Fracture Mechanics(Anderson)

Second Edition

and

T.L.Department of Mechanical Engineering

Texas A&M UniversityCollege Station, Texas

CRC PressBoca Raton London New York Washington, D.C.

Cover Art: Nonlinear 3-dimensional analysis of a compact tension, C(T), specimen. VonMises equivalent stress shown on a deformed model. Courtesy of Arne Gullerud and ProfessorRobert Dodds, Department of Civil Engineering, University of Illinois, Champaign-Urbana.

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Da ic of Hiscei&i. ji

Library of Congress Cataloging-in-Publication Data

Anderson, T. L.Fracture mechanics : fundamentals and applications / T. L.

Anderson. --2nd ed.p. cm.

Includes bibliographical references and index.ISBN 0-8493-4260-0 (acid-free paper)1. Fracture mechanics. L Title.

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PREFACE

The field of fracture mechanics was virtually nonexistent prior to World War II, but hassince matured into an established discipline. Most universities with an engineeringprogram offer at least one fracture mechanics course on the graduate level, and anincreasing number of undergraduates have been exposed to this subject. Applications offracture mechanics in industry are relatively common, as knowledge that was onceconfined to a few specialists is becoming more widespread.

While there are a number of books on fracture mechanics, most are geared to aspecific audience. Some treatments of this subject emphasize material testing, whileothers concentrate on detailed mathematical derivations. A few books address themicroscopic aspects of fracture, but most consider only continuum models. Many booksare restricted to a particular material system, such as metals or polymers. Currentofferings include advanced, highly specialized books, as well as introductory texts.While the former are valuable to researchers in this field, they are unsuitable for studentswith no prior background. On the other hand, introductory treatments of the subject aresometimes simplistic and misleading.

This book provides a comprehensive treatment of fracture mechanics that shouldappeal to a relatively wide audience. Theoretical background and practical applications areboth covered in detail. This book is suitable as a graduate text, as well as a reference forengineers and researchers. Selected portions of this book would also be appropriate for anundergraduate course in fracture mechanics.

This is the second edition of this text. The first edition was published in 1991.While the overwhelming response to the first edition was positive, I received a fewconstructive criticisms from several colleagues whose opinions I respect. I have tried toincorporate their comments in this revision, and I hope the final product meets with theapproval of readers who are acquainted with the first edition, as well as those who areseeing this text for the first time.

Many sections have been revised and expanded in the second edition. In a few cases,material from the first edition was dropped because it had become obsolete or did not fitwithin the context of the revised material. Chapter 2, which covers linear elastic fracture,has been expanded to include weight functions and mixed-mode fracture. (Weightfunctions were previously covered in Chapter 11, but the subject is too important to placeat the back of the book.) A new section on crack tip constraint has been added to Chapter3, and Chapter 6 now includes a section on fracture in concrete and rock. Chapter 7 wasupdated to account for recent developments in fracture toughness testing standards.Additional stress intensity solutions for part-through cracks have been added to Chapter12. A number of problems have been added to Chapter 13, and several problems from thefirst edition have been modified or deleted.

The basic organization and underlying philosophy are unchanged in the secondedition. The book is intended to be readable without being superficial. The fundamentalconcepts are first described qualitatively, with a minimum of higher level mathematics.

111

This enables a student with a reasonable grasp of undergraduate calculus to gain physicalinsight into the subject. For the more advanced reader, appendices at the end of certainchapters give the detailed mathematical background.

In outlining the basic principles and applications of fracture mechanics, I haveattempted to integrate materials science and solid mechanics to a much greater extent thanprevious texts. Although continuum theory has proved to be a very powerful tool infracture mechanics, one cannot ignore microstructural aspects. Continuum theory canpredict the stresses and strains near a crack tip, but it is the material's microstructure thatdetermines the critical conditions for fracture.

The first chapter introduces the subject of fracture mechanics and provides anoverview; this chapter includes a review of dimensional analysis, which proves to be auseful tool in later chapters. Chapters 2 and 3 describe the fundamental concepts oflinear elastic and elastic-plastic fracture mechanics, respectively. One of the mostimportant and most often misunderstood concepts in fracture mechanics is the singleparameter assumption, which enables the prediction of structural behavior from smallscale laboratory tests. When a single parameter uniquely describes the crack tipconditions, fracture toughness, which is a critical value of this parameter, is independentof specimen size. When the single-parameter assumption breaks down, fracture toughnessbecomes size dependent, and a small scale fracture toughness test may not be indicative ofstructural behavior. Chapters 2 and 3 describe the basis of the single-parameterassumption in detail, and outline the requirements for its validity. Chapter 3 includes theresults of recent research that extends fracture mechanics beyond the limits of single-parameter theory. The main bodies of Chapters 2 and 3 are written in such a way as to beaccessible to the beginning student. Appendices 2 and 3, which follow Chapters 2 and 3,respectively, give mathematical derivations of several important relationships in linearelastic and elastic-plastic fracture mechanics. Most of the material in these appendicesrequires a graduate-level background in solid mechanics.

Chapter 4 introduces dynamic and time-dependent fracture mechanics. The section ondynamic fracture includes a brief discussion of rapid loading of a stationary crack, as wellas rapid crack propagation and arrest. The C*, C(t), and Q parameters for characterizingcreep crack growth are introduced, together with analogous quantities that characterizefracture in viscoelastic materials.

Chapter 5 outlines micromechanisms of fracture in metals and alloys, while Chapter6 describes fracture mechanisms in polymers, ceramics, composites, and concrete. Thesechapters emphasize the importance of microstructure and material properties on thefracture behavior.

The applications portion of this book begins with Chapter 7, which gives practicaladvice on fracture toughness testing in metals. This chapter describes standard testmethods, such as Kjc, //c, and CTOD, as well as recent research results. Chapter 7includes a section on weldment testing, which has yet to be standardized in the U.S.Chapter 8 describes fracture testing of nonmetallic materials. Most of these test methodsare still experimental in nature, since this is a relatively new field. Currently, a numberof researchers are characterizing fracture behavior of plastics with test methods that wereoriginally developed for metals; Chapter 8 discusses the validity of such tests forpolymers, and suggests improvements in current methodology. Chapter 9 outlines the

I V

available methods for applying fracture mechanics to structures, including linear elasticapproaches, the EPRI J estimation scheme, the R-6 method, and the British Standards PD6493 approach. A brief description of probabilistic fracture mechanics is also included, aswell as a discussion of the shortcomings of existing analyses. Chapter 10 describes thefracture mechanics approach to fatigue crack propagation, and discusses some of thecritical issues in this area, including crack closure and the behavior of short cracks.Chapter 11 outlines some of the most recent developments in computational fracturemechanics. Procedures for determining stress intensity and the J integral in structures aredescribed, with particular emphasis on the energy domain integral approach.

Chapter 12 provides reference material that is usually found in fracture mechanicshandbooks. This material includes stress intensity factors for common configurations, aswell as limit load, elastic compliance, and fully plastic J solutions. Chapter 13 containsa series of practice problems that correspond to material in Chapters 1 to 11.

If this book is used as a college text, it is unlikely that all of the material can becovered in a single semester. Thus the instructor should select the portions of the bookthat suit the needs and background of the students. The first three chapters, excludingappendices, should form the foundation of any course. In addition, I strongly recommendthe inclusion of at least one of the materials chapters (5 or 6), regardless of whether or notmaterials science is the students' major field of study. A course that is oriented towardapplications could include Chapters 7 to 10, in addition to the earlier chapters. A graduatelevel course in a solid mechanics curriculum might include Appendices 2 and 3, Chapter4, Appendix 4, and Chapter 11.

As with the first edition, I produced this book in camera-ready form on a Macintoshpersonal computer. The appearance of this edition is, I believe, superior to the firstedition. This cosmetic enhancement can be attributed to a combination of better software,a higher resolution printer, and an incremental improvement in the desk-top publishingskills of the author.

I am pleased to acknowledge all those individuals who helped make this bookpossible. I am grateful to Bob Stern and Joel Claypool at CRC Press for their continualsupport. A number of colleagues and friends reviewed portions of the draft manuscriptand/or provided photographs and homework problems, including W.L. Bradley, M.Cayard, R Chona, M.G. Dawes, R.H. Dodds Jr., A.G. Evans, S.J. Garwood, J.P. Gudas,E.G. Guynn, A.L. Highsmith, R.E. Jones Jr., Y.W. Kwon, J.D. Landes, E.J. Lavernia,A. Letton, R.C. McClung, D.L. McDowell, J.G. Merkle, M.T. Miglin, D.M. Parks,P.T. Purtscher, R.A. Schapery, and C.F. Shih. I apologize to anyone whose name I haveinadvertently omitted from this list. All of these individuals contributed to the firstedition, and some offered more recent comments which I incorporated into the currentedition. I received valuable editorial assistance from Victoria Stolarski, who completelyreformatted the text for this edition. Mr. Sun Yongqi produced a number of SEMfractographs especially for this book. I would like to express my appreciation to BudPeterson, the Head of the Mechanical Engineering Department at Texas A&M University,and his predecessor, Walter Bradley, for providing an environment conducive to thepreparation of this book. Last but certainly not least, Russ Hall, formerly with CRCPress and now with IEEE Press, deserves special mention. In 1989, Russ used variousmethods (legitimate and otherwise) to convince me to write this book. When I agreed to

embark on this project at that time, I must have either been temporarily insane or wasbeing blackmailed (or both). When Russ' first novel is made into a movie, I hope heinvites me to the premier.

T.L. AndersonOctober 1994

VI

ToMolly and Tom

CONTENTS

PREFACE iii

PART I:INTRODUCTION 1

1. HISTORY AND OVERVIEW 31.1 WHY STRUCTURES FAIL....:.." 31.2 HISTORICAL PERSPECTIVE 7

1.2.1 Early Fracture Research 101.2.2 The Liberty Ships 101.2.3 Post-War Fracture Mechanics Research 111.2.4 Fracture Mechanics from 1960 to 1980 121.2.5 Recent Trends in Fracture Research 14

1.3 THE FRACTURE MECHANICS APPROACH TO DESIGN 141.3.1 The Energy Criterion 151.3.2 The Stress Intensity Approach 171.3.3 Time-Dependent Crack Growth and Damage Tolerance 18

1.4 EFFECT OF MATERIAL PROPERTIES ON FRACTURE 191.5 A BRIEF REVIEW OF DIMENSIONAL ANALYSIS 21

1.5.1 The Buckingham Il-Theorem 211.5.2 Dimensional Analysis in Fracture Mechanics 22

REFERENCES 25

PART II:FUNDAMENTAL CONCEPTS 29

2. LINEAR ELASTIC FRACTURE MECHANICS 312.1 AN ATOMIC VIEW OF FRACTURE 312.2 STRESS CONCENTRATION EFFECT OF FLAWS 332.3 THE GRIFFITH ENERGY BALANCE 36

2.3.1 Comparison with the Critical Stress Criterion 382.3.2 Modified Griffith Equation 39

2.4 THE ENERGY RELEASE RATE 412.5 INSTABILITY AND THE R CURVE 46

2.5.1 Reasons for the R Curve Shape 482.5.2 Load Control Versus Displacement Control 482.5.3 Structures with Finite Compliance 50

2.6 STRESS ANALYSIS OF CRACKS 512.6.1 The Stress Intensity Factor 522.6.2 Relationship between K and Global Behavior 552.6.3 Effect of Finite Size 58

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2.6.4 Principle of Superposition 642.6.5 Weight Functions 67

2.7 RELATIONSHIP BETWEEN £ AND §. 692.8 CRACK TIP PLASTICITY 72

2.8.1 The Irwin Approach 722.8.2 The Strip Yield Model 752.8.3 Comparison of Plastic Zone Corrections 782.8.4 Plastic Zone Shape 78

2.9 PLANE STRESS VERSUS PLANE STRAIN 822.10 K AS A FAILURE CRITERION 84

2.10.1 Effect of Loading Mode 872.10.2 Effect of Specimen Dimensions 872.10.3 Limits to the Validity of LEFM 89

2.11 MIXED-MODE FRACTURE 912.11.1 Propagation of an Angled Crack 912.11.2 Equivalent Mode I Crack 93

REFERENCES 96

APPENDIX 2: MATHEMATICAL FOUNDATIONS OF LINEARELASTIC FRACTURE MECHANICS 101

A2.1 PLANE ELASTICITY 101A2.1.1 Cartesian Coordinates 101A2.1.2 Polar Coordinates 103

A2.2 CRACK GROWTH INSTABILITY ANALYSIS 104A2.3 CRACK TIP STRESS ANALYSIS 105

A2.3.1 Generalized In-Plane Loading 105A2.3.2 The Westergaard Stress Function 109

A2.4 ELLIPTICAL INTEGRAL OF THE SECOND KIND 115

3. ELASTIC-PLASTIC FRACTURE MECHANICS 1173.1 CRACK TIP OPENING DISPLACEMENT. : 1173.2 THE J CONTOUR INTEGRAL 122

3.2.1 Nonlinear Energy Release Rate 1233.2.2 J as a Path-Independent Line Integral 1263.2.3 J as a Stress Intensity Parameter 1273.2.4 The Large Strain Zone 1293.2.5 Laboratory Measurement of J 131

3.3 RELATIONSHIPS BETWEEN/AND CTOD 1383.4 CRACK GROWTH RESISTANCE CURVES 142

3.4.1 Stable and Unstable Crack Growth 1433.4.2 Computing J for a Growing Crack 146

3.5 J-CONTROLLED FRACTURE 1483.5.1 Stationary Cracks 1493.5.2 /-Controlled Crack Growth 152

3.6 CRACK-TIP CONSTRAINT UNDER LARGE-SCALEYIELDING 155

3.6.1 The Elastic T Stress 1603.6.2 J-Q Theory 163

3.6.3 Scaling Model for Cleavage Fracture 1693.6.4 Limitations of Two-Parameter Fracture Mechanics 173

REFERENCES 178

APPENDIX 3: MATHEMATICAL FOUNDATIONS OFELASTIC-PLASTIC FRACTURE MECHANICS 183

A3.1 DETERMINING CTOD FROM THE STRIP YIELD MODEL 183A3.2 THE/CONTOUR INTEGRAL 186A3.3 / AS A NONLINEAR ENERGY RELEASE RATE 188A3.4 THE HRR SINGULARITY 190A3.5 ANALYSIS OF STABLE CRACK GROWTH IN SMALLSCALE YIELDING 194

A3.5.1 The Rice-Drugan-Sham Analysis 194A3.5.2 Steady State Crack Growth 198

A3.6 NOTES ON THE APPLICABILITY OF DEFORMATIONPLASTICITY TO CRACK PROBLEMS 201

4. DYNAMIC AND TIME-DEPENDENT FRACTURE 2054.1 DYNAMIC FRACTURE AND CRACK ARREST 205

4.1.1 Rapid Loading of a Stationary Crack 2064.1.2 Rapid Crack Propagation and Arrest 2114.1.3 Dynamic Contour Integrals 222

4.2 CREEP CRACK GROWTH 2254.2.1 The C* Integral 2264.2.2 Short Time Versus Long Time Behavior 229

4.3 VISCOELASTIC FRACTURE MECHANICS 2334.3.1 Linear Viscoelasticity 2344.3.2 The Viscoelastic J Integral 2384.3.3 Transition from Linear to Nonlinear Behavior 242

A4.1 ELASTODYNAMIC CRACK TIP FIELDS 251A4.2 DERIVATION OF THE

GENERALIZED 255

PART III:MATERIAL BEHAVIOR 261

5. FRACTURE MECHANISMS IN METALS 2655.1 DUCTILE FRACTURE 265

5.1.1 Void Nucleation 2675.1.2 Void Growth 2685.1.3 Ductile Crack Growth 279

5.2 CLEAVAGE 2825.2.1 Fractography 2835.2.2 Mechanisms of Cleavage Initiation 2855.2.3 Mathematical Models of Cleavage Fracture Toughness 289

5.3 THE DUCTILE-BRITTLE TRANSITION 2975.4 INTERGRANULAR FRACTURE 299

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REFERENCES 302

APPENDIX 5: STATISTICAL MODELING OF CLEAVAGEFRACTURE 307

A5.1 WEAKEST LINK FRACTURE 307A5.2 INCORPORATING A CONDITIONAL PROBABILITY OFPROPAGATION 309

6. FRACTURE MECHANISMS IN NONMETALS 3136.1 ENGINEERING PLASTICS 314

6.1.1 Structure and Properties of Polymers 3146.1.2 Yielding and Fracture in Polymers 3226.1.3 Fiber-Reinforced Plastics 329

6.2 CERAMICS AND CERAMIC COMPOSITES 3436.2.1 Microcrack Toughening 3476.2.2 Transformation Toughening 3486.2.3 Ductile Phase Toughening 3506.2.4 Fiber and Whisker Toughening 350

6.3 CONCRETE AND ROCK 354REFERENCES 357

FART IV:APPLICATIONS 363

7. FRACTURE TOUGHNESS TESTING OF METALS 3657.1 GENERAL CONSIDERATIONS 365

7.1.1 Specimen Configurations 3667.1.2 Specimen Orientation 3687.1.3 Fatigue Precracking 3707.1.4 Instrumentation 3727.1.5 Side Grooving 373

7.2 KIC TESTING 3757.3 £-.7? CURVE TESTING 380

7.3.1 Specimen Design 3817.3.2 Experimental Measurement of K-R Curves 382

7.4 J TESTING OF METALS 3857.4.1 Jjc Measurements 3857.4.2 J-R Curve Testing 3897.4.3 Critical / Values for Cleavage 392

7.5 CTOD TESTING 3927.6 DYNAMIC AND CRACK ARREST TOUGHNESS 396

7.6.1 Rapid Loading 3967.6.2 Kja Measurements 397

7.7 FRACTURE TESTING OF WELDMENTS 4027.7.1 Specimen Design and Fabrication 4037.7.2 Notch Location and Orientation 403

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7.7.3 Fatigue Precracking 4067.7.4 Post-Test Analysis 406

7.8 TESTING AND ANALYSIS OF STEELS IN THE DUCTILE-BRITTLE TRANSITION REGION 4077.9 QUALITATIVE TOUGHNESS TESTS 409

7.9.1 Charpy and Izod Impact Test 4097.9.2 Drop Weight Test 4117.9.3 Drop Weight Tear and Dynamic Tear Tests 412

REFERENCES 413

APPENDIX 7: EXPERIMENTAL ESTIMATES OFDEFORMATION J 419

8. FRACTURE TESTING OF NONMETALS 4238.1 FRACTURE TOUGHNESS MEASUREMENTS INENGINEERING PLASTICS 423

8.1.1 The Suitability of K and J for Polymers 4238.1.2 Precracking and Other Practical Matters 4328.1.3 Klc Testing 4348.1.4/Testing 4398.1.5 Experimental Estimates of Time-Dependent FractureParameters 4408.1.6 Qualitative Fracture Tests on Plastics 444

8.2 INTERLAMINAR TOUGHNESS OF COMPOSITES 4458.3 CERAMICS 451

8.3.1 Chevron-Notched Specimens 4518.3.2 Bend Specimens Precracked by Bridge Indentation 454

REFERENCES 456

9. APPLICATION TO STRUCTURES 4599.1 LINEAR ELASTIC FRACTURE MECHANICS 460

9 . l . lK j fo r Part-Through Cracks 4619.1.2 Primary and Secondary Stresses 4679.1.3 Plasticity Corrections 4689.1.4 KIC from Jcny.Advantages and Pitfalls 4699.1.5 A Warning About LEFM -. 469

9.2 THE ASME REFERENCE CURVES 4709.3 THE CTOD DESIGN CURVE 4729.4 FAILURE ASSESSMENT DIAGRAMS 4749.5 THE EPRI J ESTIMATION SCHEME 478

9.5.1 Theoretical Background 4789.5.2 Estimation Equations 4809.5.3 Comparison with Experimental Estimates 4849.5.4 J-Based Failure Assessment Diagrams 4879.5.5 Ductile Instability Analysis 4889.5.6 Some Practical Considerations 493

9.6 THE REFERENCE STRESS APPROACH 493

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9.7 COMPARISON OF DRIVING FORCE EQUATIONS 4969.8 THE PD 6493 METHOD 498

9.8.1 Level 1 5009.8.2 Level 2 5019.8.3 Level 3 504

9.9 THE R6 METHOD 5059.10 PROBABILISTIC FRACTURE MECHANICS 5069.11 LIMITATIONS OF EXISTING APPRO ACHES 507

9.11.1 Driving Force in Weldments 5089.11.2 Residual Stresses 5089.11.3 Three-Dimensional Effects 5099.11.4 Crack Tip Constraint 509

REFERENCES 509

10. FATIGUE CRACK PROPAGATION 51310.1 SIMILITUDE IN FATIGUE 51310.2 EMPIRICAL FATIGUE CRACK GROWTH EQUATIONS 51710.3 CRACK CLOSURE 520

10.3.1 A Simplistic View of Closure and AKth 52510.3.2 Effects of Loading Variables on Closure 52610.3.3 The Fatigue Threshold 52810.3.4 Pitfalls and Limitations of AKeff. 530

10.4 VARIABLE AMPLITUDE LOADING ANDRETARDATION 53110.4.1 Reverse Plasticity at the Crack Tip 53110.4.2 The Effect of Overloads 53410.4.3 Analysis of Variable Amplitude Fatigue 537

10.5 GROWTH OF SHORT CRACKS 53910.5.1 Microstructurally Short Cracks 54010.5.2 Mechanically Short Cracks 541

10.6 MICROMECHANISMS OF FATIGUE 54310.6.1 Fatigue in Region II 54310.6.2 Micromechanisms Near the Threshold ....54610.6.3 Fatigue at High DK Values 546

10.7 EXPERIMENTAL MEASUREMENT OF FATIGUE CRACKGROWTH 547

10.7.1 ASTM Standard E 647 54710.7.2 Closure Measurements 549

10.8 DAMAGE TOLERANCE 552REFERENCES 555

APPENDIX 10: APPLICATION OF THE J CONTOURINTEGRAL TO CYCLIC LOADING 559

A10.1 DEFINITION OF Af 559A10.2 PATH INDEPENDENCE OF AJ 561Al0.3 SMALL SCALE YIELDING LIMIT 562

xiv

11. COMPUTATIONAL FRACTURE MECHANICS 56511.1 OVERVIEW OF NUMERICAL METHODS 565

11.1.1 The Finite Element Method 56611.1.2 The Boundary Integral Equation Method 569

11.2 TRADITIONAL METHODS IN COMPUTATIONALFRACTURE MECHANICS 571

11.2.1 Stress and Displacement Matching 57211.2.2 Elemental Crack Advance 57211.2.3 Contour Integration 57311.2.4 Virtual Crack Extension: Stiffness DerivativeFormulation 57311.2.5 Virtual Crack Extension: Continuum Approach 575

11.3 THE ENERGY DOMAIN INTEGRAL 57711.3.1 Theoretical Background 57811.3.2 Generalization to Three Dimensions 58111.3.3 Finite Element Implementation 583

11.4 MESH DESIGN 58611.5 LIMITATIONS OF NUMERICAL FRACTURE ANALYSIS 591REFERENCES 592

APPENDIX 11PROPERTIES OF SINGULARITY ELEMENTS 595Al 1.1 QUADRILATERAL ELEMENT 596A11.2 TRIANGULAR ELEMENT 598

PART V:REFERENCE MATERIAL 599

12. COMPILATION OF K, J, COMPLIANCE AND LIMITLOAD SOLUTIONS 601

12.1 THROUGH-THICKNESS CRACKS-FLAT PLATES 60112.1.1 Stress Intensity and Elastic Compliance 60112.1.2 Limit Load 60912.1.3 Fully Plastic J and Displacement 610

12.2 PART-THROUGH CRACKS-FLAT PLATES 62712.3 FLAWED CYLINDERS 633

12.3.1 Stress Intensity Factor 63412.3.2 Limit Load 64112.3.3 Fully Plastic J and Displacement 641

REFERENCES 655

13. PRACTICE PROBLEMS 65713.1 CHAPTER 1 65713.2 CHAPTER 2 65813.3 CHAPTER 3 66213.4 CHAPTER 4 66413.5 CHAPTER 5 666

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13.6 CHAPTER 6 66713.7 CHAPTER 7 66713.8 CHAPTER 8 67113.9 CHAPTER 9 67313.10 CHAPTER 10 67613.11 CHAPTER 11 679

INDEX 681

xvi

PARTI: INTRODUCTION

1. HISTORY AND OVERVIEW

Fracture is a problem that society has faced for as long as there have been man-madestructures. The problem may actually be worse today than in previous centuries, becausemore can go wrong in our complex technological society. Major airline crashes, for in-stance, would not be possible without modern aerospace technology.

Fortunately, advances in the field of fracture mechanics have helped to offset someof the potential dangers posed by increasing technological complexity. Our understandingof how materials fail and our ability to prevent such failures has increased considerablysince World War II. Much remains to be learned, however, and existing knowledge offracture mechanics is not always applied when appropriate.

While catastrophic failures provide income for attorneys and consulting engineers,such events are detrimental to the economy as a whole. An economic study [1] estimatedthe cost of fracture in the United States in 1978 at $119 billion (in 1982 dollars), about4% of the gross national product. Furthermore, this study estimated that the annual costcould be reduced by $35 billion if current technology were applied, and that further frac-ture mechanics research could reduce this figure by an additional $28 billion.

1.1 WHY STRUCTURES FAIL

The cause of most structural failures generally falls into one of the following categories:

(1) Negligence during design, construction or operation of the structure.

(2) Application of a new design or material, which produces an unexpected(and undesirable) result.

In the first instance, existing procedures are sufficient to avoid failure, but are notfollowed by one or more of the parties involved, due to human error, ignorance, or willfulmisconduct. Poor workmanship, inappropriate or substandard materials, errors in stressanalysis, and operator error are examples of where the appropriate technology and experi-ence are available, but not applied.

The second type of failure is much more difficult to prevent. When an "improved"design is introduced, there are invariably factors that the designer does not anticipate.New materials can offer tremendous advantages, but also potential problems.Consequently, a new design or material should be placed into service only after extensivetesting and analysis. Such an approach will reduce the frequency of failures, but not elim-inate them entirely; there may be important factors that are overlooked during testing andanalysis.

One of the most famous Type 2 failures is the brittle fracture of the World War HLiberty ships (see Section 1.2.2). These ships, which were the first to have an all-weldedhull, could be fabricated much faster and cheaper than earlier riveted designs, but a signif-

Chapter 1

icant number of these vessels sustained serious fractures as a result of the design change.Today, virtually all steel ships are welded, but sufficient knowledge was gained from theLiberty ship failures to avoid similar problems in present structures.

Knowledge must be applied in order to be useful, however. Figure 1.1 shows anexample of a Type 1 failure, where poor workmanship in a seemingly inconsequentialstructural detail caused a more recent fracture in a welded ship. In 1979, the Kurdistan oiltanker broke completely in two while sailing in the north Atlantic [2]. The combinationof warm oil in the tanker with cold water in contact with the outer hull produced substan-tial thermal stresses. The fracture initiated from a bilge keel that was improperly welded.The weld failed to penetrate the structural detail, resulting in a severe stress concentration.Although the hull steel had adequate toughness to prevent fracture initiation, it failed tostop the propagating crack.

Polymers, which are becoming more common in structural applications, provide anumber of advantages over metals, but also have the potential for causing Type 2 failures.For example, polyethylene (PE) is currently the material of choice in natural gas trans-portation systems in the United States. One advantage of PE piping is that maintenancecan be performed on a small branch of the line without shutting down the entire system;a local area is shut down by applying a clamping tool to the PE pipe and stopping theflow of gas. The practice of pinch clamping has undoubtedly saved vast sums of money,but has also led to an unexpected problem.

In 1983 a section of 4 in diameter PE pipe developed a major leak. The gas col-lected beneath a residence where it ignited, resulting in severe damage to the house.Maintenance records and a visual inspection of the pipe indicated that it had been pinchclamped 6 years earlier in the region where the leak developed. A failure investigation [3]concluded that the pinch clamping operation was responsible for the failure. Microscopicexamination of the pipe revealed that a small flaw apparently initiated on the inner sur-face of the pipe and grew through the wall. Figure 1.2 shows a low magnification pho-tograph of the fracture surface. Laboratory tests simulated the pinch clamping operationon sections of PE pipe; small thumbnail-shaped flaws (Fig. 1.3) formed on the innerwall of the pipes, as a result of the severe strains that were applied. Fracture mechanicstests and analyses [3, 4] indicated that stresses in the pressurized pipe were sufficient tocause the observed time-dependent crack growth; i.e., growth from a small thumbnail flawto a through-thickness crack over a period of 6 years.

The introduction of flaws in PE pipe by pinch clamping represents a Type 2 failure.The pinch clamping process was presumably tested thoroughly before it was applied inservice, but no one anticipated that the procedure would introduce damage in the materialthat could lead to failure after several years in service. Although specific data are notavailable, pinch clamping has undoubtedly led to a significant number of gas leaks. Thepractice of pinch clamping is still widespread in the natural gas industry, but many com-panies and some states now require that a sleeve be fitted to the affected region in order torelieve the stresses locally. In addition, newer grades of PE pipe material have lower den-sity and are less susceptible to damage by pinch clamping.

Introduction and Overview

(a) Fractured vessel in dry dock.

(b) Bilge keel front which the fracture initiated.

FIGURE 1.1 The MSV Kurdistan oil tanker, which sustained a brittle fracture while sailing in the northAtlantic in 1979. Photographs provided by S.J. Garwood

Chapter 1

FIGURE 1.2 Fracture surface of a PE pipe that sustained time-dependent crack growth as a result ofpinch clamping [3]. (Photograph provided by R.E. Jones Jr.)

FIGURE 1.3 Thumbnail crack produced in a PE pipe after pinch clamping for 72 hours. (Photographprovided by R.E. Jones Jr.)


Some catastrophic events include elements both of Types 1 and 2 failures. OnJanuary 28, 1986, the Challenger Space Shuttle exploded because an O-ring seal in one ofthe main boosters did not respond well to cold weather. The Shuttle represents relativelynew technology, where service experience is limited (Type 2), but engineers from thebooster manufacturer suspected a potential problem with the O-ring seals and recom-mended that the launch be delayed (Type 1). Unfortunately, these engineers had little orno data to support their position and were unable to convince their managers or NASA of-ficials. The tragic results of the decision to launch are well known.

Over the past few decades, the field of fracture mechanics has undoubtedly preventeda substantial number of structural failures. We will never know how many lives havebeen saved or how much property damage has been avoided by applying this technology,because it is impossible to quantify disasters that don't happen. When applied correctly,fracture mechanics not only helps to prevent Type 1 failures but also reduces the fre-quency of failures of the second type, because designers can rely on rational analysis ratherthan trial and error.

1.2 HISTORICAL PERSPECTIVE

Designing structures to avoid fracture is not a new idea. The fact that many structurescommissioned by the Pharaohs of ancient Egypt and the Caesars of Rome are still stand-ing is a testimony to the ability of early architects and engineers. In Europe, numerousbuildings and bridges constructed during the Renaissance Period are still used for their in-tended purpose.

The ancient structures that are still standing today obviously represent successful de-signs. There were undoubtedly many more unsuccessful designs that endured a muchshorter life span. Since mankind's knowledge of mechanics was limited prior to the timeof Isaac Newton, workable designs were probably achieved largely by trial and error. TheRomans supposedly tested each new bridge by requiring the design engineer to stand un-derneath while chariots drove over it. Such a practice would not only provide an incentivefor developing good designs, but would also result in a Darwinian natural selection, wherethe worst engineers are "removed" from the profession.

The durability of ancient structures is particularly amazing when one considers thatthe choice of building materials prior to the Industrial Revolution was rather limited.Metals could not be produced in sufficient quantity to be formed into load-bearing mem-bers for buildings and bridges. The primary construction materials prior to the 19th cen-tury were timber, brick, and mortar; only the latter two materials were usually practicalfor large structures such as cathedrals, because trees of sufficient size for support beamswere rare.

Brick and mortar are relatively brittle and are unreliable for carrying tensile loads.Consequently, pre-Industrial Revolution structures were usually designed to be loaded incompression. Figure 1.4 schematically illustrates a Roman bridge design. The archshape causes compressive rather than tensile stresses to be transmitted through the struc-ture.

Chapter 1

FIGURE 1.4 Schematic Roman bridge design. The arch shape of the bridge causes loads to be transmit-ted through the structure as compressive stresses.

The arch is the predominate shape in pre-Industrial Revolution architecture.Windows and roof spans were arched in order to maintain compressive loading. For ex-ample, Fig. 1.5 shows two windows and a portion of the ceiling in Kings CollegeChapel in Cambridge, England. Although these shapes are aesthetically pleasing, theirprimary purpose is more pragmatic.

Compressively loaded structures are obviously stable, since some have lasted formany centuries. The pyramids in Egypt are the epitome of a stable design.

With the Industrial Revolution came mass production of iron and steel. (Or, con-versely, one might argue that mass production of iron and steel fueled the IndustrialRevolution.) The availability of relatively ductile construction materials removed the ear-lier restrictions on design. It was finally feasible to build structures that carried tensilestresses. Note the difference between the design of the Tower Bridge in London (Fig. 1.6)and the earlier bridge design (Fig. 1.4).

The change from brick and mortar structures loaded in compression to steel struc-tures in tension brought problems, however. Occasionally, a steel structure would failunexpectedly at stresses well below the anticipated tensile strength. One of the most fa-mous of these failures was the rupture of a molasses tank in Boston in January 1919 [5].Over 2 million gallons of molasses were spilled, resulting in 12 deaths, 40 injuries, mas-sive property damage, and several drowned horses.

The cause of failures as the molasses tank was largely a mystery at the time. In thefirst edition of his elasticity text published in 1892, Love [6] remarked that "the condi-tions of rupture are but vaguely understood." Designers typically applied safety factors of10 or more (based on the tensile strength) in an effort to avoid these seemingly randomfailures.


FIGURE 1.5 Kings College Chapel in Cambridge, England. This structure was completed in 1515.

FIGURE 1.6 The Tower Bridge in London, completed in 1894. Note the modern beam design, madepossible by the availability of steel support girders.

10 Chapter 1

1.2.1 Early Fracture Research

Experiments performed by Leonardo da Vinci several centuries earlier provided some cluesas to the root cause of fracture. He measured the strength of iron wires and found that thestrength varied inversely with wire length. These results implied that flaws in the mate-rial controlled the strength; a longer wire corresponded to a larger sample volume and ahigher probability of sampling a region containing a flaw. These results were only quali-tative, however.

A quantitative connection between fracture stress and flaw size came from the workof Griffith, which was published in 1920 [7]. He applied a stress analysis of an ellipticalhole (performed by Inglis [81 seven years earlier) to the unstable propagation of a crack.Griffith invoked the First Law of Thermodynamics to formulate a fracture theory based ona simple energy balance. According to this theory, a flaw becomes unstable, and thusfracture occurs, when the strain energy change that results from an increment of crackgrowth is sufficient to overcome the surface energy of the material (See Section 2.3).Griffith's model correctly predicted the relationship between strength and flaw size inglass specimens. Subsequent efforts to apply the Griffith model to metals wereunsuccessful. Since this model assumes that the work of fracture comes exclusively fromthe surface energy of the material, the Griffith approach only applies to ideally brittlesolids. A modification to Griffith's model that made it applicable to metals did not comeuntil 1948.

1.2.2 The Liberty Ships

The mechanics of fracture progressed from being a scientific curiosity to an engineeringdiscipline, primarily because of what happened to the Liberty ships during World War II[9].

In the early days of World War II, the United States was supplying ships and planesto Great Britain under the Lend-Lease Act. Britain's greatest need at the time was forcargo ships to carry supplies. The German Navy was sinking cargo ships at three timesthe rate at which they could be replaced with existing ship-building procedures.

Under the guidance of Henry Kaiser, a famous construction engineer whose previousprojects included the Hoover Dam, the United States developed a revolutionary procedurefor fabricating ships quickly. These new vessels, which became known as the Libertyships, had an all-welded hull, as opposed to the riveted construction of traditional ship de-signs.

The Liberty ship program was a resounding success, until one day in 1943, whenone of the vessels broke completely in two while sailing between Siberia and Alaska.Subsequent fractures occurred in other Liberty ships. Of the roughly 2700 liberty shipsbuild during World War II, approximately 400 sustained fractures, of which 90 were con-sidered serious. In 20 ships the failure was essentially total, and about half of these brokecompletely in two.

Investigations revealed that the Liberty ship failures were caused by a combinationof three factors:

Introduction and Overview 11

0 The welds, which were produced by a semi-skilled work force, contained crack-like flaws.

• Most of the fractures initiated on the deck at square hatch corners, where therewas a local stress concentration.

8 The steel from which the Liberty ships were made had poor toughness, as mea-sured by Charpy impact tests.

The steel in question had always been adequate for riveted ships because fracturecould not propagate across panels that were joined by rivets. A welded structure, how-ever, is essentially a single piece of metal; propagating cracks in the Liberty ships en-countered no significant barriers, and were sometimes able to traverse the entire hull.

Once the causes of failure were identified, the remaining Liberty ships were retro-fit-ted with rounded reinforcements at the hatch corners. In addition, high toughness steelcrack arrester plates were riveted to the deck at strategic locations. These corrections pre-vented further serious fractures.

In the longer term, structural steels were developed with vastly improved toughness,and weld quality control standards were developed. Also, a group of researchers at theNaval Research Laboratory in Washington D.C. studied the fracture problem in detail.The field we now know as fracture mechanics was born in this lab during the decade fol-lowing the War.

1.2.3 Post-War Fracture Mechanics Research*

The fracture mechanics research group at the Naval Research Laboratory was led by Dr.G.R. Irwin. After studying the early work of Inglis, Griffith, and others, Irwin concludedthat the basic tools needed to analyzed fracture were already available. Irwin's first majorcontribution was to extend the Griffith approach to metals by including the energy dissi-pated by local plastic flow [10]. Orowan independently proposed a similar modificationto the Griffith theory [11]. During this same period, Mott [12] extended the Griffith the-ory to a rapidly propagating crack.

In 1956, Irwin [13] developed the energy release rate concept, which is related to theGriffith theory but is in a form that is more useful for solving engineering problems.Shortly afterward, several of Irwin's colleagues brought to his attention a paper byWestergaard [14] that was published in 1938. Westergaard had developed a semi-inversetechnique for analyzing stresses and displacements ahead of a sharp crack. Irwin [15] usedthe Westergaard approach to show that the stresses and displacements near the crack tipcould be described by a single constant that was related to the energy release rate. Thiscrack tip characterizing parameter later became known as the stress intensity factor.

1 For an excellent summary of early fracture mechanics research, refer to Fracture MechanicsRetrospective: Early Classic Papers (1913-1965), John M. Barsom, ed., American Society of Testing andMaterials (RPS 1), Philadelphia, 1987. This volume contains reprints of 17 classic papers, as well as acomplete bibliography of fracture mechanics papers published up to 1965.

12 Chapter 1

During this same period of time, Williams [16] applied a somewhat different technique toderive crack tip solutions that were essentially identical to Irwin's results.

A number of successful early applications of fracture mechanics bolstered the stand-ing of this new field in the engineering community. In 1956, Wells [17] used fracturemechanics to show that the fuselage failures in several Comet jet aircraft resulted from fa-tigue cracks reaching a critical size. These cracks initiated at windows and were caused byinsufficient reinforcement locally, combined with square corners which produced a severestress concentration. (Recall the unfortunate hatch design in the Liberty ships.) A secondearly application of fracture mechanics occurred at General Electric in 1957. Winne andWundt [18] applied Irwin's energy release rate approach to the failure of large rotors fromsteam turbines. They were able to predict the bursting behavior of large disks extractedfrom rotor forgings, and applied this knowledge to the prevention of fracture in actual ro-tors.

It seems that all great ideas encounter stiff opposition initially, and fracture mechan-ics is no exception. Although the U.S. military and the electric power generating indus-try were very supportive of the early work in this field, such was not the case in allprovinces of government and industry. Several government agencies openly discouragedresearch in this area.

In 1960, Paris and his co-workers [19] failed to find a receptive audience for theirideas on applying fracture mechanics principles to fatigue crack growth. Although Pariset al. provided convincing experimental and theoretical arguments for their approach, itseems that design engineers were not yet ready to abandon their S-N curves in favor of amore rigorous approach to fatigue design. The resistance to this work was so intense thatParis and his colleagues were unable to find a peer-reviewed technical journal that waswilling to publish their manuscript. They finally opted to publish their work in aUniversity of Washington periodical entitled The Trend in Engineering.

1.2.4 Fracture Mechanics from 1960 to 1980

The Second World War obviously separates two distinct eras in the history of fracture me-chanics. There is, however, some disagreement as to how the period between the end ofthe War and the present should be divided. One possible historical boundary occursaround 1960, when the fundamentals of linear elastic fracture mechanics were fairly wellestablished, and researchers turned their attention to crack tip plasticity.

Linear elastic fracture mechanics (LEFM) ceases to be valid when significant plasticdeformation precedes failure. During a relatively short time period (1960-61) several re-searchers developed analyses to correct for yielding at the crack tip, including Irwin [20],Dugdale [21], Barenblatt [22], and Wells [23]. The Irwin plastic zone correction [20] wasa relatively simple extension of LEFM, while Dugdale [21] and Barenblatt [22] each de-veloped somewhat more elaborate models based on a narrow strip of yielded material atthe crack tip.

Wells [23] proposed the displacement of the crack faces as an alternative fracture cri-terion when significant plasticity precedes failure. Previously, Wells had worked withIrwin while on sabbatical at the Naval Research Laboratory. When Wells returned to hispost at the British Welding Research Association, he attempted to apply LEFM to low-


and medium-strength structural steels. These materials were too ductile for LEFM to ap-ply, but Wells noticed that the crack faces moved apart with plastic deformation. Thisobservation led to the development of the parameter now known as the crack tip openingdisplacement (CTOD}.

In 1968, Rice [24] developed another parameter to characterize nonlinear materialbehavior ahead of a crack. By idealizing plastic deformation as nonlinear elastic, Rice wasable to generalize the energy release rate to nonlinear materials. He showed that this non-linear energy release rate can be expressed as a line integral, which he called the J integral,evaluated along an arbitrary contour around the crack. At the time his work was beingpublished, Rice discovered that Eshelby [25] had previously published several so-calledconservation integrals, one of which was equivalent to Rice's J integral. Eshelby, how-ever, did not apply his integrals to crack problems.

That same year, Hutchinson [26] and Rice and Rosengren [27] related the / integralto crack tip stress fields in nonlinear materials. These analyses showed that J can beviewed as a nonlinear stress intensity parameter as well as an energy release rate.

Rice's work might have been relegated to obscurity had it not been for the active re-search effort by nuclear power industry in the United States in the early 1970s. Becauseof legitimate concerns for safety, as well as political and public relations considerations,the nuclear power industry endeavored to apply state-of-the-art technology, including frac-ture mechanics, to the design and construction of nuclear power plants. The difficultywith applying fracture mechanics in this instance was that most nuclear pressure vesselsteels were too tough to be characterized with LEFM without resorting to enormous labo-ratory specimens. In 1971, Begley and Landes [28], who were research engineers atWestinghouse, came across Rice's article and decided, despite skepticism from their co-workers, to characterize fracture toughness of these steels with the J integral. Their exper-iments were very successful and led to the publication of a standard procedure for / testingof metals ten years later [29].

Material toughness characterization is only one aspect of fracture mechanics. In or-der to apply fracture mechanics concepts to design, one must have a mathematical rela-tionship between toughness, stress and flaw size. Although these relationships were wellestablished for linear elastic problems, a fracture design analysis based on the J integralwas not available until Shih and Hutchinson [30] provided the theoretical framework forsuch an approach in 1976. A few years later, the Electric Power Research Institute(EPRI) published a fracture design handbook [31] based on the Shih and Hutchinsonmethodology.

In the United Kingdom, Well's CTOD parameter was applied extensively to fractureanalysis of welded structures, beginning in the late 1960s. While fracture research in theU.S. was driven primarily by the nuclear power industry during the 1970s, fracture re-search in the UK was motivated largely by the development of oil resources in the NorthSea. In 1971, Burdekin and Dawes [32] applied several ideas proposed by Wells [33] sev-eral years earlier and developed the CTOD design curve, a semiempirical fracture mechan-ics methodology for welded steel structures. The nuclear power industry in the UK devel-oped their own fracture design analysis [34], based on the strip yield model of Dugdale[21]andBarenblatt[22].

14 Chapter 1

Shih [35] demonstrated a relationship between the J integral and CTOD, implyingthat both parameters are equally valid for characterizing fracture. The J-based materialtesting and structural design approaches developed in the U.S. and the British CTODmethodology have begun to merge in recent years, with positive aspects of each approachcombined to yield improved analyses. Both parameters are currently applied throughoutthe world to a range of materials.

Much of the theoretical foundation of dynamic fracture mechanics was developed inthe period between 1960 and 1980. Significant contributions were made by a number ofresearchers, as discussed in Chapter 4.

1.2.5 Recent Trends in Fracture Research

It is difficult to discuss fracture mechanics research performed since 1980 in a historicalcontext. Identifying major breakthroughs usually requires the passage of time; whatseems important today may be obsolete later, while a major discovery may be overlookedwhen it is first published. It is possible, however, to identify a few trends in recent work.

The field of fracture mechanics has matured in recent years. Current research tendsto result in incremental advances rather than major gains.

More sophisticated models for material behavior are being incorporated into fracturemechanics analyses. While plasticity was the important concern in 1960, more recentwork has gone a step further, incorporating time-dependent nonlinear material behaviorsuch as viscoplasticity and viscoelasticity. The former is motivated by the need fortough, creep-resistant high temperature materials, while the latter reflects the increasingproportion of plastics in structural applications. Fracture mechanics has also been used(and sometimes abused) in the characterization of composite materials.

Another trend in recent research is the development of microstuctural models forfracture and models to relate local and global fracture behavior of materials. A relatedtopic is the efforts to characterize and predict geometry dependence of fracture toughness.Such approaches are necessary when traditional, so-called single-parameter fracture me-chanics break down.

1.3 THE FRACTURE MECHANICS APPROACH TO DESIGN

Figure 1.7 contrasts the fracture mechanics approach with the traditional approach tostructural design and material selection. In the latter case, the anticipated design stress iscompared to the flow properties of candidate materials; a material is assumed to be ade-quate if its strength is greater than the expected applied stress. Such an approach may at-tempt to guard against brittle fracture by imposing a safety factor on stress, combinedwith minimum tensile elongation requirements on the material. The fracture mechanicsapproach (Fig. 1.7(b)) has three important variables, rather than two as in Fig. 1.7(a).The additional structural variable is flaw size, and fracture toughness replaces strength asthe relevant material property. Fracture mechanics quantifies the critical combinations ofthese three variables.


APPLIEDSTRESS

YIELD OR TENSILESTRENGTH

(a) The strength of materials approach.

FRACTURETOUGHNESS

(b) The fracture mechanics approach

FIGURE 1.7 Comparison of the fracture mechanics approach to design with the traditional strength ofmaterials approach.

There are two alternative approaches to fracture analysis: the energy criterion andthe stress intensity approach. These two approaches are equivalent in certain circum-stances. Both are discussed briefly below.

1.3.1 The Energy Criterion

The energy approach states that crack extension (i.e. fracture) occurs when the energyavailable for crack growth is sufficient to overcome the resistance of the material. Thematerial resistance may include the surface energy, plastic work, or other type of energydissipation associated with a propagating crack.

Griffith [7] was the first to propose the energy criterion for fracture, but Irwin [13]is primarily responsible for developing the present version of this approach: the energyrelease rate, §, which is defined as the rate of change in potential energy with crack areafor a linear elastic material. At the moment of fracture, § - Qc, the critical energy releaserate, which is a measure of fracture toughness.

For a crack of length 2a in an infinite plate subject to a remote tensile stress (Fig.1.8), the energy release rate is given by

7TCT a(1.1)

16 Chapter 1

FIGURE 1.8 Through-thickness crack in an infinite plate subject to a remote tensile stress. In practicalterms, "infinite" means that the width of the plate is » 2a.

where E is Young's modulus, cris the remotely applied stress, and a is the half cracklength. At fracture, §= £e, and Eq. (1.1) describes the critical combinations of stress andcrack size for failure:

E(1.2)

Note that for a constant §c value, failure stress, <jf, varies with 7/V a. The energy releaserate, g, is the driving force for fracture, while gc is the material's resistance to fracture.To draw an analogy to the strength of materials approach of Fig. 1.7(a), the applied stresscan be viewed as the driving force for plastic deformation, while the yield strength is ameasure of the material's resistance to deformation.

The tensile stress analogy is also useful for illustrating the concept of similitude. Ayield strength value measured with a laboratory specimen should be applicable to a largestructure; yield strength does not depend on specimen size, provided the material is rea-sonably homogeneous. One of the fundamental assumptions of fracture mechanics is thatfracture toughness (£c in this case) is independent of the size and geometry of the cracked


body; a fracture toughness measurement on a laboratory specimen should be applicable toa structure. As long as this assumption is valid, all configuration effects are taken intoaccount by the driving force, £. The similitude assumption is valid as long as the mate-rial behavior is predominantly linear elastic.

1.3.2 The Stress Intensity Approach

Figure 1.9 schematically shows an element near the tip of a crack in an elastic material,together with the in-plane stresses on this element. Note that each stress component isproportional to a single constant, Kj. If this constant is known, the entire stress distribu-tion at the crack tip can be computed with the equations in Fig. 1.9. This constant,which is called the stress intensity factor, completely characterizes the crack tip conditionsin a linear elastic material. (The meaning of the subscript on K is explained in Chapter2.) If one assumes that the material fails locally at some critical combination of stressand strain, then it follows that fracture must occur at a critical stress intensity, KIC. ThusKjc is an alternate measure of fracture toughness.For the plate illustrated in Fig. 1.8, the stress intensity factor is given by

(1.3)Kr = cr-v na

Failure occurs when Kj = Kjc. In this case, Kj is the driving force for fracture and Kjc isa measure of material resistance. As with £c, the property of similitude should apply toKjc. That is, Kjc is assumed to be a size-independent material property.

Comparing Eqs. (1.1) and (1.3) results in a relationship between K] and £:

!±LE

(1.4)

•cos - 'i - (0} • w1 —sin — sin —UJ U J

•cos 0^1 . (e\ . (30Y— 1 + sm — sin —2)[ UJ I 2 J.

(36^cosl — |sm| — | cos —

UJ V 2

FIGURE 1.9 Stresses near the tip of a crack in an elastic material.

18 Chapter 1

This same relationship obviously holds for £c and Kjc. Thus the energy and stress in-tensity approaches to fracture mechanics are essentially equivalent for linear elastic mate-rials.

1.3.3 Time-Dependent Crack Growth and Damage Tolerance

Fracture mechanics often plays a role in life prediction of components that are subject totime-dependent crack growth mechanisms such as fatigue or stress corrosion cracking. Therate of cracking can be correlated with fracture mechanics parameters such as the stress in-tensity factor, and the critical crack size for failure can be computed if the fracture tough-ness is known. For example, the fatigue crack growth rate in metals can usually be de-scribed by the following empirical relationship:

da~dN

m (1-5)

where da/dN is the crack growth per cycle, AK is the stress intensity range, and C and mare material constants.

Damage tolerance, as its name suggests, entails allowing subcritical flaws to remainin a structure. Repairing flawed material or scrapping a flawed structure is expensive andis often unnecessary. Fracture mechanics provides a rational basis for establishing flawtolerance limits.

Consider a flaw in a structure that grows with time (e.g. a fatigue crack or a stresscorrosion crack) as illustrated schematically in Fig. 1.10. The initial crack size is inferredfrom nondestructive examination (NDE), and the critical crack size is computed from theapplied stress and fracture toughness. Normally, an allowable flaw size would be definedby dividing the critical size by a safety factor. The predicted service life of the structurecan then be inferred by calculating the time required for the flaw to grow from its initialsize to the maximum allowable size.

FLAWSIZE

Failure

TIME

FIGURE 1.10 The damage tolerance approach to design.


1.4 EFFECT OF MATERIAL PROPERTIES ON FRACTURE

Figure 1.11 shows a simplified family tree for the Field of fracture mechanics. Most earlywork was applicable only to linear elastic materials under quasistatic conditions, whilesubsequent advances in fracture research incorporated other types of material behavior.Elastic-plastic fracture mechanics considers plastic deformation under quasistatic condi-tions, while dynamic, viscoelastic, and viscoplastic fracture mechanics include time as avariable. A dashed line is drawn between linear elastic and dynamic fracture mechanicsbecause some early research considered dynamic linear elastic behavior. The chapters thatdescribe the various types of fracture behavior are shown in Fig. 1.11. Elastic-plastic,viscoelastic, and viscoplastic fratcture behavior are sometimes included in the more generalheading of nonlinear fracture mechanics. The branch of fracture mechanics one should ap-ply to a particular problem obviously depends on material behavior.

Consider a cracked plate (Fig. 1.8) that is loaded to failure. Figure 1.12 is aschematic plot of failure stress versus fracture toughness (Kfc)- For low toughness ma-terials, brittle fracture is the governing failure mechanism, and critical stress varies lin-early with Kjc, as predicted by Eq. (1.3). At very high toughness values, LEFM is nolonger valid, and failure is governed by the flow properties of the material. At intermedi-ate toughness levels, there is a transition between brittle fracture under linear elastic con-ditions and ductile overload. Nonlinear fracture mechanics bridges the gap between LEFMand collapse. If toughness is low, LEFM is applicable to the problem, but if toughnessis sufficiently high, fracture mechanics ceases to be relevant to the problem because fail-ure stress is insensitive to toughness; a simple limit load analysis is all that is required topredict failure stress in a material with very high fracture toughness.

Table 1.1 lists various materials, together with the typical fracture regime for eachmaterial.

4

//

/ ^_p.DYNAMICFRACTURE

MECHANICS

/

/

X

LINEAR ELASTICFRACTURE

MECHANICS

,™J™™

\ELASTIC-PLASTIC

FRACTUREMECHANICS

1 , , ,\

VISCOELASTICFRACTURE

MECHANICS

LinearTime-Independent

Materials(Chapter 2)

NonlinearTime-Independent

Materials(Chapter 3)

N^k

"^xTVISCOPLASTIC Time-Dependent

FRACTURE MaterialsMECHANICS (Chapter 4)

FIGURE 1.11 Simplified family tree of fracture mechanics.

20 Chapter 1

FAILURESTRESS 2a

"7* ^-HL—^

Nonlinear FractureMechanics

FRACTURE TOUGHNESS (KIC)

FIGURE 1.12 Effect of fracture toughness on the governing failure mechanism.

TABLE 1.1Typical fracture behavior of selected materials. Temperature is ambient unless otherwise specified.

Material Typical Fracture Behavior

High strength steelLow- and medium-strength steel

Austenitic stainless steelPrecipitation-hardened aluminum

Metals at high temperaturesMetals at high strain rates

Linear elasticElastic-plastic/Fully plastic

Fully plasticLinear elasticViscoplastic

Dynamic- viscoplastic

Polymers (below Tg)*Polymers (above Tg)*

Linear elastic/ViscoelasticViscoelastic

Monolithic ceramicsCeramic composites

Ceramics at high temperatures

Linear elasticLinear elasticViscoplastic

*Tg - Glass transition temperature.

Introduction and Overview 2 1

1.5 A BRIEF REVIEW OF DIMENSIONAL ANALYSIS

At first glance, a section on dimensional analysis may seem out of place in the introduc-tory chapter of a book on fracture mechanics. However, dimensional analysis is an im-portant tool for developing mathematical models of physical phenomena, and it can helpus understand existing models. Many difficult concepts in fracture mechanics become rel-atively transparent when one considers the relevant dimensions of the problem. For ex-ample, dimensional analysis gives us a clue as to when a particular model, such as linearelastic fracture mechanics, is no longer valid.

Let us review the fundamental theorem of dimensional analysis and then look at afew simple applications to fracture mechanics.

1.5.1 The Buckingham n -Theorem

The first step in building a mathematical model of a physical phenomenon is to identifyall of the parameters that may influence the phenomenon. Assume that a problem, or atleast an idealized version of it, can be described by the following set of scalar quantities:{u, Wj, W2, . . . , Wn}. The dimensions of all quantities in this set is denoted by {[M],[Wj], [W2\, • • • , [Wfl]}. Now suppose that we wish to express the first variable, u, as afunction of the remaining parameters:

...,Wn) (1.6)

Thus the process of modeling the problem is reduced to finding a mathematical relation-ship that represents / as best as possible. We might accomplish this by performing a setof experiments in which we measure u while varying each Wi independently. The num-ber of experiments can be greatly reduced, and the modeling processes simplified, throughdimensional analysis. The first step is to identify all of the fundamental dimensionalunits (fdu's) in the problem: (Lj, L2, • . . Lm}. For example, a typical mechanics prob-lem may have {Li = length, L2 = mass, Lj = time}. We can express the dimensions ofeach quantity in our problem as the product of powers of the fdu's; i.e. for any quantityX, we have

...!^* (1.7)

The quantity X is dimensionless if [X] = I,In the set of W s, we can identify ra primary quantities that contain all of the fdu' s

in the problem. The remaining variables are secondary quantities, and their dimensionscan be expressed in terms of the primary quantities:

.[Wmm+ (j = l,2,...,ii - m) (1.8)

Thus we can define a set of new quantities, TTJ, that are dimensionless:

22 Chapter 1

(1.9)

Similarly, the dimensions of « can be expressed in terms of the dimensions of the pri-mary quantities:

(1.10)

and we can form the following dimensionless quantity:

u(i.ii)

According to the Buckingham Il-theorem, K depends only on the other dimensionlessgroups:

(1.12)

This new function, F, is independent of the system of measurement units. Note that thenumber of quantities in F has been reduced from the old function by m, the number offdu's. Thus dimensional analysis has reduced the degrees of freedom in our model, and weneed only vary n-m quantities in our experiments or computer simulations.

The Buckingham EE-theorem gives guidance on how to scale a problem to differentsizes or to other systems of measurement units. Each dimensionless group, (nf) must bescaled in order to obtain equivalent conditions at two different scales. Suppose, for exam-ple, that we wish to perform wind tunnel tests on a model of a new airplane design.Dimensional analysis tells us that we should reduce all length dimensions in the sameproportion; thus we would build a "scale" model of the airplane. The length dimensionsof the plane are not the only important quantities in the problem, however. In order tomodel the aerodynamic behavior accurately, we would need to scale the wind velocity andthe viscosity of the air in accordance with the reduced size of the airplane model.(Modifying the viscosity of the air is not practical in most cases. In real wind tunneltests, the size of the model is usually close enough to full scale that the errors introducedby not scaling viscosity are minor.)

1.5.2 Dimensional Analysis in Fracture Mechanics

Dimensional analysis proves to be a very useful tool in fracture mechanics. Later chap-ters describe how dimensional arguments play a key role in developing mathematical de-scriptions for important phenomena. For now, let us explore a few simple examples.


Consider a series of cracked plates under a remote tensile stress, a00, as illustrated inFig. 1.13. Assume that each is a two-dimensional problem; that is, the thickness dimen-sion does not enter into the problem. The first case, Fig. 1.13(a), is an edge crack oflength a in an elastic, semi-infinite plate. In this case infinite means that the plate widthis much larger than the crack size. Suppose that we wish to know how one of the stresscomponents, Oy, varies with position. We will adopt a polar coordinate system with theorigin at the crack tip, as illustrated in Fig. 1.9. A generalized functional relationship canbe written as

(1.13)

where v is Poisson' s ratio, Okl represents the other stress components, and %/ representsall nonzero components of the strain tensor. We can eliminate Ofc/ and e&/ from// bynoting that for a linear elastic problem, strain is uniquely defined by stress throughHooke's law and the stress components at a point increase in proportion to one another.Let o°° and a be the primary quantities. Invoking the Buckingham Ft - theorem gives

a a(1.14)

When the plate width is finite (Fig. 1.13(b)), an additional dimension is required to de-scribe the problem:

r Wy IT .L=F 2 —,-, — , V,0CT a a

(1-15)

Thus, one might expect Eq. (1.14) to give erroneous results when the crack extends acrossa significant fraction of the plate width. Consider a large plate and a small plate made ofthe same material (same E and v), with the same a/W ratio, loaded to the same remotestress. The local stress at an angle 0 from the crack plane in each plate would dependonly on the r/a ratio, as long as both plates remained elastic.

When a plastic zone forms ahead of the crack tip (Fig. 1.13(c)), the problem iscomplicated further. If we assume that the material does not strain harden, the yieldstrength is sufficient to define the flow properties. The stress field is given by

,v,0a a a

(1.16)

The first two functions, Fj and F2, correspond to linear elastic fracture mechanics(LEFM), while Fj is an elastic-plastic relationship. Thus, dimensional analysis tells us

24 Chapter 1

that LEFM is only valid when ry « a and a00 « oys- In Chapter 2, the same conclu-sion is reached through a somewhat more complicated argument.

cr5

• ' "w * A

(a) Edge crack m a wide elastic plate. (b) Edge crack in a finite width elastic plate.

FIGURE 1.13 Edge cracked plates subject to aremote tensile stress.

(c) Edge crack with a plastic zone at the cracktip.

Introduction and Overview 2 5

REFERENCES

1. Duga, J.J., Fisher, W.H., Buxbaum, R.W., Rosenfield, A.R., Burh, A.R., Honton, E.J.,and McMillan, S.C., "The Economic Effects of Fracture in the United States." NBSSpecial Publication 647-2, United States Department of Commerce, Washington, DCMarch 1983.

2. Garwood, S.J., Private Communication, 1990.

3. Jones, R.E. and Bradley, W.L., "Failure Analysis of a Polyethylene Natural GasPipeline." Forensic Engineering, Vol. 1, 1987, pp. 47-59.

4. Jones, R.E. and Bradley, W.L., "Fracture Toughness Testing of Polyethylene PipeMaterials." ASTM STP 995, Vol. 1, 1989, American Society for Testing and Materials,Philadelphia, pp. 447-456.

5. Shank, M.E., "A Critical Review of Brittle Failure in Carbon Plate Steel Structures Otherthan Ships." Ship Structure Committee Report SSC-65, National Academy of Science-National Research Council, Washington, DC, December, 1953.

6. Love A.E.H, A Treatise on The Mathematical Theory of Elasticity. Dover Publications,New York, 1944.

7. Griffith, A.A. "The Phenomena of Rupture and Flow in Solids." PhilosophicalTransactions, Series A, Vol. 221, 1920, pp. 163-198.

8. Inglis, C.E., "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners."Transactions of the Institute of Naval Architects, Vol. 55, 1913, pp. 219-241.

9. Bannerman, D.B. and Young, R.T., "Some Improvements Resulting from Studies ofWelded Ship Failures." Welding Journal, Vol. 25, 1946.

10. Irwin, G.R., "Fracture Dynamics." Fracturing of Metals, American Society for Metals,Cleveland, 1948, pp. 147-166.

11. Orowan, E., "Fracture and Strength of Solids." Reports on Progress in Physics, Vol.XII, 1948, p. 185-232.

12. Mott, N.F., "Fracture of Metals: Theoretical Considerations." Engineering, Vol. 165,1948, pp. 16-18.

13. Irwin, G.R., "Onset of Fast Crack Propagation in High Strength Steel and AluminumAlloys." Sagamore Research Conference Proceedings, Vol. 2, 1956, pp. 289-305.

14. Westergaard, H.M., "Bearing Pressures and Cracks." Journal of Applied Mechanics,Vol. 6, 1939, pp. 49-53.

15. Irwin, G.R., "Analysis of Stresses and Strains near the End of a Crack Traversing aPlate." Journal of Applied Mechanics, Vol. 24, 1957, pp. 361-364.

16. Williams, M.L., "On the Stress Distribution at the Base of a Stationary Crack." Journalof Applied Mechanics, Vol. 24, 1957, pp. 109-114.

2 6 Chapter 1

17. Wells, A.A., "The Condition of Fast Fracture in Aluminum Alloys with ParticularReference to Comet Failures." British Welding Research Association Report, April1955.

18. Winne, D.H. and Wundt, B.M., "Application of the Griffith-Irwin Theory of CrackPropagation to the Bursting Behavior of Disks, Including Analytical and ExperimentalStudies." Transactions of the American Society of Mechanical Engineers, Vol. 80,1958, pp. 1643-1655.

19. Paris, P.C., Gomez, M.P., and Anderson, W.P., "A Rational Analytic Theory ofFatigue." The Trend in Engineering, Vol. 13, 1961, pp. 9-14.

20. Irwin, G.R., "Plastic Zone Near a Crack and Fracture Toughness." Sagamore ResearchConference Proceedings, Vol. 4, 1961.

21. Dugdale, D.S., "Yielding in Steel Sheets Containing Slits." Journal of the Mechanicsand Physics of Solids, Vol 8, pp. 100-104.

22. Barenblatt, G.I., "The Mathematical Theory of Equilibrium Cracks in Brittle Fracture."Advances in Applied Mechanics, Vol VII, Academic Press, 1962, pp. 55-129.

23. Wells, A.A., "Unstable Crack Propagation in Metals: Cleavage and Fast Fracture."Proceedings of the Crack Propagation Symposium, Vol 1, Paper 84, Cranfield, UK,1961.

24. Rice, J.R. "A Path Independent Integral and the Approximate Analysis of StrainConcentration by Notches and Cracks." Journal of Applied Mechanics, Vol. 35, 1968,pp. 379-386.

25. Eshelby, J.D., "The Continuum Theory of Lattice Defects." Solid State Physics, Vol. 3,1956.

26. Hutchinson, J.W., "Singular Behavior at the End of a Tensile Crack Tip in a HardeningMaterial." Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp. 13-31.

27. Rice, J.R. and Rosengren, G.F., "Plane Strain Deformation near a Crack Tip in a Power-Law Hardening Material." Journal of the Mechanics and Physics of Solids, Vol. 16,1968, pp. 1-12.

28. Begley, J. A. and Landes, J.D., "The ./-Integral as a Fracture Criterion." ASTM STP 514,American Society for Testing and Materials, Philadelphia, 1972, pp. 1-20.

29. E 813-81, "Standard Test Method for Jjc, a Measure of Fracture Toughness." AmericanSociety for Testing and Materials, Philadelphia, 1981.

