Multiaxial Fatigue

Other SAE books of interest:

Fatigue Design Handbook(Order No. AE-22)

Recent Developments in Fatigue TechnologyEdited by Russell A. Chernenkoff and John J. Bonnen

(Order No. PT-67)

For more information or to order this book, contact SAE at 400 CommonwealthDrive, Warrendale, PA 15096-0001; phone (724) 776-4970; fax (724) 776-0790;e-mail: publications@sae.org.

Multiaxial Fatigue

Darrell F. Socie

Gary B. Marquis

Society of Automotive Engineers, Inc.Warrendale, Pa.

Library of Congress Cataloging-in-Publication Data

Socie, Darrell.Multiaxial fatigue / Darrell F. Socie, Gary B. Marquis.

p. cm.Includes bibliographical references.ISBN 0-7680-0453-51. Materials—Fatigue. 2. Axial loads. I. Marquis, G.

(Gary) II. Title.TA418.38.S64 2000620.1' 126—dc21 99-32460

CIP

Copyright © 2000 Society of Automotive Engineers, Inc.400 Commonwealth DriveWarrendale, PA 15096-0001 U.S.A.Phone: (724) 776-4841Fax: (724) 776-5760E-mail: publications@sae.orghttp://www.sae.org

ISBN 0-7680-0453-5

All rights reserved. Printed in the United States of America.

Permission to photocopy for internal or personal use, or the internal or personal use ofspecific clients, is granted by SAE for libraries and other users registered with the Copy-right Clearance Center (CCC), provided that the base fee of $.50 per page is paid directlyto CCC, 222 Rosewood Dr., Danvers, MA 01923. Special requests should be addressedto the SAE Publications Group. 0-7680-0453-5/00-$.50.

SAE Order No. R-234

Contents

Preface ix

Acknowledgments xiii

Nomenclature xv

Chapter 1—State of Stress and Strain 11.1 Introduction 11.2 Stresses and Strains Acting on a Plane 21.3 Maximum Stress and Strain 81.4 Common States of Stress and Strain 111.5 Effective, Hydrostatic, and Deviatoric Stresses 171.6 Cyclic Stresses 211.7 Using the Ideas 241.8 Summary 311.9 References 33

Chapter 2—Stress-Strain Relationships 352.1 Introduction 352.2 Elastic Stress and Strain 352.3 Plastic Stress and Strain 372.4 Cyclic Deformation 442.5 Cyclic Plasticity Models 532.6 Using the Ideas 672.7 Summary 722.8 References 75

Chapter 3—Fatigue Damage Mechanisms 773.1 Introduction 773.2 Crack Nucleation and Early Growth 783.3 Tensile Mechanisms—Mode I Growth 823.4 Shear Mechanisms—Mode II Growth 903.5 Damage Maps 923.6 Summary 983.7 References 99

v

Multiaxial Fatigue

Chapter 4—Multiaxial Testing 1014.1 Introduction 1014.2 Torsion-Bending 1014.3 Plates 1054.4 Disks 1094.5 Cruciform 1114.6 Tubes 1144.7 Fracture Mechanics Specimens 1214.8 Summary 1254.9 References 126

Chapter 5—Stress-Based Models 1295.1 Introduction 1295.2 Models 1305.3 Comparison of Models 1465.4 Using the Ideas 1515.5 Summary 1675.6 References 169

Chapter 6—Strain-Based and Energy-Based Models 1716.1 Introduction 1716.2 Static Yield Criteria 1716.3 Energy Models 1736.4 Critical Plane Models 1816.5 Combined Critical Plane and Energy Models 1906.6 Comparison of Models 1976.7 Using the Ideas 2096.8 Summary 2266.9 References 230

Chapter 7—Fracture Mechanics Models 2337.1 Introduction 2337.2 Plastic Zones in Multiaxial Loading 2337.3 Mode I Growth 2367.4 Crack Growth in Torsion 2387.5 Mixed Mode I and Mode II Crack Growth 2457.6 Mixed-Mode Growth Rate Models 2487.7 Using the Ideas 256

vi

Contents

7.8 Summary 2687.9 References 270

Chapter 8—Nonproportional Loading 2738.1 Introduction 2738.2 Definition of Nonproportionality 2738.3 Nonproportional Loading Histories 2828.4 Variable Amplitude Multiaxial Loading 2948.5 Comparison of the Methods 3138.6 Using the Ideas 3258.7 Summary 3348.8 References 336

