fx = 3. fxy/fx = 0.25; e fxy/fx = 2; f fxy/fx = 8.

25CrMo4

HSF49

a

34Cr4

HSF48

0 0 0 0

0 0 0 0

205.0 175.0 181.0 195.0 192.0 180.0 225.0 225.0 190.0

0 0 0 0

0 0 0 0

250.0 214.0 221.0 238.0 234.0 220.0 275.0 275.0 232.0

140.6 148.1 117.5 123.5

105.0 110.0 121.0 98.0

96.0 82.0 85.0 92.0 90.0 85.0 0 0 0

0.0 0.0 117.5 123.5

0 0 0 0

0 0 85.0 92.0 90.0 85.0 0 0 0

0 0 0 0

0 0 0 0

0 180 0 0 60 180 0 60 180

0 90 0 90

0 90 0 0

0 0 0 90 90 90 0 0 0

Experimental results generated under multiaxial fatigue loading 305

Material

25CrMo4

Cast iron

Grey cast iron

Code

HSF51

HSF52

HSF53

Table B.11

130.1 161.7 110.7

83.4 95.2 56.3 93.7 67.6 101.9 73.2

261.0 275.0 240.0 196.0 270.0 261.0 277.0 220.0 233.0 155.0 159.0

sx,a MPa)

0 0 0

0 0 0 0 0 0 0

340.0 340.0 340.0 340.0 0 0 0 340.0 340.0 340.0 340.0

sx,m MPa)

0 0 0

0 0 0 0 0 0 0

261.0 275.0 240.0 196.0 0 0 0 0 0 0 0

sy,a MPa)

0 0 0

0 0 0 0 0 0 0

170.0 170.0 170.0 170.0 0 0 0 170.0 170.0 170.0 170.0

sy,m MPa)

65.1 46.7 95.8

41.6 19.7 68.0 46.9 81.6 21.1 88.4

0 0 0 0 135.0 131.0 139.0 110.0 117.0 155.0 159.0

txy,a MPa)

0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

txy,m MPa)

0 0 0

0 0 0 0 0 0 0

0 60 90 180 0 0 0 0 0 0 0

dy,x (°)

0 0 0

0 0 0 60 60 90 90

0 0 0 0 0 60 90 60 90 60 90

dxy,x (°)

306 Multiaxial notch fatigue

Cast iron

FGS 800-2

FeE 460

FGS 700/2

Ti-6Al-4V

HSF54

HSF55

HSF56

HSF57

HSF58

442.0 567.0

229.0 246.3

210.0 200.0

228.0 199.0 245.0

400.0 0.0 277.0 285.0 292.0 304.0 250.0 288.0

0 0

0 0

0 0

0 0 0

0 417.0 277.0 0 0 0 250.0 0

0 0

0 0

0 0

0 0 0

0 0 0 0 0 0 0 0

0 0

0 0

0 0

0 0 0

0 0 0 0 0 0 0 0

255.0 328.0

132.2 142.2

121.2 115.5

132.0 147.0 142.0

0 241.0 159.0 163.0 167.0 174.0 144.0 165.0

0 0

0 0

0 0

0 0 0

200.0 0 0 163.0 0 0 0 165.0

0 0

0 0

0 0

0 0 0

0 0 0 0 0 0 0 0

0 90

0 90

90 0

0 0 90

0 0 0 0 60 90 90 90

Experimental results generated under multiaxial fatigue loading 307

Material

High-strength steel

High-strength steel

High-strength steel

R7T (axial)

Code

HSF59

HSF60

HSF61

HSF62

Table B.12

100.0 200.0

310.0 300.0 280.0 253.0 265.0 215.0 198.0 167.0

0.0 0.0 0.0

152.0 293.0 467.0 570.0

sx,a MPa)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

−100.0 −200.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0

sy,a MPa)

75.0 150.0 200.0 300.0 75.0 150.0 200.0 300.0

166.0 162.0 148.0

0.0 0.0 0.0 0.0

sx,m MPa)

0.0 0.0

0.0 0.0 0.0 0.0 150.0 300.0 400.0 600.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0

sy,m MPa)

287.0 267.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

166.0 162.0 148.0

352.0 314.0 234.0 134.0

txy,a MPa)

0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0

txy,m MPa)

0 0

0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

dy,x (°)

90 90

0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

dxy,x (°)

308 Multiaxial notch fatigue

39NiCrMo3

HSF65

fxy/fx = 1; b fxy/fx = 2; c fxy/fx = 3.

42CrMo4

HSF64

a

R7T (circum.)

HSF63

0.0 400.6 719.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

294.5a 259.5b 266.0c

−100.0 −200.0

525.7 400.0 240.0 0.0 0.0 0.0 163 414.7

100.0 200.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0

170.0 149.8 153.6

0.0 0.0 0.0 336.3 318.4 282.4 326.7 205.9

301.0 293.0

0.0 0.0 0.0

0.0 0.0 0.0 0.0 100.1 279.5 0.0 0.0

0.0 0.0

0 0 0

0 0 0 0 0 0 0 0

0 0

0 0 0

0 0 0 0 0 0 0 0

90 90

Experimental results generated under multiaxial fatigue loading 309

High-cycle fatigue strength of notched specimens

Material

0.4% C steel (normalised) 3% Ni steel 3/3.5% Ni steel Cr-Va steel 3.5% NiCr steel (n. impact) 3.5% NiCr steel (l. impact) NiCrMo steel (75–80 tons) CrMo steel CrMo steel CrMo steel

Code

N-HSF1 N-HSF2 N-HSF3 N-HSF4 N-HSF5 N-HSF6 N-HSF7 N-HSF8 N-HSF9 N-HSF10

Table B.13

Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Gough, 1949 Frith, 1956 Frith, 1956 Frith, 1956

Reference 179.1 209.9 302.6 216.1 268.6 247.0 271.7 450.9 223.4 424.7

s0n (MPa)

176.0 151.3 183.7 160.6 236.2 182.2 240.8 295.5 162.1 305.5

t0n (MPa)

Fig. Fig. Fig. Fig. Fig. Fig. Fig. OH OH OH

B.3 B.3 B.3 B.3 B.3 B.3 B.3

Geometry

B.4.1 Selected materials and uniaxial and torsional notch fatigue limits (or endurance limits)

B.4

310 Multiaxial notch fatigue

N-HSF11 N-HSF12 N-HSF13 N-HSF14 N-HSF15 N-HSF16 N-HSF17 N-HSF18 N-HSF19 N-HSF20 N-HSF21 N-HSF22

CrMo steel CrMo steel NiCrMoVa steel NiCrMoVa steel CrMoVa steel DTD551 CrMoVa steel DTD551 CrMoVa steel DTD551 CrMoVa steel DTD551 NiCr steel S65A steel S65A steel S65A steel

Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Frith, 1956 Gough et al., 1951 Gough et al., 1951 Gough et al., 1951

423.4 423.4 271.7 288.7 300.7 297.8 471.3 448.1 312.5 259.3 347.3 563.5

300.3 288.4 240.8 211.5 235.4 220.9 354.6 354.9 225.5 180.6 270.1 185.2

OH OH VN OH OH OH OH OH OH Fig. B.4 Fig. B.5 Fig. B.6

Experimental results generated under multiaxial fatigue loading 311

Material

0.4% C steel (normalised)

3% Ni steel

3/3.5% Ni steel

Cr-Va steel

Code

N-HSF1

N-HSF2

N-HSF3

N-HSF4

Table B.14

77.3 137.6 172.0 210.3 210.9

97.6 175.5 214.4 265.4 286.2

72.6 126.7 171.5 195.6 209.2

76.7 120.1 148.4 166.1 180.5

sx,a (MPa)

B.4.2 Notch multiaxial fatigue limits (or endurance limits)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

sx,m (MPa)

147.3 117.0 88.6 58.0 27.9

172.6 153.5 108.4 78.7 40.1

132.8 106.4 83.5 56.8 25.3

140.0 107.3 74.7 51.7 27.6

txy,a (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

txy,m (MPa)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

dxy,x (°)

312 Multiaxial notch fatigue

3.5% NiCr steel (normal impact)

3.5% NiCr steel (low impact)

NiCrMo steel (75–80 tons)

Cr-Mo steel

N-HSF5

N-HSF6

N-HSF7

N-HSF8

449.2 442.9 337.5 260.3 143.6

104.2 169.8 237.3 252.9 265.7

857.1 168.1 198.7 223.8 235.4

102.8 170.0 228.2 261.0 290.8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

60.2 83.2 168.7 225.4 268.0

194.8 147.6 123.8 75.5 36.6

160.6 147.7 99.4 67.2 31.8

188.2 145.3 113.8 77.0 36.9

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 313

Material

Cr-Mo steel

Cr-Mo steel

Cr-Mo steel

Code

N-HSF9

N-HSF10

N-HSF11

Table B.15

397.8 389.5 370.7 320.8 247.5 141.7

364.3 326.3 245.8

219.8 216.7 210.6 177.7 136.2 74.3

sx,a (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sx,m (MPa)

53.3 73.2 107.0 160.4 214.4 264.4

105.1 163.2 212.9

29.5 40.8 60.8 88.8 117.9 138.6

txy,a (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dxy,x (°)

314 Multiaxial notch fatigue

Cr-Mo steel

NiCrMoVa steel

NiCrMoVa steel

CrMoVa steel DTD551

N-HSF12

N-HSF13

N-HSF14

N-HSF15

297.8 288.2 280.2 251.0 185.3 111.8

265.5 198.7 95.9

265.5 253.2 239.3 169.8 103.4

390.4 381.9 353.5 305.5 243.9 140.8

0 0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

39.8 54.2 80.9 125.5 160.4 208.6

75.6 132.1 190.2

35.5 74.1 120.4 146.7 193.0

52.3 71.8 102.0 152.7 211.2 262.6

0 0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 315

Material

CrMoVa steel DTD551

CrMoVa steel DTD551

CrMoVa steel DTD551

Code

N-HSF16

N-HSF17

N-HSF18

Table B.16

417.6 438.7 409.9 362.3 289.3 162.4

409.9 371.6 278.0

287.4 274.0 259.8 236.4 174.8 103.9

sx,a (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

sx,m (MPa)

55.9 82.4 118.3 181.2 250.6 302.9

118.4 185.9 240.8

38.4 51.6 75.0 118.1 151.4 193.9

txy,a (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0

0 0 0

0 0 0 0 0 0

dxy,x (°)

316 Multiaxial notch fatigue

NiCr steel

S65A

N-HSF19

N-HSF20

259.4 237.8 259.4 236.2 0 0 0 0 174.5 159.0 166.8 142.0 219.2 86.5

298.7 296.4 268.9 235.6 194.1 108.8 0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 0

0 0 0 0 0 0 0 0 0 0 180.6 169.8 173.7 172.9 115.8 106.5 111.2 94.2 61.8 171.4

40.1 55.6 77.7 117.8 167.8 202.9 0 0 169.8 169.8 0 0 169.8 169.8 0 0 169.8 169.8 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

Experimental results generated under multiaxial fatigue loading 317

Material

S65A

S65A

Code

N-HSF21

N-HSF22

Table B.17

563.6 537.3 534.2 540.4 0 0 0 0 264.0 247.0 253.2 250.1 501.8 95.7

347.4 347.4 361.3 318.0 0 0 0 0 253.2 250.1 234.7 216.2 308.8 123.5

sx,a (MPa)

0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 0

0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 266.3 0 0

sx,m (MPa)

0 0 0 0 185.3 188.4 177.6 173.7 176.0 163.7 168.3 166.8 142.0 189.9

0 0 0 0 270.2 243.2 276.4 230.1 168.3 166.8 155.9 143.6 88.0 245.5

txy,a (MPa)

0 0 169.8 169.8 0 0 169.8 169.8 0 0 169.8 169.8 0 0

0 0 169.8 169.8 0 0 169.8 169.8 0 0 169.8 169.8 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

318 Multiaxial notch fatigue

s′f = 1000 MPa b = −0.114 0.171 e ′f = c = −0.402 K′ = 1660 MPa n′ = 0.287 Remarks: Nf = complete failure

Material: AISI 304 E = 183000 G = 82800 325 sy = 650 sUTS = K = 1210 n = 0.193 Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Socie, 1987

Code: SCT2

s′f = – MPa b = – – e′f = c = – K′ = – MPa n′ = – Remarks: Nf = life to 1 mm crack

Material: Inconel 718 E = 208500 G = 77800 1160 sy = 1420 sUTS = K = 1910 n = 0.08 Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Socie et al., 1989

Code: SCT1

Table B.18

t ′f = b0 = g ′f = c0 = K′0 = n′0 =

t ′f = b0 = g ′f = c0 = K′0 = n′0 =

Fatigue results generated by testing plain and notched specimens under strain control

B.5.1 Static and fatigue properties of the selected materials

B.5

709 −0.121 0.413 −0.353 785 0.296

2146 −0.148 18 −0.922 860 0.079

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 319

s′f = 691 MPa b = −0.1692 0.1014 e′f = c = −0.3768 K′ = 1932 MPa n′ = 0.4492 Remarks: tests run at 550°C Nf = 25% load drop

MPa MPa MPa MPa MPa

Material: AISI 304 E= 167500 G= 56000 – sy = – sUTS = K= – n= – Geom.: plain, tubular

Nitta et al., 1989

Ref.:

Code: SCT4

s′f = 948 MPa b = −0.092 0.26 e ′f = c = −0.445 K′ = 1258 MPa n′ = 0.208 Remarks: 10% load drop Nf =

Material: SAE 1045 E= 204000 G= 80300 380 sy = 621 sUTS = K= 1185 n= 0.23 Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Kurath et al., 1989

Code: SCT3

Table B.18 Continued

t ′f = b0 = g ′f = c0 = K′0 = n′0 =

t ′f = b0 = g ′f = c0 = K′0 = n′0 =

1137 −0.2149 1.055 −0.5664 1113 0.3795

505 −0.097 0.413 −0.445 614 0.217

MPa

MPa

MPa

MPa

320 Multiaxial notch fatigue

Ref.: Lin et al., 1993 s′f = – MPa b = – – e′f = c = – K′ = – MPa n′ = 0.069 Remarks: anisotropic material Nf = 10% load drop

Code: SCT6

Material: 6061-T6 E = 69000 MPa G = – MPa 276 MPa sy = 310 MPa sUTS = K = – MPa n = – Geom.: plain, cylindrical

s′f = – MPa b = – – e′f = c = – K′ = 854 MPa n′ = 0.0149 Remarks: Nf = specimen separation

Material: Ti-6Al-4V E = 116000 G = – 930 sy = 977 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Kallmeyer et al., 2002

Code: SCT5

Table B.19

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′f = b0 = g′f = c0 = K′0 = n′0 =

– – – – – –

– – – – – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 321

Ref.: Lin et al., 1992 s′f = 369 MPa b = −0.311 0.0899 e′f = c = −0.452 K′ = 436 MPa n′ = 0.069 Remarks: anisotropic material Nf = 10% load drop

Code: SCT8a

Material: 6061-T6 (Rod 1) E = 71500 MPa G = 28200 MPa 300 MPa sy = 330 MPa sUTS = K = – MPa n = – Geom.: plain, cylindrical

s′f = 923 MPa b = −0.099 0.359 e′f = c = −0.519 K′ = 1215 MPa n′ = 0.217 Remarks: Nf = 15–20% load drop

Material: S45C E = 186000 G = 70600 496 sy = 770 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Kim et al., 1999

Code: SCT7

Table B.19 Continued

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′f = b0 = g′f = c0 = K′0 = n′0 =

285 −0.0497 0.3881 −0.6423 292 0.068

685 −0.12 0.198 −0.36 1174 0.334

MPa

MPa

MPa

MPa

322 Multiaxial notch fatigue

Ref.: Hua and Socie, 1984 s′f = 980 b = −0.11 0.2 e′f = c = −0.43 K′ = – n′ = – Remarks: Nf = 10% load drop

Code: SCT9

Material: SAE 1045 E = 205000 G = – 150 sy = – sUTS = K = 185 n = 0.23 Geom.: plain, tubular MPa

MPa

s′f = 373 MPa b = −0.0326 0.1043 e′f = c = −0.473 K′ = 436 MPa n′ = 0.069 Remarks: anisotropic material Nf = 10% load drop

Material: 6061-T6 (Rod 2) E = 71500 MPa G = 28200 MPa 300 MPa sy = 330 MPa sUTS = K = – MPa n = – Geom.: plain, cylindrical

MPa MPa MPa MPa MPa

Ref.: Lin et al., 1992

Code: SCT8b

Table B.20

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′ = b0 = g′f = c0 = K′0 = n′0 =

– – – – – –

245 −0.048 1.4746 −0.6745 249 0.071

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 323

Ref.: Chen et al., 2004 s′f = 1124 b = −0.0905 0.8072 e′f = c = −0.6652 K′ = 1115 n′ = 0.1304 Remarks: Nf = 25% stress drop

Code: SCT11

Material: 1Cr-18Ni-9Ti E = 193000 G = 74300 sy = 310 605 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

s′f = 847 b = −0.0832 1.201 e′f = c = −0.6399 K′ = 827 n′ = 0.13 Remarks: Nf = 10% load drop

Material: A533B E = 198000 MPa G = 76100 MPa 482 MPa sy = 576 MPa sUTS = K = – MPa n = – Geom.: plain, cylindrical

MPa

MPa

MPa

MPa

Ref.: Nelson and Rostami, 1997

Code: SCT10

Table B.20 Continued

t′f b0 g′f c0 K′0 n′0

= = = = = =

t′f = b0 = g′f = c0 = K′0 = n′0 =

644 −0.0876 0.8118 −0.5334 1053 0.13

– –

586 −0.1148 1.554 −0.6152

MPa

MPa

MPa

MPa

324 Multiaxial notch fatigue

s′f = 854 MPa b = −0.1054 0.3151 e′f = c = −0.5464 K′ = 1067 MPa n′ = 0.193 Remarks: Nf = life to through-wall crack

Material: A-516 Gr. E = 204000 G = – 325 sy = – sUTS = K = – n = – Geom.: plain, tubular

70 MPa MPa MPa MPa MPa

Ref.: Ellyin et al., 1991; Ellyin and Xia, 1993

MPa

MPa

Code: SCT13

s′f = 728 b = −0.0514 0.095 e′f = c = −0.3273 K′ = 1053 n′ = 0.1304 Remarks: Nf = 25% stress drop

Material: EN8 E = 205000 G = 79000 479 sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Petrone, 1996

Code: SCT12

Table B.21

t′f b0 g′f c0 K′0 n′0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

– – – – – –

466 −0.0582 0.303 −0.3221 – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 325

Ref.: Shang et al., 2007 s′f = 843 b = −0.1047 0.3269 e′f = c = −0.5458 K′ = – n′ = – Remarks: Nf =

Code: SCT15

Material: 45 Steel E = 190000 G = 79000 370 sy = 610 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

s′f = – b = – – e′f = c = – K′ = – n′ = – Remarks: Nf =

Material: 6061T6/Alumina E = – MPa G = – MPa – MPa sy = – MPa sUTS = K = – MPa n = – Geom.: plain, tubular

MPa

MPa

MPa

MPa

Ref.: Xia and Ellyin, 1998

Code: SCT14

Table B.21 Continued

t′f b0 g′f c0 K′0 n′0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

559 −0.1078 0.496 −0.469 – –

– – – – – –

MPa

MPa

MPa

MPa

326 Multiaxial notch fatigue

s′f = 1616 MPa b = −0.12 1.5675 e′f = c = −0.9 K′ = 1040 MPa n′ = 0.085 Remarks: fatigue constants determined under Re = Nf = 50% diametral stress drop

Material: 1-Cr-Mo-V E = 201000 G = 72000 630 sy = 805 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Lohr and Ellison, 1980

Code: SCT17

s′f = – MPa b = – – e′f = c = – K′ = – MPa n′ = – Remarks: Nf = 50% diametral stress drop

Material: En15R E = – G = – – sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Shatil et al., 1994

Code: SCT16

Table B.22

t′f b0 g′f c0 K′0 n′0 0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

740 −0.16 2.389 −0.9 1070 0.134

– – – – – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 327

Geom.: plain, tubular

Nf = load drop

s′f = – b = – – e′f = c = – K′ = – n′ = – Remarks: Batch A

MPa MPa MPa MPa MPa

Material: Waspaloy E = – G = – – sy = – sUTS = K = – n = – MPa

MPa

Jayaraman and Ditmars, 1989

Ref.:

Code: SCT19

s′f = 834 MPa b = −0.0793 0.1572 e′f = c = −0.4927 K′ = 1115 MPa n′ = 0.161 Remarks: life to 0.5 mm crack Nf =

Material: S460N E = 208500 G = 80200 500 sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Jiang et al., 2007

Code: SCT18

Table B.22 Continued

t′f b0 g′f c0 K′0 n′0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

– – – – – –

– –

529 −0.0955 0.213 −0.418

MPa

MPa

MPa

MPa

328 Multiaxial notch fatigue

s′f = 969.6 MPa b = 0.086 0.281 e′f = c = 0.493 K′ = 1115 MPa n′ = 0.161 Remarks: Nf = life to 250 μm crack

Material: S460N E = 208500 G = – 500 sy = 643 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Hoffmeyer et al., 2006

Code: SCT21

MPa

MPa

s′f = – b = – – e′f = c = – K′ = – n′ = – Remarks: Batch F Nf = load drop

Material: Waspaloy E = – G = – – sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Jayaraman and Ditmars, 1989

Code: SCT20

Table B.23

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′f = b0 = g′f = c0 = K′0 = n′0 =

– – – – – –

– – – – – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 329

s′f = 865 MPa b = −0.097 0.119 e′f = c = −0.359 K′ = 1329 MPa n′ = 0.244 Remarks: Nf = life to 250 μm crack

Material: 347SS E = 200000 G = – 251 sy = 588 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Hoffmeyer et al., 2006

Code: SCT23

s′f = 780.3 MPa b = −0.114 1.153 e′f = c = −0.8614 K′ = 544 MPa n′ = 0.075 Remarks: Nf = life to 250 μm crack

Material: Al 5083 E = 68000 G = – 169 sy = 340 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Hoffmeyer et al., 2006

Code: SCT22

Table B.23 Continued

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′f = b0 = g′f = c0 = K′0 = n′0 =

– – – – – –

– – – – – –

MPa

MPa

MPa

MPa

330 Multiaxial notch fatigue

s′f = – b = – – e′f = c = – K′ = – n′ = – Remarks: Nf = 5% stress drop

Material: AISI 304 E = – G = – – sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Itoh et al., 1995

Code: SCT25

MPa

MPa

s′f = 1640 MPa b = −0.06 2.67 e′f = c = −0.82 K′ = 1530 MPa n′ = 0.07 Remarks: Nf = life to 1 mm crack

Material: Inconel 718 E = 209000 G = – 1160 sy = – sUTS = K = 1910 n = 0.08 Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Socie et al., 1985

Code: SCT24

Table B.24

t′f b0 g′f c0 K′0 n′0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

– – – – – –

– – – – – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 331

s′f = – MPa b = – – e′f = c = – K′ = – MPa n′ = – Remarks: Nf = life to 0.05–0.25 mm crack

Material: AISI 316 E = 179600 G = 57.8 247.1 sy = – sUTS = K = – n = – Geom.: Fig. B.7

MPa MPa MPa MPa MPa

Ref.: Yip and Jen, 1997

Code: SCT27

– MPa s′f = b = – – e′f = c = – K′ = – MPa n′ = – Remarks: Nf = 25% stress drop

Material: 45 Steel E = 207000 G = – 351.9 sy = 598.7 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Chen et al., 1999

Code: SCT26

Table B.24 Continued

t′f b0 g′f c0 K′0 n′0

t′f b0 g′f c0 K′0 n′0

= = = = = =

= = = = = =

– – – – – –

– – – – – –

MPa

MPa

MPa

MPa

332 Multiaxial notch fatigue

s′f = – MPa b = – – e′f = c = – K′ = – MPa n′ = – Remarks: Nf = life to 2–3 mm crack

Material: Type 304 E = – G = – – sy = – sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Wu and Yang, 1987

MPa

MPa

Code: SCT29

s′f = 859.6 b = −0.1047 0.3269 e′f = c = −0.5458 K′ = – n′ = – Remarks: Nf = 25% stress drop

Material: 42CrMo E = 210000 G = – 868 sy = 955 sUTS = K = – n = – Geom.: plain, tubular

MPa MPa MPa MPa MPa

Ref.: Chen et al., 1996

Code: SCT28

Table B.25

t′f = b0 = g′f = c0 = K′0 = n′0 =

t′f = b0 = g′f = c0 = K′0 = n′0 =

– – – – – –

817 −0.1017 3.212 −0.8518 – –

MPa

MPa

MPa

MPa

Experimental results generated under multiaxial fatigue loading 333

Code, material

Table B.26

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.01 0.01 0.005 0.005 0 0 0 0 0 0 0 0 0.0071 0.0071 0.0035 0.0035 0.0015 0.0015

0 0 0 0 0.0176 0.0176 0.0087 0.0087 0.0054 0.0054 0.0043 0.0038 0.0123 0.0123 0.0061 0.0061 0.0027 0.0027

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 1090 1100 924 931 0 0 0 0 0 0 0 0 752 758 633 633 334 318

sx,a (MPa)

B.5.2 Experimental results generated under strain control

SCT1 – Inconel 718

−30 −23 −19 −30 0 0 0 0 0 0 0 0 −29 −26 −87 −26 8 −10

sx,m (MPa) 0 0 0 0 605 596 538 535 416 414 324 285 435 427 376 400 199 202

txy,a (MPa) 0 0 0 0 5 0 1 −12 10 −20 −18 −40 34 −3 34 −2 −9 −2

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1000 1050 13500 11000 890 800 7200 7000 34000 35700 105000 114000 1000 1200 8000 7000 160000 290000

Nf (cycles)

334 Multiaxial notch fatigue

0.01 0.01 0.005 0.005 0 0 0 0 0 0 0.0071 0.0071 0.0035 0.0035 0.0015 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035