30. Shih, C.F. and Hutchinson, J.W., "Fully Plastic Solutions and Large-Scale YieldingEstimates for Plane Stress Crack Problems." Journal of Engineering Materials andTechnology, Vol. 98, 1976, pp. 289-295.

31. Kumar, V., German, M.D., and Shih, C.F., "An Engineering Approach for Elastic-Plastic Fracture Analysis." EPRI Report NP-1931, Electric Power Research Institute,Palo Alto, CA, 1981.


32. Burdekin, P.M. and Dawes, M.G., "Practical Use of Linear Elastic and Yielding FractureMechanics with Particular Reference to Pressure Vessels." Proceedings of the Instituteof Mechanical Engineers Conference, London, May 1971, pp. 28-37.

33. Wells, A.A., "Application of Fracture Mechanics at and Beyond General Yielding."British Welding Journal, Vol 10, 1963, pp. 563-570.

34. Harrison, R.P., Loosemore, K., Milne, I, and Dowling, A.R., "Assessment of theIntegrity of Structures Containing Defects." Central Electricity Generating BoardReport R/H/R6-Rev 2, April 1980.

35. Shih, C.F. "Relationship between the J-Integral and the Crack Opening Displacementfor Stationary and Extending Cracks." Journal of the Mechanics and Physics of Solids,Vol 29, 1981, pp. 305-326.

PART II: FUNDAMENTAL CONCEPTS

2. LINEAR ELASTIC FRACTURE MECHANICS

The concepts of fracture mechanics that were derived prior to 1960 are applicable only tomaterials that obey Hooke's law. Although corrections for small scale plasticity wereproposed as early as 1948, these analyses are restricted to structures whose global behav-ior is linear elastic.

Since 1960, fracture mechanics theories have been developed to account for varioustypes of nonlinear material behavior (i.e. plasticity, viscoplasticity, and viscoelasticity) aswell as dynamic effects. All of these more recent results, however, are extensions of lin-ear elastic fracture mechanics (LEFM). Thus a solid background in the fundamentals ofLEFM is essential to an understanding of more advanced concepts in fracture mechanics.

This chapter describes both the energy and stress intensity approaches to linear frac-ture mechanics. The early work of Inglis and Griffith is summarized, followed by an in-troduction to the energy release rate and stress intensity parameters. The appendix at theend of this chapter includes mathematical derivations of several important results inLEFM.

Subsequent chapters also address linear elastic fracture mechanics. Chapters 7 and 8discuss laboratory testing of linear elastic materials, Chapter 9 addresses application ofLEFM to structures, Chapter 10 applies LEFM to fatigue crack propagation, and Chapter1 1 outlines numerical methods for computing stress intensity factor and energy releaserate.

2.1 AN ATOMIC VIEW OF FRACTURE

A material fractures when sufficient stress and work are applied on the atomic level tobreak the bonds that hold atoms together. The bond strength is supplied by the attractiveforces between atoms.

Figure 2.1 shows schematic plots of the potential energy and force versus separationdistance between atoms. The equilibrium spacing occurs where the potential energy is ata minimum. A tensile force is required to increase the separation distance from the equi-librium value; this force must exceed the cohesive force to sever the bond completely.The bond energy is given by

Eb = \Pdx (2.1)xo

where x0 is the equilibrium spacing and p is the applied force.It is possible to estimate the cohesive strength at the atomic level by idealizing the

interatomic force-displacement relationship as one half the period of a sine wave:

31

PART II: FUNDAMENTAL CONCEPTS

2. LINEAR ELASTIC FRACTURE MECHANICS

The concepts of fracture mechanics that were derived prior to 1960 are applicable only tomaterials that obey Hooke's law. Although corrections for small scale plasticity wereproposed as early as 1948, these analyses are restricted to structures whose global behav-ior is linear elastic.

Since 1960, fracture mechanics theories have been developed to account for varioustypes of nonlinear material behavior (i.e. plasticity, viscoplasticity, and viscoelasticity) aswell as dynamic effects. All of these more recent results, however, are extensions of lin-ear elastic fracture mechanics (LEFM). Thus a solid background in the fundamentals ofLEFM is essential to an understanding of more advanced concepts in fracture mechanics.

This chapter describes both the energy and stress intensity approaches to linear frac-ture mechanics. The early work of Inglis and Griffith is summarized, followed by an in-troduction to the energy release rate and stress intensity parameters. The appendix at theend of this chapter includes mathematical derivations of several important results inLEFM.

Subsequent chapters also address linear elastic fracture mechanics. Chapters 7 and 8discuss laboratory testing of linear elastic materials, Chapter 9 addresses application ofLEFM to structures, Chapter 10 applies LEFM to fatigue crack propagation, and Chapter11 outlines numerical methods for computing stress intensity factor and energy releaserate.

2.1 AN ATOMIC VIEW OF FRACTURE

A material fractures when sufficient stress and work are applied on the atomic level tobreak the bonds that hold atoms together. The bond strength is supplied by the attractiveforces between atoms.

Figure 2.1 shows schematic plots of the potential energy and force versus separationdistance between atoms. The equilibrium spacing occurs where the potential energy is ata minimum. A tensile force is required to increase the separation distance from the equi-librium value; this force must exceed the cohesive force to sever the bond completely.The bond energy is given by

= \Pdx (2.1)

where x0 is the equilibrium spacing and p is the applied force.It is possible to estimate the cohesive strength at the atomic level by idealizing the

interatomic force-displacement relationship as one half the period of a sine wave:

31

32 Chapter 2

= Pr sin (2.2a)

where the distance A is defined in Fig. 2.1. For the sake of simplicity, the origin is de-fined at x0. For small displacements, the force-displacement relationship is linear:

Repulsion

POTENTIALENERGY

Attraction

Tension

APPLIED

FORCE

Compression

I*-*

I

DISTANCE

FIGURE 2.1 Potential energy and force as afunction of atomic separation. At the equilib-rium separation, x0, the potential energy isminimized, and the attractive and repellingforces are balanced.

Linear Elastic Fracture Mechanics 3 3

and the bond stiffness (i.e., the spring constant) is given by

n_I

= Pc- (2.3)

Multiplying both sides of this equation by the number of bonds per unit area and the gagelength, x0, converts k to Young's modulus, £, and Pc to the cohesive stress, <7C.Solving for ac gives

Elac = (2.4)

nx0or

E<7C~— (2.5)

71

if A is assumed to be approximately equal to the atomic spacing.The surface energy can be estimated as follows:

I ,T~ I I J • - ,— ,-»

2 QC V A J c TT

The surface energy per unit area, %., is equal to one half the fracture energy because twosurfaces are created when a material fractures. Substituting Eq. (2.4) into Eq. (2.6) andsolving for <JC gives

(2-7)

2.2 STRESS CONCENTRATION EFFECT OF FLAWS

The derivation in the previous section showed that the theoretical cohesive strength of amaterial is approximately E/n, but experimental fracture strengths for brittle materials aretypically three or four orders of magnitude below this value. As discussed in Chapter 1,experiments by Leonardo da Vinci, Griffith, and others indicated that the discrepancy be-tween the actual strengths of brittle materials and theoretical estimates was due to flaws inthese materials. Fracture cannot occur unless the stress at the atomic level exceeds thecohesive strength of the material. Thus the flaws must lower the global strength bymagnifying the stress locally.

34 Chapter 2

The first quantitative evidence for the stress concentration effect of flaws was pro-vided by Inglis [1], who analyzed elliptical holes in flat plates. His analyses included anelliptical hole 2a long by 2b wide with an applied stress perpendicular to the major axisof the ellipse (see Fig. 2.2). He assumed that the hole is not influenced by the plateboundary; i.e., the plate width » 2a and the plate height » 2b . The stress at the tip ofthe major axis (Point A) is given by

2ab

(2.8)

The ratio o^/cris defined as the stress concentration factor, kt. When a - b, the hole iscircular and kt = 3.0, a well-known result that can be found in most strength of materialstext books.

As the major axis, a , increases relative to b, the elliptical hole begins to take onthe appearance of a sharp crack. For this case, Inglis found it more convenient to expressEq. (2.8) in terms of the radius of curvature, p:

(2.9)

FIGURE 2.2 Elliptical hole in a flat plate.


where

b2

p-— (2.10)a

When a » b, Eq. (2.9) becomes

(2.11)

Inglis showed that Eq. (2.11) gives a good approximation for the stress concentration dueto a notch that is not elliptical except at the tip.

Equation (2.11) predicts an infinite stress at the tip of an infinitely sharp crack,where p=0. This result caused concern when it was first discovered, because no materialis capable of withstanding infinite stress. A material that contains a sharp crack theoreti-cally should fail upon the application of an infinitesimal load. The paradox of a sharpcrack motivated Griffith [2] to develop a fracture theory based on energy rather than localstress (Section 2.3).

An infinitely sharp crack in a continuum is a mathematical abstraction that is notrelevant to real materials, which are made of atoms. Metals, for instance, deform plasti-cally, which causes an initially sharp crack to blunt. In the absence of plastic deforma-tion, the minimum radius a crack tip can have is on the order of the atomic radius. Bysubstituting p=x0 into Eq. [2.11] we obtain an estimate of the local stress concentrationat the tip of an atomically sharp crack:

(2.12)

If it is assumed that fracture occurs when <JA=(JC, Eq. (2.12) can be set equal to Eq. (2.7),resulting in the following expression for the remote stress at failure:

/2

Equation (2.13) must be viewed as a rough estimate of failure stress, because the contin-uum assumption upon which the Inglis analysis is based is not valid at the atomic level.However, Gehlen and Kanninen [3] obtained similar results from a numerical simulationof a crack in a- two-dimensional lattice, where discrete "atoms" were connected by nonlin-ear springs:

/2(Jf = a —- (2.14)J \ a

3 6 Chapter 2

where a is a constant, on the order of unity, which depends slightly on the assumedatomic force-displacement law (Eq. (2.2)).

2.3 THE GRIFFITH ENERGY BALANCE

According to the First Law of Thermodynamics, when a system goes from a nonequilib-rium state to equilibrium, there will be a net decrease in energy. In 1920 Griffith appliedthis idea to the formation of a crack [2] :

It may be supposed, for the present purpose, that the crack is formed by the suddenannihilation of the tractions acting on its surface. At the instant following thisoperation, the strains, and therefore the potential energy under consideration, havetheir original values; but in general, the new state is not one of equilibrium. If it isnot a state of equilibrium, then, by the theorem of minimum potential energy, thepotential energy is reduced by the attainment of equilibrium; if it is a state ofequilibrium the energy does not change.

A crack can form (or an existing crack can grow) only if such a process causes thetotal energy to decrease or remain constant. Thus the critical conditions for fracture canbe defined as the point where crack growth occurs under equilibrium conditions, with nonet change in total energy.

Consider a plate subjected to a constant stress, cr, which contains a crack la long(Fig. 2.3). Assume that the plate width » la and that plane stress conditions prevail.(Note that the plates in Figs. 2.2 and 2.3 are identical when a » b). In order for thiscrack to increase in size, sufficient potential energy must be available in the plate to over-come the surface energy of the material. The Griffith energy balance for an incrementalincrease in the crack area, dSi , under equilibrium conditions can be expressed in the fol-lowing way:

dE dn dW, n— = — + —^" = 0 (2.15a)dA d%.

ordR dWs

— — = — -$- (2.15b)d& dA

where E is the total energy, /I is the potential energy supplied by the internal strain en-ergy and external forces, and Ws is the work required to create new surfaces. For thecracked plate illustrated in Fig. 2.3, Griffith used the stress analysis of Inglis [1] to showthat

where U0 is the potential energy of an uncracked plate and B is the plate thickness. Sincethe formation of a crack requires the creation of two surfaces, Ws is given by

Linear Elastic Fracture Mechanics 37

FIGURE 2.3 A through-thickness crack in an infinitely wide plate subjected to a remote tensile stress.

Ws=4aB7s

where % is the surface energy of the material. Thus:

(2.17)

andE

= 27,

(2.18a)

(2.18b)

Equating (2.18a) and (2.18b) and solving for fracture stress gives

na(2.19)

It is important to note the distinction between crack area and surface area. The crackarea is defined as the projected area of the crack (2aB in the present example), but since acrack includes two matching surfaces, the surface area =2>?.

38 Chapter 2

TTTT7FIGURE 2.4 A penny-shaped(circular) crack embedded in a solidsubjected to a remote tensile stress.

The Griffith approach can be applied to other crack shapes. For example, the frac-ture stress for a penny-shaped flaw embedded in the material (Fig. 2.4) is given by

(2.20)

where a is the crack radius and v is Poisson's ratio.

2.3.1 Comparison with the Critical Stress Criterion

The Griffith model is based on a global energy balance: for fracture to occur, the energystored in the structure must be sufficient to overcome the surface energy of the material.Since fracture involves breaking bonds, the stress on the atomic level must be > the co-hesive stress. This local stress intensification can be provided by flaws in the material, asdiscussed in Section 2.2.

The similarity between Eqs. (2.13), (2.14), and (2.19) is obvious. Predictions ofthe global fracture stress from the Griffith approach and the local stress criterion differ byless than 40 percent. Thus these two approaches are consistent with one another, at leastin the case of a sharp crack in an ideally brittle solid.

An apparent contradiction emerges when the crack tip radius is significantly greaterthan the atomic spacing. The change in stored energy with crack formation (Eq. (2.16)) isinsensitive to the notch radius as long as a»b; thus the Griffith model implies that thefracture stress is insensitive to p. According to the Inglis stress analysis, however, in or-der for <JC to be attained at the tip of the notch, c^must vary with l/-\/P •


Consider a crack with p = 5 x 10"" m. Such a crack would appear sharp under alight microscope, but p would be four orders of magnitude larger than the atomic spacingin a typical crystalline solid. Thus the local stress approach would predict a global frac-ture strength 100 times larger than the Griffith equation. Actual material behavior issomewhere between these extremes; fracture stress does depend on notch root radius, butnot to the extent implied by the Inglis stress analysis.

The apparent discrepancy between the critical stress criterion and the energy criterionbased on thermodynamics can be resolved by viewing fracture as a nucleation and growthprocess. When the global stress and crack size satisfy the Griffith energy criterion, thereis sufficient thermodynamic driving force to grow the crack, but fracture must first be nu-cleated. This situation is analogous to the solidification of liquids. Water, for example,is in equilibrium with ice at 0°C, but the liquid-solid reaction requires ice crystals to benucleated, usually on the surface of another solid (e.g., your car windshield on a Januarymorning). When nucleation is suppressed, liquid water can be supercooled (at least mo-mentarily) to as much as 30°C below the equilibrium freezing point.

Nucleation of fracture can come from a number of sources. For example, micro-scopic surface roughness at the tip of the flaw could produce sufficient local stress concen-tration to nucleate failure. Another possibility, illustrated in Fig. 2.5, involves a sharpmicrocrack near the tip of a macroscopic flaw with a finite notch radius. The macroscopiccrack magnifies the stress in the vicinity of the microcrack, which propagates when it sat-isfies the Griffith equation. The microcrack links with the large flaw, which then propa-gates if the Griffith criterion is satisfied globally. This type of mechanism controlscleavage fracture in ferritic steels, as discussed in Chapter 5.

2.3.2 Modified Griffith Equation

Equation (2.19) is valid only for ideally brittle solids. Griffith obtained good agree-ment between Eq. (2.19) and experimental fracture strength of glass, but the Griffith equa-tion severely underestimates the fracture strength of metals.

Irwin [4] and Orowan [5] independently modified the Griffith expression to accountfor materials that are capable of plastic flow. The revised expression is given by

(2.21)

where jp is the plastic work per unit area of surface created, and is typically much largerthan ys,

In an ideally brittle solid, a crack can be formed merely by breaking atomic bonds;7S reflects the total energy of broken bonds in a unit area. When a crack propagatesthrough a metal, however, dislocation motion occurs in the vicinity of the crack tip, re-sulting in additional energy dissipation.

Although, Irwin and Orowan originally derived Eq. (2.21) for metals, it is possibleto generalize the Griffith model to account for any type of energy dissipation:

40 Chapter 2

local _» r

t t t t t

inn•ai-

FIGURE 2.5 A sharp microcrack at the tip of a macroscopic crack.

EXAMPLE 2.1

A flat plate made from a brittle material contains a macroscopic through-thicknesscrack with half length aj and notch tip radius p. A sharp penny-shaped microcrackwith radius a2 is located near the tip of the larger flaw, as illustrated in Fig. 2.5. Esti-mate the minimum size of the microcrack to cause failure in the plate when theGriffith equation is satisfied by the global stress and aj.

Solution: The nominal stress at failure is obtained by substituting aj into Eq. (2.19).The stress in the vicinity of the microcrack can be estimated from Eq. (2.11), which isset equal to the Griffith criterion for the penny-shaped microcrack (Eq. 2.20):

/2

2(l-v2}a2

Solving for #2 gives

a? =16(l-v2)

For v = 0.3, a2 = 0,68 p. Thus the nucleating microcrack must be approximately thesize of the macroscopic crack tip radius.


EXAMPLE 2.1 (cont.)

This derivation contains a number of simplifying assumptions. The notch tipstress computed from Eq. (2.11) is assumed to act uniformly ahead of the notch, in theregion of the microcrack; the actual stress would decay away from the notch tip.Also, this derivation neglects free boundary effects from the tip of the macroscopicnotch.

Of = (2.22)

where wy is the fracture energy, which could include plastic, viscoelastic, or viscoplasticeffects, depending on the material. The fracture energy can also be influenced by crackmeandering and branching, which increase the surface area. Figure 2.6 illustrates varioustypes of material behavior and the corresponding fracture energy.

A word of caution is necessary when applying Eq. (2.22) to materials that exhibitnonlinear deformation. The Griffith model, in particular Eq. (2.16), applies only to linearelastic material behavior. Thus the global behavior of the structure must be elastic. Anynonlinear effects, such as plasticity, must be confined to a small region near the crack tip.In addition, Eq. (2.22) assumes that wy is constant; in many ductile materials, the frac-ture energy increases with crack growth, as discussed in Section 2.5.

2.4 THE ENERGY RELEASE RATE

In 1956, Irwin [6] proposed an energy approach for fracture that is essentially equivalentto the Griffith model, except that Irwin's approach is in a form that is more convenientfor solving engineering problems. Irwin defined an energy release rate, </, which is ameasure of the energy available for an increment of crack extension:

(2.23)

The term rate, as it is used in this context, does not refer to a derivative with respect totime; £Fis the rate of change in potential energy with crack area. Since § is obtainedfrom the derivative of a potential, it is also called the crack extension force or the crackdriving force. According to Eq. (2.18a), the energy release rate for a wide plate in planestress with a crack of length la (Fig. 2.3) is given by

(2.24)

42 Chapter 2

CRACK Broken Bonds

(a) Ideally brittle material.

Crack Propagation Plastic Deformation j

Wf =

Wf = YS + Jp

(b) Quasi-brittle elastic-plastic material.

/ True Area \V's [Projected Area/

(c) Brittle material with crack meandering and branching.

FIGURE 2.6 Crack propagation in various types of materials, with the corresponding fracture energy.


Referring to the previous section, crack extension occurs when § reaches a critical value;i.e.,

C2-25)

where @c is a measure of tiie fracture toughness of the material.The potential energy of an elastic body, II, is defined as follows:

U = U-F (2.26)

where U is the strain energy stored in the body and F is the work done by external forces.Consider a cracked plate that is dead loaded, as illustrated in Fig. 2.7. Since the load

is fixed at P, the structure is said to be load controlled. For this case,

and

Therefore,

and

A

0

n = -u

i-2

I f d U } P f d__ - = - —B\da)p 2B\da

When displacement is fixed (Fig. 2.8), the plate is displacement controlled; F = 0 andn=U. Thus

l(dU\ A f d P= -- - = -- - (2.28)

It is convenient at this point to introduce the compliance, which is the inverse of theplate stiffness:

C = — (2.29)P

By substituting Eq. (2.29) into Eqs. (2.27) and (2.28) it can be shown that

g-*-* (2.30)2B da

44 Chapter 2

LOAD -

DISPLACEMENT

(a) (b)

FIGURE 2.7 Cracked plate at a fixed load, P.

-Tmiiin. I

LOAD

.1DISPLACEMENT

(a) (b)

FIGURE 2.8 Cracked plate at a fixed displacement, D.


for both load control and displacement control. Therefore, the energy release rate, as de-fined in Eq. (2.23), is the same for load control and displacement control. Also,

(2.31)dU\ =

da Jp V da

Equation (2.31) is demonstrated graphically in Figs. 2.7b and 2.8b. In load control,a crack extension da results in a net increase in strain energy because of the contributionof the external force P:

PdA PdA

When displacement is fixed, dF=0 and the strain energy decreases:

where dP is negative. As can be seen in Figs 2.7b and 2.8b, the absolute values of theseenergies differ by the amount dPdk 1 2 , which is negligible. Thus

(dU)P =

for an increment of crack growth at a given P and A.

EXAMPLE 2.2Determine the energy release rate for a double cantilever beam (DCS) specimen (Fig.2.9)

Solution: From beam theory,

A Pa3

where / =2 3 E I

The elastic compliance is given by

4 _ 2 a3

P ~ 3 E I

Bh3

12

46 Chapter 2

EXAMPLE 2.2 (cont.)

Substituting C into Eq. 2.30 gives

P2a2 12 P2 a2

^^B E I = B2 h3 E

Thickness = B

FIGURE 2.9 Double cantilever beam (DCB) specimen.

2.5 INSTABILITY AND THE R CURVE

Crack extension occurs when g=2wf, but crack growth may be stable or unstable, depend-ing on how £?and wf, vary with crack size. To illustrate stable and unstable behavior, itis convenient to replace 2wf, with R, the material resistance to crack extension. A plotof R versus crack extension is called a resistance curve or R curve. The correspondingplot of § versus crack extension is the driving force curve.

Consider a wide plate with a through crack of initial length 2ao (Fig. 2.3). At afixed remote stress, cr, the energy release rate varies linearly with crack size (Eq. (2.24).Figure 2.10 shows schematic driving force//? curves for two types of material behavior.

The first case, Fig. 2.10a, shows a flat R curve, where the material resistance isconstant with crack growth. When the stress = 07, the crack is stable. Fracture occurswhen the stress reaches 0*2 , the crack propagation is unstable because the driving forceincreases with crack growth, but the material resistance remains constant.


Figure 2.10b illustrates a material with a rising R curve. The crack grows a smallamount when the stress reaches 02 > but cannot grow further unless the stress increases.When stress is fixed at 02 »the driving force increases at a slower rate than R. Stablecrack growth continues as the stress increases to Oj. Finally, when the stress reaches (74,the driving force curve is tangent to the R curve. The plate is unstable with further crackgrowth because the rate of change in driving force exceeds the slope of the R curve.

The conditions for stable crack growth can be expressed as follows:

and

da da

(2.32a)

(2.32b)

Unstable crack growth occurs when

dG-/

da

dR_da

(2.33)

When the resistance curve is flat, as in Fig 2.10a, one can define a critical value ofenergy release rate, gc, unambiguously. A material with a rising R curve, however, can-

not be uniquely characterized with a single toughness value. According to Eq. (2.33) aflawed structure fails when the driving force curve is tangent with the R curve, but thispoint of tangency depends on the shape of the driving force curve, which depends on con-figuration of the structure. The driving force curve for the through crack configuration islinear, but </in the DCB specimen (Example 2.2) varies with a^\ these two configura-tions would have different Rvalues for a given R curve.

a.0 a.oCRACK SIZE CRACK SIZE

(a) Flat R curve (b) Rising R curve

FIGURE 2.10 Schematic driving force/R curve diagrams.

48 Chapter 2

Materials with rising R curves can be characterized by the value of ^"at initiation ofcrack growth. Although the initiation toughness is usually not sensitive to structural ge-ometry, there are other problems with this measurement. It is virtually impossible to de-termine the precise moment of crack initiation in most materials; an engineering defini-tion of initiation, analogous to the 0.2 percent offset yield strength in tensile tests, isusually required. Another limitation of initiation toughness is that it characterizes onlythe onset of crack growth; it provides no information on the shape of the R curve.

2.5.1 Reasons for the R Curve Shape

Some materials exhibit a rising R curve, while the R curve for other materials is flat.The shape of the R curve depends on material behavior and, to a lesser extent, on the con-figuration of the cracked structure.

The R curve for an ideally brittle material is flat because the surface energy is an in-variant material property. When nonlinear material behavior accompanies fracture, how-ever, the R curve can take on a variety of shapes. For example, ductile fracture in metalsusually results in a rising R curve; a plastic zone at the tip of the crack increases in sizeas the crack grows. The driving force must increase in such materials to maintain crackgrowth. If the cracked body is infinite (i.e. if the plastic zone is small compared to rele-vant dimensions of the body) the plastic zone size and R eventually reach steady-state val-ues, and the R curve becomes flat with further growth (see Section 3.5.2).

Some materials can display a falling R curve. When a metal fails by cleavage, forexample, the material resistance is provided by the surface energy and local plastic dissipa-tion, as illustrated in Fig. 2.6b. The R curve would be relatively flat if the crack growthwere stable. However, cleavage propagation is normally unstable; the material near thetip of the growing crack is subject to very high strain rates, which suppress plastic de-formation. Thus the resistance of a rapidly growing cleavage crack is less than the initialresistance at the onset of fracture.

The size and geometry of the cracked structure can exert some influence on the shapeof the R curve. A crack in a thin sheet tends to produce a steeper R curve than a crack ina thick plate because the thin sheet is loaded predominantly in plane stress, while materialnear the tip of the crack in the thick plate may be in plane strain. The R curve can alsobe affected if the growing crack approaches a free boundary in the structure; Thus a wideplate may exhibit a somewhat different crack growth resistance behavior than a narrowplate of the same material

Ideally, the R curve, as well as other measures of fracture toughness, should be aproperty only of the material and not depend on the size or shape of the cracked body.Much of fracture mechanics is predicated on the assumption that fracture toughness is amaterial property. Configurational effects can occur, however; a practitioner of fracturemechanics should be aware of these effects and their potential influence on the accuracy ofan analysis. This issue is explored in detail in Sections 2.10, 3.5, and 3.6.

2.5.2 Load Control Versus Displacement Control

According to Eqs. (2.32) and (2.33), the stability of crack growth depends on the rate ofchange in £ i.e., the second derivative of potential energy. Although the driving force(£) is the same for both load control and displacement control, the rate of change of thedriving force curve depends on how the structure is loaded.


Displacement control tends to be more stable than load control. With some config-urations, the driving force actually decreases with crack growth in displacement control.A typical example is illustrated in Fig. 2.11.

Referring to Fig. 2.11, consider a cracked structure subjected to a load Py and a dis-placement A-$. If the structure is load controlled, it is at the point of instability, wherethe driving force curve is tangent to the R curve. In displacement control, however, thestructure is stable because the driving force decreases with crack growth; the displacementmust be increased for further crack growth.

When an R curve is determined experimentally, the specimen is usually tested indisplacement control (or as near to pure displacement control as is possible in the testmachine). Since most of the common test specimen geometries exhibit falling drivingforce curves in displacement control, it is possible to obtain a significant amount of sta-ble crack growth. If an instability occurs during the test, the R curve cannot be definedbeyond the point of ultimate failure.

CRACK SIZE

FIGURE 2.11 Schematic driving forceAR curve diagram which compares load control and displacementcontrol.

EXAMPLE 2.3

Evaluate the relative stability of a DCB specimen (Fig. 2.9) in load control and dis-placement control.

Solution: From the result derived in Example 2.2, the slope of the driving force curvein load control is given by

50 Chapter 2

EXAMPLE 2.3 (cont.).

da

2P a

BEI

In order to evaluate displacement control, it is necessary to expressand a. From beam theory, load is related to displacement as follows:

n terms of A

P =3AEI

substituting the above equation into expression for energy release rate gives

9A2EI

4Ba

Thus

Ba4$a

Therefore, the driving force increases with crack growth in load control and decreasesin displacement control. For a flat R curve, crack growth in load control is always un-stable, while displacement control is always stable.

2.5.3 Structures with Finite Compliance

Most real structures are subject to conditions between pure load control and pure dis-placement control. This intermediate situation can be schematically represented by aspring in series with the flawed structure (Fig. 2.12). The structure is fixed at a constantremote displacement, Aj\ the spring represents the system compliance, Cm. Pure dis-placement control corresponds to an infinitely stiff spring, where Cm. = 0. Load control

(dead loading) implies an infinitely soft spring; i.e., Cm. = °°.When the system compliance is finite, the point of fracture instability obviously

lies somewhere between the extremes of pure load control and pure displacement control.However, determining the precise point of instability requires a rather complex analysis.

At the moment of instability, the following conditions are satisfied:

(2.34a)and

Linear Elastic Fracture Mechanics

f-(.daj da

51

(2.34b)

The left side of Eq. (2.34b) is given by [7]:

dg\a? a _

(2.35)

Equation (2.35) is derived in Appendix 2.2.

2.6 STRESS ANALYSIS OF CRACKS

For certain cracked configurations subjected to external forces, it is possible to deriveclosed-form expressions for the stresses in the body, assuming isotropic linear elastic ma-terial behavior. Westergaard [8], Irwin [9], Sneddon [10] and Williams [11] were amongthe first to publish such solutions. If we define a polar coordinate axis with the origin atthe crack tip (Fig. 2.13), it can be shown that the stress field in any linear elastic crackedbody is given by

(2.36)

A

FIGURE 2.12 A cracked structurewith finite compliance, representedschematically by a spring in series.

l-*(r\ > > ''is! *V -^ '

' ~V v*\-î*»«y..-;j;-"

52 Chapter 2

where Oy is the stress tensor, r and 6 are as defined in Fig. 2.13, k is a constant, and fjj isa dimensionless function of 6. The higher order terms depend on geometry, but the solu-tion for any given configuration contains a leading term that is proportional to 1/Vr . Asr—» 0, the leading term approaches infinity, but the other terms remain finite or approach

zero. Thus stress near the crack tip varies with 1/Vr, regardless of the configuration ofthe cracked body. It can also be shown that displacement near the crack tip varies with*Jr. Equation (2.36) describes a stress singularity, since stress is asymptotic to r= 0.The basis of this relationship is explored in more detail in Appendix 2.3.

FIGURE 2.13 Definition of the coordi-nate axis ahead of a crack tip. The z di-rection is normal to the page.

There are three types of loading that a crack can experience, as Fig. 2.14 illustrates..Mode I loading, where the principal load is applied normal to the crack plane, tends toopen the crack. Mode II corresponds to in-plane shear loading and tends to slide one crackface with respect to the other. Mode III refers to out-of-plane shear. A cracked body canbe loaded in any one of these modes, or a combination of two or three modes.

2.6.1 The Stress Intensity Factor

Each mode of loading produces the 1/Vr" singularity at the crack tip, but the proportion-ality constant, k, and fy depend on mode. It is convenient at this point to replace k by

the stress intensity factor, K, where K = k-^2n. The stress intensity factor is usuallygiven a subscript to denote the mode of loading; i.e., Kj ,,Kj /, or Km. Thus the stressfields ahead of a crack tip in an isotropic linear elastic material can be written as

lim

lim

limr-»0

(2.37a)

(2.37b)

(2.37c)


Mode I(Opening)

Mode II(In-Plane Shear)

Mode III(Out-of-Plane Shear)

FIGURE 2.14 The three modes of loading that can be applied to a crack.

for Modes I, II, and III, respectively. In a mixed-mode problem (i.e., when more than oneloading mode is present), the individual contributions to a given stress component are ad-ditive:

(2.38)

Equation (2.38) stems from the principle of linear superposition.Detailed expressions for the singular stress fields for Modes I and II are given in

Table 2.1. Displacement relationships for Modes I and II are listed in Table 2.2. Table2.3 lists the nonzero stress and displacement components for Mode EL

Consider the Mode I singular field on the crack plane, where 0=0. According toTable 2.1, the stresses in the x and y direction are equal:

(2.39)

When 6=0, the shear stress is zero, which means that the crack plane is a principalplane for pure Mode I loading. Figure 2.15 is a schematic plot of CT-y-y, the stress normalto the crack plane, versus distance from the crack tip. Equation (2.39) is only valid near

the crack tip, where the 1/Vr singularity dominates the stress field. Stresses far from

the crack tip are governed by the remote boundary conditions. For example, if the crackedstructure is subjected to a uniform remote tensile stress, <jyy, approaches a constantvalue, o°°. We can define a singularity dominated zone as the region where the equationsin Tables 2.1 to 2.3 describe the crack tip fields.

54 Chapter 2

TABLE 2.1Stress fields ahead of a crack tip for Mode I and Mode II in a linear elastic, isotropic material.

cxx

CJyy

^

a,

£

Model

KI ~Je~\

KI J6]V2^C°\2j

L UJ UJJ

" . re^ . /^36>V

K! (Q\ . (6} f30^

0 (Plane Stress)V (orxx + <?yy) (Plane Strain)

0

Modell

KH (o^ re} fsayi

KU . f e \ (e\ wi sin cos cos —

V f £}\ /" A\ / ^Q£l \~1Jv I f 1 v I . f t / ) . f - ) C / | l, - cos — 1 — sin — sift —

•y27Cr \2 ) \2j \2 yj

0 (Plane Stress)v (axx + °yy) (Plane Strain)

0

t) is Poisson's ratio.

J TABLE 2.2Crack tip displacement fields for Mode I and Mode II (linear elastic, isotropic material).

Model ModeH

Kcos —

2sm —

sm —2

?f+l-2cos2 — cos — /c-

p. is the shear modulusK = 3 - 4v (plane strain)K = (3 - v)/(l -f v) (plane stress)


K,

Km (oTV7 = -7=^COS —Lyz

r . (0sinn nn v-

TABLE 2.3 Non-zero stress and displacementcomponents in Mode III (linear elastic, isotropicmaterial).

/\ 0

Singularity dominatedzone

FIGURE 2.15 Stress normal tothe crack plane in Mode I.

The stress intensity factor defines the amplitude of the crack tip singularity. Thatis, stresses near the crack tip increase in proportion to K. Moreover, the stress intensityfactor completely defines the crack tip conditions; if K is known, it is possible to solvefor all components of stress, strain, and displacement as a function of r and 9. This sin-gle-parameter description of crack tip conditions turns out to be one of the most impor-tant concepts in fracture mechanics.

2.6.2 Relationship between K and Global Behavior

In order for the stress intensity factor to be useful, one must be able to determine K. fromremote loads and the geometry. Closed-form solutions for K have been derived for anumber of simple configurations. For more complex situations the stress intensity factorcan be estimated by experiment or numerical analysis (see Chapter 11).

One configuration for which a closed-form solution exists is a through crack in aninfinite plate subjected to a remote tensile stress (Fig. 2.3). Since the remote stress, <7, isperpendicular to the crack plane, the loading is pure Mode I. Linear elastic bodies must

56 Chapter 2

undergo proportional stressing; i.e., all stress components at all locations increase in pro-portion to the remotely applied forces. Thus the crack tip stresses must be proportionalto the remote stress, and Kjaa. According to Eq. (2.37), stress intensity has units of

stress* v length. Since the only relevant length scale in Fig. 2.3 is the crack size, the re-lationship between Kf and the global conditions must have the following form:

(2.40)

The actual solution, which is derived in Appendix 2.3, is given by

«•,= (2.41)

Thus the amplitude of the crack tip singularity for this configuration is proportional tothe remote stress and the square root of crack size. The stress intensity factor for Mode IIloading of the plate in Fig. 2.3 can be obtained by replacing a in Eq. (2.41) by the re-motely applied shear stress (see Fig 2.18 and Eq. (2.43) below).

A related solution is that for a semi-infinite plate with an edge crack (Fig. 2.16).Note that this configuration can be obtained by slicing the plate in Fig. 2.3 through themiddle of the crack. The stress intensity factor for the edge crack is given by

(2.42)

nrmFIGURE 2.16 Edge crack in a semi-infinite plate subject to a remote tensile stress.


FIGURE 2.17 Comparison of crack opening displacements for an edge crack and through crack. Theedge crack opens wider at a given stress, resulting in a stress intensity that is 12% higher.

which is similar to Eq. (2.41). The 12% increase in Kf for the edge crack is caused bydifferent boundary conditions at the free edge. As Fig. 2.17 illustrates, the edge crackopens more because it is less restrained than the through crack, which forms an ellipticalshape when loaded.

Consider a through crack in an infinite plate where the normal to the crack plane isoriented at an angle j3 with the stress axis (Fig. 2.18a). If fl =£ 0, the crack experiencescombined Mode I and Mode II loading; AT/// = 0 as long as the stress axis and the cracknormal both lie in the plane of the plate. If we redefine the coordinate axis to coincidewith the crack orientation (Fig. 2.18b), we see that the applied stress can be resolved intonormal and shear components. The stress normal to the crack plane, <Jy'y' , producespure Mode I loading, while ix'y' applies Mode II loading to the crack. The stress inten-sity factors for the plate in Fig. 2.18 can be inferred by relating Gyy' and tx'y' to dand/3 through Mohr's circle:

K, = CTVV

= CTCOS (2.43a)

and

(2.43b)

Note that Eq. (2.43) reduces to the pure Mode I solution when /3 = 0. The maximum KJJoccurs at /3 = 45°, where the shear stress is also at a maximum. Section 2.11 addressesfracture under mixed mode conditions.

The penny-shaped crack in an infinite medium (Fig. 2.4) is another configuration forwhich a closed-form Kj solution exists [11]:

58 Chapter 2

\

1>

\ \ \(a) (b)

FIGURE 2.18 Through crack in an infinite platefor the general case where the principal stress isnot perpendicular to the crack plane.

(2.44)

where a is crack radius. Note that Eq. (2.44) has the same form as the previous relation-ships for a through crack, except that the crack radius is the characteristic length in theabove equation. The more general case of an elliptical or semi-elliptical flaw is illustratedin Fig. 2.19. In this instance, two length dimensions are needed to characterize the cracksize: 2c and 2a, the major and minor axes of the ellipse, respectively (see Fig. 2.19).Also, when a < c, the stress intensity factor varies along the crack front, with the maxi-mum Kj at 0= 90°. The flaw shape parameter, Q, is obtained from an elliptic integral,as discussed in Appendix 2.4 Figure 2.19 gives an approximate solution for Q.

2.6.3 Effect of Finite Size

Most configurations for which there is a closed-form K solution consist of a crack with asimple shape (e.g. a rectangle or ellipse) in an infinite plate. Stated another way, thecrack dimensions are small compared to the size of the plate; the crack tip conditions arenot influenced by external boundaries. As the crack size increases, or as the plate dimen-sions decrease, the outer boundaries begin to exert an influence on the crack tip. In suchcases, a closed-form stress intensity solution is usually not possible.


Embedded Flaw: Surface Flaw:

A =

/ \1.65(2=1 + 1.464 -

\CJ

.13-0.09f-l fl + 0.1(1-sin0)21\cj i J

9 I asinz 0 +1 -

FIGURE 2.19 Mode I stress intensity factors for elliptical and semi-elliptical cracks. These solutions arevalid only as long as the crack is small compared to the plate dimensions and a <c.

! s t

60 Chapter 2

Consider a cracked plate subjected to a remote tensile stress. Figure 2.20 schemati-cally illustrates the effect of finite width on the crack tip stress distribution, which is rep-resented by lines of force; the local stress is proportional to the spacing between lines offorce. Since a tensile stress cannot be transmitted through a crack, the lines of force arediverted around the crack, resulting in a local stress concentration. In the infinite plate,the line of force at a distance W from the crack center line has force components in the xand y directions. If the plate width is restricted to 2W, the x force must be zero on thefree edge; this boundary condition causes the lines of force to be compressed, which re-sults in a higher stress intensification at the crack tip.

One technique to approximate the finite width boundary condition is to assume a pe-riodic array of collinear cracks in an infinite plate (Fig. 2.21). The Mode I stress inten-sity factor for this situation is given by

'2W (natanl •

na \2W(2-45)

The stress intensity approaches the infinite plate value as a/W approaches zero; Kf isasymptotic to a/W= I .

More accurate solutions for a through crack in a Finite plate have been obtained fromfinite element analysis; solutions of this type are usually fit to a polynomial expression.One such solution [12] is given by

secna

l - 0 . 0 2 5 — + 0 . 0 6 f —{w

(2.46)

Figure 2.22 compares the finite width corrections in Eqs. (2.45) and (2.46). The secantterm (without the polynomial term) in Eq. (2.46) is also plotted. Equation (2.45) agreeswith the finite element solution to within 7% for a/W < 0.6. The secant correction ismuch closer to the finite element solution; the error is less than 2% for a/W < 0.9. Thusthe polynomial term in Eq. [2.46] contributes little and can be neglected in most cases.

Table 2.4 lists stress intensity solutions for several common configurations. TheseKI solutions are plotted in Fig. 2.23. Chapter 12 contains a more extensive collection ofK solutions. Several handbooks devoted solely to stress intensity solutions have alsobeen published [12-14].

Although stress intensity solutions are given in a variety of forms, K can always berelated to the through crack (Fig. 2.4) through the appropriate correction factor:

(2.47)

where a is a characteristic stress, a is a characteristic crack dimension, and Y is a dimen-sionless constant that depends on geometry and mode of loading.


(a) Infinite plate (b) Finite plate

FIGURE 2.20 Stress concentration effects due to a through crack in finite and infinite width plates.

r -2W- *1

r J [J& m '"|P"'1

FIGURE 2.21 Collinear cracks in an infinite plate subject to remote tension.

EXAMPLE 2.4

Show that the Kj solution for the single edge notched tensile panel reduces to Eq.(2.42) when a « W.

Solution: All of the Kj expressions in Table 2.4 are of the form:

P f(w)

62 Chapter 2

EXAMPLE 2.4 (cent.)

where P is the applied force, B is plate thickness, andffaAV) is a dimensionless func-tion. The above equation can be expressed in the form of Eq. (2.47):

where

In the limit of a small flaw, the geometry correction factor in Table 2.4 becomes

Thus

lim /(t) =a/W -> 0 V /

lim (Y) = 1.12a/W -*0

W [0.752 + 0.37]

4 -

3 -

0.2 0.4 0.6

a / W0.8

FIGURE 2.22 Comparison of finite width corrections for a center cracked plate in tension.


TABLE 2.4Kj solutions for common test specimens [12].

GEOMETRY f(a/w)Single Edge Notched Tension (SENT)

*..

2 tan

cos-na

2W

Single Edge Notched Bend (SENS)P/Zw s »a.P/2

i

1.99

w w

Center Cracked Tension (CCT) •sec-2W

1-0.025

w

Double Edge Notched Tension (DENT)i

3

II"

1w

Compact Specimen 2+

1,25 W

w

5V Wf ( a I W) where B is the specimen thickness.

64 Chapter 2

IfS

0.4 0.6

a / W

FIGURE 2.23 Plot of stress intensity solutions from Table 2.4..

2.6.4 Principle of Superposition

For linear elastic materials, individual components of stress, strain, and displacement areadditive. For example, two normal stresses in the x direction imposed by different exter-nal forces can be added to obtain the total crxr, but a normal stress cannot be summedwith a shear stress. Similarly, stress intensity factors are additive as long as the mode ofloading is consistent. That is,

but- r — JV r ~r JL\. j ~r JL\. r

'(total) * &

In many instances, the principle of superposition allows stress intensity solutionsfor complex configurations to be built from simple cases for which the solutions are wellestablished. Consider, for example, an edge cracked panel (Table 2.4) subject to combinedmembrane (axial) loading, Pm, and three-point bending, P^. Since both types of loadingimpose pure Mode I conditions, the Kj values can be added:

_ -rAmembrane) , ^(bending}


(2.48)

where fm and//? are the geometry correction factors for membrane and bending loading, re-spectively, listed in Table 2.4 and plotted in Fig. 2.23.

EXAMPLE 2.5

Determine the stress intensity factor for a semi-elliptical surface crack subjected to aninternal pressure, p (Fig. 2.24(a)).

Solution: The principle of superposition enables us to construct the solution fromknown cases. One relevant case is the semi-elliptical surface flaw under uniform re-mote tension, p (Fig. 2.24(b)). If we impose a uniform compressive stress, -p, on thecrack surface (Fig. 2.24(c)), K]=Q because the crack faces close and the plate behavesas if the crack were not present. The loading configuration of interest is obtained bysubtracting the stresses in Fig. 2.24(c) from those of Fig. 2.24(b):

(b) (c)

FIGURE 2.24 Determination of Kj for a semi-elliptical surface crack under internal pressure, p, bymeans of the principle of superposition.

66 Chapter 2

Example 2.5 is a simple illustration of a more general concept: namely, stresses act-ing on the boundary (i.e., tractions) can be replaced with tractions that act on the crackface, such that the two loading configurations (boundary tractions versus crack face trac-tions) result in the same stress intensity factor. Consider an uncracked body subject to aboundary traction P(x), as illustrated in Figure 2.25. This boundary traction results in anormal stress distribution p(x) on Plane A-B. In order to confine the problem to Mode I,let us assume that no shear stresses act on Plane A-B. (This assumption is made only forthe sake of simplicity; the basic principle can be applied to all three modes of loading.)Now assume that a crack that forms on Plane A-B and the boundary traction, P(x), re-mains fixed, as Fig. 2.26(a) illustrates. If we remove the boundary traction and apply atraction p(x) on the crack face (Fig 2.26(b)), the principle of superposition indicates thatthe applied KI will be unchanged. That is,

(since K^ = 0)

P(x)

FIGURE 2.25 Uncracked body subject to anarbitrary boundary traction P(x), which resultsin a normal stress distribution p(x) acting onPlane A-B.

FIGURE 2.26 Application of superposition to replace a boundary traction P(x) with a crack face trac-tion p(x) that results in the same Kj.


2.6.5 Weight Functions

When one performs an analysis to infer a stress intensity factor for a cracked body, the Kvalue that is computed applies only to one particular set of boundary conditions; differentloading conditions result in a different stress intensity factors for that geometry. It turnsout, however, that the solution to one set of boundary conditions contains sufficient in-formation to infer K for any other boundary conditions on that same geometry.

Consider two arbitrary loading conditions on an isotropic elastic cracked body inplane stress or plane strain. For now, we assume that both loadings are symmetric withrespect to the crack plane, such that pure Mode I loading is achieved in each case.Suppose that we know the stress intensity factor for loading (1) and we wish to solve for/Jf/2), the stress intensity factor for the second set of boundary conditions. Rice [15]

showed that /£/•*/ and Kp^' are related as follows:

• ~ ' '" • " ' " (2.49)/

where F and A are the perimeter and area of the body, respectively, and M/ are the dis-placements in the x and y directions. Since loading systems (1) and (2) are arbitrary, itfollows that Kf2> cannot depend on Kp') and u/'). Therefore, the function

E du; '(2.50)

where xi represents the x and y coordinates, must be independent of the nature of loadingsystem (1). Bueckner [16] derived a similar result to Eq. (2.50) two years before Rice,and referred to h as a weight function.

Weight functions are first order tensors that depend only on the geometry of thecracked body. Given the weight function for a particular configuration, it is possible tocompute K{ from Eq. (2.49) for any boundary conditions. Moreover, the previous sectioninvoked the principle of superposition to show that any loading configuration can be rep-resented by appropriate tractions applied directly to the crack face. Thus KI for a two-di-mensional cracked body can be inferred from the following expression.

Kt = \p(x}h(x}dx (2.51)

where p(x) is the crack face traction (equal to the normal stress acting on the crack planewhen the body is uncracked) and Tc is the perimeter of the crack. The weight function,h(x), can be interpreted as the stress intensity resulting from a unit force applied to thecrack face at x.

68 Chapter 2

EXAMPLE 2.6

Derive an expression for Kj for an arbitrary traction on the face of a through crack in

an infinite plate.

Solution: We already know Kf for this configuration when a uniform tensile stress is

applied:

= o\m

where a is the half crack length. From Eq. (A2.43), the opening displacement of thecrack faces in this case is given by

(2a-x)

where the x-y coordinate axis is defined in Fig 2.27(a). The since the crack length is2a we must differentiate Uy with respect to 2a rather than a:

duy

d(2a)

Thus the weight function for this crack geometry is given by

h(x) = ±

If we apply a surface traction of ±p(x) on the crack faces, the Mode I stress intensityfactor for the two crack tips is as follows:

2a

0

2a

2a-x dx

Jrnd

2a-xdx


ayfc

(a) Definition of coordinate axes (b) Arbitrary traction applied to crack faces.

FIGURE 2.27 Through crack configuration analyzed in Example 2.6.

The weight function concept is not restricted to two-dimensional bodies, Mode Iloading, or isotropic elastic materials. In their early work on weight functions, Rice [15]extended the theory to three dimensions, Bueckner [16] considered combined Mode I/IIloading, and both allowed for anisotropy in the elastic properties. Subsequent researchers[17-22] have shown that the theory applies to all linear elastic bodies that contain an arbi-trary number of cracks.

For mixed-mode problems, separate weight functions are required for each mode: h/,/z//, and /z///. Since the stress intensity factors can vary along a three-dimensional crackfront, the weight functions also vary along the crack front. That is,

}i = hry(xm 7?) (2.52)

where a (= 1,2,3) indicates the mode of loading and 77 is the crack front position.Given that any loading configuration in a cracked body can be represented by equiva-

lent crack face tractions, the general mixed-mode three-dimensional formulation of theweight function approach can be expressed in the following form:

(2-53)sc

where 7/ are the tractions assumed to act on the crack surface, Sc.

2.7 RELATIONSHIP BETWEEN K AND Q

Two parameters that describe the behavior of cracks have been introduced so far: the en-ergy release rate and the stress intensity factor. The former parameter quantifies the netchange in potential energy that accompanies an increment of crack extension; the latterquantity characterizes the stresses, strains, and displacements near the crack tip. The en-

7 Q Chapter 2

ergy release rate describes global behavior, while K is a local parameter. For linear elasticmaterials, K and Q are uniquely related.

For a through crack in an infinite plate subject to a uniform tensile stress (Fig. 2.3),§ and Kj are given by Eqs. (2.24) and (2.41), respectively. Combining these two equa-tions leads to the following relationship between £and Kj for plane stress:

(2.54)

For plane strain conditions, E must be replaced by E/(l-v2). To avoid writing separateexpressions for plane stress and plane strain, the following notation will be adoptedthroughout this book:

E' — E for plane stress (2.55a)and

£E' = - =- for plane strain (2.55b)

1- V

Thus the § - Kj relationship for both plane stress and plane strain becomes

(2.56)K2

Since Eqs. (2.24) and (2.41) apply only to a through crack in an infinite plate, wehave yet to prove that Eq. (2.56) is a general relationship that applies to all configura-tions. Irwin [9] performed a crack closure analysis that provides such a proof.

Consider a crack of initial length a -f Aa subject to Mode I loading, as illustrated inFig. 2.28(a). It is convenient in this case to place the origin a distance Aa behind thecrack tip. Assume that the plate has unit thickness. Let us now apply a compressivestress field to the crack faces between x = 0 and x = Aa of sufficient magnitude to closethe crack in this region. The work required to close the crack at the tip is related to theenergy release rate:

(2.57)

where AU is the work of crack closure, which is equal to the sum of contributions towork from x = 0 to x = Aa :

A/7= J dU(x) (2.58)jt=0

and the incremental work at x is equal to the area under the force-displacement curve:


FIGURE 2.28 Application of closurestresses which shorten a crack by Aa.

dU(x) = 2—Fy(x)Uy(x) = <jyy(x)Uy(x}dx (2.59)

The factor of 2 on work is required because both crack faces are displaced an absolute dis-tance Uy(x). The crack opening displacement, uy, for Mode I is obtained from Table 2.2by setting Q = it.

(2.60)

where Kj(a + Aa) denotes the stress intensity factor at the original crack tip. The normalstress required to close the crack is related to Kj for the shortened crack:

ff» ~ V27K(2.61)

Combining Eqs. (2.57) to (2.61) gives

= lim

7 2 Chapter 2

(K + l)Kj K]- \ - L-L = __L (2.62)8/1 £'

Thus Eq. (2.56) is a general relationship for Mode I. The above analysis can be repeatedfor other modes of loading; the relevant closure stress and displacement for Mode II is iyx

and ux and the corresponding quantities for Mode III are Tyz and uz. When all threemodes of loading are present, the energy release rate is given by

Ef E'(2>63)

Contributions to § from the three modes are additive because energy release rate, like en-ergy, is a scalar quantity. Equation (2.63), however, assumes self similar crack growth;i.e., a planar crack is assumed to remain planar and maintain a constant shape as it grows.Such is usually not the case for mixed-mode fracture. See Section 2.1 1 for further discus-sion of energy release rate in mixed-mode problems

2.8 CRACK TIP PLASTICITY

Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. Inreal materials, however, stresses at the crack tip are finite because the crack tip radiusmust be finite (Section 2.2). Inelastic material deformation, such as plasticity in metalsand crazing in polymers, leads to further relaxation of crack tip stresses.

The elastic stress analysis becomes increasingly inaccurate as the inelastic region atthe crack tip grows. Simple corrections to linear elastic fracture mechanics (LEFM) areavailable when moderate crack tip yielding occurs. For more extensive yielding, onemust apply alternative crack tip parameters that take nonlinear material behavior into ac-count (see Chapter 3).

The size of the crack tip yielding zone can be estimated by two methods: the Irwinapproach, where the elastic stress analysis is used to estimate the elastic-plastic boundary,and the strip yield model. Both approaches lead to simple corrections for crack tip yield-ing. The term plastic zone usually applies to metals, but will be adopted here to describeinelastic crack tip behavior in a more general sense. Differences in the yielding behaviorbetween metals and polymers are discussed in Chapter 6.

2.8.1 The Irwin Approach

On the crack plane (6=0) the normal stress, oyv, in a linear elastic material is given byEq. (2.39). As a first approximation, we can assume that the boundary between elasticand plastic behavior occurs when the stresses given by Eq. (2.39) satisfy a yield criterion.For plane stress conditions, yielding occurs when <jyy = crys, the uniaxial yield strengthof the material. Substituting yield strength into the left side of Eq. (2.39) and solving forr gives a first order estimate of plastic zone size:


(2.64)

If we neglect strain hardening, the stress distribution for r < ry can be represented by ahorizontal line at oy-y = ays, as Fig. 2.29 illustrates; the stress singularity is truncatedby yielding at the crack tip.

The simple analysis in the preceding paragraph is not strictly correct because it wasbased on an elastic crack tip solution. When yielding occurs, stresses must redistribute inorder to satisfy equilibrium. The cross-hatched region in Fig. 2.29 represents forces thatwould be present in an elastic material but cannot be carried in the elastic-plastic materialbecause the stress cannot exceed yield. The plastic zone must increase in size in order toaccommodate these forces. A simple force balance leads to a second order estimate of theplastic zone size, r.

Integrating and solving for rp gives

r = !(-£.

(2.65)

K\.ayS)

which is twice as large as rv, the first order estimate.

(2.66)

FIGURE 2,29 First-order and second-order estimates of plastic zone size (ry and rp, respectively). Thecross-hatched area represents load that must be redistributed, resulting in a larger plastic zone.

74 Chapter 2

Referring to Fig. 2.29, note that the redistributed stress in the elastic region ishigher than Eq. (2.39) predicts, implying a higher effective stress intensity factor. Irwin[23] accounted for this increase in K by defining an effective crack length that is slightlylonger than the actual crack size. He found that a good approximation of Kejycan be ob-tained by placing the tip of the effective crack in the center of the plastic zone, as Fig.2.30 illustrates. Thus the effective crack length is defined as the sum of the actual cracksize and a plastic zone correction:

a*ff (2.67)

where ry for plane stress is given by Eq. (2.64). In plane strain, yielding is suppressed bythe triaxial stress state, and the approximate plastic zone correction is smaller by a factorof three:

(2.68)

The effective stress intensity is obtained by inserting aeff\n\.Q the K expression forthe geometry of interest:

(2.69)

Since the effective crack size is taken into account in the geometry correction factor, 7,an iterative solution is usually required to solve for Keff. That is, K is first determined inthe absence of a plasticity correction; a first order estimate of aeff\s then obtained fromEq. (2.64) or (2.68), which in turn used to estimate Keff. A new aeffis computed fromthe Keff estimate, and the process is repeated until successive Keff estimates converge.Typically, no more than three or four iterations are required for reasonable convergence.

FIGURE 2.30 The Irwin plastic zone correction. The increase in the effective stress intensity is takeninto account by assuming the crack is longer by ry.


In certain cases, this iterative procedure is unnecessary because a closed-form solu-tion is possible. For example, the effective Mode I stress intensity factor for a throughcrack in an infinite plate in plane stress is given by

(2.70)

Elliptical and semi-elliptical flaws (Fig. 2.20) also have an approximate closed form plas-tic zone correction, provided the flaw is small compared to plate dimensions. In the caseof the embedded elliptical flaw, Keff_ is given by

Tiasin (f) + — | cos

where Qeffis the effective flaw shape parameter, defined as

/ \2

(2.71)

= 2-0.212 -a

(2.72)

Equation (2.72) must be multiplied by surface correction factor for a semi-elliptical sur-face flaw (see Fig. 2.20).

2.8.2 The Strip Yield Model

The strip yield model, which is illustrated in Fig 2.31, was first proposed by Dugdale[24] and Barenblatt [25]. They assumed a long, slender plastic zone at the crack tip in anonhardening material in plane stress. These early analyses considered only a throughcrack in an infinite plate. The strip yield plastic zone is modeled by assuming a crack oflength la + 2p, where p is the length of the plastic zone, with a closure stress equal to<7ys applied at each crack tip (Fig. 2.3l(b)).

(a) (b)

FIGURE 2.31 The strip yield model. The plastic zone is modeled by yield magnitude compressive stressesat each crack tip (b).

76 Chapter 2

This model approximates elastic-plastic behavior by superimposing two elastic so-lutions: a through crack under remote tension and a through crack with closure stresses atthe tip. Thus the strip yield model is a classical application of the principle of superposi-tion.

Since the stresses are finite in the strip yield zone, there cannot be a stress singular-ity at the crack tip. Therefore, the leading term in the crack tip field that varies with1/Vr (Eq. 2.36) must be zero. The plastic zone length, p, must be chosen such that thestress intensity factors from the remote tension and closure stress cancel one another.

The stress intensity due to the closure stress can be estimated by considering a nor-mal force P applied to the crack at a distance x from the center line of the crack (Fig.2.32). The stress intensities for the two crack tips are given by

K

Ki (2.73)

assuming the plate is of unit thickness. The closure force at a point within the strip yieldzone is equal to

(2.74)

Thus the total stress intensity at each crack tip resulting from the closure stresses is ob-tained by replacing a with a + p in Eq. (2.73) and summing the contribution from bothcrack tips:

^closurea + p + x la + p — x .

dx

dx(2.75)

FIGURE 2.32 Crack opening forceapplied at a distance x from the centerline.

Linear Elastic Fracture Mechanics

Solving this integral gives

77

K.closure = COS

The stress intensity from the remote tensile stress, K0 =

with Kciosure. Therefore,

a \ KG \= cos

a + p {2aYSJ(2.77)

Note that p approaches infinity as a —> <Jys- Let us explore the strip yield model further

by performing a Taylor series expansion on Eq. (2.77):

a + p

_ 1 no~ \( KG -If. na2l2oYS

(2.78)

Neglecting all but the first two terms and solving for the plastic zone size gives

9 9 f_ K a a _K(~ ~ (2.79)

YS<

for a « OYS- N°te the similarity between Eqs. (2.79) and (2.66); since 1/n =0.318 and7Z/S = 0.392, the Irwin and strip yield approaches predict similar plastic zone sizes.

One way to estimate the effective stress intensity with the strip yield model is to setaeffêqual to a+p:

Keff = (Jna

(2.80)

However, Eq. (2.80) tends to overestimate Keff, the actual aeff is somewhat less thana+p because the strip yield zone is loaded to dy^. Burdekin and Stone [26] obtained amore realistic estimate of Kefffor the strip yield model:

Keff =8

_... ....

n2

no

2(J YS

(2.81)

Refer to Appendix 3.1 for a derivation of Eq. (2.81).

78 Chapter 2

2.8.3 Comparison of Plastic Zone Corrections

Figure 2.33 shows a comparison between a pure LEFM analysis (Eq. (2.41)), the Irwincorrection for plane stress (Eq. (2.70)), and the strip yield correction on stress intensity(Eq. (2.81)). The effective stress intensity, nondimensionalized by (Jy^V^, is plottedagainst the normalized stress. The LEFM analysis predicts a linear relationship betweenK and stress. Both the Irwin and strip yield corrections deviate from LEFM theory atstresses greater than 0.5 (Jys- The two plasticity corrections agree with each other up toapproximately 0.85 ffyS- According to the strip yield model, Keff

ls infinite at yield; thestrip yield zone extends completely across the plate, which has reached its maximum loadcapacity.

The plastic zone shape predicted by the strip yield model bears little resemblance toactual plastic zones in metals (see below), but many polymers produce crack tip crazezones which look very much like Fig. 2.31. Thus although Dugdale originally proposedthe strip yield model to account for yielding in thin steel sheets, this model is bettersuited to polymers (see Chapter 6).

2.8.4 Plastic Zone Shape

The estimates of plastic zone size that have been presented so far consider only the crackplane 6=0. It is possible to estimate the extent of plasticity at all angles by applying anappropriate yield criterion to the equations in Tables 2.1 and 2.3. Consider the von Misesequation:

rimLEFM

Irwin Correction

Strip Yield Correction

FIGURE 233 Comparison of plastic zone corrections for a through crack in plane strain.


(2.82)

where oe is the effective stress, and 07, (72, and 03 are the three principal normalstresses. According to the von Mises criterion, yielding occurs when O"e=ay£, the uniax-ial yield strength. For plane stress or plane strain conditions, the principal stresses can becomputed from the two-dimensional Mohr's circle relationship:

0i,02 =xx 'xx yy (2.83)

For plane stress, 03 = 0, and (J3 = V((J1 + <72) for plane strain. Substituting the Mode

I stress fields into Eq. (2.83) gives

KI (6<ji = .—L- cos — || i + smi —V27rr 12

02 = - ii 1 • i ^-Ill-sm^-

(2.84a)

(2.84b)

(73 = 0 (plane stress)

—r=^= COS — (plane strain)V2flr UJ

(2.84c)

By substituting Eq. (2.84) into Eq. (2.82), setting oôys, and solving for r, we obtainestimates of the Mode I plastic zone radius as a function of 6:

(2.85a)

for plane stress, and

3 .2S11 (2.85b)

for plane strain. Equations (2.85a) and (2.85b), which are plotted in Fig. 2.34(a), definethe approximate boundary between elastic and plastic behavior. The corresponding equa-tions for Modes II and III are plotted in Figs 2.34(b) and 2.34(c), respectively.

80 Chapter 2

2 r

ifKnl2 1E-"ane

*LffYsJ „ = I/ x

(a) Model (b) Mode II

ryKm'

1.521

0.5

0

0.5

1

1.5

2

: MODE III

{ /""

i /: i' „

i

-\ \~_ \

\i

^yFIGURE 2.34 Crack tip plastic zone shapes esti-mated from the elastic solutions (Tables 2.1 and2.3) and the von Mises yield criterion.

(c) Modelll

Note the significant difference in the size and shape of the Mode I plastic zones forplane stress and plane strain. The latter condition suppresses yielding, resulting in asmaller plastic zone for a given Kj value.

Equations (2.85a) and (2.85b) are not strictly correct because they are based on apurely elastic analysis. Recall Fig. 2.29, which schematically illustrates how crack tipplasticity causes stress redistribution, which is not taken into account in Fig. 2.34. TheIrwin plasticity correction, which accounts for stress redistribution by means of an effec-tive crack length, is also simplistic and not totally correct.

Figure 2.35 compares the plane strain plastic zone shape predicted from Eq. (2.85b)with a detailed elastic-plastic crack tip stress solution obtained from finite element analy-sis. The latter, which was published by Dodds, et al. [27], assumed a material with thefollowing uniaxial stress-strain relationship:

(2.86)

where e0, (J0, a, and n are material constants. We will examine the above relationshipin more detail in Chapter 3; for now it is sufficient to note that the exponent, n, charac-


terizes the strain hardening rate of a material. Dodds, et al. analyzed materials with n = 5,10, and 50, which corresponds to high, medium, and low strain hardening, respectively.Figure 2.35 shows contours of constant <je for n = 50. The definition of the elastic-plas-tic boundary is somewhat arbitrary, since materials that can be described by Eq. (2.86) donot have a definite yield point. When the plastic zone boundary is defined at ae = <Jys(the 0.2% offset yield strength), the plane strain plastic zone is considerably smaller thanpredicted by Eq. (2.85b). Defining the boundary at a slightly lower effective stress resultsin a much larger plastic zone. Given the difficulties of defining the plastic zone unam-biguously with a detailed analysis, the estimates of plastic zone size and shape from theelastic analysis (Fig. 2.34) appear to be reasonable.

Figure 2.33 illustrates the effect of strain hardening on the plastic zone. A highstrain hardening rate results in a smaller plastic zone because the material inside of theplastic zone is capable of carrying higher stresses, and less stress redistribution is neces-sary.

FIGURE 2.35 Contours of constant ef-fective stress in Mode I, obtained fromfinite element analysis [27]. The elas-tic-plastic boundary estimated fromEq. (2.85a) is shown for comparison.

0.35

I 0.25Kil2

0.15

0.05

0.05

0.15

0.25

0.35

Plane Strain

n:

FIGURE 2.36 Effect of strain harden-ing on the Mode I plastic zone; n = 5corresponds to a high strain hardeningmaterial, while n = 50 corresponds tovery low hardening (cf. Eq. (2.86)).

82 Chapter 2

2.9 PLANE STRESS VERSUS PLANE STRAIN

Most of the classical solutions in fracture mechanics reduce the problem to two dimen-sions. That is, at least one of the principal stresses or strains is assumed to equal zero(plane stress or plane strain, respectively).

In general, the conditions ahead of a crack are neither plane stress nor plane strain,but are three-dimensional. There are, however, limiting cases where a two-dimensionalassumption is valid, or at least provides a good approximation.