Chapter 9—Notches 3419.1 Introduction 3419.2 Stresses and Strains in Notches 3419.3 Stress-Based Approaches 3569.4 Strain-Based Approaches 3609.5 Crack Growth Approaches 3799.6 Using the Ideas 3879.7 Summary 4059.8 References 407

Chapter 10—Applications 41110.1 Introduction 41110.2 Nonproportional Stressing and Loading 41510.3 Analysis Guidelines 41610.4 Recognizing Nonproportional Stressing 42210.5 Stress and Strain Concentration Factors 42510.6 Case Studies and Applications 42610.7 Summary 44410.8 References 445

Index 447

About the Authors 483

vii

Preface

Fatigue evaluation of components and structures has become an integral part ofthe design process in many industries. However, multiaxial fatigue continues tobe largely the domain of a limited number of specialists. During the analysis ofcomponents subject to multiaxial loading, the problem often is reduced to an"equivalent" uniaxial fatigue case without thought as to whether the simplifyingassumptions are valid for the specific load sequence or component being consid-ered. Several international conferences on multiaxial fatigue have been held inrecent years, and these have provided a wealth of test data and insight into thechallenge of fatigue under multiaxial load conditions. However, for the non-expert in the field, these volumes often are difficult to digest and complex toapply.

We have written this book primarily for practicing engineers, researchers, andstudents. Our goal was to provide working knowledge of the fatigue damageprocesses and models under multiaxial states of stress and strain. Readers areintroduced to the important considerations of multiaxial fatigue that differentiateit from uniaxial fatigue. We assumed that the reader has a basic background inengineering mechanics and is familiar with the fatigue damage process underuniaxial loading, but expertise in these fields is not required. In some cases,those wishing to implement the ideas should refer to other resources. Exampleproblems are included in most chapters to illustrate how to use the ideas andconcepts.

This book is not intended to be a comprehensive summary of all publishedresearch in the field of multiaxial fatigue. Instead, an interpretive summary ofvarious classes of models is presented and compared. We attempted to lend physi-cal interpretation to observed results and provide an explanation as to why cer-tain models work for one type of problem and not for another. Our focus is on acomplete treatment of the subject from many perspectives. The reference lists atthe end of each chapter, while complete, are by no means exhaustive and admit-tedly are biased by our own experiences. Several excellent surveys give otherhistorical perspectives of the field.

ix

Multiaxial Fatigue

Models and test data in this book were generated primarily for metallic materialsfor which the vast majority of test data is available. However, many of the con-cepts are material independent and will be useful in designing test programs oranalysis procedures for ceramic, composite, or other materials.

The first four chapters provide the background for the subsequent chapters. Chap-ter 1 reviews states of stress and strain. Terms, definitions, and equations formultiaxial stresses and strains are included. This is followed in Chapter 2 by adescription of stress-strain relationships, with emphasis placed on describing cyclicplastic deformation. Constitutive equations are reviewed briefly. Chapter 3 laysthe foundation for later chapters by providing a review of fatigue damage mecha-nisms, especially those under multiaxial loading. Fatigue damage models are anattempt to describe mathematically the complex fatigue damage behavior dis-cussed in this chapter. Chapter 4 surveys various test methods and specimensused in multiaxial fatigue research. Each type of specimen and test technique hasa limited number of stress and strain states that can be produced; therefore, acomprehensive view of multiaxial fatigue requires many testing methods.

The next three chapters describe fatigue damage models. Stress-based modelscommon for high-cycle fatigue analysis are discussed in Chapter 5. The low-cycle fatigue strain and energy-based models are covered in Chapter 6. Crackgrowth-based approaches are described in Chapter 7. As in uniaxial fatigue, nosingle analysis method is appropriate for all component and loading situations.These chapters present a comprehensive treatment of all basic analysis methods.