0.01 0.01 0.005 0.005 0 0 0 0 0 0 0.0071 0.0071 0.0035 0.0035 0.0015 −0.004 −0.004 0 0 0 0

0 0 0 0 0.0173 0.0173 0.0087 0.0087 0.0043 0.0037 0.0123 0.0123 0.0063 0.0063 0.0026 0.0063 0.0063 0.0063 0.0063 0.0063 0.0063

0 0 0 0 0.0173 0.0173 0.0087 0.0087 0.0043 0.0037 0.0123 0.0123 0.0063 0.0063 0.0026 −0.006 −0.006 0.0063 0.0063 0.0063 0.0063

1083 1089 965 944 0 0 0 0 0 0 751 779 646 632 311 598 594 631 632 637 631

19 22 215 215 0 0 0 0 0 0 119 8 170 68 327 −190 −213 −216 −189 84 97

0 0 0 0 580 568 507 502 315 285 419 432 396 383 190 381 375 386 381 389 398

0 0 0 0 14 14 76 81 305 235 −71 7 52 102 198 −43 −29 139 138 126 120

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

800 800 7000 6000 1000 800 4500 6500 48000 110000 1050 1000 4800 7500 53000 6000 7000 5000 5000 4000 4000

Experimental results generated under multiaxial fatigue loading 335

ex,m

0.0035 0.0035 0.001 0.001 0.0015 −0.001 −0.001 −0.002 0 0 0 0 0 0 0.0035 0.0035

ex,a

0.0035 0.0035 0 0 0 0 0 0 0.0035 0.0035 0.0071 0.0071 0.0035 0.0035 0.0035 0.0035

Code, material

Table B.27

SCT1 – Inconel 718

0.0063 0.0063 0.0086 0.0086 0.0038 0.0086 0.0086 0.0038 0.0062 0.0062 0.0123 0.0123 0.0062 0.0062 0.0062 0.0062

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 638 637 0 0 0 0 0 0 520 510 999 999 758 758 724 724

sx,a (MPa) 218 279 75 63 335 −172 −161 −329 −10 −28 −29 −32 −19 −34 283 450

sx,m (MPa) 394 413 504 505 287 507 513 274 495 496 552 567 463 463 450 450

txy,a (MPa)

dxy,x (°) 0 0 0 0 0 0 0 0 45 45 90 90 90 90 90 90

txy,m (MPa) −71 −85 −42 −4 −10 0 0 −33 31 25 1 −8 −10 1 −7 −4

3000 3000 6500 7470 52000 3890 9090 462000 5550 6080 430 450 4370 3350 3550 3330

Nf (cycles)

336 Multiaxial notch fatigue

0.0035 0.0046 0.01 0.01 0.006 0.0035 0.0035 0.0035 0.002 0.002 0 0 0 0 0 0 0 0 0 0.0025 0.0025 0.0015 0.0015

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0.017 0.008 0.0079 0.008 0.006 0.0061 0.0034 0.0034 0.0034 0.0043 0.0043 0.0023 0.0023

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

240 279 383 426 379 261 258 266 230 206 0 0 0 0 0 0 0 0 0 184 176 161 157

−2.2 0 2 −1.4 −21.9 5.8 −9.1 8.4 16.5 29 0 0 0 0 0 0 0 0 0 2.7 −3.7 0.6 0.8 0 0 0 0 0 0 0 0 0 0 248 191 156 157 157 140 138 125 127 109 101 89 90

0 0 0 0 0 0 0 0 0 0 1.2 1.9 0.4 0 0 0 0 0.6 −0.6 0.3 0.7 1.2 −2.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

38500 10300 1070 1167 6080 30700 33530 29000 286400 333100 4090 48500 32100 33900 133000 83400 1000000 824200 1100000 53000 52900 440000 356000

Experimental results generated under multiaxial fatigue loading 337

SCT2 – AISI 304

ex,m

0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0035 0.0035 0.002 0.002 0.0025 0.0025 0.0014 0.0014 0.0025 0.0025 0.0014 0.0014

Code, material

Table B.28

SCT2 – AISI 304

0.0061 0.0061 0.0033 0.0035 0.0043 0.0043 0.0024 0.0024 0.0043 0.0043 0.0024 0.0024

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 477 457 307 300 420 380 230 234 365 347 210 210

sx,a (MPa) 2.6 −4.5 29.3 18.6 0 0 20.6 17.1 12 35 47.7 44.5

sx,m (MPa) 267 256 176 168 226 218 139 137 199 209 114.7 115.6

txy,a (MPa)

dxy,x (°) 90 90 90 90 E-A E-A E-A E-A E-B E-B E-B E-B

txy,m (MPa) −4.4 1.3 −3.5 −2.1 0 0 1.8 2.2 −13.2 −21.8 −17 −15.7

3560 3730 45000 50000 5110 6200 89312 100000 11080 9800 200000 205000

Nf (cycles)

338 Multiaxial notch fatigue

0.01 0.01 0.0051 0.0034 0.0022 0.0022 0.0021 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0.025 0.0251 0.025 0.0173 0.0173 0.015 0.015 0.015 0.0082 0.0082 0.0082 0.0072 0.005 0.005 0.0041 0.0038 0.005 0.004 0.0038 0.0039

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

458 450 374 351 279 270 274 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3.1 3.4 −0.8 2.3 1.7 3.7 −0.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 272 270 259 248 251 232 237 232 198 200 194 196 165 168 160 164 161 159 155 154

0 0 0 0 0 0 0 0 0 0 0.8 −2 0 0 0 0 0 0 −1.6 0 0 0 −0.9 0 0 −4.1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1107 1137 4959 7839 78270 94525 142500 470 495 541 889 890 1269 1379 1467 5505 7130 8360 8710 35020 36120 41840 57370 60750 72950 93050 95250

Experimental results generated under multiaxial fatigue loading 339

SCT3 – SAE 1045

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0 0 0 0 0.0014 0.0087 0.0094 0.0087 0.0064 0.0037 0.0037 0.0042 0.0026 0.0014 0.0026 0.0019 0.0004 0.0021

Code, material

Table B.29

SCT3 – SAE 1045

0.0038 0.0039 0.003 0.0026 0.0014 0.0087 0.0047 0.0087 0.0129 0.0037 0.0037 0.0021 0.0052 0.0056 0.0052 0.0021 0.0037 0.001

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 0 0 0 0 224 394 425 385 289 304 305 337 234 138 229 224 45.2 261

sx,a (MPa) 0 0 0 0 −0.3 −4.6 2.3 −0.9 −1.5 1.5 0 0.1 7.3 5.2 −1.2 32 1.8 3.1

sx,m (MPa) 163 160 148 147 224 135 78.9 136 197 104 107 55.5 154 173 152 85 160 48.5

txy,a (MPa)

dxy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) −0.7 0 0 −0.5 −0.3 −0.3 1.2 1.2 −1.8 1.9 −1.2 −0.5 −6.1 −3.1 0 6 0.6 −1.6

102100 111400 546000 1000000 546 1229 1258 1616 1758 10380 11610 11780 16890 19770 20030 40380 66810 80000

Nf (cycles)

340 Multiaxial notch fatigue

0.0015 0.0019 0.0015 0.0021 0.0019 0.0014 0.0013 0.001 0.0014 0.0027 0.0041 0.0041 0.0026 0.0015 0.0015 0.0015 0.0019 0.0021 0.0019 0.001 0.001 0.0014

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0029 0.0019 0.0029 0.001 0.0019 0.0015 0.002 0.0019 0.0014 0.0053 0.0037 0.0019 0.0051 0.0029 0.0028 0.0028 0.0019 0.0011 0.0019 0.0019 0.002 0.0014

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

176 234 179 266 238 194 202 147 212 380 348 353 300 274 250 272 289 281 285 207 190 232

0.3 2.9 −0.6 −1.3 −0.3 34 4.5 −4.4 −1.6 0 0 0 0 0 0 0 0 0 0 0 0 0

127 87.2 126 52.9 87.7 82 113 110 79.6 208 151 111 180 168 157 157 133 82 134 142 124 100

−0.5 −2 1.6 −3 −0.4 7 0.5 0.5 −1.1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90

87500 90000 98780 115500 123500 171300 393600 545800 595600 4350 5119 5260 5262 18330 34720 38930 49140 58530 64650 91455 613600 1000000

Experimental results generated under multiaxial fatigue loading 341

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.007 0.007 0.005 0.0035 0.0053 0.0053 0.0038 0.0027 0.005 0.005 0.0036 0.0025 0.0025 0.004 0.0029 0.002 0.003 0.0022 0.0015 0.0015

Code, material

Table B.30

SCT4 – AISI 304

0 0 0 0 0.008 0.008 0.0057 0.004 0.0085 0.0085 0.0061 0.0043 0.0043 0.01 0.0071 0.005 0.011 0.0079 0.0055 0.0055

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

822 1310 2270 7612 1210 1680 3415 6400 1541 2182 3392 8170 8842 1980 3820 10150 2350 4147 12488 14370

Nf (cycles)

342 Multiaxial notch fatigue

0 0 0 0.0053 0.0038 0.0027 0.0053 0.0038 0.0027 0.0053 0.0038 0.0027 0.0053 0.0038 0.0027 0.005 0.0036 0.0025 0.004

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0122 0.0087 0.0061 0.008 0.0057 0.004 0.008 0.0057 0.004 0.008 0.0057 0.004 0.008 0.0057 0.004 0.0085 0.0061 0.0043 0.01

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 30 30 30 45 45 45 60 60 60 90 90 90 90 90 90 90

2252 5156 10540 1100 2600 7700 950 2200 4150 720 2400 5430 540 1620 3988 532 1023 3450 654

Experimental results generated under multiaxial fatigue loading 343

0 0 0 0 0 0 0 0 0 0 0 0

0.0078 0.0065 0.006 0.005 0.0046 0 0 0 0 0 0 0

ex,m

0 0 0 0 0 0 0 0

ex,a

0.0029 0.002 0.003 0.0022 0.0015 0.0015 0.0011 0.0008

Code, material

Table B.31

SCT4 – AISI 304

SCT5 – Ti-6Al-4V

0 0 0 0 0 0.0087 0.0061 0.0056 0.0082 0.0056 0.0061 0.0044

0.0071 0.005 0.011 0.0079 0.0055 0.012 0.0085 0.006

gxy,a

0 0 0 0 0 0 0 0 0.0099 0.0065 0.0076 0.0052

0 0 0 0 0 0 0 0

gxy,m

772.4 744.7 700.5 578.8 529.1 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0 375.7 265.4 243.4 353.0 240.8 259.9 184.3

txy,a (MPa)

0 0 0 0 0 0.2 0.7 −0.7 101 202.2 188.2 231.5

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90

dxy,x (°)

5246 6608 9640 20515 49518 72141 241250 961806 30007 150293 151598 814753

2013 4237 592 1420 4080 1280 3260 9900

Nf (cycles)

344 Multiaxial notch fatigue

SCT6 – 6061-T6

0 0 0 0 0 0.005 0.006

0 0.0031 0.0021 0.0021 0.003 0.0037 0.0037 0.0036 0.0036 0.0000 0.0000 0.0000

0 0 0 0 0 0 0

0 0 0.0026 0.0026 0 0 0 0 0 0.0009 0.0009 −0.0009 0.0073 0.009 0.0106 0.0124 0.014 0 0

0.0057 0.0042 0.0028 0.0028 0.0041 0.0047 0.0047 0.0024 0.0024 0.0067 0 0.0066 0 0 0 0 0 0 0

0.0139 0 0.0034 0.0034 0 0 0 0.0024 −0.0024 0 0.0067 0

0 365.4 241.7 243.8 351.7 430.0 431.0 426.0 426.1 0.7 0.7 1.4

0 0 298.9 299.6 0.8 −6 −4.9 0 1.4 105.5 104.8 −105.5

246.1 181.0 121.5 122.5 175.9 204.0 205.5 102.1 102.1 288.6 288.9 288.9

209.6 −0.4 145.6 146.6 −0.7 0 −1.4 104.2 −104.2 0.4 0 0

0 0 0 0 0 0 0

0 0 0 0 90 E-G E-G 0 0 0 0 0

3320 1730 990 590 400 2420 1420

141229 67965 60514 87920 111783 38355 43009 71358 79367 72124 73728 329058

Experimental results generated under multiaxial fatigue loading 345

ex,m

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0.007 0.008 0.009 0.0035 0.0045 0.0055 0.0045 0.0055 0.0065

0.025 0.01 0.005 0.004 0.015 0.004 0.003 0.015 0 0 0 0

Code, material

ex,a

Table B.32

SCT6 – 6061-T6

SCT7 – S45C

0 0 0 0 0 0 0 0 0.015 0.015 0.015 0.009

0 0 0 0.007 0.009 0.0111 0.0037 0.0045 0.0052

gxy,a

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

gxy,m

595.2 480.2 389.1 361.9 509.0 365.5 325.9 500.2 0 0 0 0

sx,a (MPa)

−17.0 −7.1 −3.7 −4.2 −11.1 −4.1 −3.1 −9.4 0 0 0 0

sx,m (MPa)

0 0 0 0 0 0 0 0 287.1 283.8 286.3 244.3

txy,a (MPa)

0 0 0 0 0 0 0 0 −3.3 0 −2.5 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

dxy,x (°)

110 852 3383 5514 421 8933 22071 407 1151 1761 1771 5644

1150 690 360 1430 740 400 1880 900 600

Nf (cycles)

346 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0.006 0.009 0.0072 0.0036 0.009 0.0077 0.006 0.006 0.009 0.018 0.018

SCT8a – 6061T6 (Rod 1)

0.0059 0.0066 0.007 0.0076 0.0081 0

0 0 0 0 0 0.0078

0.008 0.0052 0.0082 0.0065 0.0065 0.0041 0.0066 0.0055 0.0055 0.0041 0.0082 0.0082 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 370.3 395.2 391.1 285.1 430.6 514.2 433.0 419.1 456.8 554.4 542.5

0 −2.8 −6.5 −4.4 −0.2 −12.0 −9.3 −1.5 −1.3 −5.7 −14.2 −12.3

229.4 103.6 111.4 108.9 164.4 57.3 267.5 204.3 214.8 156.3 184.9 189.0

−5.3 −1.2 −0.5 −2.1 −1.5 −0.3 −7.5 −0.2 −1.7 −2.2 −4.9 −6.8 0 0 0 0 0 0

0 0 0 0 0 0 E-A 90 45 90 22.5 45

2160 1470 1140 670 480 3360

14930 2278 568 1366 4647 1181 435 1617 1631 678 215 191

Experimental results generated under multiaxial fatigue loading 347

ex,m

0 0 0 0 0 0 0 0 0 0 0

ex,a

0 0 0 0 0 0.0045 0.0055 0.007 0.0035 0.0045 0.0055

Code, material

Table B.33

SCT8a – 6061-T6 (Rod 1)

0.0085 0.0092 0.0102 0.0109 0.0129 0.0045 0.0055 0.007 0.007 0.009 0.011

gxy,a 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

Nf (°) 1980 1310 960 600 370 2810 1020 390 1680 930 280

348 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0059 0.0066 0.0075 0 0 0 0 0 0.0052 0.0056 0.0067 0.0041 0.005 0.0061

0.0022 0.0021 0.0021 0.0019 0.0019 0.0014 0.0014 0 0 0.01 0.01 0.0096 0.0087 0.0065 0

0 0.0011 0.0011 0.0019 0.0019 0.0029 0.0029 0.0038 0.0038 0 0 0.0048 0.0087 0.0131 0.0173

0 0 0 0.0098 0.0119 0.0137 0.0154 0.0174 0.0053 0.0057 0.0067 0.0082 0.0101 0.0123 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

142541 115462 80000 123544 90000 98778 87500 102083 98052 1137 1107 1258 1616 1758 890

2200 1450 690 3260 1900 1250 940 700 1810 1320 780 1580 920 500

Experimental results generated under multiaxial fatigue loading 349

SCT8b – 6061-T6 (Rod 2)

SCT9 – SAE 1045

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0168 0.0126 0.0079 0.0075 0.0071 0.005 0.005 0.0047 0.0041 0.0025 0.0019 0 0 0 0 0 0 0

Code, material

Table B.34

SCT10 – A533B

0 0 0 0 0 0 0 0 0 0 0 0.02 0.0194 0.0143 0.0103 0.0095 0.008 0.0066

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

520 900 2700 2300 2300 4500 7800 4060 13500 47300 127500 800 850 1600 3350 3500 6220 6950

Nf (cycles)

350 Multiaxial notch fatigue

0 0 0 0 0 0.0033 0.0033 0.0033 0.0034 0.0034 0.0022 0.0022 0.0022 0.0022 0.0011 0.0011 0.0011 0.0011 0.0043 0.0043 0.0043 0.0043 0.003

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.005 0.005 0.004 0.0031 0.0026 0.0066 0.0066 0.0066 0.0069 0.0069 0.0045 0.0045 0.0044 0.0044 0.002 0.002 0.0021 0.0021 0.0041 0.0041 0.0041 0.0041 0.0029

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 90 90 0 0 90 90 0 0 90 90 0 0 90 90 0

13150 11940 38800 60000 Run out 6300 9600 8000 5010 4400 19800 17700 18000 18000 272000 324000 Run out 225600 3500 3500 3430 4060 11550

Experimental results generated under multiaxial fatigue loading 351

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.003 0.003 0.003 0.003 0.0023 0.0023 0.0023 0.0024 0.0024 0.0024 0.002 0.002 0.002 0.002 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0036 0.0036 0.0012 0.0018 0.0018 0.0018 0.0018

Code, material

SCT10 – A533B

Table B.35

0.0029 0.003 0.003 0.003 0.0021 0.0021 0.0021 0.0022 0.0023 0.0023 0.002 0.002 0.0021 0.0021 0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0025 0.0025 0.002 0.0055 0.0055 0.0055 0.0055

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 90 90 90 0 0 0 90 90 90 0 0 90 90 0 0 0 90 90 90 90 90 0 90 90 90 90

dxy,x (°)

10570 9060 12090 12460 46000 42700 42500 28700 30500 27500 78800 82200 65200 74500 225500 472600 478400 115600 68900 127000 6500 6100 280000 17700 9750 12400 14500

Nf (cycles)

352 Multiaxial notch fatigue

0.002 0.003 0.004 0.005 0.01 0.0014 0.002 0 0 0 0 0 0.0028 0.0071

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0.0024 0.0034 0.0043 0.0069 0.0104 0.0173 0.026 0.0024 0.0034

0 0 0 0 0 0 0 0 0 0 0 0 0 0

350 448 472 535 568 258.1 338 0 0 0 0 0 438.4 497.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 149 195.1 221.7 271.4 282.9 329.1 352.2 253.1 287

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 90 90 0 0 0 0 0 90 90

200000 12410 5500 3100 950 30028 3648 81376 12188 5283 1500 376 646 184

Experimental results generated under multiaxial fatigue loading 353

SCT11 – 1Cr-18Ni-9Ti

ex,m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.01 0.0086 0.0068 0 0 0 0 0.015 0.01 0.008 0.007 0.0076 0.01 0.006 0.0047 0.016

Code, material

Table B.36

SCT12 – EN8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ey,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ey,m 0 0 0 0.017 0.015 0.012 0.01 0.03 0.015 0.012 0.01 0.006 0.007 0.0042 0.0033 0

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

6234 2260 5087 6550 15680 27750 54300 327 875 3200 4750 6420 1672 6122 17502 375

Nf (cycles)

354 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.012 0.012 0.01 0.008 0.006 0.005 0.0065 0.005 0.003 0.007 0.008

SCT13 – A-516 Gr. 70

0.0005 0.0008 0.0011 0.0015 0.0015 0.0018 0.002 0.001 0.001 0.0011 0.0015

0.0005 0.0007 0.001 0.0015 0.0015 0.0018 0.002 0.0005 0.0005 0.0006 0.0007

0 0 0 0 0 0 0.0065 0.005 0.003 0.004 0.002 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

346500 115000 49980 11300 8802 4180 1417 159500 171000 109318 64000

1325 310 867 1850 3637 6238 1735 2287 3756 2200 1350

Experimental results generated under multiaxial fatigue loading 355

ex,m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0015 0.002 0.0019 0.0025 0.0015 0.002 0.002 0.0023 0.003 0.0031 0.0035 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0015 0.001

Code, material

Table B.37

SCT13 – A-516 Gr. 70

0.0008 0.001 0.001 0.0013 1E-05 2E-05 0 0.0001 3E-05 2E-05 4E-05 0.0003 0.0007 0.0011 0.0016 0.0022 0.0026 0.0031 0.0015 0.001

ey,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ey,m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 0 0 0 0 0 0 0 0 0 0 0 180 180 180 180 180 180 180 90 90

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

33519 14420 21314 4895 127800 26650 27450 21856 12400 11821 4060 Run out 111000 18000 7300 4525 2700 1350 1009 10800

Nf (cycles)

356 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0008 0.0005 0.0005 0.0008 0.0015 0.0008 0.001 0.0015 0.0008 0.001 0.0015 0.002

SCT14 – 6061T6/Alumina

0.002 0.0025 0.0015 0.003 0.0028 0.0033 0.0015

0.002 0.0025 0.0015 0.003 0.0028 0.0033 0.0015

0.0008 0.0005 0.0005 0.0008 0.0015 0.0008 0.001 0.0015 0.0008 0.001 0.0015 0.002 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 180 180 180 180 180 180 180

90 90 90 0 0 60 60 60 90 90 90 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

53475 23053 86551 5013 2786 855 178295

34500 159000 99605 225660 16290 219480 142050 13030 239600 136380 10246 2933

Experimental results generated under multiaxial fatigue loading 357

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.001 0.002 0.003 0.002 0.0025 0.0015 0.001 0.0025 0.002 0.0015 0.001 0.002 0.0025 0.0015 0.001 0.0025 0.0015 0.002 0.003 0.002 0.003 0.0015 0.004 0.0021 0.0041 0.0028

Code, material

SCT14 – 6061-T6/Alumina

Table B.38

0.001 0.002 0.003 0.002 0.0025 0.0015 0.001 0.0025 0.002 0.0015 0.001 0.002 0.0025 0.0015 0.001 0.0025 0.0015 0.002 0 0 0 0 0 0 0 0

ey,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ey,m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 180 180 180 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

Run out 49178 5542 2325 357 10911 358912 291 3940 37645 426452 16523 586 43218 147957 1561 38301 29538 1574 35000 1472 999033 6154 99001 3029 11882

Nf (cycles)

358 Multiaxial notch fatigue

0 0 0.0057 0.0057 0.008 0.008 0.008 0.004 0.004 0.004 0.004 0.0057 0.008

0 0 0 0.0057 0 0.008 0 0 0 0.004 0 0 0

0.0104 0.0139 0.0098 0.0098 0.0069 0.0069 0.0069 0.0139 0.0139 0.0139 0.0139 0.0098 0.0139

0 0 0 0.0098 0 0 0.0069 0 0 0 0.0139 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 90 90 90 90 90 90 90 45 45

3715 1671 1085 1279 448 440 450 622 560 670 650 681 278

Experimental results generated under multiaxial fatigue loading 359

SCT15 – 45 steel

ex,m

0.0057 0 0 0.008 0 0.008 0 0 0

0.0029 0.0026 0.0021 0.0074 0.0056 0.0046 0.004 0.0064 0.0037 0.0031 0.0062 0.0051 0.0039 0.0036

0.0057 0.0057 0.008 0.008 0.008 0.008 0.0048 0.0059 0.0069

0.0029 0.0026 0.0021 0.0074 0.0056 0.0046 0.004 0.0064 0.0037 0.0031 0.0062 0.0051 0.0039 0.0036

Code, material

ex,a

Table B.39

SCT15 – 45 steel

SCT16 – En15R

0.0064 0.0037 0.0031 0* 0* 0* 0*

0.0029 0.0026 0.0021

ey,a

0.0064 0.0037 0.0031 0* 0* 0* 0*

0.0029 0.0026 0.0021

ey,m 0.0098 0.0098 0.0139 0.0139 0.0139 0.0139 0 0 0

gxy,a 0.0098 0 0 0 0.0139 0.0139 0 0 0

gxy,m 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 180 180 180 0 0 0 0

45 60 90 90 90 90 0 0 0

dxy,x (°)

3104 4598 11711 737 1689 2557 4087 1814 7723 13383 894 1874 3968 5681

664 544 384 426 406 455 3881 2830 1962

Nf (cycles)

360 Multiaxial notch fatigue

SCT17 – 1-Cr-Mo-V

* Plane strain.