Consider a cracked plate with thickness B subject to in-plane loading, as illustratedin Fig. 2.37. For the moment, assume that the plastic zone is small; the effect of cracktip plasticity is considered later. If there were no crack, the plate would be in a state ofplane stress. Thus, regions of the plate that are sufficiently far from the crack tip mustalso be loaded in plane stress. Material near the crack tip is loaded to higher stresses thanthe surrounding material. Because of the large sfre^s_npjm^tojhe_crack plane, the cracktip material tries to contract in the x and z directions, but is prevented from doing so bythe surrounding material (Fig. 2.37 (b)). This constraint causes a triaxial state of stressnear the crack tip. For r « B, plane strain conditions exist in the interior of the plate.Material on the plate surface is in a state of plane stress, however, because there are nostresses normal to the free surface.

Figure 2.38 schematically illustrates the through-thickness variation of stress andstrain in the z direction for r « B. At the plate surface, crzz = 0 and £zz is at its maxi-mum (absolute) value. At the midplane (z=0), plane strain conditions exist and crzz =V(&xx+ &yy) (assuming r » ry). There is a region near the plate surface where thestress state is neither plane stress nor plane strain.

3%v *" •" X •* -u "* -, * ""

FIGURE 2.37 Three-dimensional deformation at the tip of a crack. The high normal stress at the cracktip causes material near the surface to contract, but material in the interior is constrained, resulting in atriaxial stress state.


Figure 2.39 is a plot of ozz as a function of z/B and rIB. These results were obtainedfrom a three-dimensional elastic-plastic finite element analysis performed by Narasimhanand Rosakis [28]. Note the transition from plane strain (at mid thickness) to plane stressas r increases relative to thickness.

'zz

FIGURE 2.38 Schematic variation of transverse stress and strain through the thickness at a point nearthe crack tip.

0.1 0.2 0.3z /B

0.4 0.5

FIGURE 2.39 Transverse stress through the thickness as a function of distance from the crack tip. [28].

84 Chapter 2

The stress state at the elastic-plastic boundary depends on the plastic zone size rela-tive to the plate thickness. Plane strain conditions exist at the boundary if the plasticzone is small compared to the thickness, but the stress state is predominantly plane stressif the plastic zone is of the same order as the thickness. Figure 2.40 shows Mode I plas-tic zones at mid-thickness computed from a three-dimensional elastic-plastic finite ele-ment analysis performed by Nakamura and Parks [29]. The elastic-plastic boundary is de-fined at ae - ays in this case- As (Kl/&YS)2 increases relative to thickness, the plasticzone grows, as one might expect. It is interesting, however, to note the change in shapeof the elastic-plastic boundary: at low Kj values, the plastic zone has a typical planestrain shape, but takes on a plane stress shape as Kj increases (cf. Fig. 2.31(a)). If thestress state remained constant, the plastic zone size would increase in proportion to(Kl/<jys)2 and would retain a constant shape; the plastic zone actually increases at afaster rate because the stress state changes from plane strain to plane stress as Kj in-creases. Although the stress state at the elastic-plastic boundary is predominately planestress when the plastic zone size is of the order of half the plate thickness (or larger), atriaxial stress state may exist deep inside the plastic zone.

2.10 K AS A FAILURE CRITERION

Section 2.6.1 introduced the concept of the singularity dominated zone and alluded to sin-gle-parameter characterization of crack tip conditions. The stresses near the crack tip in a

linear elastic material vary as 1/V r, the stress intensity factor defines the amplitude of thesingularity. Given the equations in Tables 2.1 to 2.3, one can completely define thestresses, strains, and displacements in the singularity dominated zone if the stress inten-sity factor is known. If we assume a material fails locally at some combination ofstresses and strains, then crack extension must occur at a critical K value. This Kc

value, which is a measure of fracture toughness, is a material constant that is independentof the size and geometry of the cracked body. Since energy release rate is uniquely relatedto stress intensity (Section 2.7), £"also provides a single-parameter description of cracktip conditions, and §c is an alternative measure of toughness.

The forgoing discussion does not consider plasticity or other types of nonlinear ma-

terial behavior at the crack tip. Recall that the 1/Vr singularity applies only to linearelastic materials. The equations in Tables 2.1 to 2.3 do not describe the stress distribu-tion inside the plastic zone. As discussed in Chapters 5 and 6, the microscopic eventsthat lead to fracture in various materials generally occur well within the plastic zone (ordamage zone, to use a more generic term). Thus even if the plastic zone is very small,fracture may not nucleate in the singularity dominated zone. This fact raises an importantquestion: is stress intensity a useful failure criterion in materials that exhibit inelastic de-formation at the crack tip?

Under certain conditions, K still uniquely characterizes crack tip conditions when aplastic zone is present. In such cases, Kc is a geometry-independent material constant, asdiscussed below.

Consider a test specimen and structure loaded to the same Kf level, as illustrated inFig. 2.41. Assume that the plastic zone is small compared to all length dimensions inthe structure and test specimen. Let us construct a free-body diagram with a small regionremoved from the crack tip of each material. If this region is sufficiently small to bewithin the singularity dominated zone, the stresses and displacements at the boundary are


defined by the relationships in Tables 2.1 and 2.2. The disk-shaped region in Fig. 2.41

can be viewed as an independent problem. Imposition of the l/\ r singularity at theboundary results in a plastic zone at the crack tip. The size of the plastic zone and thestress distribution within the disc-shaped region are a function only of the boundary condi-tions and material properties. Therefore, even though we do not know the actual stressdistribution in the plastic zone, we can argue that it is uniquely characterized by the

boundary conditions; i.e., Aj characterizes crack tip conditions even though the 1/vr sin-gularity does not apply to the plastic zone. Since the structure and test specimen in Fig.2.41 are loaded to the same Kj value, the crack tip conditions must be identical in thetwo configurations. Furthermore, as load is increased, both configurations will fail at thesame critical stress intensity, provided the plastic zone remains small in each case.Similarly, if both structures are loaded in fatigue at the same AK, the crack growth rateswill be similar as long as the cyclic plastic zone is embedded within the singularity dom-inated zone in each case (see Chapter 10).

Figure 2.42 schematically illustrates the stress distributions in the structure and testspecimen from the previous figure. In the singularity dominated zone, a log-log plot ofthe stress distribution is linear with a slope of - */2- Inside of the plastic zone, thestresses are lower than predicted by the elastic solution, but are identical for the two con-figurations. Outside of the singularity dominated zone, higher order terms become signif-icant (Eq. 2.36) and the stress fields are different for the structure and test specimen; Kdoes not uniquely characterize the magnitude of the higher order terms.

0.5

0.4 -

0.3 -

0.2 -

0.1 -

-0.1 0.1 0.2 0.3 0.4 0.5

x / B0.6

Figure 2.40 Effect of Kj, relative to thickness, of the plastic zone size and shape [291.

86Chapter 2

s-e stress [ntens,y. The crack tip

relevant dimensions. Thus both will PlaSt'C Z°ne IS Sma11 comPared to a11

Logayy

A) -Plasticzone

B) - Singularity dominated zone

Logr

FIGURE 2.42 Crack tip stress fields for the specimen and structure in Fig. 2.41.


2.10.1 Effect of Loading Mode

A brief word of caution is necessary with respect to the mode of loading. Although thecritical stress intensity factor for a given mode is a material constant (when crack tip plas-ticity is limited), Kc generally varies with loading mode. That is,

Most materials are more susceptible to fracture by normal tensile stresses than byshear stresses. Consequently, Mode I loading has the most practical importance. Mode IIand Mode III loading usually do not lead to fracture. Stated another way, KHC and KUJC

are generally greater than KIC.The vast majority of practical applications of fracture mechanics consider only the

Mode I component of loading. (The reader may have noticed that most of the examples inthis chapter are pure Mode I problems.) Other modes of loading become important whenthey are applied to a weak interface in the material. For example, Mode II fiber/matrixdebonding and Mode II delamination can occur in composite materials (see Chapter 6).

Refer to Section 2.11 for further discussion of fracture under mixed-mode conditions.

2.10.2 Effect of Specimen Dimensions

The critical stress intensity factor is only a material constant when certain conditions aremet. Otherwise, Kc values can be geometry dependent.

As stated in Section 2.9, the plastic zone must be small compared to the specimenthickness in order to achieve plane strain conditions at the elastic-plastic interface. Whenthe plastic zone reaches a significant fraction of the plate thickness, the stress state at theedge of the plastic zone is plane stress, but plane strain conditions may persist at thecrack tip, deep inside the plastic zone. With further plastic deformation, however, thelevel of stress triaxiality at the crack tip relaxes. A lower degree of stress triaxiality usu-ally results in higher toughness. Figure 2.43 illustrates the effect of thickness on thecritical Mode I stress intensity factor. Small thickness (relative to the plastic zone size)corresponds to nominally plane stress fracture. Fracture toughness decreases with thick-ness until a plateau is reached; further increases in thickness have little or no effect ontoughness (see Appendix 5.2 for an exception to this rule). The critical Kj value at theplateau is defined as KIC, the plane strain fracture toughness. (Critical KI values corre-sponding to less than plane strain constraint are not called Kjc values. These are some-times designated as Kc values, but this convention is avoided here because it can lead toconfusion when other modes of loading are present.)

The through- thickness constraint can influence the shape of the R curve, particularlyfor ductile materials. Section 2.5.1 alluded to this effect. The R curve for a material inplane stress is often much steeper than the plane strain R curve for the same material.Some materials have a relatively flat plane strain R curve, resulting in toughness that issingle valued, while the plane stress R curve rises with crack growth. Refer to Fig. 2.10for an illustration of flat and rising R curves.

The in-plane dimensions of a specimen or structure are as important as the thick-ness. In order for the stress intensity factor to have any meaning, there must be a singu-larity dominated zone near the crack tip. When the plastic zone becomes too large, the

88 Chapter 2

singularity dominated zone is destroyed, and K no longer characterizes crack tip condi-tions. Thus the plastic zone must be embedded within the singularity dominated zone. Ingeneral, the singularity zone is small relative to in-plane length scales in the structure(see Example 2.6).

CRITICALKl

Plane Stress Plane Strain

THICKNESS

FIGURE 2.43 Effect of specimen thickness on Mode I fracture toughness.

EXAMPLE 2.7

Estimate the relative size of the singularity dominated zone ahead of a through crack in an infinite

plate subject to remote uniaxial tension (Fig. 2.3), The full solution for the stresses on the crack

plane (6=0) for this geometry are as follows (see Appendix 2.3.2):

a(a+r)

OW =O(a+r)

where cr is the remotely applied tensile stress. Also, estimate the value of Kj where the plane

strain plastic zone engulfs the singularity dominated zone.

Solution: As r-»0 both of the above relationships reduce to the expected result:


EXAMPLE 2.7 (cont.)

cA K,<7w = Or? -

Figure 2.44 is a plot of the ratio of the total stress to the singular stress given by the above equation.

Note that the stress in the y direction is close to the singular limit to relatively large distances from

the crack tip but the x stress diverges considerably from the near-tip limit. When r/a = 0.02, thesingularity approximation results in roughly a 2% underestimate of crvv and a 20% overestimate of

axx. Let us arbitrarily define this point as the limit of the singularity zone:

50

By setting the plane strain plastic zone correction (Eq. 2.63) equal to a/50, we obtain an esti-mate of the Kj value at which the singularity zone is engulfed by crack tip plasticity:

= 0.35 ays

Therefore, when the nominal stress exceeds approximately 35% of yield in this case, the accuracy

of Kj as a crack tip characterizing parameter in this particular geometry is suspect..

2.10.3 Limits to the Validity of LEFM

According to the American Society for Testing and Materials (ASTM) standard for Kjc

testing [30], the following specimen size requirements must be met to obtain a valid Kjc

result in metals:

a,B,(W-a)>2.5°YS )

(2.87)

Note the similarity between the crack length requirement and the result derived inExample 2.7. Thus Eq. (2.82) implies that rv must be < -1/50 times specimen dimen-sions in order to obtain a size-independent critical K[ value. Equation (2.82) was basedon experimental observations of the size dependence of fracture toughness in steel and

*The singularity zone is small in this geometry because of a significant transverse compressive stress. Incracked geometries loaded in bending, this transverse stress (also called the T stress) is near zero or slightlypositive; consequently, the singularity zone is larger in these configurations. See Section 3.6 for furtherdiscussion on the effect of the T stress.

90 Chapter 2

o

aluminum [31]. The thickness requirement ensures plane strain conditions^, while the re-quirement on in-plane dimensions ensures that the nominal behavior is linear elastic andthat KI characterizes crack tip conditions.

Equation (2.87) gives the requirements tot plane strain, linear elastic fracture. Avalid Kjc result is a material property that does not depend on the size or geometry of thecracked body. While plane strain conditions are necessary to measure a valid Kjc, the lackof plane strain does not necessarily invalidate LEFM. As long as the in-plane dimensionsare sufficiently large to confine the plastic zone to the singularity dominated zone, thestress intensity factor is a valid crack tip characterizing parameter. A fracture toughnessvalue obtained from a laboratory specimen in plane stress or mixed conditions isapplicable to a structure made of the same material, as long as the specimen and structureare the same thickness and the in-plane dimensions of both are large compared to theplastic zone. An example application of non-plane strain LEFM is fracture toughnesstesting of thin aluminum sheet used in aerospace structures.

Plasticity corrections such as those described in Section 2.8 can extend LEFM be-yond its normal validity limits. One must remember, however, that the Irwin and stripyield corrections are only rough approximations of elastic-plastic behavior. When non-linear material behavior becomes significant, one should discard stress intensity and adopta crack tip parameter that takes the material behavior into account. Two such parameters,the crack tip opening displacement (CTOD) and the J integral, are the subject of Chapter3.

1.05 -

0.95

THROUGH-THICKNESS CRACKINFINITE PLATE

= 0

0.85 -

r / aFIGURE 2.44 Ratio of actual stresses on the crack plane to the singularity limit in an infinite plate with athrough-thickness crack.

^Recent three-dimensional finite element results from several investigators, combined with experimentaldata, indicate that the Kj<^ thickness requirements are overly conservative. In standard test specimens,nominally plane strain conditions exist at the crack tip to relatively high deformation levels. The in-planerequirements (i.e. a and W) appear to be reasonable.


2.11 MIXED-MODE FRACTURE

When two or more modes of loading are present, Eq. (2.63) indicates that energy releaserate contributions from each mode are additive. This equation assumes self-similar crackgrowth, however. Consider the angled crack problem depicted in Fig. 2.18. Equation(2.63) gives the energy release rate for planar crack growth at an angle 90 - /3 degrees fromthe applied stress. Figure 2.45 illustrates a more typical scenario for an angled crack.When fracture occurs, the crack tends to propagate orthogonal to the applied normalstress; i.e., the mixed-mode crack becomes a Mode I crack.

A propagating crack seeks the path of least resistance (or the path of maximum driv-ing force) and need not be confined to its initial plane. If the material is isotropic and ho-mogeneous, the crack will propagate in such a way as to maximize the energy releaserate. What follows is an evaluation of the energy release rate as a function of propagationdirection in mixed-mode problems. Only Modes I and II are considered here, but the basicmethodology can, in principle, be applied to a more general case where all three modes arepresent. This analysis is based on similar work in Refs. [32-34].

2.11.1 Propagation of an Angled Crack

We can generalize the angled through-thickness crack of Fig. 2.18 to any planar crack ori-ented 90 - /3 degrees from the applied normal stress. For uniaxial loading, the stress inten-sity factors for Modes I and II are given by

(2.88a)

*// = (2.88b)

where Kj(Q) is the Mode I stress intensity when /3 = 0. The crack tip stress fields (in po-lar coordinates) for the Mode I portion of the loading are given by

'5 (6} 1 (36—cos — —cos —4 UJ 4 [.2

(2.89a)

1 (36•cos| — | + —cos —

4 V 21 . (6\ 1 . (39

sin — +—sin2) 4

(2.89b)

(2.89c)

As stated earlier, these singular fields only apply as r —> 0. The singular stress fields forMode II are given by

5 . (6—sin —

3 . (30\— sm —4 I 2 )

(2.90a)

92 Chapter 2

®ee ~3 . (9\ 3 . (30

—sin — sin —. 4 UJ 4 V 2

(2.90b)

"1 (e\ 3 (36—cos — + —cos —.4 U; 4 12 (2.90c)

Suppose that the crack in question forms an infinitesimal kink at an angle a from theplane of the crack, as Fig. 2.46 illustrates. The local stress intensity factors at the tip ofthis kink differ from the nominal K values of the main crack. If we define a local x-y co-ordinate system at the tip of the kink and assume that Eqs. (2.89) and (2.90) define the lo-cal stress fields, the local Mode I and Mode II stress intensity factors at the tip are ob-tained by summing the normal and shear stresses, respectively, at ex:

ka(a) =

Cl2Kn

C22Kn

(2.9 la)

(2.91b)

FIGURE 2.45 Typical propagation from an initialcrack that is not orthogonal to the applied normalstress. The loading for the initial angled crack is acombination of Modes I and II, but the cracktends to propagate normal to the applied stress,resulting in pure Mode I loading.

(X

FIGURE 2.46 Infinitesimal kink at the tip of amacroscopic crack.


where Iq and kn are the local stress intensity factors at the tip of the kink and KI and Knare the stress intensity factors for the main crack, which are given by Eq. (2.88) for thetilted crack. The coefficients CM are given by

_ 3 f a } 1 (3aC, =—cos — -f —cos —11 4 {2J 4 I 2

c — ——-12" 4' . f a } . (3asm — +sm —

UJ I 2

, } . (3asin — + sin —

2) \ 2

1 fa\ 3 (3a= —cos — H-—COS —4 UJ 4 V 2

The energy release rate for the kinked crack is given by

(2.92a)

(2.92b)

(2.92c)

(2.92d)

E(2.93)

Figure 2.47 is a plot of £(a) normalized by <j((x=0). The peak in $ a) at each ft cor-responds to the point where kj exhibits a maximum and &// = 0. Thus the maximum en-ergy release rate is given by

=z/max (2.94)

= 0. Crackwhere a* is the angle at which both § and kj exhibit a maximum andgrowth in a homogeneous material should initiate along a*.

Figure 2.48 shows the effect of (3 on the optimum propagation angle. The dashedline corresponds to propagation perpendicular to the remote principal stress. Note that the^max criterion implies an initial propagation plane that differs slightly from the normalto the remote stress.

2.11.2 Equivalent Mode I Crack

Let us now introduce an effective Mode I crack that results in the same stress intensityand energy release rate as a crack oriented at an angle (3 and propagating at an angle a*:

(2.95)

.jJiJ

94 Chapter 2

-120 -60 0 60 120KINK ANGLE (a), DEGREES

FIGURE 2.47 Local energy release rate at the tip of a kinked crack.

90

75

W 60>-)

§

30

Ogogg 15

0

Normal toRemote Stress

Maximum EnergyRelease Rate

Criterion

180

0 15 7530 45 60(3, DEGREES

FIGURE 2.48 Optimum propagation angle for a crack oriented at an angle P from the stress axis.

90


For the special case of a through-thickness crack in an infinite plate (Fig 2.18), Eq. (2.95)becomes

(2.96)

Solving for aeq gives

- = [cos2 pC} j (a *) 4- sin£ cos/3C12 (a *) (2.97)

2.11.3 Biaxial Loading

Figure 2.49 illustrates a cracked plate subject to principal stresses 07 and <J2, where 07 isthe greater of the two stresses; /3 is defined as the angle between the crack and the 07plane. Applying superposition leads to the following expressions for Kj and KIJ:

(2.98a)

(2.98b)

where B is the biaxiality ratio, defined as

(2.99)

The local Mode I stress intensity for a kinked crack is obtained by substituting Eq.(2.98) into Eq. (2.91a):

FIGURE 2.49 Cracked plane subject to a bi-axial stress state.

96 Chapter 2

(a) - B)Cn(a)] (F14)

The maximum local stress intensity factor and energy release rate occurs at the optimumpropagation angle, a* which depends on the biaxiality ratio. Figure 2.50 illustrates theeffect of B and /? on the propagation angle. Note that when B > 0 and f3 = 90°, propaga-tion occurs in the crack plane (a* = 0), since the crack lies on a principal plane and issubject to pure Mode I loading.

30 45 60J3, DEGREES

FIGURE 2.50 Optimum propagation angle as a function of p and biaxialty.

REFERENCES

1. Inglis, C.E., "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners."Transactions of the Institute of Naval Architects, Vol. 55, 1913, pp. 219-241.

2. Griffith, A.A. "The Phenomena of Rupture and Flow in Solids." PhilosophicalTransactions, Series A, Vol. 221, 1920, pp. 163-198.

3. Gehlen, P.C. and Kanninen, M.F., "An Atomic Model for Cleavage Crack Propagationin a iron." Inelastic Behavior of Solids, McGraw-Hill, New York, 1970, pp. 587-603.


4. Irwin, G.R., "Fracture Dynamics." Fracturing of Metals, American Society for Metals,Cleveland, 1948, pp. 147-166.

5. Orowan, E., "Fracture and Strength of Solids." Reports on Progress in Physics, Vol.XII, 1948, p. 185.

6. Irwin, G.R., "Onset of Fast Crack Propagation in High Strength Steel and AluminumAlloys." Sagamore Research Conference Proceedings, Vol. 2, 1956, pp. 289-305.

7. Hutchinson, J.W. and Paris, P.C., "Stability Analysis of J-Controlled Crack Growth."ASTM STP 668, American Society for Testing and Materials, Philadelphia, 1979, pp.37-64.


9. Irwin, G.R., "Analysis of Stresses and Strains near the End of a Crack Traversing aPlate," Journal of Applied Mechanics, Vol. 24, 1957, pp. 361-364.

10. Sneddon, I.N., 'The Distribution of Stress in the Neighbourhood of a Crack in an ElasticSolid." Proceedings, Royal Society of London, Vol. A-187, 1946, pp. 229-260.


12. Tada, H., Paris, P.C., and Irwin, G.R. The Stress Analysis of Cracks Handbook. (2ndEd.) Paris Productions, Inc., St. Louis, 1985.

13. Murakami, Y. Stress Intensity Factors Handbook. Pergamon Press, New York, 1987.

14. Rooke, D.P. and Cartwright, D.J., Compendium of Stress Intensity Factors. HerMajesty's Stationary Office, London, 1976.

15. Rice, J.R., "Some Remarks on Elastic Crack-Tip Stress Fields." International Journalof Solids and Structures, Vol. 8, 1972, pp. 751-758.

16. Bueckner, H.F., "A Novel Principle for the Computation of Stress Intensity Factors."Zeitschrift fur Angewandie Mathematik und Mechanik, Vol. 50, 1970, pp. 529-545.

17. Rice, J.R., "Weight Function Theory for Three-Dimensional Elastic Crack Analysis."ASTM STP 1020, American Society for Testing and Materials, Philadelphia, 1989, pp.29-57.

18. Parks, D.M. and Kamentzky, E.M., "Weight Functions from Virtual Crack Extension."International Journal for Numerical Methods in Engineering, Vol. 14, 1979, pp. 1693-1706.

19. Vainshtok, V.A., "A Modified Virtual Crack Extension Method of the Weight FunctionsCalculation for Mixed Mode Fracture Problems." International Journal of Fracture, Vol.19, 1982, pp. R9-R15.

20. Sha, G.T. and Yang, C.-T., "Weight Function Calculations for Mixed-Mode FractureProblems with the Virtual Crack Extension Technique." Engineering FractureMechanics. Vol. 21, 1985, pp. 1119-1149.

98 Chapter 2

21. Atluri, S.N. and Nishoika, T., "On Some Recent Advances in Computational Methods inthe Mechanics of Fracture." Advances in Fracture Research: Seventh InternationalConference on Fracture, Pergamon Press, Oxford, 1989 pp. 1923-1969.

22. Sham, T.-L., "A Unified Finite Element Method for Determining Weight Functions inTwo and Three Dimensions." International Journal of Solids and Structures, Vol. 23,1987, pp. 1357-1372.

23. Invin, G.R., "Plastic Zone Near a Crack and Fracture Toughness." Sagamore ResearchConference Proceedings, Vol. 4, 1961.



26. Burdekin, P.M. and Stone, D.E.W., "The Crack Opening Displacement Approach toFracture Mechanics in Yielding Materials." Journal of Strain Analysis, Vol. 1, 1966,pp. 145-153.

27. Dodds, R.H. Jr., Anderson T.L. and Kirk, M.T., "A Framework to Correlate a/W Effectson Elastic-Plastic Fracture Toughness (Jc)-" International Journal of Fracture, Vol. 48,1991, pp. 1-22.

28. Narasimhan, R. and Rosakis A.J., "Three Dimensional Effects Near a Crack Tip in aDuctile Three Point Bend Specimen - Part I: A Numerical Investigation." CaliforniaInstitute of Technology, Division of Engineering and Applied Science, Report SM 88-6, Pasadena, CA, January 1988.

29. Nakamura, T and Parks, D.M. "Conditions of J-Dominance in Three-Dimensional ThinCracked Plates." Analytical, Numerical, and Experimental Aspects of Three-Dimensional Fracture Processes, ASME AMD-91, American Society of MechanicalEngineers, New York, 1988, pp. 227-238.

30. E 399-90, "Standard Test Method for Plane-Strain Fracture Toughness of MetallicMaterials." American Society for Testing and Materials, Philadelphia, 1990.

31. Brown W.F. Jr. and Srawley, J.E., Plane Strain Crack Toughness Testing of HighStrength Metallic Materials. ASTM STP 410, American Society for Testing andMaterials, Philadelphia, PA, 1966.

32. Erdogan, F. and Sih, G.C., "On the Crack Extension in Plates under Plane Loading andTransverse Shear." Journal of Basic Engineering, Vol. 85, 1963, pp. 519-527.

33. Williams, J.G. and Ewing, P.D., "Fracture under Complex Stress-The Angled CrackProblem." International Journal of Fracture Mechanics, Vol. 8, 1972, pp. 441-446.

34. Cottrell, B. and Rice, J.R., "Slightly Curved or Kinked Cracks." International Journalof Fracture, Vol. 16, 1980, pp. 155-169.

35. Williams, M.L., "Stress Singularities Resulting from Various Boundary Conditions inAngular Corners of Plates in Extension." Journal of Applied Mechanics, Vol. 74,1952, pp. 526-528.

36. Irwin, G.R., discussion of Ref. 29, 1958.


37. Sih, G.C., "On the Westergaard Method of Crack Analysis." International Journal ofFracture Mechanics, Vol. 2, 1966, pp. 628-631.

38. Eftis, J. and Liebowitz, H., "On the Modified Westergaard Equations for Certain PlaneCrack Problems." International Journal of Fracture Mechanics, Vol. 8, p. 383.

39. Sanford, R.J., "A Critical Re-Examination of the Westergaard Method for SolvingOpening Mode Crack Problems." Mechanics Research Communications, Vol. 6, 1979,pp. 289-294.

40. Wells, A.A. and Post, D., "The Dynamic Stress Distribution Surrounding a RunningCrack—A Photoelastic Analysis." Proceedings of the Society for Experimental StressAnalysis, Vol. 16, 1958, pp. 69-92.

41. Muskhelishvili, N.I., Some Basic Problems in the Theory of Elasticity. Noordhoff,Ltd., Netherlands, 1953.

42. Green, A.E. and Sneddon, I.N., "The Distribution of Stress in the Neighbourhood of aFlat Elliptical Crack in an Elastic Solid." Proceedings, Cambridge PhilosophicalSociety, Vol. 46, 1950, pp. 159-163.

APPENDIX 2: MATHEMATICALFOUNDATIONS OF LINEAR ELASTIC

FRACTURE MECHANICS(Selected Results)

A2.1 PLANE ELASTICITY

This section catalogs the governing equations from which linear fracture mechanics is de-rived. The reader is encouraged to review the basis of these relationships by consultingone of the many textbooks on elasticity theory.3

The equations that follow are simplifications of more general relationships in elas-ticity and are subject to the following restrictions:

Two-dimensional stress state (plane stress or plane strain).

• Isotropic material.

• Quasistatic, isothermal deformation.

e Body forces are absent from the problem. (In problems where body forces arepresent, a solution can first be obtained in the absence of body forces, and thenmodified by superimposing the body forces.)

Imposing these restrictions simplifies crack problems considerably, and permits closed-form solutions in many cases.

The governing equations of plane elasticity are given below for rectangular Cartesiancoordinates. Section A2.1.2 lists the same relationships in terms of polar coordinates.

A2.1.1 Cartesian Coordinates

Strain-displacement relationships:

c —fcyy ~~"-xx -> yy -N jty

dx " dy J

where x and y are the horizontal and vertical coordinates, respectively, EXX, £yy, etc. arethe strain components, and ux and uy are the displacement components.

-'Appendix 2 is intended only for more advanced readers, who have at least taken one graduate-level coursein the theory of elasticity.

101

102 Appendix 2

Stress-strain relationships:

1. Plane strain.

(l+v)(l-2v)

+ ve

(A2.2)

£ . = P. = £ = T = T =0

where a and i are the normal and shear stress components, respectively, E is Young'smodulus, jl is the shear modulus, and D is Poisson's ratio.

2. Plane stress.

i _

(A2.3)

^ 1-V J

Equilibrium equations:

„. -t~ _ -- — n /> i •*•! _ A / A O /n-l- -U — h— -U (A2.4)ox dy By dx

Mathematical Foundations ofLEFM 103

Compatibility equation:

0 (A2.5)\ •> j iwhere

a*2 a/Airy stress function:

For a two-dimensional continuous elastic medium, there exists a function 3>(x,y)from which the stresses can be derived:

-.9 , -,9 , -.9(j has the following property:

or

V2V2O = 0 (A2.7)

A2.1.2 Polar Coordinates

Strain-displacement relationships:

_ a«r _ wr i a^ _ i ( i a«r aw^ UQ ê =__ e^= h — erQ=-\ ^- + -3 (A2.8)

or r r 06 2\r 06 or r J

where wrand UQ are the radial and tangential displacement components, respectively.

Stress-strain relationships:The stress-strain relationships in polar coordinates can be obtained by substituting r

and 6 for x and y in Eqs. (A2.2) and (A2.3). For example, the radial stress is given by

+ v e ] (A2'9a)

for plane strain, and

104 Appendix 2

for plane stress.

Equilibrium equations:

[ r | <Jrr - = Q

(A2.9b)

3r r c)6 r(A2.10)

1 d<300 , o^rfl ,. 2Trg _ n___ -|- -p ———— — ^_/r 30 or r

Compatibility equation:

Trr + O~QQ) = 0 (A2.11)

where

y •«-• >* |" *" " I >s A

Airy stress function:

V2V2O = 0 . (A2.12)

where = Or, 0 and

<Trr — —s-r—«--f r— CT^d — 0 ^m — +______ (A2.13)

A2.2 CRACK GROWTH INSTABILITY ANALYSIS

Figure 2.12 schematically illustrates the general case of a cracked structure with finitesystem compliance, CM- The structure is held at a fixed remote displacement, AT givenby

P (A2.14)

where A is the local load line displacement and P is the applied load. Differentiating Eq.(A2. 14) gives

dP+CMdP = 0 (A2.15)daj \oP


assuming A depends only on load and crack length. We can make this same assumptionabout the energy release rate:

dP (A2.16)Ja

Dividing both sides of Eq. (A2.16) by da and fixing AT yields

P\(A2.17)

' iiy

which, upon substitution of Eq. (A2.15), leads to

c +\*±^•*M ' _m

9'-a

(A2.18)

A virtually identical expression for the / integral (Eq. 3.52) can be derived by assuming Jdepends only on P and a, and expanding dJ into its partial derivatives.

Under dead-loading conditions, CM = °°, and all but the first term in Eq. (A2.18)vanish. Conversely, C^/ = 0 corresponds to an infinitely stiff system, and Eq. (A2.18)reduces to the pure displacement control case.

A2.3 CRACK TIP STRESS ANALYSIS

A variety of techniques are available for analyzing stresses in cracked bodies. This sectionfocuses on two early approaches developed by Williams [11,35] and Westergaard [8].These two analyses are complementary; the Williams approach considers the local cracktip fields under generalized in-plane loading, while Westergaard provided a means for con-necting the local fields to global boundary conditions in certain configurations.

Space limitations preclude listing every minute step in each derivation. Moreover,stress, strain, and displacement distributions are not derived for all modes of loading. Thederivations that follow serve as illustrative examples. The reader who is interested in fur-ther details should consult the original references.

A2.3.1 Generalized In-Plane Loading

Williams [11,35] was the first to demonstrate the universal nature of the 1/Vr singular-ity for elastic crack problems, although Inglis [1], Westergaard [8] and Sneddon [10] hadearlier obtained this result in specific configurations. Williams actually began by con-sidering stresses at the corner of a plate with various boundary conditions and included an-gles; a crack is a special case where the included angle of the plate corner is 2n and thesurfaces are traction free (Fig. A2.1).

106 Appendix 2

For the configuration shown in Fig. A2.1(b), Williams postulated the followingstress function:

r [q sin(A + 1) 9 * +c2 cos( A + 1) 0 *+c3 sin(A - 1)9 * 4-c4 cos(A - 1)0*]

= r 4>(0*,/l) (A2.19)

where cj, C2, 03, and 04 are constants, and 6* is defined in Fig. A2.1(b). Invoking Eq.(A2.13) gives the following expressions for the stresses:

A - 1 / /<*„ = r-[F / /(0*) + (A + 1)F(0*)]

(A2.20)

where the primes denote derivatives with respect to 0*. Williams also showed that Eq.(A2.19) implies that the displacements vary with A In order for displacements to be fi-nite in all regions of the body, A must be > 0. If the crack faces are traction free,<700(0) = <700(27T)= Tr0(0)= rr0(27r) = 0, which implies the following boundary con-

ditions:

= F(2?r) = F'CO) = F'(2;r) = 0 (A2.21)

(a) Plate corner with included angle y. (b) Special case of a sharp crack.

FIGURE A2.1 Plate comer configuration analyzed by Williams [351. A crack is formed when yr= 2it.


Assuming the constants in Eq. (A2.19) are nonzero in the most general case, the bound-ary conditions can only be satisfied when sin (2nk)=Q. Thus

A=- , where n= 1, 2, 3, . . .2

There are an infinite number of A values that satisfy the boundary conditions; the mostgeneral solution to a crack problem, therefore, is a polynomial of the form

(A2.22)

and the stresses are given by

(A2.23)

where F is a function that depends on F and its derivatives. The order of the stress func-tion polynomial, N, must be sufficient to model the stresses in all regions of the body.When r —» 0, the first term in Eq. (A2.23) approaches infinity, while the higher orderterms remain finite (when m = 0) or approach zero (for m > 0). Thus the higher order

terms are negligible close to the crack tip, and stress exhibits an l/^ singularity. Notethat this result was obtained without assuming a specific configuration; thus it can beconcluded that the inverse square-root singularity is universal for cracks in isotropic elas-tic media.

Further evaluation of Eqs. (A2.19) and (A2.20) with the appropriate boundary condi-tions reveals the precise nature of the function F. Recall that Eq. (A2.19) contains four,as yet unspecified, constants; by applying Eq. (A2.21), it is possible to eliminate two ofthese constants, resulting in

n + 2 V2

(A2.24)

for a given value of n. For crack problems it is more convenient to express the stressfunction in terms of, 9, the angle from the symmetry plane (Fig. A2.1). Substituting 6=9* - n into Eq. (A2.24) yields, after some algebra, the following stress function for thefirst few values of n:

e i 30}_rnc --- -•cos COS -

3 2 J. 0 . 3 6

-sin sm —2 2

108 Appendix 2

where j?/ and f; are constants to be defined. The stresses are given by

I I - 9 361 r « • e a • 3e'-5cos—I-cos— + ?i —5 sin— + 3sm —2 2

(A2.25)

' e 36-3 cos cos—

2 2 [ /a 0/3"-3sin 3sin —

2 2.(A2.26)

. a .391 r 0 _ 39'-sin sin— H-fi cos—+ 3cos—

2 2 J !L 2 2.

Note that the constants Sj in the stress function (Eq. (A2.25)) are multiplied by cosineterms while the t{ are multiplied by sine terms. Thus the stress function contains sym-metric and antisymmetric components, with respect to 9 = 0. When the loading is sym-metric about 9=0, ti = 0, while j/ = 0 for the special case of pure antisymmetric loading.Examples of symmetric loading include pure bending and pure tension; in both cases theprincipal stress is normal to the crack plane. Therefore, symmetric loading corresponds toMode I (Fig. 2.14); antisymmetric loading is produced by in-plane shear on the crackfaces and corresponds to Mode H

It is convenient in most cases to treat the symmetric and antisymmetric stressesseparately. The constants sj and tj can be replaced by the Mode I and Mode II stress in-tensity factors, respectively:

K,(A2.27)

The crack tip stress fields for symmetric (Mode I) loading (assuming the higher orderterms are negligible) are given by


"5 re} i 30—cos — —cos4 U

'3 f < 9 ^ 1 (30—cos — +—cos —4 U; 4 V 2

(A2.28)

' i . (e\ i . (30—sin — +—sin —4 U; 4 V 2

The singular stress fields for Mode II are given by

5 . (8} 3 . (30—sin — +~sm —4 UJ 4 I 2

3 . (8} 3 . (30—sm — —sin —4 UJ 4 1 2

(A2.29)

Tr0 ='1 (8} 3 3d—cos — +—cos4 UJ 4

The relationships in Table 2.1 can be obtained by converting Eqs (A2.28) and (A2.29) toCartesian coordinates.

The stress intensity factor defines the amplitude of the crack tip singularity; allstress and strain components at points near the crack tip increase in proportion to K, pro-vided the crack is stationary. The precise definition of the stress intensity factor is arbi-trary, however; the constants sj and tj would serve equally well for characterizing the sin-gularity. The accepted definition of stress intensity stems from the early work of Irwin[9], who quantified the amplitude of the Mode I singularity with -i/iÊ , where </ is the

energy release rate. It turns out that the in the denominators of Eqs. (A2.28) and(A2.29) is superfluous (see Eqs. (A2.34) to (A2.36), below), but convention establishedover the last 35 years precludes redefining AT in a more convenient form.

Williams also derived relationships for radial and tangential displacements near thecrack tip. We will postpone evaluation of displacements until the next section, however,because the Westergaard approach for deriving displacements is somewhat more compact.

A2.3.2 The Westergaard Stress Function

Westergaard showed that a limited class of problems could be solved by introducing acomplex stress function Z(z), where z~x-\-iy and i = V-T- The Westergaard stress func-tion is related to Airy stress function as follows:

= ReZ+)>ImZ (A2.30)

110 Appendix 2

where Re and Im denote real and imaginary parts of the function, respectively, and thebars over Z represent integrations with respect to z; i.e.,

dz

Applying Eq. (A2.6) gives

and —dz

(A2.31)

T =-yReZ'

Note that the imaginary part of the stresses vanishes when y=0. In addition, the shearstress vanishes when y = 0, implying that the crack plane is a principal plane. Thus thestresses are symmetric about 0=0 and Eq. (A2.31) implies Mode I loading.

The Westergaard stress function, in its original form, is suitable for solving a lim-ited range of Mode I crack problems. Subsequent modifications [36-39] generalized theWestergaard approach to be applicable to a wider range of cracked configurations.

Consider a through crack in an infinite plate subject to biaxial remote tension (Fig.A2.2). If the origin is defined at the center of the crack, the Westergaard stress function isgiven by

/"-:

FIGURE A2.2 Through-thicknesscrack in an infinite plate loaded in biax-ial tension.


~~ (A2.32)

where a is the remote stress and a is the half crack length, as defined in Fig. A2.2.Consider the crack plane where y=0. For -a < x < a, Z is pure imaginary, while Z is realfor \x\ > \a\. The normal stresses on the crack plane are given by

_ _ OX°xx = G = Re Z = (A233)

Let us now consider the horizontal distance from each crack tip, x* - x-a\ Eq. (A2.33)becomes

for ** « a. Thus the Westergaard approach leads to the expected inverse square-root sin-gularity. One advantage of this analysis is that it relates the local stresses to the globalstress and crack size. From Eq. (A2.28), the stresses on the crack plane (0=0) are givenby

Comparing Eqs. (A2.34) and (A2.35) gives

(A2.36)

for the configuration in Fig. A2.2. Note that -îi appears in Eq. (A2.36) because K wasoriginally defined in terms of the energy release rate; an alternative definition of stress in-tensity might be

K*< j ( 0 = 0) = r-1— where Ki - <jVa for the plate in Fig. A2.2.

Substituting Eq. (A3. 3 6) into Eq. (A2.32) results in an expression of theWestergaard stress function in terms of Kj:

(A2.37)

where z* = z-a. It is possible to solve for the singular stresses at other angles by makingthe following substitution in Eq. (A2.37):

112 Appendix 2

where= reiQ

r2=(x-a)2+y2 and 0 = tan *' y

x-a

which leads to

0COS — 1 • •l-sm — sinsin

) (2 2

K eCOSI ~

2nr 12, • •1 + sm — sm (A2.38)

Kr 9} . (0— sm —

assuming r»a. Equation (A2.38) is equivalent to Eq. (A2.28), except that the latter isexpressed in terms of polar coordinates.

Westergaard published the following stress function for an array of collinear cracksin a plate in biaxial tension (Fig 2.21):

1 —

r . ( na\oin5>m

\2WJ. ( m\

bUl\. \2WJ_

t !/2 (A2.39)

where a is the half crack length and 2W is the spacing between the crack centers. Thestress intensity for this case is given in Eq. (2.45); early investigators used this solutionto approximate the behavior of center cracked tensile panel with finite width.

Irwin [9] published stress functions for several additional configurations, including apair of crack opening forces located a distance X from the crack center (Fig. 2.30):

Pa h-(X/a)2

(A2.40)

where P is the applied force. When there are matching forces at ±X, the appropriatestress function can be obtained by superposition:


(A2.41)

In each case, the stress function can be expressed in the form of Eq. (A2.37) and the neartip stresses are given by Eq. (A2.38). This is not surprising, since all of the above casesare pure Mode I and the Williams analysis showed that the inverse square root singularityis universal.

For plane strain conditions, the in-plane displacements are related to the Westergaardstress function as follows:

1[(l-2v)ReZ-;yImZ]

My = —[2(1 - v)ImZ- jReZ]

(A2.42)

For the plate in Fig. A2.2, the crack opening displacement is given by

l - v T =, 2( l-v2)T -7ImZ = — ImZ =fi E

•-\Ja2 - x2

assuming plane strain, and

(A2.43a)

(A2.43b)

for plane stress. Eq. (A2.43) predicts that a through crack forms an elliptical opening pro-file when subjected to tensile loading.

The near-tip displacements can be obtained by inserting Eq. (A2.37) into Eq.(A2.42):

ux ~r (0

cos —U

sin

:-2[*

2(A2.44)

for r « a , where

and3-v1 + v

for plane strain

for plane stress

(A2.45)

114 Appendix 2

Although the original Westergaard approach correctly describes the singular Mode Istresses in certain configurations, it is not sufficiently general to apply to all Mode Iproblems; this shortcoming has prompted various modifications to the Westergaard stressfunction. Irwin [36] noted that photoelastic fringe patterns observed by Wells and Post[40] on center cracked panels did not match the shear strain contours predicted by theWestergaard solution. Irwin achieved good agreement between theory and experiment bysubtracting a uniform horizontal stress:

cr^ = Re Z - ylmZ -0OXX (A2.46)

where ffoxx depends on the remote stress. The other two stress components remain thesame as in Eq. (A2.31). Subsequent analyses have revealed that when a center crackedpanel is loaded in uniaxial tension, a transverse compressive stress develops in the plate.Thus Irwin' s modification to the Westergaard solution has a physical basis in the case ofa center cracked panel . Equation (A2.46) has been used to interpret photoelastic fringepatterns in a variety of configurations.

Sih [37] provided a theoretical basis for the Irwin modification. A stress functionfor Mode I must lead to zero shear stress on the crack plane. Sih showed that theWestergaard function was more restrictive than it needed to be, and was thus unable to ac-count for all situations. Sih generalized the Westergaard approach by applying a complexpotential formulation for the Airy stress function [41]. He imposed the condition lyy - 0at p=0, and showed that the stresses could be expressed in terms of a new function /fz):

GXX = 2 Re 0' (z) - 2y Im f (z) - A

oyy = 2 Re 0' (z) + 2ylm(j)"(z) + A (A2.47)

where A is a real constant. Equation (A2.47) is equivalent to the Irwin modification ofthe Westergaard approach if

A (A2.48)

Substituting Eq. (A2.48) into Eq. (A2.47) gives

cr^ = ReZ-)>ImZ-2A

ayy = Re Z + y Im Z ( A2.49)

4Recall that the stress functions in Eqs. (2.32) and (2.39) are strictly valid only for biaxial loading. Althoughthis restriction was not imposed in Westergaard's original work, a transverse tensile stress is necessary inorder to cancel with -ooxx. However, the transverse stresses, whether compressive or tensile, do not affectthe singular term; thus the stress intensity factor is the same for uniaxial and biaxial tensile loading and isgiven by Eq. (A2.36).


Comparing Eq. (A2.49) with Eqs. (A2.31) and (A2.46), it is obvious that the Sih andIrwin modifications are equivalent and 2A=(JOXX.

Sanford [39] showed that the Irwin-Sih approach is still too restrictive, and he pro-posed replacing A with a complex function r](z):

(A2.50)

The modified stresses are given by

a xx = ReZ - ylmZ +ylm?7' -2 Re 7]

(Jyy = ReZ + 3>ImZ+yImr/ (A2.51)

T - -y ReZ +>>Re ?]' +ImT]

Equation (A2.51) represents the most general form of Westergaard-type stress functions.When r](z)=a real constant for all z, Eq. (A2.51) reduces to the Irwin-Sih approach, whileEq. (A2.51) reduces to the original Westergaard solution when rj(z) = 0 for all z.

The function 77 can be represented as a polynomial of the form

Mn(Z)= 2>mzm/2 (A2.52)

m=0

Combining Eqs. (A2.37), (A2.50), and (A2.52) and defining the origin at the crack tipgives

v Mm/2 (A2.53)

which is consistent with the Williams [11,35] asymptotic expansion.

A2.4 ELLIPTICAL INTEGRAL OF THE SECOND KIND

The solution of stresses in the vicinity of elliptical and semielliptical cracks in elasticsolids [10,42] involves an elliptic integral of the second kind:

*/2 I r2_ 2l- sin20^ (A2.54)

0

116 Appendix 2

where 2c and 2a are the major and minor axes of the elliptical flaw, respectively. Seriesexpansion of Eq. (A2.54) gives

(A2.55)", lc2 -a2

4 c23

64 fc2~*2TI <2 JMost stress intensity solutions for elliptical and semi-ellipical cracks in the published lit-erature are written in terms of a flaw shape parameter, Q, which can be approximated by

Q = l.464 -1.65

(A2.56)

3. ELASTIC-PLASTIC FRACTUREMECHANICS

Linear elastic fracture mechanics (LEFM) is valid only as long as nonlinear materialdeformation is confined to a small region surrounding the crack tip. In many materials, itis virtually impossible to characterize the fracture behavior with LEFM, and an alternativefracture mechanics model is required.

Elastic-plastic fracture mechanics applies to materials that exhibit time-independent,nonlinear behavior (i.e., plastic deformation). Two elastic-plastic parameters areintroduced in this chapter: the crack tip opening displacement (CTOD) and the J contourintegral. Both parameters describe crack tip conditions in elastic-plastic materials, andeach can be used as a fracture criterion. Critical values of CTOD or / give nearly size-independent measures of fracture toughness, even for relatively large amounts of crack tipplasticity. There are limits to the applicability of /and CT0D(Sections 3.5 & 3.6), butthese limits are much less restrictive than the validity requirements of LEFM.

3.1 CRACK TIP OPENING DISPLACEMENT

When Wells [1] attempted to measure KIC values in a number of structural steels, hefound that these materials were too tough to be characterized by LEFM. This discoverybrought both good news and bad news: high toughness is obviously desirable todesigners and fabricators, but Wells' experiments indicated that existing fracturemechanics theory was not applicable to an important class of materials. While ex-amining fractured test specimens, Wells noticed that the crack faces had moved apart priorto fracture; plastic deformation blunted an initially sharp crack, as illustrated in Fig. 3.1.The degree of crack blunting increased in proportion to the toughness of the material.This observation led Wells to propose the opening at the crack tip as a measure of fracturetoughness. Today, this parameter is known as the crack tip opening displacement(CTOD).

In his original paper, Wells [1] performed an approximate analysis that relatedCTOD to the stress intensity factor in the limit of small scale yielding. Consider a crackwith a small plastic zone, as illustrated in Fig. 3.2. Irwin [2] showed that crack tipplasticity makes the crack behave as if it were slightly longer. Thus, we can estimateCTOD by solving for the displacement at the physical crack tip, assuming an effectivecrack length of a+ry. From Table 2.2, the displacement rv behind the effective crack tipis given by

(3-D

and the Irwin plastic zone correction for plane stress is

117

118 Chapter 3

FIGURE 3.1 Crack tip openingdisplacement (CTOD). An initiallysharp crack blunts with plasticdeformation, resulting in a finite dis-placement (5) at the crack tip.

FIGURE 3.2 Estimation of CTODfrom the displacement of the effectivecrack in the Irwin plastic zonecorrection.

v —v —

Substituting Eq. (3.2) into Eq. (3.1) gives

-2

(3.2)

C/ ™"™* ~

n (3.3)

where S is the CTOD. Alternatively, CTOD can be related to the energy release rate byapplying Eq. (2.51): y

Elastic-Plastic Fracture Mechanics 119

n(3.4)

Thus in the limit of small scale yielding, CTOD is related to Q and Kj. Wells postulatedthat CTOD is an appropriate crack tip characterizing parameter when LEFM is no longervalid. This assumption was shown to be correct several years later when a uniquerelationship between CTOD and the / integral was established (Section 3.3).

The strip yield model provides an alternate means for analyzing CTOD[3]. RecallSection 2.8.2, where the plastic zone was modeled by yield magnitude closure stresses.The size of the strip yield zone was defined by the requirement of finite stresses at thecrack tip. The CTOD can be defined as the crack opening displacement at the end of thestrip yield zone, as Fig. 3.3 illustrates. According to this definition, CTOD in a throughcrack in an infinite plate subject to a remote tensile stress (Fig. 2.3) is given by [3]

c SCVca, | 71 O"8 = —^— In sec

7CE 1(3-5)

Equation (3.5) is derived in Appendix 3.1. Series expansion of the In sec term gives

I f n a \ I \ n <jnE

_ A/<7YSE

1+ l(n o- Y(3-6)

FIGURE 3.3 Estimation of CTODfrom the strip yield model [3].

120 Chapters

5 = (3.7)

which differs slightly from Eq. (3.3).The strip yield model assumes plane stress conditions and a nonhardening material.

The actual relationship between CTOD and Kj and £ depends on stress state and strainhardening. The more general form of this relationship can be expressed as follows:

8 = d (3.8)

where m is a dimensionless constant that is approximately 1.0 for plane stress and 2.0 forplane strain.

There are a number of alternative definitions of CTOD. The two most commondefinitions, which are illustrated in Fig. 3.4, are the displacement at the original crack tipand the 90° intercept. The latter definition was suggested by Rice [4] and is commonlyused to infer CTOD in finite element measurements. Note that these two definitions areequivalent if the crack blunts in a semicircle.

Most laboratory measurements of CTOD have been made on edge-cracked specimensloaded in three-point bending (see Table 2.4). Early experiments utilized a flat paddle-shaped gage that was inserted into the crack; as the crack opened, the paddle gage rotated,and an electronic signal was sent to an x-y plotter. This method was inaccurate, however,because it was difficult to reach the crack tip with the paddle gage. Today, thedisplacement, V, at the crack mouth is measured, and the CTOD is inferred by assumingthe specimen halves are rigid and rotate about a hinge point, as illustrated in Fig. 3.5.Referring to this figure, we can estimate CTOD from a similar triangles construction:

X

(a) Displacement at the original crack tip. (b) Displacement at the intersection of a 90° ver-tex with the crack flanks.

FIGURE 3.4 Alternative definitions of CTOD.


FIGURE 3.5 The hinge model for estimating CTOD from three-point bend specimens.

Vr(W-a) r(W-a)

Therefore,

r(W-a)Vr(W~a) + t

(3.9)

where r is the rotational factor, a dimensionless constant between 0 and 1.The hinge model is inaccurate when displacements are primarily elastic.

Consequently, standard methods for CTOD testing [5,6] typically adopt a modified hingemodel, in which displacements are separated into elastic and plastic components; thehinge assumption is applied only to plastic displacements. Figure 3.6 illustrates atypical load (P) versus displacement (V) curve from a CTOD test. The shape of the load-displacement curve is similar to a stress-strain curve: it is initially linear but deviatesfrom linearity with plastic deformation. At a given point on the curve, the displacementis separated into elastic and plastic components by constructing a line parallel to theelastic loading line. The dashed line represents the path of unloading for this specimen,assuming the crack does not grow during the test. The CTOD in this specimen isestimated by

r(W-a}VD

maYSE' rp(W-d)(3.10)

The subscripts "el" and "p" denote elastic and plastic components, respectively. Theelastic stress intensity factor is computed by inserting the load and specimen dimensionsinto the appropriate expression in Table 2.4. The plastic rotational factor, rp, isapproximately 0.44 for typical materials and test specimens. Note that Equation (3.10)

122 Chapter 3

reduces to the small scale yielding result (Eq. 3.8) for linear elastic conditions, but thehinge model dominates when V ~ Vp.

Further details of CTOD testing are given in Chapter 7. Chapter 9 outlines howCTOD is used in design.

LOAD

(V,P)

FIGURE 3.6 Determination of the plasticcomponent of the crack mouth openingdisplacement.

MOUTH OPENING DISPLACMENT

3.2 THE J CONTOUR INTEGRAL

The / contour integral has enjoyed great success as a fracture characterizing parameter fornonlinear materials. By idealizing elastic-plastic deformation as nonlinear elastic, Rice[4] provided the basis for extending fracture mechanics methodology well beyond thevalidity limits of LEFM.

Figure 3.7 illustrates the uniaxial stress-strain behavior of elastic-plastic andnonlinear elastic materials. The loading behavior for the two materials is identical, butthe material responses differ when each is unloaded. The elastic-plastic material follows alinear unloading path with the slope equal to Young's modulus, while the nonlinearelastic material unloads along the same path as it was loaded. There is a unique relation-ship between stress and strain in an elastic material, but a given strain in an elastic-plasticmaterial can correspond to more than one stress value if the material is unloaded orcyclically loaded. Consequently, it is much easier to analyze an elastic material than amaterial that exhibits irreversible plasticity.

As long as the stresses in both materials in Fig. 3.7 increase monotonically, themechanical response of the two materials is identical. When the problem is generalized tothree dimensions, it does not necessarily follow that the loading behavior of the nonlinearelastic and elastic-plastic materials is identical, but there are many instances where this isa good assumption (see Appendix 3.6). Thus an analysis that assumes nonlinear elasticbehavior may be valid for an elastic-plastic material, provided no unloading occurs. The


deformation theory of plasticity, which relates total strains to stresses in a material, isequivalent to nonlinear elasticity.

Rice [4] applied deformation plasticity (i.e., nonlinear elasticity) to the analysis of acrack in a nonlinear material. He showed that the nonlinear energy release rate, J, couldbe written as a path-independent line integral. Hutchinson [7] and Rice and Rosengren [8]also showed that J uniquely characterizes crack tip stresses and strains in nonlinearmaterials. Thus the / integral can be viewed as both an energy parameter and a stressintensity parameter.

STRESS1Nonlinear Elastic

Material

FIGURE 3.7 Schematic comparison of thestress-strain behavior of elastic-plastic andnonlinear elastic materials.

STRAIN

3.2.1 Nonlinear Energy Release Rate

Rice [4] presented a path-independent contour integral for analysis of cracks. He thenshowed that the value of this integral, which he called /, is equal to the energy release ratein a nonlinear elastic body that contains a crack. In this section, however, the energyrelease rate interpretation is discussed first because it is closely related to concepts intro-duced in Chapter 2. The /contour integral is outlined in Section 3.2.2. Appendix 3.2gives a mathematical proof, similar to what Rice [4] presented, that shows that this lineintegral is equivalent to the energy release rate in nonlinear elastic materials.

Equation (2.23) defines the energy release rate for linear materials. The samedefinition holds for nonlinear elastic materials, except that £is replaced by J:

dn(3.11)

where 77 is the potential energy and & is crack area. The potential energy is given by

U = U-F (3.12)

124 Chapter 3

where U is the strain energy stored in the body and F is the work done by external forces.Consider a cracked plate which exhibits a nonlinear load-displacement curve, as illustratedin Fig. 3.8. If the plate has unit thickness, M - a.J For load control,

where U* is the complimentary strain energy, defined as

P

0

Thus if the plate in Fig. 3.8 is in load control, / is given by

\j ~"~*

da //>

If the crack advances at a fixed displacement, F = 0, and J is given by

'--f-1\da y

(3.13)

(3.14)

(3.15)

According to Fig. 3.8, dU* for load control differs from -dU for displacement control by

the amount - dPdA, which is vanishingly small compared to dU. Therefore, J for load

control is equal to J for displacement control. Recall that we obtained this same result forg in Section 2.4.

By invoking the definitions for U and U*, we can express J in terms of load anddisplacement:

J =

= ^~ dP (3.16)

0

It is important to remember that the energy release rate is defined in terms of crack area, not crack length.Failure to recognize this can lead to errors and confusion when computing Q or J for configurations other thanedge cracks; examples include a through crack, where dJL = 2da (assuming unit thickness), and a penny-shaped crack, where dJ3 = 2rcada.


\

LOAD _ __

dU* = -dU

DISPLACEMENT

FIGURE 3.8 Nonlinear energy release rate.

or

0

(3.17)

Integrating Eq. (3.17) by parts leads to a rigorous proof of what we have already inferredfrom Fig. 3.8. That is, Eqs. (3.16) and (3.17) are equal, and J is the same for fixed loadand fixed grip conditions.

Thus, J is a more general version of the energy release rate. For the special case ofa linear elastic material, J-<^. Also,

E(3.18)

for linear elastic Mode I loading. (For mixed mode loading, refer to Eq. (2.58).)A word of caution is necessary when applying J to elastic-plastic materials. The

energy release rate is normally defined as the potential energy that is released from astructure when the crack grows in an elastic material. However, much of the strain

126 Chapters

energy absorbed by an elastic-plastic material is not recovered when the crack grows or thespecimen is unloaded; a growing crack in an elastic-plastic material leaves a plastic wake(Fig. 2.6(b)). Thus the energy release rate concept has a somewhat different interpretationfor elastic-plastic materials. Rather than defining the energy released from the body whenthe crack grows, Eq. (3.15) relates /to the difference in energy absorbed by specimenswith neighboring crack sizes. This distinction is important only when the crack grows(Section 3.4.2). See Appendix 4.2 and Chapter 11 for further discussion of the energyrelease rate concept.

The energy release rate definition of J is useful for elastic-plastic materials whenapplied in an appropriate manner. For example, Section 3.2.5 describes how Eqs. (3.15)to (3. 17) can be exploited to measure / experimentally.

3.2.2 J as a Path-Independent Line Integral

Consider an arbitrary counter-clockwise path (F) around the tip of a crack, as il-lustrated in Fig. 3.9. The 7 integral is given by:

(3.19)

where w is the strain energy density, TI are components of the traction vector, u{ are thedisplacement vector components, and ds is a length increment along the contour 7~1 Thestrain energy density is defined as

w= \Cijdetj (3.20)o

where <7y and eij are the stress and strain tensors, respectively. The traction is a stressvector normal to the contour. That is, if we were to construct a free body diagram on thematerial inside of the contour, 7*f would define the normal stresses acting at theboundaries. The components of the traction vector are given by

FIGURE 3.9 Arbitrary contour aroundthe tip of a crack.


Ti = Gijtij (3.21)

where nj are the components of the unit vector normal to F.Rice [4] showed that the value of the J integral is independent of the path of in-

tegration around the crack. Thus J is called & path-independent integral. Appendix 3.2demonstrates this path independence, and shows that Eq. (3.19) is equal to the energyrelease rate.

3.2.3 J as a Stress Intensity Parameter

Hutchinson [7] and Rice and Rosengren [8] independently showed that J characterizescrack tip conditions in a nonlinear elastic material. They each assumed a power lawrelationship between plastic strain and stress. If elastic strains are included, thisrelationship for uniaxial deformation is given by

(3.22)

where <jo is a reference stress value that is usually equal to the yield strength, £0 = <J(/E,a is a dimensionless constant, and n is the strain hardening exponent^. Eq. (3.22) isknown as the Ramberg-Osgood equation, and is widely used for curve-fitting stress-straindata. Hutchinson, Rice, and Rosengren showed that in order to remain path independent,stress«strain must vary as 1/r near the crack tip. At distances very close to the crack tip,well within the plastic zone, elastic strains are small in comparison to the total strain,and the stress-strain behavior reduces to a simple power law. These two conditions implythe following variation of stress and strain ahead of the crack tip:

1

lnrj

(3.23D)r

where k] and &2 are proportionality constants, which are defined more precisely below.

For a linear elastic material, n=l, and Eq. (3.23) predicts a Jf/\ r singularity, which isconsistent with LEFM theory.

f\Although Eq. (3.22) contains four material constants, there are only two fitting parameters. The choice of

ao, which is arbitrary, defines e0; a linear regression is then performed on a log-log plot of stress versusplastic strain to determine a and n.

128 Chapters

The actual stress and strain distributions are obtained by applying the appropriateboundary conditions (see Appendix 3.4):

and

EJ\

n+l(3.24a)

E(3.24b)

where In is an integration constant that depends on n, and Oij and EIJ are dimensionlessfunctions of n and 6. These parameters also depend on the stress state (i.e. plane stress orplane strain). Equations (3.24a) and (3.24b) are called the HRR singularity, named afterHutchinson, Rice, and Rosengren [7,8]. Figure 3.10 is a plot of In versus n for plane

stress and plane strain. Figures 3.11 shows the angular variation of <Jij(n,6) [1]. Thestress components in Fig. 3.11 are defined in terms of polar coordinates rather than x andy.

5.5

2.5

4.5

4

3.5 f

3

Plane Strain

Plane Stress

10 12 14

nFIGURE 3.10 Effect of the strain hardening exponent on the HRR integration constant.


-0.5

(b) Plane strain

FIGURE 3.11 Angular variation of dimensionless stress for n = 3 and n = 13 [7].

The J integral defines the amplitude of the HRR singularity, just as the stress in-tensity factor characterizes the amplitude of the linear elastic singularity. Thus Jcompletely describes the conditions within the plastic zone. A structure in small-scaleyielding has two singularity-dominated zones: one in the elastic region, where stress

varies as 7/V r, and one in the plastic zone where stress varies as r" (n+l). The latteroften persists long after the linear elastic singularity zone has been destroyed by crack tipplasticity.

3.2.4 The Large Strain Zone

The HRR singularity contains the same apparent anomaly as the LEFM singularity;namely, both predict infinite stresses as r —> 0. The singular field does not persist all theway to the crack tip, however. The large strains at the crack tip cause the crack to blunt,which reduces the stress triaxiality locally. The blunted crack tip is a free surface; thus<JXX must vanish at r = 0.

130 Chapter 3

The analysis that leads to the HRR singularity does not consider the effect of theblunted crack tip on the stress fields, nor does it take account of the large strains that arepresent near the crack tip. This analysis is based on small strain theory, which is themulti-axial equivalent of engineering strain in a tensile test. Small strain theory breaksdown when strains are greater than -0.10 (10%).

McMeeking and Parks [9] performed crack tip finite element analyses that incor-porated large strain theory and finite geometry changes. Some of their results are shownin Fig. 3.12, which is a plot of stress normal to the crack plane versus distance. TheHRR singularity (Eq. (3.24a)) is also shown on this plot. Note that both axes arenondimensionalized in such a way that both curves are invariant, as long as the plasticzone is small compared to specimen dimensions.

The solid curve in Fig. 3.12 reaches a peak when the ratio x(J(/J is approximatelyunity, and decreases as x —> 0. This distance corresponds approximately to twice theCTOD. The HRR singularity is invalid within this region, where the stresses are in-fluenced by large strains and crack blunting.

4.5

Go

3.5

2.5

= 10

Large Strain Analysis

HRR Singularity

Stress field influencedby crack blunting

FIGURE 3.12 Large-strain crack tip finite element results of McMeeking and Parks [9]. Bluntingcauses the stresses to deviate from the HRR solution close to the crack tip


The break-down of the HRR solution at the crack tip leads to a similar question toone that was posed in Section 2.10: is the J integral a useful fracture criterion when ablunting zone forms at the crack tip? The answer is also similar to the argument offeredin Section 2.10. That is, as long as there is a region surrounding the crack tip that can bedescribed by Eq. (3.24), the /integral uniquely characterizes crack tip conditions, and acritical value of J is a size-independent measure of fracture toughness. The question of Jcontrolled fracture is explored further in Section 3.5.

3.2.5 Laboratory Measurement of J

When the material behavior is linear elastic, calculation of the J integral in a testspecimen or structure is relatively straightforward because / = £, and Q is uniquely relatedto the stress intensity factor. The latter quantity can be computed from the load and cracksize, assuming a K solution for that particular geometry is available. Table 2.4 andChapter 12 give several examples of stress intensity solutions.

Computing the J integral is somewhat more difficult when the material is nonlinear.The principle of superposition no longer applies, and J is not proportional to the appliedload. Thus a simple relationship between J, load, and crack length is usually notavailable.

One option for determining J is to apply the line integral definition Eq. (3.19) to theconfiguration of interest. Read [10] has measured the J integral in test panels byattaching an array of strain gages in a contour around the crack tip. Since J is pathindependent and the choice of contour is arbitrary, he selected a contour in such a way asto simplify the calculation of J as much as possible. This method can also be applied tofinite element analysis; i.e. stresses, strains and displacements can be determined along acontour and J can then calculated according to Eq. (3.19). The contour method fordetermining J is impractical in most cases, however. The instrumentation required forexperimental measurements of the contour integral is highly cumbersome, and the con-tour method is also not very attractive in numerical analysis (see Chapter 11). A muchbetter method for determining / numerically is outlined in Chapter 11. More practicalexperimental approaches are developed below and are explored further in Chapter 7.

