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1、MULTIAXIAL FATIGUE DARRELL F. SOCIE GARY B. MARQUISMultiaxial FatigueOther SAE books of interest: Fatigue Design Handbook (Order No. AE-22) Recent Developments in Fatigue Technology Edited by Russell A. Chernenkoff and John J. Bonnen (Order No. PT-67) For more information or to order this book, cont

2、act SAE at 400 Commonwealth Drive, Warrendale, PA 15096-0001; phone (724) 776-4970; fax (724) 776-0790; e-mail: publicationssae.org.Multiaxial Fatigue Darrell F. Socie Gary B. Marquis Society of Automotive Engineers, Inc. Warrendale, Pa. Copyright 2000 Society of Automotive Engineers, Inc. eISBN: 97

3、8-0-7680-6510-7Library of Congress Cataloging-in-Publication Data Socie, Darrell. Multiaxial fatigue / Darrell F. Socie, Gary B. Marquis. p. cm. Includes bibliographical references. ISBN 0-7680-0453-5 1. MaterialsFatigue. 2. Axial loads. I. Marquis, G. (Gary) II. Title. TA418.38.S64 2000 620.1 126dc

4、21 99-32460 CIP Copyright 2000 Society of Automotive Engineers, Inc. 400 Commonwealth Drive Warrendale, PA 15096-0001 U.S.A. Phone: (724) 776-4841 Fax: (724) 776-5760 E-mail: publicationssae.org http:/www.sae.org ISBN 0-7680-0453-5 All rights reserved. Printed in the United States of America. Permis

5、sion to photocopy for internal or personal use, or the internal or personal use of specific 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 directly to CCC, 222 Rosewood Dr., Danvers,

6、 MA 01923. Special requests should be addressed to the SAE Publications Group. 0-7680-0453-5/00-$.50. SAE Order No. R-234Contents Preface ix Acknowledgments xiii Nomenclature xv Chapter 1State of Stress and Strain 1 1.1 Introduction 1 1.2 Stresses and Strains Acting on a Plane 2 1.3 Maximum Stress a

7、nd Strain 8 1.4 Common States of Stress and Strain 11 1.5 Effective, Hydrostatic, and Deviatoric Stresses 17 1.6 Cyclic Stresses 21 1.7 Using the Ideas 24 1.8 Summary 31 1.9 References 33 Chapter 2Stress-Strain Relationships 35 2.1 Introduction 35 2.2 Elastic Stress and Strain 35 2.3 Plastic Stress

8、and Strain 37 2.4 Cyclic Deformation 44 2.5 Cyclic Plasticity Models 53 2.6 Using the Ideas 67 2.7 Summary 72 2.8 References 75 Chapter 3Fatigue Damage Mechanisms 77 3.1 Introduction 77 3.2 Crack Nucleation and Early Growth 78 3.3 Tensile MechanismsMode I Growth 82 3.4 Shear MechanismsMode II Growth

9、 90 3.5 Damage Maps 92 3.6 Summary 98 3.7 References 99 vMultiaxial Fatigue Chapter 4Multiaxial Testing 101 4.1 Introduction 101 4.2 Torsion-Bending 101 4.3 Plates 105 4.4 Disks 109 4.5 Cruciform 111 4.6 Tubes 114 4.7 Fracture Mechanics Specimens 121 4.8 Summary 125 4.9 References 126 Chapter 5Stres

10、s-Based Models 129 5.1 Introduction 129 5.2 Models 130 5.3 Comparison of Models 146 5.4 Using the Ideas 151 5.5 Summary 167 5.6 References 169 Chapter 6Strain-Based and Energy-Based Models 171 6.1 Introduction 171 6.2 Static Yield Criteria 171 6.3 Energy Models 173 6.4 Critical Plane Models 181 6.5

11、Combined Critical Plane and Energy Models 190 6.6 Comparison of Models 197 6.7 Using the Ideas 209 6.8 Summary 226 6.9 References 230 Chapter 7Fracture Mechanics Models 233 7.1 Introduction 233 7.2 Plastic Zones in Multiaxial Loading 233 7.3 Mode I Growth 236 7.4 Crack Growth in Torsion 238 7.5 Mixe

12、d Mode I and Mode II Crack Growth 245 7.6 Mixed-Mode Growth Rate Models 248 7.7 Using the Ideas 256 viContents 7.8 Summary 268 7.9 References 270 Chapter 8Nonproportional Loading 273 8.1 Introduction 273 8.2 Definition of Nonproportionality 273 8.3 Nonproportional Loading Histories 282 8.4 Variable