0.0049 0.0039 0.0034 0.0030 0.0024 0.0025 0.0022 0.0017 0.0050 0.0039 0.0029 0.0025 0.0050 0.0039 0.0029 0.0024

0.0049 0.0039 0.0034 0.0030 0.0024 0.0025 0.0022 0.0017 0.0050 0.0039 0.0029 0.0025 0.0050 0.0039 0.0029 0.0024 0.0025 0.0022 0.0017 0* 0* 0* 0* 0.0050 0.0039 0.0029 0.0024

0.0025 0.0022 0.0017 0* 0* 0* 0* 0.0050 0.0039 0.0029 0.0024

0 0 0 0 0 0 0 0 0 0 0 0 180 180 180 180

989 1679 2700 5585 10036 1922 2713 9544 643 1305 3703 6781 1272 1793 4259 9403

Experimental results generated under multiaxial fatigue loading 361

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0050 0.0033 0.0022 0.0022 0.0050 0 0 0 0 0.0017 0.0010 0.0014 0.0017 0.0017 0.0023 0.0014 0.0014 0.0012 0.0010 0.0014

Code, material

Table B.40

SCT18 – S460N

0 0 0 0 0 0.0043 0.0100 0.0045 0.0045 0.0030 0.0018 0.0025 0.0030 0.0030 0.0040 0.0025 0.0025 0.0020 0.0018 0.0025

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 450.2 385.0 333.8 338.1 467.7 0 0 0 0 244.2 172.2 216.5 318.8 325.1 391.5 284.3 295.0 228.6 215.0 295.3

sx,a (MPa)

sx,m (MPa) 0 0 0 0 0 204.0 257.5 212.7 210.6 147.3 137.4 147.3 184.2 199.6 230.9 195.5 193.4 161.6 149.2 183.0

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

dxy,x (°)

1630 7690 50100 33100 1600 38250 1820 23000 30000 31100 521000 130300 39670 22800 6570 47140 51900 90700 574600 30000

Nf (cycles)

362 Multiaxial notch fatigue

0.0040 0.0012 0.0017 0.0014 0.0014 0.0014 0.0012 0.0012 0.0017 0.0017 0.0014 0.0014 0.0023 0.0023 0.0017 0.0026 0.0017 0.0030 0.0017

0 0 0 0 0.0014 0.0014 0.0012 0.0012 0 0 0 0 0 0 0 0 0 0 0

0.0070 0.0020 0.0030 0.0025 0.0025 0.0025 0.0020 0.0020 0.0030 0.0030 0.0025 0.0025 0.0040 0.0040 0.0030 0.0045 0.0013 0.0052 0.0030

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

480.9 216.2 282.1 280.4 293.0 289.0 237.8 246.3 338.0 361.8 289.8 276.8 413.9 419.3 343.7 442.5 318.3 436.0 308.5

269.3 146.3 282.1 164.8 191.3 193.2 154.8 160.3 218.3 227.2 190.8 187.9 245.5 251.2 214.8 265.8 200.3 277.0 204.0

90 45 45 45 90 90 90 90 E-A E-A E-A E-A E-C E-C E-C E-D E-D E-E E-E

540 218400 43000 55000 43300 48500 386800 325100 6730 4565 18000 18300 9600 6000 26800 4700 41600 1050 10050

Experimental results generated under multiaxial fatigue loading 363

Code, material

Table B.41

SCT18 – S460N

ex,m

0 0 0 0 0.0014 0.0017 0.0017 0.0014 0 0 0.0014 0.0023

0 0

ex,a

0.0020 0.0017 0.0017 0.0030 0.0014 0.0017 0.0017 0.0014 0.0017 0.0014 0.0014 0

0.0033 0.0022

0.0035 0.0030 0.0030 0.0052 0.0025 0.0030 0.0030 0.0025 0.0030 0.0025 0.0025 0.0040 0.0030 0.0030 0.0025 0.0025 0.0030 0.0030 0.0025 0.0030 0.0030 0.0025 0.0025 0.0030

gxy,a 0 0 0 0 0 0 0 0 0.0030 0.0025 0.0025 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

0 0 0 0 0 0 0 0 0 0 0 0 405.5 334.5

341.1 294.5 289.8 440.2 222.2 237.0 233.5 242.0 242.6 233.7 230.0

sx,a (MPa)

35 35 80 80 90 90 140 150 150 200 200 220 0 0

sx,m (MPa) 225.5 193.5 197.7 273.4 148.7 160.2 156.6 150.5 171.7 152.6 149.1 199.6 180.7 195.4 174.8 175.5 189.4 184.4 171.4 177.7 181.9 164.4 178.1 176.9

txy,a (MPa)

120 80

txy,m (MPa) E-E E-F E-F E-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

6550 12750 8600 767 76600 20000 22700 106300 17000 106300 67100 46040 320300 60000 196000 579500 58600 91170 300000 80000 78160 126530 108000 78300 5440 48200

Nf (cycles)

364 Multiaxial notch fatigue

0 0 0 0.0052 0.0096 0.0025 0.003 0.0055 0.01 0.0025 0.003

0 0 0 0 0 0 0 0 0 0 0

0.015 0.0069 0.0121 0.0044 0.0085 0.005 0.003 0.006 0.0085 0.005 0.003

0 0 0 0 0 0 0 0 0 0 0

0 0 0 700 786 458.5 576 803.5 1031 541 634.5

0 0 0 −24 −55 −24.5 −31 −17.5 −17 −38 −34.5

441 372 431 224 224.5 303 207 393 513.5 348.5 224.5

7 0 3 −17 3.5 7 0 −7 −10.5 −3.5 −3.5

0 0 0 0 0 0 0 0 0 0 0

3351 19761 3705 7311 1429 24214 33350 2620 429 11029 24218

Experimental results generated under multiaxial fatigue loading 365

SCT19 – Waspaloy

ex,m

0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0 0 0 0 0.0087 0.0046 0.0025 0.003 0.0049 0.0094 0.0025 0.003

Code, material

Table B.42

SCT20 – Waspaloy

0.015 0.011 0.0066 0.005 0.008 0.004 0.005 0.003 0.0055 0.0099 0.005 0.003

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 0 0 0 0 744.5 662 427.5 586 765.5 1013.5 558.5 634.5

sx,a (MPa) 0 0 0 0 −27.5 −14 −13.5 −14 −13.5 −20.5 −20.5 −27.5

sx,m (MPa) 435 421 372 138 228 200 290 210.5 389.5 548.5 362 227.5

txy,a (MPa)

0 0 0 0 0 0 0 −3.5 −3.5 −10.5 −3 −6.5

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2497 3462 10326 21613 1520 5117 12573 11423 2439 520 5727 12573

Nf (cycles)

366 Multiaxial notch fatigue

0.0082 0.0082 0.0041 0.0041 0.0023 0.0023 0.0017 0.0017 0.0017 0.0014 0.0014 0.0014 0.0011 0.0011 0.001 0.0017 0.0014 0.0011 0.004 0.0023 0.0017 0.0014 0.0014 0.0014 0.0017 0.0011 0.001

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0142 0.0142 0.0071 0.007 0.004 0.004 0.003 0.003 0.003 0.0025 0.0025 0.0025 0.0019 0.0019 0.0018 0.003 0.0025 0.0019 0.007 0.004 0.0029 0.0025 0.0025 0.0025 0.0029 0.0019 0.0018

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45 45 90 90 90 90 90 90 90 90 90

332 353 1762 1969 11403 12433 31047 48425 50250 130142 149082 172895 298028 609886 513335 42802 54170 216195 532 6543 22804 29949 46139 50929 39259 88877 566662

Experimental results generated under multiaxial fatigue loading 367

SCT21 – S460N

ex,m

0 0 0 0 0 0 0 0

ex,a

0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

Code, material

Table B.43

SCT21 – S460N

0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030

gxy,a 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 90 45 E-C E-E E-F E-A E-D

dxy,x (°)

43400 31250 43000 27000 10050 10000 5700 41700

Nf (cycles)

368 Multiaxial notch fatigue

0.0059 0.0059 0.0059 0.0035 0.0035 0.0035 0.0023 0.0023 0.0023 0.002 0.002 0.002 0.0017 0.0017 0.0023 0.0023 0.0017 0.0034 0.0035 0.0035 0.0023 0.0023 0.0017 0.0017 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0101 0.0102 0.0102 0.006 0.006 0.006 0.004 0.004 0.004 0.0035 0.0035 0.0035 0.003 0.003 0.004 0.004 0.003 0.006 0.006 0.006 0.0039 0.004 0.003 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.004

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45 45 90 90 90 90 90 90 90 0 90 45 E-C E-E E-F E-A

212 522 794 2320 3276 4405 17524 38579 46416 51863 54485 60133 239248 443186 18410 20826 106032 1252 2723 3119 26001 28345 182408 533221 37600 28150 20450 40400 8000 10450 4050

Experimental results generated under multiaxial fatigue loading 369

SCT22- Al 5083

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0058 0.0058 0.0035 0.0035 0.0023 0.0023 0.0017 0.0017 0.0058 0.0058 0.0035 0.0035 0.0023 0.0023 0.0058 0.0058 0.0035 0.0035 0.0023

Code, material

Table B.44

SCT23 – 347SS

0.01 0.0101 0.0061 0.0061 0.004 0.004 0.003 0.003 0.01 0.01 0.0061 0.0061 0.004 0.004 0.01 0.01 0.006 0.006 0.004

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 45 45 45 45 45 45 90 90 90 90 90

dxy,x (°)

865 1094 2464 3314 6362 12859 63911 93712 521 932 2687 2823 11088 22136 450 628 1997 2205 11940

Nf (cycles)

370 Multiaxial notch fatigue

0.01 0.01 0.005 0.007 0.007 0.0035 0.0035 0 0 0 0.01 0.01 0.005 0.005 0.007 0.007 0.0035 0.0035 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0.01 0.01 0.005 0.005 0.007 0.007 0.0035 0.0035 0 0 0 0

0 0 0 0.012 0.012 0.0063 0.0063 0.0017 0.0017 0.0085 0 0 0 0 0.012 0.012 0.0063 0.0063 0.017 0.017 0.0087 0.0087

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.012 0.012 0.0063 0.0063 0.017 0.017 0.0087 0.0087

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1000 1050 13500 1000 1200 7000 8000 890 800 7000 800 800 7000 6000 1050 1000 4800 7500 1000 800 4500 6500

Experimental results generated under multiaxial fatigue loading 371

SCT24 – Inconel 718

ex,m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0025 0.0033 0.004 0.005 0.0057 0.006 0.0075 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025

Code, material

Table B.45

SCT25 – AISI 304

0 0 0 0 0 0 0 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044 0.0044

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m 265 290 315 365 365 402.5 412.5 342.5 335 335 395 242.5 250 265 380 390 382.5 285 330 327.5

sx,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa) 0 0 0 0 0 0 0 197.5 177.5 210 197.5 92.5 120 142.5 205 185 200 140 180 180

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 E-H E-D E-K E-I 0 E-M E-L M-O E-N E-A E-P E-C E-Q

dxy,x (°)

49000 23400 7100 1500 1700 690 540 9500 20000 24000 3400 17500 9700 18000 2050 2950 2600 14400 4750 3200

Nf (cycles)

372 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004

SCT26 – 45 steel

0.0017 0.0024 0.0017 0.0022 0.0024

0.001 0.0007 0.001 0.0006 0.0014

0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

475 430 487.5 505 295 335 367.5 527.5 537.5 530 425 470 482.5

0 0 0 0 0 0 0 0 0 0 0 0 0

265 245 272.5 260 125 160 195 280 300 277.5 250 255 275

0 0 0 0 0 0 0 0 0 0 0 0 0

90° E-R E-A E-S E-C

E-H E-D E-K E-I 0 E-M E-L M-O E-N E-A E-P E-C E-Q

1100 988 820 863 1173

1400 2100 820 900 3200 2600 1700 470 660 320 1200 710 1000

Experimental results generated under multiaxial fatigue loading 373

ex,m

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

ex,a

0.0024 0.003 0.0027 0.003 0.0024 0.0024

0.0035 0.0035 0.0035 0.0030 0.0030 0.0030 0.0018 0.0018 0.0018 0.0030 0.0030 0.0030 0.0017

Code, material

Table B.46

SCT26 – 45 steel

SCT27 – AISI316*

0 0 0 0 0 0 0 0 0 0.0030 0.0030 0.0030 0.0017

0.0014 0.0017 0.0015 0.0017 0.0017 0.0017

gxy,a

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 0 E-C E-C E-D

dxy,x (°)

832 574 765 966 953 1614 8172 5655 6177 908 1253 923 4342

762 414 3305 617 743 1160

Nf (cycles)

374 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

0.0017 0.0017 0.0016 0.0016 0.0016 0.0078 0.0078 0.0078 0.0069 0.0069 0.0069 0.0061 0.0061 0.0061 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

* Strains measured by biaxial extensometer with 10 mm gauge length including the notch.

0.0017 0.0017 0.0016 0.0016 0.0016 0 0 0 0 0 0 0 0 0 0.0030 0.0030 0.0030 0.0025 0.0025 0.0025

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3814 4676 5414 7810 6309 959 1090 940 1675 1544 1494 5225 5309 4752 838 914 832 1547 1919 1325

Experimental results generated under multiaxial fatigue loading 375

ex,m

0.0017 0.0017 0.0017 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009

ex,a

0.0020 0.0020 0.0020 0.0030 0.0030 0.0030 0.0020 0.0020 0.0020 0.0018 0.0018 0.0018 0.0030 0.0030 0.0030 0.0025 0.0025 0.0025 0.0020

Code, material

Table B.47

SCT27 – AISI316*

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,a 0.0017 0.0017 0.0017 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2228 2332 2811 947 973 843 3773 3456 3174 4782 4453 4232 2010 1502 1685 3274 3052 3776 9798

Nf (cycles)

376 Multiaxial notch fatigue

0.0009 0.0009 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

−0.0009 −0.0009 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

* Strains measured by biaxial extensometer with 10 mm gauge length including the notch.

0.0020 0.0020 0.0030 0.0030 0.0030 0.0020 0.0020 0.0020 0.0018 0.0018 0.0018 0.0035 0.0035 0.0035 0.0030 0.0030 0.0030 0.0020 0.0020 0.0020

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8112 12453 952 1505 1742 3970 4618 4490 7703 8563 7572 634 709 875 1243 1111 1056 4007 3438 4715

Experimental results generated under multiaxial fatigue loading 377

ex,a

0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0017 0.0017 0.0017

Code, material

Table B.48

SCT27 – AISI316*

0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

ex,m 0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0017 0.0017 0.0017

gxy,a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

718 1093 866 1785 1442 1834 4402 4225 4198 1063 1090 1182 3813 1767 2569 6072 7299 8033

Nf (cycles)

378 Multiaxial notch fatigue

−0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 −0.0010 0 0 0 0 0 0 0 0 0 0 0 0 0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0017 0.0017 0.0017 0.0026 0.0026 0.0026

0 0 0 0 0 0 0 0 0 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0035 0.0017 0.0017 0.0017

* Strains measured by biaxial extensometer with 10 mm gauge length including the notch.

0.0030 0.0030 0.0030 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0026 0.0026 0.0026 0.0022 0.0022 0.0022 0.0017 0.0017 0.0017 0.0026 0.0026 0.0026

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1529 1103 1529 2100 1800 1899 5190 6070 6113 1864 1647 1661 2616 2502 2559 5410 3638 4194 1662 1318 1317

Experimental results generated under multiaxial fatigue loading 379

ex,a

0.0022 0.0022 0.0022 0.0017 0.0017 0.0017 0 0 0 0 0 0 0 0 0 0 0 0

Code, material

Table B.49

SCT27 – AISI316*

0 0 0 0 0 0 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.001 0.001

ex,m 0.0022 0.0022 0.0022 0.0017 0.0017 0.0017 0.0061 0.0061 0.0061 0.0052 0.0052 0.0052 0.0043 0.0043 0.0043 0.0078 0.0078 0.0078

gxy,a 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0 0 0 0 0 0 0 0 0 0 0 0

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

3090 2689 2731 4099 4219 4272 1530 1762 2097 3253 3662 3462 5929 6494 6158 1607 1179 1433

Nf (cycles)

380 Multiaxial notch fatigue

0.001 0.001 0.001 0.001 0.001 0.001 −0.001 −0.001 −0.001 −0.001 −0.001 −0.001 −0.001 −0.001 −0.001 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

0.0069 0.0069 0.0069 0.0043 0.0043 0.0043 0.0087 0.0087 0.0087 0.0078 0.0078 0.0078 0.0061 0.0061 0.0061 0.0078 0.0078 0.0078 0.0069 0.0069 0.0069

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017

* Strains measured by biaxial extensometer with 10 mm gauge length including the notch.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1596 1791 1779 4712 3814 2774 1517 1681 1189 2025 2349 2405 6880 6567 10275 762 711 750 1583 1146 1630

Experimental results generated under multiaxial fatigue loading 381

ex,m

0.0017 0.0017 0.0017 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009 −0.0009

ex,a

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Code, material

Table B.50

SCT27 – AISI316*

0.0061 0.0061 0.0061 0.0078 0.0078 0.0078 0.0069 0.0069 0.0069 0.0061 0.0061 0.0061 0.0104 0.0104 0.0104 0.0087 0.0087 0.0087 0.0078 0.0078 0.0078

gxy,a 0.0017 0.0017 0.0017 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

3597 3743 3254 889 1183 1159 2609 1475 1703 3175 3320 3738 947 955 920 1939 1456 1422 3261 3468 3602

Nf (cycles)

382 Multiaxial notch fatigue

SCT28 – 42CrMo

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0058 0.0081 0.0036 0.0073 0.0058 0.0081 0.0081 0.0058 0.0052 0.0052 0.0087 0.0045 0.0073 0.0147 0.0031 0.0073

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

* Strains measured by biaxial extensometer with 10 mm gauge length including the notch.

0.0033 0 0.0042 0.0021 0.0033 0.0047 0.0047 0.0033 0.0060 0.0060 0.0100 0.0026 0.0042 0.0085 0.0043 0.0042

0 0 E-A E-A E-A E-H E-D 90 E-H E-D E-R E-A E-A E-A 0 0

2074 1530 1120 1066 738 690 916 375 780 506 161 1434 220 188 2450 621

Experimental results generated under multiaxial fatigue loading 383

ex,m

0.005 0.004 0.003 0.0025 0.002 0 0 0 0 0 0.006 0.0045 0.0060 0.0045 0.0038 0.0038 0.0038 0.0030 0.0023 0.0015 0.006 0.0045 0.0060 0.0045 0.0038 0.0038 0.0038 0.0030 0.0023 0.0015

ex,a

0.005 0.004 0.003 0.0025 0.002 0 0 0 0 0 0.006 0.0045 0.006 0.0045 0.0038 0.0038 0.0038 0.003 0.0023 0.0015 0.006 0.0045 0.006 0.0045 0.0038 0.0038 0.0038 0.003 0.0023 0.0015

Code, material

SCT29 – Type 304 stainless steel

Table B.51

0 0 0 0 0 0.0140 0.0120 0.0090 0.0075 0.0100 0.0020 0.0015 0.0040 0.0030 0.0023 0.0023 0.0023 0.0020 0.0038 0.0025 0.0020 0.0015 0.0040 0.0030 0.0023 0.0023 0.0023 0.0020 0.0038 0.0025

gxy,a 0 0 0 0 0 0.014 0.012 0.009 0.0075 0 0.002 0.0015 0.004 0.003 0.0023 0.0023 0.0023 0.002 0.0038 0.0025 0.002 0.0015 0.004 0.003 0.0023 0.0023 0.0023 0.002 0.0038 0.0025

gxy,m

sx,a (MPa)

sx,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 E-Sa E-Sa E-Sa E-Sa E-Sa E-Sa E-Sa E-Sa E-Sa E-Sa E-Sb E-Sb E-Sb E-Sb E-Sb E-Sb E-Sb E-Sb E-Sb E-Sb

dxy,x (°) 3053 4390 19410 31000 107433 6911 14697 59980 125632 20180 9936 34692 954 2867 2577 6820 5057 45979 4711 28865 11345 65998 1700 4956 4541 9444 8994 71435 5466 24643

Nf (cycles)

384 Multiaxial notch fatigue

Material

SM45C 6082-T6 Z12CNDV12-2 7010 BS250A53 316L 5% Cr Al-LY12CZ 1045 steel S45C AlCu4Mg1 JIS SGV410 42CrMo4 42CrMo4 SAE 1045 SAE 1045 Low-carbon steel

Code

USC1 USC2 USC3 USC4 USC5 USC6 USC7 USC8 USC9 USC10 USC11 USC12 USC13 USC14 USC15 USC16 USC17

Table B.52

Lee, 1985 Susmel and Petrone, 2003 Chaudonneret, 1993 Chaudonneret, 1993 Zhang and Akid, 1997 Zhang and Akid, 1997 Kim et al., 2004 Wang and Yao, 2006 Verreman and Guo, 2007 Cheng et al., 2001 Kardas et al., 2004 Cheng et al., 2004 Reis et al., 2007 Froeschl et al., 2007 Kurath et al., 1989 Yip and Jen, 1996 Qilafku et al., 2001

Reference

– 500

1610 647 1400 545 703 591 545 470 1100 996 621

– – 1440 320 900 400 387 371 395 275 980 – 380 – 312

– –

301

– 343

–

sy (MPa)

sUTS (MPa) B-T B-T Te-T Te-T Te-T Te-T Te-T Te-T Te-T Te-T B-T Te-T Te-T Te-T B-T Te-T Te-T

Loading path

Low/medium-cycle fatigue results generated under stress control

B.6.1 Materials, static properties, loading paths and geometries

B.6

Plain Plain Plain Plain Plain Plain Plain Plain Plain Plain Plain Plain Plain Plain Fig. B.8 Fig. B.9 Plain

Geometry

Experimental results generated under multiaxial fatigue loading 385

Material

Low-carbon steel Low-carbon steel C40 steel C40 steel C40 steel BS04A12 BS04A12 BS04A12 BS04A12 BS04A12 BS04A12 En3B En3B En3B En3B SGV410 SGV410 SGV410 SUS316NG SUS316NG SUS316NG

Code

USC18 USC19 USC20 USC21 USC22 USC23 USC24 USC25 USC26 USC27 USC28 USC29 USC30 USC31 USC32 USC33 USC34 USC35 USC36 USC37 USC38

Table B.52 Continued

Qilafku et al., 2001 Qilafku et al., 2001 Atzori et al., 2006 Atzori et al., 2006 Atzori et al., 2006 Susmel and Taylor, 2003 Susmel and Taylor, 2003 Susmel and Taylor, 2003 Susmel and Taylor, 2003 Susmel and Taylor, 2003 Susmel and Taylor, 2003 Susmel and Taylor, 2008 Susmel and Taylor, 2008 Susmel and Taylor, 2008 Susmel and Taylor, 2008 Ohkawa et al., 2007 Ohkawa et al., 2007 Ohkawa et al., 2007 Ohkawa et al., 2007 Ohkawa et al., 2007 Ohkawa et al., 2007

Reference 312 312 – – – – – – – – – 653 653 653 653 275 275 275 260 260 260

sy (MPa) 500 500 – – – 410 410 410 410 410 410 676 676 676 676 470 470 470 591 591 591

sUTS (MPa) Te-T Te-T Te-T Te-T Te-T I/II I/II I/II I/II I/II I/II Te-T Te-T Te-T Te-T Te-T Te-T Te-T Te-T Te-T Te-T

Loading path

Fig. B.10 Fig. B.10 Plain Fig. B.11 Fig. B.11 Fig. B.12 Fig. B.12 Fig. B.12 Fig. B.12 Fig. B.12 Fig. B.12 Plain Fig. B.13 Fig. B.13 Fig. B.13 Plain Fig. B.14 Fig. B.14 Plain Fig. B.14 Fig. B.14

Geometry

386 Multiaxial notch fatigue

Material

Al 1070 6082-T6 18G2A EN24T GRS 500/ISO 1083 1045 (BHN 456) 1045 (BHN 203) EN24T S460N SAE 1045 Ck45 Ck45

Code

USC39 USC40 USC41 USC42 USC43 USC44 USC45 USC46 USC47 USC48 USC49 USC50

Table B.53

Yamamoto et al., 2007 Pinho da Cruz, 2001 Gasiak and Pawliczek, 2001 McDiarmid, 1991 Marquis and Karjalainen-Roikonen, 2003 Kaufman and Topper, 2003 Kaufman and Topper, 2003 McDiarmid, 1989 Hoffmeyer et al., 2006 Kurath et al., 1989 Simbürger, 1975 Simbürger, 1975

Reference – 290 357 680 340 – – – 500 380 704 704

sy (MPa) – – 535 850 620 – – – 643 621 850 850

sUTS (MPa) Te-T B-T B-T IEP-Te IP-Te-T IEP-Te IEP-Te IEP-Te Te-T B-T Te-T B-T

Loading path

Plain Fig. B.15 Plain Plain Plain Plain Plain Plain Fig. B.16 Plain Plain Fig. B17

Geometry

Experimental results generated under multiaxial fatigue loading 387

Code, material

Table B.54

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

413.4 389.1 373.5 366.8 354.1 337.3 324.1 314.9 313 295.2 294.2 0 0 0 0 0 0

B.6.2 Experimental results

USC1 – SM45C

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 280.4 257.7 270.2 256.8 249.5 231.8

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

14921 25972 52288 73480 91209 101076 163940 210131 323068 446301 714643 10370 19918 23486 30169 108937 140230

Nf (cycles)

388 Multiaxial notch fatigue

0 0 0 0 390 349 325 372 309 265 392 417 346 245 245 304 304 314 286 167 265

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

247.7 233.1 227.9 220.4 151 148 153 93 134 225 118 78 173 216 211 186 152 127 143 211 132

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90

163940 332893 402734 1127312 8500 24000 32000 38000 100000 12000 12400 13000 16000 20000 25000 26000 57000 100000 120000 29000 350000

Experimental results generated under multiaxial fatigue loading 389

Code, material

Table B.55

USC2 – 6082-T6

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa) −1 0 −1 −4 0 −2 −1 −1 1 3 1 1 3 0 2 1 −3

sx,a (MPa)

224 190 188 180 162 165 145 145 14 18 15 16 13 24 15 15 70

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 4 5 4 4 3 4 4 4 138 139 111 111 99 98 86 87 118

txy,a (MPa) 0 7 0 −1 1 1 1 0 0 0 0 0 0 0 1 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

52990 159000 197275 244403 421560 437636 1060730 1235690 14695 23052 67690 113455 196555 449997 497990 1100000 71255

Nf (cycles)

390 Multiaxial notch fatigue

71 59 61 53 52 79 79 69 68 68 60 147 151 163 147 146 118 119 188 189 189

−1 −1 0 −1 −2 −1 −4 1 2 2 3 −2 −4 −2 1 −3 −3 1 0 −5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

117 100 98 83 82 129 116 110 99 99 94 106 104 81 90 76 82 72 106 106 106

1 1 0 1 0 1 0 0 0 0 0 1 0 0 −1 −1 1 −1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 −7 −18 −2 2 129 125 126 128 125 126 −4 −3 −5 −8 −6 −5 0 89 94 88

78730 230750 516985 1018775 1289550 20730 41490 188882 234725 368080 1016280 31000 64090 124460 132215 232370 315795 694062 5590 27420 34015

Experimental results generated under multiaxial fatigue loading 391

0 0 0 0 0 0

−4 −4 0 0 1 −1

0 0 0 0 0 0 0 0 0 0

171 190 149 151 155 152

870 775 641 627 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

sx,m (MPa)

Code, material

sx,a (MPa)