Chapter 8 presents some of the additional complications introduced into fatigueanalysis when the loading is nonproportional. Nonproportionality first is defined;then, topics of nonproportional hardening, multiaxial cycle counting, and dam-age models are discussed. Regions of stress concentrations cannot be avoided inreal structures; therefore, Chapter 9 gives attention to combining stress-strainanalysis, damage models, and nonproportionality effects to analyze fatigue ofcomponents containing notches.

The final chapter, Chapter 10, presents several case studies and illustrates howand when multiaxial fatigue analysis should be used in the design or optimizationof engineering components.

x

Preface

References

Conferences

Multiaxial Fatigue, ASTM STP 853, K.J. Miller and M.W. Brown, eds., Ameri-can Society for Testing and Materials, West Conshohocken, PA, 1985, 741 pp.

Advances in Multiaxial Fatigue, ASTM STP 1191, D.L. McDowell and R. Ellis,eds., American Society for Testing and Materials, West Conshohocken, PA, 1993,455 pp.

Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280,S. Kalluri and P.J. Bonacuse, eds., American Society for Testing and Materials,West Conshohocken, PA, 1993, 309 pp.

Biaxial and Multiaxial Fatigue, European Group on Fracture, EGF Publica-tion 3, M.W. Brown and K.J. Miller, eds., Mechanical Engineering Publications,London, 1989, 686 pp.

Fatigue Under Biaxial and Multiaxial Loading, European Structural IntegritySociety, ESIS Publication 10, K.F. Kussmaul, D.L. McDiarmid, and D.F. Socie,eds., Mechanical Engineering Publications, London, 1991, 480 pp.

Multiaxial Fatigue and Design, European Structural Integrity Society, ESISPublication 21, A. Pineau, G. Cailletaud, and T.C. Lindley, eds., Mechanical Engi-neering Publications, London, 1996, 532 pp.

Proceedings of the Fifth International Conference on Biaxial/Multiaxial Fatigueand Fracture, E. Macha and Z. Mróz, eds., Technical University of Opole, Poland,Sept. 5–8, 1997, 1411 pp.

Literature/Historical Surveys

Krempl, E., "The Influence of State of Stress on Low-Cycle Fatigue of StructuralMaterials: A Literature Survey and Interpretive Report," ASTM STP 549, Ameri-can Society for Testing and Materials, West Conshohocken, PA, 1974, 46 pp.

Garud, Y.S., "Multiaxial Fatigue: A Survey of the State of the Art," Journal ofTesting and Evaluation, Vol. 9, No. 3, 1981, pp. 165–178.

xi

Multiaxial Fatigue

Jordan, E.H., "Fatigue-Multiaxial Aspects," Pressure Vessel and Piping: DesignTechnology—1982—A Decade of Progress, S.Y. Zamrik and D. Dietrich, eds.,American Society of Mechanical Engineers, New York, 1982, pp. 507–518.

Brown, M.W., and Miller, K.J., "Two Decades of Progress in the Assessment ofMultiaxial Low-Cycle Fatigue," Low-Cycle Fatigue and Life Prediction, ASTMSTP 770, C. Amzallag, B.N. Leis, and P. Rabbe, eds., American Society for Test-ing and Materials, West Conshohocken, PA, 1982, pp. 482–499.

Ellyin, F. and Valaire, B., "Development of Fatigue Failure Theories for Multi-axial High Strain Conditions," Solid Mechanics Archives, Vol. 10, MartinusNijhoff Publishers, Dordrecht, 1985, pp. 45–85.

xii

Acknowledgments

This book started as a set of seminar notes for the Fracture Control Program atthe University of Illinois at Urbana-Champaign. The first portion of the bookwas completed when Prof. Socie was a Visiting Professor at VTT ManufacturingTechnology, Finland. Financial support from VTT during this time is gratefullyacknowledged.

The second portion of the book was completed when Dr. Marquis was a VisitingScientist at the University of Illinois at Urbana-Champaign. His research workwas supported by the U.S. Air Force Office of Scientific Research and VTTManufacturing Technology.

The final portion of the book was completed when Prof. Socie was a VisitingProfessor at Kyushu University under a Japan Society for the Promotion of Sci-ence Fellowship.