Landes and Begley [11,12], who were among the first to measure / experimentally,invoked the energy release rate definition of J (Eq. 3.11). Figure 3.13 schematically il-lustrates their approach. They obtained a series of test specimens of the same size,geometry, and material and introduced cracks of various lengths^. They deformed eachspecimen and plotted load versus displacement (Fig. 3.13 (a)). The area under a givencurve is equal to U, the energy absorbed by the specimen. Landes and Begley plotted Uversus crack length at various fixed displacements (Fig. 3.13 (b)). For an edge crackedspecimen of thickness B, the J integral is given by

?See Chapter 7 for a description of fatigue precracking procedures for test specimens.

132 Chapter 3

(c)

FIGURE 3.13 Schematic of early experimental measurements of /, performed by Landes and Begley

[11,12]

Thus / can be computed by determining the slope of the tangent to the curves in Fig.3.13 (b). Applying Eq. (3.25) leads to Fig. 3.13 (c), a plot of/versus displacement atvarious crack lengths. The latter is a calibration curve, which only applies to thematerial, specimen size, specimen geometry, and temperature for which it was obtained.The Landes and Begley approach has obvious disadvantages, since multiple specimensmust be tested and analyzed to determine / in a particular set of circumstances.

Rice, et. al. [13] showed that it was possible, in certain cases, to determine/directly from the load displacement curve of a single specimen. Their derivations of/


relationships for several specimen configurations demonstrate the usefulness of di-mensional analysis4.

Consider a double edge notched tension panel of unit thickness (Fig. 3.14). Cracksof length a on opposite sides of the panel are separated by a ligament of length 2 b. Forthis configuration, dJ% = 2 da = -2 db (see Footnote 1); Eq. (3.16) must be modifiedaccordingly:

(3.26)

In order to compute 7 from the above expression, it is necessary to determine the re-lationship between load, displacement, and panel dimensions. Assuming an isotropicmaterial that obeys a Ramberg-Osgood stress-strain law (Eq. 3.22) dimensional analysisgives the following functional relationship for displacement:

a cr—;—-\v\a\n (3.27)

FIGURE 3.14 Double edge notched

tension (DENT) panel.

See Section 1.5 for a review of the fundamentals of dimensional analysis.

134 Chapter 3

where 0 is a dimensionless function. For fixed material properties, we need onlyconsider load and specimen dimensions. For reasons described below, we can simplify thefunctional relationship for displacement by separating A into elastic and plasticcomponents:

(3.28)

Substituting Eq. (3.28) into Eq. (3.26) leads to a relationship for elastic and plasticcomponents of J:

7 — dP

v 2 ^_ Kj 1 r

" " # 2 - 1

toiiafe dP (3.29)

where E' = E for plane stress and E' = £7(7 - v^J for plane strain, as defined in Chapter 2.Thus we need only be concerned about plastic displacements because a solution for theelastic component of/is already available (Table 2.4). If plastic deformation is confinedto the ligament between the crack tips (Fig. 3.14 (b)), we can assume that b is the onlylength dimension that influences Ap. This is a reasonable assumption, provided the panelis deeply notched so that the average stress in the ligament is substantially higher thanthe remote stress in the gross cross section. We can define a new function for Ap:

(3.30)

Note that the net-section yielding assumption has eliminated the dependence on the a/bratio. Taking a partial derivative with respect to the ligament length gives

PîP

where //'denotes the first derivative of the function H. We can solve for H' by taking apartial derivative of Eq. (3.30) with respect to load:


Therefore,

apj, u

aA,(3.31)

Substituting Eq. (3.31) into Eq. (3.29) and integrating by parts gives

K1 1 I" A'J=ir+Yb2l (3.32)

Recall that we assumed a unit thickness at the beginning of this derivation. In general,the plastic term must be divided by the plate thickness; the term in square brackets, whichdepends on the load displacement curve, is normalized by the net cross-sectional area ofthe panel. The J integral has units of energy/area.

Another example from the Rice, et al. article [13] is an edge cracked plate in bending(Fig. 3.15). In this case they chose to separate displacements along somewhat differentlines from the previous problem. If the plate is subject to a bending moment M, itwould displace by an angle £lnc if no crack were present, and an additional amount, £lc,when the plate is cracked. Thus the total angular displacement can be written as

Q. = £lnc + Q

If the crack is deep, £lc » £lnc- The energy absorbed by the plate is given by

(3.33)

(3.34)

0

When we differentiate U with respect to crack area in order to determine J, only Qc

contributes to the energy release rate because Qnc is not a function of crack size, bydefinition. By analogy with Eq. (3.16), / for the cracked plate in bending can be writtenas

M

dMM -Jf§ dM (3.35)

M

136 Chapters

FIGURE 3.15 Edge cracked plate in pure bending.

If material properties are fixed, dimensional analysis leads to

(3.36)

assuming the ligament length is the only relevant length dimension, which is reasonableif the crack is deep. When Eq. (3.36) is differentiated with respect to b and inserted intoEq, (3.35), the resulting expression for J is as follows:

(3.37)

The decision to separate £2 into "crack" and "no-crack" components was somewhatarbitrary. The angular displacement could have been divided into elastic and plasticcomponents as in the previous example. If the crack is relatively deep, Q.nc should beentirely elastic, while jQc may contain both elastic and plastic contributions. Therefore,Eq. (3.37) can be written as

or

(3.38)


Conversely, the prior analysis on the double edged cracked plate in tension could havebeen written in terms of Ac and Anc. Recall, however, that the dimensional analysis wassimplified in each case (Eqs. (3.30) and (3.36)) by assuming a negligible dependence ona/b. This turns out to be a reasonable assumption for plastic displacements in deeply-notched DENT panels, but less so for elastic displacements. Thus while elastic andplastic displacements due to the crack can be combined to compute / in bending (Eq.(3.37)), it is not advisable to do so for tensile loading. The relative accuracy and thelimitations of Eqs. (3.32) and (3.37) are evaluated in Section 9.5.

In general, the J integral for a variety of configurations can be written in the fol-lowing form:

, r\U,J (339)

where 77 is a dimensionless constant. Note that Eq. (3.39) contains the actual thickness,while the above derivations assumed unit thickness for convenience. Equation (3.39)expresses J as the energy absorbed, divided by the cross-sectional area, times adimensionless constant. For a deeply cracked plate in pure bending, 77 = 2. Equation(3.39) can be separated into elastic and plastic components:

rlelUc(el) [ TlpU

Bb Bb

UpV (3.40)E Bb

EXAMPLE 3.1

Determine the plastic TJ factor for the DENT configuration, assuming the load-plasticdisplacement curve follows a power law:

P = CApN

Solution: The plastic energy absorbed by the specimen is given by

AnP C A"*1 PAp

0

Comparing Eqs. (3.32) and (3.40) and solving for T]p gives

138 Chapters

EXAMPLE 3.1 (cont)

- ')Up = P Ap

N+l

= 1-N

For a nonhardening material^ N= 0 and r]p = 1.

3.3 RELATIONSHIPS BETWEEN J AND CTOD

For linear elastic conditions, the relationship between CTOD and £is given by Eq. (3.8).Since /= § for linear elastic material behavior, these equations also describe therelationship between CTOD and / in the limit of small scale yielding. That is,

where m is a dimensionless constant that depends on stress state and material properties.It can be shown that Eq. (3.41) applies well beyond the validity limits of LEFM.

Consider, for example, a strip yield zone ahead of a crack tip, as illustrated in Fig.3.16. Recall (from Chapter 2) that the strip yield zone is modeled by surface tractionsalong the crack face. Let us define a contour, F, along the boundary of this zone. If thedamage zone is long and slender, i.e., if p » 8, the first term in the / contour integral(Eq. 3.19) vanishes because dy = 0. Since the only surface tractions within p are in the ydirection, ny = 1 and nx = nz = 0. Thus the / integral is given by

ds (3.42)

Let us define a new coordinate system with the origin at the tip of the strip yield zone: X= p-x. For a fixed S, Oyy and uy depend only on X, provided p is small compared to thein-plane dimensions of the cracked body. The / integral becomes


FIGURE 3.16 Contour along the boundary of the strip yield zone ahead of a crack tip.

= \Cfyy(S}dd (3.43)

where 8 = 2 Uy (X = p). Since the strip yield model assumes o-y-y = ays within theplastic zone, the J-CTOD relationship is given by

J = (3.44)

Note the similarity between Eqs. (3.44) and (3.7). The latter was derived from the stripyield model by neglecting the higher order terms in a series expansion; no suchassumption was necessary to derive Eq. (3.44). Thus the strip yield model, which as-sumes plane stress conditions and a nonhardening material, predicts that m = 1 for bothlinear elastic and elastic-plastic conditions.

Shih [14] provided further evidence that a unique J-CTOD relationship applies wellbeyond the validity limits of LEFM. He evaluated the displacements at the crack tipimplied by the HRR solution and related the displacement at the crack tip to / and flowproperties. According to the HRR solution, the displacements near the crack tip are asfollows:

_aa0

EEJ n+l

(3.45)

140 Chapter 3

FIGURE 3.17 Estimation of CTOD from a90° intercept construction and HRRdisplacements.

where• ui is a dimensionless function of 0 and n, analogous to cr/y and e/y (Eq. 3.24).Shin [14] invoked the 90° intercept definition of CTOD, as illustrated in Fig. 3.4(b).This 90° intercept construction is examined further in Fig. 3.17. The CTOD is obtainedby evaluating ux and My at r = r* and 9 = IT.

y— = uy (r*, TT) = r * -ux (r*, n)

Substituting Eq. (3.46) into Eq. (3.45) and solving for r* gives

E

Setting 5=2 uy(r*; it) leads to-a TT

where dn is a dimensionless constant, given by

n-H /<u

(3.46)

(3.47)

(3.48)

" 1 £ (3.49)


Figure 3.18 shows plots of dn for a= 1.0, which exhibits a strong dependence on thestrain hardening exponent and a mild dependence on O.GQ/E. A comparison of Eqs. (3.41)and (3.48) indicates that dn = 1/m, assuming <J0 = uys (see Footnote 2). According toFig. 3.18(a), dn = 1.0 for a nonhardening material (n = °°) in plane stress, which agreeswith the strip yield model (Eq. (3.44)).

The Shih analysis shows that there is a unique relationship between J and CTOD fora given material. Thus these two quantities are equally valid crack tip characterizingparameters for elastic-plastic materials. The fracture toughness of a material can bequantified either by a critical value of J or CTOD.

dn

1.0

0.8

0.6

0.4

0.2

0

dn

1.0

0.8

0.6

0.4

0.2

0.0080.0040.002|0.001

0 0.1 0.2 0.3 0.4 0.5

1/n

T

(U O3 O4 (X51/n

FIGURE 3.18 Predicted J-CTOD relationships for plane stress and plane strain, assuming a = 1 [14].For a# 1, the above values should be multiplied by or .

142 Chapters

The above analysis contains an apparent inconsistency. Equation (3.48) is based onthe HRR singularity, which does not account for large geometry changes at the crack tip.Figure 3.12 indicates that the stresses predicted by the HRR theory are inaccurate for r <28, but the Shih analysis uses the HRR solution to evaluate displacements well withinthe large strain region. Crack tip finite element analyses [14], however, are in generalagreement with Eq. (3.48). Thus the displacement fields predicted from the HRR theoryare reasonably accurate, despite the large plastic strains at the crack tip.

3.4 CRACK GROWTH RESISTANCE CURVES

Many materials with high toughness do not fail catastrophically at a particular value of/or CTOD. Rather, these materials display a rising R curve, where / and CTOD increasewith crack growth. In metals, a rising R curve is normally associated with growth andcoalescence of microvoids. See Chapter 5 for a discussion of microscopic fracturemechanisms in ductile metals.

Figure 3.19 schematically illustrates a typical /resistance curve for a ductile ma-terial. In the initial stages of deformation, the R curve is nearly vertical; there is a smallamount of apparent crack growth due to blunting. As J increases, the material at thecrack tip fails locally and the crack advances further. Because the R curve is rising, theinitial crack growth is usually stable, but an instability can be encountered later, asdiscussed below.

CRACK EXTENSION

FIGURE 3.19 Schematic /resistance curve for a ductile material.


One measure of fracture toughness, Jjc is defined near the initiation of stable crackgrowth. The precise point at which crack growth begins is usually ill-defined.Consequently, the definition of J]c is somewhat arbitrary, much like a 0.2% offset yieldstrength. The corresponding CTOD near the initiation of stable crack growth is denoted<5/ by U.S. and British testing standards. Chapter 7 describes experimental measurementsof JIG and Si in more detail.

While initiation toughness provides some information about the fracture behavior ofa ductile material, the entire R curve gives a more complete description. The slope of theR curve at a given amount of crack extension is indicative of the relative stability of thecrack growth; a material with a steep R curve is less likely to experience unstable crackpropagation. For J resistance curves, the slope is usually quantified by a dimensionlesstearing modulus:

E dJRTR=-^-K (3.49)

where the subscript R indicates a value of J on the resistance curve.

3.4.1 Stable and Unstable Crack Growth

The conditions that govern the stability of crack growth in elastic-plastic materials arevirtually identical to the elastic case presented in Section 2.5. Instability occurs when thedriving force curve is tangent to the jR curve. As Fig. 3.20 indicates, load control isusually less stable than displacement control. The conditions in most structures aresomewhere between the extremes of load control and displacement control. Theintermediate case can be represented by a spring in series with the structure, where remotedisplacement is fixed (Fig. 2.12). Since the R curve slope has been represented by adimensionless tearing modulus (Eq. 3.49), it is convenient to express the driving force interms of an applied tearing modulus:

T - . (350)a ~ da J (3'50)

where Ay is the total remote displacement defined as

144 Chapter 3

JJa

CRACK SIZE

FIGURE 3.20 Schematic / driving force//? curve diagram which compares load control anddisplacement control.

Ar = A + CmP (3.51)

and Cm is the system compliance. The slope of the driving force curve for a fixed AT isidentical to the linear elastic case (Eq. 2.35), except that § is replaced by J:

C"

-I

(3.52)

For load control, Cm = °°, and the second term in Eq. (3.52) vanishes:

-di\ ran

For displacement control, Cm = 0, and Aj = A. Equation (3.52) is derived in Appendix2.1 for the linear elastic case.

The conditions during stable crack growth can be expressed as follows:

'R (3.53a)and


Tapp<TR (3.53b)

Unstable crack propagation occurs when

Tapp > TR (3.54)

Chapter 9 gives practical guidance on assessing structural stability with Eqs. (3.50) to(3.54). A simple example is presented below.

EXAMPLE 3.2

Derive an expression for the applied tearing modulus in the double cantilever beam(DCB) specimen with a spring in series (Fig. 3.21), assuming linear elasticconditions.

Solution: From Example 2.1, we have the following relationships:

P2a2 J A 2 Pa3

J — § — n "f J ant^ A = 3 p ,

Therefore, the relevant partial derivatives are given by

A 2 P2 a

dajp B E I

'dJ\ 2 Pa2

dPJa = BEI

_2Pa2

dap " El

2 a3

dP)a~ 3EI

Substituting the above relationships into Eqs. (3.50) and (3.52) gives

rCMEl I 3EI r

146 Chapters

Thickness = B

FIGURE 3.21 Double cantilever beam specimen with a spring in series.

As discussed in Section 2.5, the point of instability in a material with a rising Rcurve depends on the size and geometry of the cracked structure; a critical value of / atinstability is not a material property if / increases with crack growth. It is usuallyassumed that the R curve, including the Jjc value, is a material property, independent ofconfiguration. This is a reasonable assumption, within certain limitations.

3.4.2 Computing J for a Growing Crack

The geometry dependence of a / resistance curve is influenced by the way in which / iscalculated. The equations derived in Section 3.2.5 are based on the pseudo energy releaserate definition of J and are valid only for a stationary crack. There are various ways tocompute J for a growing crack, including the deformation J and the far-field /, which aredescribed below. The former method is typically used to obtain experimental J resistancecurves.

Figure 3.22 illustrates the load-displacement behavior in a specimen with a growingcrack. Recall that the J integral is based on a deformation plasticity (or nonlinear elastic)assumption for material behavior. Consider point A on the load-displacement curve inFig. 3.22. The crack has grown to a length aj from an initial length ao. The cross-hatched area represents energy that would be released if the material were elastic. In anelastic-plastic material, only the elastic portion of this energy is released; the remainder isdissipated in a plastic wake that forms behind the growing crack (see Figs 2.6(b) and3.25).


In an elastic material, all quantities, including strain energy, are independent of theloading history. The energy absorbed during crack growth in an elastic-plastic material,however, exhibits a history dependence. The dashed curve in Fig. 3.22 represents theload-displacement behavior when the crack size is fixed at aj. The area under this curve isthe strain energy in an elastic material; this energy depends only on the current load andcrack length:

(3.55)

where the subscript D refers to deformation theory. Thus the / integral for a nonlinearelastic body with a growing crack is given by

B

orBb

J = L + 1£D E' Bb

(3.56a)

(3.56b)

i Crack Growth

FIGURE 3.22 Schematic load-displacementcurve for a specimen with a crack thatgrows to aj from an initial length ao. Uj)represents the strain energy in a nonlinearelastic material.

148 Chapters

where b is the current ligament length. When the / integral for an elastic-plastic materialis defined by Eq. (3.56), the history dependence is removed and the energy release rateinterpretation of J is restored. The deformation /is usually computed from Eq. (3.56b)because no correction is required on the elastic term as long as Kj is determined from thecurrent load and crack length. The calculation of f/Df/?) is usually performedincrementally, since the deformation theory load-displacement curve (Fig. 3.22 and Eq.(3.55)) depends on crack size. Specific procedures for computing the deformation / areoutlined in Chapter 7.

One can determine a far-field J from the contour integral definition of Eq. (3.19),which may differ from //> For a deeply cracked bend specimen, Rice, et. al. [15] showedthat the far-field J contour integral in a rigid, perfectly plastic material is given by

ajy = 0.73cr0J bd& (3.57)

0

where the variation in b during the loading history is taken into account. Deformationtheory leads to the following relationship for J in this specimen:

JD = Q.13cr0b& (3.58)

The two expressions are obviously identical when the crack is stationary.Finite element calculations of Dodds, et. al. [16,17] for a three-point bend specimen

made from a strain hardening material indicate that Jy-and Jrj are approximately equal formoderate amounts of crack growth. The / integral obtained from a contour integration ispath-dependent in an elastic-plastic material, however, and tends to zero as the contourshrinks to the crack tip. See Appendix 4.2 for a theoretical explanation of the pathdependence of J for a growing crack in an inelastic material.

There is no guarantee that either the deformation Jj) or Jf will uniquely characterizecrack tip conditions for a growing crack. Without this single parameter characterization,the J-R curve becomes geometry dependent. The issue of J validity and geometrydependence is explored in detail in Sections 3.5 and 3.6.

3.5 J-CONTROLLED FRACTURE

The term J-controlled fracture corresponds to situations where / completely characterizescrack tip conditions. In such cases, there is a unique relationship between / and CTOD(Section 3.3); thus /-controlled fracture implies CTOD-controlled fracture, and vice versa.Just as there are limits to LEFM, fracture mechanics analyses based on / and CTODbecome suspect when there is excessive plasticity or significant crack growth. In suchcases, fracture toughness and the J-CTOD relationship depend on the size and geometry ofthe structure or test specimen.

The required conditions for/-controlled fracture are discussed below. Both fractureinitiation from a stationary crack and stable crack growth are considered.


3.5.1 Stationary Cracks

Figure 3.23 schematically illustrates the effect of plasticity on the crack tip stresses; log(o-y-y) is plotted against normalized distantce from the crack tip. The characteristic lengthscale L corresponds to the size of the structure; for example, L could represent theuncracked ligament length. Figure 3.23(a) shows the small scale yielding case, whereboth K and J characterize crack tip conditions. At a short distance from the crack tip,

relative to L, the stress is proportional to 7/v r, this area is called the K-dominated region.Assuming monotonic, quasistatic loading, a J-dominated region occurs in the plasticzone, where the elastic singularity no longer applies. Well inside of the plastic zone, the

HRR solution is approximately valid and the stresses vary as r '«+-/. The finite strainregion occurs within approximately 28 from the crack tip, where large deformationinvalidates the HRR theory. In small scale yielding, K uniquely characterizes crack tip

conditions, despite the fact that the 7/V r singularity does not exist all the way to thecrack tip. Similarly, J uniquely characterizes crack tip conditions even though thedeformation plasticity and small strain assumptions are invalid within the finite strain re-gion.

Figure 3.23(b) illustrates elastic-plastic conditions, where / is still approximatelyvalid, but there is no longer a K field. As the plastic zone increases in size (relative toL), the K dominated zone disappears, but the J dominated zone persists in somegeometries. Thus although K has no meaning in this case, the J integral is still anappropriate fracture criterion. Since J dominance implies CTOD dominance, the latterparameter can also be applied in the elastic-plastic regime.

With large scale yielding (Fig. 3.23(c)), the size of the finite strain zone becomessignificant relative to L, and there is no longer a region uniquely characterized by /.Single-parameter fracture mechanics is invalid in large scale yielding, and critical J valuesexhibit a size and geometry dependence.

In certain configurations, the K and / zones are vanishingly small, and a single-parameter description is not possible except at very low loads. For example, a plate loadedin tension with a through-thickness crack is not amenable to a single-parameterdescription, either by K or /. Example 2.4 and Figure 2.44 indicate that the stress in thex direction in this geometry deviates significantly from the elastic singularity solution assmall distances from the crack tip because of a compressive transverse (T) stress.Consequently the ^-dominated zone is virtually nonexistent. The T stress influencesstresses inside the plastic zone, so a highly negative T stress also invalidates a single-parameter description in terms of J. See Section 3.61 for further details about the T stress.

150 Chapters

(a) Small scale yielding

LOGcryy

(b) Elastic-plastic conditions

LOG r/Lrs/L

(c) Large scale yielding LOG a

LOG r/LLEGEND:

Large strain region

J-dominated zoneK-dominated zone FIGURE 3.23 Effect of plasticity on the

li as;:sw:si crack tip stress fields.No single parameter characterization


Recall Fig. 2.38, in which a free-body diagram was constructed from a disk-shapedregion removed from the crack tip of a structure loaded in small scale yielding. Since the

stresses on the boundary of this disk exhibit a 7/v r singularity, KI uniquely defines thestresses and strains within the disk. For a given material5, dimensional analysis leads tothe following functional relationship for the stress distribution within this region:

(for 0<r<r.(0)) (3.59)

where rs is the radius of the elastic singularity dominated zone, which may depend on 6.

Note that the 7/v r singularity is a special case of F, which exhibits a different dependenceon r within the plastic zone. Invoking the relationship between J and Kf for small scaleyielding (Eq. 3.18) gives

( forO<r<r y (0)) (3.60)

where rj is the radius of the /-dominated zone. The HRR singularity (Eq. (3.24a) is a

special case of Eq. (3.60), but stress exhibits a r 'n+1 dependence only over a limitedrange of r.

For small scale yielding, rs = rj, but rs vanishes when the plastic zone engulfs theelastic singularity dominated zone. The / dominated zone usually persists longer than theelastic singularity zone, as Fig. 3.23 illustrates.

It is important to emphasize that J dominance at the crack tip does not require theexistence of an HRR singularity. In fact, J dominance requires only that Eq. (3.60) isvalid in the process zone near the crack tip, where the microscopic events that lead to frac-ture occur. The HRR singularity is merely one possible solution to the more generalrequirement that J uniquely define crack tip stresses and strains. The flow properties ofmost materials do not conform to the idealization of a Ramberg-Osgood power law, uponwhich the HRR analysis is based. Even in a Ramberg-Osgood material, the HRRsingularity is valid over a limited range; large strain effects invalidate the HRR singu-larity close to the crack tip, and the computed stress lies below the HRR solution atgreater distances. The latter effect can be understood by considering the analyticaltechnique employed by Hutchinson [7], who represented the stress solution as an infinite

series and showed that the leading term in the series was proportional to r /n+l (SeeAppendix 3.4). This singular term dominates as r —> 0; higher order terms are significantfor moderate values of r. When the computed stress field deviates from HRR, it still

-1 A complete statement of the functional relationship of ajj should include all material flow properties (e.g. aand n for a Ramberg-Osgood material). These quantities were omitted from Eqs. (3.65) and (3.66) for thesake of clarity, since material properties are assumed to be fixed in this problem.

152 Chapters

scales with J/(a0 r), as required by Eq. (3.60). Thus J dominance does not necessarilyimply agreement with the HRR Fields.

Equations (3.59) and (3.60) gradually become invalid as specimen boundaries in-teract with the crack tip. We can apply dimensional arguments to infer when a single-parameter description of crack tip conditions is suspect. As discussed in Chapter 2, theLEFM solution breaks down when the plastic zone size is a significant fraction of in-plane dimensions. Moreover, the crack tip conditions evolve from plane strain to planestress as the plastic zone size grows to a significant fraction of the thickness. The Jintegral becomes invalid as a crack tip characterizing parameter when the large strainregion reaches a finite size relative to in-plane dimensions. Section 3.6 providesquantitative information on size effects.

3.5.2 J-ControlIed Crack Growth

According to the dimensional argument in the previous section, J controlled conditionsexist at the tip of a stationary crack (loaded monotonically and quasistatically), providedthe large strain region is small compared to in-plane dimensions of the cracked body.Stable crack growth, however, introduces another length dimension; i.e., the change incrack length from its original value. Thus / may not characterize crack tip conditionswhen the crack growth is significant compared to in-plane dimensions. Prior crackgrowth should not have any adverse effects in a purely elastic material, because the localcrack tip fields depend only on current conditions. Prior history does influence thestresses and strains in an elastic-plastic material, however. Therefore, we might expect Jintegral theory to break down when there is a combination of significant plasticity andcrack growth. This heuristic argument based on dimensional analysis agrees withexperiment and with more complex analyses.

Figure 3.24 illustrates crack growth under /-controlled conditions. Material behindthe growing crack tip has unloaded elastically. Recall Fig. 3.7, which compares theunloading behavior of nonlinear elastic and elastic-plastic materials; the material in theunloading region of Fig. 3.24 obviously violates the assumptions of deformationplasticity. The material directly in front of the crack also violates the single-parameterassumption because the loading is highly nonproportional; i.e., the various stresscomponents increase at different rates and some components actually decrease. In order forthe crack growth to be J controlled, the elastic unloading and nonproportional plasticloading regions must be embedded within a zone of J dominance. When the crack growsout of the zone of/dominance, the measured R curve is no longer uniquely characterizedby/.

In small scale yielding, there is always a zone of / dominance because the crack tipconditions are defined by the elastic stress intensity, which depends only on current valuesof load and crack size. The crack never grows out of the /-dominated zone as long as allspecimen boundaries are remote from the crack tip and the plastic zone.

Figure 3.25 illustrates three distinct stages of crack growth resistance behavior insmall scale yielding. During the initial stage the crack is essentially stationary; the finiteslope of the R curve is caused by blunting. The crack tip fields for Stage 1 are given by


FIGURE 3.24 /-controlled crackgrowth.

(1) Crack blunting

(2) Fracture initiation

(3) Steady state crack growth

Plastic Wake

Aa

FIGURE 3.25 Three stages of crack growth in an infinitebody.

154 Chapters

(3.61)

which is a restatement of Eq. (3.60). The crack begins to grow in Stage 2. The crack tipstresses and strains are probably influenced by the original blunt crack tip during the earlystages of crack growth. Dimensional analysis implies the following relationship:

0.62)

where <% is the CTOD at initiation of stable tearing. When the crack grows well beyondthe initial bluntedt tip, a steady-state condition is reached, where the local stresses andstrains are independent of the extent of crack growth:

(3.63)

Although Eqs. (3.61) and (3.63) would predict identical conditions in the elasticsingularity zone, material in the plastic zone at the tip of a growing crack is likely toexperience a different loading history from material in the plastic zone of a bluntingstationary crack; thus F^/ ?*F^ as r-*0. During steady-state crack growth, a plasticzone of constant size sweeps through the material, leaving a plastic wake, as illustrated inFig. 3.25. The R curve is flat; /does not increase with crack extension, provided thematerial properties do not vary with position. Appendix 3.5.2 presents a formalmathematical argument for a flat R curve during steady-state growth; a heuristicexplanation is given below.

If Eq. (3.63) applies, J uniquely describes crack tip conditions, independent of crackextension. If the material fails at some critical combination of stresses and strains, thenit follows that local failure at the crack tip must occur at a critical J value, as in thestationary crack case. This critical J value must remain constant with crack growth. Arising or falling R curve would imply that the local material properties varied withposition.

The second stage in Fig. 3.25 corresponds to the transition between blunting of astationary crack and crack growth under steady state conditions. A rising R curve ispossible in Stage 2. For small scale yielding conditions the R curve depends only oncrack extension:

JR = /fl(Afl) (3.64)

That is, the J-R curve is a material property.


The steady-state limit is usually not observed in laboratory tests on ductile mate-rials. In typical test specimens, the ligament is fully plastic during crack growth, therebyviolating the small scale yielding assumption. Moreover, the crack approaches a finiteboundary while still in Stage 2 growth. Enormous specimens would be required toobserve steady state crack growth in tough materials.

3.6 CRACK-TIP CONSTRAINT UNDER LARGE-SCALE YIELDING

Under small scale yielding conditions, a single parameter (e.g. K, J or CTOD) charac-terizes crack tip conditions and can be used as a geometry-independent fracture criterion.Single-parameter fracture mechanics breaks down in the presence of excessive plasticity,and fracture toughness depends on the size and geometry of the test specimen.

McClintock [18] applied slip line theory to estimate the stresses in a variety of con-figurations under plane strain, fully plastic conditions. Figure 3.26 summarizes some ofthese results. For small scale yielding (Fig. 3.26(a)), the maximum stress at the crack tipis approximately 3a0 in a nonhardening material. According to the slip line analysis, adeeply notched double-edged notched tension (DENT) panel, illustrated in Fig. 3.26(b),maintains a high level of triaxiality under fully plastic conditions, such that the crack tipconditions are similar to the small scale yielding case. An edge cracked plate in bending(Fig. 3.26(c)) exhibits slightly less stress elevation, with the maximum principal stressapproximately 2.5<JO, A center-cracked panel in pure tension (Fig. 3.26(d)) is incapableof maintaining significant triaxiality under fully plastic conditions.

The results in Fig. 3.26 indicate that, for a nonhardening material under fully yieldedconditions, the stresses near the crack tip are not unique, but depend on geometry.Traditional fracture mechanics approaches recognize that the stress and strain fields remotefrom the crack tip may depend on geometry, but it is assumed that the near-tip fields havea similar form in all configurations that can be scaled by a single parameter. The single-parameter assumption is obviously not valid for nonhardening materials under fullyplastic conditions, because the near tip fields depend on the configuration. Fracturetoughness, whether quantified by J, K, or CTOD, must also depend on the configuration.

The prospects for applying fracture mechanics in the presence of large scale yieldingare not quite as bleak as the McClintock analysis indicates. The configurational effectson the near-tip fields are much less severe when the material exhibits strain hardening.Moreover, single-parameter fracture mechanics may be approximately valid in thepresence of significant plasticity, provided the specimen maintains a relatively high levelof triaxiality. Both the DENT specimen and the edge cracked plate in bending apparentlysatisfy this requirement. Most laboratory measurements of fracture toughness areperformed with bend-type specimens, such as the compact and three-point bendgeometries, because these specimens present the fewest experimental difficulties.

Figure 3.27 compares the cleavage fracture toughness for bending and tensileloading. Although the scatter bands overlap, the average toughness for the single edgenotched bend (SENB) specimens is considerably lower than that of the center crackedtension (CCT) panels or the surface cracked panels.

156 Chapter 3

(a) Small scale yielding (b) DENT panel

(c) Edge crack in bending (d) Center cracked panel

FIGURE 3.26 Comparison of the plastic deformation pattern in small scale yielding (a) with slip patternsunder fully plastic conditions in three configurations. The estimated local stresses are based on the slipline analyses of McClintock [IS], and apply only to nonhardening materials.

Crack depth and specimen size can also have an effect on fracture toughness, as Fig.3.28 illustrates. Note that the SENB specimens with shallow cracks tend to have highertoughness than deep cracked specimens, and the specimens with 50mm x 50mm crosssections have lower average toughness than smaller specimens with the same a/W ratio.


DOHU

yH2 0.5U

BS 4360 SOD STEEL-65°C

B = 52 mm

87.5 Percentile 3

9 Median (74 values) tj

12.5 PercentileI I I

SENBSPECIMENS

CCT PANELS SURFACECRACKED PANELS

FIGURE 3.27 Critical CTOD values for cleavage fracture in bending and tensile loading for a low-alloystructural steel [19].

500

SPS

400

300

200

100

i i i i i i i i i i i i r

A 515 Grade 70 Steel(Kirk, et al., 1991)

O

Specimen Thicknessand Width:

—O— 50 mm

- 0- -25 mm

4- 10 mm

o1 1 1 1 1 1 1 !

0.1 0.2 0.3 0.4

a / W

0.5 0.6

FIGURE 3.28 Critical J values for cleavage as a function of crack depth and specimen size of single edgenotched bend (SENB) specimens [20]

158 Chapter3

Figures 3.27 and 3.28 illustrate the effect of specimen size and geometry oncleavage fracture toughness. Specimen configuration can also influence the R curve ofductile materials. Figure 3.29 shows the effect of crack depth on crack growth resistancebehavior. Note that the trend is the same as in Fig. 3.28. Joyce and Link [21] measuredJ-R curves for several geometries and found that the initiation toughness, Jjc, is relativelyinsensitive to geometry (Fig. 3.30), but the tearing modulus, as defined in Eq. (3.49), is astrong function of geometry (Fig. 3.31). Configurations that have a high level ofconstraint under full plastic conditions, such as the compact and deep-notched SENDspecimens, have low TR values relative to low constraint geometries, such as single edgenotched tension (SENT) panels.

Note that the DENT specimens have the highest tearing modulus in Fig. 3.31, butMcClintock's slip line analysis indicates that this configuration should have a high levelof constraint under fully plastic conditions. Joyce and Link presented elastic-plastic finiteelement results for the DENT specimen that indicated significant constraint loss in thisgeometry6, which is consistent with the observed elevated tearing modulus. Thus the slipline analysis apparently does not reflect the actual crack tip conditions of this geometry.

A number of researchers have attempted to extend fracture mechanics theory beyondthe limits of the single-parameter assumption. Most of these new approaches involve theintroduction of a second parameter to characterize crack tip conditions. Several suchmethodologies are described below.

1.0 2.0 3.0 4.0

CRACK EXTENSION, mm

FIG. 3.29 Effect of crack length/specimen width ratio on J-R curves for HY130 steel single edgenotched bend (SENS) specimens [22].

"Joyce and Link quantified crack tip coastraint with the T and Q parameters, which are described in Sections3.6.1 and 3.6.2, respectively.


600

500

400

300

200

100

A533 Grade B Steel

Shallow NotchedSENB Specimens

Deeply NotchedSENB Specimens

gENT Specimens

CompactSpecimens DENT Specimens _

!

SPECIMEN TYPE

FIGURE 3.30 Effect of specimen geometry on critical J values for initiation of ductile tearing [21].

CO

DDO

<3

3wH

JvW

300

250

200

150

inn

i i i i; A533 Grade B Steel DENT Specimens .

SENT Specimens©

Shallow Notched @^ SENB Specimens

: • :~ Deeply NotchedI SENB Specimens ^ !

Compact @- Specimens ft

: § :

SPECIMEN TYPE

FIGURE 3.31 Effect of specimen geometry on tearing modulus at Aa = 1 mm [22].

160 Chapters

3.6.1 The Elastic T Stress

Williams [23] showed that the crack tip stress fields in an isotropic elastic material can be

expressed as an infinite power series, where the leading term exhibits a 7/V r singularity,

the second term is constant with r, the third term is proportional to vr, and so on.Classical fracture mechanics theory normally neglects all but the singular term, whichresults in a single-parameter description of the near-tip fields (see Chapter 2). Althoughthe third and higher terms in the Williams solution, which have positive exponents on r,vanish at the crack tip, the second (uniform) term remains finite. It turns out that thissecond term can have a profound effect on the plastic zone shape and the stresses deepinside the plastic zone [24,25].

For a crack in an isotropic elastic material subject to plane strain Mode I loading,the first two terms of the Williams solution are as follows:

T 00 00 0

0 "0vT

(3.65)

where T is a uniform stress in the x direction (which induces a stress vT in the z directionin plane strain).

We can assess the influence of the T stress by constructing a circular model thatcontains a crack, as illustrated in Fig. 3.32. On the boundary of this model, let us applyin-plane tractions that correspond to Eq. (3.65). A plastic zone develops at the crack tip,but its size must be small relative to the size of the model in order to ensure the validityof the boundary conditions, which are inferred from an elastic solution. Thisconfiguration, often referred to as a modified boundary layer analysis, simulates the near-tip conditions in an arbitrary geometry, provided the plasticity is well contained withinthe body. It is equivalent to removing a core region from the crack tip and constructing afree body diagram, as in Fig. 2.41.

FIGURE 3.32 Modified boundary layer analysis. The first two terms of the Williams series are appliedas boundary conditions.


Figure 3.33 is a plot of finite element results from a modified boundary layeranalysis [26] that show the effect of the T stress on stresses deep inside the plastic zone.The special case of T = 0 corresponds to the small-scale yielding limit, where the plasticzone is a negligible fraction of the crack length and size of the body7, and the singularterm uniquely defines the near-tip fields. The single-parameter description is rigorouslycorrect only for T — 0. Note that negative T values cause a significant downward shift inthe stress fields. Positive T values shift the stresses to above the small-scale yieldinglimit, but the effect is much less pronounced than it is for negative T stress.

Note that the HRR solution does not match the T - 0 case. The stresses deep insidethe plastic zone can be represented by a power series, where the HRR solution is theleading term. Figure 3.33 indicates that the higher order plastic terms are not negligiblewhen T = 0. A single-parameter description in terms of / is still valid, however, asdiscussed in Section 3.5.1.

In a cracked body subject to Mode I loading, the T stress, like Kj, scales with theapplied load. The biaxiality ratio relates T to stress intensity:

(3.66)

J2L70

Modified Boundary Layer Analysis= 10

- rt.

rd,

FIGURE 3.33 Stress fields obtained from modified boundary layer analysis [26].

'In this case, "body" refers to the global configuration, not the modified boundary layer model. A modifiedboundary layer model with T = 0 simulates an infinite body with an infinitely long crack.

162 Chapter 3

For a through-thickness crack in an infinite plate subject to a remote normal stress (Fig.2.3), (3 = -1. Thus a remote stress cr induces a T stress of -cr in the x direction. RecallExample 2.7, where a rough estimate of the elastic singularity zone and plastic zone sizeled to the conclusion that K breaks down in this configuration when the applied stressexceeds 35% of yield, which corresponds to T/ao = -0.35. From Fig. 3.33, we see thatsuch a T stress leads to a significant relaxation in crack tip stresses, relative to the small-scale yielding case.

For laboratory specimens with Kf solutions of the form in Table 2.4, the T stress isgiven by

(3.67)

Figure 3.34 is a plot of /J for several geometries. Note that ft is positive for deeplynotched SENT and SENB specimens, where the uncracked ligament is subjectpredominately to bending stresses. As discussed above, such configurations maintain ahigh level of constraint under fully plastic conditions. Thus a positive T stress in theelastic case generally leads to high constraint under fully plastic conditions, whilegeometries with negative T stress loose constraint rapidly with deformation.

a 0.5oP

NJ•<

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.5 -

FIGURE 3.34 Biaxiality ratio for single edge notched bend, single edge notched tension, double edgenotched tension, and center cracked tension geometries.


The biaxiality ratio can be used as a qualitative index of the relative crack tipconstraint of various geometries. The T stress, combined with the modified boundarylayer solution (Fig. 3.33) can also be used quantitatively to estimate the crack tip stressfield in a particular geometry [26-28]. For a given load level, the T stress can be inferredfrom Eq. (3.66) or (3.67), and the corresponding crack tip stress field for the same T stresscan be estimated from the modified boundary layer solution with the same applied T. Thismethodology has limitations, however, because T is an elastic parameter. A T stressestimated from load through Eq. (3.67) has no physical meaning under fully plasticconditions. Errors in stress fields inferred from T stress and the modified boundary layersolution increase with plastic deformation. This approximate procedure works fairly wellfor 1/31 > 0.9 but breaks down when IjSI < 0.4 [26].

3.6.2 J-Q Theory

Assuming small-strain theory, the crack tip fields deep inside the plastic zone can berepresented by a power series, where the HRR solution is the leading term. The higherorder terms can be grouped together into a difference field:

CT, =<7 , + <7, (3.68a)'J \ V> HRR \ 'J/Diff

Alternatively, the difference field can be defined as the deviation from the T= 0 referencesolution:

cr, =(cr..) +(cr..) (3.68b)y V v /r=o V v/Diff

Note from Figure 3.33 that non-zero T stresses cause the near-tip field at 9 = 0 to shift upor down uniformly; i.e., the magnitude of the shift is constant with distance from thecrack tip. O'Dowd and Shih [29,30] observed that the difference field is relatively constantwith both distance and angular position in the forward sector of the crack tip region (101 <7i/2). Moreover, they noted that

( c r ) ~(crjn. >>(cr ) for |0|<-V yy/Diff \ xx'Diff \ -9/Di/f ' ' 2

Thus the difference field corresponds approximately to a uniform hydrostatic shift of thestress field in front of the crack tip. O'Dowd and Shih designated the amplitude of thisapproximate difference field by the letter Q. Equation (3.68b) then becomes

164 Chapter 3

where <5// is tne Kronecker delta. The Q parameter can be inferred by subtracting the stressfield for the T= 0 reference state from the stress field of interest. O'Dowd and Shih andmost subsequent researchers defined Q as follows:

(3.70)J

Referring to Fig. 3.33, we see that Q is negative when T is negative. For the modifiedboundary layer solution, T and Q are uniquely related. Figure 3.35 is a plot of Q versus Tfor a wide range of hardening exponents.

In a given cracked body, Q = 0 in the limit of small scale yielding, but Q generallybecomes increasingly negative with deformation. Figure 3.36 shows the evolution of Qfor a deeply crack bend (SENB) specimen and a center cracked panel. Note that the SENDspecimen stays close to the Q = 0 limit to fairly high deformation levels, but Q for thecenter cracked panel becomes highly negative at relatively small / values.

Q-0.5 -

Modified Boundary Layer Analysis

-1.5

FIGURE 335 Relationship between Q and T as a function of strain hardening exponent. [29]


Q

-0.5

-1

-1.5

\\

\

Center Cracked Panel, a/W = 0.1Edge Cracked Bend Specimen, a/W = 0.5

0.0001 0.001 0.01 0.1

FIGURE 3.36 Evolution of the Q parameter with deformation in two geometries [29].

The J-Q Toughness LocusClassical single-parameter fracture mechanics assumes that fracture toughness is a

material constant. With J-Q theory, however, an additional degree of freedom as beenintroduced, which implies that the critical J value for a given material depends on Q:

(3.71)

Thus fracture toughness is no longer viewed as a single value; rather, it is a curve thatdefines a critical locus of J and Q values.

Figure 3.37 is a plot of critical J values (for cleavage fracture) as a function of Q[29]. Although there is some scatter, the trend in Fig. 3.37 is clear. The critical /increases as Q becomes more negative. This trend is consistent with Figs. 3.27 to 3.31.That is, fracture toughness tends to increase as constraint decreases. The Q parameter is adirect measure of the relative stress triaxiality (constraint) at the crack tip.

Since the T stress is also an indication of the level of crack tip constraint, a J-Tfailure locus can be constructed [27,28]. Such plots have similar trends to J-Q plots, butthe ordering of data points sometimes differs. That is, the relative ranking of geometriescan be influenced by whether constraint is quantified by T or Q. Under well-containedyielding, T and Q are uniquely related (Fig. 3.35), but the T stress loses its meaning for

166 Chapter 3

large-scale yielding. Thus a J-T toughness locus is unreliable when significant yieldingprecedes fracture.

Single-parameter fracture mechanics theory assumes that toughness values obtainedfrom laboratory specimens can be transferred to structural applications. Two-parameterapproaches such as J-Q theory imply that the laboratory specimen must match theconstraint of the structure; i.e., the two geometries must have the same Q at failure inorder for the respective Jc values to be equal. Figure 3.38 illustrates the application ofthe J-Q approach to structures. The applied / versus Q curve for the configuration ofinterest is obtained from finite element analysis and plotted with the J-Q toughness locus.Failure is predicted when the driving force curve passes through the toughness locus.Since toughness data are often scattered, however, there is not a single unambiguouscross-over point. Rather, there is a range of possible Jc values for the structure.

Effect of Failure Mechanism on the J-0 LocusThe J-Q approach is descriptive but not predictive. That is, Q quantifies the crack tip

constraint, but it gives no indication as the to effect of constraint on fracture toughness.A J-Q failure locus, such as Fig. 3.37, can be inferred from a series of experiments on arange of geometries. Alternatively, a micromechamcal failure criterion can be invoked.

300

250

200

150

100

50

A515 Grade 70 Steel

\\ ©\

\ oX\ A A

-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25

QFIGURE 3.37 J-Q toughness locus for SENS specimens of A515 Grade 70 steel [31].


Structure

IncreasingDeformation

QFIGURE 3.38 Application of a J-Q toughness locus. Failure occurs when the applied J-Q curve passesthrough the toughness locus.

Consider, for example, the Ritchie-Knott-Rice (RKR) [32] model for cleavagefracture, which states that fracture occurs when a critical fracture stress, Of, is exceededover a characteristic distance, rc. As an approximation, let us replace the T = 0 referencesolution with the HRR field in Eq. (3.69):

(3.72)

Setting the stress normal to the crack plane equal to Of and r = rc, and relating theresulting equation to the Q = 0 limit leads to

°vGn

i\n+l \n+l

(3.73)

where J0 is the critical J value for the Q = 0 small-scale yielding limit. Rearranging gives

n+l

(3.74)

which is a prediction of the J-Q toughness locus. Equation (3.74) predicts that toughnessis highly sensitive to Q, since the quantity is brackets is raised to the n+l power.

The shape of the J-Q locus depends on the failure mechanism. Equation (3.74) refersto stress-controlled fracture, such as cleavage in metals, but strain-controlled fracture is

168 Chapter 3

less sensitive to crack dp constraint. A simple parametric study illustrates the influenceof the local failure criterion.

Suppose that fracture occurs when a damage parameter, , reaches a critical value rc

ahead of the crack tip, where is given by

(0 < y < 1) (3.75)

where am is the mean (hydrostatic) stress and Bpi is the equivalent plastic strain. When y= 1, Eq. (3.75) corresponds to stress controlled fracture, similar to the RKR model. Theother limit, y= 0, corresponds to strain-controlled failure. By varying y and applying Eq.(3.75) to the finite element results of O'Dowd and Shih [29,30], we obtain a family of 7-Q toughness locii, which are plotted in Fig. 3.39. The J-Q locus for stress-controlledfracture is highly sensitive to constraint, as expected. For strain-controlled fracture, thelocus has a slight negative slope, indicating that toughness decreases as constraint relaxes.As Q decreases (i.e., becomes more negative), crack tip stresses relax, but the plasticstrain fields at a given J value increase with constraint loss. Thus as constraint relaxes, asmaller Jc is required for failure for a purely strain-controlled mechanism. The predictedJc is nearly constant for y = 0.5. Microvoid growth in metals is governed by acombination of plastic strain and hydrostatic stress (see Chapter 6). Consequently, criticalJ values for initiation of ductile crack growth are relatively insensitive to geometry, asFig. 3.30 indicates.

2.5

JcritJo

1.5

0.5

-1.5 -1 -0.5

Q0.5

FIGURE 3.39 Effect of failure criterion on the J-Q locus [33]. Fracture is assumed to occur whenreaches a critical value at a specific distance from the crack tip.


According to Fig. 3.31, the slope of the J resistance curve is influenced by specimenconfiguration. However, the stress and strain fields ahead of a growing crack are differentfrom the stationary crack case [16,17], and J-Q theory is not applicable to a growingcrack.

3.6.3 Scaling Model for Cleavage Fracture

Both the J-Q and T stress methodologies are based on continuum theory. As stated above,these approaches characterize the crack tip fields but they cannot predict the effect of thesefields on a material's fracture resistance. A micromechanical failure criterion must beintroduced to relate crack tip fields to fracture toughness. The RKR model provides asimple means for such predictions. Anderson and Dodds [34-36] have developed asomewhat more sophisticated model for cleavage, which is described below.

Failure CriterionCleavage initiation usually involves a local Griffith instability of a microcrack

which forms from a microstructural feature such as a carbide or inclusion. The Griffithenergy balance is satisfied when a critical stress is reached in the vicinity of themicrocrack. The size and location of the critical rnicrostructural feature dictate the fracturetoughness; thus cleavage toughness is subject to considerable scatter. See Chapter 5 for amore detailed description of the micromechanisms of cleavage fracture.

The Griffith instability criterion implies fracture at a critical normal stress near thetip of the crack; the statistical sampling nature of cleavage initiation (i.e., the probabilityof finding a critical microstructural feature near the crack tip) suggests that the volume ofthe process zone is also important. Thus the probability of cleavage fracture in a crackedspecimen can be expressed in the following general form:

F = F(V(ai)] (3.76)

where F is the failure probability, 07 is the maximum principle stress at a point, andV(<7j) is the cumulative volume sampled where the principal stress is > 07. For aspecimen subjected to plane strain conditions, V = BA, where B is the specimen thicknessand A is cumulative area on the x-y plane.

The J0 ParameterFor small scale yielding, dimensional analysis shows that the principal stress ahead

of the crack tip can be written as

(3.77)

Equation (3.77) implies that the crack tip stress fields depend only on J. When J domi-nance is lost, there is a relaxation in triaxiality; the principal stress at a fixed r and 0 isless than the small scale yielding value.

170 Chapters

Equation (3.77) can be inverted to solve for the radius corresponding to a givenstress and angle:

r(0i / (7<,, 0) = — £(0i / (70, 6») (3.78)

Solving for the area inside a specific principal stress contour gives

A(0i / 00) = h(0i I 0-0) (3.79)

where1 ^

01 °o -2J

Thus for a given stress, the area scales with ft in the case of small scale yielding. Underlarge scale yielding conditions, the test specimen or structure experiences a loss in con-straint, and the area inside a given principal stress contour (at a given J value) is less thanpredicted from small scale yielding:

00) = 0 - h(<?i I GO) (3.81)

where 0 is a constraint factor that is < 1. Let us define an effective J in large scaleyielding that relates the area inside the principal stress contour to the small scale yieldingcase:

A(OJ / a0) = -k(ff1 / cr0) (3.82)

where Jo is the effective small scale yielding /; i.e., the value of J that would result in thearea A(aj/<jo) if the structure were large relative to the plastic zone, and T = Q = 0.Therefore, the ratio of the applied J to the effective J is given by

(3-83)

The small scale yielding J value (/#) can be viewed as the effective driving force forcleavage, while / is the apparent driving force.

, . 171Elastic-Plastic Fracture Mechanics

The J/J0 ratio quantifies the size dependence of cleavage fracture toughness.Consider, for example, a finite size test specimen that fails at Jc = 200 kPa m If the.VJO

ratio were 2.0 in this case, a very large specimen made from the f^^^ât Jc = 100 kPa m. An equivalent toughness ratio m terms of crack tip opening dis-

placement (CTOD) can also be defined.

Three-Dimensional Effects , nft^The constraint model described above considers only stressed areas in front of the

crack tip. This model is incomplete, because it is the volume of matenal sampled aheadof the crack tip that controls cleavage fracture. The stressed volume obviously sea eswith specimen thickness (or crack front length in the more generaj case). Moreover, thestressed volume is a function of the constraint parallel to the crack front; tagher constraintresults in a larger volume, as is the case for in-plane constraint.

One way to treat tee-dimensional constraint effects ,s to define an effectivethickness based on an equivalent two-dimensional case. Consider a three-dimensionalspecimen that is loaded to a given J value. If we choose a principal stress value andconstruct contours at two-dimensional slices on the x-y plane, the area mside of thesecontours will vary along the crack front because the center of the specimen is more highlyconstrained than the free surface, as Fig. 3.40 illustrates The volume can be obtained bysumming the areas in these two-dimensional contours. This volume can then be relatedto an equivalent 2-D specimen loaded to the same J value:

V = 20

where Ac is the area inside the a, contour on the center plane of the 3-D specimen and

Beff is the effective thickness.The effective thickness influences the cleavage driving force through a sample vol-

ume effect: longer crack fronts have a higher probability of cleavage fracture because morevolume is sampled along the crack front. This effect can be characterized by a three-

parameter Weibull distribution (See Ch. 5):

F - 1 - exp (3.85)

Where B is the thickness (or crack front length), Bo is a reference thickness, Kmm is thethreshold toughness, and 9K is the 63rd percentile toughness when B = BO.

Consider two samples with effective crack front lengths-.Bj and B2. If a val»e ofJP/Cd) is measured for Specimen 1, the expected toughness for Specimen 2 can be in-ferred from Eq. (10) by equating failure probabilities:

172 ChapterB

>yy

-Beff/2-

FIGURE 3.40 Schematic illustration of•oi 2 the effective thickness, Bejjm

KJC(2) (3.86)

Equation (3.86) is a statistical thickness adjustment that can be used to relate two sets ofdata with different thicknesses.

Application of the ModelAs with the J-Q approach, implementation of the scaling model requires detailed

elastic-plastic finite element analysis of the configuration of interest. Principal stresscontours must be constructed and the areas compared with the T = 0 reference solutionobtained from a modified boundary layer analysis. The effective driving force, J0, is thenplotted against the applied /, as Figure 3.41 schematically illustrates. At low deformationlevels, the J0-J curves follow the 1:1 line, but deviate from the line with furtherdeformation. When J ~ J0, the crack tip stress fields are close to the Q = 0 limit, andfracture toughness is not significantly influenced by specimen boundaries. At highdeformation levels, J > J0, and fracture toughness is artificially elevated by constraintloss. Constraint loss occurs more rapidly in specimens with shallow cracks, as Fig. 3.28illustrates. A specimen with a/W = 0.15 would tend to fail at a higher Jc value than aspecimen with a/W = 0.5. Given the J0-J curve, however, the Jc values for bothspecimens can be corrected to J0, as Fig. 3.41 illustrates.

Figure 3.42 is a nondimensional plot of Jo at the midplane versus the average Jthrough the thickness of SENB specimens with various W/B ratios [36]. These curveswere inferred from a 3-D elastic-plastic analysis. The corresponding curve from a 2-Dplane strain analysis is shown for comparison. Note that for W/B = 1 and 2, Jo at themidplane lies well above the plane strain curve. For W/B = 4, J0 at the midplane followsthe plane strain curve initially, but falls below the plane strain results at high deformationlevels. The three-dimensional nature of the plastic deformation apparently results in a


high level of constraint at the midplane when the uncracked ligament length is < thespecimen thickness.

Figure 3.43 is a plot of effective thickness, Beff, as a function of deformation. Thetrends in this plot are consistent with Fig. 3.42; namely, the constraint increases withdecreasing W/B. Note that all three curves reach a plateau. Recall that Beff is defined insuch a way as to be a measure of the through-thickness relaxation of constraint, relativeto the in-plane constraint at the midplane. At low deformation levels there is negligiblerelation at the midplane and J ~ Jo, but through-thickness constraint relation occurs,resulting in a falling Beff/B ratio. At high deformation levels, the Beff/B ratio isessentially constant, indicating that the constraint relaxation is proportional in threedimensions. Figures 3.44 and 3.45 show data that has been corrected with the scalingmodel.

Critical J0 Jf FIGURE 3.41 Schematic illustration ofthe scaling model. A specimen with a/W= 0.15 will fail at a higher Jc value thana specimen with a/W = 0.5, but both Jc

values can be corrected down to thesame critical /„ value.

3.6.4 Limitations of Two-Parameter Fracture Mechanics

The T stress approach, J-Q theory, and the cleavage scaling model are examples of two-parameter fracture theories, where a second quantity (e.g., T, Q or J0) has been introducedto characterize the crack tip environment. Thus these approaches assume that the cracktip fields contain two degrees of freedom. When single-parameter fracture mechanics isvalid, the crack tip fields have only one degree of freedom. In such cases, any one ofseveral parameters (e.g., J, K, or CTOD) will suffice to characterize the crack tipconditions, provided the parameter can be defined unambiguously; K is a suitablecharacterizing only when an elastic singularity zone exists ahead of the crack tip**.Similarly, the choice of a second parameter in the case of two-parameter theory is mostlyarbitrary, but the T stress has no physical meaning under large-scale yielding conditions.

Just as plastic flow invalidates single-parameter fracture mechanics in manygeometries, two-parameter theories eventually break down with extensive deformation. Ifwe look at the structure of the crack tip fields in the plastic zone, we can evaluate therange of validity of both single- and two-parameter methodologies.

0

"An effective K can be inferred from / through Eq.(3.18). Such a parameter has units of K, but it looses itsmeaning as the amplitude of the elastic singularity when such a singularity no longer exists.

174 Chapter 3

0.03

0.025 -

SENB SPECIMENSn = 10

a/W = 0.5

0.05

FIGURE 3.42 Effective driving force for cleavage, J0, for deeply notched SENB specimens.

PQ 0.6 -

SENB SPECIMENSn = 10

a/W = 0.5

0.01 0.02 0.03 0.04 0.05

FIGURE 3.43 Effective thickness for deeply notched SENB specimens.


0.4

0.3

0O °-Hu

0.1 -

1 1

-

-A 36 STEEL

B=W = 31.8mm

^ Experimental Data

-|- Corrected for Constraint

1 '

-

•o•

4- 76°C - 43 °C $

1 ^ ii i i

aAV = 0.5a/W = 0.15

a/W = 0.5= 0.15

FIGURE 3.44 Fracture toughness data for a mild steel, corrected for constraint loss [34,37]-

500

400

§3

200

100

0

A 515 Grade 70 Steel(Kirk, et al., 1991)

Specimen Thicknessand Width:

—O—50 mm

- 0- -25 mm

-|- 10 mm

- O

IO

4*o

0.1 0.2 0.5 0.60.3 0.4

a / W

FIGURE 3.45 Experimental data from Fig. 3.28 corrected for constraint loss [38] (Jssy in [38] =J0)

176 Chapters

A number of investigators [39-43] have performed asymptotic analyses of the cracktip fields for elastic-plastic materials. These analyses utilize deformation plasticity andsmall-strain theory. In the case of plane strain, these analyses assume incompressiblestrain. Consequently, asymptotic analyses are not valid close to the crack tip (in the largestrain zone) nor remote from the crack tip, where elastic strains are a significant fractionof the total strain. Despite these limitations, asymptotic analysis provides insights intothe range of validity of both single- and two-parameter fracture theories.

In the case of a plane strain crack in a power-law hardening material, asymptoticanalysis leads to the following power series:

(3.87)

The exponents, %, and the angular functions for each term in the series can be determinedfrom the asymptotic analysis. The amplitudes for the first five terms are as follows:

"A"A

3

A4

.Aj

1

ft)""*1

(unspecified)

A2

A(unspecified)

A3

A2 J

The two unspecified coefficients, A 2 and A^, are governed by the far-field boundaryconditions. The first five terms of the series have three degrees of freedom, where /, A2and A4 are independent parameters. For low and moderate strain hardening materials,Crane [43] showed that a fourth independent parameter does not appear in the series formany terms. For example, when n = 10, the fourth independent coefficient appears inapproximately the 100th term. Thus for all practical purposes, the crack tip stress fieldinside the plastic zone has three degrees of freedom.

Since two-parameter theories assume two degrees of freedom, they cannot berigorously correct in general. There are, however, situations where two-parameterapproaches provide a good engineering approximation.

Consider the modified boundary layer model in Fig. 3.32. Since the boundaryconditions have only two degrees of freedom (K and T), the resulting stresses and strainsinside the plastic zone must be two-parameter fields. Thus there must be a uniquerelationship between A2 and A^ in this case. That is,

(3.88)


Two-parameter theory is approximately valid for other geometries to the extent thatthe crack tip fields obey Eq. (3.88). Figure 3.46 schematically illustrates the ^2-^-4relationship. This relationship can be established by varying the boundary conditions onthe modified boundary layer model. When a given cracked geometry is loaded, ^2 and A4initially will evolve in accordance with Eq. (3.88) because the crack tip conditions in thegeometry of interest can be represented by the modified boundary layer model when theplastic zone is relatively small. Under large-scale yielding conditions, however, the A2~A4 relationship may deviate from the modified boundary layer solution, in which casetwo-parameter theory is no longer valid.

T=0

Modified BoundaryLayer Model

t

FIGURE 3.46 Schematic relationshipbetween the two independent ampli-tudes in the asymptotic power series.

Figure 3.47 is a schematic three-dimensional plot of J, A2 and A^. Single-parameterfracture mechanics can be represented by a vertical line, since A2 and A^ must be constantin this case. Two-parameter theory, where Eq. (3.88) applies, can be represented by asurface in this three-dimensional space. The loading path for a cracked body initiallyfollows the vertical single-parameter line. When it deviates from this line, it may remainin the two-parameter surface for a time before diverging from the surface.

The loading path in J-A.2-&4 space depends on geometry [43]. Low constraintconfigurations like the center cracked panel and shallow notched bend specimens divergefrom single-parameter theory almost immediately, but follow Eq. (3.88) to fairly highdeformation levels. Deeply notched bend specimens maintain high constraint to relativelyhigh J values, but they do not follow Eq. (3.88) when constraint loss eventually occurs.Thus low-constraint geometries should be treated with two-parameter theory, and highconstraint geometries can be treated with single-parameter theory in many cases. Whenhigh constraint geometries violate the single-parameter assumption, however, two-parameter theory is of little value.

178 Chapter 3

Two-Parameter Theory

FIGURE 3.46 Single- and two-parameter assumptions in terms of thethree independent variables in theelastic-plastic crack tip field. Theloading path initially lies in the two-parameter surface and then diverges,as indicated by the dashed line,

REFERENCES

1. Wells, A.A., "Unstable Crack Propagation in Metals: Cleavage and Fast Fracture."Proceedings of the Crack Propagation Symposium, Vol. 1, Paper 84, Cranfield, UK,1961.


3. Burdekin, P.M. and Stone, D.E.W., "The Crack Opening Displacement Approach toFracture Mechanics in Yielding Materials." Journal of Strain Analysis, Vol. 1, 1966,pp. 145-153.

4. Rice, J.R. " A Path Independent Integral and the Approximate Analysis of StrainConcentration by Notches and Cracks." Journal of Applied Mechanics, Vol. 35, 1968,pp. 379-386.

5. BS 5762: 1979, "Methods for Crack Opening Displacement (COD) Testing." BritishStandards Institution, London, 1979.

6. E 1290-89 "Standard Test Method for Crack Tip Opening Displacement Testing."American Society for Testing and Materials, Philadelphia, 1989.

7. Hutchinson, J.W., "Singular Behavior at the End of a Tensile Crack Tip in a HardeningMaterial." Journal of the Mechanics and Physics of Solids, Vol. 16, 1968, pp. 13-31.

8. Rice, J.R. and Rosengren, G.F., "Plane Strain Deformation near a Crack Tip in a Power-Law Hardening Material. Journal of the Mechanics and Physics of Solids, Vol. 16,1968, pp. 1-12.


9. McMeeking, R.M. and Parks, D.M., "On Criteria for J-Dominance of Crack Tip Fieldsin Large-Scale Yielding." ASTM STP 668, American Society for Testing and Materials,Philadelphia, 1979, pp. 175-194.

10. Read, D.T., "Applied J-Integral in HY130 Tensile Panels and Implications for Fitnessfor Service Assessment." Report NBSIR 82- 1670, National Bureau of Standards,Boulder, CO, 1982.

11. Begley, J. A. and Landes, J.D., "The J-Integral as a Fracture Criterion." ASTM STP 514,American Society for Testing and Materials, Philadelphia, 1972, pp. 1-20.

12. Landes, J.D. and Begley, J.A., "The Effect of Specimen Geometry on Jic." ASTM STP514, American Society for Testing and Materials, Philadelphia, 1972, pp. 24-29.

13. Rice, J.R., Paris, P.C., and Merkle, J.G., "Some Further Results of J-Integral Analysisand Estimates." ASTM STP 536, American Society of Testing and Materials,Philadelphia, 1973, pp. 231-245.

14. Shih, C.F. "Relationship between the J-Integral and the Crack Opening Displacementfor Stationary and Extending Cracks." Journal of the Mechanics and Physics of Solids,Vol. 29, 1981, pp. 305-326.

15. Rice, J.R., Drugan, W.J., and Sham, T.-L., "Elastic-Plastic Analysis of GrowingCracks." ASTM STP 700, American Society of Testing and Materials, Philadelphia,1980, pp. 189-221.

16. Dodds, R.H. Jr. and Tang, M., "Numerical Techniques to Model Ductile Crack Growth inFracture Test Specimens." Engineering Fracture Mechanics, Vol. 46, 1993, pp. 253-246.

17. Dodds, R.H. Jr., Tang, M., and Anderson, T.L. "Effects of Prior Ductile Tearing onCleavage Fracture Toughness in the Transition Region." Second Symposium onConstraint Effects in Fracture, American Society of Testing and Materials, Philadelphia,(in press).

18. McClintock, F.A., "Plasticity Aspects of Fracture." Fracture: An Advanced Treatise,Vol. 3, Academic Press, New York, 1971, pp. 47-225.

19. Anderson, T.L., "Ductile and Brittle Fracture Analysis of Surface Flaws Using CTOD."Experimental Mechanics, June 1988, pp. 188-193.

20. Kirk, M.T., Koppenhoefer, K.C., and Shih, C.F., "Effect of Constraint on SpecimenDimensions Needed to Obtain Structurally Relevant Toughness Measures." ConstraintEffects in Fracture, ASTM STP 1171, American Society for Testing and Materials,Philadelphia, 1993, pp. 79-103.

21 Joyce, J.A. and Link, R.E., Effect of Constraint on Upper Shelf Fracture Toughness."Fracture Mechanics: 26th Volume, ASTM STP 1256, American Society for Testing andMaterials, Philadelphia, (in press).

22. Towers, O.L. and Garwood, S.J., "Influence of Crack Depth on Resistance Curves forThree-Point Bend Specimens in HY130." ASTM STP 905 American Society for Testingand Materials, Philadelphia, 1986, pp. 454-484.