Table B.56

USC2 – 6082-T6

USC3 – Z12CNDV12-2

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

sy,m (MPa)

0 0 0 0 470 432 410 373 160 323

99 105 68 67 72 47

txy,a (MPa)

0 0 0 0 0 0 0 0 0 180

1 0 0 0 1 2

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0

91 91 93 94 92 91

dxy,x (°)

477 1556 15951 16929 754 3800 6415 42070 1570000 21796

44750 47020 114845 273325 445560 456725

Nf (cycles)

392 Multiaxial notch fatigue

0 0 759 747 664 463 556 219 122 402 791 779 778 897 316 340 400 629 716 691

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

343 388 0 0 0 263 312 425 440 440 441 437 438 480 159 195 230 302 300 388

136 126 57 115 228 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 8 6 90 90 90 90 90 90 90 63 106 75 82 64 62

21591 6692 2510 2728 7457 21673 1732 2088 1612 780 725 724 706 130 277040 76715 43316 5240 1147 330

Experimental results generated under multiaxial fatigue loading 393

Code, material

Table B.57

USC4 – 7010

USC5 – BS250A53

0 0 0 0 0

260 220 0 0 0 0 180 260 260 180

sx,a (MPa)

600 300 0 −300 −600

0 0 0 0 0 0 0 0 0 0

sx,m (MPa)

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

525 525 525 525 525

0 0 150 149 160 160 104 149 149 104

txy,a (MPa)

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0

0 0 0 0 0 0 0 90 90 90

dxy,x (°)

18000 44000 125000 128000 245000

27800 80600 54000 49400 16000 45800 50450 7200 7100 40000

Nf (cycles)

394 Multiaxial notch fatigue

220 150 0 −220 0 0 0 0 0 0 300 300 300 300 −100 −100 −100 −100 −100 −300 −300 −300 −300 0

0 0 0 0

USC6 – 316L

USC7 – 5% Cr

950 850 800 700 650 600 600 500 400 350 900 850 800 750 720 900 850 820 800 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000

189 189 189 189 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

290 1658 4516 24434 102032 128175 3479 65657 519592 2388905 3422 2320 112038 308211 81438 8486 192110 45819 2615767 8595

271300 197000 260000 1010000

Experimental results generated under multiaxial fatigue loading 395

Code, material

Table B.58

USC7 – 5% Cr

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 747 748 735 589 589 594 423 434 430 674 690 680 551 550 549 642 396

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 900 850 800 700 650 432 433 432 598 605 597 747 751 745 390 394 390 548 549 543 791 499

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 14 58 94 14 58 101 7 50 101 14 72 108 14 101 94 79 14

dxy,x (°)

18454 13096 95313 367012 1017325 3044 46794 45358 5147 6040 20609 2139 13727 6890 12130 45731 13842 20174 279968 110207 8889 124234

Nf (cycles)

396 Multiaxial notch fatigue

247.5 247.5 247.5 247.5 247.5 212.6 212.6 212.6 212.6 212.6 176.8 176.8 176.8 176.8 176.8 247.5 247.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

142.9 142.9 142.9 142.9 142.9 122.5 122.5 122.5 122.5 122.5 102.1 102.1 102.1 102.1 102.1 142.9 142.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45

13824 13329 12173 24725 24360 51494 33391 44203 81622 53386 405312 381483 348696 424156 336097 10432 7900

Experimental results generated under multiaxial fatigue loading 397

USC8 – Al-LY12CZ

Code, material

Table B.59

USC8 – Al-LY12CZ

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

247.5 247.5 247.5 212.6 212.6 212.6 212.6 176.8 176.8 176.8 176.8 176.8 247.5 247.5 247.5 247.5 247.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 142.9 142.9 142.9 122.5 122.5 122.5 122.5 102.1 102.1 102.1 102.1 102.1 142.9 142.9 142.9 142.9 142.9

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

45 45 45 45 45 45 45 45 45 45 45 45 90 90 90 90 90

dxy,x (°)

11487 12715 9305 18800 34334 29048 51813 198490 619530 109071 623110 618572 7297 13947 11190 6658 21898

Nf (cycles)

398 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

212.6 212.6 212.6 212.6 212.6 176.8 176.8 176.8 176.8 176.8

USC9 – 1045 steel

0 0 0 0 0 275 285 300 290 300 300 220

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 166.5 180 180 175 180 0 0 0 0 0 0 110

122.5 122.5 122.5 122.5 122.5 102.1 102.1 102.1 102.1 102.1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90 90

4917324 497000 440920 317943 257679 5417503 5148389 379178 366512 238236 214908 573978

21911 30558 43640 33002 25934 104315 74855 88715 64209 103859

Experimental results generated under multiaxial fatigue loading 399

sx,m (MPa)

0 0 0 0 0 0 0 0 0 90 150 200

0 0 0 96 176 0 0 0 96 176

220 220 250 230 240 80 82.5 85 87.5 0 0 0

0 0 0 0 0 0 0 0 0 0

Code, material

sx,a (MPa)

Table B.60

USC9 – 1045 steel

USC10 – S45C

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

165 165 165 165 165 180 180 180 180 180

110 110 125 115 120 160 165 170 175 150 150 150

txy,a (MPa)

0 55 110 0 0 0 55 110 0 0

0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0

0 90 90 90 90 90 90 90 90 0 0 0

dxy,x (°)

285900 371800 191300 211600 242100 137000 117900 107000 122000 120100

177263 2420570 397940 377718 151109 Run out 1490671 727149 345774 2855000 616542 462000

Nf (cycles)

400 Multiaxial notch fatigue

242.5 226.4 227.8 242.8 242.8 226.7 204.9 192.3 177.6 203.9 192.4 204.0 177.7 178.7 168.6 191.6 169.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

63602 97373 157229 163634 202469 222239 253878 331307 444016 438146 462104 520910 679782 776557 935617 1490769 1971504

Experimental results generated under multiaxial fatigue loading 401

USC11 – AlCu4Mg1

Code, material

Table B.61

USC11 – AlCu4Mg1

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

168.9 155.2 155.2 155.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 137.4 138.2 137.4 138.2 138.2 128.9 128.9 128.9 102.4 93.5 82.7 102.4 102.4 83.2 93.0 93.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

4939029 6276071 Run out Run out 108364 127248 137891 143541 145476 180225 262183 829096 1494234 1778279 1778279 2841112 3291864 4078168 5548634 8515939

Nf (cycles)

402 Multiaxial notch fatigue

0 0 0 0 0 0 170.5 169.6 168.7 161.6 160.8 160.8 146.9 146.1 145.3 132.7 132.7 133.4 121.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

83.2 75.9 75.9 75.9 70.1 69.7 85.2 84.8 84.3 80.8 80.4 80.4 73.4 73.0 72.6 66.4 66.4 66.7 60.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8864912 8402695 8071918 6873996 Run out Run out 358127 529217 558510 751157 814374 847946 792909 1220009 1802849 1270596 1396186 3048729 1709038

Experimental results generated under multiaxial fatigue loading 403

Code, material

Table B.62

USC11 – AlCu4Mg1

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

121.9 105.0 105.5 106.1 93.9 114.9 114.9 114.9 105.0 105.0 96.4 88.5 89.0 89.0 105.0 95.9 96.4 78.3 77.9 68.2 77.9 67.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 60.9 52.5 52.8 53.0 46.9 114.9 114.9 114.9 105.0 105.0 96.4 88.5 89.0 89.0 105.0 95.9 96.4 78.3 77.9 68.2 77.9 67.8

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1852847 2299127 4157836 4329180 Run out 232423 284111 307873 466235 554863 851594 1027136 1238863 1222389 1307010 2552565 2997396 1754632 2841112 4245286 8515939 Run out

Nf (cycles)

404 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 72.5 72.5 72.5 135 145 165 67.5 70 72.5 77.5 125 135 145

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

140 145 155 165 180 145 145 145 135 145 165 135 140 145 155 125 135 145

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

690000 301000 103000 33000 20000 128000 305000 162000 230000 160000 38000 384000 198000 111000 45000 478000 217000 57000

Experimental results generated under multiaxial fatigue loading 405

USC12 – JIS SGV410

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

629.7 610.2 580.9 567.0 513.3 496.5 489.6 488.1 476.9 467.1 451.7

498.1 497.8 497.8 463.0 462.6 444.1 444.2 444.5 430.0

Code, material

sx,a (MPa)

Table B.63

UCS13 – 42CrMo4

UCS14 – 42CrMo4

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

249.1 248.9 248.9 231.5 231.3 222.1 222.1 222.3 215.0

363.6 352.3 335.4 327.3 296.3 286.7 282.7 281.8 275.3 269.7 260.8

txy,a (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0

0 0 0 0 90 90 90 90 90 90 90

dxy,x (°)

71200 98054 106495 115664 264159 129573 338427 399207 856970

237814 312644 996748 1492707 54565 95543 107428 195601 316743 615532 996748

Nf (cycles)

406 Multiaxial notch fatigue

430.0 430.4 414.4 414.4 414.4 493.9 494.3 494.0 462.7 462.4 449.2 449.6 449.2 429.7 429.7 514.8 514.8 514.9 494.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

215.0 215.2 207.2 207.2 207.2 247.0 247.2 247.0 231.4 231.2 224.6 224.8 224.6 214.9 214.9 257.4 257.4 257.5 247.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 45 45 45 45 45 45 45 45 45 45 90 90 90 90

911728 1481093 Run out Run out Run out 93121 137852 352694 597102 1064426 734033 822303 960024 Run out Run out 42934 47603 60986 230984

Experimental results generated under multiaxial fatigue loading 407

sx,m (MPa)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

494.3 494.0 472.3 472.4 472.3 453.8 454.0

499.4 499.1 499.4 499.1 499.1 499.4 499.1 478.7 478.7 478.7 456.1 456.1

Code, material

Table B.64

UCS14 – 42CrMo4

UCS14 – 42CrMo4a

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

sy,m (MPa)

249.7 249.6 249.7 249.6 249.6 249.7 249.6 239.4 239.4 239.4 228.1 228.1

247.2 247.0 236.2 236.2 236.2 226.9 227.0

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90

dxy,x (°)

10565 15556 17039 22905 26861 34114 70669 48468 71367 82746 104905 187436

461280 567064 1255593 1998017 675847 1268622 Run out

Nf (cycles)

408 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0

498.5 498.8 485.4 484.7 459.7 459.7 450.6 440.9 440.2 431.4 440.5 450.2 440.4

fxy/fx = 3; b fxy/fx = 1.

UCS14 – 42CrMo4b

a

0 0 0 0 0 0 0

433.3 433.3 433.3 433.3 408.3 433.4 408.2 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 249.3 249.4 242.7 242.4 229.9 229.9 225.3 220.5 220.1 215.7 220.3 225.1 220.2

216.7 216.7 216.7 216.7 204.2 216.7 204.1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

44824 49660 76449 189993 266800 320083 249018 308898 597653 1434556 2368538 Run out Run out

145670 166985 180832 216946 887928 Run out Run out

Experimental results generated under multiaxial fatigue loading 409

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

445.6 413.8 413.8 413.8 411.6 298.4 298.4 275.3 275.3 275.3 275.3 271.8 234.8 234.8 232.4 0.0 0.0 0.0 0.0

Code, material

Table B.65

USC15 – SAE 1045

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 159.2 159.2 159.2 191.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

8262 13060 13760 18310 17450 106700 117700 83600 184300 132300 228300 249900 403800 709000 764000 1293000 2238000 2000000 101100

Nf (cycles)

410 Multiaxial notch fatigue

0.0 0.0 0.0 12.7 90.7 90.7 124.1 124.1 133.7 134.5 135.4 157.6 157.6 183.0 194.2 194.2 194.2 198.9 198.9 198.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

191.0 238.7 238.7 201.6 173.5 173.5 173.5 173.5 214.9 143.2 214.9 110.6 110.6 214.9 135.3 136.1 136.1 70.0 70.0 214.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

164070 9528 14720 33330 177800 186700 142700 169500 24540 396800 17730 587000 1194000 12700 124500 163700 158100 722500 747000 10420

Experimental results generated under multiaxial fatigue loading 411

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

206.9 215.7 246.7 246.7 267.4 267.4 273.7 273.7 294.4 294.4 294.4 318.3 370.0 370.0 183.0 294.4 286.5 270.2 366.1 122.5 206.1 194.2 194.2

Code, material

Table B.66

USC15 – SAE 1045

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 111.4 202.9 86.7 86.7 71.6 76.4 107.4 107.4 167.1 167.1 202.9 167.1 107.4 107.4 214.9 167.1 167.1 178.4 105.4 173.5 136.1 136.1 136.1

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90

dxy,x (°)

226000 11630 159900 220500 153800 65049 51780 65800 11500 11630 5113 12050 11380 12090 13110 27470 24620 10840 23980 157100 45580 213800 266200

Nf (cycles)

412 Multiaxial notch fatigue

177.4 177.4 177.4 162.6 162.6 162.6 147.8 147.8 147.8 133.0 133.0 133.0 0.0 0.0 0.0 0.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 136.5 136.5 136.5 125.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1084 1423 1444 1859 2186 2178 4282 4134 4699 10643 8785 9776 847 1104 915 2034

Experimental results generated under multiaxial fatigue loading 413

USC16 – SAE 1045

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 148.5 148.5 148.5 127.3 127.3 127.3 120.2 120.2 120.2 113.2

Code, material

Table B.67

USC16 – SAE 1045

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 125.1 125.1 113.7 113.7 113.7 102.3 102.3 102.3 74.3 74.3 74.3 63.7 63.7 63.7 60.1 60.1 60.1 58.6

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2306 2961 4934 4373 3141 9205 9611 9797 1121 1014 1196 3577 3797 3602 5400 5031 4291 8039

Nf (cycles)

414 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

113.2 113.2 58.2 58.2 58.2 53.4 53.4 53.4 48.5 48.5 48.5 43.7 43.7 43.7

USC17 – low carbon steel

298.9 298.9 278.8 281.2 281.2 260.1 260.1

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

58.6 58.6 116.4 116.4 116.4 106.7 106.7 106.7 97.0 97.0 97.0 87.3 87.3 87.3 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

18375 20195 29461 34122 41213 61398 64029

9193 8500 1830 1556 1702 3768 3684 3176 6717 6054 5097 12531 17340 15382

Experimental results generated under multiaxial fatigue loading 415

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

269.3 242.6 236.4 240.5 240.5 198.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Code, material

Table B.68

USC17 – low carbon steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 0.0 0.0 0.0 225.1 225.1 224.1 225.1 209.9 209.9 209.9 209.9 200.8 200.0 200.8 189.7 190.6

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

93409 156179 785627 1182750 Run out Run out 18350 21025 38580 53367 115781 119474 145759 189353 343846 460931 693281 949014 1194738

Nf (cycles)

416 Multiaxial notch fatigue

USC18 – low carbon steel

180.6 145.3 165.5 140.3 120.0

0.0 0.0 303.2 297.5 297.6 292.1 268.2 243.8 270.9 241.8 244.1 242.0 215.9 218.1 216.1 185.8

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.0 0.0 0.0 0.0

189.7 180.0 151.6 148.8 148.8 146.0 134.1 121.9 135.5 120.9 122.1 121.0 108.0 109.1 108.0 92.9 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

45294 55284 71111 74158 93409

1873817 Run out 9795 11151 13028 21047 39728 39728 82113 147161 167533 415187 847087 1820267 1942175 Run out

Experimental results generated under multiaxial fatigue loading 417

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

157.1 140.3 159.9 119.0 100.0 95.8 100.0 79.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Code, material

Table B.69

USC18 – low carbon steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 200.8 200.8 188.1 174.5 174.5 149.7 168.5 140.8 140.2 120.2 119.7

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

98439 121418 132047 302439 380948 598088 700007 Run out 17783 19539 25119 38580 67892 84581 151991 586384 811131 3195549 Run out

Nf (cycles)

418 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

179.6 172.9 177.9 155.7 140.3 147.2 140.4 112.8 113.9 126.5 135.2 106.6 96.9 85.6 92.5

USC19 – low carbon steel

177.4 177.4 158.5 159.9 137.9 140.3

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.0 0.0 0.0 0.0 0.0

89.8 86.4 89.0 77.9 70.2 73.6 70.2 56.4 56.9 63.2 67.6 53.3 48.5 42.8 46.3 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

31706 58875 64704 146652 198795 315400

20244 21322 23961 33566 74989 85370 101043 149081 214329 234690 243999 442993 697378 836175 3018073

Experimental results generated under multiaxial fatigue loading 419

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

118.0 118.0 113.9 98.3 99.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 183.1

Code, material

Table B.70

USC19 – low carbon steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 0.0 0.0 208.9 200.0 200.0 189.7 189.7 189.7 179.2 179.2 179.2 169.3 169.3 158.5 149.1 91.6

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

414297 527346 909909 1489775 Run out 21246 22154 29389 58028 63760 76176 128556 336723 419495 442033 574237 1383277 Run out 26927

Nf (cycles)

420 Multiaxial notch fatigue

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

183.1 183.2 154.3 128.8 137.7 140.5 102.7 113.0

USC20 – C40 steel

367.5 283.5 257.3 315.0 294.0 273.0 283.5 336.0 262.5 315.0 273.0 294.0 283.5

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

91.6 91.6 77.2 64.4 68.9 70.2 51.3 56.5 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

12737 399559 1366308 64626 248465 1982335 1528731 41238 4709486 41265 1946108 166490 608853

35353 44070 64185 130954 147161 355366 1028934 1479262

Experimental results generated under multiaxial fatigue loading 421

Code, material

Table B.71

USC20 – C40 steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 180 250 200 225 240 195 210 245 205 200 240 195 210 200

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

Run out 27287 875512 93305 42442 4080000 949862 31500 619000 1307000 70600 2208000 492000 1240000

Nf (cycles)

422 Multiaxial notch fatigue

260.0 208.0 182.0 130.0 156.0 312.0 260.0 208.0 143.0 182.0 156.0 169.0 143.0 312.0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 245 245 215 185 160 140 140 185 215 160 150 160

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

56163 92698 321849 Run out 558192 24547 47292 98319 1326510 160790 625577 285707 1046739 34439 26090 39970 141924 208184 600000 2596176 2934683 405061 239112 2209136 5681742 2430875

Experimental results generated under multiaxial fatigue loading 423

USC21 – C40 steel

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 140 120 80 160

sx,a (MPa)

100 140 160 120 120 120 180 130 130 180 220 200 110 140 120 80 160

Code, material

Table B.72

USC21 – C40 steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 100 140 160 120 120 120 180 130 130 180 220 200 110 140 120 80 160

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 140 120 80 160

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

Run out 156497 72939 1691679 443513 1794135 229471 634302 183838 193076 12475 27183 836305 47578 157461 Run out 26228

Nf (cycles)

424 Multiaxial notch fatigue

100 90 90 80 80 120 130 200 180 140 140 130 160 160 180 110 100 140 120 70 80 90

100 90 90 80 80 0 0 0 0 0 0 0 0 0 0 0 0 140 120 70 80 90

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

100 90 90 80 80 120 130 200 180 140 140 130 160 160 180 110 100 140 120 70 80 90

100 90 90 80 80 0 0 0 0 0 0 0 0 0 0 0 0 140 120 70 80 90

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

347734 432360 498803 801747 Run out 941705 687276 11514 14537 19603 282612 117684 47168 42125 18003 1112637 1427380 35575 103229 Run out Run out 339421

Experimental results generated under multiaxial fatigue loading 425

sx,m (MPa)

90 100 90 100 100 120 100 140 160 160 90

0 0 0 0 0 0 0 0 0 0

90 100 90 100 100 120 100 140 160 160 90

0 0 0 0 0 0 0 0 0 0

Code, material

sx,a (MPa)

Table B.73

USC21 – C40 steel

USC22 – C40 steel

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

160 200 200 180 140 225 250 160 150 225

90 100 90 100 100 120 100 140 160 160 90

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0

90 100 90 100 100 120 100 140 160 160 90

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90 90 90

dxy,x (°)

1827607 271283 219301 651809 2106893 87123 10929 2142267 3507075 92905

Run out 211017 Run out 86976 426688 84420 577554 33082 22623 21368 Run out

Nf (cycles)

426 Multiaxial notch fatigue

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 116.7 103.3 80.0 66.7 66.7 73.3 80.0 73.3

100.0 72.0 100.0 88.0 80.0 88.0 76.0 76.0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

19046 35288 223225 1660181 1405877 479980 462696 840286

33060 Run out 46888 127582 574377 138354 607433 992992

Experimental results generated under multiaxial fatigue loading 427

USC23 – BS04A12

USC24 – BS04A12

sx,m (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

50.0 50.0 38.0 38.0 33.0 35.0 33.0 35.0 33.0

40.0 33.3 33.3 40.0 30.0 31.7 31.7 48.3 48.3

Code, material

sx,a (MPa)

Table B.74

USC25 – BS04A12

USC26 – BS04A12

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

sy,m (MPa)

69.3 57.7 57.7 69.3 52.0 54.8 54.8 83.7 83.7

86.6 86.6 65.8 65.8 57.2 60.6 57.2 60.6 57.2

txy,a (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

dxy,x (°)

173467 674133 552233 365249 Run out 994062 1568092 53815 63742

52749 47024 340645 369109 2077381 659765 575553 745632 534420

Nf (cycles)

428 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0

56.6 70.7 50.9 56.6 70.7 50.9 84.9 56.6 84.9

82.5 75.4 56.6 47.1 56.6 47.1 51.9 51.9

229.2 458.4 407.4 407.4 376.9

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

82.5 75.4 56.6 47.1 56.6 47.1 51.9 51.9

56.6 70.7 50.9 56.6 70.7 50.9 84.9 56.6 84.9

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Run out 3283 40081 28931 411741

32770 54433 349054 1212560 602101 1535352 1115318 551675

614898 164588 1572192 1627475 291357 1338746 56444 1067819 38131

Experimental results generated under multiaxial fatigue loading 429

USC28 – BS04A12

USC29 – En3B

USC27 – BS04A12

sx,m (MPa)

0 0 0 0 0 0 0 0 0 345.9 280.1 331.0 310.7 323.4 300.5 323.4 300.5 310.7 331.0 338.7

sx,a (MPa)

376.9 432.9 432.9 356.5 356.5 458.4 387.1 387.1 356.5 345.9 280.1 331.0 310.7 323.4 300.5 323.4 300.5 310.7 331.0 338.7

Code, material

Table B.75

USC29 – En3B

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

249199 10583 9473 208067 Run out 4967 99596 171937 605146 5330 Run out 49170 349294 361035 776909 42600 1173770 707412 73824 6420

Nf (cycles)

430 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

USC30 – En3B

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 114.1 134.5 171.1 264.8 264.8 171.1 171.1 203.7 203.7 244.5 297.4

215.9 285.2 317.8 285.2 317.8 301.5 273.0 301.5 273.0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Run out Run out 1025331 31705 30028 306633 904657 344380 614536 70232 7700

Run out 707338 15765 276550 72802 67062 613038 276412 471108

Experimental results generated under multiaxial fatigue loading 431

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

331.0 203.7 137.5 137.5 107.0 331.0 119.7 331.0 203.7 127.3 300.5 132.4 137.5 142.6 142.6 203.7 259.6 216.3 183.9

Code, material

Table B.76

USC30 – En3B

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 155.9 129.9 110.4

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

10852 96785 236230 314269 Run out 18762 Run out 22505 148533 Run out 26164 Run out 1313155 224361 237312 52908 14743 30837 87177

Nf (cycles)

432 Multiaxial notch fatigue

146.0 135.2 135.2 183.9 216.3 146.0 259.6 135.2 129.8 194.7 194.7 162.2 108.2 97.3 129.8 129.8 91.9 146.0 108.2 119.0 135.2

0 0 0 0 0 0 0 0 0 194.7 194.7 162.2 108.2 97.3 129.8 129.8 91.9 146.0 108.2 119.0 135.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

87.7 81.2 81.2 110.4 129.9 87.7 155.9 81.2 77.9 202.5 202.5 168.7 112.5 101.2 135.0 135.0 95.6 151.9 112.5 123.7 140.6

0 0 0 0 0 0 0 0 0 202.5 202.5 168.7 112.5 101.2 135.0 135.0 95.6 151.9 112.5 123.7 140.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 90 0 0 0 0 0 0 90 90 90 90 90 90

460400 227391 924890 67416 30764 210914 9202 189952 1646841 6364 6680 11899 396137 626504 83907 62951 685098 45580 1397923 203792 59130

Experimental results generated under multiaxial fatigue loading 433

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 150.0 180.0 160.0 165.0 190.0 170.0

sx,a (MPa)

259.6 200.0 180.0 275.0 230.0 190.0 260.0 200.0 285.0 285.0 230.0 270.0 230.0 250.0 150.0 180.0 160.0 165.0 190.0 170.0

Code, material

Table B.77

USC31 – En3B

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 155.9 115.5 103.9 158.8 132.8 109.7 150.1 115.5 164.5 164.5 132.8 155.9 132.8 144.3 150.0 180.0 160.0 165.0 190.0 170.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 150.0 180.0 160.0 165.0 190.0 170.0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 90 90 90 90 90 90 90 90 0 0 0 0 0 0

dxy,x (°)

82952 437907 2174897 46254 188480 1400006 314817 Run out 31700 36976 150125 59622 245935 79328 844615 28108 370618 249286 34298 110056

Nf (cycles)

434 Multiaxial notch fatigue

170.0 155.0 180.0 145.0 150.0 160.0 190.0 200.0 160.0 155.0 235.0 160.0 175.0

0 0 0 0 0 0 0

170.0 155.0 180.0 145.0 150.0 160.0 190.0 200.0 160.0 155.0 235.0 160.0 175.0

USC32 – En3B

290.0 270.0 270.0 330.0 370.0 370.0 350.0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 167.4 155.9 155.9 190.5 213.6 213.6 202.1

170.0 155.0 180.0 145.0 150.0 160.0 190.0 200.0 160.0 155.0 135.7 160.0 175.0 0 0 0 0 0 0 0