13、Amplitude Multiaxial Loading 294 8.5 Comparison of the Methods 313 8.6 Using the Ideas 325 8.7 Summary 334 8.8 References 336 Chapter 9Notches 341 9.1 Introduction 341 9.2 Stresses and Strains in Notches 341 9.3 Stress-Based Approaches 356 9.4 Strain-Based Approaches 360 9.5 Crack Growth Approaches

14、379 9.6 Using the Ideas 387 9.7 Summary 405 9.8 References 407 Chapter 10Applications 411 10.1 Introduction 411 10.2 Nonproportional Stressing and Loading 415 10.3 Analysis Guidelines 416 10.4 Recognizing Nonproportional Stressing 422 10.5 Stress and Strain Concentration Factors 425 10.6 Case Studie

15、s and Applications 426 10.7 Summary 444 10.8 References 445 Index 447 About the Authors 483 viiPreface Fatigue evaluation of components and structures has become an integral part of the design process in many industries. However, multiaxial fatigue continues to be largely the domain of a limited num

16、ber of specialists. During the analysis of components subject to multiaxial loading, the problem often is reduced to an “equivalent“ uniaxial fatigue case without thought as to whether the simplifying assumptions are valid for the specific load sequence or component being consid- ered. Several inter

17、national conferences on multiaxial fatigue have been held in recent years, and these have provided a wealth of test data and insight into the challenge of fatigue under multiaxial load conditions. However, for the non- expert in the field, these volumes often are difficult to digest and complex to a

18、pply. We have written this book primarily for practicing engineers, researchers, and students. Our goal was to provide working knowledge of the fatigue damage processes and models under multiaxial states of stress and strain. Readers are introduced to the important considerations of multiaxial fatig

19、ue that differentiate it from uniaxial fatigue. We assumed that the reader has a basic background in engineering mechanics and is familiar with the fatigue damage process under uniaxial loading, but expertise in these fields is not required. In some cases, those wishing to implement the ideas should

20、 refer to other resources. Example problems are included in most chapters to illustrate how to use the ideas and concepts. This book is not intended to be a comprehensive summary of all published research in the field of multiaxial fatigue. Instead, an interpretive summary of various classes of mode

21、ls 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 a complete treatment of the subject from many perspectives. The reference lists at

22、the end of each chapter, while complete, are by no means exhaustive and admit- tedly are biased by our own experiences. Several excellent surveys give other historical perspectives of the field. ixMultiaxial Fatigue Models and test data in this book were generated primarily for metallic materials fo

23、r 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 or analysis procedures for ceramic, composite, or other materials. The first four chapters provide the background for the subsequent chapters. C

24、hap- ter 1 reviews states of stress and strain. Terms, definitions, and equations for multiaxial stresses and strains are included. This is followed in Chapter 2 by a description of stress-strain relationships, with emphasis placed on describing cyclic plastic deformation. Constitutive equations are

25、 reviewed briefly. Chapter 3 lays the foundation for later chapters by providing a review of fatigue damage mecha- nisms, especially those under multiaxial loading. Fatigue damage models are an attempt to describe mathematically the complex fatigue damage behavior dis- cussed in this chapter. Chapte

26、r 4 surveys various test methods and specimens used in multiaxial fatigue research. Each type of specimen and test technique has a limited number of stress and strain states that can be produced; therefore, a comprehensive view of multiaxial fatigue requires many testing methods. The next three chap

27、ters describe fatigue damage models. Stress-based models common for high-cycle fatigue analysis are discussed in Chapter 5. The low- cycle fatigue strain and energy-based models are covered in Chapter 6. Crack growth-based approaches are described in Chapter 7. As in uniaxial fatigue, no single anal

28、ysis 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 fatigue analysis when the loading is nonproportional. Nonproportionality first

29、is defined; then, topics of nonproportional hardening, multiaxial cycle counting, and dam- age models are discussed. Regions of stress concentrations cannot be avoided in real structures; therefore, Chapter 9 gives attention to combining stress-strain analysis, damage models, and nonproportionality

30、effects to analyze fatigue of components containing notches. The final chapter, Chapter 10, presents several case studies and illustrates how and when multiaxial fatigue analysis should be used in the design or optimization of engineering components. xPreface References Conferences Multiaxial Fatigu

31、e, 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. Mult

32、iaxial 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., Me

33、chanical Engineering Publications, London, 1989, 686 pp. Fatigue Under Biaxial and Multiaxial Loading, European Structural Integrity Society, ESIS Publication 10, K.F. Kussmaul, D.L. McDiarmid, and D.F. Socie, eds., Mechanical Engineering Publications, London, 1991, 480 pp. Multiaxial Fatigue and De

34、sign, European Structural Integrity Society, ESIS Publication 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 Fatigue and Fracture, E. Macha and Z. Mrz, eds., Te