180 Chapters

23. Williams, M.L., "On the Stress Distribution at the Base of a Stationary Crack," Journalof Applied Mechanics, Vol. 24, 1957, pp. 109-114.

24. Bilby, B.A., Cardew, G.E., Goldthorpe, M.R., and Howard, I.C., "A Finite ElementInvestigation of the Effects of Specimen Geometry on the Fields of Stress and Strain atthe Tips of Stationary Cracks." Size Effects in Fracture, Institute of MechanicalEngineers, London, 1986, pp. 37-46.

25. Betegon, C. and Hancock, J.W., "Two Parameter Characterization of Elastic-PlasticCrack Tip Fields." Journal of Applied Mechanics, Vol. 58, 1991, pp. 104-110.

26. Kirk, M.T., Dodds, R.H., Jr., and Anderson, T.L., "Approximate Techniques forPredicting Size Effects on Cleavage Fracture Toughness." Fracture Mechanics: 24thVolume, ASTM STP 1207, American Society for Testing and Materials, Philadelphia,(in press).

27. Hancock, J.W., Reuter, W.G., and Parks, D.M., "Constraint and ToughnessParameterized by T." Constraint Effects in Fracture, ASTM STP 1171, American Societyfor Testing and Materials, Philadelphia, 1993, pp. 21-40.

28. Sumpter, J.D.G., "An Experimental Investigation of the T Stress Approach."Constraint Effects in Fracture, ASTM STP 1171, American Society for Testing andMaterials, Philadelphia, 1993, pp. 492-502.

29. O'Dowd, N.P. and Shih, C.F., "Family of Crack-Tip Fields Characterized by aTriaxiality Parameter-I. Structure of Fields." Journal of the Mechanics and Physics ofSolids, Vol. 39, 1991, pp. 898-1015.

30. "O'Dowd, N.P. and Shih, C.F., "Family of Crack-Tip Fields Characterized by aTriaxiality Parameter-II. Fracture Applications." Journal of the Mechanics and Physicsof Solids, Vol. 40, 1992, pp. 939-963.

31. Shih, C.F., O'Dowd , N.P., and Kirk, M.T., "A Framework for Quantifying Crack TipConstraint." Constraint Effects in Fracture, ASTM STP 1171, American Society forTesting and Materials, Philadelphia, 1993, pp. 2-20.

32. Ritchie, R.O., Knott, J.F., and Rice, J.R. "On the Relationship between Critical TensileStress and Fracture Toughness in Mild Steel." Journal of the Mechanics and Physics ofSolids, Vol. 21, 1973, pp. 395-410.

33. Anderson, T.L., Vanaparthy, N.M.R., and Dodds, R.H. Jr., "Predictions of SpecimenSize Dependence on Fracture Toughness for Cleavage and Ductile Tearing." ConstraintEffects in Fracture, ASTM STP 1171, American Society for Testing and Materials,Philadelphia, 1993, pp. 473-491.

34. Anderson, T.L. and Dodds, R.H., Jr., "Specimen Size Requirements for FractureToughness Testing in the Ductile-Brittle Transition Region." Journal of Testing andEvaluation, Vol. 19, 1991, pp. 123-134.

35. Dodds, R.H. Jr., Anderson T.L. and Kirk, M.T., "A Framework to Correlate aAV Effectson Elastic-Plastic Fracture Toughness (Jc)." International Journal of Fracture, Vol. 48,1991, pp. 1-22.


36. Anderson, T.L. and Dodds, R.H., Jr., "An Experimental and Numerical Investigation ofSpecimen Size Requirements for Cleavage Fracture Toughness." NUREG/CR-6272,Nuclear Regulatory Commission, Washington DC, (in press).

37. Sorem, W.A., "The Effect of Specimen Size and Crack Depth on the Elastic-PlasticFracture Toughness of a Low-Strength High-Strain Hardening Steel." Ph.D.Dissertation, The University of Kansas, Lawrence, KS, May 1989.

38. Anderson, T.L., Stienstra, D.I.A., and Dodds, R.H. Jr., "A Theoretical Framework forAddressing Fracture in the Ductile-Brittle Transition Region." Fracture Mechanics: 24thVolume, ASTM STP 1207, American Society for Testing and Materials, Philadelphia,(in press).

39. Li, W.C. and Wang, T.C., "Higher-Order Asymptotic Field of Tensile Plane StrainNonlinear Crack Problems." Scientia, Sinica (Series A), Vol. 29, 1986, pp. 941-955.

40. Sharma, S.M. and Aravas, N., "Determination of Higher-Order Terms in AsymptoticElastoplastic Crack Tip Solutions." Journal of the Mechanics and Physics of Solids,Vol. 39, 1991, pp. 1043-1072.

41. Yang, S., Chao, Y.J, and Sutton, M.A., "Higher Order Asymptotic Crack Tip Fields in aPower Law Hardening Material." Engineering Fracture Mechanics, Vol. 45, 1993, pp. 1-20.

42. Xia, L., Wang, T.C., and Shih, C.F., "Higher Order Analysis of Crack-Tip Fields inElastic-Plastic Power-Law Hardening Materials." Journal of the Mechanics and Physicsof Solids, Vol. 41, 1993, pp. 665-687.

43. Crane, D.L., "Deformation Limits on Two-Parameter Fracture Mechanics in Terms ofHigher Order Asymptotics." Ph.D. Dissertation, Texas A&M University, CollegeStation, TX, December 1994.


45. Bilby, B.A., Cottrell, A.H., and Swindon, K. H., "The Spread of Plastic Yield from aNotch." Proceedings, Royal Society of London, Vol. A-272, 1963, pp. 304-314,

46. Smith, E., "The Spread of Plasticity from a Crack: an Approach Based on the Solution ofa Pair of Dual Integral Equations." CEGB Research Laboratories, Lab. Note No.RD/L/M31/62, July 1962.

47. Rice, J.R. and Tracey, D.M., Journal of the Mechanics and Physics of Solids, Vol. 17,1969, pp. 201-217.

48. Budiansky, B. "A Reassessment of Deformation Theories of Plasticity." Journal ofApplied Mechanics. Vol. 81, 1959, pp. 259-264.

APPENDIX 3: MATHEMATICALFOUNDATIONS OF ELASTIC-PLASTIC

FRACTURE MECHANICS(Selected Results)

A3.1 DETERMINING CTOD FROM THE STRIP YIELD MODEL

Burdekin and Stone [3] applied the Westergaard [44] complex stress function approach tothe strip yield model. They derived an expression for CTOD by superimposing a stressfunction for closure forces on the crack faces in the strip yield zone. Their result wassimilar to previous analyses based on the strip yield model performed by Bilby, et al. [45]and Smith [46].

Recall from Appendix 2.3 that the Westergaard approach expresses the in-planestresses (in a limited number of cases) in terms of Z:

cr^ = ReZ->>ImZ' (A3. la)

Gyy = ReZ + ylmZ' (A3.1b)

(A3.1c)

where Z is an analytic function of the complex variable z = x + iy, and the prime denotesa first derivative with respect to z. By invoking the equations of elasticity for the planeproblem, it can be shown that the displacement in the y direction is as follows:

and

_uv = — [2 Im Z - y(l + V) Re Z] for plane stress (A3.2a)y E

—uv = — [2(1 - V ) Im Z - y(l + V) Re Z] for plane strain (A3.2b)

* E

where Z is the integral of Z with respect to z, as discussed in Appendix 2. For a throughcrack of length 2aj in an infinite plate under biaxial tensile stress d, the Westergaardfunction is given by

Z= . °* (A3.3)

183

184 Appendix 3

where the origin is defined at the crack center.The stress function for a pair of splitting forces, P, at ± x within a crack of length 2

a i (see Fig. 2.29) is given by

Z = (A3.4)

For a uniform compressive stress = crys along the crack surface between a and a] (Fig.A3.1), the Westergaard stress function is obtained by substituting P = - ays dx into Eq.(A3.4) and integrating:

2 9 2- flf (z - &i

•cos — -cot\a\)

-1 .2 2(A3.5)

The stress functions of Eqs. (A3.3) and (A3.5) can be superimposed, resulting in the stripyield solution for the through crack. Recall from Section 2.8.2 that the size of the stripyield zone was chosen so that the stresses at the tip would be finite. Thus

. a [ 7Wk = — = cos (A3.6)

When Eq. (A3.6) is substituted into Eq. (A3.5) and Eq. (A3.3) is superimposed, the firstterm in Eq. (A3.5) cancel with Eq. (A3.3), which leads to

FIGURE A3.1 Strip yield model for a through crack.

Mathematical Foundations ofEPFM 185

y

n z V l-k'(A3.7)

Integrating Eq. (A3.7) gives

where

and

—Z =

71(A3.8)

= COt-ll1-

=cot

On the crack plane, y = 0 and the displacement in the y direction (Eq. (A3.2))reduces to

2 _wv = — Im Zy E

(A3.9)

for plane stress. Solving for the imaginary part of Eq. (A3.8) gives

nE-1

2 _21 a - z -zcoth-l.2 _2

for \z\ ^ ay. Setting z = a leads to

1

v — iny nE

(A3.10)

which is identical to Eq. (3.5).Recall the J-CTOD relationship (Eq. (3.44)) derived from the strip yield model. Let

us define an effective stress intensity for elastic-plastic conditions in terms of the /integral:

186 Appendix 3

(A3.11)

Combining Eqs. (3.44) (A3.10) and (A3.11) gives

K e f f ~ • In secVZOKS

(A3.12)

which is the strip yield plastic zone correction given in Eq. (2.76) and plotted in Fig.2.31. Thus the strip yield correction to Kj is equivalent to a /-based approach for anonhardening material in plane stress.

A3.2 THEJ CONTOUR INTEGRAL

Rice [4] presented a mathematical proof of the path independence of the J contour integral.He began by evaluating / along a closed contour, F* (Fig A3.2):

/*=r*

(A3.13)

where the various terms in this expression are defined in Section 3.2.2. Rice theninvoked the divergence theorem to convert Eq. (A3.13) into an area integral:

FIGURE A3.2 Closed contour, P*,in a two-dimensional solid.


J*=A* djc,-

dxdy (A3.14)

where A * is the area enclosed by F*. By invoking the definition of strain energy densitygiven by Eq. (3.20), we can evaluate the first term in square brackets in Eq. (A3.14):

;;,- 3*(A3.15)

Note that Eq. (A3.15) applies only when w exhibits the properties of an elastic potential.Applying the strain-displacement relationship (for small strains) to Eq. (A3.15) gives

1. reVlj

2 IJaa*

( -cla

ty]xj )

1

a^

(A3.16)

/y = Cj[. Invoking the equilibrium condition:

(A3.17)

leads to

, (A3.18)

which is identical to the second term in square bracket in Eq. (A3.14). Thus the integrandin Eq. (A3.14) vanishes and J= 0 for any closed contour.

Consider now two arbitrary contours, F] and /2 around a crack tip, as illustrated inFig. A3.3. If F] and F2 are connected by segments along the crack face (Fj and F/), aclosed contour is formed. The total J along the closed contour is equal to the sum ofcontributions from each segment:

J — (A3.19)

188 Appendix 3

FIGURE A3.3 Two arbitrarycontours, Tj and r?, around the tip ofa crack. When these contours areconnected by F3 and F4, a closedcontour is formed, and the total J = 0.

On the crack face, Tt = dy = 0. Thus, J3 = J4 = Q and Jj = - J2. Therefore, any arbitrary(counter-clockwise) path around a crack will yield the same value of /; / is path-independent '

A3.3 J AS A NONLINEAR ELASTIC ENERGY RELEASE RATE

Consider a two-dimensional cracked body bounded by the curve r (Fig A3 4) Let A'denote the area of the body. The coordinate axis is attached to the crack tip ' Underquasistatic conditions and in the absence of body forces, the potential energy is given by

11= J wdA- I T^ds (A3-20)/( t T""»tlf\ I

where F" is the portion of the contour on which tractions are defined Let us nowconsider the change in potential energy resulting from a virtual extension of the crack:

^S (A3'21>

The line integration in Eq. (A3.21) can be performed over the entire contour, P, becauseduj/da - 0 over the region where displacements are specified; also, dTt/da = 0 over theregion the tractions are specified. When the crack grows, the coordinate axis moves Thusa derivative with respect to crack length can be written as


=da da da dx

(A3.22)

since obc/Az = -1. Applying this result to Eq. (A3.21) gives

f \ V w . r ^ u ; w/- = I - -- — \dA - \ TA — - - — L Idsda ., \oa ox j p. V oa ax

(A3.23)

By applying the same assumptions as in Eqs. (A3. 15) and (A3. 16), we obtain:

da de da V d(A3.24)

Invoking the principle of virtual work gives

(A3.25)

FIGURE A3.4 A two-dimensional cracked body bounded by the curve F'.

190 Appendix 3

which cancels with one of the terms in the line integral in Eq. (A3. 23), resulting in thefollowing:

(A3.26)da

Applying the divergence theorem and multiplying both sides by -1 leads to

dH f

~~7~= Jda

(A3.26)ox

since nx ds = dy. Therefore, the J contour integral is equal to the energy release rate for alinear or nonlinear elastic material under quasistatic conditions.

A3.4 THE HRR SINGULARITY

Hutchinson [7] and Rice and Rosengren [8] independently evaluated the character of cracktip stress fields in the case of power-law hardening materials. Hutchinson evaluated bothplane stress and plane strain, while Rice and Rosengren considered only plane strainconditions. Both articles, which were published in the same issue of the Journal of theMechanics and Physics of Solids, argued that stress times strain varies as 1/r near thecrack tip, although only Hutchinson was able to provide a mathematical proof of thisrelationship.

The Hutchinson analysis is outlined below. Some of the details are omitted forbrevity. We focus instead on his overall approach and the ramifications of this analysis.

Hutchinson began by defining a stress function, <P, for the problem. The governingdifferential equation for deformation plasticity theory for a plane problem in a Ramberg-Osgood material is more complicated than the linear elastic case:

/* A Q T7\

A4 + 7(, ae , r, n, a) = 0

where the function 7 differs for plane stress and plane strain. For the Mode I crackproblem, Hutchinson chose to represent 4> in terms of an asymptotic expansion in thefollowing form:


C2(0)/-f... (A3.28)

where C; and €2 are constants that depend on 0, the angle from the crack plane.Equation (A3.28) is analogous to the Williams expansion for the linear elastic case(Appendix 2.3). If s < t, and t is less than all subsequent exponents on r, then the firstterm dominates as r —> 0. If the analysis is restricted to the region near the crack tip, thenthe stress function can be expressed as follows:

(A3.29)

where K is the amplitude of the stress function and 0 is a dimensionless function of 6.Although Eq. (A3. 27) is different from the linear elastic case, the stresses can still bederived from «2> through Eqs. (A2.6) or (A2. 13). Thus the stresses, in polar coordinates,are given by

orr = K00rs~2arr(Q} - Ka0r

s

GQQ = KG0r

(A3.30)

KG0rs~2(02

r

The boundary conditions for the crack problem are as follows:

±7T = &±ri = 0

In the region close to the crack tip where Eq. (A3. 29) applies, elastic strains arenegligible compared to plastic strains; only the second term in Eq. (A3.27) is relevant inthis case. Hutchinson substituted the boundary conditions and Eq. (A3. 29) into Eq.(A3.27) and obtained a nonlinear eigenvalue equation for s. He then solved this equationnumerically for a range of n values. The numerical analysis indicated that s could bedescribed quite accurately (for both plane stress and plane strain) by a simple formula:

. . _ _(A3.31)

n + l

192 Appendix 3

which implies that the strain energy density varies as J/r near the crack tip. This

numerical analysis also yielded relative values for the angular functions oyy. Theamplitude, however, cannot be obtained without connecting the near-tip analysis with theremote boundary conditions. The J contour integral provides a simple means for makingthis connection in the case of small scale yielding. Moreover, by invoking the path-independent property of J, Hutchinson was able to obtain a direct proof of the validity ofEq. (A3.31).

Consider two circular contours of radius rj and r2 around the tip of a crack in smallscale yielding, as illustrated in Fig. A3.5. Assume that rj is in the region described bythe elastic singularity, while 77 is we^ inside of the plastic zone, where the stresses aredescribed by Eq. (A3.30). When the stresses and displacements in Tables 2.1 and 2.2 areinserted into Eq. (A3.26), and the /integral is evaluated along rj, one finds that / =Kp/E', as expected from the previous section. Since the connection between Kj andglobal boundary conditions is well established for a wide range of configurations, and /ispath-independent, the near-tip problem for small scale yielding can be solved byevaluating J at r2 and relating /to the amplitude (K).

Solving for the integrand in the /integral at r2 leads to

W =n + 1

and

sm 0[arr(ue - u'r} - dre(ur + u'e}]

+ cos 6[n (s - 2) + l][drrur + OQQUQ ]} (A3.32b)

FIGURE A3.5 Two circular contoursaround the crack tip. rj is in the zonedominated by the elastic singularity,while T2 is in the plastic zone wherethe leading term of the Hutchinsonasymptotic expansion dominates.


where ur and ugare dimensionless displacements, defined by

(A3.33)

ur and UQ can be derived from the strain-displacement relationships. Evaluating the Jintegral at r^ gives

J = ae0aoKn+lr(

2n+ms-2)+l!n (A3.34)

where In is an integration constant, given by

+7rr „f I It' —• f t [ I < ^ - > ^ W f W ' ^ ' V < W »

U0 (A3.35)

In order for / to be path independent, it cannot depend on r2, which was defined arbitrarily.The radius vanishes in Eq. (A3.34) only when (n+l)(s-2)+l - 0, or

_2n + ls —

n + l

which is identical to the result obtained numerically (Eq. (A3.31)). Thus the amplitude ofthe stress function is given by

1

/ n+lXT= - - (A3.36)

Substituting Eq. (A3.36) into Eq. (A3.30) yields the familiar form of the HRR singular-ity:

EJ

1Y

n+l „(A3.37)

194 Appendix 3

since eo = Oo/E. The integration constant is plotted in Fig. 3.10 for both plane stress

and plane strain, while Fig. 3.11 shows the angular variation of afj forn = 3 and n = 13.Rice and Rosengren [8] obtained essentially identical results to Hutchinson (for

plane strain), although they approached the problem in a somewhat different manner.Rice and Rosengren began with a heuristic argument for the 1/r variation of strain energydensity, and then introduced an Airy stress function of the form of Eq. (A3.29) with theexponent on r given by Eq. (A3.31). They computed stresses, strains and displacementsin the vicinity of the crack tip by applying the appropriate boundary conditions.

The HRR singularity was an important result because it established J as a stressamplitude parameter within the plastic zone, where the linear elastic solution is invalid.The analyses of Hutchinson, Rice and Rosengren demonstrated that the stresses in theplastic zone are much higher in plane strain than in plane stress; recall that the elasticsolution predicts identical in-plane stresses for both cases. These results provided atheoretical explanation for empirically observed thickness effects in fracture toughnesstests.

One must bear in mind the limitations of the HRR solution. Since the singularityis merely the leading term in an asymptotic expansion, and elastic strains were assumedto be negligible, this solution dominates only valid near the crack tip, well within theplastic zone. For very small r values, however, the HRR solution is invalid because itneglects finite geometry changes at the crack tip. When the HRR singularity dominates,the loading is proportional, which implies a single parameter description of crack tipfields. When the higher order terms in the series are significant, the loading is oftennonproportional and a single-parameter description may no longer be possible (SeeSection 3.6).

A3.5 ANALYSIS OF STABLE CRACK GROWTH IN SMALL SCALEYIELDING

A3.5.1 The Rice-Drugan-Sham Analysis

Rice, Drugan and Sham (RDS) [15] performed an asymptotic analysis of a growing crackin an elastic-plastic solid in small-scale yielding. They assumed crack extension at aconstant crack opening angle, and predicted the shape of J resistance curves. They alsospeculated about the effect of large scale yielding on the crack growth resistance behavior.

Small scale yieldingRice et al. analyzed the local stresses and displacements at a growing crack by modi-

fying the classical Prandtl slip line field to account for elastic unloading behind the cracktip. They assumed small scale yielding conditions and a nonhardening material; thedetails of the derivation are omitted for brevity. The RDS crack growth analysis resultedin the following expression;


r,o ..8 = a — + p— Q-ain — fo r r ->o (A3.38)

G E

where 8 is the rate of crack opening displacement at a distance R behind the crack tip, / isthe rate of change in the / integral, a is the crack growth rate, and a, ft, and R areconstants9. The asymptotic analysis indicated that j3 = 5.083 for v = 0.3 and ft = 4.385for v = 0.5. The other constants, a and R, could not be inferred from the asymptoticanalysis. Rice, et. al. [15] performed elastic-plastic finite element analysis of a growingcrack and found that R, which has units of length, scales approximately with plastic zonesize, and can be estimated by

AE7« Q.2 (A3.39)

The dimensionless constant a can be estimated by considering a stationary crack (a = 0):

r /§ = a — (A3 .40)

Referring to Eq. (3.48), a obviously equals dn when 8 is defined by the 90° interceptmethod. The finite element analysis performed by Rice, et al. indicated that a for agrowing crack is nearly equal to the stationary crack case.

Rice et. al. performed an asymptotic integration of (Eq. A3. 38) for the case wherecrack length increases continuously with /, which led to

c ar dJ 0 G0. ( eR\8 = -- + Br— £-ln — (A3.41)

P<J0 da E r

where 8, in this case, is the crack opening displacement at a r from the crack tip, and e (=2.718) is the natural logarithm base. Equation (A3.41) can be rearranged to solve for thenondimensional tearing modulus:

„ E dJ E8 B, feR}T = — T — = -- —In — (A3.42)

OQ da acr0r a \ r J

Rice, et al. proposed a failure criterion that corresponds approximately to crackextension at a constant crack tip opening angle (CTOA). Since dS/dr = °° at the crack

^The constant a in the RDS analysis should not be confused with the dimensionless constant in the Ramberg-Osgood relationship (Eq. 3.22), for which the same symbol is used.

196 Appendix 3

tip, CTOA is undefined, but an approximate CTOA can be inferred a finite distance fromthe tip. Figure A3.6 illustrates the RDS crack growth criterion. They postulated thatcrack growth occurs at a critical crack opening displacement, Sc, at a distance rm behindthe crack tip. That is,

• m da E r- constant (A3.43)

m

Rice, et al. found that it was possible to define the micromechanical failure parameters,8C and rm, in terms of global parameters that are easy to obtain experimentally. Setting

J = Jjc and combining Eqs. (A3.39), (A3.42), and (A3.43) gives

T =<*G0

rm a \ rma>

(A3.44)

where To is the initial tearing modulus. Thus for, J> Jjc, the tearing modulus is givenby

a JIc(A3.45)

FIGURE A3.6 The RDS crack growthcriterion. The crack is assumed toextend with a constant openingdisplacement, Sc, at distance rm behindthe crack tip. This criterioncorresponds approximately to crackextension at a constant crack tipopening angle (CTOA).


Rice, et al. computed normalized R curves (J/7/c v. Aa/R) for a range of T0 values andfound that T = To in the early stages of crack growth, but the R curve slope decrease untilsteady state plateau is reached. The steady state / can easily be inferred from Eq. (A3. 45)by setting 7=0:

(A3.46)

Large scale yieldingAlthough the RDS analysis was derived for small scale yielding conditions, Rice, et

al. speculated that the form of Eq. (A3.38) might also be valid for fully plastic condi-tions. The numerical values of some of the constants, however, probably differ for thelarge scale yielding case.

The most important difference between small scale yielding and fully plasticconditions is the value of R. Rice, et. al. argued that R would no longer scale withplastic zone size, but should be proportional to the ligament length. They made a roughestimate of R ~ b/4 for the fully plastic case.

The constant a depends on crack tip triaxiality and thus may differ for small scaleyielding and fully yielded conditions. For highly constrained configurations, such as bendspecimens, a for the two cases should be similar.

In small scale yielding, the definition of J is unambiguous, since it is related to theelastic stress intensity factor. The J integral for a growing crack under fully plasticconditions can be computed in a number of ways, however, and not all definitions of J areappropriate in the large scale yielding version of Eq. (A3.38).

Assume that the crack growth resistance behavior is to be characterized by a ./-likeparameter, Jx. Assuming Jx depends on crack length and displacement, the rate of

change in Jx should be linearly related to displacement rate and a:

(A3.47)

where £ and # are functions of displacement and crack length. Substituting Eq. (A3.47)into Eq. (A3.38) gives

a,-.\^n.(R\ a . (A348)

)

In the limit of a rigid-ideally plastic material, GQ/E = 0. Also, the local crack openingrate must be proportional to the global displacement rate for a rigid-ideally plasticmaterial:

5 = \/A (A3.49)

198 Appendix 3

Therefore, the term in square brackets in Eq. (A3.48) must vanish, which implies that^ =0, at least in the limit of a rigid-ideally plastic material. Thus in order for the RDSmodel to apply to large scale yielding, the rate of change in the ./-like parameter must notdepend on the crack growth rate:

jx*jx(a) (A3.50)

Rice, et al. showed that neither the deformation theory JOT the far-field J satisfy Eq.(A3.50) for all configurations.

Satisfying Eq. (A3.50) does not necessarily imply that a JX-R curve is geometryindependent. The RDS model suggests that a resistance curve obtained from a fullyyielded specimen will not, in general, agree with the small scale yielding R curve for thesame material. Assuming R = b/4 for the fully plastic case, the RDS model predicts thefollowing tearing modulus:

r= b/4assy

11 ssy

a,(A3.51)

where the subscripts ssy and fy denote small scale yielding and fully yielded conditions,respectively. According to Eq. (A3.51), the crack growth resistance curve under fullyyielded conditions has a constant initial slope, but this slope is not equal to To (theinitial tearing modulus in small scale yielding) unless afy = assy and b = 4X, E J}c/cro^.Equation (A3.51) does not predict a steady state limit where 7=0; rather this relationshippredicts that T actually increases as the ligament becomes smaller.

The forgoing analysis implies that crack growth resistance curves obtained fromspecimens with fully yielded ligaments are suspect. One should exercise extreme cautionwhen applying experimental results from small specimens to predict the behavior of largestructures.

A3.5.2 Steady State Crack Growth

The RDS analysis, which assumed a local failure criterion based on crack opening angle,indicated crack growth in small scale yielding reaches a steady state, where dJ/da —> 0.The derivation that follows shows that the steady state limit is a general result for smallscale yielding; the R curve must eventually reach a plateau in an infinite body, regardlessof the failure mechanism.

Generalized Damage Integral

Consider a material element a small distance from a crack tip, as illustrated in Fig.A3.7. This material element will fail when it is deformed beyond its capacity. The crackwill grow as consecutive material elements at the tip fail. Let us define a generalizeddamage integral, 6, which characterizes the severity of loading at the crack tip:


-eq

0= J0

(A3.52)

where eeq is the equivalent (von Mises) plastic strain and Q is a function of the stress andstrain tensors (aij and £/y, respectively). The above integral is sufficiently general that itcan depend on the current values of all stress and strain components, as well as the entiredeformation history. Referring to Fig. A3.7, the material element will fail at a criticalvalue of 0. At the moment of crack initiation or during crack extension, material near thecrack tip will be close to the point of failure. At a distance r* from the crack tip, wherer* is arbitrarily small, we can assume that 0 - &c.

The precise form of the damage integral depends on the micromechanism of fracture.For example, a modified Rice and Tracey [47] model for ductile hole growth (see Chapter5) can be used to characterize ductile fracture in metals:

( R \ eq M0 = ln — =0.283 j exp -

o cr.deeq (A3.53)

where R is the void radius, Ro is the initial radius, om is the mean (hydrostatic) stress,and oe is the effective (von Mises) stress. Failure, in this case, is assumed when the voidradius reaches a critical value.

FIGURE A3.7 Material point adistance r* from the crack tip.

200 Appendix 3

Stable crack growthConsider an infinite body^ that contains a crack that is growing in a stable, self-

similar, quasistatic manner. If the crack has grown well beyond the initial blunted tip,dimensional analysis indicates that the local stresses and strains are uniquely characterizedby the far-field / integral, as stated in Eq. (3.69). In light of this single-parametercondition, the integrand of Eq. (A3.52) becomes

Q = d - ,0 (A3.54)

We can restrict this analysis to & - 0 by assuming that the material on the crack planefails during Mode I crack growth. For a given material point on the crack plane, rdecreases as the crack grows, and the plastic strain increases. If strain increasesmonotonically as this material point approaches the crack tip, Eq. (A3. 54) permitswriting D as a function of the von Mises strain:

) (A3.55)

Therefore the local failure criterion is given by

£*(A3.56)

0

where £ * is the critical strain (i.e., the von Mises strain at r = r*). Since the integrand isa function only of £eq, the integration path is the same for all material points ahead of thecrack tip, and e* is constant during crack growth. That is, the equivalent plastic strain atr* will always equal e* when the crack is growing. Based on Eqs. (3.69) and (A3.54), e*is a function only of r* and the applied J:

£*=£*(/,r*) (A3 .57)

Solving for the differential of £* gives

d£* = ~-dJ + —dr * (A3.58)dJ dr*

Since e* and r* are both fixed, d£* = dr* = dl=0.

iuln practical terms "infinite" means that external boundaries are sufficiendy far from the crack tip so thatthe plastic zone is embedded within an elastic singularity zone.


Thus the / integral remains constant during crack extension (df/da = 0} when Eq.(3.69) is satisfied. Steady state crack growth is usually not observed experimentallybecause large scale yielding in finite sized specimens precludes characterizing a growingcrack with /. Also, a significant amount of crack growth may be required before a steadystate is reached (Fig. 3.25); the crack tip in a typical laboratory specimen approaches afree boundary well before the crack growth is sufficient to be unaffected by the initialblunted tip.

A3.6 NOTES ON THE APPLICABILITY OF DEFORMATIONPLASTICITY TO CRACK PROBLEMS

Since elastic-plastic fracture mechanics is based on deformation plasticity theory, it maybe instructive to take a closer look at this theory and assess its validity for crackproblems.

Let us begin with the plastic portion of the Ramberg-Osgood equation for uniaxialdeformation, which can be expressed in the following form:

/ \n-l . . . , , - , '<7 I <T

Differentiating Eq. (A3. 59) gives

f Y7"2 jI a } o da v -," • * •den = an\ — -- (A3.60)

for an increment of plastic strain. For the remainder of this section, the subscript onstrain is suppressed for brevity; only plastic strains are considered, unless statedotherwise.

Equations (A3. 59) and (A3. 60) represent the deformation and incremental flowtheories, respectively, for uniaxial deformation in a Ramberg-Osgood material. In thissimple case, there is no difference between the incremental and deformation theories,provided no unloading occurs. Equation (A3. 60) can obviously be integrated to obtainEq. (A3. 59). Stress is uniquely related to strain when both increase monotonically. Itdoes not necessarily follow that deformation and incremental theories are equivalent in thecase of three-dimensional monotonic loading, but there are many cases where this is agood assumption.

Equation (A3. 59) can be generalized to three dimensions by assuming deformationplasticity and isotropic hardening:

(A3.61)

202 Appendix 3

where <je is the effective (von Mises) stress and Sij is the deviatoric component of thestress tensor, defined by

where 8y is the Kronecker delta. Equation (A3.61) is the deformation theory flow rule fora Ramberg-Osgood material. The corresponding flow rule for incremental plasticitytheory is given by

(A3.63){<?„ E a0

By comparing Eqs. (A3.61) and (A3. 63), one sees that the deformation and incrementaltheories of plasticity coincide only if the latter equation can be integrated to obtain theformer. If the deviatoric stress components are proportional to the effective stress:

S^ = 0>ijCre (A3.64)

where coij is a constant tensor that does not depend on strain, then integration of Eq.(A3.63) results in Eq. (A3.61). Thus deformation and incremental theories of plasticityare identical when the loading is proportional in the deviatoric stresses. Note that thetotal stress components need not be proportional in order for the two theories to coincide;the flow rule is not influenced by the hydrostatic portion of the stress tensor.

Proportional loading of the deviatoric components does not necessarily mean thatdeformation plasticity theory is rigorously correct; it merely implies that deformationtheory is no more objectionable than incremental theory. Classical plasticity theory,whether based on incremental strain or total deformation, contains simplifying assump-tions about material behavior. Both Eqs. (A3. 61) and (A3. 63) assume that the yieldsurface expands symmetrically and that its radius does not depend on hydrostatic stress.For monotonic loading ahead of a crack in a metal, these assumptions are probablyreasonable; the assumed hardening law is of little consequence for monotonic loading, andhydrostatic stress effects on the yield surface are relatively small for most metals.

Budiansky [48] showed that deformation theory is still acceptable when there aremodest deviations from proportionality. Low work hardening materials are the leastsensitive to nonproportional loading.

Since most of classical fracture mechanics assumes either plane stress or planestrain, it is useful to examine plastic deformation in the two-dimensional case, anddetermine under what conditions the requirement of proportional deviatoric stresses is atleast approximately satisfied. Consider, for example, plane strain. When elastic strainsare negligible, the in-plane deviatoric normal stresses are given by

Mathematical Foundations of EPFM 203

(A3.65)

assuming incompressible plastic deformation, where crzz = (axx + Oyy)/2. The ex-pression for von Mises stress in plane strain reduces to

^xy\ L (A3.66)

where S^y = T^y. Alternatively, oe can be written in terms of principal normal stresses:

<7e = [(Jj — 0*2] where 07 > 02

= V35] (A3.67)

Therefore, the principal deviatoric stresses are proportional to Ge in the case of planestrain. It can easily be shown that the same is true for plane stress. If the principal axesare fixed, Sxx, 5VV, and SXy must also be proportional to oe. If, however, the principalaxes rotate during deformation, the deviatoric stress components defined by a fixedcoordinate system will not increase in proportion to one another.

In the case of Mode I loading of a crack, TXy is always zero on the crack plane,implying that the principal directions on the crack plane are always parallel to the x-y-zcoordinate axes. Thus, deformation and incremental plasticity theories should be equallyvalid on the crack plane, well inside the plastic zone (where elastic strains are negligible).At finite angles from the crack plane, the principal axes may rotate with deformation,which will produce nonproportional deviatoric stresses. If this effect is small,deformation plasticity theory should be adequate to analyze stresses and strains near thecrack tip in either plane stress or plane strain.

The validity of deformation plasticity theory does not automatically guarantee thatthe crack tip conditions can be characterized by a single parameter, such as J or K.Single-parameter fracture mechanics requires that the total stress components beproportional near the crack tip11, a much more severe restriction. Proportional totalstresses imply that the deviatoric stresses are proportional, but the reverse is not nec-essarily true. In both the linear elastic case (Appendix 2.3) and the nonlinear case(Appendix 3.4) the stresses near the crack tip were derived from a stress function of theform

(A3.68)

The proportional loading region need not extend all the way to the crack tip, but the nonproportional zoneat the tip must be embedded within the proportional zone in order for a single loading parameter tocharacterize crack tip conditions.

204 Appendix 3

where Kisa constant. The form of Eq. (A3.68) guarantees that all stress components areproportional to K, and thus proportional to one another. Therefore any monotonicfunction of K uniquely characterizes the stress fields in the region where Eq. (A3.68) isvalid. Nonproportional loading automatically invalidates Eq. (A3.68) and the single pa-rameter description that it implies.

As stated earlier, the deviatoric stresses are proportional on the crack plane, wellwithin the plastic zone. The hydrostatic stress may not be proportional to oe, however.For example, the loading is highly nonproportional in the large strain region, as Fig.3.12 indicates. Consider a material point at a distance x from the crack tip, where x is inthe current large strain region. At earlier stages of deformation the loading on this pointwas proportional, but <jyy reached a peak when the ratio x Gg/J was approximately unity,and the normal stress decreased with subsequent deformation. Thus the most recentloading on this point was nonproportional, but the deviatoric stresses are stillproportional to ae.

When the crack grows, material behind the crack tip unloads elastically and de-formation plasticity theory is no longer valid. Deformation theory is also suspect nearthe elastic-plastic boundary. Equations (A3.65) to (A3.67) were derived assuming theelastic strains were negligible, which implies crzz = 0.5(axx + Oyy) in plane strain. Atthe onset of yielding, however, cr^ = v (axx + &yy)> and the proportionality constantsbetween ae and the deviatoric stress components are different than for the fully plasticcase. Thus when elastic and plastic strains are of comparable magnitude, the deviatoricstresses are nonproportional, as (Dy (Eq. (A3.64)) varies from its elastic value to the fullyplastic limit. The errors in deformation theory that may arise from the transition fromelastic to plastic behavior should not be appreciable in crack problems, because the straingradient ahead of the crack tip is relatively steep, and the transition zone is small.

4. DYNAMIC AND TIME-DEPENDENTFRACTURE

In certain fracture problems, time is an important variable. At high loading rates, for ex-ample, inertia effects and material rate dependence can be significant. Metals and ceramicsalso exhibit rate-dependent deformation (creep) at temperatures that are close to the melt-ing point of the material. The mechanical behavior of polymers is highly sensitive tostrain rate, particularly above the glass transition temperature. In each of these cases, lin-ear elastic and elastic-plastic fracture mechanics, which assume quasistatic, rate-indepen-dent deformation, are inadequate.

Early fracture mechanics researchers considered dynamic effects, but only for the spe-cial case of linear elastic material behavior. More recently, fracture mechanics has beenextended to include time-dependent material behavior such as viscoplasticity and viscoelas-ticity. Most of these newer approaches are based on generalizations of the J contour inte-gral.

This chapter gives an overview of time-dependent fracture mechanics. The treat-ment of this subject is far from exhaustive, but should serve as an introduction to a com-plex and rapidly developing field. The reader is encouraged to consult the published litera-ture for further background.

4.1 DYNAMIC FRACTURE AND CRACK ARREST

As any undergraduate engineering student knows, dynamics is more difficult than statics.Problems become more complicated when the equations of equilibrium are replaced by theequations of motion.

In the most general case, dynamic fracture mechanics contains three complicatingfeatures that are not present in LEFM and elastic-plastic fracture mechanics: inertiaforces, rate-dependent material behavior, and reflected stress waves. Inertia effects are im-portant when the load changes abruptly or the crack grows rapidly; a portion of the workthat is applied to the specimen is converted to kinetic energy. Most metals are not sensi-tive to moderate variations in strain rate near ambient temperature, but the flow stress canincrease appreciably when strain rate increases by several orders of magnitude. The effectof rapid loading is even more pronounced in rate sensitive materials such as polymers.When the load changes abruptly or the crack grows rapidly, stress waves propagatethrough the material and reflect off of free surfaces, such as the specimen boundaries andthe crack plane. Reflecting stress waves influence the local crack tip stress and strainfields which, in turn, affect the fracture behavior.

In certain problems, one or more of the above effects can be ignored. If all three ef-fects are neglected, the problem reduces to the quasistatic case.

The dynamic version of LEFM is termed elastodynamic fracture mechanics, wherenonlinear material behavior is neglected, but inertia forces and reflected stress waves areincorporated when necessary. The theoretical framework of elastodynamic fracture me-

205

206 Chapter 4

chanics is fairly well established, and practical applications of this approach are becomingmore common. Extensive reviews of this subject have been published by Freund [1-5],Kanninen and Poplar [6], Rose [7], and others. Elastodynamic fracture mechanics haslimitations, but is approximately valid in many cases. When the plastic zone is restrictedto a small region near the crack tip in a dynamic problem, the stress intensity approach,with some modifications, is still applicable.

Dynamic fracture analyses that incorporate nonlinear, time-dependent material behav-ior are a relatively recent innovation. A number of researchers have generalized the / in-tegral to account for inertia and viscoplasticity [8-13].

There are two major classes of dynamic fracture problems: (1) fracture initiation asa result of rapid loading, and (2) rapid propagation of a crack. In the latter case, the crackpropagation may initiate either by quasistatic or rapid application of a load; the crack mayarrest after some amount of unstable propagation. Dynamic initiation, propagation, andcrack arrest are discussed below.

4.1.1 Rapid Loading of a Stationary Crack

Rapid loading of a structure can come from a number of sources, but most often occurs asthe result of impact with a second object (e.g. a ship colliding with an offshore platformor a missile striking its target). Impact loading is often applied in laboratory tests when ahigh strain rate is desired. The Charpy test [14], where a pendulum dropped from a fixedheight fractures a notched specimen, is probably the most common dynamic mechanicaltest. Dynamic loading of a fracture mechanics specimen can be achieved through impactloading [15,16], a controlled explosion near the specimen [17], or servohydraulic testingmachines that are specially designed to impart high displacement rates. Chapter 7 de-scribes some of the practical aspects of high rate fracture testing.

Figure 4.1 schematically illustrates a typical load-time response for dynamic load-ing. The load tends to increase with time, but oscillates at a particular frequency that de-pends on specimen geometry and material properties. Note that the loading rate is finite;i.e., a finite time is required to reach a particular load. The amplitude of the oscillationsdecreases with time, as kinetic energy is dissipated by the specimen. Thus inertia effectsare most significant at short times, and are minimal after sufficiently long times, wherethe behavior is essentially quasistatic.

Determining a fracture characterizing parameter, such as the stress intensity factor orthe /integral, for rapid loading can be very difficult. Consider the case where the plasticzone is confined to a small region surrounding the crack tip. The near-tip stress fields forhigh rate Mode I loading are given by

Dynamic and Time-Dependent Fracture 207

LOAD

FIGURE 4.1 Schematic load-time responseof a rapidly loaded structure.

TIME

where (?) denotes a function of time. The angular functions, fij, are identical to the qua-sistatic case and are given in Table 2.1. The stress intensity factor, which characterizesthe amplitude of the elastic singularity, varies erratically in the early stages of loading.Reflecting stress waves that pass through the specimen constructively and destructivelyinterfere with one another, resulting in a highly complex time-dependent stress distribu-tion. The instantaneous Kj depends on the magnitude of the discrete stress waves thatpass through the crack tip region at that particular moment in time. When the discretewaves are significant, it is not possible to infer Kj from the remote loads.

Recent work by Nakamura et al. [18,19] quantified inertia effects in laboratory speci-mens and showed that these effects can be neglected in many cases. They observed thatthe behavior of a dynamically loaded specimen can be characterized by a short-time re-sponse, dominated by discrete waves, and a long-time response that is essentially qua-sistatic. At intermediate times, global inertia effects are significant but local oscillationsat the crack are small, because kinetic energy is absorbed by the plastic zone. To distin-guish short-time response from long-time response, Nakamura et al. defined a transitiontime, t<t, when the kinetic energy and the deformation energy (the energy absorbed by thespecimen) are equal. Inertia effects dominate prior to the transition time, but the deforma-tion energy dominates at times significantly greater than t^. In the latter case, a ./-domi-nated field should exist near the crack tip and quasistatic relationships can be used to infer/ from global load and displacement.

Since it is not possible to measure kinetic and deformation energies separately dur-ing a fracture mechanics experiment, Nakamura et al. developed a simple model to esti-mate kinetic energy and transition time in a three-point bend specimen (Fig. 4.2). Thismodel was based on the Bernoulli-Euler beam theory and assumed that the kinetic energyat early times was dominated by the elastic response of the specimen. Incorporating theknown relationship between load line displacement and strain energy in a three-point bendspecimen leads to an approximate relationship for the ratio of kinetic to deformation en-ergy:

208 Chapter 4

=uc0A(t)J

(4.2)

where £# is the kinetic energy, U is the deformation energy, W is the specimen width, A

is the load line displacement, A is the displacement rate, c0 is the longitudinal wave speed(i.e. the speed of sound) in a one-dimensional bar, and A is a geometry factor, which forthe bend specimen is given by

(4.3)

where S is the span of the specimen. The advantage of Eq. (4.2) is that the displacementand displacement rate can be measured experimentally. The transition time is defined atthe moment in the test when the ratio Efc/U = 1. In order to obtain an explicit expressionfor t?, it is convenient to introduce a dimensionless displacement coefficient, D:

tA(t)A(t)

(4.4)

If, for example, the displacement varies with time as a power law: A - fit?, then D = y.Combining Eqs. (4.2) and (4.4) and setting £&/£/= 1 leads to

w

jFIGURE 4.2 Three-point bend specimen.


W(4.5)

Nakamura et al. [18,19] performed dynamic finite element analysis on a three pointbend specimen in order to evaluate the accuracy of Eqs. (4.2) and (4.5). Figure 4.3 com-pares the Efc/U ratio computed from finite element analysis with that determined from ex-periment and Eq. (4.2). The horizontal axis is a dimensionless time scale, and cj is thelongitudinal wave speed in an unbounded solid. The ratio W/c] is an estimate of the timerequired for a stress wave to traverse the width of the specimen. Based on Eq. (4.2) andexperiment, tT cj/W ~ 28 (or t? CQ/H ~ 24), while the finite element analysis estimatedt^cj/W- 27. Thus the simple model agrees quite well with a more detailed analysis.

The simple model was based on the global kinetic energy and did not consider dis-crete stress waves. Thus the model is only valid after stress waves have traversed thewidth of the specimen several times. This limitation does not affect the analysis of tran-sition time, since stress waves have made approximately 27 passes when f-j- is reached.Note, in Fig. 4.3, that the simple model agrees very well with the finite element analysiswhen t c]/W > 20.

When t » ti, inertia effects are negligible and quasistatic models should apply tothe problem. Consequently, the J integral for a deeply cracked bend specimen at longtimes can be estimated by

2.0

1.5

1.0

0.5

Eq. (4.2)Finite elementanalysis

0 20 40 60tG/W

80

FIGURE 4.3 Ratio of kinetic to stress work energy in a dynamically loaded three-point bend specimen[19].

210 Chapter 4

9

-~Bb(4.6)

where £ is the plate thickness, b is the uncracked ligament length, M is the applied mo-ment on the ligament, Q. is the angle of rotation, and f* is the current time. Equation(4.6), which was originally published by Rice, et al. [20], is derived in Section 3.2.5.

Nakamura, et al. [19] performed a three-dimensional dynamic elastic-plastic finite el-ement analysis on a three-point bend specimen in order to determine the range of applica-bility of Eq. (4.6). They evaluated a dynamic J integral (see Section 4.1.3) at variousthickness positions and observed a through-thickness variation of J that is similar to Fig.3.36. They computed a nominal J that averaged the through-thickness variations andcompared this value with Jfa. The results of this exercise are plotted in Fig. 4.4. Atshort times, the average dynamic J is significantly lower than the J computed from thequasistatic relationship. For t > 2^, the Jrfc/Jave reaches a constant value that is slightlygreater than 1. The modest discrepancy between Jrfc and Jave at long times is probablydue to three-dimensional effects rather than dynamic effects (Eq. (4.6) is essentially a two-dimensional formula).

According to Fig. 4.4, Eq. (4.6) provides a good estimate of / in a high rate test attimes greater than approximately twice the transition time. It follows that if fracture ini-tiation occurs after 2t^, the critical value of J obtained from Eq. (4.6) is a measure of frac-ture toughness for high rate loading. If small-scale yielding assumptions apply, the criti-cal/can be converted to an equivalent KIC through Eq. (3.18).

1.4

1.2

1.0

0.8 -

0.60 20 40 60 80 100 120 140

tG/W

FIGURE 4.4 Ratio of / computed from Eq. (4.6) to the through-thickness average / computed fromthree-dimensional dynamic finite element analysis.


Given the difficulties associated with defining a fracture parameter in the presence ofinertia forces and reflected stress waves, it is obviously preferable to apply Eq. (4.6)whenever possible. For a three point bend specimen with W = 50 mm, the transitiontime is approximately 300 (is [19]. Thus the quasistatic formula can be applied as longas fracture occurs after ~ 600 )is. This requirement is relatively easy to meet in impacttests on ductile materials [15,16]. For more brittle materials, the transition time require-ment can be met by decreasing the displacement rate or the width of the specimen.

The transition time concept can be applied to other configurations by adjusting thegeometry factor in Eq. (4.2). Duffy and Shih [17] have applied this approach to dynamicfracture toughness measurement in notched round bars. Small round bars have proved tobe suitable for dynamic testing of brittle materials such as ceramics, where the transitiontime must be small.

If the effects of inertia and reflected stress waves can be eliminated, one is left withthe rate-dependent material response. The transition time approach allows material rate ef-fects to be quantified independent of inertia effects. High strain rates tend to elevate theflow stress of the material. The effect of flow stress on fracture toughness depends on thefailure mechanism. High strain rates tend to decrease cleavage resistance, which is stresscontrolled. Materials whose fracture mechanisms are strain controlled often see an in-crease in toughness at high loading rates because more energy is required to reach a givenstrain value.

Figure 4.5 shows fracture toughness data for a structural steel at three loading rates[21]. The critical Kj values were determined from quasistatic relationships. For a givenloading rate, fracture toughness increases rapidly with temperature at the onset of the duc-tile-brittle transition. Note that increasing the loading rate has the effect of shifting thetransition to higher temperatures. Thus at a constant temperature, fracture toughness ishighly sensitive to strain rate.

The effect of loading rate on fracture behavior of a structural steel on the upper shelfof toughness is illustrated in Fig. 4.6. In this instance, strain rate has the opposite effectfrom Fig. 4.5, because ductile fracture of metals is primarily strain controlled. The J in-tegral at a given amount of crack extension is elevated by high strain rates.

4.1.2 Rapid Crack Propagation and Arrest

When the driving force for crack extension exceeds the material resistance, the structure isunstable, and rapid crack propagation occurs. Figure 4.7 illustrates a simple case, wherethe (quasistatic) energy release rate increases linearly with crack length and the material re-sistance is constant. Since the first law of thermodynamics must be obeyed even by anunstable system, the excess energy, denoted by the shaded area in Fig. 4.7, does not sim-ply disappear, but is converted into kinetic energy. The magnitude of the kinetic energydictates the crack speed.

In the quasistatic case, a crack is stable if the driving force is less than or equal tothe material resistance. Similarly, if the energy available for an incremental extension ofa rapidly propagating crack falls below the material resistance, the crack arrests. Figure4.8 illustrates a simplified scenario for crack arrest. Suppose that cleavage fracture initi-ates when KI = Kfc. The resistance encountered by a rapidly propagating cleavage crack

212 Chapter 4

100

80

IsIs 6°

* 40

20

I I T

A 572 Grade 50 Steel

Nominal Strain Rate (sec"1):

ol L-300 0-200 -100

TEMPERATURE, °F

FIGURE 4.5 Effect of loading rate on the cleavage fracture toughness of a structural steel [21].

800

1 2 3 4 5CRACK EXTENSION, mm

FIGURE 4.6 Effect of loading rate on the J-R curve behavior of HY80 steel [15].


is less than for cleavage initiation, because plastic deformation at the moving crack tip issuppressed by the high local strain rates. If the structure has a falling driving force curve,it eventually crosses the resistance curve. Arrest does not occur at this point, however,because the structure contains kinetic energy that can be converted to fracture energy.Arrest occurs below the resistance curve, after most of the available energy has been dis-sipated. The apparent arrest toughness, Kja, is less than the true material resistance, KJA-The difference between Kia and KJA is governed by the kinetic energy created during crackpropagation; KJA is a material property, but Kja depends on geometry.

FIGURE 4.7 Unstable crack propaga-tion, which results in the generation ofkinetic energy.

CRACK SIZE

aoCRACK SIZE

FIGURE 4.8 Unstable crack propaga-tion and arrest with a falling drivingforce curve. The apparent arresttoughness, Kj^ is slightly below the truematerial resistance, Kj^, due to excesskinetic energy.

214 Chapter 4

Figures 4.7 and 4.8 compare material resistance with quasistatic driving forcecurves. That is, these curves represent K/ and Cj values computed with the procedures de-scribed in Chapter 2. Early researchers [22-26] realized that the crack driving force shouldincorporate the effect of kinetic energy. The Griffith-Irwin energy balance (Sections 2.3and 2.4) can be modified to include kinetic energy, resulting in a dynamic definition ofenergy release rate:

where F is the work done by external forces and Jl is the crack area. Equation (4.7) isconsistent with the original Griffith approach, which is based on the first law of thermo-dynamics. The kinetic energy must be included in a general statement of the first law;Griffith implicitly assumed that Efc = 0.

Crack speedMott [22] applied dimensional analysis to a propagating crack in order to estimate

the relationship between kinetic energy and crack speed. For a through crack of length 2ain an infinite plate in tension, the displacements must be proportional to crack size, sincea is the only relevant length dimension. Assuming the plate is elastic, displacementsmust also be proportional to the nominal applied strain; thus

a crux = axa — and «v = aâ — (4.8)

E y y E

where cxx and ay are dimensionless constants. (Note that quantitative estimates for ax

and ay near the crack tip in the quasistatic case can be obtained by applying the relation-ships in Table 2.2.) The kinetic energy is equal to one half the mass times the velocitysquared. Therefore, E% for the cracked plate (assuming unit thickness) is given by

Ek = pa (a + adxdy (4.9)JL \tLt J

•where p is the mass density of the material and V(= a) is the crack speed. Assuming theintegrand depends only on position1, Ek can be written in the following form:

Ek=-kpa2V2(—} (4.10)* 2 H (EJwhere £ is a constant. Applying the modified Griffith energy balance (Eq. (4.7)) gives

In a rigorous dynamic analysis, ax and ay , and thus k, depend on crack speed.


E (4.11)

where wf is the work of fracture, defined in Chapter 2; in the limit of an ideally brittlematerial, wf= ys, the surface energy. Note that Eq. (4.11) assumes a flat R curve(constant wf). At initiation, the kinetic energy term is not present, and the initial cracklength, ao, can be inferred from Eq. (2.22):

(4.12)KG

Substituting Eq. (4.12) into Eq. (4.11) and solving for V leads to

\2n(4.13)

where co = v E/p , the speed of sound for one-dimensional wave propagation. Mott [22]actually obtained a somewhat different relationship from Eq. (4.13), because he solved Eq.(4.11) by making the erroneous assumption that dV/da = 0. Dulaney and Brace [27] andBerry [28] later corrected the Mott analysis and derived Eq. (4.13).

Roberts and Wells [29] obtained an estimate for k by applying the Westergaardstress function (Appendix 2.3) for this configuration. After making a few assumptions,

they showed that V2 n/k - 0.38.According to Eq. (4.13) and the Roberts and Wells analysis, the crack speed reaches

a limiting value of 0.38 c0 when a » a0. This estimate compares favorably with mea-sured crack speeds in metals, which typically range from 0.2 to 0.4 co [30].

Freund [2-4] performed a more detailed numerical analysis of a dynamically propa-gating crack in an infinite body and obtained the following relationship

(4.14)

where cr is the Raleigh (surface) wave speed. For Poisson's ratio = 0.3, the Cj/c0 ratio =0.57. Thus the Freund analysis predicts a larger limiting crack speed than the Roberts andWells analysis. The limiting crack speed in Eq. (4.14) can be argued on physical grounds[26]. For the special case where wf= 0, a propagating crack is merely a disturbance on afree surface, which must move at the Raleigh wave velocity. In both Eq. (4.13) and(4.14), the limiting velocity is independent of fracture energy; thus the maximum crackspeed should be cr for all wf.

216 Chapter 4

Experimentally observed crack speeds do not usually reach cr. Both the simpleanalysis that resulted in Eq. (4.13) and Freund's more detailed dynamic analysis assumedthat the fracture energy does not depend on crack length or crack speed. The material re-sistance actually increases with crack speed, as discussed below. The good agreement be-tween experimental crack velocities and the Roberts and Wells estimate of 0.38 co islargely coincidental.

Elastodynamic crack tip parametersThe governing equation for Mode I crack propagation under elastodynamic condi-

tions can be written as

(4.15)

where KI is the instantaneous stress intensity and KID ls the material resistance to crackpropagation, which depends on crack velocity. In general, Kj(t) is not equal to the staticstress intensity factor, as defined in Chapter 2. A number of researchers [8-10,31-33]have obtained a relationship for the dynamic stress intensity of the form

(4.16)

where £is a universal function of crack speed and Kj(0) is the static stress intensity factor.The function ftV) = 1.0 when V = 0, and decreases to zero as V approaches the Raleighwave velocity. An approximate expression for £was obtained by Rose [34]:

(4.17)

where h is a function of the elastic wave speeds and can be approximated by

(4.18)

where cj and C2 are the longitudinal and shear wave speeds, respectively.Equation (4.16) is valid only at short times or in infinite bodies. This relationship

neglects reflected stress waves, which can have a significant effect on the local crack tipfields. Since the crack speed is proportional to the wave speed, Eq. (4.16) is valid as longas the length of crack propagation (a - a0} is small compared to specimen dimensions, be-cause reflecting stress waves will not have had time to reach the crack tip (Example 4.1).In finite specimens where stress waves reflect back to the propagating crack tip, the dy-namic stress intensity must be determined experimentally or numerically on a case-by-case basis.


EXAMPLE 4.1

Rapid crack propagation initiates in a deeply-notched specimen with initial ligamentb0 (Fig. 4.9). Assuming the average crack speed = 0.2 cj, estimate how far the crack

will propagate before it encounters a reflected longitudinal wave.

Solution: At the moment the crack encounters the first reflected wave, the crack hastraveled a distance Aa, while the wave has traveled 2 bo - Aa. Equating travel times

gives

Aa0.2 c]

2 b0 - Aa

Thus

Equation (4.16) is valid in this case as long as the crack extension is less than bf/3and the plastic zone is small compared to b0.

ReflectedStressWave

FIGURE 4.9 Propagating crack encounter-ing a reflected stress wave.

For an infinite body or short times, Freund [10] showed that the dynamic energy re-lease rate could be expressed in the following form:

(4.19)

where g is a universal function of crack speed that can be approximated by

218 Chapter 4

I (4.20)

Combining Eqs. (4.16) to (4.20) with Eq. (2.51) gives

(4.21)

where

-i-l

(4-22)

Thus the relationship between Kj and § depends on crack speed. A more accurate (andmore complicated) relationship for A(V), is given in Appendix 4.1.

When the plastic zone ahead of the propagating crack is small, Kj(t) uniquely definesthe crack tip stress, strain, and displacement fields, but the angular dependence of thesequantities is different from the quasistatic case. For example, the stresses in the elasticsingularity zone are given by [32,33,35]

Q... = ^>(0 f..(e y^ (423}°v~F^fv(^y) (4-2JSj

The function jy reduces to the quasistatic case (Table 2.1) when V = 0. Appendix 4.1outlines the derivation of Eq. (4.23) and gives specific relationships forfij in the case ofrapid crack propagation. The displacement functions also display an angular dependencethat varies with V. Consequently, ax and ay in Eq. (4.9) must depend on crack velocityas well as position, and the Mott analysis is not rigorously correct for dynamic crackpropagation.

Dynamic toughnessAs Eq. (4.15) indicates, the dynamic stress intensity is equal to KID, the dynamic

material resistance, which depends on crack speed. This equality permits experimentalmeasurements of KID.

Dynamic propagation toughness can be measured as a function of crack speed bymeans of high speed photography and optical methods, such as photoelasticity [36,37]and the method of caustics [38]. Figure 4.10 shows photoelastic fringe patterns for dy-namic crack propagation in Homalite 100 [37]. Each fringe corresponds to a contour ofmaximum shear stress. Sanford and Dally [36] describe procedures for inferring stress in-tensity from photoelastic patterns.


FRAME I FRAME 120 i FRAME 9 158 M

FIGURE 4.10 Photoelastic fringe patterns for a rapidly propagating crack in Homalite 100 [37].(Photograph provided by R. Chona)

Figure 4.11 illustrates the typical variation of KID with crack speed. At lowspeeds, KID is relatively insensitive to V, but KID increases asymptotically as V ap-proaches a limiting value. Figure 4.12 shows KID data for 4340 steel published byRosakis and Freund [39].

In the limit of V- 0, KID — KjA-> the arrest toughness of the material. In general,KJA < KIC, the quasistatic initiation toughness. When a stationary crack in an elastic-plastic material is loaded monotonically, the crack tip blunts and a plastic zone forms. Apropagating crack, however, tends to be sharper and has a smaller plastic zone than a sta-tionary crack. Consequently, more energy is required to initiate fracture from a stationarycrack than is required to maintain propagation of a sharp crack.

The crack speed dependence of KID can be represented by an empirical equation ofthe form

KID ~K IA (4.24)

where V/is the limiting crack speed in the material and m is an experimentally determinedconstant. As Fig. 4.1 l(b) illustrates, KJA increases and V/"decreases with increasing ma-terial toughness. The trends in Figs. 4.11 (a) and 4.1 l(b) have not only been observed ex-perimentally, but have also been obtained by numerical simulation [40,41]. The upturnin propagation toughness at high crack speeds is apparently caused by local inertia forcesin the plastic zone.

220 Chapter 4

Increasing Toughness

CRACK SPEED CRACK SPEED

(a) Effect of crack speed on Kjj). (b) Effect of material toughness.

FIGURE 4.11 Schematic Kjj}-crack speed curves.

200

FIGURE 4.12 Experimental KID v.crack speed data for 4340 steel [39]

400 800

CRACK SPEED, m/ sec1200

Crack arrestEquation (4.15) defines the conditions for rapid crack advance. If, however, Kj(t)

falls below the minimum KID value for a finite length of time, propagation cannotcontinue, and the crack arrests. There are a number of situations that might lead to crackarrest. Figure 4.8 illustrates one possibility: if the driving force decreases with crackextension, it may eventually be less than the material resistance. Arrest is also possiblewhen material resistance increases with crack extension. For example, a crack thatinitiates in a brittle region of a structure, such as a weld, may arrest when it reaches a


material with higher toughness. A temperature gradient in a material that exhibits aductile-brittle transition is another case where the toughness can increase with position: acrack may initiate in a cold region of the structure and arrest when it encounters warmermaterial with a higher toughness. An example of this latter scenario is a pressurizedthermal shock event in a nuclear pressure vessel [42].

In many instances, it is not possible to guarantee with absolute certainty that an un-stable fracture will not initiate in a structure. Transient loads, for example, may occurunexpectedly. In such instances crack arrest can be the second line of defense. Thus thecrack arrest toughness, KJA, is an important material property.

Based on Eq. (4.16), one can argue that Kj(t) at arrest is equivalent to the quasistaticvalue, since V = 0. Thus it should be possible to infer KJA from a quasistatic calculationbased on the load and crack length at arrest. This quasistatic approach to arrest is actuallyquite common, and it is acceptable in many practical situations. Chapter 7 describes astandardized test method for measuring crack arrest toughness that is based on quasistaticassumptions.

The quasistatic arrest approach must be used with caution, however. Recall that Eq.(4.16) is only valid for infinite structures or short crack jumps, where reflected stresswaves do not have sufficient time to return to the crack tip. When reflected stress waveeffects are significant, Eq. (4.16) is no longer valid, and a quasistatic analysis tends togive misleading estimates of the arrest toughness. Quasistatic estimates of arrest tough-ness are sometimes given the designation Kja; for short crack jumps, Kja = KIA-

The effect of stress waves on the apparent arrest toughness (Kja) was demonstrateddramatically by Kalthoff, et al. [43], who performed dynamic propagation and arrest exper-iments on wedge loaded double cantilever beam (DCB) specimens. Recall from Example2.3 that the DCB specimen exhibits a falling driving force curve in displacement control.Kalthoff, et al. varied the Kj at initiation by varying the notch root radius. When thecrack was sharp, fracture initiated slightly above KIA an^ arrested after a short crack jump;the length of crack jump increased with notch tip radius.

Figure 4.13 is a plot of the Kalthoff, et al. results. For the shortest crack jump, thetrue arrest toughness and the apparent quasistatic value coincide, as expected. As thelength of crack jump increases, the discrepancy between the true arrest and the quasistaticestimate increases, with KIA > Kja. Note that KIA appears to be a material constant butKja varies with the length of crack propagation. Also note that the dynamic stress inten-sity during crack growth is considerably different from the quasistatic estimate of Kj.Kobayashi, et al. [44] obtained similar results.

A short time after arrest, the applied stress intensity reaches Kja, the quasistaticvalue. Figure 4.14 shows the variation of Kj after arrest in one of the Kalthoff, et al. ex-periments. When the crack arrests, K{ = KIA, which is greater than Kfa. Figure 4.14shows that the specimen "rings down" to Kja after -2000 |ls. The quasistatic value,however, is not indicative of the true material arrest properties.

Recall the schematic in Fig. 4.8, where it was argued that arrest, when quantified bythe quasistatic stress intensity, would occur below the true arrest toughness, KJA, becauseof kinetic energy in the specimen. This argument is a slight oversimplification, but itleads to the correct qualitative conclusion.

222 Chapter 4

2.5

2.0

g 1.5

1.0

0.5

Symbol

p o tDAV

X

Ko,MPam172

2.321.761331.030.74

MaximumCrack Speed,

m/sec

29527220710815

I

Statically Interpreted Arrest

80 100 140 180

CRACK LENGTH, mm

220

FIGURE 4.13 Crack arrest experiments on wedge-loaded DCS Araldite B specimens [43]. The stati-cally interpreted arrest toughness underestimates the true K/A of the material; this effect ismost pronounced for long crack jumps.

The DCB specimen provides an extreme example of reflected stress wave effects; thespecimen design is such that stress waves can traverse the width of the specimen and re-turn to the crack tip in a very short time. In many structures, the quasistatic approach isapproximately valid, even for relatively long crack jumps. In any case, Kja gives a lowerbound estimate of KfA, and thus is conservative in most instances.

4.1.3 Dynamic Contour Integrals

The original formulation of the / contour integral is equivalent to the nonlinear elasticenergy release rate for quasistatic deformation. By invoking a more general definition ofenergy release rate, it is possible to incorporate dynamic effects and time-dependent mate-rial behavior into the / integral.


-500 500 1000

TIME, |is

1500 2000

FIGURE 4.14 Comparison of dynamic measurements of stress intensity with static calculations for awedge loaded DCB Araldite B specimen [43].

The energy release rate is usually defined as the energy released from the body perunit crack advance. A more precise definition [11] involves the work input into the cracktip. Consider a vanishingly small contour, F, around the tip of a crack in a two-dimen-sional solid (Fig. 4.15). The energy release rate is equal to the energy flux into the cracktip, divided by the crack speed:

v (4.25)

where 7 is the energy flux into the area bounded by JT. The generalized energy releaserate, including inertia effects, is given by

r dx(4.26)

where w and Tare the stress work and kinetic energy densities defined as

224 Chapter 4

FIGURE 4.15 Energy flux into a small con-tour at the tip of a propagating crack.

and

T = 0 i i2 dt dt

(4.27)

(4.28)

Equation (4.26) has been published in a variety of forms by several researchers [8-12J.Appendix 4.2 gives a derivation of this relationship.