170.0 155.0 180.0 145.0 150.0 160.0 190.0 200.0 160.0 155.0 135.7 160.0 175.0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90 90 90 90 90

282833 136165 Run out 38446 11543 15717 12200

112944 367445 49200 Run out Run out Run out 52000 67873 304439 Run out 59243 77755 111250

Experimental results generated under multiaxial fatigue loading 435

sx,m (MPa)

0 0 0 0 0 0 0 0 0 250.0 240.0 230.0 245.0 220.0 200.0 245.0 200.0 250.0 240.0 230.0 245.0 240.0 230.0 235.0 228.0 224.0

sx,a (MPa)

280.0 280.0 290.0 350.0 270.0 230.0 215.0 345.0 245.0 250.0 240.0 230.0 245.0 220.0 200.0 245.0 200.0 250.0 240.0 230.0 245.0 240.0 230.0 235.0 228.0 224.0

Code, material

Table B.78

USC32 – En3B

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 161.7 161.7 167.4 202.1 155.9 132.8 124.1 199.2 141.5 144.3 138.6 132.8 141.5 127.0 115.5 141.5 115.5 144.3 138.6 132.8 141.5 138.6 132.8 135.7 131.6 129.3

txy,a (MPa) 0 0 0 0 0 0 0 0 0 144.3 138.6 132.8 141.5 127.0 115.5 141.5 115.5 144.3 138.6 132.8 141.5 138.6 132.8 135.7 131.6 129.3

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90

dxy,x (°)

57847 90140 88682 7427 51504 181609 Run out 6725 266899 60384 340599 316599 422875 488018 231414 48347 721275 49000 36446 Run out 106129 31641 72734 60399 407794 165543

Nf (cycles)

436 Multiaxial notch fatigue

0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 190.4 180.4 170.3 160.4 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 200.3 190.4 180.4 170.3

200.2 190.0 179.9 169.8 190.4 180.4 170.3 160.4 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0

60356 156740 191394 407041

23501 153300 236316 1155183 33894 75354 101678 134187

Experimental results generated under multiaxial fatigue loading 437

USC33 – SGV410

USC34 – SGV410

Code, material

Table B.79

USC34 – SGV410

USC35 – SGV410

USC36 – SUS316NG

sx,m (MPa)

190.6 180.7 170.9 160.9

0 0 0 0 0 189.8 180.0 159.7 139.7

0 0 0 0 0 210.3 190.1 180.1

sx,a (MPa)

0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0

sy,m (MPa)

210.3 200.3 190.2 185.4 180.1 210.3 190.1 180.1

189.7 179.8 170.1 160.2 149.7 189.8 180.0 159.7 139.7

190.6 180.7 170.9 160.9

txy,a (MPa)

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0

dxy,x (°)

22232 48343 81441 118768 376620 68953 179066 724838

24568 32422 124158 200081 1474602 9356 18616 70501 209162

24296 61711 109891 402549

Nf (cycles)

438 Multiaxial notch fatigue

0 0 0 0 0 210.6 200.7 190.6 185.5 180.5

0 0 0 0 209.8 199.5 189.9 179.8

0 0 0 0 0 0 0 0 0 0

USC37 – SUS316NG

USC38 – SUS316NG

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 210.1 200.0 190.2 179.8 209.8 199.5 189.9 179.8

210.6 200.7 190.6 185.7 180.3 210.6 200.7 190.6 185.5 180.5 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

51102 78774 269976 333343 16477 29999 45734 74523

51672 75354 179066 169401 508188 48883 62399 100556 95129 156740

Experimental results generated under multiaxial fatigue loading 439

sx,m (MPa)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

sx,a (MPa)

125.0 120.0 115.1 105.2 100.0 0.0 0.0 0.0 0.0 88.2 81.3 88.5 81.5 74.4 70.9

Code, material

Table B.80

USC39 – Al 1070

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0.0 0.0 0.0 0.0 0.0 72.2 63.6 66.6 60.6 50.9 46.9 51.1 47.1 42.9 40.9

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 90 90 90 90

dxy,x (°)

249 731 830 2914 19953 2139 6867 9080 29138 395 9080 102 662 651 2863

Nf (cycles)

440 Multiaxial notch fatigue

72.2 83.9 68.6 61.1 78.6 69.9 61.2 61.2 56.4 86.3 75.4 69.9 70.0

0 0 0 0 0 0 0 0 0 123.2 123.2

65.3 75.9 62.1 55.3 71.1 63.2 55.3 55.3 51.1 78.1 68.2 63.3 63.3

399.6 399.8 399.8 364.1 365.3 367 275 275.4 275.5 369.5 369.6

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

32.7 37.9 31.0 27.7 53.3 47.4 41.5 41.5 38.3 26.0 22.7 21.1 21.1 0 0 0 0 0 0 0 0 0 0 0

36 42 34 31 59 52 46 46 42 29 25 23 23 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

128630 153881 168309 327016 375852 415067 1306495 2869140 3364694 57172 65725

38397 50168 165119 454590 70780 150889 420279 433681 642477 47499 75764 460664 621082

Experimental results generated under multiaxial fatigue loading 441

USC40 – 6082-T6

USC41 – 18G2A

sx,m (MPa)

123.0 99.4 99.5 99.5 89.6 89.6 89.6 82.4 82.4 290.2 288.6 265.7 274.3 255.0 254.5 235.3 235.4

sx,a (MPa)

369.1 298.3 298.5 298.5 268.8 268.9 268.9 247.3 247.1 290.2 288.6 265.7 274.3 255 254.5 235.3 235.4

Code, material

Table B.81

USC41 – 18G2A

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

72614 131431 168583 178962 578316 671481 712823 976665 1356769 95759 133071 157011 173047 241679 422196 887342 1126890

Nf (cycles)

442 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

223.0 222.6 222.7 195.7 195.4 196.2 183.9 184.0 183.0 175.0 174.5 151.6 151.3 151.3 138.4 138.4 138.2 137.8 135.4 135.2 131.4 131.2

0 0 0 0 0 0 0 0 0 58.3 58.2 50.5 50.4 50.4 46.1 46.1 138.2 137.8 135.4 135.2 131.4 131.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

224794 278939 310708 637980 785106 874370 2296066 3095310 4173870 157758 229202 551271 640123 689753 2176479 3187921 356608 526773 710704 860208 1123272 1501895

Experimental results generated under multiaxial fatigue loading 443

sx,m (MPa)

0 0 0 0 0 0 0 0 0 48.8 48.9 48.8 43.3 43.3 40.5 40.5 39.5 39.4

sx,a (MPa)

199.7 199.7 199.5 180.2 180.2 180.0 164.5 164.6 164.4 146.5 146.7 146.5 129.8 129.8 121.4 121.4 118.5 118.3

Code, material

Table B.82

USC41 – 18G2A

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 199.7 199.7 199.5 180.2 180.2 180.0 164.5 164.6 164.4 146.5 146.7 146.5 129.8 129.8 121.4 121.4 118.5 118.3

txy,a (MPa) 0 0 0 0 0 0 0 0 0 48.8 48.9 48.8 43.3 43.3 40.5 40.5 39.5 39.4

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

168497 192588 226543 609639 615487 737954 1303741 1407201 2101434 87304 93327 121935 382788 417127 1408351 1505671 2317350 3056739

Nf (cycles)

444 Multiaxial notch fatigue

fy/fx = 1.

USC42 – EN24Ta

a

139.4 139.4 129.2 129.5 123.2 123.2 120.9 120.9 118.0

0 0 0 0 0 0 0 0 191.0 183.1 179.4 175.4

139.4 139.4 129.2 129.5 123.2 123.2 120.9 120.9 118.0

233.1 239.1 231.2 229.4 216.5 222.8 221.8 214.9 191.0 183.1 179.4 175.4

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

139.4 139.4 129.2 129.5 123.2 123.2 120.9 120.9 118.0 0 0 0 0 0 0 0 0 0 0 0 0

139.4 139.4 129.2 129.5 123.2 123.2 120.9 120.9 118.0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

33450 51331 61949 79405 236472 351504 Run out Run out 63626 82332 139651 Run out

115525 138497 211846 233041 257221 366179 681921 794444 1165727

Experimental results generated under multiaxial fatigue loading 445

sx,m (MPa)

292.4 282.0 269.0 0.0 0.0 0.0 192.7 183.5 177.1 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

146.2 141.0 134.5 245.3 238.2 231.0 192.7 183.5 177.1 0 0 0 0 0 0 0 0 0 0 0 0

Code, material

Table B.83

USC42 – EN24Ta

0 0 0 0 0 0 0 0 0 201.3 192.9 185.6 178.6 173.9 156.2 147.3 137.7 137.7 129.5 103.4 99.6

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 156.2 147.3 137.7 137.7 129.5 206.8 199.2

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

69795 244155 Run out 81309 151937 Run out 134378 178475 255420 107729 325334 318307 284830 Run out 58000 105316 105150 196540 Run out 175077 Run out

Nf (cycles)

446 Multiaxial notch fatigue

a

fy/fx = 1.

0 0 0 199.6 183.8 190.9 177.5 199 177.8 183.5 190.6 168.7 145.6 137 146.1 129.2 137.8 131.2 133.4

177.5 173.0 169.8 0 0 0 0 199 177.8 183.5 190.6 168.7 0 0 0 0 137.8 131.2 133.4

177.5 173.0 169.8 199.6 183.8 190.9 177.5 199 177.8 183.5 190.6 168.7 145.6 137 146.1 129.2 137.8 131.2 133.4

0 0 0 0 0 0 0 0 0 0 0 0 145.6 137 146.1 129.2 137.8 131.2 133.4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

265863 304815 Run out 108261 207936 308017 727210 100378 354165 447929 837657 Run out 51791 84011 100696 Run out 75307 337035 783525

Experimental results generated under multiaxial fatigue loading 447

sx,m (MPa)

0 0 0 0 269.4 269.5 261.2 261.1 0 0 246.1 239.0 231.6

sx,a (MPa)

293.4 285.9 278.7 278.6 269.4 269.5 261.2 261.1 245.8 238.7 246.1 239.0 231.6

Code, material

Table B.84

USC42 – EN24Ta

293.4 285.9 278.7 278.6 269.4 269.5 261.2 261.1 245.8 238.7 246.1 239.0 231.6

sy,a (MPa) 0 0 0 0 0 0 0 0 245.8 238.7 246.1 239.0 231.6

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 180 180 180 180 180 180 180 180 180 180 180 180

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

23285 29752 156237 Run out 19272 49219 536486 Run out 111468 234004 85201 278053 Run out

Nf (cycles)

448 Multiaxial notch fatigue

fy/fx = 1; b fy/fx = 2.

USC42 – EN24Tb

a

285.6 285.1 271.9 257.9 251.0 257.9 244.5 238.2 244.8 257.9 244.7 230.6 223.5 203.2 216.3 230.6 216.8 209.6 253.5 240.4 228.4 228.2 228.1 222.4 215.3 228.3 215.3

0 0 0 0 251.0 257.9 244.5 238.2 244.8 0 0 0 0 0 0 230.6 216.8 209.6 0 0 0 228.2 228.1 222.4 215.3 0 0

285.6 285.1 271.9 257.9 251.0 257.9 244.5 238.2 244.8 257.9 244.7 230.6 223.5 203.2 216.3 230.6 216.8 209.6 253.5 240.4 228.4 228.2 228.1 222.4 215.3 228.3 215.3

0 0 0 0 0 0 0 0 0 257.9 244.7 230.6 223.5 203.2 216.3 230.6 216.8 209.6 0 0 0 0 0 0 0 228.3 215.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

50638 97288 389982 895857 46629 122058 192124 323927 908257 22349 36659 36912 116325 162907 895857 53870 69469 908257 71707 186915 393329 83018 254964 312408 Run out 39743 80850

Experimental results generated under multiaxial fatigue loading 449

sx,m (MPa)

0 0 0 203.9 193.1

0 0 0 261.1 254.7 246.7 246.8 239.6 232.5 0 0 0 0 0 0

sx,a (MPa)

203.2 203.2 191.6 203.9 193.1

278.9 271.1 262.3 261.1 254.7 246.7 246.8 239.6 232.5 261.2 246.2 232 216.1 215.9 208.4

Code, material

Table B.85

USC42 – EN24Tb

USC42 – EN24Tc

278.9 271.1 262.3 261.1 254.7 246.7 246.8 239.6 232.5 261.2 246.2 232 216.1 215.9 208.4

203.2 203.2 191.6 203.9 193.1

sy,a (MPa)

0 0 0 0 0 0 0 0 0 261.2 246.2 232 216.1 215.9 208.4

203.2 203.2 191.6 203.9 193.1

sy,m0 (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

dxy,x (°)

66150 550613 Run out 33634 31900 54391 315487 Run out Run out 20758 64633 41779 53473 71434 Run out

51718 64020 Run out 59324 Run out

Nf (cycles)

450 Multiaxial notch fatigue

b

fy/fx = 2; c fy/fx = 3.

200.8 230.5 224.5 216.6 235.9 223.6 211.6 200.1 223.1 216.4 211.6 210.9 205.7 187.2 175.1 164.3 156.0 198.9 188.1 176.8

0 230.5 224.5 216.6 0 0 0 0 223.1 216.4 211.6 210.9 205.7 0 0 0 0 198.9 188.1 176.8

200.8 230.5 224.5 216.6 235.9 223.6 211.6 200.1 223.1 216.4 211.6 210.9 205.7 187.2 175.1 164.3 156.0 198.9 188.1 176.8

200.8 230.5 224.5 216.6 0 0 0 0 0 0 0 0 0 187.2 175.1 164.3 156.0 198.9 188.1 176.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Run out 41801 52951 488799 111621 245857 141868 Run out 72442 95616 136111 Run out Run out 50270 86353 63432 Run out 37266 79273 Run out

Experimental results generated under multiaxial fatigue loading 451

sx,m (MPa)

182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 182.0 220.2 199.2 176.0 176.7 165.5

sx,a (MPa)

227.6 228.1 228.1 227.6 227.6 172.2 172.2 172.5 172.5 172.2 137.5 137.5 137.5 137.5 137.2 180.2 163.0 144.0 144.6 135.4

Code, material

Table B.86

USC43 – GRS 500/ISO 1083

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 176.6 159.7 141.1 141.7 132.7

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 215.8 195.2 172.5 173.2 162.2

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

29468 35226 37544 39005 46624 316977 342167 374119 403850 924640 979339 1170641 Run out Run out Run out 74231 149211 106755 209796 Run out

Nf (cycles)

452 Multiaxial notch fatigue

165.5 0 0 0 0 0 0 0 0

0 0 −12 −36 23 −5 −13 −10 3 2

135.4 0 0 0 0 0 0 0 0

USC44 – 1045 (BHN 456)

335 344 352 387 387 438 433 465 464 437

335 344 352 387 387 438 433 465 464 437

132.7 0 0 0 0 0 0 0 0 0 0 −12 −36 23 −5 −13 −10 3 2

162.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 132.2 132.2 124.9 117.5 117.5 124.9 109.8 102.4 0 0 0 0 0 0 0 0 0 0

0 161.6 161.6 152.7 143.6 143.6 152.7 134.2 125.2 180 180 180 180 180 180 180 180 180 180

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Run out Run out Run out Run out 100467 76839 72602 61620 45125 34925

Run out 179596 287802 234340 379701 453871 523065 1208688 Run out

Experimental results generated under multiaxial fatigue loading 453

sx,m (MPa)

6 −5 0 5 0 491 390 360 500 500 502 392 500 497 350 665 360 635 405 360

sx,a (MPa)

515 519 497 609 642 152 175 210 160 148 167 197 176 175 265 199 241 194 251 305

Code, material

Table B.87

USC44 – 1045 (BHN 456)

515 519 497 609 642 152 175 210 160 148 167 197 176 175 265 199 241 194 251 305

sy,a (MPa) 6 −5 0 5 0 491 390 360 500 500 502 392 500 497 350 665 360 635 405 360

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

20691 20299 15000 12191 8222 Run out Run out Run out 274363 233000 107036 62381 60000 51000 48624 46477 37000 35000 31500 31435

Nf (cycles)

454 Multiaxial notch fatigue

192 323 238 271 268 358 483 310 251 237 308 371 446 266 288 345 287 301 295

584 369 684 405 547 374 106 411 450 449 711 270 212 500 390 414 428 646 741

192 323 238 271 268 358 483 310 251 237 308 371 446 266 288 345 287 301 295

584 369 684 405 547 374 106 411 450 449 711 270 212 500 390 414 428 646 741

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

29606 29559 25640 24517 17913 17577 15956 15320 15277 13148 12818 12683 12292 12104 10000 9132 6578 6477 5730

Experimental results generated under multiaxial fatigue loading 455

sx,m (MPa)

495 350 403 456 566 382 390 364 379 395 701 523 414 −382 −318 −150 −119 −292 −185

sx,a (MPa)

261 420 456 254 327 429 450 500 506 580 408 432 652 556 505 420 431 540 610

Code, material

Table B.88

USC44 – 1045 (BHN 456)

261 420 456 254 327 429 450 500 506 580 408 432 652 556 505 420 431 540 610

sy,a (MPa) 495 350 403 456 566 382 390 364 379 395 701 523 414 −382 −318 −150 −119 −292 −185

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

4971 4798 4533 4507 4468 3742 3742 3317 2150 646 458 340 255 Run out Run out Run out 350000 264192 142000

Nf (cycles)

456 Multiaxial notch fatigue

−123 −388 −138 −275 −97 −305 −141 −324 −316

0 0 0 0 0 0 0 100 40 40 100

448 574 483 547 451 591 553 623 730

USC45 – 1045 (BHN 203)

242 231 222 192 184 177 164 125 140 147 130

242 231 222 192 184 177 164 125 140 147 130

448 574 483 547 451 591 553 623 730 0 0 0 0 0 0 0 100 40 40 100

−123 −388 −138 −275 −97 −305 −141 −324 −316 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 180 180 180 180 180 180 180 180 180 180 180

180 180 180 180 180 180 180 180 180 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

4111 4960 10519 57067 273262 1393112 Run out Run out Run out 228285 166234

104000 76907 55000 38789 30041 26000 22000 17000 12643

Experimental results generated under multiaxial fatigue loading 457

sx,m (MPa)

40 100 100 −60 −60 −60 −60 −60

0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

160 137 150 186 187 190 200 220

232.9 240.8 232.3 230.9 217.4 224.2 216.1 223.7 246.7 241 232.6 223.6 279.0 271.1

Code, material

Table B.89

USC45 – 1045 (BHN 203)

USC46 – EN24Ta

0 0 0 0 0 0 0 0 0 0 0 0 69.8 67.8

160 137 150 186 187 190 200 220

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

40 100 100 −60 −60 −60 −60 −60

sy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

180 180 180 180 180 180 180 180

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

dxy,x (°)

34835 53009 62672 84677 245882 381227 Run out Run out 86440 166099 Run out Run out 99980 170846

88845 58719 47935 Run out 941969 372048 99217 52041

Nf (cycles)

458 Multiaxial notch fatigue

a

fy/fx = 1.

254.0 264.3 255.6 232.2 244.1 241.3 241.0 192.9 179.8 178.6 200.1 193.0 184.4 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

63.5 132.2 127.8 116.1 183.1 181.0 180.8 192.9 179.8 178.6 400.2 386.0 368.8 193.1 185.0 178.5 174.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Run out 102924 447184 Run out 98740 142269 347154 379354 235489 772697 173688 322021 777835 349496 345932 334498 994277

Experimental results generated under multiaxial fatigue loading 459

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

313.1 312.6 299.9 293.4 287.1 277.6 276.7 278.7 268.7 255.5 187.0

284.7 285.0 272.9 258.0 229.7 242.6 228.2

Code, material

sx,a (MPa)

Table B.90

USC46 – EN24Ta

USC46 – EN24Tb

284.7 285.0 272.9 258.0 229.7 242.6 228.2

156.6 156.3 150.0 293.4 287.1 277.6 276.7 557.4 537.4 511.0 561.0

sy,a (MPa)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 90 90 90

180 180 180 180 180 180 180 180 180 180 180

dy,x (°)

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

51364 105266 413481 Run out 80928 204153 418950

22841 100108 Run out 24898 31485 167708 Run out 32926 76694 320137 172393

Nf (cycles)

460 Multiaxial notch fatigue

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0

278.7 278.9 271.0 235.3 211.8 224.1 199.5

300.3 225.4 150.3 150.3 125.1 300.3 225.4 150.3 125.6

318.3 413.8 0.0 0.0 254.6

USC47 – S460N

USC48 – SAE 1045

fy/fx = 1; b fy/fx = 2; c fy/fx = 3.

USC46 – EN24Tc

a

278.7 278.9 271.0 235.3 211.8 224.1 199.5

0 0 0 0 0 0 0

216.5 0.0 238.7 238.7 173.2

262.2 196.8 131.2 131.2 109.2 262.2 196.8 131.2 109.6

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 180 180 180 180

0 0 0 0 0

0 0 0 0 0 90 90 90 90

0 0 0 0 0 0 0

5471 22800 19500 18550 50660

225 4317 57677 90360 903597 1156 4990 218586 Run out

64494 979425 Run out 117920 150331 260150 Run out

Experimental results generated under multiaxial fatigue loading 461

Code, material

Table B.91

USC48 – SAE 1045

USC49 – CK45

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

254.6 350.1 350.1 0.0 218.0 218.0 212.0 212.0 310.4 310.4 0.0

491.0 491.0 491.0 491.0 491.0 491.0 451.0 451.0 451.0 451.0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

0 0 0 0 0 0 0 0 0 0

173.2 0.0 0.0 202.9 148.8 148.8 143.2 143.2 0.0 0.0 159.2

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

15200 20250 21200 21900 26850 37300 40100 51550 52900 63320

55050 121000 196700 391900 300000 280000 366400 404900 225000 459800 350000

Nf (cycles)

462 Multiaxial notch fatigue

451.0 451.0 392.0 392.0 392.0 392.0 392.0 0 0 0 0 0 0 0 0 373.0 373.0 373.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 373.0 373.0 373.0

0 0 0 0 0 0 0 451.0 451.0 451.0 451.0 392.0 392.0 392.0 392.0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

67700 78550 141780 154700 165390 210520 221610 51992 54118 57857 68832 109734 130550 197542 216910 21312 27457 28578

Experimental results generated under multiaxial fatigue loading 463

Code, material

Table B.92

USC49 – CK45

sx,m (MPa)

373.0 373.0 373.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 0 0 0 0 0 0

sx,a (MPa)

373.0 373.0 373.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 344.0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 373.0 373.0 373.0 373.0 344.0 344.0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 373.0 373.0 373.0 373.0 344.0 344.0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

34444 39357 67099 130791 170775 195136 337122 360366 380110 395625 509708 639404 730618 15557 17545 24504 25849 64171 102430

Nf (cycles)

464 Multiaxial notch fatigue

0 0 0 334.0 320.0 320.0 320.0 320.0 320.0 290.0 290.0 290.0 314.0 314.0 314.0 314.0 314.0 290.0 290.0 290.0

0 0 0 334.0 320.0 320.0 320.0 320.0 320.0 290.0 290.0 290.0 314.0 314.0 314.0 314.0 314.0 290.0 290.0 290.0

344.0 344.0 344.0 367.4 352.0 352.0 352.0 352.0 352.0 319.0 319.0 319.0 345.4 345.4 345.4 345.4 345.4 319.0 319.0 319.0

344.0 344.0 344.0 367.4 352.0 352.0 352.0 352.0 352.0 319.0 319.0 319.0 345.4 345.4 345.4 345.4 345.4 319.0 319.0 319.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

128551 146932 159196 27133 61180 66277 74728 77778 93743 159673 166196 202999 45910 49076 50402 59144 64936 169380 176294 185967

Experimental results generated under multiaxial fatigue loading 465

Code, material

Table B.93

USC49 – CK45

sx,m (MPa)

290.0 275.0 245.0 245.0 245.0 245.0 234.0 234.0 234.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

sx,a (MPa)

290.0 275.0 245.0 245.0 245.0 245.0 234.0 234.0 234.0 345.0 345.0 345.0 345.0 310.0 310.0 310.0 300.0 300.0

319.0 302.5 269.5 269.5 269.5 269.5 257.4 257.4 257.4 379.5 379.5 379.5 379.5 341.0 341.0 341.0 330.0 330.0

sy,a (MPa) 319.0 302.5 269.5 269.5 269.5 269.5 257.4 257.4 257.4 379.5 379.5 379.5 379.5 341.0 341.0 341.0 330.0 330.0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 180 180 180 180 180 180 180 180 180 0 0 0 0 0 0 0 90 90

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1033 4132 65248 130536 194742 228538 350041 416302 433294 48204 66386 92652 104467 155456 159658 247915 61147 108495

Nf (cycles)

466 Multiaxial notch fatigue

300.0 300.0 290.0 290.0 242.0 235.0 235.0 235.0 235.0 235.0 230.0 230.0 230.0 222.0 305.0 305.0 305.0 305.0 305.0 305.0 294.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 −305.0 −305.0 −305.0 −305.0 −305.0 −305.0 −294.0

330.0 330.0 319.0 319.0 266.2 258.5 258.5 258.5 258.5 258.5 253.0 253.0 253.0 244.2 335.5 335.5 335.5 335.5 335.5 335.5 323.4

330.0 330.0 319.0 319.0 266.2 258.5 258.5 258.5 258.5 258.5 253.0 253.0 253.0 244.2 335.5 335.5 335.5 335.5 335.5 335.5 323.4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 180 180 180 180 180 180 180 180 180 180 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

114440 134295 151384 197649 20825 50256 58986 68324 95417 98001 193585 212562 227246 852003 76892 81113 86717 97798 104554 107386 99113

Experimental results generated under multiaxial fatigue loading 467

Code, material

Table B.94

USC49 – CK45

sy,a (MPa) 323.4 323.4 291.5 291.5 291.5 291.5 275.0 275.0 236.5 236.5 225.5 225.5 225.5 225.5 470.0 450.0 390.0 390.0 330.0 330.0 306.0 280.0

sx,m (MPa)