35、chnical University of Opole, Poland, Sept. 58, 1997, 1411 pp. Literature/Historical Surveys Krempl, E., “The Influence of State of Stress on Low-Cycle Fatigue of Structural Materials: A Literature Survey and Interpretive Report,“ ASTM STP 549, Ameri- can Society for Testing and Materials, West Consh

36、ohocken, PA, 1974, 46 pp. Garud, Y.S., “Multiaxial Fatigue: A Survey of the State of the Art,“ Journal of Testing and Evaluation, Vol. 9, No. 3, 1981, pp. 165178. xiMultiaxial Fatigue Jordan, E.H., “Fatigue-Multiaxial Aspects,“ Pressure Vessel and Piping: Design Technology1982A Decade of Progress, S

37、.Y. Zamrik and D. Dietrich, eds., American Society of Mechanical Engineers, New York, 1982, pp. 507518. Brown, M.W., and Miller, K.J., “Two Decades of Progress in the Assessment of Multiaxial Low-Cycle Fatigue,“ Low-Cycle Fatigue and Life Prediction, ASTM STP 770, C. Amzallag, B.N. Leis, and P. Rabb

38、e, eds., American Society for Test- ing and Materials, West Conshohocken, PA, 1982, pp. 482499. Ellyin, F. and Valaire, B., “Development of Fatigue Failure Theories for Multi- axial High Strain Conditions,“ Solid Mechanics Archives, Vol. 10, Martinus Nijhoff Publishers, Dordrecht, 1985, pp. 4585. xi

39、iAcknowledgments This book started as a set of seminar notes for the Fracture Control Program at the University of Illinois at Urbana-Champaign. The first portion of the book was completed when Prof. Socie was a Visiting Professor at VTT Manufacturing Technology, Finland. Financial support from VTT

40、during this time is gratefully acknowledged. The second portion of the book was completed when Dr. Marquis was a Visiting Scientist at the University of Illinois at Urbana-Champaign. His research work was supported by the U.S. Air Force Office of Scientific Research and VTT Manufacturing Technology.

41、 The final portion of the book was completed when Prof. Socie was a Visiting Professor 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 original pictures, data, and reviews of the manuscript. D

42、r. Chin-Chan Chu of Ford Motor Company and Prof. Ali Fatemi deserve special mention for their exten- sive reviews of the text and the equations. xiiiNomenclature We have attempted to use a standard set of nomenclature throughout this book for stresses and strains. Many of the fatigue damage models c

43、ontain adjustable constants 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 in various constants and parameters (such as C, k, f, etc.) that are duplicated through- out the book. Each of these constan

44、ts has a specific meaning when used with a particular model, and a detailed description of the constant or parameter is given in the text associated with the various models. Stresses, Strains, and Energy Nonproportional hardening coefficient ij Backstress tensor Normal strain range Shear strain rang

45、e Normal strain range Shear stress range Normal strain n Normal strain on a plane n,max Maximum normal strain on a plane x, y, z Normal strains in X-Y-Z coordinate system 1, 2, 3 Principal strains Strain on a plane ij Strain tensor eq Equivalent strain Equivalent or effective strain Shear strain xy,

46、 yz, xz Shear strains in X-Y-Z coordinate system 13, 23, 12 Principal shear strains Shear strain on a plane oct Octahedral shear strain Stress ratio 3/1 Strain ratio 3/1 xvMultiaxial Fatigue * Residual stress tensor Normal stress x, y, z Normal stresses in X-Y-Z coordinate system 1, 2, 3 Principal s

47、tresses Stress on a plane h Hydrostatic stress ij Stress tensor eq Equivalent stress max Maximum stress n Normal stress on a plane n,max Maximum stress acting normal to a plane Equivalent or effective stress Shear stress xy, yz, xz Shear stresses in X-Y-Z coordinate system 13, 23, 12 Principal shear

48、 stress Shear stress on a plane oct Octahedral shear stress Cijkl Stiffness tensor f Bending stress q Shear stress I1, I2, I3 Stress invariants J1, J2, J3 Deviatoric stress invariants Sij Deviatoric stress tensor Sijkl Compliance tensor U Strain energy WI Axial work WII Shear work W Work Wc Work per

49、 cycle Many of these variables may be combined with superscripts e, p, and t to indicate elastic, plastic, and total, respectively. Material Properties uts Ultimate strength y Yield strength xviNomenclature fl Fatigue limit fl Shear fatigue limit f Fatigue ductility coefficient f Shear fatigue ductility coefficient f Fatigue strength coefficient f Shear fatigue strength coefficient b Fatigue strength exponent b Shear fatigue strength exponent c Fatigue duct