Equation (4.26) is valid for time-dependent as well as history-dependent material be-havior. When evaluating J for a time-dependent material, it may be convenient to expressw in the following form:

=/ (4.29)

where E{J is the strain rate.Unlike the conventional J integral, the contour in Eq. (4.26) cannot be chosen arbi-

trarily. Consider, for example, a dynamically loaded cracked body with stress waves re-flecting off of free surfaces. If the integral in Eq. (4.26) were computed at two arbitrarycontours a finite distance from the crack tip, and a stress wave passed through one contourbut not the other, the values of these integrals would normally be different for the twocontours. Thus the generalized J integral is not path independent, except in the immediatevicinity of the crack tip. If, however, T= 0 at all points in the body, the integrand in Eq.(4.26) reduces to the form of the original J integral. In the latter case, the path-indepen-


dent property of J is restored if w displays the property of an elastic potential (seeAppendix 4.2).

The form of Eq. (4.26) is not very convenient for numerical calculations, since it isextremely difficult to obtain adequate numerical precision from a contour integration veryclose to the crack tip. Fortunately, Eq. (4.26) can be expressed in a variety of other formsthat are more conducive to numerical analysis. The energy release rate can also be gener-alized to three dimensions. The results in Figs. 4.3 and 4.4 are obtained from finite ele-ment analysis that utilized alternate forms of Eq. (4.26). Chapter 11 discusses numericalcalculations of J for both quasistatic and dynamic loading.

4.2 CREEP CRACK GROWTH

Components that operate at high temperatures relative to the melting point of the mate-rial may fail by slow, stable extension of a macroscopic crack. Traditional approaches todesign in the creep regime apply only when creep and material damage are uniformly dis-tributed. Time-dependent fracture mechanics approaches are required when creep failure iscontrolled by a dominant crack in the structure.

Figure 4.16 illustrates the typical creep response of a material subject to constantstress. Deformation at high temperatures can be divided into four regimes: instantaneous(elastic) strain, primary creep, secondary (steady state) creep, and tertiary creep. The elas-tic strain occurs immediately upon application of the load. As discussed in the previoussection on dynamic fracture, the elastic stress-strain response of a material is not instanta-neous (i.e., it is limited by the speed of sound in the material), but it can be viewed assuch in creep problems, where the time scale is usually measured in hours. Primary creepdominates at short times after application of the load; the strain rate decreases with time,as the material strain hardens. In the secondary creep stage, the deformation reaches asteady state, where strain hardening and strain softening are balanced; the creep rate is con-stant in the secondary stage. In the tertiary stage, the creep rate accelerates, as the mate-rial approaches ultimate failure. Microscopic failure mechanisms, such as grain boundarycavitation, nucleate in this final stage of creep.

During growth of a macroscopic crack at high temperatures, all four types of creepresponse can occur simultaneously in the most general case (Fig. 4.17). The material atthe tip of growing crack is in the tertiary stage of creep, since the material is obviouslyfailing locally. The material may be elastic remote from the crack tip, and in the primaryand secondary stages of creep at moderate distances from the tip.

Most analytical treatments of creep crack growth assume limiting cases, where oneor more of these regimes are not present or are confined to a small portion of the compo-nent. If, for example, the component is predominantly elastic, and the creep zone is con-fined to a small region near the crack tip, the crack growth can be characterized by thestress intensity factor. In the other extreme, when the component deforms globally insteady state creep, elastic strains and tertiary creep can be disregarded. A parameter thatapplies to the latter case is described below, followed by a brief discussion of approachesthat consider the transition from elastic to steady state creep behavior.

226 Chapter 4

STRAIN

Failure

TIME

FIGURE 4.16 Schematic creep behavior of a material subject to a constant stress.

FIGURE 4.17 Creep zones at the tip of a crack.

4.2.1 The C* Integral

A formal fracture mechanics approach to creep crack growth was developed soon after the/integral was established as an elastic-plastic fracture parameter. Landes and Begley [45],Ohji, et al. [46], and Nikbin, et al. [47] independently proposed what became known asthe C* integral to characterize crack growth in a material undergoing steady state creep.They applied Hoffs analogy [48], which states that if there exists a nonlinear elastic body

that obeys the relationship EIJ - f((7ij) and a viscous body that is characterized by EIJ -ij), where the function of stress is the same for both, then both bodies develop identi-


cal stress distributions when the same load is applied. Hoff s analogy can be applied tosteady state creep, since the creep rate is a function only of the applied stress.

The C* integral is defined by replacing strains with strain rates, and displacementswith displacement rates in the /contour integral:

C* = f wdy - cr/7-w j-^-ds] (4.30)rV

J J d x J

where w is the stress work rate (power) density, defined as

%W = J OijdBij (4.31)

0

Hoff s analogy implies that the C* integral is path-independent, because J is path-inde-pendent. Also, if secondary creep follows a power law:

£ij = Aaj (4.32)

where A and n are material constants, then it is possible to define an HRR-type singular-ity for stresses and strain rates near the crack tip:

(4.33a)v \Âlnr J "J

andn

. ( C* "AInr

(4.33b)

where the constants In, Oy, and EIJ are identical to the corresponding parameters in theHRR relationship (Eq. (3.24)). Note that in the present case, n is a creep exponent ratherthan a strain hardening exponent.

Just as the / integral characterizes the crack tip fields in an elastic or elastic-plasticmaterial, the C* integral uniquely defines crack tip conditions in a viscous material.Thus the time-dependent crack growth rate in a viscous material should depend only onthe value of C*. Experimental studies [45-49] have shown that creep crack growth ratescorrelate very well with C*, provided steady state creep is the dominant deformationmechanism in the specimen. Figure 4.18 shows typical creep crack growth data. Notethat the crack growth rate follows a power law:


1 M0B - > (436)

The J integral can be related to the energy absorbed by a laboratory specimen, divided bythe ligament area—

7?(437)

BbQ

where 7] is a dirnensionless constant that depends on geometry. Therefore, C* is givenby

77(4.38)

For a material that creeps according to a power law (Eq. (4.32)), the displacement rate isproportional to Pn, assuming global creep in the specimen. In this case Eq. (4.38) re-duces to

C* = — -- p (4.39)IBb

The geometry factor r\ has been determined for a variety of test specimens. For example7] = 2.0 for a deeply notched bend specimen (Eqs. (3.37) and (4.6)).

4.2.2 Short Time Versus Long Time Behavior

The C* parameter only applies to crack growth in the presence of global steady statecreep. Stated another way, C* applies to long time behavior, as discussed below.

Consider a stationary crack in a material that is susceptible to creep deformation. Ifa remote load is applied to the cracked body, the material responds almost immediatelywith the corresponding elastic strain distribution. Assuming the loading is pure Mode I,

the stresses and strains exhibit a In r singularity near the crack tip and are uniquely de-fined by Kf. Large scale creep deformation does not occur immediately, however. Soon

n•'The load line displacement, A, in Eqs. (4.37) to (4.39) corresponds to the portion of the displacement due tothe presence of the crack, as discussed in Section 3.2.5. This distinction is not necessary in Eqs. (4.35) and(4.36), because the displacement component attributed to the uncracked configuration vanishes whendifferentiated with respect to a.

230 Chapter 4

after the load is applied, a small creep zone, analogous to a plastic zone, forms at thecrack tip. The crack tip conditions can be characterized by Kj as long as the creep zone isembedded within the singularity dominated zone. The creep zone grows with time, even-tually invalidating Kj as a crack tip parameter. At long times, the creep zone spreadsthroughout the entire structure.

When the crack grows with time, the behavior of the structure depends on the crackgrowth rate relative to the creep rate. In brittle materials, the crack growth rate is so fastthat it overtakes the creep zone; crack growth can be characterized by Kj because the creepzone at the tip of the growing crack remains small. At the other extreme, if the crackgrowth is sufficiently slow that the creep zone spreads throughout the structure, C* isthe appropriate characterizing parameter.

Riedel and Rice [50] analyzed the transition from short time elastic behavior to longtime viscous behavior. They assumed a simplified stress-strain rate law that neglectsprimary creep:

£ = — -t-A<7n (4.40)

for uniaxial tension. If a load is suddenly applied and then held constant, a creep zonegradually develops in an elastic singularity zone, as discussed above. Riedel and Rice ar-gued that the stresses well within the creep zone can be described by

1j.

.F1n+l „

\AInr)

where C(t) is a parameter that characterizes the amplitude of the local stress singularity inthe creep zone; C(t) varies with time and is equal to C* in the limit of long time behav-ior. If the remote load is fixed, the stresses in the creep zone relax with time, as creepstrain accumulates in the crack tip region. For small scale creep conditions, C(t) decaysas 1/t according to the following relationship:

K?(l-V2)1 (4.42)

And the approximate size of the creep zone is given by

,n ,rc(8,t) =c E

(n + l}AInEnt

271(1- 2n-l

rc(0,n) (4.43)


At 6 = 90°, rc is a maximum and ranges from 0.2 to 0.5, depending on n. As rc in-creases in size, C(t) approaches the steady state value C*. Riedel and Rice defined a char-acteristic time for the transition from short time to long time behavior:

(4.44a)

or

t. = - (4Mb)1 (n + l)C*

When significant crack growth occurs over time scales much less than ti, the behaviorcan be characterized by Kj, while C* is the appropriate parameter when significant crackgrowth requires times » tj. Based on finite element analysis, Riedel [51] suggestedthe following simple formula to interpolate between small scale creep and extensive creep(short and long time behavior, respectively):

(4.45)

Note the similarity to the transition time concept in dynamic fracture (Section 4.1.1). Inboth instances, a transition time characterizes the interaction between two competingphenomena.

The Cf parameterUnlike Kj and C*, direct experimental measurement of C(t) under transient condi-

tions is usually not possible. Consequently Saxena [52] defined an alternate parameter,Q, which was originally intended as an approximation of C(t). The advantage of Q isthat it can be measured relatively easily.

Saxena began by separating global displacement into instantaneous elastic and time-dependent creep components:

A = Ae + Ar (4.46)

The creep displacement, A{, increases with time as the creep zone grows. Also, if load is* *

fixed, Af = A. The Q parameter is defined as the creep component of the power releaserate:

r_ 1C'~~B

(4.47)

232 Chapter 4

Note the similarity between Eqs. (4.36) and (4.47).For small scale creep (ssc) conditions, Saxena defined an effective crack length,

analogous to the Irwin plastic zone correction described in Chapter 2:

asjf=a + f}rc (4.48)

where (3~ ^3 and rc is defined at 9 = 90°. The displacement due to the creep zone isgiven by

j/~»A, = A - Ae = P—prc (4.49)

da

where Cis the elastic compliance, defined in Chapter 2. Saxena showed that the smallscale creep limit for Q can be expressed as follows

/ \P&t--•• (4-50)BW

where f(a/w) is tf16 geometry correction factor for Mode I stress intensity (see Table 2.4):

/("

and/ is the first derivative of/ Equation (4.50) predicts that (Cf)ssc is proportional toKp\ thus Q does not coincide with C(t) in the limit of small scale creep (Eq. (4.42)).

Saxena proposed the following interpolation between small scale creep and extensivecreep:

where C* is determined from Eq. (4.38) using the total displacement rate. In the limit oflong time behavior, C*/Q = 1.0, but this ratio is less than unity for small scale creepand transient behavior.

Bassani, et al. [53] applied the Q parameter to experimental data with variousC*/Cf ratios and found that Q characterized crack growth rates much better than C* orKj. They state that Q, when defined by Eqs. (4.50) and (4.51), characterizes experimentaldata better than C(t), as defined by Riedel's approximation (Eq. (4.45)).

Although Cf was originally intended as an approximation of C(t), it has becomeclear that these two parameters are distinct from one another. The C(t) parameter charac-


terizes the stresses ahead of a stationary crack, while Q is related to the rate of expansionof the creep zone. The latter quantity appears to be better suited to materials that experi-ence relatively rapid creep crack growth. Both parameters approach C* in the limit ofsteady-state creep.

Primary creepThe analyses introduced so far do not consider primary creep. Referring to Fig.

4.17, which depicts the most general case, the outer ring of the creep zone is in the pri-mary stage of creep. Primary creep may have an appreciable effect on the crack growthbehavior if the size of the primary zone is significant.

Recently, researchers have begun to develop crack growth analyses that include theeffects of primary creep. One such approach [54] considers a strain hardening model forthe primary creep deformation, resulting in the following expression for total strain rate:

e = - + Al(jn+A2cjm(l+^£-P (4.51)

E

Riedel [54] introduced a new parameter, Cfr*, which is the primary creep analog to C*.The characteristic time that defines the transition from primary to secondary creep is de-fined as

Jt = - « - - (4>52)2

The stresses within the steady state creep zone are still defined by Eq. (4.41), but the in-terpolation scheme for C(t) is modified when primary creep strains are present [54]:

C(r)t J

£+1

P C* (4.53)

Equation (4.53) has been applied to experimental data in a limited number of cases. Thisrelationship appears to give a better description of experimental data than Eq. (4.45),where the primary term is omitted.

Chun-Pok and McDowell [55] have recently incorporated the effects of primary creepinto the estimation of the Q parameter.

4.3 VISCOELASTIC FRACTURE MECHANICS

Polymeric materials have seen increasing service in structural applications in recent years.Consequently, the fracture resistance of these materials has become an important consider-

234 Chapter 4

ation. Much of the fracture mechanics methodology that was developed for metals is notdirectly transferable to polymers, however, because the latter behave in a viscoelasticmanner.

Theoretical fracture mechanics analyses that incorporate viscoelastic material re-sponse are relatively new, and practical applications of viscoelastic fracture mechanics arerare, as of this writing. Most current applications to polymers utilize conventional, time-independent fracture mechanics methodology (see Chapters 6 and 8). Approaches that in-corporate time dependence should become more widespread, however, as the methodologyis developed further and is validated experimentally.

This section introduces viscoelastic fracture mechanics and outlines a number of re-cent advances in this area. The work of Schapery [56-61] is emphasized, because he hasformulated the most complete theoretical framework, and his approach is related to the /and C* integrals, which were introduced earlier in this text.

4.3,1 Linear Viscoelasticity

Viscoelasticity is perhaps the most general (and complex) type of time-dependent materialresponse. From a continuum mechanics viewpoint, viscoplastic creep in metals is actu-ally a special case of viscoelastic material behavior. While creep in metals is generallyconsidered permanent deformation, the strains can recover with time in viscoelastic mate-rials. In the case of polymers, time-dependent deformation and recovery is a direct resultof their molecular structure, as discussed in Chapter 6.

Let us introduce the subject by considering linear viscoelastic material behavior. Inthis case, linear implies that the material meets two conditions: superposition and pro-portionality. The first condition requires that stresses and strains at time t be additive.For example, consider two uniaxial strains, £/ and £2, at time t, and the correspondingstresses, G(£i) and cr(£2)- Superposition implies

+ Cfe (OJ = 0[£i (0 + £2 (f )] (4.54)

If each stress is multiplied by a constant, the proportionality condition gives

(4-55)

If a uniaxial constant stress creep test is performed on a linear viscoelastic material,such that <J= 0 for / < 0 and a= a0 for t > 0, the strain increases with time according to

e(t) = D(t)(J0 (4.56)

where D(t) is the creep compliance. The loading in this case can be represented morecompactly as <jo H(t), where H(t) is the Heaviside step function, defined as


JO fort<Q:|l for f > 0

In the case of a constant uniaxial strain, i.e., e = £o H(t), the stress is given by

(4.57)

where E(t) is the relaxation modulus. When e0 is positive, the stress relaxes with time.Figure 4.19 schematically illustrates creep at a constant stress, and stress relaxation at afixed strain.

e(t)

TIME TIME(a) Creep at a constant stress.

TIME TIME

(b) Stress relaxation at a constant strain

FIGURE 4.19 Schematic uniaxial viscoelastic deformation.

236 Chapter 4

When stress and strain both vary, the entire deformation history must be taken intoaccount. The strain at time t is obtained by summing strain increments from earliertimes. The incremental strain at time T, where 0 < r < t, that results from an incremen-tal stress daH(t - r) is given by

de( r) = D(t - i)do( T} (4.58)

Integrating this expression with respect to time t gives

S(t] = D(t- < t ) d T (4.59)o dr

where it is assumed that £ = <J = 0 at t = 0. In order to allow for a discontinuous changein stress at t = 0, the lower integration limit is assumed to be 0", an infinitesimal timebefore t = 0. Relationships such as Eq. (4.59) are called hereditary integrals because theconditions at time t depend on prior history. The corresponding hereditary integral forstress in given by the inverse of Eq. (4.59):

<j(r) = f E(t - T) — — dl (4.60)0 dT

By performing a Laplace transform on Eqs. (4.59) and (4.60), it can be shown that thecreep compliance and the relaxation modulus are related as follows:

(4.61)di

For deformation in three dimensions, the generalized hereditary integral for strain isgiven by

(4.62)0

but symmetry considerations reduce the number of independent creep compliance con-stants. In the case of a linear viscoelastic isotropic material, there are two independentconstants, and the mechanical behavior can be described by E(t) or D(t), which areuniquely related, plus vc(t), the Poisson's ratio for creep.

Following an approach developed by Schapery [59], it is possible to define a pseudoelastic strain, which for uniaxial conditions is given by


(4.63)R

where ER is a reference modulus. Substituting Eq. (4.63) into Eq. (4.59) gives

(4.64)

The pseudo strains in three dimensions are related to the stress tensor through Hooke'slaw, assuming isotropic material behavior:

ej = ER-i [(1 + V)C7y - VOfaSij] (4.65)

where 8ij is the Kronecker delta, and the standard convention of summation on repeatedindices is followed. If vc = v = constant with time, it can be shown that the three-dimen-sional generalization of Eq. (4.64) is given by

' deg(r)£0(0 = ER\D(t - t)—j - dT (4.66)

0

and the inverse of Eq. (4.66) is as follows.

(4.67)0

The advantage of introducing pseudo strains is that they can be related to stressesthrough Hooke's law. Thus if a linear elastic solution is known for a particular geome-try, it is possible to determine the corresponding linear viscoelastic solution through ahereditary integral. Given two identical configurations, one made from a linear elasticmaterial and the other made from a linear viscoelastic material, the stresses in both bodiesmust be identical, and the strains are related through Eqs. (4.66) or (4.67), provided bothconfigurations are subject to the same applied loads. This is a special case of a corre-spondence principle, which is discussed in more detail below; note the similarity toHoff s analogy for elastic and viscous materials (Section 4.2).

23S Chapter 4

4.3.2 The Viscoelastic J Integral

Constitutive EquationsSchapery [59] developed a generalized J integral that is applicable to a wide range of vis-coelastic materials. He began by assuming a nonlinear viscoelastic constitutive equation

in the form of a hereditary integral:

£-(0 = ER D(t- T,t)~^—dr (4.68)

o 3r

where the lower integration limit is taken as 0". The pseudo elastic strain, ee. ,, is related

to stress through a linear or nonlinear elastic constitutive law. The similarity betweenEqs. (4.66) and (4.68) is obvious, but the latter relationship also applies to certain typesof nonlinear viscoelastic behavior. The creep compliance, D(t), has a somewhat differentinterpretation for the nonlinear case.

The pseudo strain tensor and reference modulus in Eq. (4.68) are analogous to thelinear case. In the previous section, these quantities were introduced to relate a linear vis-coelastic problem to a reference elastic problem. This idea is generalized in the presentcase, where the nonlinear viscoelastic behavior is related to a reference nonlinear elasticproblem through a correspondence principle, as discussed below.

The inverse of Eq. (4.68) is given by

(4.69)

Since hereditary integrals of the form of Eqs. (4.68) and (4.69) are used extensively in theremainder of this discussion, it is convenient to introduce an abbreviated notation:

(4.70a)

T,t-T (4.70b)0 dr

where /is a Action of time. In each case, it is assumed that integration begins at 0"Thus Eqs. (4.68) and (4.69) become, respectively:

and e


Correspondence PrincipleConsider two bodies with the same instantaneous geometry, where one material is

elastic and the other is viscoelastic and is described by Eq. (4.68). Assume that at time t,a surface traction TI = GIJ nj is applied to both configurations along the outer boundaries.If the stresses and strains in the elastic body are Oif and £y£, respectively, while the cor-responding quantities in the viscoelastic body are cr/y and £/y, the stresses, strains, anddisplacements are related as follows [59]:

Equation (4.71) defines a correspondence principle, introduced by Schapery [59], which al-lows the solution to a viscoelastic problem to be inferred from a reference elastic solu-tion. This correspondence principle stems from the fact that the stresses in both bodiesmust satisfy equilibrium, and the strains must satisfy compatibility requirements in bothcases. Also, the stresses are equal on the boundaries by definition:

rp _ 6

Schapery [59] gives a rigorous proof of Eq. (4.71) for viscoelastic materials that satisfyEq. (4.68).

Applications of correspondence principles in viscoelasticity, where the viscoelasticsolution is related to a corresponding elastic solution, usually involve performing aLaplace transform on a hereditary integral in the form of Eq. (4.62), which contains actualstresses and strains. The introduction of pseudo quantities makes the connection betweenviscoelastic and elastic solutions more straightforward.

Generalized J integralThe correspondence principle in Eq. (4.71) makes it possible to define a generalized

time-dependent J integral by forming an analogy with the nonlinear elastic case:

( 3 e \Jv = f wedy- aijni-^-ds (4.72)

A a* Jwhere we is the pseudo strain energy density:

'iidefi (4.73)

The stresses in Eq. (4.72) are the actual values in the body, but the strains and displace-ments are pseudo elastic values. The actual strains and displacements are given by Eq.

240 Chapter 4

(4.71). Conversely, if £/y and u{ are known, Jv can be determined by computing pseudovalues, which are inserted into Eq. (4.73). The pseudo strains and displacements are givenby

eeij-{Edeij] and ue

i~{Edui] (4.74}

Consider a simple example, where the material exhibits steady state creep at t > to. Thehereditary integrals for strain and displacement reduce to

efj = £ij and uf = «/

By inserting the above results into Eq. (4.73), we see that Jv = C*. Thus C* is a specialcase of Jv. The latter parameter is capable of taking account of a wide range of time-de-pendent material behavior, and includes viscous creep as a special case.

Near the tip of the crack, the stresses and pseudo strains are characterized by Jv

through an HRR-type relationship in the form of Eq. (4.33). The viscoelastic / can alsobe determined through a pseudo energy release rate:

/ g

J- f Pdlf (4.75)

where Ae is the pseudo displacement in the loading direction, which is related to the ac-tual displacement by

A = {DdAe} (4.76)

Finally, for Mode I loading of a linear viscoelastic material in plane strain, Jv is related tothe stress intensity factor as follows:

, JL\. I \i — V IJv = -^ (4.77)

The stress intensity factor is related to specimen geometry, applied loads, and crack di-mensions through the standard equations outlined in Chapter 2.

Crack initiation and growthWhen characterizing crack initiation and growth, it is useful to relate Jv to physical

parameters such as CTOD and fracture work, which can be used as local failure criteria.Schapery [59] derived simplified relationships between these parameters by assuming astrip yield-type failure zone ahead of the crack tip, where a closure stress <Jm acts over p,


as illustrated in Fig. 4.20. While material in the failure zone may be severely damagedand contain voids and other discontinuities, it is assumed that the surrounding materialcan be treated as a continuum. If csm does not vary with x, applying Eq. (3.44) gives

r _Jv ~ (4.78)

where Se is the pseudo crack tip opening displacement, which is related to the actualCTOD through a hereditary integral of the form of Eq. (4.77). Thus the CTOD is givenby

(4-79)

Although <7m was assumed to be independent of x at time t, Eq. (4.79) permits <Jm tovary with time. The CTOD can be utilized as a local failure criterion: if crack initiationoccurs at 5j, the Jv at initiation can be inferred from Eq. (4.79). If 8} is assumed to beconstant, the critical Jv would, in general, depend on the strain rate. A more general ver-sion of Eq. (4.79) can be derived by allowing <Jm to vary with x.

An alternative local failure criterion is the fracture work, wf. Equating the workinput to the crack tip to the energy required to advance the crack tip by da results in thefollowing energy balance at initiation:

(4.80)

Failure Zose

-H

(a) Failure zone (b) Strip yield model

FIGURE 4.20 Failure zone at the crack tip in a viscoelastic material. This zone is modeled by surfacetractions within 0 <x < p.

242 Chapter 4

assuming unit thickness and Mode I loading. This energy balance can also be written interms of a time integral:

0

Inserting Eq. (4.79) into Eq. (4.81) gives

d{Dd(Jv I am)} ,- - - —J

.= 2wf

0

If om is independent of time, it cancels out of Eq. (4.82), which then simplifies to

TI a/ER J D(tt - T, % ) — x dr = 2 W/ (4.83)

For an elastic material, D = £/? , and Jv = 2 wy. If the failure zone is viscoelastic andthe surrounding continuum is elastic, Jv may vary with time. If the surrounding contin-uum is viscous, D = (tv ER)~* (t - r), where tv is a constant with units of time.Inserting this latter result into Eq. (4.83) and integrating by parts gives

(4.84)

4.3.3 Transition from Linear to Nonlinear Behavior

Typical polymers are linear viscoelastic at low stresses and nonlinear at high stresses. Aspecimen that contains a crack may have a zone of nonlinearity at the crack tip, analogousto a plastic zone, that is surrounded by linear viscoelastic material. The approach de-scribed in the previous section applies only when one type of behavior (linear or nonlin-ear) dominates.

Schapery [61] has recently modified the Jv concept to cover the transition fromsmall stress to large stress behavior. He introduced a modified constitutive equation,where strain is given by the sum of two hereditary integrals: one corresponding to linearviscoelastic strains and the other describing nonlinear strains. For the latter term, he as-sumed power-law viscoelasticity. For the case of uniaxial constant tensile stress, (J0, thecreep strain in this modified model is given by


\n

(4.85)

where D and DL are the nonlinear and linear creep compliance, respectively, and <Jrefi& areference stress.

At low stresses and short times, the second term in Eq. (4.85) dominates, while thenonlinear term dominates at high stresses or long times. In the case of a viscoelasticbody with a stationary crack at a fixed load, the nonlinear zone is initially small but nor-mally increases with time, until the behavior is predominantly nonlinear. Thus there is adirect analogy between the present case and the transition from elastic to viscous behaviordescribed in Section 4.2.

Close to the crack tip, but outside of the failure zone, the stresses are related to apseudo strain through a power law:

£ c-—

f \«Or

a(4.86)

ref

In the region dominated by Eq. (4.86), the stresses are characterized by Jv, regardless ofwhether the global behavior is linear or nonlinear:

1

= °refn+1

(4.87)

y

If the global behavior is linear, there is a second singularity further away from the cracktip:

(4.88)

Let us define a pseudo strain tensor that, when inserted into the path-independent in-tegral of Eq. (4.72), yields a value /£. Also suppose that this pseudo strain tensor is re-lated to the stress tensor by means of linear and power law pseudo complementary strainenergy density functions (wci and wcn, respectively):

•(fwcn+wci) (4.89)

244 Chapter 4

where fit) is an as yet unspecified aging function, and the complementary strain energydensity is defined by

For uniaxial deformation, Eq. (4.89) reduces to

^=/l /T/*/

+<J

£*(4.90)

Comparing Eqs (4.85) and (4.90), it can be seen that

/f = ffor constant stress creep.

The latter relationship for pseudo strain agrees with the conventional definition in thelimit of linear behavior.

Let us now consider the case where the inner and outer singularities, Eqs. (4.87) and(4.88), exist simultaneously. For the outer singularity, second term in Eq. (4.90) domi-nates, the stresses are given by Eq. (4.88), and JL is related to Kj as follows:

Closer to the crack tip, the stresses are characterized by Jv through Eq. (4.87), but Jj_, isnot necessarily equal to Jv, because/appears in the first term of the modified constitutiverelationship (Eq. (4.90)), but not in Eq. (4.86). These two definitions of J coincide if

ffrefm Eq. (4.90) is replaced with (Jreff ^n- Thus, the near-tip singularity in terms ofJL is given by

(Tft = (7refJL

171+1

(4.92)

Therefore,

(4.93)


Schapery showed that/= 1 in the limit of purely linear behavior; thus JL is the limit-ing value of Jv when the nonlinear zone is negligible. The function/is indicative of theextent of nonlinearity. In most cases, /increases with time, until Jv reaches Jn, the lim-iting value when the specimen is dominated by nonlinear viscoelasticity. Schapery alsoconfirmed that

,f = — — (4.94)

for small scale nonlinearity. Equations (4.93) and (4.94) provide a reasonable descriptionof the transition to nonlinear behavior. Schapery defined a transition time by setting Jv =Jn in Eq. (4.93):

Jn = — — (4.95a)

or

(4.95b)

For the special case of linear behavior followed by viscous creep, Eq. (4.95b) becomes

t = £ (4.96)* (» + l)C*

which is identical to the transition time defined by Riedel and Rice [50].

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53. Bassani, J.L., Hawk, D.E., and Saxena, A., "Evaluation of the Q Parameter forCharacterizing Creep Crack Growth Rate in the Transient Regime." ASTM STP 995,Vol. I, American Society for Testing and Materials, Philadelphia, 1990, pp. 112-130.

54. Riedel, H., "Creep Deformation at Crack Tips in Elastic-Viscoplastic Solids." Journalof the Mechanics and Physics of Solids, Vol 29, 1981, pp. 35-49.

55. Chun-Pok, L. and McDowell, D.L., "Inclusion of Primary Creep in the Estimation of theCt Parameter." International Journal of Fracture, Vol. 46, 1990, pp. 81-104.

56. Schapery, R.A. "A Theory of Crack Initiation and Growth in Viscoelastic Media—I.Theoretical Development." International Journal of Fracture, Vol 11, 1975, pp. 141-159.

57. Schapery, R.A. "A Theory of Crack Initiation and Growth in Viscoelastic Media—II.Approximate Methods of Analysis." International Journal of Fracture, Vol 11, 1975,pp. 369-388.

58. Schapery, R.A. "A Theory of Crack Initiation and Growth in Viscoelastic Media—III.Analysis of Continuous Growth." International Journal of Fracture, Vol 11, 1975, pp.549-562.

59. Schapery, R.A., "Correspondence Principles and a Generalized J Integral for LargeDeformation and Fracture Analysis of Viscoelastic Media." International Journal ofFracture, Vol. 25, 1984, pp. 195-223.

60. Schapery, R.A., "Time-Dependent Fracture: Continuum Aspects of Crack Growth."Encyclopedia of Materials Science and Engineering, Pergamon Press, Oxford, 1986, pp.5043-5054.

61. Schapery, R.A., "On Some Path Independent Integrals and Their Use in Fracture ofNonlinear Viscoelastic Media." International Journal of Fracture, Vol. 42, 1990, pp.189-207.

62. Irwin, G.R., "Constant Speed Semi-Infinite Tensile Crack Opened by a Line Force."Lehigh University Memorandum, 1967.

APPENDIX 4: DYNAMIC FRACTUREANALYSIS

(Selected Results)

A4.1 ELASTODYNAMIC CRACK TIP FIELDS

Rice [31], Sih [35], and Irwin [62] each derived expressions for the stresses ahead of a

crack propagating at a constant speed. They found that the moving crack retained the 7/v rsingularity, but that the angular dependence of the stresses, strains and displacements de-pends on crack speed. Freund and Clifton [32] and Nilsson [33] later showed that the so-lution for a constant speed crack was valid in general; the near-tip quantities depend onlyon instantaneous crack speed. The following derivation presents the more general case,where the crack speed is allowed to vary.

For dynamic problems, the equations of equilibrium are replaced by the equations ofmotion, which, in the absence of body forces, are given by

where xj denotes the orthogonal coordinates and each dot indicates a time derivative. Forquasistatic problems, the term on the right side of Eq. (A4.1) vanishes. For a linear elas-tic material, it is possible to write the equations of motion in terms of displacements andelastic constants by invoking the strain-displacement and stress-strain relationships:

(A4.2)

where n and A are the Lame' constants; (J, is the shear modulus and

l-2v

Consider rapid crack propagation in a body subject to plane strain loading. Let usdefine a fixed coordinate axis, X-Y, with an origin on the crack plane at a(t) = 0, as illus-trated in Fig. A4.1. It is convenient at this point to introduce two displacement poten-tials, defined by

251

252 Appendix 4

FIGURE A4.1 Definition of coordinate axesfor a rapidly propagating crack. The X-Yaxes are fixed in space and the x-y axes areattached to the crack tip.

ux -OA 01

Substituting Eq. (A4.3) into Eq. (A4.2) leads to

32y/i 32i^ __ i ..

and

37(A4.3)

(A4.4)

ax2 ay2 cf

since the wave speeds are given by

for plane strain. Thus ^7 and y2 are the longitudinal and shear wave potentials respec-tively. The stresses can be written in terms of \f/l and y/2 by invoking Eqs. (A2.1) and(A2.2):

37'

Dynamic Fracture Analysis 253

(A4.5)dx ar

Let us now introduce a moving coordinate system, jc-y, attached to the crack tip,where x = X - a(t) and y = Y. The rate of change of each wave potential can be written as

dti =1,2) (A4.6)

where V (= - dx/dt) is the crack speed. Differentiating Eq. (A4.6) with respect to timegives

T17 • ... * * / A/I '7^^ -1

According to Eq. (A4.5) the first term on the right-hand side of Eq. (A4.7) is propor-tional to the stress tensor. This term should dominate close to the crack tip, assumingthere is a stress singularity. Substituting the first term of Eq. (A4.7) into Eq. (A4.4)leads to

and (A4.8)

R2^2¥2 , aV2_ f tL/9 O ' o — "2 a*2 a/

where

Note that the governing equations depend only on instantaneous crack speed; the term thatcontains crack acceleration in Eq. (A4.7) is negligible near the crack tip.

254 Appendix 4

If we scale y by defining new coordinates, yj - fi] y and y>2 = @2 >'. Eq. (A4.8) be-comes the Laplace equation. Freund and Clifton [32] applied a complex variable methodto solve Eq. (A4.8). The general solutions to the wave potentials are as follows:

and (A4.9)

where F and G are as yet unspecified complex functions, z] = x + iyj, and Z2 = x + iy>2-The boundary conditions are the same as for a stationary crack: Gyy = %y = 0 on

the crack surfaces. Freund and Clifton showed that these boundary conditions can be ex-pressed in terms of second derivatives for F and G at y - 0 and x < 0:

(A4.10)

where the subscripts + and - correspond to the upper and lower crack surfaces, respec-tively. The following functions satisfy the boundary conditions and lead to integrablestrain energy density and finite displacement at the crack tip:

(A4.ll)

where C is a constant. Making the substitution zi = rj e^J and z2 =?2 e*®2 leads to thefollowing expressions for the Mode I crack tip stress fields:


Kj(t) 2ft (1

where

sin (A4.12)

Equation (A4.12) reduces to the quasistatic relationship (Table 2.1) when V= 0.Craggs [25] and Freund [10] obtained the following relationship between Kj(t) and

energy release rate for crack propagation at a constant speed:

(A4.13)

for plane strain, where

A(V) =

It can be shown that

and Eq. (A4.13) reduces to the quasistatic result. Equation (A4.13) can be derived by sub-stituting the dynamic crack tip solution (Eq. (A4.12) and the corresponding relationshipsfor strain and displacement) into the generalized contour integral given by Eq. (4.26).

The derivation that led to Eq. (A4.12) implies that Eq. (A4.13) is a general relation-ship that applies to accelerating cracks as well as constant speed cracks.

A4.2 DERIVATION OF THE GENERALIZED ENERGY RELEASERATE

Equation (4.26) will now be derived. The approach closely follows that of Moran andShih [11], who applied a general balance law to derive a variety of contour integrals, in-cluding the energy release rate. Other authors [8-10] have derived equivalent expressionsusing slightly different approaches.

Beginning with the equation of motion, Eq. (A4.1), taking an inner product of both«

sides with displacement rate, MJ, and rearranging gives

256 Appendix 4

(A4.14)

where !Tand w are the kinetic energy and stress work densities, respectively, as defined inEqs. (4.27) to (4.29). Equation (A4.14) is a general balance law that applies to all mate-rial behavior. Integrating this relationship over an arbitrary volume, and applying the di-vergence and transport theorems gives

j ji id_dt

(A4.15)

where fis volume, mj is the outward normal to the surface dlS, and V{ is the instanta-neous velocity of dtf

Consider now the special case of a crack in a two-dimensional body, where the crackis propagating along the x axis and the origin is attached to the crack tip. (Fig. A4.2).Let us define a contour, Co, fixed in space, that contains the propagating crack andbounds the area ^L The crack tip is surrounded by a small contour, F, that is fixed in sizeand moves with the crack. The balance law in Eq. (A4.15) becomes

Co

—dt

(A4.16)

FIGURE A4.2 Conventions for the en-ergy balance for a propagating crack.The outer contour, C& is fixed in space,and the inner contour, F, and the x-yaxes are attached to the moving cracktip.

Co


where V is the crack speed. The integral on the left side of Eq. (A4.16) is the rate atwhich energy is input into the body. The first term on the right side of this relationshipis the rate of increase in internal energy in the body. Consequently, the second integralon the right side of Eq. (A4.16) corresponds to the rate at which energy is lost from thebody due to flux through F. By defining nj = -my on F, we obtain the following expres-sion for the energy flux into F:

= J[(w + T)VSij + ffjiu^njdr (A4.17)F

In the limit of a vanishingly small contour, the flux is independent of the shape of F.Thus the energy flux to the crack tip is given by

f= limr_»0 J[(w + T)V8ij + (JjiU^njdT (A4.18)r

In an increment of time dt, the crack extends by da = V dt and the energy expended is ydt.Thus the energy release rate is given by

ffJ = — (A4.19)

Substituting Eq. (A4.18) into Eq. (A4.19) will yield a generalized expression for the J in-tegral. First, however, we must express displacement rate in terms of crack speed. Byanalogy with Eq. (A4.6), displacement rate can be written as

(A4.20)

Under steady state conditions, the second term in Eq. (A4.20) vanishes; the displacementat a fixed distance from the propagating crack tip remains constant. Close to the cracktip, displacement changes rapidly with position (at a fixed time) and the first term in Eq.(A4.20) dominates in all cases. Thus the J integral is given by

J = limr_»0 J

(A4.21)

258 Appendix 4

Equation (A4.21) applies to all types of material response (e.g. elastic, plastic, viscoplas-tic, and viscoelastic behavior), because it was derived from a generalized energy balance.3

In the special case of an elastic material (linear or nonlinear), w is the strain energy den-sity, which displays the properties of an elastic potential:

(A4.22)

Recall from Appendix 3 that Eq. (A4.22) is necessary to demonstrate path independence ofJ in the quasistatic case. In general, Eq. (A4.21) is not path independent except in a localregion near the crack tip. For an elastic material, however, / is path independent in thedynamic case when the crack propagation is steady state (3ui/dt = 0) [8].

Although Eq. (A4.22) is, in principle, applicable to all types of material response,special care must be taken when J is evaluated for a growing crack. Figure A4.3(a) illus-trates a growing crack under small scale yielding conditions. A small plastic zone (orprocess zone) is embedded within an elastic singularity zone. The plastic zone leaves be-hind a wake as it sweeps through the material. Unrecoverable work is performed on mate-rial inside the plastic wake, as Fig. A4.3(b) illustrates. The work necessary to form theplastic wake comes from the energy flux into the contour F. In an ideally elastic body,the energy flux is released from the body through the crack tip, but in an elastic-plasticmaterial, the majority of this energy is dissipated in the wake.

Recall the modified Griffith model (Section 2.3.2), where the work required to in-crease the crack area a unit amount is equal to 2(ys + Yp), where ys is the surface energyand Yp is the plastic work. The latter term corresponds to the energy dissipated in theplastic wake (Fig. 2.6(b)).

(a) Growing crack (b) Schematic stress-strain curve for material inthe plastic wake

FIGURE A4.3 Crack growth in small scale yielding. The plastic wake, which forms behind the growingcrack, dissipates energy

Since the divergence and transport theorems were invoked, there is an inherent assumption that the materialbehaves as a continuum with smoothly varying displacement fields.


The energy release rate computed from Eq. (A4.21) must therefore be interpreted asthe energy flow to the plastic zone and plastic wake, rather than to the crack tip. That is,F cannot shrink to zero; rather, the contour must have a small, but finite radius. The /integral is path-independent as long as F is defined within the elastic singularity zone, butJ becomes path-dependent when the contour is taken inside the plastic zone. In the limitas F shrinks to the crack tip, the computed energy release rate would approach zero (in acontinuum analysis), since the calculation would exclude the work dissipated by the plas-tic wake. The actual energy flow to the crack tip is not zero, since a portion of the en-ergy is required to break bonds at the tip. In all but the most brittle materials, however,the bond energy (ys) is a small fraction of the total fracture energy.

As long as the plastic zone or process zone is embedded within an elastic singular-ity, the energy release rate can be defined unambiguously for a growing crack. In largescale yielding conditions, however, / is path dependent. Consequently, an unambiguousdefinition of energy release rate does not exist for a crack growing in an elastic-plastic orfully plastic body. Recall from Chapter 3 that there are several definitions of / for grow-ing cracks. The so-called deformation J, which is based on a pseudo energy release rateconcept, is the most common methodology. The deformation J is not, in general, equalto the J integral inferred from a contour integration.

PART III: MATERIAL BEHAVIOR

Chapters 5 and 6 give an overview of the micromechanisms of fracture in various materialsystems. This subject is of obvious importance to materials scientists, because an under-standing of microstructural events that lead to fracture is essential to the development ofmaterials with optimum toughness. Those who approach fracture from a solid mechanicsviewpoint, however, often sidestep microstructural issues and consider only continuummodels.

In certain cases, classical fracture mechanics provides some justification for disre-garding microscopic failure mechanisms. Just as it is not necessary to understand disloca-tion theory to apply tensile data to design, it may not be necessary to consider the micro-scopic details of fracture when applying fracture mechanics on a global scale. When asingle parameter (i.e., K, J, or CTOD) uniquely characterizes crack tip conditions, a criti-cal value of this parameter is a material constant that is transferable from a test specimento a structure made from the same material (see Sections 2.10 and 3.5). A laboratoryspecimen and a flawed structure experience identical crack tip conditions at failure whenthe single parameter assumption is valid, and it is not necessary to delve into the detailsof microscopic failure to characterize global fracture.

The situation becomes considerably more complicated when the single parameter as-sumption ceases to be valid. A fracture toughness test on a small scale laboratory speci-men is no longer a reliable indicator of how a large structure will behave. The fracturetoughness of the structure and test specimen are likely to be different, and the two config-urations may even fail by different mechanisms. A number of researchers are currently at-tempting to develop alternatives to single parameter fracture mechanics (see Section 3.6).Such approaches cannot succeed with continuum theory alone, but must also consider mi-croscopic fracture mechanisms. Thus the next two chapters should be of equal value tomaterials scientists and solid mechanicians.

263

5. FRACTURE MECHANISMS INMETALS

Figure 5.1 schematically illustrates three of the most common fracture mechanisms inmetals and alloys. (A fourth mechanism, fatigue, is discussed in Chapter 10.) Ductilematerials (Fig. 5.1 (a)) usually fail as the result of nucleation, growth and coalescence ofmicroscopic voids that initiate at inclusions and second phase particles. Cleavage fracture(Fig. 5.1(b)) involves separation along specific crystallographic planes. Note that thefracture path is transgranular. Although cleavage is often called brittle fracture, it can bepreceded by large scale plasticity and ductile crack growth. Intergranular fracture (Fig.5.1(c)), as its name implies, occurs when the grain boundaries are the preferred fracturepath in the material.

5.1 DUCTILE FRACTURE

Figure 5.2 schematically illustrates the uniaxial tensile behavior in a ductile metal. Thematerial eventually reaches an instability point, where strain hardening cannot keep pacewith loss in cross sectional area, and a necked region forms beyond the maximum load.In very high purity materials, the tensile specimen may neck down to a sharp point, re-sulting in extremely large local plastic strains and nearly 100% reduction in area.Materials that contain impurities, however, fail at much lower strains. Micro voids nucle-ate at inclusions and second phase particles; the voids grow together to form a macro-scopic flaw, which leads to fracture.

The commonly observed stages in ductile fracture are [1-5]:

(1) Formation of a free surface at an inclusion or second phase particle by eitherinterface decohesion or particle cracking.

(2) Growth of the void around the particle, by means of plastic strain and hydro-static stress.

(3) Coalescence of the growing void with adjacent voids.

In materials where the second phase particles and inclusions are well bonded to the matrix,void nucleation is often the critical step; fracture occurs soon after the voids form. Whenvoid nucleation occurs with little difficulty, the fracture properties are controlled by thegrowth and coalescence of voids; the growing voids reach a critical size, relative to theirspacing, and a local plastic instability develops between voids, resulting in failure.

265

266 ChapterS

1 J, 1(a) Ductile fracture.

,t t , t

t I '*(b) Cleavage

FIGURE 5.1 Three micromechanisms of fracturein metals.

(c) Intergranular fracture.

ENGINEERINGSTRESS

EngineeringPure Material

Materia Material

ENGINEERING STRAIN

FIGURE 5.2 Uniaxial tensile deformation of ductile materials.

Fracture Mechanisms in Metals 267

5.1.1 Void Nucleation

A void forms around a second phase particle or inclusion when sufficient stress is appliedto break the interfacial bonds between the particle and the matrix. A number of modelsfor estimating void nucleation stress have been published, some of which are based oncontinuum theory [6,7] while others incorporate dislocation-particle interactions [8,9].The latter models are required for particles < 1 (am in diameter.

The most widely used continuum model for void nucleation is due to Argon, et al.[6]. They argued that the interfacial stress at a cylindrical particle is approximately equalto the sum of the mean (hydrostatic) stress and the effective (von Mises) stress. The de-cohesion stress is defined as a critical combination of these two stresses:

<j = & + a (5.1)

where ae is the effective stress, given by

-<^2)2+( (Tl~<T3)2+(cr3 ~<J2)2 | / 2 (5-2)

<3m is the mean stress, defined as

and 07, O"2, and <7j are the principal normal stresses. According to the Argon, et almodel, the nucleation strain decreases as the hydrostatic stress increases. That is, voidnucleation occurs more readily in a triaxial tensile stress field, a result that is consistentwith experimental observations.

The Beremin research group in France [7] applied the Argon et al. criterion to exper-imental data for a carbon manganese steel, but found that the following semi-empirical re-lationship gave better predictions of void nucleation at MnS inclusions that were elon-gated in the rolling direction:

ac = <TOT + C(ae - cjy5) (5.4)

where ays is me yield strength and C is a fitting parameter that is approximately 1 .6 forlongitudinal loading and 0.6 for loading transverse to the rolling direction.

Goods and Brown [9] have developed a dislocation model for void nucleation atsubmicron particles. They estimated that dislocations near the particle elevate the stressat the interface by the following amount:

268 Chapter5

(5.5)

where or is a constant that ranges from 0.14 to 0.33, \JL is the shear modulus, e/ is themaximum remote normal strain, b is the magnitude of the Burger's vector, and r is theparticle radius. The total maximum interface stress is equal to the maximum principalstress plus Aa^. Void nucleation occurs when the sum of these stresses reaches a criticalvalue:

<JC = A<Jd + <Jj (5.6)

An alternative but equivalent expression can be obtained by separating 07 into deviatoricand hydrostatic components:

am (5.7)

where Sj is the maximum deviatoric stress.The Goods and Brown dislocation model indicates that the local stress concentration

increases with decreasing particle size; void nucleation is more difficult with larger parti-cles. The continuum models (Eqs. (5.1) and (5.4)), which apply to particles with r> I\im, imply that ac is independent of particle size.

Experimental observations usually differ from both continuum and dislocation mod-els, in that void nucleation tends to occur more readily at large particles [10]. Recall,however, that these models only considered nucleation by particle-matrix debonding.Voids can also be nucleated when particles crack. Larger particles are more likely to crackin the presence of plastic strain, because they are more likely to contain small defectswhich can act like Griffith cracks (see Section 5.2). In addition, large nonmetallic inclu-sions, such as oxides and sulfides, are often damaged during fabrication; some of theseparticles may be cracked or debonded prior to plastic deformation, making void nucleationrelatively easy. Further research is obviously needed to develop void nucleation modelsthat are more in line with experiment.

5.1.2 Void Growth and Coalescence

Once voids form, further plastic strain and hydrostatic stress cause the voids to grow andeventually coalesce. Figures 5.3 and 5.4 are scanning electron microscope (SEM) frac-tographs which show dimpled fracture surfaces that are typical of microvoid coalescence.Figure 5.4 shows an inclusion that nucleated a void.

Figure 5.5 schematically illustrates the growth and coalescence of microvoids. Ifthe initial volume fraction of voids is low (< 10%), each void can be assumed to grow in-dependently; upon further growth, neighboring voids interact. Plastic strain is concen-trated along a sheet of voids, and local necking instabilities develop. The orientation ofthe fracture path depends on the stress state [11].


FIGURE 53 Scanning electron microscope (SEM) fractograph which shows ductile fracture in a lowcarbon steel. (Photograph provided by Mr. Sun Yongqi.)

FIGURE 5.4 High magnification fractograph of the steel ductile fracture surface. Note the sphericalinclusion which nucleated a microvoid. (Photograph provided by Mr. Sun Yongqi.)

270 Chapters

t t t t t0

0g

Q

(a) Inclusions in a ductile matrix.

Py *iitpiri,

P

(b) Void nucleation.

t t t

on

(c) Void growth.

I I t(d) Strain localization between voids.

(e) Necking between voids. (f) Void coalescence and fracture.

FIGURE 5.5 Void nucleation, growth, and coalescence in ductile metals.


Many materials contain a bimodal or trimodal distribution of particles. For exam-ple, a precipitation-hardened aluminum alloy may contain relatively large intermetallicparticles, together with a fine dispersion of submicron second phase precipitates. Thesealloys also contain micron-size dispersoid particles for grain refinement. Voids formmuch more readily in the inclusions, but the smaller particles can contribute in certaincases. Bimodal particle distributions can lead to so-called "shear" fracture surfaces, as de-scribed below.

Figure 5.6 illustrates the formation of the "cup and cone" fracture surface that iscommonly observed in uniaxial tensile tests. The neck produces a triaxial stress state inthe center of the specimen, which promotes void nucleation and growth in the larger par-ticles. Upon further strain, the voids coalesce, resulting in a penny-shaped flaw. Theouter ring of the specimen contains relatively few voids, because the hydrostatic stress islower than in the center. The penny-shaped flaw produces deformation bands at 45° fromthe tensile axis. This concentration of strain provides sufficient plasticity to nucleatevoids in the smaller more numerous particles. Since the small particles are closelyspaced, an instability occurs soon after these smaller voids form, resulting in total frac-ture of the specimen and the cup and cone appearance of the matching surfaces. The cen-tral region of the fracture surface has a fibrous appearance at low magnifications, but theouter region is relatively smooth. Because the latter surface is oriented 45° from the ten-sile axis and there is little evidence (at low magnifications) of microvoid coalescence,many refer to this type of surface as shear fracture. The 45° angle between the fractureplane and the applied stress results in a combined Mode I/Mode n loading.

Figure 5.7 is a photograph of the cross section of a fractured tensile specimen; notethe high concentration of microvoids in the center of the necked region, compared withthe edges of the necked region.

Figure 5.8 shows SEM fractographs of a cup and cone fracture surface. The centralportion of the specimen exhibits a typical dimpled appearance, but the outer region ap-pears to be relatively smooth, particularly at low magnification (Fig. 5.8(a)). At some-what higher magnification (Fig. 5.8(b)), a few widely spaced voids are evident in the outerregion. Figure 5.9 shows a representative fractograph at higher magnification of the"shear" surface. Note the dimpled appearance, that is characteristic of microvoid coales-cence. The average void size and spacing, however, are much smaller than in the centralregion of the specimen.

There are a number of mathematical models for void growth and coalescence. Thetwo most widely referenced models were published by Rice and Tracey [12] and Gurson[13]. The latter approach was actually based on the work of Berg [14], but it is com-monly known as the Gurson model. Both the Gurson and Rice and Tracey models havebeen modified in more recent investigations [15,16].

Rice and Tracey considered a single void in an infinite solid, as illustrated in Fig.5.10. The void is subject to remote normal stresses 07, 03, CTJ, and remote normal

• » *strain rates £/, £2, £3- The initial void is assumed to be spherical, but it becomesellipsoidal as it deforms. Rice and Tracey analyzed both rigid plastic material behaviorand linear strain hardening. They showed that the rate of change of radius in eachprincipal direction has the form:

272 Chapter 5

4 4 4 4

(a) Void growth in a triaxial stress state.(b) Crack and deformation band

formation.

J I I(c) Nucleation at smaller particles alone the de- ,,. „formation bands. w Cup and cone fracture.

FIGURE 5.6 Formation of the cup and cone fracture surface in uniaxial tension.

' = (1 + (i 7 = 1, 2, 3) (5-8)

where £> and G are constants that depend on stress state and strain hardening, and Ro is theradius of the initial spherical void. The standard notation, where repeated indices implies

summation, is followed here. Invoking the incompressibility condition (EJ + £2 +*£3 =0) reduces the number of independent principal strain rates to two. Rice and Tracey, • • .

cnose to express €2 and ej in terms of £j and a second parameter:


FIGURE 5.7 Metallographic cross section (unetched) of a ruptured austenitic stainless steel tensilespecimen. The dark areas in the necked region are microvoids. (Photograph provided by P.T.Purtscher.)

where

_ -20 .~3 + 0 l

0-3 .mHtm T f*— ci

3 + 0

3e2

(5-9)

0=--

274 Chapter5

(b)

FIGURE 5.8 Cup and cone fracture in an austenitic stainless steel [17]. (Photographs provided by P.T.Purtscher.)


FIGURE 5.9 High magnification fractograph of the "shear" region of a cup and cone fracture surface inaustenitic stainless steel [17]. (Photograph provided by P.T. Purtscher.)

FIGURE 5.10 Spherical void in a solid, subject to a triaxial stress state.

276 Chapter 5

Substituting Eq. (5.9) into Eq. (5.8) and making a few simplifying assumptions leads tothe following expressions for the radial displacements of the ellipsoidal void:

A +

v

/

A- (5.10)

A + -

where

A =

D __ -DD

and e/ is the total strain, integrated from the undeformed configuration to the currentstate.

Rice and Tracey solved Eq. (5.10) for a variety of stress states and found that thevoid growth in all cases could be approximated by the following semi-empirical relation-ship:

" "••

In — = 0.283 Jexp - \deeq (5.11)'YS

where R = (Rj +- R2 + Rs)/3 and eeq is the equivalent (von Mises) plastic strain.Subsequent investigators found that Eq. (5.11) could be approximately modified for strainhardening by replacing the yield strength with ae, the effective stress [18].

Since the Rice and Tracey model is based on a single void, it does not take accountof interactions between voids, nor does it predict ultimate failure. A separate failure crite-rion must be applied to characterize microvoid coalescence. For example, one might as-sume that fracture occurs when the nominal void radius reaches a critical value.


The Gurson model [13] analyzes plastic flow in a porous medium by assuming thatthe material behaves as a continuum. Voids appear in the model indirectly through theirinfluence on the global flow behavior. The effect of the voids is averaged through thematerial, which is assumed to be continuous and homogeneous (Fig. 5.11). The maindifference between the Gurson model and classical plasticity is that the yield surface in theformer exhibits a weak hydrostatic stress dependence, while classical plasticity assumesthat yielding is independent of hydrostatic stress. This modification to conventional plas-ticity theory has the effect of introducing a strain softening term.

Unlike the Rice and Tracey model, the Gurson model contains a failure criterion.Ductile fracture is assumed to occur as the result of a plastic instability that produces aband of localized deformation. Such an instability occurs more readily in a Gurson mate-rial because of the strain softening induced by hydrostatic stress. However, because themodel does not consider discrete voids, it is unable to predict necking instability betweenvoids.

t t t

I(a) Porous medium (b) Continuum.

FIGURE 5.11 The continuum assumption for modeling a porous medium. The material is assumed to behomogeneous, and the effect of the voids is averaged through the solid.

The original Gurson model describes the yield surface as follows:

(5.12)

where/is the void volume fraction, Si is the deviatoric stress, defined as

(5.13)

278 Chapters

and $ij is the Kronecker delta:

'I i = j. . (5-14)

When/= 0, Eq. (5.12) reduces to the classical von Mises yield surface with isotropichardening. Equation (5.12) greatly overpredicts failure strains in real materials. Tvergaard[15] attempted to correct the Gurson model by adding two adjustable parameters, qj and

12'-

2 °"KS J

Tvergaard calibrated the revised equation with experimental data and found that reasonablepredictions of failure could be obtained when qj = 2 and #2 = 1 • This modification hasthe effect of amplifying the influence of hydrostatic stress at all strain levels. In real ma-terials, the behavior deviates only slightly from classical plasticity theory through mostof the deformation; at incipient failure, the deviation is rather abrupt.

Tvergaard and Needleman [16] have modified the Gurson model further by replacing/with an effective void volume fraction,/*:

farf<fc

where fc,fu* and/p are fitting parameters. This most recent modification introduces anabrupt failure point, which more closely matches experimental observation. The effect ofhydrostatic stress is amplified when/>/c, which accelerates the onset of a plastic insta-bility. A major disadvantage of the revised Gurson model is that it contains numerousadjustable parameters.

Although the Gurson model (and its subsequent modifications) may adequately char-acterize plastic flow in the early stages of the ductile fracture process, it does not provide agood description of the events that lead to final failure. Ductile failure results from localnecking instabilities between voids. Since the Gurson model does not contain discretevoids, it is incapable of predicting void interactions that lead to failure.

Thomason [11] developed a simple limit load model for internal necking betweenmicrovoids. This model states that failure occurs when the net section stress betweenvoids reaches a critical value, ffn(c)- Figure 5.12 illustrates a two-dimensional case,where cylindrical voids are growing in a material subject to plane strain loading (£3 = 0).If the in-plane dimensions of the voids are 2a and 2b, and the spacing between voids is 2d,the row of voids illustrated in Fig. 5.12 is stable if


d + b(5.17a)

and fracture occurs when

rn(c)d

d + b(5.17b)

where cry is the maximum remote principal stress. Thomason applied the Rice andTracey void growth model to predict the size and shape of growing voids, and utilized Eq.(5.17b) as a failure criterion. He predicted failure strains that were relatively close to ex-perimental observations and were an order of magnitude lower than estimated by Eq.(5.12).

FIGURE 5.12 The limit load model for void in-stability. Failure is assumed to occur when thenet section stress between voids reaches a criti-cal value.

5.1.3 Ductile Crack Growth

Figure 5.13 schematically illustrates microvoid initiation, growth, and coalescence at thetip of a pre-existing crack. As the cracked structure is loaded, local strains and stresses atthe crack tip become sufficient to nucleate voids. These voids grow as the crack blunts,and they eventually link with the main crack. As this process continues, the crack grows.

Figure 5.14 is a plot of stress and strain near the tip of a blunted crack [19]. Thestrain exhibits a singularity near the crack tip, but the stress reaches a peak at approxi-

280 Chapter 5

mately two times the crack tip opening displacement (CTOD)^. In most materials, thetriaxiality ahead of the crack tip provides sufficient stress elevation for void nucleation;thus growth and coalescence of microvoids are usually the critical steps in ductile crackgrowth. Nucleation typically occurs when a particle is ~ 28 from the crack tip, whilemost of the void growth occurs much closer to the crack tip, relative to CTOD. (Notethat although a void remains approximately fixed in absolute space, its distance from thecrack tip, relative to CTOD, decreases as the crack blunts; the absolute distance from thecrack tip also decreases as the crack grows)

Ductile crack growth is usually stable because it produces a rising resistance curve,at least during the early stages of crack growth. Stable crack growth and R curves are dis-cussed in detail in Chapters 3 and 7.

(a) Initial state. (b) Void growth at the crack tip.

FIGURE 5.13growth.

Mechanism for ductile crack

(c) Coalescence of voids with the crack tip.

1 Finite element analysis and slip line analysis of blunted crack tips predict a stress singularity very close tothe crack tip (-0.1 CTOD), but it is not clear whether or not this actually occurs in real materials because thecontinuum assumptions break down at such fine scales.


CJ4.5

yy

2.5

n = 10Small Scale Yielding

>O 4 -

3.5

Stress (0 = 0°)

0.25

0.2

0.15

0.1

0.05

<eq

FIGURE 5.14 Stress and strain ahead of a blunted crack tip, determined by finite element analysis [19].

When an edge crack in a plate grows by microvoid coalescence, the crack exhibits atunneling effect, where it grows faster in the center of the plate, due to the higher stresstriaxiality. The through-thickness variation of triaxiality also produces shear lips, wherethe crack growth near the free surface occurs at a 45° angle from the maximum principalstress, as illustrated in Fig. 5.15. The shear lips are very similar to the cup and cone fea-tures in fractured tensile specimens. The growing crack in the center of the plate producesdeformation bands which nucleate voids in small particles (Fig. 5.6). Thus the so-calledshear lips are caused by a tensile (Mode I) fracture, despite the fact that the preferred frac-ture path is not perpendicular to the tensile axis.

Plane strain crack growth in the center of a plate appears to be relatively flat, butcloser examination reveals a more complex structure. For a crack subject to plane strainMode I loading, the maximum plastic strain occurs at 45° from the crack plane, as illus-trated in Fig. 5.16(a). On a local level, this angle is the preferred path for void coales-cence, but global constraints require that the crack propagation remain in its originalplane. One way to reconcile these competing requirements is for the crack to grow in azig-zag pattern (Fig. 5.16(b)), such that the crack appears flat on a global scale, but ori-ented ± 45° from the crack propagation direction when viewed at higher magnification.This zig-zag pattern is often observed in ductile materials (20,21). Figure 5.17 shows ametallographic cross section of a growing crack that exhibits this behavior.

282 Chapters

FIGURE 5.15. Ductile growth of an edge crack. The so-called shear lips are produced by the samemechanism as the cup and cone in uniaxial tension (Fig. 5.7).

Plane of maximumClastic strain

4S*

FIGURE 5.16. Ductile crack growth in a 45° zig-zag pattern.

5.2 CLEAVAGE

Cleavage fracture can be defined as rapid propagation of a crack along a particular crystal-lographic plane. Cleavage may be brittle, but it can be preceded by large scale plasticflow and ductile crack growth (see Section 5.3) The preferred cleavage planes are thosewith the lowest packing density, since fewer bonds must be broken and the spacing be-tween planes is greater. In the case of body centered cubic (BCC) materials, cleavage oc-curs on {100} planes. The fracture path is transgranular in polycrystalline materials, asFig. 5.1(b) illustrates. The propagating crack changes direction each time it crosses agrain boundary; the crack seeks the most favorably oriented cleavage plane in each grain.The nominal orientation of the cleavage crack is perpendicular to the maximum principalstress.


100 /zm

FIGURE 5.17 Optical micrograph (unetched) of ductile crack growth in an A 710 high strength-low alloysteel [21]. (Photograph provided by J.P. Gudas.)

Cleavage is most likely when plastic flow is restricted. Face centered cubic (FCC)metals are usually not susceptible to cleavage because there are ample slip systems forductile behavior at all temperatures. At low temperatures, BCC metals fail by cleavagebecause there are a limited number of active slip systems. Polycrystalline hexagonalclose packed (HCP) metals, which have only three slip systems per grain, are also suscep-tible to cleavage fracture.

This section and Section 5.3 focus on ferritic steel, because it is the most techno-logically important (and the most extensively studied) material that is subject to cleavagefracture. This class of materials has a BCC crystal structure, which undergoes a ductile-brittle transition with decreasing temperature. Many of the mechanisms described belowalso operate in other material systems that fail by cleavage.

5.2.1 Fractography

Figure 5.18 shows SEM fractographs of cleavage fracture in a low alloy steel. Theinultifaceted surface is typical of cleavage in a polycrystalline material; each facet corre-sponds to a single grain. The "river patterns" on each facet are also typical of cleavagefracture. These markings are so named because multiple lines converge to a single line,much like tributaries to a river.

284 Chapter 5

(a)

(b)

Sun YmSqi) 8 &EM fract°graphs of deava8e in an A 508 Class 3 alloy. (Photographs provided by Mr.


Figure 5.19 illustrates how river patterns are formed. A propagating cleavage crackencounters a grain boundary, where the nearest cleavage plane in the adjoining grain isoriented at a finite twist angle from the current cleavage plane. Initially, the crack ac-commodates the twist mismatch by forming on several parallel planes. As the multiplecracks propagate, they are joined by tearing between planes. Since this process consumesmore energy than crack propagation on a single plane, there is a tendency for the multiplecracks to converge into a single crack. Thus the direction of crack propagation can be in-ferred from river patterns. Figure 5.20 shows a fractograph of river patterns in a low al-loy steel, where tearing between parallel cleavage planes is evident.

5.2.2 Mechanisms of Cleavage Initiation

Since cleavage involves breaking bonds, the local stress must be sufficient to over-come the cohesive strength of the material. In Chapter 2, we learned that the theoreticalfracture strength of a crystalline solid is approximately E/n. Figure 5.14, however, indi-cates that the maximum stress achieved ahead of the crack tip is 3 to 4 times the yieldstrength. For a steel with &YS = 400 MPa and E = 210,000 MPa, the cohesive strengthwould be -50 times higher than the maximum stress achieved ahead of the crack tip.Thus a macroscopic crack provides insufficient stress concentration to exceed the bondstrength.

In order for cleavage to initiate, there must be a local discontinuity ahead of themacroscopic crack that is sufficient to exceed the bond strength. A sharp microcrack isone way to provide sufficient local stress concentration. Cottrell [22] postulated that mi-crocracks form at intersecting slip planes by means of dislocation interaction. A far morecommon mechanism for microcrack formation in steels, however, involves inclusionsand second phase particles [1,23,24].