−294.0 −294.0 −265.0 −265.0 −265.0 −265.0 −250.0 −250.0 −215.0 −215.0 −205.0 −205.0 −205.0 −205.0 235.0 225.0 195.0 195.0 165.0 165.0 −153.0 −140.0

sx,a (MPa)

294.0 294.0 265.0 265.0 265.0 265.0 250.0 250.0 215.0 215.0 205.0 205.0 205.0 205.0 235.0 225.0 195.0 195.0 165.0 165.0 153.0 140.0

323.4 323.4 291.5 291.5 291.5 291.5 275.0 275.0 236.5 236.5 225.5 225.5 225.5 225.5 470.0 450.0 390.0 390.0 330.0 330.0 306.0 280.0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 90 90 90 90 90 90 180 180 180 180 180 180 0 0 0 0 0 0 180 180

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

126061 138420 97987 123008 128048 135091 140342 169269 75947 131189 258667 287786 299532 412505 1920 9442 67480 94299 113574 243599 76510 208818

Nf (cycles)

468 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 427.0 427.0 427.0 392.0 392.0 392.0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 441.2 441.2 441.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

294.0 294.0 294.0 294.0 275.0 275.0 275.0 275.0 0 0 0 0 0 0 255.0 255.0 255.0

0 0 0 0 0 0 0 0 213.5 213.5 213.5 196.0 196.0 196.0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

33300 36570 52220 59240 266100 306830 333130 349980 18769 24862 29195 106715 157334 187241 12892 13783 14933

Experimental results generated under multiaxial fatigue loading 469

Code, material

Table B.95

USC49 – CK45

sx,m (MPa)

406.6 406.6 406.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 344.0 344.0 344.0 344.0 314.0 314.0 314.0 314.0 344.0 344.0 344.0 344.0 314.0 314.0 314.0 314.0 344.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 235.0 235.0 235.0 197.8 197.8 197.8 197.8 180.6 180.6 180.6 180.6 197.8 197.8 197.8 197.8 180.6 180.6 180.6 180.6 197.8

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 60 60 60 60 60 60 60 60 90

dxy,x (°)

156717 188952 289748 27483 30178 36386 40490 183255 190750 212269 419589 22795 28992 41587 68180 54326 55798 73871 75871 26085

Nf (cycles)

470 Multiaxial notch fatigue

344.0 344.0 344.0 310.0 310.0 310.0 310.0 304.0 304.0 304.0 304.0 275.0 275.0 275.0 275.0 304.0 304.0 304.0 304.0

0 0 0 0 0 0 0 304.0 304.0 304.0 304.0 275.0 275.0 275.0 275.0 304.0 304.0 304.0 304.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

197.8 197.8 197.8 178.3 178.3 178.3 178.3 174.8 174.8 174.8 174.8 158.1 158.1 158.1 158.1 174.8 174.8 174.8 174.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 90 90 90 90

30223 41674 55202 64640 74894 89130 191186 10937 12332 15675 17209 97155 102478 108092 159124 14986 43551 47812 67626

Experimental results generated under multiaxial fatigue loading 471

Code, material

Table B.96

USC49 – CK45

sx,m (MPa)

275.0 275.0 275.0 275.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

275.0 275.0 275.0 275.0 304.0 304.0 304.0 304.0 275.0 275.0 275.0 275.0 304.0 304.0 304.0 304.0 284.0 284.0 284.0 284.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 158.1 158.1 158.1 158.1 174.8 174.8 174.8 174.8 158.1 158.1 158.1 158.1 174.8 174.8 174.8 174.8 163.3 163.3 163.3 163.3

txy,a (MPa) 0 0 0 0 174.8 174.8 174.8 174.8 158.1 158.1 158.1 158.1 174.8 174.8 174.8 174.8 163.3 163.3 163.3 163.3

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 90 90 90 90 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

dxy,x (°)

33803 41843 62426 140815 17431 20736 21298 22467 138026 149558 280221 459408 51129 71407 80528 95808 82654 90758 113902 174670

Nf (cycles)

472 Multiaxial notch fatigue

490.0 490.0 490.0 490.0 490.0 490.0 490.0 490.0 442.0 442.0 442.0 442.0 442.0 442.0 442.0 442.0 480.0 480.0 480.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 480 480 480

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

44334 51367 55661 61129 67137 72749 79896 86582 96452 117906 134789 138447 144120 150026 162573 190904 19099 19617 27395

Experimental results generated under multiaxial fatigue loading 473

USC50 – CK45

Code, material

Table B.97

USC50 – CK45

sx,m (MPa)

480.0 480.0 480.0 480.0 480.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 0 0 0 0 0 0

sx,a (MPa)

480.0 480.0 480.0 480.0 480.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 0 0 0 0 0 0

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 296.0 296.0 296.0 296.0 296.0 296.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

30895 32158 33029 38780 60265 43937 85689 107557 133190 144305 154274 191039 201525 25325 35371 37815 43220 44988 48096

Nf (cycles)

474 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

296.0 296.0 255.0 255.0 255.0 255.0 255.0 255.0 255.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0

0 0 0 0 0 0 0 0 0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

49399 55711 123877 200395 238427 346576 375504 429182 575839 13627 13996 16645 17325 20604 22025 24504 32423 34659 37049 41220

Experimental results generated under multiaxial fatigue loading 475

Code, material

Table B.98

USC50 – CK45

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 0 0 442.0 442.0 442.0 442.0 442.0 442.0 442.0 442.0 343.0 343.0 343.0 343.0

sy,a (MPa)

sy,m (MPa) 255.0 255.0 255.0 255.0 255.0 255.0 255.0 254.2 254.2 254.2 254.2 254.2 254.2 254.2 254.2 197.2 197.2 197.2 197.2

txy,a (MPa) 255.0 255.0 255.0 255.0 255.0 255.0 255.0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

70100 72956 124370 136539 180666 198359 292013 7651 11272 11891 13232 16386 17993 21985 25128 94530 139272 143043 179519

Nf (cycles)

476 Multiaxial notch fatigue

343.0 343.0 343.0 343.0 286.0 286.0 286.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 343.0 343.0 343.0 343.0 343.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

197.2 197.2 197.2 197.2 164.5 164.5 164.5 225.4 225.4 225.4 225.4 225.4 225.4 225.4 225.4 197.2 197.2 197.2 197.2 197.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 30 30 30 30 30 30 30 30 30 30 30 30 30

186861 202457 231398 257501 930146 1007782 1608690 13537 17674 18152 18643 19147 19404 19928 33975 28983 31397 64513 67146 71773

Experimental results generated under multiaxial fatigue loading 477

Code, material

Table B.99

USC50 – CK45

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

343.0 343.0 343.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 392.0 315.0 315.0 315.0 315.0 315.0 315.0 315.0 315.0 392.0

sy,a (MPa)

sy,m (MPa) 197.2 197.2 197.2 225.4 225.4 225.4 225.4 225.4 225.4 225.4 225.4 181.1 181.1 181.1 181.1 181.1 181.1 181.1 181.1 225.4

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 30 30 30 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 90

dxy,x (°)

77752 97539 108516 8623 13765 15115 17742 18223 18968 21108 21971 65475 68158 77889 97758 134715 147923 166824 101756 13072

Nf (cycles)

478 Multiaxial notch fatigue

392.0 392.0 392.0 392.0 392.0 392.0 343.0 343.0 343.0 343.0 343.0 343.0 343.0 343.0 335.0 335.0 335.0 335.0 335.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 335.0 335.0 335.0 335.0 335.0

225.4 225.4 225.4 225.4 225.4 225.4 197.2 197.2 197.2 197.2 197.2 197.2 197.2 197.2 192.6 192.6 192.6 192.6 192.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90 90 90 90 90 90 0 0 0 0 0

13973 15965 16840 17764 19763 27218 42365 47767 50380 59122 65783 69387 94300 98149 23264 24215 29589 32059 33370

Experimental results generated under multiaxial fatigue loading 479

Code, material

Table B.100

USC50 – CK45

sx,m (MPa)

335.0 335.0 335.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0

sx,a (MPa)

335.0 335.0 335.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0

sy,a (MPa)

sy,m (MPa) 192.6 192.6 192.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

dxy,x (°)

35202 36642 44179 77339 98367 108011 113941 130229 142991 152870 210662 9609 10000 14135 14711 17030 19197 23129 24071

Nf (cycles)

480 Multiaxial notch fatigue

295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 295.0 295.0 295.0 295.0

295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

169.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 169.6 169.6 169.6 169.6

0 0 0 0 0 0 0 0 192.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 169.6 169.6 169.6 169.6

90 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0

29391 34480 40452 42664 48093 49390 50723 91104 21049 28988 33123 39922 46230 48117 52125 57227 155756 170995 206085 241849

Experimental results generated under multiaxial fatigue loading 481

Code, material

Table B.101

USC50 – CK45

sx,m (MPa)

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

sx,a (MPa)

295.0 295.0 295.0 295.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 335.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0 295.0

sy,a (MPa)

sy,m (MPa) 169.6 169.6 169.6 169.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6

txy,a (MPa) 169.6 169.6 169.6 169.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 192.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6 169.6

txy,m (MPa)

dy,x (°) 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

dxy,x (°)

342088 380600 510364 666333 23432 25048 25384 27869 35429 50108 55013 64560 37827 56437 64487 102844 137908 166216 197687 291015

Nf (cycles)

482 Multiaxial notch fatigue

Material

StE 460 StE 460 StE 460 A519 A519-A36 BS4360 Gr. 50E Fe 52 steel BS4360 Domex 350 JIS SM400B AlSi1MgMn 6060-T6

Code

WFR1 WFR2 WFR3 WFR4 WFR5 WFR6 WFR7 WFR8 WFR9 WFR10 WFR11 WFR12

Table B.102

Sonsino, 1995, 2001 Yousefi et al., 2001 Amstutz et al., 2001 Young and Lawrence, 1986 Siljander et al., 1992 Razmjoo, 1996 Bäckström et al., 1997 Archer, 1987 Dhale et al., 1997 Takahashi et al., 1999, 2003 Kueppers and Sonsino, 2003 Costa et al., 2005

Reference 520 520 – 552 – 415 – – 397 283 315 215

sy (MPa) 670 670 – 700 – 577 – – 486 432 332 240

sUTS (MPa)

B-T B-T B-T B-T B-T Te-T B-T B-T B-T Te-Te B-T B-T

Loading path

Fig. B.18 Fig. B.19 Tube to plate Fig. B.20 Fig. B.21 Fig. B.22 Fig. B.23 Fig. B.24 Fig. B.25 Fig. B.26 Fig. B.27 Fig. B.28

Geometry

Fatigue results generated by testing steel and aluminium welded specimens under multiaxial fatigue loading

B.7.1 Materials, static properties, geometries and applied loading

B.7

Experimental results generated under multiaxial fatigue loading 483

Code, material

Table B.103

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

286.0 281.0 289.0 284.0 183.0 223.0 196.0 206.0 164.0 191.0 143.0 152.0 113.0 87.0 120.0 101.0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 228.0

txy,a (MPa)

B.7.2 Experimental results generated by testing welded samples

WFR1 – StE460

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

23400 27100 29600 32300 71400 76800 160000 208000 279000 374000 403000 590000 735000 3190000 3380000 6630000 16400

Nf (cycles)

484 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 174.0 175.0 168.0 164.0 127.0 128.0 127.0 128.0 164.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

219.0 217.0 181.0 109.0 136.0 103.0 118.0 84.5 103.0 88.8 103.0 79.6 79.6 100.9 101.5 97.4 95.1 73.7 74.2 73.7 74.2 95.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90

19900 21800 66300 374000 473000 826000 839000 929000 1460000 1490000 2020000 3430000 3580000 54000 71400 83900 118000 314000 314000 353000 851000 24100

Experimental results generated under multiaxial fatigue loading 485

Code, material

Table B.104

WFR1 – StE460

sx,m (MPa)

0 0 0 0 0 0 0 0

sx,a (MPa)

167.0 168.0 127.0 125.0 127.0 123.0 81.3 81.3

0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0

sy,m (MPa) 96.9 97.4 73.7 72.5 73.7 71.3 47.2 47.2

txy,a (MPa) 0 0 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0

dy,x (°)

90 90 90 90 90 90 90 90

dxy,x (°)

25900 29200 51700 77900 98500 343000 724000 2050000

Nf (cycles)

486 Multiaxial notch fatigue

408.5 356.1 278.6 281.1 204.9 199.8 200.2 155.9 150.9 151.5 123.8 120.1 120.5 120.3 120.6 119.6 103.5 309.5 253.9 206.4 127.8 82.3. 104.2 93.8 72.4 0.0 0.0 0.0 0.0 0.0 0.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 309.5 253.9 206.4 127.8 82.3 104.2 93.8 72.4 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 198.5 199.2 160.4 151.4 151.4 120.4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5299 7390 22677 32221 86631 108600 208033 261833 448755 598986 390614 520730 582009 1012579 1053905 1901275 1856649 7459 15806 26956 364095 1147067 1612412 2077033 1381351 15031 21197 196408 202643 225925 730694

Experimental results generated under multiaxial fatigue loading 487

WFR2 – StE460

Code, material

Table B.105

WFR2 – StE460

sx,m (MPa)

0 0 0 0 0 0 0 0 177.8 152.5 126.9 103.2 80.9 91.9 71.2 0 0 0 0

sx,a (MPa)

0.0 156.1 113.7 135.1 103.5 93.9 83.6 72.6 177.8 152.5 126.9 103.2 80.9 91.9 71.2 109.2 99.5 109.2 99.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 120.2 156.1 113.7 135.1 103.5 93.9 83.6 72.6 177.8 152.5 126.9 103.2 80.9 91.9 71.2 109.2 99.5 109.2 99.8

txy,a (MPa) 0 0 0 0 0 0 0 0 177.8 152.5 126.9 103.2 80.9 91.9 71.2 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90

dxy,x (°)

816681 23339 23426 48190 125547 332442 479493 1385779 15378 40280 79890 108581 389463 572605 641432 43386 123369 178162 238997

Nf (cycles)

488 Multiaxial notch fatigue

0 0 0 0 0 155.3 155.1 133.9 114.5 103.6 93.5 82.0

0 0 0 0 0 0 313.9 263.4

69.2 69.5 69.4 80.0 69.2 155.3 155.1 133.9 114.5 103.6 93.5 82.0

WFR3 – StE460

203.5 148.5 120.5 120.2 119.8 120.1 313.9 263.4

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

69.2 69.5 69.4 80.0 69.2 155.3 155.1 133.9 114.5 103.6 93.5 82.0 0 0 0 0 0 0 0 0

0 0 0 0 0 155.3 155.1 133.9 114.5 103.6 93.5 82.0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90 90 90 90

44000 128951 553212 858330 1984854 145070 2240 2939

143319 393064 921961 961991 2149055 23238 41144 47787 23180 123313 330339 473713

Experimental results generated under multiaxial fatigue loading 489

Code, material

Table B.106

WFR3 – StE460

sx,m (MPa)

153.3 127.8 104.1 92.9 0 0 0 0 0 0 0 0 0 0 172.3 120.9 96.7 78.9 69.8 89.2 0

sx,a (MPa)

153.3 127.8 104.1 92.9 0 0 0 0 0 0 0 98.7 97.3 93.8 172.3 120.9 96.7 78.9 69.8 89.2 96.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 147.2 147.7 119.6 154.4 148.3 133.4 117.7 98.7 97.3 93.8 172.3 120.9 96.7 78.9 69.8 89.2 96.8

txy,a (MPa) 0 0 0 0 0 0 0 154.4 148.3 133.4 117.7 0 0 0 172.3 120.9 96.7 78.9 69.8 89.2 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90

dxy,x (°)

8929 64229 141889 2330957 26830 51744 104028 18354 36191 43939 155047 73391 97516 255044 2480 15300 30696 50562 65558 187052 22636

Nf (cycles)

490 Multiaxial notch fatigue

0 0 150.9 130.4 101.4 91.4 81.8 62.0 72.2 0 0 0 0 0 0 0 0 0

106.4 105.7 150.9 130.4 101.4 91.4 81.8 62.0 72.2

WFR4 – A519

114.5 108.8 100.2 86.2 79.2 114.5 108.8 108.8 86.2

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 39.8 62.1 41.4 49.8

106.4 105.7 150.9 130.4 101.4 91.4 81.8 62.0 72.2 0 0 0 0 0 0 0 0 0

0 0 150.9 130.4 101.4 91.4 81.8 62.0 72.2 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90

76660 198000 145690 560850 624330 27100 47090 78620 93690

24294 31243 7534 12275 34787 82608 278614 317467 637773

Experimental results generated under multiaxial fatigue loading 491

Code, material

Table B.107

WFR4 – A519

WFR5 – A519-A36

sx,m (MPa) 0 0 0 0 0

0 0 0 0 220.0 140.0 103.9 110.0 70.0 0 0 0 0 0

sx,a (MPa)

79.2 64.7 64.7 0 0

220.0 158.5 140.0 110.0 220.0 140.0 103.9 110.0 70.0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

sy,m (MPa)

0 0 0 0 0 0 0 0 0 110.0 85.0 70.0 70.0 110.0

45.8 37.4 37.4 69.7 59.7

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 110.0

0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

dxy,x (°)

76800 276400 395900 729900 55700 270390 1036000 1300140 Run out 132000 1605000 3303000 1989000 374000

220030 788370 1641740 466860 827560

Nf (cycles)

492 Multiaxial notch fatigue

0 130.2 130.2 108.1 108.1 68.0 87.0 87.0 87.0 87.0 40.2 0.0 87.0 87.0 87.0 87.0 87.0 87.0 87.0

0 130.2 130.2 108.1 108.1 68.0 87.0 87.0 87.0 87.0 40.2 170.0 87.0 87.0 87.0 87.0 87.0 87.0 87.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

85.0 54.5 54.5 14.7 14.7 68.0 36.3 36.3 36.3 36.3 40.2 85.0 72.6 72.6 72.6 72.6 36.3 36.3 36.3

85.0 54.5 54.5 14.7 14.7 68.0 36.3 36.3 36.3 36.3 40.2 85.0 0 0 0 0 36.3 36.3 36.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 W-A W-A W-A W-A W-B W-B W-B

919600 260200 274700 577000 748600 852900 1196100 1201400 1699800 Run out Run out 981200 111500 142600 167900 206300 352500 354200 644500

Experimental results generated under multiaxial fatigue loading 493

Code, material

Table B.108

WFR6 – BS4360 Gr. 50E

sx,m (MPa)

90 80 70 70 60 50 0 0 0 0 0 0 0 0 53.5 53.5 41 40 40

sx,a (MPa)

90 80 70 70 60 50 0 0 0 0 0 0 0 0 53.5 53.5 41 40 40

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 0 0 0 0 0 0 59 78.5 69 59 65.5 52.5 72.5 79 59 56.5 59 57.5 40

txy,a (MPa) 0 0 0 0 0 0 59 78.5 69 59 65.5 52.5 72.5 79 59 56.5 59 57.5 40

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

200000 258000 306000 392000 731000 1040000 2980000 605000 491000 1280000 455000 6750000 1800000 840000 355000 433000 1170000 1080000 1550000

Nf (cycles)

494 Multiaxial notch fatigue

50 27 105.5 40 53.5 40 80 27

230.8 195.3 284.0 301.8 303.5 3.5 3.0 3.5 6.5 0 0 0

50 27 105.5 40 53.5 40 80 27

WFR7 – Fe 52 steel

124.3 159.8 71.0 53.3 53.6 3.5 3.0 3.5 6.5 207.0 178.0 100.0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 163.0 134.0 76.0 110.5 70.0 78.0 110.0

25 80 97.5 57.5 57.5 40 57.5 80 0 0 0 0 0 0 0 0 110.5 0 0 0

25 80 97.5 57.5 57.5 40 57.5 80 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 90 90 90 90 90 90

45000 127000 274000 692000 1110000 40000 160000 627000 520000 44000 122000 274000

1030000 180000 12000 330000 195000 812000 75600 65400

Experimental results generated under multiaxial fatigue loading 495

sx,m (MPa)

13.4 13.6 175.5 118.5 127.5 105.5 127.5 128.5 105.5 105.5

97.5 0 0 0 0 0 0 0 0 0 0

120.6 122.4 175.5 118.5 127.5 105.5 127.5 128.5 105.5 105.5

97.5 0 0 0 0 0 0 0 0 0 0

Code, material

sx,a (MPa)

Table B.109

WFR7 – Fe 52 steel

WFR8 – BS4360

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

0 51.5 67.5 54.5 65 65.5 55.5 54 60 37.5 75

64.9 64.4 79.0 72.1 56.5 48.0 55.5 55.5 45.5 46.0

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0

5.9 8.4 79.0 59.0 56.5 48.0 55.5 55.5 45.5 46.0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 90 90 90 90

dxy,x (°)

453000 3560000 766000 2980000 794000 864000 685000 1080000 Run out Run out 600000

1081000 1467000 11000 95000 120000 345000 100000 148000 413000 529000

Nf (cycles)

496 Multiaxial notch fatigue

0 0 61 63.5 51 40.5 47 63 51 41.5 46 0 0 0 0 0 0 0

0 0 61 63.6 51 40.5 47 63 51 41.5 46

WFR9 – highstrength steel

97.2 121.5 158.0 89.8 76.0 69.1 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 63.8

80.5 93 58 62 50.5 40.5 45.5 61.5 52.5 38 45.5 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 90 90 90 90

1540000 915000 300000 455000 478000 887000 780000

662000 347000 460000 698000 783000 Run out 793000 404000 944000 Run out Run out

Experimental results generated under multiaxial fatigue loading 497

Code, material

Table B.110

WFR9 – high-strength steel

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 91.8 57.3 77.6 77.0 61.4 76.1 60.4 55.5 72.8 69.3 58.3 59.8 72.6 101.3 98.1 53.9 91.1 71.8 65.7

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

140000 2220000 397000 975000 295500 343000 840000 1860000 304000 1050000 2144000 385000 870000 40000 51500 1670000 32000 62000 146000

Nf (cycles)

498 Multiaxial notch fatigue

0 0 53.4 62.7 47.5 71.8 75.2 49.5 42.0 53.2 72.6 58.6 66.0 41.4 53.2 67.4 35.5 69.4 61.8 61.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

66.2 65.7 72.9 80.2 60.8 43.2 51.8 34.5 55.3 69.1 82.9 43.2 51.8 34.5 92.1 110.5 68.9 69.1 57.6 50.7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

360000 160000 363000 285000 1490000 150000 137000 433000 850000 460000 120000 760000 180000 815000 930000 125000 224000 210000 320000 660000

Experimental results generated under multiaxial fatigue loading 499

sx,m (MPa)

90 0 0 0 0 0 0

94.9 79.2 67.3 58.3 48.9 40.4 72.5 62.9 54.4 47.9 82.2 72.5

90.0 67.5 67.5 63.3 84.4 67.5 63.3

94.9 79.2 67.3 58.3 48.9 40.4 64.2 54.6 46.0 39.6 73.9 64.2

Code, material

sx,a (MPa)

Table B.111

WFR9 – highstrength steel

WFR10 – JIS SM400B

0 0 0 0 0 0 62.9 53.6 45.2 38.8 21.9 62.9

0 0 0 0 0 0 0

sy,a (MPa)

0 0 0 0 0 0 71.2 61.9 53.5 47.1 30.2 71.2

0 0 0 0 0 0 0

sy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

92.1 69.1 69.1 69.1 92.1 73.7 69.1

txy,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

92.1 0 0 0 0 0 0

txy,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 180 180 180 180 0 180

0 0 0 90 90 90 90

dxy,x (°)

13030 45510 63220 115500 317100 329900 58395 110878 166634 211233 192201 58395

101500 772000 554000 842000 700000 133000 490000

Nf (cycles)

500 Multiaxial notch fatigue

62.9 54.4 47.9 74.4 63.6 52.6 73.2 61.1 51.8 44.7

0 0 0 0 0 0 0 0 0 0

54.6 46.0 39.6 66.1 55.3 44.3 64.9 52.8 43.5 36.4

WFR11 – AlSi1MgMn

70.9 72.3 61.1 59.5 48.5 40.3 44.2 52.7 40.6 0

0 0 0 0 0 0 0 0 0 0

53.6 45.2 38.8 65.0 64.9 65.1 42.4 42.2 42.2 42.5 0 0 0 0 0 0 0 0 0 0

61.9 53.5 47.1 73.3 73.2 73.4 −50.8 −50.5 −50.5 −50.8 0 0 0 0 0 0 0 0 0 60.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

180 180 180 180 180 180 0 0 0 0

16400 147000 174000 325000 832000 1250000 1470000 1670000 5650000 48800

110878 166634 211233 97655 170769 138571 117818 215000 186923 903333

Experimental results generated under multiaxial fatigue loading 501

Code, material

Table B.112

WFR11 – AlSi1MgMn

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

0 0 0 0 0 0 0 49.9 49.9 49.9 38.7 29.7 29.7 29.7 24.9 49.9 49.9 49.9 39.7 39.7 29.7 29.7 24.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 61.7 50.5 41 49.2 40.2 36.4 29 28.9 28.9 28.9 22.4 17.2 17.2 17.2 14.4 28.9 28.9 28.9 23.0 23.0 17.2 17.2 14.4

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

dxy,x (°)

221000 384000 631000 745000 2050000 3250000 Run out 47000 83200 94600 363000 879000 1670000 2090000 Run out 49700 54500 67900 344000 377000 941000 1910000 Run out

Nf (cycles)

502 Multiaxial notch fatigue

85.8 74.0 74.0 63.8 53.3 60.6 55.0 48.4 42.4 37.4 68.2 61.2 58.3 52.8 52.8 47.7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

42.9 37.0 37.0 31.9 26.6 45.4 41.3 36.3 31.8 28.1 34.1 30.6 29.1 26.4 26.4 23.9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

45243 46813 55519 253340 681292 49271 130261 236632 321668 857696 39473 100856 204697 246941 210001 335680