During this period, many colleagues and friends helped with copies of originalpictures, data, and reviews of the manuscript. Dr. Chin-Chan Chu of FordMotor Company and Prof. Ali Fatemi deserve special mention for their exten-sive reviews of the text and the equations.

xiii

Nomenclature

We have attempted to use a standard set of nomenclature throughout this bookfor stresses and strains. Many of the fatigue damage models contain adjustableconstants and parameters. We have chosen to use the nomenclature of the origi-nal authors rather than inventing an entirely new set of symbols. This results invarious constants and parameters (such as C, k, f, etc.) that are duplicated through-out the book. Each of these constants has a specific meaning when used with aparticular model, and a detailed description of the constant or parameter is givenin the text associated with the various models.

Stresses, Strains, and Energy

α Nonproportional hardening coefficientαij Backstress tensorΔε Normal strain rangeΔγ Shear strain rangeΔσ Normal strain rangeΔτ Shear stress rangeε Normal strainεn Normal strain on a planeε n , m a x Maximum normal strain on a planeεx, εy, εz Normal strains in X-Y-Z coordinate system

ε1, ε2, ε3 Principal strainsεθ Strain on a plane θεij Strain tensorεeq Equivalent strain

ε Equivalent or effective strainγ Shear strain

γxy, γyz, γxz Shear strains in X-Y-Z coordinate systemγ13, γ 2 3 , γ12 Principal shear strainsγθ Shear strain on a plane θγ o c t Octahedral shear strainλ Stress ratio σ3/σ1

Φ Strain ratio ε3/ε1

xv

Multiaxial Fatigue

ρ* Residual stress tensorσ Normal stressσx, σy, σz Normal stresses in X-Y-Z coordinate systemσ1, σ2, σ3 Principal stressesΣΘ Stress on a plane θσh Hydrostatic stressσij Stress tensorσeq Equivalent stressσ m a x Maximum stressσn Normal stress on a planeσ n , m a x Maximum stress acting normal to a planeσ Equivalent or effective stressτ Shear stressτ x y , τ y z , τxz Shear stresses in X-Y-Z coordinate system

τ13, τ23, τ12 Principal shear stressτθ Shear stress on a plane θτ o c t Octahedral shear stressC i j k l Stiffness tensorf Bending stressq Shear stressI1, I2, I3 Stress invariantsJ1, J2, J 3 Deviatoric stress invariantsSij Deviatoric stress tensorS i j k l Compliance tensorU Strain energyΔWI Axial workΔWII Shear workW WorkWc Work per cycle

Many of these variables may be combined with superscripts e, p, and t toindicate elastic, plastic, and total, respectively.

Material Properties

σ u t s Ultimate strengthσy Yield strength

xvi

Nomenclature

σfl Fatigue limitτfl Shear fatigue limitε f Fatigue ductility coefficientγ f Shear fatigue ductility coefficientσ f Fatigue strength coefficientτ f Shear fatigue strength coefficientb Fatigue strength exponentbγ Shear fatigue strength exponentc Fatigue ductility exponentcγ Shear fatigue ductility exponentC Paris equation interceptm Paris equation slopeE Elastic modulusE* Effective modulusG Shear modulusH Plastic modulusK Strength coefficientK Cyclic strength coefficientK* Effective strength coefficientK np Out-of-phase strength coefficientn Strain hardening exponentn Cyclic strain hardening exponentν Poisson's ratiot Fatigue limit in torsionb Fatigue limit in bendingtA,B Fatigue limit for Case A and Case B cracksΔKO Long crack threshold stress intensityΔK th Threshold stress intensity

Fracture Mechanics

ΔK e q Equivalent stress intensityΔKI, ΔKII, ΔKIII Cyclic stress intensity in Modes I, II, and IIIKI, K I I , KIII Stress intensity in Modes I, II, and IIIJI, J I I , JIII J integral for Modes I, II, and IIIR Stress ratioS Strain energy density

xvii

Multiaxial Fatigue

a Flaw sizeao Intrinsic flaw sizeY Geometry factorΔK(ε) Strain intensity factor

area Projected area of a flaw

General

N f Cycles to failure2N f Reversals to failureKt Stress concentration factorK f Fatigue notch factor

KB

t Stress concentration in bending

KT

t Stress concentration in torsion

Keq

t Equivalent stress concentrationF Yield functionHV Vickers hardness

xviii