Figure 5.21 illustrates the mechanism of cleavage nucleation in ferritic steels. Themacroscopic crack provides a local stress and strain concentration. A second phase parti-cle, such as a carbide or inclusion, cracks because of the plastic strain in the surroundingmatrix. At this point the microcrack can be treated as a Griffith crack (Section 2.3). Ifthe stress ahead of the macroscopic crack is sufficient, the microcrack propagates into theferrite matrix, causing failure by cleavage. For example, if the particle is spherical and itproduces a penny-shaped crack, the fracture stress is given by

(5.18)(1-V2) o J

where Yp is me plastic work required to create a unit area of fracture surface in the ferriteand C0 is the particle diameter. It is assumed that Yp » Ys> where 7^ is the surfaceenergy (c.f. Eq. (2.21)). Note that the stress ahead of the macrocrack is treated as a remotestress in this case.

286 Chapters

FIGURE 5.19 Formation of river patterns, as a result of a cleavage crack crossing a twist boundary be-tween grains.

iearto8 ("


DISTANCE

FIGURE 5.21 Initiation of cleavage at a microcrack that forms in a second phase particle ahead of amacroscopic crack.

Consider the hypothetical material described earlier, where OYS - 400 MPa and E =

210,000 MPa. Knott [1] has estimated yp = 14 J/m2 for ferrite. Setting cy= 3 ays and

solving for critical particle diameter yields Co = 7.05 |_im. Thus the Griffith criterion canbe satisfied with relatively small particles.

The nature of the microstructural feature that nucleates cleavage depends on the alloyand heat treatment. In mild steels, cleavage usually initiates at grain boundary carbides[1,23,24]. In quenched and tempered alloy steels, the critical feature is usually either aspherical carbide or an inclusion [1,25]. Various models have been developed to explainthe relationship between cleavage fracture stress and microstructure; most of these modelsresulted in expressions similar to Eq. (5.18). Smith [24] proposed a model for cleavagefracture that considers stress concentration due to a dislocation pile-up at a grain boundarycarbide. The resulting failure criterion is as follows:

v2)(5.19)

where Q», in this case, is the carbide thickness, and TI and ky are the friction stress andpile-up constant, respectively, as defined in the Hall-Petch equation:

1' =Tt+k ,-l/2

where iy is the yield strength in shear. The second term on the left side of Eq. (5.19)contains the dislocation contribution to cleavage initiation. If this term is removed, Eq.(5.19) reduces to the Griffith relationship for a grain boundary microcrack.

288 Chapter 5

(a) Initiation at a grain boundary carbide.

(b) Initiation at an inclusion near the center or a gram.

FIGURE 5.22 SEM fractographs of cleavage initiation in an A 508 Class 3 alloy. (Photographs providedby M.T. Miglin.)


Figure 5.22 shows SEM fractographs which give examples of cleavage initiationfrom a grain boundary carbide (a) and an inclusion at the interior of a grain (b). In bothcases, the fracture origin was located by following river patterns on the fracture surface.

Susceptibility to cleavage fracture is enhanced by almost any factor that increasesthe yield strength, such as low temperature, a triaxial stress state, radiation damage, highstrain rate, and strain aging. Grain size refinement increases the yield strength but alsoincreases Of. There are a number of reasons for the grain size effect. In mild steels, a de-crease in grain size implies an increase in grain boundary area, which leads to smallergrain boundary carbides and an increase in Of. In fine grained steels, the critical eventmay be propagation of the microcrack across the first grain boundary it encounters. Insuch cases the Griffith model implies the following expression for fracture stress:

(5.20)

where Ygb 1S me plastic work per unit area required to propagate into the adjoining grains.Since there tends to be a high degree of mismatch between grains in a polycrystalline ma-terial, Ygb > 7p- Equation (5.20) assumes an equiaxed grain structure. For martensiticand bainitic microstructures, Dolby and Knott [26] derived a modified expression for Ofbased on the packet diameter.

In some cases cleavage nucleates, but total fracture of the specimen or structure doesnot occur. Figure 5.23 illustrates three examples of unsuccessful cleavage events. Part(a) shows a microcrack that has arrested at the particle/matrix interface. The particlecracks due to strain in the matrix, but the crack is unable to propagate because the appliedstress is less than the required fracture stress. This microcrack does not re-initiate be-cause subsequent deformation and dislocation motion in the matrix causes the crack toblunt. Microcracks must remain sharp in order for the stress on the atomic level to ex-ceed the cohesive strength of the material. If a microcrack in a particle propagates intothe ferrite matrix, it may arrest at the grain boundary, as illustrated in Fig. 5.23(b). Thiscorresponds to a case where Eq. (5.20) governs cleavage. Even if a crack successfullypropagates into the surrounding grains, it may still arrest if there is a steep stress gradientahead of the macroscopic crack (Fig. 5.23(c)). This tends to occur at low applied Kj val-ues. Locally, the stress is sufficient to satisfy Eqs. (5.18) and (5.20) but there is insuffi-cient global driving force to continue crack propagation. Figure 5.24 shows an exampleof arrested cleavage cracks in front of a macroscopic crack in a spherodized 1008 steel[27].

5.2.3 Mathematical Models of Cleavage Fracture Toughness

A difficulty emerges when trying to predict fracture toughness from Eqs. (5.18) to (5.20).The maximum stress ahead of a macroscopic crack occurs at approximately 28 from thecrack tip, but the absolute value of this stress is constant in small scale yielding (Fig.5.14); the distance from the crack tip at which this stress occurs increases with increasing

290 Chapters

K, J, and 8. Thus if attaining a critical fracture stress were a sufficient condition forcleavage fracture, the material might fail upon application of an infinitesimal load, be-cause the stresses would be high near the crack tip. Since ferritic materials have finitetoughness, attainment of a critical stress ahead of the crack tip is apparently necessary butnot sufficient.

Ritchie, Knott and Rice (RKR) [28] introduced a simple model to relate fracturestress to fracture toughness, and to explain why steels did not spontaneously fracture uponapplication of minimal load. They postulated that cleavage failure occurs when the stressahead of the crack tip exceeds cyover a characteristic distance, as illustrated in Fig. 5.25.They inferred Gfin a mild steel from blunt notched four-point bend specimens and mea-sured K/c with conventional fracture toughness specimens. They inferred the crack tipstress field from a finite element solution published by Rice and Tracey [29]. They foundthat the characteristic distance was equal to two grain diameters for the material theytested. Ritchie, et al. argued that if fracture initiates in a grain boundary carbide and prop-agates into a ferrite grain, the stress must be sufficient to propagate the cleavage crackacross the opposite grain boundary and into the next grain; thus ovrnust be exceeded over1 or 2 grain diameters. Subsequent investigations [25,30,31], however, revealed no con-sistent relationship between the critical distance and grain size.

Curry and Knott [32] provided a statistical explanation for the RKR critical dis-tance. A finite volume of material must be sampled ahead of the crack tip in order to finda particle that is sufficiently large to nucleate cleavage. Thus a critical sample volume,over which ayy > Of, is required for failure. The critical volume, which can be relatedto a critical distance, depends on the average spacing of cleavage nucleation sites.

The statistical argument also explains why cleavage fracture toughness data tend tobe widely scattered. Two nominally identical specimens made from the same materialmay display vastly different toughness values because the locadon of the critical fracture-triggering particle is random. If one specimen samples a large fracture-triggering particlenear the crack tip, while the fracture trigger in the other specimen is further from the cracktip, the latter specimen will display a higher fracture toughness, because a higher load isrequired to elevate the stress at the particle to a critical value. The statistical nature offracture also leads to an apparent thickness effect on toughness. A thicker specimen ismore likely to sample a large fracture trigger along the crack front and therefore, will havea lower toughness than a thin specimen, on average [33-35].

The Curry and Knott approach was followed by more formal statistical models forcleavage [27,35-38]. These models all treated cleavage as a weakest link phenomenon,where the probability of failure is equal to the probability of sampling at least one criticalfracture-triggering particle. For a volume of material V, with p critical particles per unitvolume, the probability of failure can be inferred from the Poisson distribution:

F = l-exp(-pV) (5.2 la)


(a) Arrest at particle/matrix interface (b) Arrest at a grain boundary.

BISTANCE

(c) Arrest due to a steep stress gradient.

FIGURE 5.23 Examples of unsuccessful cleavage events.

FIGURE 5.24 Arrested cleavage cracks ahead of a macroscopic crack in a spherodized 1008 steel [27].

292 Chapter 5

yy

DISTANCE

FIGURE 5.25 The Ritchie-Knott-Rice model for cleavage fracture [28]. Failure is assumed to occurwhen the fracture stress is exceeded over a characteristic distance.

The second term is the probability of finding zero critical particles in V, so F is the prob-ability of sampling one or more critical particles. The Poisson distribution can be derivedfrom the binomial distribution by assuming that p is small and V is large, an assumptionthat is easily satisfied in the present problem.2 Since the critical particle size depends onstress, which varies ahead of the crack tip, p must vary with position. Therefore, forcrack problems, the failure probability must be integrated over individual volume ele-ments ahead of the crack tip:

= l-exp[ -J pdV (5.21b)

Assuming p depends only on the locally applied stress, and the crack tip conditions areuniquely defined by K or /, it can be shown (Appendix 5) that critical values of K and /follow a characteristic distribution when failure is controlled by a weakest link mecha-

,3.msm-

= l-exp (5.22a)

For a detailed discussion of the Poisson assumption, consult any textbook on probability and statistics.o-"Equations (5.22a) and (5.22b) apply only when the thickness (i.e. the crack front length) is fixed. Theweakest link model predicts a thickness effect, which is described in Appendix 5.2 but is omitted here forbrevity.


or

F = 1 - exp n , (5-22b)0

where 0% and Qj are material properties that depend on microstructure and temperature.Equations (5.22a) and (5.22b) have the form of a two-parameter Weibull distribution [39].The Weibull shape parameter, which is sometimes called the Weibull slope, is equal to4.0 for KIC data and (because of the relationship between K and J) 2.0 for Jc values forcleavage. The Weibull scale parameters, 0% and 0j, are the 63rd percentile values ofKIC and Jc, respectively. If 0% or 0j are known, the entire fracture toughness distribu-tion can be inferred from Eq. (5.22a) or (5.22b).

The prediction of a fracture toughness distribution that follows a two parameterWeibull function with a known slope is an important result. The Weibull slope is ameasure of the relative scatter; a prior knowledge of the Weibull slope enables the relativescatter to be predicted a priori, as Example 5.1 illustrates.

EXAMPLE 5.1

Determine the relative size of the 90% confidence bounds of K/c and Jc data, assumingEqs. (5.22a) and (5.22b) describe the respective distributions.

Solution: The median, 5% lower bound and 95% upper bound values are obtained bysetting F = 0.5, 0.05 and 0.95, respectively, in Eqs. (5.22a) and (5.22b). Both equa-tions have the form:

F = 7 - exp(-fy

Solving for A at each probability level gives

A0.J0 = 0.693 A0.0J = 0.0573 Afl.pj = 2.996

The width of the 90% confidence band in Kjc data, normalized by the median, is givenby

(2.996)0-25 - (0 rt513}°-25

K0.50 = (0.693)0-25

and the relative width of the Jc scatter band is

= 0.920

294 Chapter5

EXAMPLE 5.1 (cont.)

JQ.95 -JQ.05 ^2.996-^0.0513 _ ,; = . - = 1.61

JO. 50 V/I*Q?

Note that QK and 0y cancel out of the above results and the relative scatter dependsonly on the Weibull slope.

There are two major problems with the weakest link model that leads to Eqs. (5.22a) and(5.22b). First, these equations predict zero as the minimum toughness in the distribu-tion. Intuition suggests that such a prediction is incorrect, and more formal argumentscan be made for a nonzero threshold toughness. A crack cannot propagate in a materialunless there is sufficient energy available to break bonds and perform plastic work. If thematerial is a polycrystal, additional work must be performed when the crack crosses ran-domly oriented grains. Thus one can make an estimate of threshold toughness in terms ofenergy release rate:

27p<t> (5-23)

where 0 is a grain misorientation factor. If the global driving force is less than ^c(min)^the crack cannot propagate. The threshold toughness can also be viewed as a crack arrestvalue: a crack cannot propagate if Kj < Kj&.

A second problem with Eqs. (5.22a) and (5.22b) is that they tend to overpredict theexperimental scatter. That is, scatter in experimental cleavage fracture toughness data isusually less severe than predicted by the weakest link model.

According to the weakest link model, failure is controlled by the initiation of cleav-age in the ferrite as the result of cracking of a critical particle; i.e., a particle that satisfiesEq. (5.18) or (5.19). While weakest link initiation is necessary, it is apparently not suf-ficient for total failure. A cleavage crack, once initiated, must have sufficient drivingforce to propagate. Recall Fig. 5.22, which gives examples of unsuccessful cleavageevents.

Both problems, threshold toughness and scatter, can be addressed by incorporating aconditional probability of propagation into the statistical model [40,41]. Figure 5.26 is aprobability tree for cleavage initiation and propagation. When a flawed structure is sub-ject to an applied K, a microcrack may or may not initiate, depending on the temperatureas well as the location of the eligible cleavage triggers. Initiation of cleavage cracksshould be governed by a weakest link mechanism, because the process involves searchingfor a large enough trigger to propagate a microcrack into the first ferrite grain. Oncecleavage initiates, the crack may either propagate in an unstable fashion or arrest (as inFig. 5.23(b) and (c)). Initiation is governed by the local stress at the critical particle,while propagation is controlled by the orientation of the neighboring grains and the


global driving force. The overall probability of failure is equal to the probability of initi-ation times the conditional probability of propagation.

This model assumes that if a microcrack arrests, it does not contribute to subsequentfailure. This is a reasonable assumption, since only a rapidly propagating crack is suffi-ciently sharp to give the stress intensification necessary to break bonds. Once a microc-rack arrests, it is blunted by local plastic flow.

Consider the case where the conditional probability of propagation is a step func-tion:

r O Kj<Kc

That is, assume that all cracks arrest when KI < K0 and that a crack propagates iKo at the time of initiation. This assumption implies that the material has a crack arresttoughness that is single valued. It can be shown (see Appendix 5.2) that such a materialexhibits the following fracture toughness distribution on K values:

K (5.24a)

F = 0 for KI < K0 (5.24b)

Stress Appliedto Structure

P2 Weakest Link MechanismIJ

Crack Propagates(Failure)

FIGURE 5.26 Probability tree for cleavage initiation and propagation.

296 Chapters

Equation (5.24) is a truncated Weibull distribution; QK can no longer be interpreted as the63rd percentile Kjc value. Note that a threshold has been introduced, which removes oneof the shortcomings of the weakest link model. Equation (5.24) also exhibits less scatterthan the two parameter distribution (Eq. 5.22a), thereby removing the other objection tothe weakest link model.

The threshold is obvious in Eq. (5.24), but the reduction in relative scatter is lessso. The latter effect can be understood by considering the limiting cases of Eq. (5.24). IfKO/QK » 1, there are ample initiation sites for cleavage, but the microcracks cannotpropagate unless Kj > Ko. Once Kj exceeds Ko, the next microcrack to initiate willcause total failure. Since initiation events are frequent in this case, K]c values will beclustered near Ko, and the scatter will be minimal. On the other hand, if KQ/&K « 1.Eq. (5.24) reduces to the weakest link case. Thus the relative scatter decreases as K{/&Kincreases.

Equation (5.24) is an oversimplification, because it assumes a single-valued crackarrest toughness. In reality, there is undoubtedly some degree of randomness associatedwith microscopic crack arrest. When a cleavage crack initiates in a single ferrite grain,the probability of propagation into the surrounding grains depends in part on their relativeorientation; a high degree of mismatch increases the likelihood of arrest at the grainboundary. Anderson, et. al. [41] performed a probabilistic simulation of microcrack prop-agation and arrest in a polycrystalline solid. Initiation in a single grain ahead of the cracktip was assumed, and the tilt and twist angles at surrounding grains were allowed to varyrandomly (within the geometric constraints imposed by assuming {100} cleavage planes).An energy-based propagation criterion, suggested by the work of Cell and Smith [42],was applied. The conditional probability of propagation was estimated over a range ofapplied K] values. The results fit an offset power law expression:

K0f (5.25)

where a and /? are material constants. Incorporating Eq. (5.25) into the overall probabil-ity analysis leads to a complicated distribution function that is very difficult to apply toexperimental data (see Appendix 5.2). Stienstra and Anderson found, however, that thisnew function could be approximated by a three-parameter Weibull distribution:

F = 1 - exp (5.26)

where Kmin is the Weibull location parameter. Stienstra [40] showed that whenexperimental data are fit to Eq. (5.26), Kmin gives a conservative estimate of K0, the truethreshold toughness of the material.

Figure 5.27 shows experimental cleavage fracture toughness data for a low alloysteel. Critical / values measured experimentally were converted to equivalent Kjc data.The data were corrected for constraint loss through an analysis developed by Anderson and


Dodds [43] (see Section 3.6.1). Equations (5.22a), (5.24), and (5.26) were fit to theexperimental data. The three parameter Weibull distribution obviously gives the best fit.The weakest link model (Eq. (5.22a)) overestimates the scatter, while the truncatedWeibull distribution does not follow the data in the lower tail, presumably because theassumption of a single valued arrest toughness is incorrect.

A 508 Class 3 Steel-75°C

200

FIGURE 5.27 Cleavage fracture toughness data for an A 508 Class 3 steel at -75°C [41]. The data havebeen fit to various statistical distributions.

5.3 THE DUCTILE-BRITTLE TRANSITION

The fracture toughness of ferritic steels can change drastically over a small temperaturerange, as Fig 5.28 illustrates. At low temperatures, steel is brittle and fails by cleavage.At high temperatures, the material is ductile and fails by microvoid coalescence. Ductilefracture initiates at a particular toughness value, as indicated by the dashed line in Fig.5.28. The crack grows as load is increased. Eventually, the specimen fails by plastic col-lapse or tearing instability. In the transition region between ductile and brittle behavior,both micromechanisms of fracture can occur in the same specimen. In the lower transi-tion region, the fracture mechanism is pure cleavage, but the toughness increases rapidlywith temperature as cleavage becomes more difficult. In the upper transition region, acrack initiates by microvoid coalescence but ultimate failure occurs by cleavage. On ini-tial loading in the upper transition region, cleavage does not occur because there are nocritical particles near the crack tip. As the crack grows by ductile tearing, however, more

298 Chapter 5

material is sampled. Eventually, the growing crack samples a critical particle and cleav-age occurs. Because fracture toughness in the transition region is governed by these sta-tistical sampling effects, the data tend to be highly scattered. Wallin [44] has developed astatistical model for the transition region that incorporates the effect of prior ductiletearing on the cleavage probability.

Recent work by Heerens and Read [25] demonstrates the statistical sampling natureof cleavage fracture in the transition region. They performed a large number of fracturetoughness tests on a quenched and tempered alloy steel at several temperatures in the tran-sition region. As expected, the data at a given temperature were highly scattered. Somespecimens failed without significant stable crack growth while other specimens sustainedhigh levels of ductile tearing prior to cleavage. Heerens and Read examined the fracturesurface of each specimen to determine the site of cleavage initiation. The measured dis-tance from the initiation site to the original crack tip correlated very well with the mea-sured fracture toughness. In specimens that exhibited low toughness, this distance wassmall; a critical nucleus was available near the crack tip. In the specimens that exhibitedhigh toughness, there were no critical particles near the crack tip; the crack had to growand sample additional material before a critical cleavage nucleus was found. Figure 5.29is a plot of fracture toughness versus the critical distance, rc, which Heerens and Readmeasured from the fracture surface; rc is defined as the distance from the fatigue crack tipto the cleavage initiation site. The resistance curve for ductile crack growth is also shownon this plot. In every case, cleavage initiated near the location of the maximum tensilestress (c.f. Fig. 5.14). Similar fractographic studies by Wantanabe et al. [31] andRosenfield and Shetty [45] also revealed a correlation between Jc, Aa, and rc.

FRACTURETOUGHNESS

Cleavage + DuctileTearing

Plastic collapseor ductile instability

Initiation of DuctileTearing

TEMPERATUREFIGURE 5.28 The ductile-brittle transition in ferritic steels. The fracture mechanism changes fromcleavage to micro void coalescence as temperature increases.


DIN 20 MnMoNi 55 Steel-60°C

O Cleavage initiation sites

0 0.5 1.0 1.5 2.0

DISTANCE FROM THE ORIGINAL CRACK TIP, mm

FIGURE 5.29 Relationship between cleavage fracture toughness and the distance between the crack tipand the cleavage trigger [25],

Cleavage propagation in the upper transition region often displays isolated islands ofductile fracture [21,46]. When specimens with arrested macroscopic cleavage cracks arestudied metallographically, unbroken ligaments are sometimes discovered behind the ar-rested crack tip. These two observations imply that a propagating cleavage crack in theupper transition region encounters barriers, such as highly misoriented grains or particles,through which the crack cannot propagate. The crack is diverted around these obstacles,leaving isolated unbroken ligaments in its wake. As the crack propagation continues andthe crack faces open, the ligaments that are well behind the crack tip rupture. Figure 5.30schematically illustrates this postulated mechanism. The energy required to rupture theductile ligaments may provide the majority of the propagation resistance a cleavage crackexperiences. The concentration of ductile ligaments on a fracture surface increases withtemperature [46], which may explain why crack arrest toughness (K[a) exhibits a steepbrittle-ductile transition, much like Kjc and Jc.

5.4 INTERGRANULAR FRACTURE

In most cases metals do not fail along grain boundaries. Ductile metals usually fail bycoalescence of voids formed at inclusions and second phase particles, while brittle metalstypically fail by transgranular cleavage. Under special circumstances, however, cracks canform and propagate along grain boundaries.

There is no single mechanism for intergranular fracture. Rather, there are a varietyof situations that can lead to cracking on grain boundaries, including:

300 Chapter 5

CRACK ES0PAGA31ON

FIGURE 5.30 Schematic illustration of cleavage crack propagation in the ductile-brittle transition re-gion. Ductile ligaments rupture behind the crack tip, resulting in increased propagation resistance.

(1) Precipitation of a brittle phase on the grain boundary.

(2) Hydrogen embrittlement and liquid metal embrittlement.

(3) Environmental assisted cracking.

(4) Intergranular corrosion.

(5) Grain boundary cavitation and cracking at high temperatures.

Space limitations preclude discussing each of these mechanisms in detail. A brief descrip-tion of the intergranular cracking mechanisms is given below.

Brittle phases can be deposited on grain boundaries of steel through improper tem-pering [47]. Tempered martensite embrittlement, which results from tempering near350°C, and temper embrittlement, which occurs when an alloy steel is tempered at -550°C, both apparently involve segregation of impurities, such as phosphorous and sul-phur, to prior austenite grain boundaries. These thin layers of impurity atoms are not re-solvable on the fracture surface, but can be detected with surface analysis techniques suchas Auger electron spectroscopy. Segregation of aluminum nitride particles on grainboundaries during solidification is a common embrittlement mechanism in cast steels[47]. Aluminum nitride, if present in sufficient quantity, can also contribute to degrada-tion of toughness resulting from temper embrittlement in wrought alloys.

Hydrogen can severely degrade the toughness of an alloy, and much has been writtenon this subject in the last 50 years [48,49]. Although the precise mechanism of hydrogenembrittlement is not completely understood, atomic hydrogen apparently bonds with the


metal atoms and reduces the cohesive strength at grain boundaries. Hydrogen can comefrom a number of sources, including moisture, hydrogen containing compounds such asH2S, and hydrogen gas. A common problem in steel weldments is moisture adsorptionduring welding, which leads to cracking in the heat affected zone. Liquid metals, whenslightly above their melting temperature, can embrittle a second metal with a higher melt-ing point. Steel, for example, can be embrittled when placed in contact with molten met-als with low melting points, such as lithium and sodium. The mechanism for liquidmetal embrittlement is believed to be similar to hydrogen embrittlement.

Environmental assisted cracking is related to hydrogen embrittlement, in that hydro-gen plays a role in the cracking process. High strength alloys are most susceptible to en-vironmental assisted cracking, and deleterious environments include H20-NaCl solutions,H2S, ammonia, and gaseous hydrogen. The cracking is time-dependent and usually fol-lows grain boundaries. Figure 5.31 shows the fracture surface of an ammonia tanker thatexperienced environmental assisted cracking. Note the smooth "rock candy" appearance ofthe intergranular fracture. The chemical and transport processes that lead to environmentalassisted cracking are as follows [51]:

(1) Transport of the deleterious environment to the crack tip.

(2) Reactions of the environment with the crack surfaces, resulting in localizeddissolution and production of hydrogen.

(3) Hydrogen absorption into the alloy.

(4) Diffusion of the hydrogen to an embrittlement site ahead of the crack tip.

(5) Hydrogen-metal interactions leading to embrittlement and crack propagation.

Intergranular corrosion involves preferential attack of the grain boundaries, as op-posed to general corrosion, where the material is dissolved relatively uniformly across thesurface. Intergranular attack is different from environmental assisted cracking, in thatthere is no embrittlement mechanism associated with the grain boundary corrosion.

At high temperatures, grain boundaries are weak relative to the matrix, and a signifi-cant portion of creep deformation is accommodated by grain boundary sliding. In suchcases void nucleation and growth (at second phase particles) is concentrated at the crackboundaries, and cracks form as grain boundary cavities grow and coalesce. Grain boundarycavitation is the dominant mechanism of creep crack growth in metals [51], and it can becharacterized with time dependent parameters such as the C* integral (Chapter 4).

302 Chapter 5

FIGURE 531 Intergranular fracture in a steel ammonia tank. (Photograph provided by W.L. Bradley)

REFERENCES

1. Knott, J.F., "Micromechanisms of Fracture and the Fracture Toughness of EngineeringAlloys." Fracture 1977, Vol 1, ICF4, Waterloo Canada, June 1977, pp. 61-91.

2. Knott, J.F. "Effects of Microstructure and Stress-State on Ductile Fracture in MetallicAlloys." In: Advances in Fracture Research, Proceedings of the Seventh InternationalConference on Fracture (ICF7)> K. Salama, et. al., Eds., Pergamon Press, Oxford, UK,1989, pp. 125-138.

3. Wilsforf, H.G.F., "The Ductile Fracture of Metals: a Microstructural Viewpoint."Materials Science and Engineering, Vol 59, 1983, pp. 1-19.

4. Garrison, W.M. Jr. and Moody, N.R., "Ductile Fracture." Journal of the Physics andChemistry of Solids, Vol. 48, 1987, pp. 1035-1074.

5. Knott, J.F., "Micromechanisms of Fibrous Crack Extension in Engineering Alloys."Metal Science, Vol. 14, 1980, pp. 327-336.

6. Argon, A.S., Im, J., Safoglu, R., "Cavity Formation from Inclusions in DuctileFracture." Metallurgical Transactions, Vol. 6A, 1975, pp. 825-837.

7. Beremin, P.M., "Cavity Formation from Inclusions in Ductile Fracture of A 508 Steel."Metallurgical Transactions, Vol 12A, 1981, pp. 723-731.


8. Brown, L.M. and Stobbs, W.M., "The Work-Hardening of Copper-Silica v. EquilibriumPlastic Relaxation by Secondary Dislocations," Philosophical Magazine, 1976, Vol.34, pp. 351-372.

9. Goods, S.H. and Brown, L.M., "The Nucleation of Cavities by Plastic Deformation."Acta Metallurgica, Vol. 27, 1979, pp. 1-15.

10. Van Stone, R.H., Cox, T.B., Low, J.R. Jr., and Psioda, P.A., "Microstructural Aspectsof Fracture by Dimpled Rupture." International Metallurgical Reviews, Vol. 30, 1985,pp. 157-179.

11. Thomason, P.P., Ductile Fracture of Metals, Pergamon Press, Oxford, UK, 1990.

12. Rice, J.R. and Tracey, D.M., "On the Ductile Enlargement of Voids in Triaxial StressFields." Journal of the Mechanics and Physics of Solids, Vol. 17, 1969, pp. 201-217.

13. Gurson, A.L., "Continuum Theory of Ductile Rupture by Void Nucleation and Growth:Part 1—Yield Criteria and Flow Rules for Porous Ductile Media." Journal ofEngineering Materials and Technology, Vol. 99, 1977, pp. 2-15.

14. Berg, C.A., "Plastic Dilation and Void Interaction." Inelastic Behavior of Solids,McGraw-Hill, New York, 1970, pp. 171-210.

15. Tvergaard, V., "On Localization in Ductile Materials Containing Spherical Voids."International Journal of Fracture, Vol. 18, 1982, pp. 237-252.

16. Tvergaard, V and Needleman, A., "Analysis of the Cup-Cone Fracture in a Round TensileBar." Acta Metallurgica, Vol 32, 1984, pp. 157-169.

17. Purtscher, P.T., "Micromechanisms of Ductile Fracture and Fracture Toughness in a HighStrength Austenitic Stainless Steel." Ph.D. Dissertation, Colorado School of Mines,Golden, CO, April, 1990.

18. d'Escata, Y. and Devaux, J.C., "Numerical Study of Initiation, Stable Crack Growth, andMaximum Load with a Ductile Fracture Criterion Based on the Growth of Holes." ASTMSTP 668, American Society of Testing and Materials, Philadelphia, 1979, pp. 229-248.

19. McMeeking, R.M. and Parks, D.M., "On Criteria for J-Dominance of Crack-Tip Fieldsin Large-Scale Yielding." ASTM STP 668, American Society of Testing and Materials,Philadelphia, 1979, pp. 175-194.

20. Beachem, C.D. and Yoder, G.R., "Elastic-Plastic Fracture by Homogeneous MicrovoidCoalescence Tearing Along Alternating Shear Planes." Metallurgical Transactions,Vol. 4A, 1973, pp. 1145-1153.

21. Gudas, J.P. "Micromechanisms of Fracture and Crack Arrest in Two High StrengthSteels." Ph.D. Dissertation, Johns Hopkins University, Baltimore, MD, 1985.

22. Cottrell, A.H., 'Theory of Brittle Fracture in Steel and Similar Metals." Transactions ofthe ASME, Vol 212, 1958, pp. 192-203.

23. McMahon, C.J. Jr and Cohen, M., "Initiation of Cleavage in Polycrystalline Iron."Acta Metallurgica, Vol 13, 1965, pp. 591-604.

304 Chapters

24. Smith, B., "The Nucleation and Growth of Cleavage Microcracks in Mild Steel."Proceedings of the Conference on the Physical Basis of Fracture, Institute of Physicsand Physics Society, 1966, pp. 36-46.

25. Heerens, J. and Read, D.T., "Fracture Behavior of a Pressure Vessel Steel in the Ductile-to-Brittle Transition Region." NISTIR 88-3099, National Institute for Standards andTechnology, Boulder, CO, December, 1988.

26. Dolby, R.E. and Knott, J.F., "Toughness of Martensitic and Martensitic-BainiticMicrostructures with Particular Reference to Heat-Affected Zones." Journal of the Ironand Steel Institute, Vol 210, 1972, p. 857-865.

27. Lin, T., Evans, A.G. and Ritchie, R.O., "Statistical Model of Brittle Fracture byTransgranular Cleavage." Journal of the Mechanics and Physics of Solids, Vol 34,1986, pp. 477-496.

28. Ritchie, R.O., Knott, J.F., and Rice, J.R. "On the Relationship between Critical TensileStress and Fracture Toughness in Mild Steel." Journal of the Mechanics and Physics ofSolids, Vol. 21, 1973, pp. 395-410.

29. Rice, J.R. and Tracey, D.M. "Computational Fracture Mechanics." Numerical ComputerMethods in Structural Mechanics, Academic Press, New York, 1973, pp. 585-623.

30. Curry, D.A. and Knott, J.F., "Effects of Microstructure on Cleavage Fracture Stress inSteel." Metal Science, 1978, pp. 511-514.

31. Watanabe, J., Iwadate, T., Tanaka, Y., Yokoboro, T. and Ando, K., "Fracture Toughnessin the Transition Region." Engineering Fracture Mechanics, Vol. 28, 1987, pp. 589-600.

32. Curry D.A. and Knott, J.F., "Effect of Microstructure on Cleavage Fracture Toughness inMild Steel." Metal Science, Vol. 13, 1979, pp. 341-345

33. Landes J.D. and Shaffer, D.H., "Statistical Characterization of Fracture in the TransitionRegion." ASTM STP 700, American Society of Testing and Materials, Philadelphia,1980, pp. 368-372.

34. Anderson, T.L. and Williams, S., "Assessing the Dominant Mechanism for Size Effectsin the Ductile-to-Brittle Transition Region.", ASTM STP 905, American Society ofTesting and Materials, Philadelphia, 1986, pp. 715-740.

35. Anderson, T.L. and Stienstra, D., "A Model to Predict the Sources and Magnitude ofScatter in Toughness Data in the Transition Region." Journal of Testing andEvaluation, Vol. 17, 1989, pp. 46-53.

36. Evans, A.G.. "Statistical Aspects of Cleavage Fracture in Steel.", MetallurgicalTransactions, Vol. 14A, 1983, pp. 1349-1355.

37. Wallin, K., Saario, T., and Torronen, K., "Statistical Model for Carbide Induced BrittleFracture in Steel." Metal Science, Vol. 18, 1984, pp. 13-16.

38. Beremin, P.M., "A Local Criterion for Cleavage Fracture of a Nuclear Pressure VesselSteel." Metallurgical Transactions, Vol. 14A, 1983, pp. 2277-2287.


39. Weibull, W., "A Statistical Distribution Function of Wide Applicability." Journal ofApplied Mechanics, Vol. 18, 1953, pp. 293-297.

40. Stienstra, D.I.A., "Stochastic Micromechanical Modeling of Cleavage Fracture in theDuctile-Brittle Transition Region." Ph.D. Dissertation, Texas A&M University,College Station, TX, August, 1990.

41. Anderson, T.L., Stienstra, D.I.A., and Dodds, R.H. Jr., "A Theoretical Framework forAddressing Fracture in the Ductile-Brittle Transition Region." Fracture Mechanics: 24thVolume, ASTM STP 1207, American Society for Testing and Materials, Philadelphia,(in press).

42. Cell, M and Smith, E., "The Propagation of Cracks Through Grain Boundaries inPolycrystalline 3% Silicon-Iron." Acta Metallurgica, Vol. 15, 1967, pp. 253-258.

43. Anderson, T.L. and Dodds, R.H., Jr., "Specimen Size Requirements for FractureToughness Testing in the Ductile-Brittle Transition Region." Journal of Testing andEvaluation, to appear, 1991.

44. Wallin, K., "Fracture Toughness Testing in the Ductile-Brittle Transition Region." In:Advances in Fracture Research, Proceedings of the Seventh International Conference onFracture (ICF7), K. Salama, et. al., Eds., Pergamon Press, Oxford, UK, 1989, pp. 267-276.

45. Rosenfield, A.R. and Shetty, D.K., "Cleavage Fracture in the Ductile-Brittle TransitionRegion." ASTM STP 856, American Society for Testing and Materials, Philadelphia,1985, pp. 196-209.

46. Hoagland, R.G., Rosenfield, A.R., and Hahn, G.T., "Mechanisms of Fast Fracture andArrest in Steels." Metallurgical Transactions, Vol. 3, 1972, pp. 123-136.

47. Krauss, G., Principles of Heat Treatment of Steel. American Society for Metals, MetalsPark, OH, 1980.

48. Thompson, A.W. and Bernstein, I.M. (Eds.), Effect of Hydrogen on Behavior ofMaterials. TMS-AIME, Warrendale, 1976.

49. Anon, Hydrogen Damage. American Society for Metals, Metals Park, OH, 1977.

50. Wei, R.P. and Gangloff, R.P., "Environmentally Assisted Crack Growth in StructuralAlloys: Perspectives and New Directions." ASTM STP 1020, American Society forTesting and Materials, Philadelphia, 1989, pp. 233-264.

51. Riedel, H., "Creep Crack Growth." ASTM STP 1020, American Society for Testing andMaterials, Philadelphia, 1989, pp. 101-126.

52. Bain, L.J., Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker,Inc. New York, 1978.

APPENDIX 5: STATISTICAL MODELINGOF CLEAVAGE FRACTURE

When one assumes that fracture occurs by a weakest link mechanism under J controlledconditions, it is possible to derive a closed-form expression for the fracture toughness dis-tribution. When weakest link initiation is necessary but not sufficient for cleavage frac-ture, the problem becomes somewhat more complicated, but it is still possible to describethe cleavage process mathematically.

A5.1 WEAKEST LINK FRACTURE

As discussed in Section 5.2, the weakest link model for cleavage assumes that failure oc-curs when at least one critical fracture-triggering particle is sampled by the crack tip.Equation (5.21) describes the failure probability in this case/* Since cleavage is stresscontrolled, the microcrack density (i.e., the number of critical microcracks per unit vol-ume) should depend only on the maximum principal stress5:

This quantity must be integrated over the volume ahead of the crack tip. In order to per-form this integration, it is necessary to relate the crack tip stresses to the volume sampledat each stress level.

Recall Section 3.5, where dimensional analysis indicated that the stresses ahead ofthe crack tip in the limit of small scale yielding are given by

(A5.2)

assuming Young's modulus is fixed in the material (and thus does not need to be includedin the dimensional analysis). Equation (A5.2) can be inverted to solve for the distanceahead of the crack tip (at a given angle) which corresponds to a particular stress value:

<T0, 0) = - g(Gi I00,0} (A5.3)

It turns out that Eq. (5.21) is valid even when the Poisson assumption is not [40]; the quantity p is not themicrocrack density in such cases but p is uniquely related to microcrack density. Thus, the derivation of thefracture toughness distribution presented in this section does not hinge on the Poisson assumption.

•'Although this derivation assumes that the maximum principal stress at a point controls the incrementalcleavage probability, the same basic result can be obtained by inserting any stress component Eq. (A5.1). For

example one might assume that the tangential stress, CTQ0, governs cleavage.

307

308 Appendix. 5

By fixing 07 and varying 6 from -TTto +n, we can construct a contour of constant princi-pal stress, as illustrated in Fig. A5.1. The area inside this contour is given by

J(A5.4)

where h is a dimensionless integration constant:

+TT

(A5.5)-7T

For plane strain conditions in an edge cracked test specimen, the volume sampled at agiven stress value is simply B A, where B is the specimen thickness. Therefore, the in-cremental volume at a fixed / and ao is given by

BJ dh(A5.6)

Inserting Eqs. (A5.1) and (A5.6) into Eq. (5.21) gives

F = 1 - expcr;

JdG (A5.7)

FIGURE A5.1 Definition of r, 6, and area for aprincipal stress contour.

Statistical Modeling of Cleavage Fracture 309

where (Jmax is the peak value of stress that occurs ahead of the crack tip and ffu is thethreshold fracture stress, which corresponds to the largest fracture-triggering particle thematerial is likely to contain.

Note that J appears outside of the integral in Eq. (A5.7). By setting J = Jc in Eq.(A5.7), we obtain an expression for the statistical distribution of critical J values, whichcan be written in the following form:

F = 1 - expB

(A5.8)

where Bo is a reference thickness. When B = Bo, Qj is the 63rd percentile Jc value.Equation (A5.8) has the form of a two-parameter Weibull distribution, as discussed inSection 5.2.3. Invoking the relationship between K and J for small scale yielding gives

F = 1 - exp (A5.9)

Equations (A5.8) and (A5.9) both predict a thickness effect on toughness. The aver-

age toughness is proportional to 7/v B for critical / values and B~®-25 for %jc (jata> jhe

average toughness does not increase indefinitely with thickness, however. There are lim-its to the validity of the weakest link model, as discussed in the next section.

All of the above relationships are valid only when weakest link failure occurs underJ controlled conditions; i.e., the single parameter assumption must apply. When con-straint relaxes, critical / values no longer follow a Weibull distribution with a specificslope, but the effective small scale yielding / values, J0 (see Section 3.6.3), follow Eq.(A5.8) if a weakest link mechanism controls failure. Actual Jc values would be morescattered than J0 values, however, because the ratio J/JO increases with /.

A5.2 INCORPORATINGPROPAGATION

A CONDITIONAL PROBABILITY OF

In many materials, weakest link initiation of cleavage appears to be necessary but not suf-ficient. Figure 5.26 schematically illustrates a probability tree for cleavage initiation andpropagation. This diagram is a slight oversimplification, because the cumulative failureprobability must be computed incrementally.

Modifying the statistical cleavage model to account for propagation requires that theprobability be expressed in terms of a hazard function [52], which defines the instanta-neous risk of fracture. For a random variable T, the hazard function, H(T), and the cumu-lative probability are related as follows:

310 Appendix 5

F=l-exp - \H(T}dT

TV o

(A5.10)

where To is the minimum value of T. By comparing Eqs. (A5.17) and (A5.18), it caneasily be shown that the hazard function for weakest link initiation, in terms of stress in-tensity, is given by

H(K} =0

(A5.ll)'K

assuming B = Bo. The hazard function for total failure is equal to Eq. (A5.11) times theconditional probability of failure:

4K(A5.12)

Thus the overall probability of failure is given by

r \

F = l-exp -\Ppr—£-dK

J0

(A5.13)

Consider the case where Ppr is a constant; i.e., it does not depend on the applied K.Suppose, for example, that half of the carbides of a critical size have a favorable orienta-tion with respect to a cleavage plane in a ferrite grain. The failure probability becomes:

F -1 - exp -0.5K

0r(A5.14)

In this instance, the finite propagation probability merely shifts the 63rd percentiletoughness to a higher value:

Statistical Modeling of Cleavage Fracture 311

The shape of the distribution is unchanged, and the fracture process still follows a weakestlink model. In this case, the weak link is defined as a particle that is greater than the crit-ical size that is also oriented favorably.

Deviations from the weakest link distribution occur when Ppr depends on the ap-plied K. If the conditional probability of propagation is a step function:

JO K,<K.

the fracture toughness distribution becomes a truncated Weibull (Eq. 5.24); failure canoccur only when K > K0. The introduction of a threshold toughness also reduces the rela-tive scatter, as discussed in Section 5.2.3.

Equation (5.24) implies that the arrest toughness is single valued; a microcrack al-ways propagates above K0, but always arrests at or below Ko. Experimental data, how-ever, indicate that arrest can occur over a range of K values. The data in Fig. 5.27 exhibita sigmoidal shape, while the truncated Weibull is nearly linear near the threshold.

A computer simulation of cleavage propagation in a polycrystalline material [40,41]resulted in a prediction of Ppr as a function of the applied K\ these results fit an offsetpower law expression (Eq. (5.25)). The absolute values obtained from the simulation arequestionable, but the predicted trend is reasonable. Inserting Eq. (5.25) into Eq. (A5.13)gives

\a A. If

- j a(K-K0Y—j-dK (A5.15)

The integral in Eq. (A5.15) has a closed-form solution, but it is rather lengthy. Theabove distribution exhibits a sigmoidal shape, much like the experimental data in Fig.5.27. Unfortunately, it is very difficult to fit experimental data to Eq. (A5.22). Note thatthere are four fitting parameters in this distribution: a, ft, Ko, and Qg. Even with fewerunknown parameters, the form of Eq. (A5.15) is not conducive to curve fitting because itcannot be linearized.

Equation (A5.15) can be approximated with a conventional three parameter Weibulldistribution with the slope fixed at 4 (Eq. 5.26). The latter expression also gives a rea-sonably good fit of experimental data (Fig. 5.27). The three parameter Weibull distribu-tion is sufficiently flexible to model a wide range of behavior. The advantage of Eq.(5.26) is that there are only two parameters to fit (the Weibull shape parameter is fixed at4.0) and it can be linearized. The apparent threshold, Kmin, obtained by curve fittingtends to be a conservative estimate of the true threshold toughness.

6. FRACTURE MECHANISMS INNONMETALS

Traditional structural metals such as steel and aluminum are being replaced with plastics,ceramics, and composites in a number of applications. Engineering plastics have anumber of advantages, including low cost, ease of fabrication, and corrosion resistance.Ceramics provide superior wear resistance and creep strength. Composites offer highstrength/weight ratios, and enable engineers to design materials with specific elastic andthermal properties. Traditional nonmetallic materials such as concrete continue to seewidespread use.

Nonmetals, like metals, are not immune to fracture. Recall from Chapter 1 the ex-ample of pinch clamping of polyethylene pipe that led to time-dependent fracture. Theso-called high toughness ceramics that have been developed in recent years (Section 6.2),have lower toughness than even the most brittle steels. Relatively minor impact (e.g. anairplane mechanic accidentally dropping his wrench on a wing) can cause microscale dam-age in a composite material, which can adversely affect subsequent performance. The lackof ductility of concrete (relative to steel) limits its range of application.

Compared with fracture of metals, research into the fracture behavior of nonmetals isin its infancy. Much of the necessary theoretical framework is not yet fully developed fornonmetals, and there are many instances where fracture mechanics concepts that apply tometals have been misapplied to other materials.

This chapter gives a brief overview of the current state of understanding of fractureand failure mechanisms in selected nonmetallic structural materials. Although the cover-age of the subject is far from complete, this chapter should enable the reader to gain anappreciation of the diverse fracture behavior that various materials can exhibit. The refer-ences listed at the end of the chapter provide a wealth of information to those who desire amore in-depth understanding of a particular material system. The reader should also referto Chapter 8, which describes current methods for fracture toughness measurements innonmetallic materials.

Section 6.1 outlines the molecular structure and mechanical properties of polymericmaterials, and describes how these properties influence the fracture behavior. This sectionalso includes a discussion of the fracture mechanisms in polymer matrix composites.Section 6.2 considers fracture in ceramic materials, including the newest generation of ce-ramic composites. Section 6.3 addresses fracture in concrete and rock.

This chapter does not specifically address metal matrix composites, but these mate-rials have many features in common with polymer and ceramic matrix composites [1].Also, the metal matrix in these materials should exhibit the fracture mechanisms de-scribed in Chapter 5.

313

314 Chapter 6

6.1 ENGINEERING PLASTICS

The fracture behavior of polymeric materials has only recently become a major concern,as engineering plastics have begun to appear in critical structural applications. In mostconsumer products made from polymers (e.g.. toys, garbage bags, ice chests, lawn furni-ture, etc.), fracture may be an annoyance, but it is not a significant safety issue. Fracturein plastic natural gas piping systems or aircraft wings, however, can have dire conse-quences.

Several books devoted solely to fracture and fatigue of plastics have been publishedin recent years [2-5]. These references proved invaluable to the author in preparingChapters 6 and 8.

Let us begin the discussion of fracture in plastics by reviewing some of the basicprinciples of polymeric materials.

6.1.1 Structure and Properties of Polymers

A polymer is defined as the union of two or more compounds called mers. The degree ofpolymerization is a measure of the number of these units in a given molecule. Typicalengineering plastics consist of very long chains, with the degree of polymerization on theorder of several thousand.

Consider polyethylene, a polymer with a relatively simple molecular structure. Thebuilding block in this case is ethylene (C2H4), which consists of two carbon atomsjoined by a double bond, with two hydrogen atoms attached to each carbon atom. If suffi-cient energy is applied to this compound, the double bond can be broken, resulting in twofree radicals which can react with other ethylene groups:

H H H HI I I IC=C + Energy -> — C~C~

I I I IH H H H

The degree of polymerization (i.e., the length of the chain) can be controlled by the heatinput, catalyst, as well as reagents that may be added to aid the polymerization process.

Molecular WeightThe molecular weight is a measure of the length of a polymer chain. Since there is

typically a distribution of molecule sizes in a polymer sample, it is convenient to quan-tify an average molecular weight, which can be defined in one of two ways. The numberaverage molecular weight is the total weight divided by the number of molecules:

Fracture Mechanisms in Nonmetals 315

(6.D

where Nj is the number of molecules with molecular weight Af/. The number averagemolecular weight attaches equal importance to all molecules, while the weight averagemolecular weight reflects the actual average weight of molecules by placing additionalemphasis on the larger molecules:

These two measures of molecular weight are obviously identical if all molecules in thesample are the same size, but the number average is usually lower that the weight averagemolecular weight. The polydispersity is defined as the ratio of these two quantities:

(6.3)Af1YJ-n

A narrow distribution of molecular weights implies a PD close to 1, while PD can begreater than 20 in materials with broadly distributed molecule sizes. Both measures ofmolecular weight, as well as the PD, influence the mechanical properties of a polymer.

Molecular StructureThe structure of polymer chains also has a significant effect on the mechanical prop-

erties. Figure 6.1 illustrates three general classifications of polymer chains: linear,branched, and cross-linked. Linear polymers are not actual straight lines; rather, the car-bon atoms in a linear molecule form a single continuous path from one end of the chainto the other. A branched polymer molecule, as the name suggests, contains a serious ofsmaller chains that branch off from a main "backbone". A cross-linked polymer consistsof a network structure rather than linear chains. A highly cross-linked structure is typicalof thermoset polymers, while thermoplastics consist of linear and branched chains.Elastomers typically have lightly cross-linked structures and are capable of large elasticstrains.

Epoxies are the most common example of thermoset polymers. Typically, twocompounds that are in the liquid state at ambient temperature are mixed together to form

316 Chapter 6

an epoxy resin, which solidifies into a cross-linked lattice upon curing. This process isirreversible; a thermoset cannot be formed into another shape once it solidifies.

Thermomechanical processes in thermoplastics are reversible, because these materi-als do not form cross-linked networks. Thermoplastics become viscous upon heating (seebelow), where they can be formed into the desired shape.

(a) Linear polymer. (b) Branched polymer. (c) Cross-linked polymer

FIGURE 6.1 Three types of polymer chains.

Crystalline and Amorphous PolymersPolymer chains can be packed tightly together in a regular pattern, or they can form

random entanglements. Materials that display the former configuration are called crys-talline polymers, while the disordered state corresponds to amorphous (glassy) polymers.Figure 6.2 schematically illustrates crystalline and amorphous arrangements of polymermolecules.

The term crystalline does not have the same meaning for polymers as for metals andceramics. A crystal structure in a metal or ceramic is a regular array of atoms with three-dimensional symmetry; all atoms in a crystal have identical surroundings (except atomsthat are adjacent to a defect, such as a dislocation or vacancy). The degree of symmetry ina crystalline polymer, however, is much lower, as Fig. 6.2 illustrates.

Figure 6.3 illustrates the volume-temperature relationships in crystalline and amor-phous thermoplastics. As a crystalline polymer cools from the liquid state, an abrupt de-crease in volume occurs at the melting temperature, Tm, and the molecular chains packefficiently in response to the thermodynamic drive to order into a crystalline state. Thevolume discontinuity at Tm resembles the behavior of crystalline metals and ceramics.An amorphous polymer bypasses Tm upon cooling, and remains in a viscous state untilit reaches the glass transition temperature, Tg, at which time the relative motion of themolecules becomes restricted. An amorphous polymer contains more free volume thanthe same material in the crystalline state, and thus has a lower density. The glass transi-tion temperature is sensitive to cooling rate; rapid heating or cooling tends to increase Tg,as Fig. 6.3 indicates.

Semicrystalline polymers contain both crystalline and glassy regions. The relativefraction of each state depends on a number of factors, including molecular structure and


cooling rate. Slow cooling provides more time for the molecules to arrange themselvesin an equilibrium crystal structure.

^

\\i4

r\

/

)

5

J

(

<

>

0

i

n

\

\

\J

(

o

r^

jx->

/_

/

L.

n

s

'~>

V

i

*~^

or~>

0

u

uA

(a) Amorphous polymer. (b) Crystalline polymer.

FIGURE 6.2 Amorphous and crystalline polymers.

W

TEMPERATURE

Tm

FIGURE 63 Volume-temperature relation-ships for amorphous (glassy) and crystallinepolymers.

Viscoelastic BehaviorPolymers exhibit rate-dependent viscoelastic deformation, which is a direct result of

their molecular structure. Figure 6.4 gives a simplified view of viscoelastic behavior onthe molecular level. Two neighboring molecules, or different segments of a singlemolecule that is folded back upon itself, experience weak attractive forces called Van derWaals bonds. These secondary bonds resist any external force that attempts to pull themolecules apart. The elastic modulus of a typical polymer is significantly lower thanYoung's modulus for metals and ceramics, because the Van der Waals bonds are muchweaker than primary bonds. Deforming a polymer requires cooperative motion amongmolecules. The material is relatively compliant if the imposed strain rate is sufficiently

318 Chapter 6

low to provide molecules sufficient time to move. At faster strain rates, however, theforced molecular motion produces friction, and a higher stress is required to deform thematerial. If the load is removed, the material attempts to return to its original shape, butmolecular entanglements prevent instantaneous elastic recovery. If the strain is suffi-ciently large, yielding mechanisms occur, such as crazing and shear deformation (seeSection 6.6.2 below), and much of the induced strain is essentially permanent.

Section 4.3 introduced the relaxation modulus, E(t), and the creep compliance, D(t),which describe the time-dependent response of viscoelastic materials. The relaxationmodulus and creep compliance can be obtained experimentally by fixing strain and stress,respectively:

oft) 0(0 = (6.4)

See Fig. 4.19 for a schematic illustration of stress relaxation and creep experiments. Forlinear viscoelastic materials1, E(t) and D(t) are related through a hereditary integral (Eq.(4.61)).

FIGURE 6.4 Schematic deformation of apolymer chain. Secondary Van der Waalsbonds between chain segments resistforces that try to extend the molecule.

Linear viscoelastic materials do not, in general, have linear stress-strain curves (since the modulus is timedependent), but display other characteristics of linear elasticity such as superposition. See Section 4.3.1 for adefinition of linear viscoelasticity.


Figure 6.5(a) is a plot of relaxation modulus versus temperature at a fixed time for athermoplastic. Below Tg, the modulus is relatively high, as molecular motion is re-stricted. At around Tg, the modulus (at a fixed time) decreases rapidly, and the polymerexhibits a "leathery" behavior. At higher temperatures, the modulus reaches a lowerplateau, and the polymer is in a rubbery state. Natural and synthetic rubbers are merelymaterials whose glass transition temperature is below room temperature2. If the tempera-ture is sufficiently high, linear polymers loose virtually all load-carrying capacity and be-have like a viscous fluid. Highly cross-linked polymers, however, maintain a modulusplateau.

Figure 6.5(b) shows a curve with the same characteristic shape as Fig. 6.5(a), butwith fixed temperature and varying time. At short times, the polymer is glassy, but ex-hibits leathery, rubbery, and liquid behavior at sufficiently long times. Of course, shorttime and long time are relative terms that depend on temperature. A polymer signifi-cantly below Tg might remain in a glassy state during the time frame of a stress relax-ation test, while a polymer well above Tg may pass through this state so rapidly that theglassy behavior cannot be detected.

The equivalence between high temperature and long times (i.e., the time-tempera-ture superposition principle) led Williams, Landel, and Ferry [6] to develop a semiempiri-cal equation that collapses data at different times onto a single modulus-temperature mas-ter curve. They defined a time shift factor, a?, as follows:

. tTlog aT = log -+- = — - (6.5)* / ^ _ i _ T T*To (-2+1-10

where fp and fp0 are the times to reach a specific modulus at temperatures T and To, re-spectively, T0 is a reference temperature (usually defined at Tg), and Cy and €2 are fittingparameters that depend on material properties. Equation (6.5), which is known as theWLF relationship, typically is valid in the range Tg < T< Tg + 100°C. Readers familiarwith creep in metals may recognize an analogy with the Larson-Miller parameter [7],which assumes a time-temperature equivalence for creep rupture.

Mechanical AnalogsSimple mechanical models are useful for understanding the viscoelastic response of

polymers. Three such models are illustrated in Fig. 6.6. The Maxwell model (Fig.6.6(a)) consists of a spring and a dashpot in series, where a dashpot is a moving piston ina cylinder of viscous fluid. The Voigt model (Fig. 6.6(b)) contains a spring and a dashpotin parallel. Figure 6.6(c) shows a combined Maxwell-Voigt model. In each case, thestress-strain response in the spring is instantaneous:

(6.6)

To demonstrate the temperature dependence of viscoelastic behavior, try blowing up a balloon after it hasbeen in a freezer for an hour.

320 Chapter 6

LOG (E)

Cross-linkedpolymer

TEMPERATURE

(a) Modulus versus temperature at a fixed time.

LOG(E)

Cross-linkedpolymer

LOG (TIME)

(b) Modulus versus time at a fixed temperature.

FIGURE 6.5 Effect of temperature and time on the modulus of and amorphous polymer.

while the dashpot response is time-dependent:

(7c — _ (6.7)

*

where £ is the strain rate and 7] is the fluid viscosity in the dashpot. The temperaturedependence of r\ can be described by an Arrhenius rate equation:

(6.8)


where Q is the activation energy for viscous flow (which may depend on temperature), Tis the absolute temperature, and R is the gas constant (= 8.314 J/(mole °K)).

In the Maxwell model, the stresses in the spring and dashpot are equal, and thestrains are additive. Therefore,

cr 1 dcr£ = 1

77 E dt(6.9)

e

For a stress relaxation experiment (Fig. 4.19(b)), the strain is fixed at eo, and £ = 0.Integrating stress with respect to time for this case leads to

(6.10)

where ao is the stress at t = 0, and //? = Tj/E is the relaxation time.When the spring and dashpot are in parallel (the Voigt model) the strains are equal

and the stresses are additive:

a(t) ==E£ + T]£

For a constant stress creep test, Eq. (6.11) can be integrated to give:

(6.11)

r\2

111

(a) Maxwell model (b) Voigt model (c) Combined model

FIGURE 6.6 Mechanical analogs for viscoelastic deformation in polymers.

322 Chapter 6

(6.12)E

Note that the limiting value of creep strain in this model is Oo/E, which corresponds tozero stress on the dash pot. If the stress is removed, the strain recovers with time:

(6.13)

where £o is the strain at t — 0, and zero time is defined at the moment the load is re-moved.

Neither model describes all types of viscoelastic response. For example, theMaxwell model does not account for viscoelastic recovery, because strain in the dashpot isnot reversed when the stress is removed. The Voigt model cannot be applied to the stressrelaxation case, because when strain is fixed in Eq. (6.11), all of the stress is carried bythe spring; the problem reduces to simple static loading, where both stress and strain re-main constant.

If we combine the two models, however, we obtain a more realistic and versatilemodel of viscoelastic behavior. Figure 6.6(c) illustrates the combined Maxwell-Voigtmodel. In this case, the strains in the Maxwell and Voigt contributions are additive, andthe stress carried by the Maxwell spring and dashpot is divided between the Voigt springand dashpot. For a constant stress creep test, combining Eqs. (6.9) and (6.13) gives

(6.14)\

All three models are oversimplifications of actual polymer behavior, but are useful forapproximating different types of viscoelastic response.

6.1.2 Yielding and Fracture in Polymers

In metals, fracture and yielding are competing failure mechanisms. Brittle fracture occursin materials in which yielding is difficult. Ductile metals, by definition, experience ex-tensive plastic deformation before they eventually fracture. Low temperatures, high strainrates, and triaxial tensile stresses tend to suppress yielding and favor brittle fracture.

From a global point of view, the forgoing also applies to polymers, but the micro-scopic details of yielding and fracture in plastics are different from metals. Polymers donot contain crystallographic planes, dislocations, and grain boundaries; rather, they con-sist of long molecular chains. Section 2.1 states that fracture on the atomic level in-volves breaking bonds, and polymers are no exception. A complicating feature for poly-mers, however, is that two types of bond govern the mechanical response: the covalentbonds between carbon atoms and the secondary van der Waals forces between moleculesegments. Ultimate fracture normally requires breaking the latter, but the secondarybonds often play a major role in the deformation mechanisms that lead to fracture.


The factors that govern the toughness and ductility of polymers include strain rate,temperature, and molecular structure. At high rates or low temperatures (relative to To)polymers tend to be brittle, because there is insufficient time for the material to respondto stress with large-scale viscoelastic deformation or yielding. Highly cross-linked poly-mers are also incapable of large scale viscoelastic deformation. The mechanism illustratedin Fig. 6.4, where molecular chains overcome van der Waals forces, does not apply tocross-linked polymers; primary bonds between chain segments must be broken for thesematerials to deform.

Chain Scission and DisentanglementFracture, by definition, involves material separation, which normally implies sever-

ing bonds. In the case of polymers, fracture on the atomic level is called chain scission.Recall from Chapter 2 that the theoretical bond strength in most materials is several

orders of magnitude larger than measured fracture stresses, but crack-like flaws can producesignificant local stress concentrations. Another factor that aids chain scission in poly-mers is that molecules are not stressed uniformly. When a stress is applied to a polymersample, certain chain segments carry a disproportionate amount of load, which can be suf-ficient to exceed the bond strength. The degree of nonunifonnity in stress is more pro-nounced in amorphous polymers, while the limited degree of symmetry in crystallinepolymers tends to distribute stress more evenly.

Free radicals form when covalent bonds in polymers are severed. Consequently,chain scission can be detected experimentally by means of electron spin resonance (ESR)and infrared spectroscopy [8,9].

In some cases, fracture occurs by chain disentanglement, where molecules separatefrom one another intact. The likelihood of chain disentanglement depends on the lengthof molecules and the degree to which they are interwoven.^

Chain scission can occur at relatively low strains in cross-linked or highly alignedpolymers, but the mechanical response of isotropic polymers with low cross link densityis governed by secondary bonds at low strains. At high strains, many polymers yield be-fore fracture, as discussed below.

Shear Yielding and CrazingMost polymers, like metals, yield at sufficiently high stresses. While metals yield

by dislocation motion along slip planes, polymers can exhibit either shear yielding orcrazing.

Shear yielding in polymers resembles plastic flow in metals, at least from a contin-uum mechanics viewpoint. Molecules slide with respect to one another when subjectedto a critical shear stress. Shear yielding criteria can either be based on the maximumshear stress or the octahedral shear stress [10,11]:

oJAn analogy that should be familiar to most Americans is the process of disentangling Christmas tree lightsthat have been stored in a box for a year. For those who are not acquainted with this holiday ritual, a similarexample is a large mass of tangled strands of string; pulling on a single strand will either free it (chaindisentanglement) or cause it to break (chain scission).

324 Chapter 6

or

(6.15a)

(6.15b)

where om is the hydrostatic stress and fis is a material constant that characterizes the sen-sitivity of the yield behavior to <jm. When fis = 0, Eqs. (6.15a) and (6.15b) reduce to theTresca and von Mises yield criteria, respectively.

Glassy polymers subject to tensile loading often yield by crazing, which is a highlylocalized deformation that leads to cavitation (void formation) and strains on the order of100% [12,13]. On the macroscopic level, crazing appears as a stress-whitened region, dueto a low refractive index. The craze zone usually forms perpendicular to the maximumprincipal normal stress.

Figure 6.7 illustrates the mechanism for crazing in homogeneous glassy polymers.At sufficiently high strains, molecular chains form aligned packets called fibrils.Microvoids form between the fibrils due to an incompatibility of strains in neighboringfibrils. The aligned structure enables the fibrils to carry very high stresses relative to theundeformed amorphous state, because covalent bonds are much stronger and stiffer thanthe secondary bonds. The fibrils elongate by incorporating additional material, as Fig.6.7 illustrates. Figure 6.8 shows an SEM fractograph of a craze zone.

FIGURE 6.7 Craze formation in glassypolymers. Voids form between fibrils, whichare bundles of aligned molecular chains. Thecraze zone grows by drawing additional ma-terial into the fibrils.

Oxborough and Bowden [14] proposed the following craze criterion:

y(t,T)h (6.16)

'm


where £] is the maximum principal normal strain, and ft and y are parameters that aretime- and temperature-dependent. According to this model, the critical strain for crazingdecreases with increasing modulus and hydrostatic stress.

Fracture occurs in a craze zone when individual fibrils rupture. This process can beunstable if, when a fibril fails, the redistributed stress is sufficient to rupture one or moreneighboring fibrils. Fracture in a craze zone usually initiates from inorganic dust parti-cles that are entrapped in the polymer [15]. There are a number of ways to neutralize thedetrimental effects of these impurities, including the addition of soft second-phase parti-cles (see below).

Crazing and shear yielding are competing mechanisms; the dominant yielding behav-ior depends on molecular structure, stress state and temperature. A large hydrostatic ten-sile component in the stress tensor is conducive to crazing, while shear yielding favors alarge deviatoric stress component. Each yielding mechanism displays a different tempera-ture dependence; thus the dominant mechanism may change with temperature.

FIGURE 6.8 Craze zone in polypropylene. (Photograph provided by M. Cayard.)

326 Chapter 6

Crack Tip BehaviorAs with metals, a yield zone typically forms at the tip of a crack in polymers. In

the case of shear yielding, the damage zone resembles the plastic zone in metals, becauseslip in metals and shear in polymers are governed by similar yield criteria. Craze yield-ing, however, produces a Dugdale-type strip yield zone ahead of the crack tip. Of the twoyielding mechanisms in polymers, crazing is somewhat more likely ahead of a crack tip,because of the triaxial tensile stress state. Shear yielding, however, can occur at cracktips in some materials, depending on the temperature and specimen geometry [16].

Figure 6.9 illustrates a craze zone ahead of a crack tip. If the craze zone is smallcompared to specimen dimensions4, we can estimate its length, p, from the Dugdale-Barenblatt [17,18] strip yield model:

(6.17)

which is a restatement of Eq. (2.74), except that we have replaced the yield strength with<TC, the crazing stress. Figure 6.10 is a photograph of a crack tip craze zone [16], whichexhibits a typical stress whitening appearance.

The crack advances when the fibrils at the trailing edge of the craze rupture. In otherwords, cavities in the craze zone coalesce with the crack tip. Figure 6. 1 1 is an SEM frac-tograph of the surface of a polypropylene fracture toughness specimen that has experi-enced craze crack growth. Note the similarity to fracture surfaces for microvoid coales-cence in metals (Figs. 5.3 and 5.8).

Craze crack growth can either be stable or unstable, depending on the relative tough-ness of the material. Some polymers with intermediate toughness exhibit sporadic, so-called stick/slip crack growth: at a critical crack tip opening displacement, the entire crazezone ruptures, the crack arrests, and the craze zone reforms at the new crack tip [3].Stick/slip crack growth can also occur in materials that exhibit shear yield zones.

FIGURE 6.9 Schematic crack tip crazezone.

Another implicit assumption of Eq. (6.17) is that the global material behavior is linear elastic or linearviscoelastic. Chapter 8 discusses the requirements for the validity of the stress intensity factor in polymers.


FIGURE 6.10 Stress-whitened zone ahead of a crack tip, which indicates crazing. (Photograph providedby M. Cayard.)