Experimental results generated under multiaxial fatigue loading 503

WFR12 – 6060-T6

Code, material

Table B.113

WFR12 – 6060-T6

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

49.1 66.0 55.3 52.8 48.6 100.8 82.1 82.1 82.1 71.5 72.2 71.9 61.2 81.0 81.0 80.2 70.5 60.9 60.9 70.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa) 24.5 22.0 18.4 17.6 16.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 81.0 81.0 80.2 70.5 60.9 60.9 70.5

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

464159 43355 95826 210001 394727 32167 95012 149305 157143 197832 215443 285467 555194 64731 69301 78089 178588 569581 615020 584341

Nf (cycles)

504 Multiaxial notch fatigue

Multiaxial fatigue strength of composite materials

Matrix

Polylite FG-284 Sirpol 8231 Sirpol 8231 BP 2785 CV BP 2785 CV BP 2785 CV Epoxy Epoxy Epoxy Epoxy Polylite FG-284 Polylite FG-284

Code

CM1 CM2 CM3 CM4 CM5 CM6 CM7 CM8 CM9 CM10 CM11 CM12

Table B.114

Glass MG-252 Glass E Glass E Marglass 266 Marglass 266 Marglass 266 Glass E Glass E Glass E Graphite Glass MG-252 Glass MG-252

Fibres [0/90]n [±45] [0/90] [0/90]13 [22.5/112.5]13 [±45]13 [±35] [±55] [±70] [±45]s [0/90] [0/90]

Lay-up Amijima et al., 1991 Aboul Wafa et al., 1997 Aboul Wafa et al., 1997 Smith and Parcoe, 1989 Smith and Parcoe, 1989 Smith and Parcoe, 1989 Qi and Cheng, 2007 Qi and Cheng, 2007 Qi and Cheng, 2007 Francis et al., 1977 Kawakami et al., 1996 Fujii et al., 1994

Reference

B.8.1 Materials, specimen geometries and loading paths

B.8

164.5 – – 238 238 238 – – – – 224 224

s1,UTS (MPa) 71.5 – – 82.5 82.5 82.5 – – – – 73.2 73.2

s6,UTS (MPa)

Te-T B-T B-T Te-Te Te-Te Te-Te Te-T Te-T Te-T Te-T Te-T Te-T

Loading path

Plain, tubular Plain, tubular Plain, tubular Plain, cruciform Plain, cruciform Plain, cruciform Plain, tubular Plain, tubular Plain, tubular Fig. B.29 Plain, tubular Fig. B.30

Geometry

Experimental results generated under multiaxial fatigue loading 505

Code, material

Table B.115

sx,m (MPa)

50.4 50.3 49.9 45.3 46.0 40.7 40.7 36.3 36.3 31.8 40.8 40.8 35.8 31.8 31.7 36.2 31.5 27.1 27.1

sx,a (MPa)

50.4 50.3 49.9 45.3 46.0 40.7 40.7 36.3 36.3 31.8 40.8 40.8 35.8 31.8 31.7 36.2 31.5 27.1 27.1

B.8.2 Experimental results

CM1 – [0/90]n

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

838 997 5239 7534 16574 18452 22219 81385 118735 315837 1154 4253 4280 5517 7350 15644 37223 295651 347293

Nf (cycles)

506 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20.2 20.3 18.7 21.0 19.1 18.8 18.7 19.6 21.1 17.6 18.0 16.7 15.3 16.3 16.3 16.3 16.4 15.0 15.3

20.2 20.3 18.7 21.0 19.1 18.8 18.7 19.6 21.1 17.6 18.0 16.7 15.3 16.3 16.3 16.3 16.4 15.0 15.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3724 4544 5597 10037 11526 16202 19712 20332 21276 26208 55668 150855 284868 33504 43366 55783 71167 831503 986879

Experimental results generated under multiaxial fatigue loading 507

Code, material

Table B.116

CM1 – [0/90]n

sx,m (MPa)

24.9 25.1 24.1 23.2 23.1 23.8 23.8 25.1 22.0 18.7 19.9 24.0 23.2 22.2 19.1 19.9 20.8 21.0

sx,a (MPa)

24.9 25.1 24.1 23.2 23.1 23.8 23.8 25.1 22.0 18.7 19.9 24.0 23.2 22.2 19.1 19.9 20.8 21.0

sy,a (MPa)

sy,m (MPa) 11.1 11.2 10.7 10.3 10.3 10.6 10.6 11.2 9.8 8.3 8.9 10.7 10.3 9.9 8.5 8.9 9.3 9.4

txy,a (MPa) 11.1 11.2 10.7 10.3 10.3 10.6 10.6 11.2 9.8 8.3 8.9 10.7 10.3 9.9 8.5 8.9 9.3 9.4

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

3300 7009 7803 9261 11290 19111 21945 24583 29480 10700 16337 47586 48497 45100 65934 65934 67448 78578

Nf (cycles)

508 Multiaxial notch fatigue

22.4 19.9 21.0 20.1 19.1 13.2 9.8 12.4 11.9 11.8 11.5 9.7 12.1 9.7 10.8 10.7 8.2 9.0 10.7 9.0 8.2

22.4 19.9 21.0 20.1 19.1 13.2 9.8 12.4 11.9 11.8 11.5 9.7 12.1 9.7 10.8 10.7 8.2 9.0 10.7 9.0 8.2

10.0 8.9 9.4 9.0 8.5 13.2 9.8 12.4 11.9 11.8 11.5 9.7 12.1 9.7 10.8 10.7 8.2 9.0 10.7 9.0 8.2

10.0 8.9 9.4 9.0 8.5 13.2 9.8 12.4 11.9 11.8 11.5 9.7 12.1 9.7 10.8 10.7 8.2 9.0 10.7 9.0 8.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

174663 108651 122722 261753 890113 2104 6180 9493 16036 18187 24993 27997 36765 78255 104689 200153 241011 250134 399622 580623 912433

Experimental results generated under multiaxial fatigue loading 509

Code, material

Table B.117

CM2 – [±45]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

50.1 49.4 47.0 46.8 45.4 44.3 42.8 40.6 38.3 41.4 40.5 38.2 35.0 33.5 30.2 29.8 33.1 27.5 0 0

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 43.5 42.5

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2489 3530 3874 4339 5177 6271 6708 7254 10218 18573 21955 35060 53334 54867 95905 117412 123921 456706 2188 6156

Nf (cycles)

510 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 23.4 22.9 20.5 23.6 15.8 15.3 17.7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

40.2 38.6 35.4 32.9 33.6 33.8 39.7 31.0 30.2 29.1 26.4 22.9 46.8 45.8 41.0 47.2 31.6 30.6 35.4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8996 12124 15926 21992 31923 44500 51140 78671 91145 131061 139271 492866 1022 2317 4421 4783 13743 26841 30611

Experimental results generated under multiaxial fatigue loading 511

Code, material

Table B.118

CM2 – [±45]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

14.3 13.6 12.7 12.2 11.2 25.4 25.2 22.5 21.8 18.7 19.9 18.0 16.3 17.7 16.7 14.5 14.0 12.2 13.0

sy,a (MPa)

sy,m (MPa) 28.6 27.2 25.4 24.4 22.4 50.8 50.4 45.0 43.6 37.4 39.8 36.0 32.6 35.4 33.4 29.0 28.0 24.4 26.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90

dxy,x (°)

51157 61509 78925 96713 221029 1125 1403 2905 3598 11353 11681 15255 32757 32846 48854 65199 154978 188625 322575

Nf (cycles)

512 Multiaxial notch fatigue

58.5 57.3 56.3 50.7 51.2 48.7 45.0 43.8 42.6 39.8 38.1 35.9 60.6 58.8 55.1 54.6 54.2 52.6 50.5 48.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

29.3 28.7 28.2 25.4 25.6 24.4 22.5 21.9 21.3 19.9 19.1 18.0 30.3 29.4 27.6 27.3 27.1 26.3 25.3 24.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90

1483 2367 3390 4476 7769 13286 16704 35676 56860 87961 99235 193020 1059 1322 3344 4071 4463 6120 8393 12133

Experimental results generated under multiaxial fatigue loading 513

Code, material

Table B.119

CM2 – [±45]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

45.0 43.9 41.6 41.5 39.6 38.0 34.1 41.3 41.0 40.5 33.8 39.2 39.6 35.5 32.1 26.3 25.5 25.5 22.0

sy,a (MPa)

sy,m (MPa) 22.5 22.0 20.8 20.8 19.8 19.0 17.1 41.3 41.0 40.5 33.8 39.2 39.6 35.5 32.1 26.3 25.5 25.5 22.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 90 90 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

28878 37156 59783 62179 105189 160971 402846 1131 2107 2568 4038 6418 6877 16324 43835 54226 73755 91115 131187

Nf (cycles)

514 Multiaxial notch fatigue

20.8 43.5 41.9 41.4 38.8 38.5 36.5 36.3 34.5 33.4 32.1 32.1 31.0 28.3 26.4 26.9 23.4 23.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20.8 43.5 41.9 41.4 38.8 38.5 36.5 36.3 34.5 33.4 32.1 32.1 31.0 28.3 26.4 26.9 23.4 23.2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

140192 1124 1302 1283 2215 3151 4194 5230 7571 13577 14352 20166 20777 60353 82534 96451 200212 370888

Experimental results generated under multiaxial fatigue loading 515

Code, material

Table B.120

CM3 – [0/90]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

89.8 72.1 63.8 64.0 72.6 74.2 65.1 65.4 58.1 50.2 59.3 57.7 56.2 49.1 49.3 45.3 33.3 0 0 0

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 38.9 42.1 41.5

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1360 1774 2442 2702 3177 12144 14029 18006 30595 36312 40559 62621 59572 64507 83577 155650 181035 1301 2090 2224

Nf (cycles)

516 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 20.2 20.9 20.2 17.6 18.6 16.1 17.5 21.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

41.1 38.0 37.8 32.7 36.5 31.9 33.5 33.0 33.2 30.8 31.9 31.5 40.4 41.8 40.4 35.2 37.2 32.2 35.0 42.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2267 7755 10810 11502 16629 21932 30573 44022 63130 88717 103340 194140 3692 5848 7591 10627 12230 24167 29637 49727

Experimental results generated under multiaxial fatigue loading 517

Code, material

Table B.121

CM3 – [0/90]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

17.1 17.3 15.5 14.2 12.6 23.1 21.5 22.7 21.7 21.0 19.0 18.9 20.0 19.6 18.4 18.5 16.7 17.2 15.3 15.8

sy,a (MPa)

sy,m (MPa) 34.2 34.6 31.0 28.4 25.2 46.2 43.0 45.4 43.4 42.0 38.0 37.8 40.0 39.2 36.8 37.0 33.4 34.4 30.6 31.6

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90

dxy,x (°)

52418 73385 98791 164201 195470 1116 1662 2193 2368 2586 8344 14229 14718 19573 22105 54883 79799 122801 300400 334694

Nf (cycles)

518 Multiaxial notch fatigue

60.5 57.6 43.9 52.2 50.0 47.9 44.6 42.5 42.4 35.7 35.3 35.5 54.7 54.1 50.5 50.3 49.4 47.1 44.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

30.3 28.8 22.0 26.1 25.0 24.0 22.3 21.3 21.2 17.9 17.7 17.8 27.4 27.1 25.3 25.2 24.7 23.6 22.4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90

1486 3639 6428 11037 11172 16376 42513 62739 85723 119204 157252 451172 2912 4380 8712 9234 11462 16984 38416

Experimental results generated under multiaxial fatigue loading 519

Code, material

Table B.122

CM3 – [0/90]

44.9 46.1 43.4 36.4 36.0 37.6 41.2 41.2 34.6 30.4 30.1 31.4 27.0 31.2 27.6 24.4 24.9 41.4 36.9 35.6 33.4

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa)

sy,a (MPa)

sy,m (MPa) 22.5 23.1 21.7 18.2 18.0 37.6 41.2 41.2 34.6 30.4 30.1 31.4 27.0 31.2 27.6 24.4 24.9 41.4 36.9 35.6 33.4

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90

dxy,x (°)

39628 42917 49126 231439 235859 1436 2069 3714 9159 12754 22680 48402 56615 72596 92426 121153 227716 1128 2436 4886 8121

Nf (cycles)

520 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

33.7 29.9 29.5 29.4 28.5 27.6 26.3 25.9 25.3

170.6 141.4 130.5 120.2 101.5 89.1 80.4 70.0 61.1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

33.7 29.9 29.5 29.4 28.5 27.6 26.3 25.9 25.3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

90 90 90 90 90 90 90 90 90

CM4 – [0/90]13

28 264 336 1592 3769 25314 45179 383676 797665

12617 68406 80123 93593 101772 121153 196264 249295 303665

Experimental results generated under multiaxial fatigue loading 521

Code, material

Table B.123

CM4 – [0/90]13

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

146.3 129.3 113.3 91.5 85.3 77.5 65.7 54.4 112.8 94.9 81.0 73.8 108.2 80.0 67.2 61.2 101.6 80.5 63.8 55.0 48.0

146.3 129.3 113.3 91.5 85.3 77.5 65.7 54.4 56.4 47.5 40.5 36.9 54.1 40.0 33.6 30.6 101.6 80.5 63.8 55.0 48.0

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 180 180 180 180 180 180 180 180 180

dy,x (°)

dxy,x (°)

19 157 385 3136 3779 24195 147624 856055 5056 34445 115122 364648 1940 28665 165290 654505 341 2906 43550 116430 452007

Nf (cycles)

522 Multiaxial notch fatigue

93.7 81.1 66.5 54.3 80.9 65.0 51.0 41.2 65.1 53.5 41.2 34.8 53.8 41.1 33.2 27.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

46.9 40.6 33.3 27.2 0 0 0 0 32.6 26.8 20.6 17.4 53.8 41.1 33.2 27.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 180 180 180 180 180 180 180 180

786 4091 26885 324757 813 2731 85797 816150 346 2501 5186 861468 423 3864 54276 450411

Experimental results generated under multiaxial fatigue loading 523

CM5 – [22.5/112.5]13

Code, material

Table B.124

CM6 – [±45]13

59.1 59.4 59.6 49.4 50.1 49.8 49.8 49.9 49.8 49.9 50.2 35.2 35.8 30.4 52.7 40.9 32.8 25.9 40.6 26.9 20.0 20.0 15.4

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26.4 20.5 16.4 13.0 40.6 26.9 20.0 20.0 15.4

sy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sy,m (MPa)

txy,a (MPa)

txy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 180 180 180 180 180 180 180 180 180

dy,x (°)

dxy,x (°)

256 800 1150 619 733 1793 1985 2525 2796 3221 4435 20159 87333 995965 185 1548 18518 306844 198 5836 61833 80451 922693

Nf (cycles)

524 Multiaxial notch fatigue

56.0 56.0 53.0 53.0 53.0 50.5 50.5 50.5 47.5 47.5 47.5 46.0 46.0 46.0 43.1 43.1

56.0 56.0 53.0 53.0 53.0 50.5 50.5 50.5 47.5 47.5 47.5 46.0 46.0 46.0 43.1 43.1

28.0 28.0 26.5 26.5 26.5 25.3 25.3 25.3 23.8 23.8 23.8 23.0 23.0 23.0 21.5 21.5

28.0 28.0 26.5 26.5 26.5 25.3 25.3 25.3 23.8 23.8 23.8 23.0 23.0 23.0 21.5 21.5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

299 415 806 971 1038 864 1348 1914 4639 5297 6337 14203 16470 22852 35507 49072

Experimental results generated under multiaxial fatigue loading 525

CM7 – [±35]

Code, material

Table B.125

CM7 – [±35]

sx,m (MPa)

43.1 41.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

43.1 41.0 62.0 62.0 62.0 56.0 60.0 60.0 60.0 56.0 56.0 50.0 53.0 53.0 50.0 50.0 44.0 44.0 44.0 40.0

sy,a (MPa)

sy,m (MPa) 21.5 20.5 31.0 31.0 31.0 28.0 30.0 30.0 30.0 28.0 28.0 25.0 26.5 26.5 25.0 25.0 22.0 22.0 22.0 20.0

txy,a (MPa) 21.5 20.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

68617 Run out 149 461 536 613 1392 1496 2532 2509 4899 7190 7826 8748 14693 16046 4595 49819 63384 Run out

Nf (cycles)

526 Multiaxial notch fatigue

31.4 31.5 28.0 28.0 25.9 27.9 25.9 26.0 24.0 24.0 24.0 21.4 21.4 21.4 18.9 18.9 18.9 17.9 40.0

31.4 31.5 28.0 28.0 25.9 27.9 25.9 26.0 24.0 24.0 24.0 21.4 21.4 21.4 18.9 18.9 18.9 17.9 0

15.7 15.8 14.0 14.0 13.0 13.9 13.0 13.0 12.0 12.0 12.0 10.7 10.7 10.7 9.5 9.5 9.5 9.0 20.0

15.7 15.8 14.0 14.0 13.0 13.9 13.0 13.0 12.0 12.0 12.0 10.7 10.7 10.7 9.5 9.5 9.5 9.0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

111 159 180 210 302 482 903 1288 2403 3258 3454 4906 7956 10092 26645 37695 59945 Run out 298

Experimental results generated under multiaxial fatigue loading 527

CM8 – [±55]

Code, material

Table B.126

CM8 – [±55]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

sx,a (MPa)

40.0 40.0 37.0 37.0 34.0 34.0 37.0 37.0 31.0 31.0 31.0 28.0 28.0 28.0 25.0 37.0 34.1 37.0 37.1 34.1 34.1 31.0

sy,a (MPa)

sy,m (MPa) 20.0 20.0 18.5 18.5 17.0 17.0 18.5 18.5 15.5 15.5 15.5 14.0 14.0 14.0 12.5 37.0 34.1 37.0 37.1 34.1 34.1 31.0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

527 740 1357 2445 2512 2673 2820 4816 14171 15562 19277 39003 43603 49623 Run out 197 160 391 452 974 1236 1150

Nf (cycles)

528 Multiaxial notch fatigue

31.0 31.0 28.0 28.0 28.0 26.0 26.0 29.5 31.0 31.0 28.0 28.0 28.0 25.0 25.0 25.0 23.6 22.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

31.0 31.0 28.0 28.0 28.0 26.0 26.0 58.9 62.0 62.0 56.0 56.0 56.0 50.0 50.0 50.0 47.1 44.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2034 5284 26649 31134 49653 86036 Run out 218 650 964 2415 2850 3544 20035 35445 40772 80012 Run out

Experimental results generated under multiaxial fatigue loading 529

Code, material

Table B.127

CM9 – [±70]

sx,m (MPa)

22.0 22.0 20.3 20.3 20.3 19.0 19.0 17.0 19.0 17.0 17.0 15.5 15.5 15.5 14.0 14.0 14.0 12.6 0 0 0

sx,a (MPa)

22.0 22.0 20.3 20.3 20.3 19.0 19.0 17.0 19.0 17.0 17.0 15.5 15.5 15.5 14.0 14.0 14.0 12.6 31.0 31.0 31.0

sy,a (MPa)

sy,m (MPa) 11.0 11.0 10.1 10.1 10.1 9.5 9.5 8.5 9.5 8.5 8.5 7.8 7.8 7.8 7.0 7.0 7.0 6.3 15.5 15.5 15.5

txy,a (MPa) 11.0 11.0 10.1 10.1 10.1 9.5 9.5 8.5 9.5 8.5 8.5 7.8 7.8 7.8 7.0 7.0 7.0 6.3 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

169 261 453 562 683 859 1244 1654 2025 3003 4199 4470 8504 9709 46905 58576 68728 Run out 107 149 225

Nf (cycles)

530 Multiaxial notch fatigue

0 0 0 0 0 0 0 0 0 0 0 0 0

115.2 105.7 102.4 94.5 83.8 72.3

26.5 26.5 28.0 26.5 28.1 28.0 25.1 25.1 25.1 23.1 23.1 23.1 21.7

CM10 – [±45]

94.2 86.5 83.8 77.3 68.6 59.1

0 0 0 0 0 0

13.3 13.3 14.0 13.3 14.1 14.0 12.6 12.6 12.6 11.6 11.6 11.6 10.9 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 25 49 197 1517

595 674 720 865 954 1419 3872 5606 7164 42458 50075 64280 Run out

Experimental results generated under multiaxial fatigue loading 531

Code, material

Table B.128

CM10 – [±45]

sx,m (MPa)

65.9 0 0 0 0 0 0 0 0 0 0 0 94.0 89.9 86.0 77.8 73.8 71.4 66.7 52.4 86.5

sx,a (MPa)

54.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 76.9 73.6 70.4 63.6 60.3 58.5 54.6 42.9 70.7

sy,a (MPa)

sy,m (MPa) 0 106.8 82.3 95.4 88.0 72.2 69.0 82.1 82.0 63.7 55.7 51.5 38.5 36.8 35.2 31.8 30.2 29.2 27.3 21.4 70.7

txy,a (MPa) 0 130.5 100.6 116.5 107.5 88.3 84.4 100.4 100.2 77.8 68.0 63.0 47.0 45.0 43.0 38.9 36.9 35.7 33.4 26.2 86.5

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

4663 1 1 3 4 463 514 823 3049 1777 3056 91540 6 23 60 153 463 2141 6640 39637 1

Nf (cycles)

532 Multiaxial notch fatigue

76.9 70.7 65.0 61.4 53.5 49.1 43.5 49.0 50.8 44.5 36.3 38.3 38.2 42.0 39.9

84.0 83.8 83.9 83.8

63.0 57.8 53.2 50.3 43.8 40.2 35.6 40.1 41.6 36.4 29.7 31.3 31.2 34.4 32.6

CM11 – [0/90]

84.0 83.8 83.9 83.8

0 0 0 0

63.0 57.8 53.2 50.3 43.8 40.2 35.6 80.2 83.2 72.8 59.4 62.6 62.5 68.8 65.3 0 0 0 0

76.9 70.7 65.0 61.4 53.5 49.1 43.5 98.0 101.6 89.0 72.6 76.6 76.3 84.0 79.8 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

585 612 832 1259

2 37 380 1056 3462 35203 57249 50 91 830 840 1055 1341 2800 6712

Experimental results generated under multiaxial fatigue loading 533

Code, material

Table B.129

CM11 – [0/90]

sx,m (MPa)

78.8 78.9 78.6 74.2 74.0 73.9 73.9 68.1 62.3 68.7 63.0 62.6 57.5 62.9 57.7 57.7 52.8 53.2 52.7 53.3 48.3 48.3

sx,a (MPa)

78.8 78.9 78.6 74.2 74.0 73.9 73.9 68.1 62.3 68.7 63.0 62.6 57.5 62.9 57.7 57.7 52.8 53.2 52.7 53.3 48.3 48.3

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1430 1726 2321 2985 3154 3797 4750 3427 3362 4992 5350 6619 6333 10829 12230 15008 11733 15175 32019 52383 30380 42453

Nf (cycles)

534 Multiaxial notch fatigue

48.4 43.6 34.9 43.8 38.5 44.1 44.0 38.9 39.1 39.4 38.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0

48.4 43.6 34.9 43.8 38.5 44.1 44.0 38.9 39.1 39.4 38.9 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 28.2 28.3 28.3 28.2 28.2 25.9 26.0

0 0 0 0 0 0 0 0 0 0 0 28.2 28.3 28.3 28.2 28.2 25.9 26.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59490 53852 89083 123459 116493 184363 404366 408864 566635 636426 960961 283 460 997 1268 1659 1077 2111

Experimental results generated under multiaxial fatigue loading 535

Code, material

Table B.130

CM11 – [0/90]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 77.4 77.3 68.7

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 77.4 77.3 68.7

sy,a (MPa)

sy,m (MPa) 25.9 25.8 25.9 23.7 23.6 22.9 23.5 23.6 23.6 25.9 21.5 21.4 21.5 21.3 21.4 19.8 19.4 19.3 11.1 11.0 9.8

txy,a (MPa) 25.9 25.8 25.9 23.7 23.6 22.9 23.5 23.6 23.6 25.9 21.5 21.4 21.5 21.3 21.4 19.8 19.4 19.3 11.1 11.0 9.8

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2614 3253 5195 4071 7614 11284 15747 26485 31887 56735 12996 30420 56735 66072 155515 204245 319574 351137 890 1253 1818

Nf (cycles)

536 Multiaxial notch fatigue

68.5 68.4 59.7 59.8 59.7 50.5 50.6 50.6 50.5 43.1 43.1 43.2 42.9 42.8 33.7 33.8 33.8 59.1 58.9

68.5 68.4 59.7 59.8 59.7 50.5 50.6 50.6 50.5 43.1 43.1 43.2 42.9 42.8 33.7 33.8 33.8 59.1 58.9

9.8 9.8 8.5 8.5 8.5 7.2 7.2 7.2 7.2 6.2 6.2 6.2 6.1 6.1 4.8 4.8 4.8 19.7 19.6

9.8 9.8 8.5 8.5 8.5 7.2 7.2 7.2 7.2 6.2 6.2 6.2 6.1 6.1 4.8 4.8 4.8 19.7 19.6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2367 2728 4243 5388 7355 6655 11729 20501 56359 10642 35144 113822 149857 578183 281577 427121 1005014 750 1132

Experimental results generated under multiaxial fatigue loading 537

Code, material

Table B.131

CM11 – [0/90]

sx,m (MPa)

59.1 59.0 51.1 51.2 51.0 43.6 43.6 43.4 43.8 36.7 36.7 36.9 36.7 36.7 36.7 30.0 29.9 29.7 29.7 29.4 29.4

sx,a (MPa)

59.1 59.0 51.1 51.2 51.0 43.6 43.6 43.4 43.8 36.7 36.7 36.9 36.7 36.7 36.7 30.0 29.9 29.7 29.7 29.4 29.4

sy,a (MPa)

sy,m (MPa) 19.7 19.7 17.0 17.1 17.0 14.5 14.5 14.5 14.6 12.2 12.2 12.3 12.2 12.2 12.2 10.0 10.0 9.9 9.9 29.5 29.5

txy,a (MPa) 19.7 19.7 17.0 17.1 17.0 14.5 14.5 14.5 14.6 12.2 12.2 12.3 12.2 12.2 12.2 10.0 10.0 9.9 9.9 29.5 29.5

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

1271 2244 2159 2829 3865 5038 8054 11756 19374 9879 36858 51901 61419 106378 441987 101502 468363 574501 763418 239 505

Nf (cycles)

538 Multiaxial notch fatigue

29.4 26.7 24.2 24.3 24.2 24.1 21.9 21.8 19.6 19.5 19.4 19.5 17.0 17.1 17.0 14.8 59.1 59.3 59.3

29.4 26.7 24.2 24.3 24.2 24.1 21.9 21.8 19.6 19.5 19.4 19.5 17.0 17.1 17.0 14.8 59.1 59.3 59.3

29.4 26.7 24.2 24.3 24.2 24.1 21.9 21.8 19.6 19.5 19.4 19.5 17.0 17.1 17.0 14.8 19.7 19.8 19.8

29.4 26.7 24.2 24.3 24.2 24.1 21.9 21.8 19.6 19.5 19.4 19.5 17.0 17.1 17.0 14.8 19.7 19.8 19.8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

986 3334 3119 4427 5929 8799 8370 15823 16542 27441 33711 55767 40956 553037 827704 961816 759 1140 1228

Experimental results generated under multiaxial fatigue loading 539

Code, material

Table B.132

CM11 – [0/90]

sx,m (MPa)

59.1 51.2 51.1 51.1 43.8 43.5 36.6 36.8 36.7 36.4 36.4 36.5 36.5 29.6 29.6 29.7 29.6 43.6 43.6 43.8 43.7

sx,a (MPa)

59.1 51.2 51.1 51.1 43.8 43.5 36.6 36.8 36.7 36.4 36.4 36.5 36.5 29.6 29.6 29.7 29.6 43.6 43.6 43.8 43.7

sy,a (MPa)

sy,m (MPa) 19.7 17.1 17.0 17.0 14.6 14.5 12.2 12.3 12.2 12.1 12.1 12.2 12.2 9.9 9.9 9.9 9.9 14.5 14.5 14.6 14.6

txy,a (MPa) 19.7 17.1 17.0 17.0 14.6 14.5 12.2 12.3 12.2 12.1 12.1 12.2 12.2 9.9 9.9 9.9 9.9 14.5 14.5 14.6 14.6

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2195 2153 2875 3892 5182 19641 10370 37715 37405 53501 58750 103872 425252 100774 461865 563117 735481 1649 2471 2582 9522

Nf (cycles)

540 Multiaxial notch fatigue

43.8 43.6 44.1 35.4 33.9 33.6 42.6 42.5 42.7 33.0 33.2 32.7 32.7 34.5

33.6 30.3 30.2 30.3 30.2

43.8 43.6 44.1 35.4 33.9 33.6 42.6 42.5 42.7 33.0 33.2 32.7 32.7 34.5

CM12 – [0/90]

33.6 30.3 30.2 30.3 30.2

0 0 0 0 0

14.6 14.5 14.7 11.8 11.3 11.2 14.2 14.2 14.2 11.0 11.1 10.9 10.9 11.5 0 0 0 0 0

14.6 14.5 14.7 11.8 11.3 11.2 14.2 14.2 14.2 11.0 11.1 10.9 10.9 11.5 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

710 874 1520 1961 2350

14350 15458 49124 17788 138688 338386 7453 15758 57155 34917 61565 78441 206728 280615

Experimental results generated under multiaxial fatigue loading 541

Code, material

Table B.133

CM12 – [0/90]

sx,m (MPa)

30.3 30.1 30.2 30.2 27.0 27.1 27.0 27.0 27.0 27.0 24.3 24.1 24.2 21.2 21.2 21.2 21.2 21.2 21.1 21.3 18.1

sx,a (MPa)

30.3 30.1 30.2 30.2 27.0 27.1 27.0 27.0 27.0 27.0 24.3 24.1 24.2 21.2 21.2 21.2 21.2 21.2 21.1 21.3 18.1

sy,a (MPa)

sy,m (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,a (MPa) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

2526 2875 8058 17533 2073 2965 4147 4749 5351 11740 8379 9060 10549 6142 17176 22212 26510 41338 60230 80659 72180

Nf (cycles)

542 Multiaxial notch fatigue

18.1 18.1 18.2 18.1 18.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

18.1 18.1 18.2 18.1 18.2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 26.8 26.7 26.7 26.9 26.8 26.7 24.2 24.3 24.2 24.2 21.7 21.7 21.6 21.7

0 0 0 0 0 26.8 26.7 26.7 26.9 26.8 26.7 24.2 24.3 24.2 24.2 21.7 21.7 21.6 21.7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

79180 175151 197749 288125 Run out 417 511 668 968 1926 5771 2848 3168 3509 4542 5083 6251 11648 14861

Experimental results generated under multiaxial fatigue loading 543

Code, material

Table B.134

CM12 – [0/90]

sx,m (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 30.2 30.2 30.2 30.1 30.2 24.6 24.5 24.5

sx,a (MPa)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 30.2 30.2 30.2 30.1 30.2 24.6 24.5 24.5

sy,a (MPa)

sy,m (MPa) 21.6 21.7 21.6 19.2 19.2 19.2 19.2 19.2 19.1 17.0 17.0 17.0 16.8 16.8 4.3 4.3 4.3 4.3 4.3 3.5 3.5 3.5

txy,a (MPa) 21.6 21.7 21.6 19.2 19.2 19.2 19.2 19.2 19.1 17.0 17.0 17.0 16.8 16.8 4.3 4.3 4.3 4.3 4.3 3.5 3.5 3.5

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

18652 37720 93038 21482 29930 35546 65157 66504 151448 79145 140400 281615 536683 Run out 176 444 696 886 1592 1665 6724 9002

Nf (cycles)

544 Multiaxial notch fatigue

19.8 19.8 19.7 19.8 19.7 19.7 14.6 29.0 29.1 29.1 29.1 29.1 29.2 29.0 29.2 24.3 24.3 24.3

19.8 19.8 19.7 19.8 19.7 19.7 14.6 29.0 29.1 29.1 29.1 29.1 29.2 29.0 29.2 24.3 24.3 24.3

2.8 2.8 2.8 2.8 2.8 2.8 2.1 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 8.1 8.1 8.1

2.8 2.8 2.8 2.8 2.8 2.8 2.1 9.7 9.7 9.7 9.7 9.7 9.7 9.7 9.7 8.1 8.1 8.1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19791 21739 37032 41942 63341 144464 Run out 222 419 442 864 1006 1196 1231 1875 3911 4126 6550

Experimental results generated under multiaxial fatigue loading 545

Code, material

Table B.135

CM12 – [0/90]

sx,m (MPa)

24.4 24.2 19.7 19.7 19.7 19.6 19.7 19.7 19.7 14.7 14.7 14.7 14.6 14.7 14.7 24.0 18.1 18.1 18.1

sx,a (MPa)

24.4 24.2 19.7 19.7 19.7 19.6 19.7 19.7 19.7 14.7 14.7 14.7 14.6 14.7 14.7 24.0 18.1 18.1 18.1

sy,a (MPa)

sy,m (MPa) 8.1 8.1 6.6 6.6 6.6 6.5 6.6 6.6 6.6 4.9 4.9 4.9 4.9 4.9 4.9 24.0 18.1 18.1 18.1

txy,a (MPa) 8.1 8.1 6.6 6.6 6.6 6.5 6.6 6.6 6.6 4.9 4.9 4.9 4.9 4.9 4.9 24.0 18.1 18.1 18.1

txy,m (MPa)

dy,x (°) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

dxy,x (°)

6741 7455 9677 10812 18217 23405 31655 43523 51507 318510 325788 388734 552326 646970 690923 230 1044 1799 2464

Nf (cycles)

546 Multiaxial notch fatigue

18.1 18.1 16.0 16.0 16.0 13.8 13.9 13.8 13.7 13.9 13.8 13.8 11.9 11.9 11.9 11.9 10.0

18.1 18.1 16.0 16.0 16.0 13.8 13.9 13.8 13.7 13.9 13.8 13.8 11.9 11.9 11.9 11.9 10.0

18.1 18.1 16.0 16.0 16.0 13.8 13.9 13.8 13.7 13.9 13.8 13.8 11.9 11.9 11.9 11.9 10.0

18.1 18.1 16.0 16.0 16.0 13.8 13.9 13.8 13.7 13.9 13.8 13.8 11.9 11.9 11.9 11.9 10.0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3486 5984 2952 6222 13115 7672 13749 14533 24138 28742 30506 49637 25884 77356 130072 632515 Run out

Experimental results generated under multiaxial fatigue loading 547

548

Multiaxial notch fatigue

B.9

Geometries of the notched/welded samples 55° rn 0.508

Code N-HSF1 N-HSF2 N-HSF3 N-HSF4 N-HSF5 N-HSF6 N-HSF7

7.62

rn (mm)

Material 0.4% C steel (Normalised) 3% Ni steel 3/3.5% Ni steel Cr-Va steel 3.5% NiCr steel (n. impact) 3.5% NiCr steel (l. impact) NiCrMo steel (75–80 tons)

R25.4

B.3 f 1.27 R0.32 Enlarged section

f 8.9 22.23 12.7

63.5 9.9

R38.1 127

B.4 22.23 64 R12.7 12.7

9.9 R0.84 127

B.5

0.005 0.005 0.010 0.011 0.022 0.022 0.031

Kt,b

Kt,t

18.0 18.0 13.3 12.1 8.7 8.7 7.5

8.6 8.6 6.7 6.2 4.7 4.7 4.2

Experimental results generated under multiaxial fatigue loading f 12.48

f 10.8

3.18 22.23

Enlarged section

R0.19 63.5 7.62 R38.1 127

B.6

R75

R1

f12

f20

30

B.7

(360) 120

f63

40

f50

100

f63

f40 R25 R5

B.8

100

549

550

Multiaxial notch fatigue 214 63 f20

f13 f20

R20

B.9

Enlarged views of the notches 35° 35°

R0.4

R0.2

USC18

f12.7

f7.62 52

B.10

Enlarged view of the notch tip

f20

R0.5

f12 90° Enlarged view

USC22 f20

f28 200

B.11

USC21

R0.5

USC19

Experimental results generated under multiaxial fatigue loading

Fa z

D y

sx,a =

Facosz An

txy,a =

Fasinz An

60° Code USC23 USC24 USC25 USC26 USC27 USC28

R0.074 D

5 x 2D

Reference section, An

D (mm) 5 3 5 3 5 3

Fa

B.12

Enlarged view of the notch tip

USC30 – rn = 0.2 mm USC31 – rn = 1.25 mm

rn

f8

f5 60° USC32

100

f5

f8

R4

B.13 Enlarged view of the notch tip rn 60°

f12

f8

40

B.14

USC34 – rn = 3.4 mm USC35 – rn = 0.8 mm USC37 – rn = 3.4 mm USC38 – rn = 0.8 mm

z (°) 90 90 60 60 45 45

551

552

Multiaxial notch fatigue 1.5

Enlarged view of the notch tip R5

120°

f26

f32

85

115

B.15

R1.4

f35

f35

f27

R120

12

B.16

185 25

85

f26

f25

R5

B.17

f39

Experimental results generated under multiaxial fatigue loading 25

Z9

f68.9

250

f88.9

240

B.18

25 Z10

250

f84.9

f68.9

f88.9

60 240

B.19

f47.6

114

7.95

254 368

B.20

553

554

Multiaxial notch fatigue Z8

f50.8

76.3

9.525

114

f15.8 9.525

230 355

B.21

Z11 7

3.2

f120

48.6 60 12

12 352 400

B.22

200 Z6

200

90

100 100

B.23

Experimental results generated under multiaxial fatigue loading 50

10

10 6 25 200

180 2040

B.24 10

8

150

120

150

B.25 200 R80

100

Z6 12

12

100

200 Z6

B.26

555

556

Multiaxial notch fatigue 25 R17 10 16

Z10

f88.9

f68.9

250

Enlarged view of the weld toe

240

B.27

70 f36

f20

f26 Z3 175

B.28

f4.8

254

B.29

100

f5 f55 f50

R30

1.5 300

B.30

Experimental results generated under multiaxial fatigue loading

557

B.10 References Aboul Wafa, M. N., Hamdy, A. H., El-Midany, A. A. (1997) Combined bending and torsional fatigue of woven roving (GRP). Transactions of the ASME, Journal of Engineering Materials and Technology 119, 181–185. DOI: 10.1115/1.2805991. Achtelik, H., Jakubowska, I., Macha, E. (1983) Actual and estimated directions of fatigue fracture plane in ZI250 grey cast iron under combined alternating bending and torsion. Studia Geotechnica et Mechanica V 2, 9–30. Akrache, R., Lu, J. (1999) Three-dimensional calculations of high cycle fatigue life under out-of-phase multiaxial loading. Fatigue and Fracture of Engineering Materials and Structures 22, 527–534. DOI: 10.1046/j.1460-2695.1999.00172.x. Altenbach, H., Zolochevsky, A. (1996) A generalised fatigue limit criterion and a unified theory of low-cycle fatigue damage. Fatigue and Fracture of Engineering Materials and Structures 19, 1207–1219. DOI: 10.1111/j.1460-2695.1996. tb00944.x. Amijima, A., Fujii, T., Hamaguchi, M. (1991) Static and fatigue tests of a woven glass fabric composite under biaxial tension–torsion loading. Composites 22(4), 281–289. Amstutz, H., Storzel, K., Seeger, T. (2001) Fatigue crack growth of a welded tube– flange connection under bending and torsional loading. Fatigue and Fracture of Engineering Materials and Structures 24, 357–368. DOI: 10.1046/j.1460-2695.2001. 00408.x. Archer, R. (1987) Fatigue of a welded steel attachment under combined direct stress and shear stress. In: Proceedings of Fatigue of Welded Constructions, edited by S. J. Maddox, Brighton, UK, 63–72. Atzori, B., Berto, F., Lazzarin, P., Quaresimin, M. (2006) Multi-axial fatigue behaviour of a severely notched carbon steel. International Journal of Fatigue 28, 485–493. DOI: 10.1016/j.ijfatigue.2005.05.010. Bäckström, M., Siljander, A., Kuitunen, R., Ilvonen, R. (1997) Multiaxial fatigue experiments of square hollow section tube-to-plate welded joints. In: Welded High-Strength Steel Structures, edited by A. F. Bloom, EMAS, London, 163–177. Bernasconi, A., Filippini, M., Foletti, S., Vaudo, D. (2006) Multiaxial fatigue behaviour of a railway wheel steel under non-proportional loading. International Journal of Fatigue 28, 663–672. DOI: 10.1016/j.ijfatigue.2005.07.045. Bernasconi, A., Foletti, M., Papadopoulos, I. V. (2008) A study of combined torsion and axial load fatigue limit tests with stresses of different frequencies. International Journal of Fatigue 30, 1430–1440. DOI: 10.1016/j.ijfatigue.2007.10.003. Chaudonneret, M. (1993) A simple and efficient multiaxial fatigue damage model for engineering applications of macro-crack initiation. Transactions of the ASME, Journal of Engineering Materials and Technology 115, 373–379. DOI: 10.1115/ 1.2904232. Chen, X., Gao, Q., Sun, X.-F. (1996) Low-cycle fatigue under non-proportional loading. Fatigue and Fracture of Engineering Materials and Structures 19, 839–854. DOI: 10.1111/j.1460-2695.1996.tb01020.x. Chen, X., Xu, S., Huang, D. (1999) A critical plane-strain energy density criterion for multiaxial low-cycle fatigue life under non-proportional loading. Fatigue and Fracture of Engineering Materials and Structures 22, 679–686. DOI: 10.1046/j.14602695.1999.t01-1-00199.x.

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Chen, X., An, K., Kim, K. S. (2004) Low-cycle fatigue of 1Cr–18Ni–9Ti stainless steel and related weld metal under axial, torsional and 90° out-of-phase-loading. Fatigue and Fracture of Engineering Materials and Structures 27, 439–448. DOI: 10.1111/ j.1460-2695.2004.00740.x. Cheng, X., Ohkawa, I., Misumi, M. (2001) Mean stress effect on torsional fatigue behaviour of a medium carbon steel. In: Proceedings of 6th International Conference on Biaxial and Multiaxial Fatigue and Fracture, edited by M. De Freitas, Lisbon, Portugal, 563–570. Cheng, X., Ohkawa, I., Misumi, M. (2004) Mean stress effect on cyclic deformation and short crack growth behaviour in torsional fatigue of steel. In: Proceedings of 7th International Conference on Biaxial and Multiaxial Fatigue and Fracture, edited by C. M. Sonsino, Berlin, 259–264. Costa, J. D. M., Abreu, L. M. P., Pinho, A. C. M., Ferreira, J. A. M. (2005) Fatigue behaviour of tubular AlMgSi welded specimens subjected to bending-torsion loading. Fatigue and Fracture of Engineering Materials and Structures 28, 399–407. DOI: 10.1111/j.1460-2695.2005.00875.x. Delahay, T., Palin-Luc, T. (2005) Estimation of the fatigue strength distribution in high-cycle multiaxial fatigue taking into account the stress–strain gradient effect. International Journal of Fatigue 28, 474–484. DOI: 10.1016/j.ijfatigue.2005.06.048. Dhale, T., Olsson, K.-E., Jonsson, B. (1997) High strength welded box beams subjected to torsion and bending-test results and proposed design criteria for torsion/bending interaction. In: Welded High-Strength Steel Structures, edited by A. F. Bloom, EMAS, London, 143–161. Ellyin, F., Xia, Z. (1993) A general fatigue theory and its application to out-of-phase cyclic loading. Transactions of the ASME, Journal of Engineering Materials and Technology 115, 411–416. DOI: 10.1115/1.2904239. Ellyin, F., Golos, K., Xia, Z. (1991) In-phase and out-of-phase multiaxial fatigue. Transactions of the ASME, Journal of Engineering Materials and Technology 113, 112–118. DOI: 10.1115/1.2903365. Findley, W. N., Coleman, J. J., Hanley, B. C. (1956) Theory for combined bending and torsion fatigue with data for SAE 4340 steel. In: Proceedings of International Conference on Fatigue of Metals, Institution of Mechanical Engineers, London, 150–157. Fogué, M., Bahuaud, J. (1985) Fatigue multiaxiale à durée de vie illimitée. In: Proceedings of Comptes Rendus 7ème Congrès Français de Mécanique, Bordeaux, France, 30–31. Francis, P. H., Walrath, D. E., Sims, D. F., Weed, D. N. (1977) Biaxial fatigue loading of notched composites. Journal of Composite Materials 11, 488–501. DOI: 10.1177/002199837701100410. Frith, P. H. (1956) Fatigue of wrought high-tensile alloy steel. Proceedings of the Institution of Mechanical Engineers, 462–499. Froeschl, J., Gerstmayr, G., Eichlseder, W., Leitner, H. (2007) Multiaxial fatigue of qt-steels: new fatigue strength criterion for anisotropic material behaviour. In: Proceedings of 8th International Conference on Multiaxial Fatigue and Fracture, edited by U. S. Fernando, Sheffield, UK, paper S3-1. Froustey, C. (1986) Fatigue multiaxiale en endurance de l’acier 30NCD16. PhD Thesis, Ecole Nationale Supérieure d’Arts et Métiers, Bordeaux, France. Froustey, C., Lasserre, S. (1989) Multiaxial fatigue endurance of 30NCD16 steel. International Journal of Fatigue 11, 169–175. DOI: 10.1016/0142-1123 (89)90436-2.

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559

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Index

Aluminium, 35, 63, 83, 143 Aluminium weldments, 153, 157, 163, 166, 169, 177, 182 Anti-plane, 26, 80, 28, 164 Area Method (AM), 64, 66–68, 72, 83, 79, 82 Basquin’s relationship, 221, 230 Biaxiality ratio, 241, 245, 247 Blunt notch, 55, 57, 58, 62, 66, 137, 202, 205 Butt weld, 173, 178, 181 Cast iron, 63, 196 Corner, 28 Crack branching, 205 Crack initiation, 10, 59, 76, 88–89, 100, 142, 160, 176, 185, 190–191, 196, 206 Crack growth, 51–52, 89, 105 Crack path, 190, 194, 195, 205 Crack propagation, 51–52, 88–89, 105, 193, 195, 197, 203, 205 Critical plane, 9–10, 18–24, 76, 88–90, 99–101, 106, 127, 159, 160, 164, 201, 224 Cruciform joint, 156, 174, 179, 180 Cruciform specimen, 243, 246–248, 252 Cyclic creep, 217 Cyclic stress-strain curve, 217 Defect, 80, 82

564

Endurance limit, 34, 37, 129, 245 Engineering stress and strain, 213 External multiaxiality, 126, 242, 255, 260, 263 Failure criterion, 36 Fatigue class (FAT), 165, 173, 176–178 Fatigue damage model, 99, 227 Fatigue life, 36, 132, 149, 161, 165, 169, 181, 184, 190, 198, 201, 216, 236 Fatigue limit, 34, 37, 63, 135, 200, 202 Fatigue strength reduction factor (Kf), 42, 44, 47, 49, 127–128, 132 Fictitious notch radius, 152–153 Finite Element (FE), 1, 61, 62, 135, 148, 154, 169, 172, 175, 185 Finite life, 35, 119, 121, 132, 143, 165, 180, 201, 210 Focus path, 136, 142, 144, 169 Fretting fatigue, 141 Full notch sensitivity, 44 Geometrical stress, 241–243, 245, 254 Glass/polyester, 245–253, 256 Gough’s formulas, 85 Graphite/epoxy, 253–254 Gross stress, 24–25, 41, 53 Gross section, 24–25 Hot-spot method, 154–155, 162–168 Hot-spot location, 155, 163, 165, 168 Hot-spot structural stress, 155, 163

Index Imaginary defect, 54 Inclusion, 192 Inherent material strength, 70 Inherent multiaxiality, 126, 167, 242, 243, 250, 255, 262 Isotropic hardening, 218 Kinematic hardening, 219 L, 54–61, 63–65, 81, 135–137, 142, 200, 203–205 LM, 69–73, 144–147 Linear Elastic Fracture Mechanics (LEFM), 41, 50, 53–63, 81 Line Method (LM), 64, 67–68, 72–73, 79, 83, 197 Load ratio (R), 6, 34, 37, 43, 52, 63, 102, 111, 129, 148, 158, 166, 216, 243, 256, 259, 263 Long crack, 53–54, 57–58, 65, 81, 203 Local, 28, 127, 152–156, 164, 242–243 Manson–Coffin’s curve, 121 Material cracking behaviour, 53, 89, 98, 142, 167, 197, 203 Material morphology, 54, 99, 113, 192, 194, 196, 201–205 Material principal stress, 241–243, 254 Mean stress, 37–40, 87, 100, 104, 106, 159, 183, 226 Mean shear stress, 10, 14, 50, 52, 75, 103, 112 Mean stress sensitivity index (m), 105, 107, 111, 116, 118, 120, 129, 185, 196, 201 Mean stress relaxation, 216 Mesoscopic approach, 90, 192 Mode I, 26–30, 89, 100, 138, 157, 164, 171, 174–175, 193, 195–197, 205 Mode II, 26–29, 89, 105, 114, 138, 157, 164, 171, 174–175, 177, 193, 197, 205 Mode III, 26, 28–29, 73, 80, 82, 164, 171, 175 Mohr’s circle, 3–5, 107–108, 200, 212, 225–226

565

Monotonic curve, 214 Multiaxial fatigue limit, 85 Net stress, 24–25, 41, 44, 49, 77, 87, 127–132 Net section, 24–25 Neuber’s formula, 44 Nominal stress, 24–25, 41, 71, 83, 127–133, 154, 159, 162, 243, 246 Non-damaging notch, 44, 78 Non-propagating crack (NPC), 58, 59, 66, 202–205 Non-proportional loading, 84, 90, 103, 112, 118, 126, 130, 134, 153, 196, 251 Non-proportional hardening, 219–221 Notch opening angle, 28–30, 45, 67, 200 Notch sensitivity, 43, 79 Notch-stress intensity factor (N-SIF), 29, 152–153, 155, 157, 174, 179 Out-of-phase, 11, 23, 83–86, 101–104, 113–114, 131–133, 180, 219, 232, 242–243, 251 Paris law, 52 Persistent slip band (PSB), 99, 190–191 Peterson’s formula, 45 Plane strain, 26, 176, 201 Plane stress, 3, 26, 200, 241 Point Method (PM), 62, 64, 70, 79, 136, 169, 199, 206 Principal stress, 3, 5, 25 Principal strain, 211 Probability of survival (PS), 34, 71, 74, 121, 123, 161, 165, 169, 173, 177, 178, 245, 259, 261 Ramberg–Osgood’s relationship, 215, 217 Ratcheting, 221 Residual stress, 158, 183 Resolved shear stress, 19–20

566

Index

Scale effect, 134, 153, 157, 174 Sharp notch, 25, 45, 57, 66, 137, 202, 205, 236 Short crack, 53–57, 196, 202–205 Short notch, 55, 57, 63 Singular stress, 25–30, 50, 157, 171, 175, 200 Stage I, 76, 88, 100–101, 192 Stage II, 76, 100, 192 Strain hardening and softening, 216–217 Stress Intensity Factor (SIF), 27, 30, 50, 56 Stress invariant, 3 Stress–strain curve, 213 Steel, 34, 40, 42, 44–46, 56, 66, 76–77, 80–81, 84–88, 117, 132, 137–140, 147, 194, 197, 214, 222 Steel weldments, 153, 157, 162, 164, 166, 169–170, 174–176, 180–182

Stress concentration factor, 24, 44, 50, 77, 129, 132, 185, 194, 235 Stress-distance curve, 136, 175, 179 Theory of Critical Distances, 61, 79, 134, 184, 199, 205 Threshold value of the stress intensity factor, 52, 63, 80, 135, 202 Torsional endurance limit, 74 Torsional fatigue limit, 74, 76 Volume Method (VM), 65, 199 Weld root radius, 171, 185 Weld bead opening angle, 156–157, 164, 171, 174 Wöhler curve, 34, 35, 36 Wöhler diagram, 34, 41, 74