FIGURE 6.11 Fracture surface of craze crack growth in polypropylene. (Photograph provided by Mr.Sun Yongqi.)

328 Chapter 6

Rubber TougheningAs stated earlier, rapture of fibrils in a craze zone can lead to unstable crack propaga-

tion. Fracture initiates at inorganic dust particles in the polymer when the stress exceedsa critical value. It is possible to increase the toughness of a polymer by lowering thecrazing stress to well below the critical fracture stress.

The addition of rubbery second-phase particles to a polymer matrix significantly in-creases toughness by making craze initiation easier [15]. The low modulus particles pro-vide sites for void nucleation, thereby lowering the stress required for craze formation.The detrimental effect of the dust particles is largely negated, because the stress in the fib-rils tends to be well below that required for fracture. Figure 6.12 is an SEM fractographthat shows crack growth in a rubber-toughened polymer. Note the high concentration ofvoids, compared to the fracture surface in Fig. 6.11.

Of course there is a trade-off with rubber toughening, in that the increase in tough-ness and ductility comes at the expense of yield strength. A similar trade-off betweentoughness and strength often occurs in metals and alloys.

Time-dependent crack growth in the presence of cyclic stresses is a problem in virtu-ally all material systems. Two mechanisms control fatigue in polymers: chain scissionand hysteresis heating [5].

FIGURE 6.12 Fracture surface of a rubber-toughened polyvinyl chloride (PVC). Note the high concen-tration of microvoids. (Photograph provided by Mr. Sun Yongqi.)


Crack growth by chain scission occurs in brittle systems, where crack tip yielding islimited. A finite number of bonds are broken during each stress cycle, and measurablecrack advance takes place after sufficient cycles.

Tougher materials exhibit significant viscoelastic deformation and yielding at thecrack tip. Figure 6.13 illustrates the stress-strain behavior of a viscoelastic material for asingle load-unload cycle. Unlike elastic materials, where the unloading and loading pathscoincide and the strain energy is recovered, a viscoelastic material displays a hysteresisloop in the stress-strain curve; the area inside this loop represents energy that remains inthe material after it is unloaded. When a viscoelastic material is subject to multiple stresscycles, a significant amount of work is performed on the material. Much of this work isconverted to heat, and the temperature in the material rises. The crack tip region in apolymer subject to cyclic loading may rise to well above To, resulting in local meltingand viscous flow of the material. The rate of crack growth depends on the temperature atthe crack tip, which is governed by the loading frequency and the rate of heat conductionaway from the crack tip. Fatigue crack growth data from small laboratory coupons maynot be applicable to structural components because heat transfer properties depend on thesize and geometry of the sample.

STRESS

Absatfeed.Eaesgy

FIGURE 6.13 Cyclic stress-strain curve in a vis-coelastic material. Hysteresis results in absorbedenergy, which is converted to heat.

STRAIN

6.1.3 Fiber-Reinforced Plastics

This section focuses on the fracture behavior of continuous fiber-reinforced plastics, asopposed to other types of polymer composites. The latter materials tend to be isotropicon the macroscopic scale, and their behavior is often similar to homogeneous materials.Continuous fiber-reinforced plastics, however, have orthotropic mechanical propertieswhich lead to unique failure mechanisms such as delamination and microbuckling.

The combination of two or more materials can lead to a third material with highlydesirable properties. Precipitation-hardened aluminum alloys and rubber-toughened plas-tics are examples of materials whose properties are superior to those of the parent con-stituents. While these materials form "naturally" through careful control of chemicalcomposition and thermal treatments, the manufacture of composite materials normallyinvolves somewhat more heavy-handed human intervention. The constituents of a com-

330 Chapter 6

posite material are usually combined on a macroscopic scale through physical rather thanchemical means [19]. The distinction between composites and multiphase materials issomewhat arbitrary, since many of the same strengthening mechanisms operate in bothclasses of material.

Composite materials usually consist of a matrix and a reinforcing constituent. Thematrix is often soft and ductile compared to the reinforcement, but this is not always thecase (see Section 6.2). Various types of reinforcement are possible, including continuousfibers, chopped fibers, whiskers, flakes, and particulates [19].

When a polymer matrix is combined with a strong, high modulus reinforcement, theresulting material can have superior strength/weight and stiffness/weight ratios comparedto steel and aluminum. Continuous fiber-reinforced plastics tend to give the best overallperformance (compared to other types of polymer composites), but can also exhibit trou-bling fracture and damage behavior. Consequently, these materials have been the subjectof extensive research over the past 20 years.

A variety of fiber-reinforced polymer composites are commercially available. Thematrix material is usually a thermoset polymer (i.e., an epoxy), although thermoplasticcomposites have become increasingly popular in recent years. Two of the most commonfiber materials are carbon, in the form of graphite, and aramid (also known by the tradename, Kevlar*), which is a high modulus polymer. Polymers reinforced by continuousgraphite or Kevlar fibers are intended for high performance applications such as fighterplanes, while fiberglass is an example of a polymer composite that appears in moredown-to-earth applications. The latter material consists of randomly oriented choppedglass fibers in a thermoset matrix.

Figure 6.14 illustrates the structure of a fiber-reinforced composite. Consider a sin-gle ply (Fig. 6.14(a)). The material has high strength and stiffness in the fiber direction,but has relatively poor mechanical properties when loaded transverse to the fibers. In thelatter case, the strength and stiffness are controlled by the properties of the matrix. Whenthe composite is subject to biaxial loading, several plies with differing fiber orientationscan be bonded to form a laminated composite (Fig. 6.14(b)). The individual plies interactto produce complex elastic properties in the laminate. The desired elastic response can beachieved through the appropriate choice of the fiber and matrix material, the fiber volume,and the lay-up sequence of the plies. The fundamentals of orthotropic elasticity and lami-nate theory are well established [20].

Overview of Failure MechanismsMany have attempted to apply fracture mechanics to fiber-reinforced composites, and

have met with mixed success. Conventional fracture mechanics methodology assumes asingle dominant crack that grows in a self-similar fashion; i.e. the crack increases in size(either through stable or unstable growth), but its shape and orientation remain the same.Fracture of a fiber-reinforced composite, however, is often controlled by numerous micro-cracks distributed throughout the material, rather than a single macroscopic crack. Thereare situations where fracture mechanics is appropriate for composites, but it is importantto recognize the limitations of theories that were intended for homogeneous materials.

^Kevlar is a trademark of the E.I. Dupont Company.


Figure 6.15 illustrates various failure mechanisms in fiber-reinforced composites.One advantage of composite materials is that fracture seldom occurs catastrophicallywithout warning, but tends to be progressive, with subcritical damage widely dispersedthrough the material. Tensile loading (Fig. 6.15 (a)) can produce matrix cracking, fiberbridging, fiber rupture, fiber pullout and fiber/matrix debonding. Ultimate tensile failureof a fiber-reinforced composite often involves several of these mechanisms. Out-of-planestresses can lead to delamination (Fig. 6.15 (b)), because the fibers do not contribute sig-nificantly to strength in this direction. Compressive loading can produce microbucklingof fibers (Fig. 6.15 (c)); since the polymer matrix is soft compared to the fibers, thefibers are unstable in compression. Compressive loading can also lead to macroscopicdelamination buckling (Fig. 6.15 (d)), particularly if the material contains a pre-existingdelaminated region.

I

T(a) Single ply.

Elastic constants exhibitortho tropic symmetry.

EL » ETET

90°

(b) [0/45/90] laminate.

FIGURE 6.14 Schematic structure of fiber-reinforced composites.

332 Chapter 6

1. Fiber Pull-Out.2. Fiber Bridging.3. Fiber/Matrix Debonding4. Fiber Failure.5. Matrix Cracking.

(a) In-plane damage. (b) Delamination

t(c) Miorobuckling

t(d) Buckling delamination.

FIGURE 6.15 Examples of damage and fracture mechanisms in fiber-reinforced composites.

DelaminationOut-of-plane tensile stresses can cause failure between plies, as Fig. 6.15 (b) illus-

trates. Stresses that lead to delamination could result from the structural geometry, suchas if two composite panels are joined in a "T" configuration. Out-of-plane stresses,however, also arise from an unexpected source. Mismatch in Poisson ratios betweenplies results in shear stresses in the x-y plane near the ply interface. These shear stressesproduce a bending moment that is balanced by a stress in the z direction. For some lay-up sequences, substantial out-of-plane tensile stresses occur at the edge of the panel,


which can lead to the formation of a delamination crack. Figure 6.16 shows a computedaz distribution for a particular lay-up [21].

Although the assumption of self-similar growth of a dominant crack often does notapply to failure of composite materials, such an assumption is appropriate in the case ofdelamination. Consequently fracture mechanics has been very successful in characterizingthis failure mechanism.

20

, kPa 15|Lie10

5

0

-5

(-25/25/90) s /z /I /

. «»H - J.

^ -^

16 12 8 4 0d/t

FIGURE 6.16. Out-of-plane stress at mid-thickness in a composite laminate, normalized by the remotelyapplied strain (ue represents microstrain) [21]. The distance from the free edge, d, is normalized bythe ply thickness, t.

Delamination can occur in both Mode I and Mode II. The interlaminar fracturetoughness, which is usually characterized by a critical energy release rate (see Chapter 8),is related to the fracture toughness of the matrix material. The matrix and compositetoughness are seldom equal, however, due to the influence of the fibers in the latter.

Figure 6.17 is a compilation of (ftc values for various matrix materials, comparedwith the interlaminar toughness of the corresponding composite [22]. For brittle ther-mosets, the composite has higher toughness than the neat resin, but the effect is reversedfor high toughness matrices. Attempts to increase the composite toughness throughtougher resins have yielded disappointing results; only a fraction of the toughness of ahigh ductility matrix is transferred to the composite.

Let us first consider the reasons for the high relative toughness of composites withbrittle matrices. Figure 6.18 shows the fracture surface in a composite specimen with abrittle epoxy resin. The crack followed the fibers, implying that fiber/matrix debondingwas the crack growth mechanism in this case. The fracture surface has a "corrugated roofappearance; more surface area was created in the composite experiment, which apparentlyresulted in higher fracture energy. Another contributing factor in the composite tough-

334 Chapter 6

ness in this case is fiber bridging. In some instances, the crack grows around a fiber,which then bridges the crack faces, and adds resistance to further crack growth.

With respect to fracture of tough matrices, one possible explanation for the lowerrelative toughness of the composite is that the latter is limited by the fiber/matrix bond,which is weaker than the matrix material. Experimental observations, however, indicatethat fiber constraint is a more likely explanation [23], In high toughness polymers, ashear or craze damage zone forms ahead of the crack tip. If the toughness is sufficient forthe size of the damage zone to exceed the fiber spacing, the fibers restrain the crack tipyielding, resulting in a smaller zone than in the neat resin. The smaller damage zoneleads to a lower fracture energy between plies.

Delamination in Mode II loading is possible, but ^jjc is typically 2 to 10 timeshigher than the corresponding §ic [23]. The largest disparity between Mode I and ModeII interlaminar toughness occurs in brittle matrices.

In-situ fracture experiments in an SEM enable one to view the fracture process dur-ing delamination [23-25]. Long, slender damage zones containing numerous microcracksform ahead of the crack tip during Mode II loading. Figure 6.19 shows a sequence ofSEM fractographs of a Mode II damage zone ahead of a interlaminar crack in a brittleresin; the same region was photographed at different damage states. Note that the micro-cracks are oriented approximately 45° from the main crack, which is subject to Mode IIshear. Thus the microcracks are oriented perpendicular to the maximum normal stress,and are actually Mode I cracks. As loading progresses, these microcracks coalesce withthe main crack tip. The high relative toughness in Mode II results from energy dissipa-tion in this damage zone.

JX

u*"*

O

seC/D

0u

oCO

iftri

oc4

iftrH

O

Ifto"

1 I 1 1 1 1 1 1

RESINS:_ A THERMOSETS

0 EXPERIMENTALO TOUGHENED

THERMOSET4- THERMOPLASTIC ^ ^

o •" *

* '±r ^ ^" t

-'""o^_fiyi

1 1 1 1 1 1 1 1

1 I 1 1 I

O **". "~"

..x-

x ^^" O

"*

_.

*

—

1 1 1 1 1

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

RESIN Gic (kj/m2)

FIGURE 6.17 Compilation of interlaminar fracture toughness data, compared with the toughness of thecorresponding neat resin [22].


FIGURE 6.18 Fracture surface resulting from Mode I delamination of a graphite-epoxy composite witha brittle resin [23]. (Photograph provided by W.L. Bradley.)

In more ductile matrices, the appearance of the Mode II damage zone is similar tothe Mode I case, and the difference between (ftc and §HC is not as larg6 as f°r brittle ma-trices [23].

Compressive FailureHigh modulus fibers provide excellent strength and stiffness in tension, but are of

limited value for compressive loading. According to the Euler buckling equation, a col-umn of length L with a cross section moment of inertia /, subject to a compressive forceP becomes unstable when

P>K2EI

L2 (6.18)

assuming the loading is applied on the central axis of the column and the ends are unre-strained. Thus a long, slender fiber has very little load-carrying capacity in compression.

Equation (6.18) is much too pessimistic for composites, because the fibers are sup-ported by matrix material. Early attempts [26] to model fiber buckling in composites in-corporated an elastic foundation into the Euler bucking analysis, as Fig. 6.20 illustrates.This led to the following compressive failure criterion for unidirectional composites:

336 Chapter 6

(a) (b)

5 Mm

FIGURE 6.19 Sequence of photographs whichshow microcrack coalescence in a Mode H delami-nation experiment. (Photographs provided by Mr.Sun YongqL)

(C)

(6.19)

where fiijis the longitudinal-transverse shear modulus of the matrix and Efis Youngsmodulus of the fibers. This model overpredicts the actual compressive strength of com-posites by a factor of ~ 4.

One problem with Eq. (6.19) is that it assumes that the response of the material re-mains elastic; matrix yielding is likely for large lateral displacements of fibers. Anothershortcoming of this simple model is that it considers global fiber instability, while fiberbuckling is a local phenomenon; microscopic kink bands form, usually at a free edge, and


propagate across the panel [27,28].6 Figure 6.21 is a photograph of local fiber bucklingin a graphite-epoxy composite.

An additional complication in real composites is fiber waviness. Fibers are seldomperfectly straight; rather they tend to have a sine wave-like profile, as Fig. 6.22 illustrates[29]. Such a configuration is less stable in compression than a straight column.

FIGURE 6.20 Compressive loading of a columnthat is supported laterally by an elastic founda-tion.

Recent investigators [29-31] have incorporated the effects of matrix nonlinearity andfiber waviness into failure models. Most failure models are based on continuum theoryand thus do not address the localized nature of rnicrobuckling. Guynn [31], however, hasrecently performed detailed numerical simulations of compression loading of fibers in anonlinear matrix.

Microbuckling is not the only mechanism for compressive failure. Figure 6.15 (d)illustrates buckling delamination, which is a macroscopic instability. This type of failureis common in composites that have been subject to impact damage, which produces mi-crocracks and delamination flaws in the material. Delamination buckling induces Mode Iloading, which causes the delamination flaw to propagate at sufficiently high loads. Thisdelamination growth can be characterized with fracture mechanics methodology [32]. Acompression after impact test is a common screening criterion for assessing the ability ofa material to withstand impact loading without sustaining significant damage.

The long, slender appearance of the kink bands led several investigators [27,28] to apply the Dugdale-Barenblatt strip yield model to the problem. This model has been moderately successful in quantifying thesize of the compressive damage zones.

338 Chapter 6

FIGURE 6.21 Kink band formation in a graphite-epoxy composite [31]. (Photograph provided by E.G.Guynn.)

FIGURE 6.22 Fiber waviness in a graphite-epoxy composite [29]. (Photograph provided by A.L.Highsmith.)


Notch StrengthThe strength of a composite laminate that contains a hole or a notch is less than the

unnotched strength because of the local stress concentration effect. A circular hole in anisotropic plate has a stress concentration factor (SCF) of 3.0, and the SCF can be muchhigher for a elliptical notch (Section 2.2). If a composite panel with a circular hole failswhen the maximum stress reaches a critical value, the strength should be independent ofhole size, since SCF does not depend on radius. Actual strength measurements, however,indicate a hole size effect, where strength decreases with increasing hole size [33].

Figure 6.23 illustrates the elastic stress distributions ahead of a large hole and asmall hole. Although the peak stress is the same for both holes, the stress concentrationeffects of the large hole act over a wider distance. Thus the volume over which the stressacts appears to be important.

STRESS

FIGURE 6.23 Effect of hole size on localstress distribution

Whitney and Nuismer [34] proposed a simple model for notch strength, where fail-ure is assumed to occur when the stress exceeds the unnotched strength over a critical dis-tance.^ This distance is a fitting parameter that must be obtained by experiment.Subsequent modifications to this model, including the work of Pipes, et al. [35], yieldedadditional fitting parameters, but did not result in a better understanding of the failuremechanisms.

Figure 6.24 shows the effect of notch length on the strength panels that contain el-liptical center notches [33]. These experimental data actually apply to a boron-aluminumcomposite, but polymer composites exhibit a similar trend. The simple Whitney andNuismer criterion gives a reasonably good fit of the data in this case.

'Note the similarity to the Ritchie-Knott-Rice model for cleavage fracture (Chapter 5).

340 Chapter 6

1.0

0.8

0.6

O§ 0.4

OH«4

H

CO

0.2

Boron-AlununumLaminate

0 0.1 0.2 0.3 0.4 0.5 0.6

NOTCH LENGTH (a/W)FIGURE 6.24 Strength of center-notched composite laminates, relative to the unnotched strength [33].

Some researchers [36] have applied fracture mechanics concepts to the failure ofcomposites panels that contain holes and notches. They assume failure at a critical K,which is usually modified with a plastic zone correction to account for subcritical damage.Some of these models are capable of fitting experimental data such as that in Fig. 6.24,because the plastic zone correction is an adjustable parameter. The physical basis of thesemodels is dubious, however. Fracture mechanics formalism gives these models the illu-sion of rigor, but they have no more theoretical basis than the simple strength-of-materi-als approaches such as the Whitney-Nuismer criterion.

That linear elastic fracture mechanics is invalid for circular holes and blunt notchesin composites should be self evident, since LEFM theory assumes sharp cracks. If, how-ever, a sharp slit is introduced into a composite panel (Fig. 6.25), the validity (or lack ofvalidity) of fracture mechanics is less obvious. This issue is explored below.

Recall Chapter 2, which introduced the concept of a singularity zone, where the

stress and strain vary as 7/V r from the crack tip. Outside of the singularity zone, higherorder terms, which are geometry dependent, become significant. For K to define uniquelythe crack tip conditions and be a valid failure criterion, all nonlinear material behaviormust be confined to a small region inside the singularity zone. This theory is based en-tirely on continuum mechanics. While metals, plastics and ceramics are often heteroge-neous, the scale of microstructural constituents is normally small compared to the size ofthe singularity zone; thus the continuum assumption is approximately valid.

For LEFM to be valid for a sharp crack in a composite panel, the following condi-tions must be met:


JUTFIGURE 6.25 Sharp notch artificially introducedinto a composite panel.

(1) The fiber spacing must be small compared to the size of the singularityzone. Otherwise, the continuum assumption is invalid.

(2) Nonlinear damage must be confined to a small region within the singularityzone.

Harris and Morris [37] showed that K characterizes the onset of damage in crackedspecimens, but not ultimate failure, because damage spreads throughout the specimen be-fore failure, and K no longer has any meaning. Figure 6.25 illustrates a typical damagezone in a specimen with a sharp macroscopic notch. The damage, which includesfiber/matrix debonding and matrix cracking, actually propagates perpendicular to themacrocrack. Thus the crack does not grow in a self-similar fashion.

One of the most significant shortcomings of tests on composite specimens withnarrow slits is that defects of this type do not occur naturally in fiber-reinforced compos-ites; therefore, the geometry in Fig. 6.25 is of limited practical concern. Holes and bluntnotches may be unavoidable in a design, but a competent design engineer would not befoolish enough to include a sharp notch in a load-bearing member of a structure.

Fatigue DamageCyclic loading of composite panels produces essentially the same type of damage as

monotonic loading. Fiber rupture, matrix cracking, fiber/matrix debonding, and delamina-tion all occur in response to fatigue loading. Fatigue damage reduces the strength andmodulus of a composite laminate, and eventually leads to total failure.

Figures 6.26 and 6.27 show the effect of cyclic stresses on the residual strength andmodulus of graphite/epoxy laminates [38]. Both strength and modulus decrease rapidly af-

342 Chapter 6

ter relatively few cycles, but remain approximately constant up to around 80% of the fa-tigue life. Near the end of the fatigue life, strength and modulus decrease further.

0.2 0.4 0.6 0.8 1.0'tC

NORMALIZED NUMBER OF CYCLES (n/N)

FIGURE 6.26 Residual strength after fatigue damage in a graphite-epoxy laminate [38].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1NORMALIZED CYCLES (n/N)

FIGURE 6.27 Residual modulus after fatigue damage in a graphite-epoxy laminate [38].


6.2 CERAMICS AND CERAMIC COMPOSITES

A number of technological initiatives have been proposed whose implementationdepend on achieving major advances in materials technology. For example, the NationalAerospace Plane will re-enter the Earth's atmosphere at speeds of up to Mach 25, creatingextremes of both temperature and stress. Also, the Advanced Turbine TechnologyApplications Program (ATTAP) has the stated goal of developing heat engines whichhave a service life of 3000 h at 1350 °C. Additional applications are on the horizon thatwill require materials that can perform at temperatures in excess of 2000 °C. All metals,including cobalt-based superalloys, are inadequate at these temperatures. Only ceramicspossess adequate creep resistance above 1000°C.

Ceramic materials include oxides, carbides, sulfides, and intermetallic compounds,which are joined either by covalent or ionic bonds. Most ceramics are crystalline but, un-like metals, they do not have close-packed planes on which dislocation motion can occur.Therefore, ceramic materials tend to be very brittle compared to metals.

Typical ceramics have very high melting temperatures, which explains their goodcreep properties. Also, many of these materials have superior wear resistance, and havebeen used for bearings and machine tools. Most ceramics, however, are too brittle forcritical load-bearing applications. Consequently, a vast amount of research has been de-voted to improving the toughness of ceramics.

Most traditional ceramics are monolithic (single phase) and have very low fracturetoughness. Because they do not yield, monolithic ceramics behave as ideally brittle mate-rials (Fig. 2.6(a)), and a propagating crack need only overcome the surface energy of thematerial. The new generation of ceramics, however, includes multiphase materials and ce-ramic composites that have vastly improved toughness. Under certain conditions, twobrittle solids can be combined to produce a material that is significantly tougher than ei-ther parent material.

The micromechanisms that lead to improved fracture resistance in modern ceramicsinclude microcrack toughening, transformation toughening, ductile phase toughening,fiber toughening, and whisker toughening. Table 6.1 lists the dominant tougheningmechanism in several materials, along with the typical fracture toughness values [39].Fiber toughening, the most effective mechanism, produces toughness values around 20

MPavm, which is below the lower shelf toughness of steels but is significantly higherthan most ceramics.

Evans [39] divides toughening mechanisms for ceramics into two categories: pro-cess zone formation and bridging. Both mechanisms involve energy dissipation at thecrack tip. A third mechanism, crack deflection, elevates toughness by increasing the areaof the fracture surface (Fig. 2.6(c)).

The process zone mechanism for toughening is illustrated in Fig. 6.28. Consider amaterial that forms a process zone at the crack tip (Fig. 6.28(a)). When this crack propa-gates, it leaves a wake behind the crack tip. The critical energy release rate for propaga-tion is equal to the work required to propagate the crack from a to a + da, divided by da:

344 Chapter 6

0 0

(6.20)

where h is the half width of the process zone and YS istne surface energy. The integral inthe square brackets is the strain energy density, which is simply the area under the stress-strain curve in the case of uniaxial loading. Figure 6.28(b) compares the stress-straincurve of brittle and toughened ceramics. The latter material is capable of higher strains,and absorbs more energy prior to failure.

Many toughened ceramics contain second-phase particles that are capable of nonlin-ear deformation, and are primarily responsible for the elevated toughness. Figure 6.28(c)illustrates the process zone for such a material. Assuming the particles provide all of theenergy dissipation in the process zone, and the strain energy density in this region doesnot depend on y, the fracture toughness is given by

(6.21)

TABLE 6.1 Ceramics with enhanced toughness [39].

Toughening Mechanism

Fiber reinforced

Whisker reinforced

Ductile network

Transformation toughened

Microcrack toughened

Material

LAS/SiCGlass/CSiC/SiC

Al2O3/SiC(0.2)Si3N4/SiC(0.2)

A1203/A1(0.2)B4C/A1(0.2)WC/Co(0.2)

PSZTZPZTA

ZTASisN^SiC

Maximum

Toughness, MPa V m

-20-20-20

1014

121420

181610

77


(a) Process zone formed by growing crack (b) Schematic stress-strain behavior

FIGURE 6.28 The process zone mechanismfor ceramic toughening.

(c) Nonlinear deformation of second-phase particles

where/is the volume fraction of second-phase particles. Thus the toughness is controlledby the width of the process zone, the concentration of second-phase particles, and the areaunder the stress-strain curve. Recall the delamination of composites with tough resins(Section 6.1.3), where the fracture toughness of the composite was not as great as theneat resin because the fibers restricted the size of the process zone (h).

The process zone mechanism often results in a rising R curve, as Fig. 6.29 illus-trates. The material resistance increases with crack growth, as the width of the processeszone grows. Eventually, h and @R reach steady-state values.

Figure 6.30 illustrates the crack bridging mechanism, where the propagating crackleaves fibers or second-phase particles intact. The unbroken fibers or particles exert a trac-tion force on the crack faces, much like the Dugdale-Barenblatt strip yield model [17,18].

346 Chapter 6

The fibers eventually rupture when the stress reaches a critical value. According to Eqs.(3.42) and (3.43), the critical energy release rate for crack propagation is given by

(6.22)0

The sections that follow outline several specific toughening mechanisms in modemceramics.

Toughened Ceramic

Brittle Ceramic

CRACK EXTENSION (Aa)

FIGURE 6.29 The process zone toughening mechanism usually results in a rising R curve.

FIGURE 6 JO The fiber bridging mechanismfor ceramic toughening.


6.2.1 Microcrack Toughening

Although the formation of cracks in a material is generally considered deleterious, microc-racking can sometimes lead to improved toughness. Consider a material sample of vol-ume V that forms N microcracks when subject to a particular stress. If these cracks arepenny shaped with an average radius a, the total work required to form these microcracksis equal to the surface energy times the total area created:

Wc = 2NnaYs (6.23)

The formation of microcracks releases strain energy from the sample, which results in anincrease in compliance. If this change in compliance is gradual, as existing microcracksgrow and new cracks form, a nonlinear stress-strain curve results. The change in strainenergy density due to the microcrack formation is given by

5 (6.24)

where p = ///Vis the microcrack density. For a macroscopic crack that produces a pro-cess zone of microcracks, the increment of toughening due to microcrack formation canbe inferred by inserting Eq. (6.24) into Eq. (6.21).

A major problem with the above scenario is that stable microcrack growth does notusually occur in a brittle solid. Pre-existing flaws in the material remain stationary untilthey satisfy the Griffith criterion, at which time they become unstable. Stable crack ad-vance normally requires either a rising R curve, where the fracture work (wf, see Fig. 2.6)increases with crack extension, or physical barriers in the material that inhibit crackgrowth. Stable microcracking occurs in concrete because aggregates act as crack arresters(see Section 6.3).

Certain multiphase ceramics have the potential for microcrack toughening. Figure6.31 schematically illustrates this toughening mechanism [39]. Second-phase particlesoften are subject to residual stress due to thermal expansion mismatch or transformation.If the residual stress in the particle is tensile and the local stress in the matrix is compres-sive°, the particle cracks. If the signs on the stresses are reversed, the matrix materialcracks at the interface. In both cases there is a residual opening of the microcracks, whichleads to an increase in volume in the sample. Figure 6.31 (b) illustrates the stress-strainresponse of such a material. The material begins to crack at a critical stress, crc, and thestress-strain curve becomes nonlinear, due to a combination of compliance increase and di-latational strain. If the material is unloaded prior to total failure, the relative contribu-tions of dilatational effects (residual microcrack opening) and modulus effects (due to therelease of strain energy) are readily apparent.

A number of multiphase ceramic materials exhibit trends in toughness with particlesize and temperature that are consistent with the microcracking mechanism, but this phe-

o°The residual stresses in the matrix and particle must balance in order to satisfy equilibrium.

348 Chapter 6

nomenon has been directly only observed in aluminum oxide toughened with monocliniczirconium dioxide [40].

This mechanism is relatively ineffective, as Table 6.1 indicates. Moreover, the de-gree of microcrack toughening is temperature dependent. Thermal mismatch and the re-sulting residual stresses tend to be lower at elevated temperatures, which implies less di-latational strain. Also, lower residual stresses may not prevent the microcracks from be-coming unstable and propagating through the particle/matrix interface.

STRESS

STRAIN

FIGURE 6.31 The microcrack toughening mechanism [39]. The formation of microcracks in or nearsecond-phase particles results in release of strain energy (modulus work) and residual microcrack open-ing (dilatational work).

6.2.2 Transformation Toughening

Some ceramic materials experience a stress-induced martensitic transformation that resultsin shear deformation and a volume change (i.e., a dilatational strain). Ceramics that con-tain second-phase particles that transform often have improved toughness. Zirconiumdioxide (ZrO2) is the most widely studied material that exhibits a stress-induced marten-sitic transformation [41].

Figure 6.32 illustrates the typical stress-strain behavior for a martensitic transforma-tion [41]. At a critical stress, the material transforms, resulting in both dilatational andshear strains. Figure 6.33(a) shows a crack tip process zone, where second-phase particleshave transformed.

The toughening mechanism for such a material can be explained in terms of thework argument: energy dissipation in the process zone results in higher toughness. Analternative explanation is that of crack tip shielding, where the transformation lowers thelocal crack driving force [41]. Figure 6.33(b) shows the stress distribution ahead of thecrack with a transformed process zone. Outside of this zone, the stress field is defined bythe global stress intensity, but the stress field in the process zone is lower, due to dilata-


tional effects. The crack tip work and shielding explanations are consistent with one an-other; more work is required for crack extension when the local stresses are reduced.Crack tip shielding due to the martensitic transformation is analogous to the stress redis-tribution that accompanies plastic zone formation in metals (Chapter 2).

The transformation stress and the dilatational strain are temperature dependent.These quantities influence the size of the process zone, h, and the strain energy densitywithin this zone. Consequently, the effectiveness of the transformation tougheningmechanism also depends on temperature. Below Ms, the martensite start temperature, thetransformation occurs spontaneously, and the transformation stress is essentially zero.Thermally transformed martensite does not cause crack tip shielding, however [41].Above Ms, the transformation stress increases with temperature. When this stress be-comes sufficiently large, the transformation toughening mechanism is no longer effective.

STRESS

(Jc

Dilatational 'Strain i FIGURE 6.32 Schematic stress-strain response

of a material that exhibits a martensitic trans-formation at a critical stress.

STRAIN

STRESS

CJc

DISTANCE

(a) Process zone. (b) Crack tip stress field

FIGURE 6.33 The martensitic toughening mechanism. Transformation of particles near the crack tipresults in a nonlinear process zone (a) and crack tip shielding (b).

350 Chapter 6

6.2.3 Ductile Phase Toughening

Ceramics alloyed with ductile particles exhibit both bridging and process zone toughen-ing, as Fig. 6.34 illustrates. Plastic deformation of the particles in the process zone con-tributes toughness, as does the ductile rupture of the particles that intersect the crackplane. Figure 6.35 is an SEM fractograph of bridging zones in A1203 reinforced withaluminum [39]. Residual stresses in the particles can also add to the material's tough-ness. The magnitude of the bridging and process zone toughening depends on the volumefraction and flow properties of the particles. The process zone toughening also dependson the particle size, with small particles giving the highest toughness [39].

This toughening mechanism is temperature dependent, since the flow properties ofthe metal particles vary with temperature. Ductile phase ceramics are obviously inappro-priate for applications above the melting temperature of the metal particles.

6.2.4 Fiber and Whisker Toughening

One of the most interesting features of ceramic composites is that the combination of abrittle ceramic matrix with brittle ceramic fibers or whiskers can result in a material withrelatively high toughness (Table 6.1). The secret to the high toughness of ceramic com-posite lies in the bond between the matrix and the fibers or whiskers. Having a brittle in-terface leads to higher toughness than a strong interface. Thus ceramic composites defy in-tuition: a brittle matrix bonded to a brittle fiber by a brittle interface results in a toughmaterial.

A weak interface between the matrix and reinforcing material aids the bridgingmechanism. When a matrix crack encounters a fiber/matrix interface, this interface expe-riences Mode II loading; debonding occurs if the fracture energy of the interface is low(Fig. 6.36(a)). If the extent of debonding is sufficient, the matrix crack bypasses thefiber, leaving it intact. Mathematical models [42] of fiber/matrix debonding predict crackbridging when the interfacial fracture energy is an order of magnitude smaller than the ma-trix toughness. If the interfacial bond is strong, matrix cracks propagate through thefiber, and the composite toughness obeys a rule of mixtures; but bridging increases thecomposite toughness (Fig. 6.36(c)).

FIGURE 6.34 Ductile phase toughening. Ductilesecond-phase particles increase the ceramictoughness by plastic deformation in the processzone, as well as by a bridging mechanism.


FIGURE 6.35 Ductile-phase bridging in ATjOs/Al [39]. (Photograph provided by A.G. Evans.)

An alternate model [42-44] for bridging in fiber-reinforced ceramics assumes that thefibers are not bonded, but that friction between the fibers and the matrix restrict the crackopening (Fig. 6.36(b)). The model that considers Mode II debonding [42] neglects fric-tion effects, and predicts that the length of the debond controls the crack opening.

Both models predict steady-state cracking, where the matrix cracks at a constantstress that does not depend on the initial flaw distribution in the matrix. Experimentaldata support the steady-state cracking theory. Because the cracking stress is independentof flaw size, fracture toughness measurements (e.g., Kjc and ^c) have little or no mean-

ing.Figure 6.37 illustrates the stress-strain behavior of a fiber-reinforced ceramic. The

behavior is linear elastic up to ac, the steady-state cracking stress in the matrix. Once the

352 Chapter^

matrix has cracked, the load is carried by the fibers. The fibers do not fail simultane-ously, because the fiber strength is subject to statistical variability [45]. Consequently,the material exhibits quasiductility, where damage accumulates gradually until final fail-ure.

Not only is fiber bridging the most effective toughening mechanism for ceramics(Table 6.1), it is also effective at high temperatures [46,47], Consequently applicationsthat require load-bearing capability at temperatures above 1000°C will undoubtedly utilizefiber-reinforced ceramics.

(a) Fiber/matrix debonding.(b) Frictional sliding along interfaces.

FIGURE 6.36 Fiber bridging in ceramic compos-ites. Mathematical models treat bridging eitherin terms of fiber/matrix debonding (a) or fric-tional sliding (b). This mechanism provides com-posite toughness well in excess of that predictedby the rule of mixtures (c).

FIBER VOLUME FRACTION'

(c) Effect of bridging on toughness.


Whisker-reinforced ceramics possess reasonably high toughness, although whiskerreinforcement is not as effective as continuous fibers. The primary failure mechanism inwhisker composites appears to be bridging [48]; crack deflection also adds an increment oftoughness. Figure 6.38 is a micrograph that illustrates crack bridging in an A1203 ce-ramic reinforced by SiC whiskers.

STRESS

STRAIN 1. Steady-State Cracking 2. Fiber Failure

FIGURE 6.37 Stress-strain behavior of fiber-reinforced ceramic composites.

FIGURE 6.38 Crack bridging inA.G. Evans.)

reinforced with SiC whiskers [39]. (Photograph provided by

354 Chapter 6

6.3 CONCRETE AND ROCK

Although concrete and rock are often considered brittle, they are actually quasibrittlematerials that are tougher than most of the so-called advanced ceramics. In fact, much ofthe research on toughening mechanisms in ceramics is aimed at trying to make ceramiccomposites behave more like concrete.

Concrete and rock derive their toughness from subcritical cracking that precedes ul-timate failure. This subcritical damage results in non-linear stress-strain response and R-curve behavior.

A traditional strength-of-materials approach to designing with concrete has provedinadequate because fracture strength is often size dependent [49]. This size dependence isdue to the fact that nonlinear deformation in these materials is caused by subcriticalcracking rather than plasticity. Initial attempts to apply fracture mechanics to concretewere unsuccessful because these early approaches were based on linear elastic fracturemechanics (LEFM) and failed to take account of the process zones that form in front ofmacroscopic cracks.

This section gives a brief overview of the mechanisms and models of fracture inconcrete and rock. Although most of the experimental and analytical work has been di-rected at concrete as opposed to rock, due to the obvious technological importance of theformer, rock and concrete behave in a similar manner. The remainder of this section willrefer primarily to concrete, with the implicit understanding that most observations andmodels also apply to geologic materials.

Figure 6.39 schematically illustrates the formation of a fracture process zone in con-crete, together with two idealizations of the process zone. Microcracks form ahead of amacroscopic crack, which consists of a bridged zone directly behind the tip and a traction-free zone further behind the tip. The bridging is a result of the weak interface between theaggregates and the matrix. (Recall Section 6.2.4, where it was stated that fiber bridging,which occurs when the fiber-matrix bonds are weak, is the most effective tougheningmechanism in ceramic composites.) The process zone can be modeled as a region of strainsoftening (Fig. 6.40(b)) or as a longer crack that is subject to closure tractions (Fig.6.40(c)). The latter is a slight modification to the Dugdale-Barenblatt strip yield model.

Figure 6.40 illustrates the typical tensile response of concrete. After a small degreeof nonlinearity caused by microcracking, the material reaches its tensile strength, o>, andthen strain softens. Once o^ is reached, subsequent damage is concentrated in a local frac-ture zone. Virtually all of the displacement following the maximum stress is due to thedamage zone. Note that Fig. 6.40 shows a schematic stress-displacement curve rather thana stress-strain curve. The latter is inappropriate because nominal strain measured over theentire specimen is a function of gage length.

There are a number of models for fracture in concrete, but the one that is mostwidely referenced is the so-called fictitious crack model of Hillerborg [50,51]. This model,which has also been called a cohesive zone model, is merely an application of theDugdale-Barenblatt approach. The Hillerborg model assumes that the stress displacementbehavior (a-5) observed in the damage zone of a tensile specimen is a material property.Figure 6.41 (a) shows a schematic stress-displacement curve, and 6.41(b) illustrates theidealization of the damage zone ahead of a growing crack.


Traction-Free Crackgjpr SS«mp

Bridging— fes>—ppy.

Microcrackin•Ifli fiftr**«lf P*

(a) Crack growth in concrete.

(b) Process zone idealized as a zone of strain softening.

(c) Process zone idealized by closure tractions.

FIGURE 6.39 Schematic illustration of crack growth in concrete, together with two simplified models.

At the tip of a the traction-free crack, the damage zone reaches a critical displace-ment, 8C. The tractions are zero at this point, but are equal to the tensile strength, o>, atthe tip of the damage zone (Fig. 6.39(c)). Assuming that the closure stress, a, and open-ing displacement, 6, are uniquely related, the critical energy release rate for crack growthis given by

(6.25)

which is virtually identical to Eqs. (3.43) and (6.22).

356 Chapter 6

STRESS

t\

DamageZone

DISPLACEMENT (A)

FIGURE 6.40. Typical tensile response of concrete.

STRESS

DISPLACEMENT (8)

(a) Schematic stress-displacement response.

(b) Damage zone ahead of a growing crack.

FIGURE 6.41 The "fictitious crack" model for concrete [50,51].


The key assumption of the Hillerborg model that the cr-6 relationship is a uniquematerial property is not strictly correct in most cases because process zones produced dur-ing fracture of concrete are often quite large, and interaction between the process zone andfree boundaries can influence the behavior. Consequently, (?c is not a material property ingeneral, but can depend on specimen size. Fracture toughness results from small-scaletests tend to be lower than values obtained from larger samples.

The phenomena of size effects in concrete fracture has been the subject of consider-able study in recent years [49,52-55]. Some of the apparent size effects can be attributedto inappropriate data reduction methods. For example, if fracture toughness is computedby substituting the load at fracture into a linear elastic K or ^/relationship, the resultingvalue will be size dependent, because the nonlinearity due to the process zone has beenneglected. Such an approach is analogous to applying linear elastic K equations to metalspecimens that exhibit significant plasticity prior to failure. Even approaches that accountfor the process zone exhibit size dependence, however [53]. A fracture parameter thatuniquely characterizes these materials would be of great benefit.

REFERENCES

1. Johnson, W.S., ed., Metal Matrix Composites: Testing, Analysis, and Failure Modes.ASTM STP 1032, American Society for Testing and Materials, Philadelphia, 1989.

2. Williams, J.G. Fracture Mechanics of Polymers, , Halsted Press, John Wiley & Sons,New York, 1984.

3. Kinloch, A.J. and Young, R.J., Fracture Behavior of Polymers, Elsevier AppliedScience Publishers, London, 1983.

4. Brostow, W. and Corneliussen, R.D., eds., Failure of Plastics, Hanser Publishers,Munich, 1986.

5. Hertzberg, R.W. and Manson, J.A., Fatigue of Engineering Plastics, Academic Press,New York, 1980.

6. Ferry, J.D., Landel, R.F., and Williams, M.L., "Extensions of the Rouse Theory ofViscoelastic Properties to Undiluted Linear Polymers." Journal of Applied Physics, Vol26, pp. 359-362, 1955.

7. Larson, F.R. and Miller, I, "A Time-Temperature Relationship for Rupture and CreepStresses", Transactions of the American Society for Mechanical Engineers, Vol 74, pp.765-775, 1952.

8. Kausch, H.H., Polymer Fracture, Springer, Heidelberg-New York, 1978.

9. Zhurkov, S.N. and Korsukov, V.E., "Atomic Mechanism of Fracture of Solid Polymers,"Journal of Polymer Science: Polymer Physics Edition, Vol. 12, pp. 385-398, 1974.

10. Ward, I.M., Mechanical Properties of Solid Polymers. John Wiley & Sons Ltd., NewYork, 1971.

358 Chapter 6

11. Sternstein, S.S. and Ongchin, L., "Yield Criteria for Plastic Deformation of Glassy HighPolymers in General Stress Fields." American Chemical Society, Polymer Preprints,Vol. 10, pp. 1117-1124, 1969.

12. Bucknall, C.B., Toughened Plastics, Applied Science Publishers, London, 1977.

13. Donald, A.M. and Kramer, E.J., "Effect of Molecular Entanglements on CrazeMicrostructure in Glassy Polymers." Journal of Polymer Science: Polymer PhysicsEdition, Vol. 27, pp. 899-909, 1982.

14. Oxborough, RJ. and Bowden, P.B., "A General Critical-Strain Criterion for Crazing inAmorphous Glassy Polymers.", Philosophical Magazine, Vol. 28, 1973, pp. 547-559.

15. Argon, A.S., "The Role of Heterogeneities in Fracture." ASTM STP 1020, AmericanSociety for Testing and Materials, Philadelphia, 1989, pp. 127-148.

16. Cayard, M., "Fracture Toughness Testing of Polymeric Materials." Ph.D. Dissertation,Texas A&M University, College Station, TX, September, 1990.



19. Engineered Materials Handbook, Volume 1: Composites. ASM International, MetalsPark, OH, 1987.

20. Vinson, J.R. and Sierakowski, R.L., The Behavior of Structures Composed ofComposite Materials. Marinus Nijhoff, Dordrecht, The Netherlands, 1987.

21. Wang, A.S.D., "An Overview of the Delamination Problem in Structural Composites."Key Engineering Materials, Vol. 37, 1989, pp. 1-20.

22. Hunston, D. and Dehl, R., "The Role of Polymer Toughness in Matrix DominatedComposite Fracture." Paper EM87-355, Society of Manufacturing Engineers, Deerborn,MI, 1987.

23. Bradley, W.L., "Understanding the Translation of Neat Resin Toughness intoDelamination Toughness in Composites." Key Engineering Materials, Vol. 37, 1989,pp. 161-198.

24. Jordan, W.M, and Bradley, W.L., "Micromechanisms of Fracture in ToughenedGraphite-Epoxy Laminates." ASTM STP 937, American Society for Testing andMaterials, Philadelphia, 1987, pp. 95-114.

25. Hibbs, M.F., Tse, M. K., and Bradley, W.L., "Interlaminar Fracture Toughness and Real-Time Fracture Mechanisms of Some Toughened Graphite/Epoxy Composites." ASTMSTP 937, American Society for Testing and Materials, Philadelphia, 1987, pp 115-130.

26. Rosen, B.W., "Mechanics of Composite Strengthening." Fiber Composite Materials,American Society for Metals, Metals Park, OH, 1965, pp. 37-75.


27. Guynn, E.G., Bradley, W.L. and Elber, W., "Micromechanics of Compression Failuresin Open Hole Composite Laminates." ASTM STP 1012, American Society for Testingand Materials, Philadelphia, 1989, pp 118-136.

28. Soutis, C., Fleck, N.A., and Smith, P.A., 'Failure Prediction Technique for CompressionLoaded Carbon Fibre-Epoxy Laminate with Open Holes." Submitted to Journal ofComposite Materials, 1990.

29. Highsmith, A.L. and Davis, J., "The Effects of Fiber Waviness on the CompressiveResponse of Fiber-Reinforced Composite Materials." Progress Report for NASAResearch Grant NAG-1-659, NASA Langley Research Center, Hampton, VA, January1990.

30. Wang, A.S.D., "A Non-Linear Microbuckling Model Predicting the CompressiveStrength of Unidirectional Composites." ASME Paper 78-WA/Aero-l, AmericanSociety for Mechanical Engineers, New York, 1978.

3 1. Guynn, E.G., "Experimental Observations and Finite Element Analysis of the Initiationof Fiber Microbuckling in Notched Composite Laminates." Ph.D. Dissertation, TexasA&M University, College Station, TX, December 1990.

32. Whitcomb, J.D., "Finite Element Analysis of Instability Related DelaminationGrowth." Journal of Composite Materials, Vol. 15, 1981, pp. 403-425.

33. Awerbuch, J. and Madhukar, M.S., "Notched Strength of Composite Laminates:Predictions and Experiments-A Review." Journal of Reinforced Plastics andComposites, Vol. 4, 1985, pp. 3-159.

34. Whitney, J.M. and Nuismer, R.J., "Stress Fracture Criteria for Laminated CompositesContaining Stress Concentrations." Journal of Composite Materials, Vol. 8, 1974, pp.253-265.

35. Pipes, R.B., Wetherhold, R.C., and Gillespie, J.W., Jr., "Notched Strength ofComposite Materials." Journal of Composite Materials, Vol. 12, 1979, pp. 148-160.

36. Waddoups, M.E., Eisenmann, J.R., and Kaminski, B.E., "Macroscopic FractureMechanics of Advanced Composite Materials." Journal of Composite Materials, Vol.5, 1971, pp. 446-454.

37. Harris, C.E. and Morris, D.H., "A Comparison of the Fracture Behavior of ThickLaminated Composites Utilizing Compact Tension, Three-Point Bend, and Center-Cracked Tension Specimens." ASTM STP 905, American Society for Testing andMaterials, Philadelphia, 1986, pp. 124-135.

38. Charewicz, A. and Daniel, I.M., "Damage Mechanisms and Accumulation inGraphite/Epoxy Laminates." ASTM STP 907, American Society for Testing andMaterials, Philadelphia, 1986, pp. 274-297.

39. Evans, A.G., "The New High Toughness Ceramics." ASTM STP 907, American Societyfor Testing and Materials, Philadelphia, 1989, pp. 267-291.

40. Hutchinson, J.W., "Crack Tip Shielding by Micro Cracking in Brittle Solids", ActaMetallurgica, Vol. 35, 1987, p. 1605-1619.

360 Chapter 6

41. A.G. Evans, ecL, Fracture in Ceramic Materials: Toughening Mechanisms, MachiningDamage, Shock. Noyes Publications, Park Ridge, NJ, 1984.

42. Budiansky, B., Hutchinson, J.W., and Evans, A.G., "Matrix Fracture in Fiber-Reinforced Ceramics." Journal of the Mechanics and Physics of Solids, Vol. 34, 1986,pp. 167-189.

43. Aveston, J., Cooper G.A., and Kelly, A., The Properties of Fiber Composites, 1971,pp 15-26.

44. Marshall, D.B., Cox, B.N. and Evans, A.G., "The Mechanics of Matrix Cracking inBrittle-Matrix Fiber Composites." Acta Metallurgica, Vol 33, 1985, pp. 2013-2021.

45. Marshall, D,B. and Ritter, I.E., "Reliability of Advanced Structural Ceramics andCeramic Matrix Composites—A Review." Ceramic Bulletin, Vol. 68, 1987, pp. 309-317.

46. Mah, T., Mendiratta, M.G., Katz, A.P., Ruh, R., and Mazsiyasni, K.S., "RoomTemperature Mechanical Behavior of Fiber-Reinforced Ceramic Composites." Journalof the American Ceramic Society, Vol. 68, 1985, pp. C27-C30.

47. Mah, T., Mendiratta, M.G., Katz, A.P., Ruh, R., and Mazsiyasni, K.S., "High-Temperature Mechanical Behavior of Fiber-Reinforced Glass-Ceramic-MatrixComposites." Journal of the American Ceramic Society., Vol. 68, 1985, pp. C248-C251.

48. Ruble, M., Dalgleish, B.J., and Evans, A.G., "On the Toughening of Ceramics byWhiskers." Scripta Metallurgica, Vol 21, pp. 681-686.

49. Bazant, Z.P., "Size Effect in Blunt Fracture: Concrete, Rock, Metal." Journal ofEngineering Mechanics, Vol. 110, 1984, pp. 518-535.

50. Hillerborg, A. , Modeer, M., and Petersson, P.E., "Analysis of Crack Formation andCrack Growth in Concrete by Means of Fracture Mechanics and Finite Elements."Cement and Concrete Research, Vol. 6, 1976, pp. 773-782.

51. Hillerborg, A. "Application of the Fictitious Crack Model to Different Materials."International Journal of Fracture, Vol. 51, 1991, pp. 95-102.

52. Bazant, Z.P. and Kazemi, M.T., "Determination of Fracture Energy, Process ZoneLength and Brittleness Number from Size Effect, with Application to Rock andConcrete." International Journal of Fracture, Vol. 44, 1990, pp. 111-131.

53. Bazant, Z.P. and Kazemi, M.T., Size Dependence of Concrete Fracture EnergyDetermined by RILEM Work-of-Fracture Method." International Journal of Fracture,Vol. 51, 1991, pp. 121-138.

54. Planas, J. and Slices, M., "Nonlinear Fracture of Cohesive Materials." InternationalJournal of Fracture, Vol. 51, 1991, pp, 139-157.


55. Mazars, J., Pijaudier-Cabot, G., and Saourdis, C., "Size Effect and Continuous Damagein Cementitious Materials." International Journal of Fracture, Vol. 51, 1991, pp.159-173.

PART IV: APPLICATIONS

7. FRACTURE TOUGHNESS TESTINGOF METALS

A fracture toughness test measures the resistance of a material to crack extension. Such atest may yield either a single value of fracture toughness or a resistance curve, where atoughness parameter such as K, J, or CTOD is plotted against crack extension. A singletoughness value is usually sufficient to describe a test that fails by cleavage, because thisfracture mechanism is typically unstable. The situation is similar to the schematic in Fig2.10(a), which illustrates a material with a flat R curve. Cleavage fracture actually has afalling resistance curve, as Fig. 4.8 illustrates. Crack growth by microvoid coalescence,however, usually yields a rising R curve, such as that shown in Fig. 2.10(b); ductilecrack growth can be stable, at least initially. When ductile crack growth initiates in a testspecimen, that specimen seldom fails immediately. Therefore, one can quantify ductilefracture resistance either by the initiation value or by the entire R curve.

A variety of organizations throughout the world publish standardized procedures forfracture toughness measurements, including the American Society for Testing andMaterials (ASTM), the British Standards Institution (BSI), the International Institute ofStandards (ISO) and the Japan Society of Mechanical Engineers (JSME). The first stan-dards for K and J testing were developed by ASTM in 1970 and 1981, respectively, whileBSI published the first CTOD test method in 1979.

Existing fracture toughness standards include procedures for Kjc, K-R curve, //c, J-Rcurve, CTOD, and Kja testing. This chapter focuses primarily on ASTM standards, sincethey are the most widely used throughout the world. Standards produced by other organi-zations, however, are generally consistent with the ASTM procedures, and usually differonly in minute details. A few of the more substantive differences between alternativeprocedures are discussed briefly in this chapter.

The ongoing work of the various standardizing bodies is also discussed. The exist-ing standards are continuously evolving, as the technology improves and more experienceis gained. Also, there are still some important applications, such as weldrnent testing,that present standards do not address. The fracture testing community has, however, ob-tained substantial experience in some of these areas, and draft standards are currently beingprepared.

The reader should not rely on this chapter alone for guidance on conducting fracturetoughness tests, but should consult the relevant standards. Also, the reader is stronglyencouraged to review Chapters 2 and 3 in order to gain an understanding of the fundamen-tal basis of K, J, and CTOD, as well as the limitations of these parameters.

7.1 GENERAL CONSIDERATIONS

Virtually all fracture toughness tests have several common features. The design of testspecimens is similar in each of the standards, and the orientation of the specimen relativeto symmetry directions in the material is always an important consideration. The cracksin test specimens are introduced by fatigue in each case, although the requirements for fa-

365

366 Chapter 7

tigue loads varies from one standard to the next. The basic instrumentation required tomeasure load and displacement is common to virtually all fracture mechanics tests, butsome tests require additional instrumentation to monitor crack growth.

7.1.1 Specimen Configurations

There are five types of specimens that are permitted in ASTM standards that characterizefracture initiation and crack growth, although no single standard allows all five configura-tions, and the design of a particular specimen type may vary between standards. The con-figurations that are currently standardized include the compact specimen, the single edgenotched bend (SENB) geometry, the arc-shaped specimen, the disk specimen, and the mid-dle tension (MT) panel. Figure 7.1 shows a drawing of each specimen type.

An additional configuration, the compact crack arrest specimen, is used for Kja mea-surements and is described in Section 7.6. Specimens for qualitative toughness measure-ments, such as Charpy and drop weight tests, are discussed in Section 7.9. Chevronnotched specimens, which are applied to brittle materials, are discussed in Chapter 8.

Each specimen configuration has three important characteristic dimensions: thecrack length (a), the thickness (B) and the width (W). In most cases, W = 2 B and a/W~ 0.5, but there are exceptions which are discussed later in this chapter.

There are number of specimen configurations that are used in research, but have yetto be standardized. Some of the more common nonstandard configurations include thesingle edge notch tensile panel, the double edge notched tensile panel, the axisymmetricnotched bar, and the double cantilever beam specimen.

The vast majority of fracture toughness tests are performed on either compact orSENB specimens. Figure 7.2 illustrates the profiles of these two specimen types, assum-ing the same characteristic dimensions (B, W, a). The compact geometry obviously con-sumes less material, but this specimen requires extra material in the width direction, dueto the holes. If one is testing plate material or a forging, the compact specimen is moreeconomical, but the SENB configuration may be preferable for weldment testing, becauseless weld metal is consumed in some orientations (Section 7.7).

The compact specimen is pin-loaded by special clevises, as illustrated in Fig. 7.3.Compact specimens are usually machined in a limited number of sizes, because a separatetest fixture must be fabricated for each specimen size. Specimen size is usually scaledgeometrically; standard sizes include: 1/2T, IT, 2T and 4T, where the nomenclature refersto the thickness in inches1. For example, a standard IT compact specimen has the di-mensions B = 1 in (25.4 mm) and W - 1 in (50.8 mm). Although ASTM has convertedto SI units, the above nomenclature for compact specimen sizes persists.

The SENB specimen is more flexible with respect to size. The standard loadingspan for SENB specimens is 4W. If the fixture is designed properly, the span can be ad-justed continuously to any value that is within its capacity. Thus SENB specimens witha wide range of thicknesses can be tested with a single fixture. An apparatus for three-point bend testing is shown in Fig. 7.4.

lAn exception to this interpretation of the nomenclature occurs in thin sheet specimens, as discussed inSection 7.3.

Fracture Toughness Testing of Metals 367

(a) Compact specimen. (b) Disk shaped compact specimen.

Si w11

NX1 ] i

T \Ia

r

(c) Single edge notched bend (SENB) specimen.

f•2W

-2a

f t(d) Arc shaped specimen (e) Middle tension (MT) specimen.

368 Chapter 7

FIGURE 7.1 Standardized fracture mechanics test specimens.

FIGURE 7.2 Comparison of the profiles of compact and SENB specimens with the same in-plane char-acteristic dimensions (W and a).

FIGURE 7.3 Apparatus for testing compactspecimens.

7.1.2 Specimen Orientation

Engineering materials are seldom homogeneous and isotropic. Microstructure, and thusmechanical properties, are often sensitive to direction. The sensitivity to orientation isparticularly pronounced in fracture toughness measurements, because a microstructurewith a preferred orientation may contain planes of weakness, where crack propagation isrelatively easy. Since specimen orientation is such an important variable in fracturetoughness measurements, all ASTM fracture testing standards require that the orientationbe reported along with the measured toughness; ASTM has adopted a notation for thispurpose [1].


FIGURE 7.4 Three-point bending apparatus for testing SENB specimens.

Figure 7.5 illustrates the ASTM notation for fracture specimens extracted from arolled plate or forging. When the specimen is aligned with the axes of symmetry in theplate, there are six possible orientations. The letters L, T, and S denote the longitudinal,transverse, and short transverse directions, respectively, relative to the rolling direction orforging axis. Note that two letters are required to identify the orientation of a fracture me-chanics specimen; the first letter indicates the direction of the principal tensile stress(which is always perpendicular to the crack plane in Mode I tests) and the second letter de-notes the direction of crack propagation. For example, the L-T orientation corresponds toloading in the longitudinal direction and crack propagation in the transverse direction.

A similar notation applies to round bars and hollow cylinders, as Fig. 7.6 illus-trates. The symmetry directions in this case are circumferential, radial, and longitudinal(C, R, and L, respectively).

Ideally, one should measure the toughness of a material in several orientations, butthis is often not practical. When choosing an appropriate specimen orientation, oneshould bear in mind the purpose of the test, as well as geometrical constraints imposed bythe material. A low toughness orientation, where the crack propagates in the rolling di-rection (T-L or S-L), should be adopted for general material characterization or screening.When the purpose of the test is to simulate conditions in a flawed structure, however, thecrack orientation should match that of the structural flaw. Geometrical constraints maypreclude testing some configurations; the S-L and S-T orientations, for example, are onlypractical in thick sections. The T-S and L-S orientations may limit the size of compactspecimen that can be extracted from a rolled plate.

370 Chapter 7

ROLLINGDIRECTION

OR FORGINGAXIS

FIGURE 7.5 ASTM notation for specimens extracted from rolled plate and forgings [1].

FIGURE 7.6 ASTM notation for specimens extracted from disks and hollow cylinders [1].

7.1.3 Fatigue Precracking

Fracture mechanics theory applies to cracks that are infinitely sharp prior to loading.While laboratory specimens invariably fall short of this ideal, it is possible to introducecracks that are sufficiently sharp for practical purposes. The most efficient way to pro-duce such a crack is through cyclic loading.


Figure 7.7 illustrates the precracking procedure in a typical specimen, where a fa-tigue crack initiates at the tip of a machined notch and grows to the desired size throughcareful control of the cyclic loads. Modern servohydraulic test machines can be pro-grammed to produce sinusoidal loading, as well as a variety of other wave forms.Dedicated fatigue precracking machines that cycle at a high frequency are also available.

The fatigue crack must be introduced in such a way as not to adversely influence thetoughness value that is to be measured. Cyclic loading produces a crack of finite radiuswith a small plastic zone at the tip, which contains strain hardened material and a compli-cated residual stress distribution (see Chapter 10). In order for a fracture toughness to re-flect true material properties, the fatigue crack must satisfy the following conditions:

• The crack tip radius at failure must be much larger than the initial radius of the fa-tigue crack.

• The plastic zone produced during fatigue cracking must be small compared to theplastic zone at fracture.

Each of the various fracture testing standards contains restrictions on fatigue loads, whichare designed to satisfy the above requirements. The precise guidelines depend on the na-ture of the test. In Kjc tests, for example, the maximum K during fatigue loading mustbe no greater than a particular fraction of Kjc. In J and CTOD tests, where the test spec-imen is typically fully plastic at failure, the maximum fatigue load is defined as a fractionof the load at ligament yielding. Of course one can always perform fatigue precrackingwell below the allowable loads in order to gain additional assurance of the validity of theresults, but the time required to produce the crack (i.e., the number of cycles) increasesrapidly with decreasing fatigue loads.

LOADMACHINEDNOTCH

CRACK

TIME

FIGURE 7.7 Fatigue precracking a fracture mechanics specimen. A fatigue crack is introduced at thetip of a machined notch by means of cyclic loading.

372 Chapter?

7.1.4 Instrumentation

At a minimum, the applied load and a characteristic displacement on the specimen mustbe measured during a fracture toughness test. Additional instrumentation is applied tosome specimens in order to monitor crack growth or to measure more than one displace-ment.

Measuring load during a conventional fracture toughness test is relatively straight-forward, since nearly all test machines are equipped with load cells. The most commondisplacement transducer in fracture mechanics tests is the clip gage [2], which is illus-trated in Fig. 7.8. The clip gage, which attaches to the mouth of the crack, consists offour resistance strain gages bonded to a pair of cantilever beams. Deflection of the beamsresults in a change in voltage across the strain gages, which varies linearly with dis-placement. A clip gage must be attached to sharp knife edges in order to ensure that theends of each beam are free to rotate. The knife edges can either be machined into the spec-imen or attached to the specimen at the crack mouth.

A linear variable differential transformer (LVDT) provides an alternative means forinferring displacements in fracture toughness tests. Figure 7.9 schematically illustratesthe underlying principle of an LVDT. A steel rod is placed inside a hollow cylinder thatcontains a pair of tightly wound coils of wire. When a current passes through the firstcoil, the core becomes magnetized and induces a voltage in the second core. When the rodmoves, the voltage drop in the second coil changes; the change in voltage varies linearlywith displacement of the rod. The LVDT is useful for measuring displacements on a testspecimen at locations other than the crack mouth.

The potential drop technique utilizes a voltage change to infer crack growth, as illus-trated in Fig. 7.10. If a constant current passes through the uncracked ligament of a testspecimen, the voltage must increase as the crack grows, because the electrical resistanceincreases and the net cross-sectional area decreases. The potential drop method can use ei-ther DC or AC current. See Refs. [3] and [4] for examples of this technique.

The disadvantage of the potential drop technique is that it requires additional instru-mentation. The unloading compliance technique [5,6], however, allows crack growth tobe inferred from the load and displacement transducers that are part of any standard fracturemechanics test. A specimen can be partially unloaded at various points during the test inorder to measure the elastic compliance, which can be related to the crack length. Section7.4 describes the unloading compliance technique in more detail.

In some cases it is necessary to measure more than one displacement on a test spec-imen. For example, one may want to measure both the crack mouth opening displace-ment (CMOD} and the displacement along the loading axis. A compact specimen can bedesigned such that the load line displacement and the CMOD are identical, but these twodisplacements do not coincide in an SENB specimen. Figure 7.11 illustrates simultane-ous CMOD and load line displacement measurement in an SENB specimen. The CMODis inferred from a clip gage attached to knife edges; the knife edge height must be takeninto account when computing the relevant toughness parameter (see Section 7.5). Theload line displacement can be inferred by a number of methods, including the comparisonbar technique [7,8] that is illustrated in Fig. 7.11. A bar is attached to the specimen attwo points which are aligned with the outer loading points. The outer coil of an LVDT


is attached to the comparison bar, which remains fixed during deformation, while the cen-tral rod is free to move as the specimen deflects.

7.1.5 Side Grooving

In certain cases, grooves are machined into the sides of a fracture toughness specimens[9], as Fig. 7.12 illustrates. The primary purpose of side grooving is to maintain astraight crack front during an R-curve test. A specimen without side grooves is subject tocrack tunneling and shear lip formation (Fig. 5.15) because the material near the outersurfaces is in a state of low stress triaxiality. Side grooves remove the free surfaces,where plane stress conditions prevail and, if done properly, lead to relatively straight crackfronts. Typical side-grooved fracture toughness specimens have a net thickness that isapproximately 80% of the gross thickness. If the side grooves are too deep, they producelateral singularities, which cause the crack to grow more rapidly at the outer edges.

FIGURE 7.8. Measurement of the crack mouth opening displacement with a clip gage.

FIGURE 7.9 Schematic of a linear variable dif-ferential transformer (LVDT). Electric currentin the first coil induces a magnetic field, whichproduces a voltage in the second coil.Displacement of the central core causes a varia-tion in the output voltage.

374 Chapter 7

CRACKLENGTH

FIGURE 7.10 Potential drop method for monitoring crack growth. As the crack grows and the netcross sectional area decreases, the effective resistance increases, resulting in an increase in voltage (V).

FIGURE 7.11 Simultaneous measurement ofcrack mouth opening displacement (CMOD)and load line displacement on an SENB speci-men. The CMOD is inferred from a clip gageattached to knife edges, while the load line dis-placement can be determined from a compar-ison bar arrangement; the bar and outer coilof the LVDT remain fixed, while the inner rodmoves with the specimen.


FIGURE 7.12 Side grooves in a fracture me-chanics test specimen.

7.2 KIC TESTING

When a material behaves in a linear elastic manner prior to failure, such that the plasticzone is small compared to specimen dimensions, a critical value of the stress intensityfactor, KIC, may be an appropriate fracture parameter. Standard methods for KIC testinginclude ASTM E 399 [2] and BS 5447 [10], the latter of which was published by theBritish Standards Institution. An ASTM combined standard for K, J, and CTOD testing[11] is in draft form as of this writing, and should appear in late 1995. This draft ASTMstandard incorporates most of the provisions of E 399, but liberalizes the specimen designprovisions. For example, the ASTM combined method permits side grooves, while E 399does not.

The ASTM standard E 399 was first published in 1970, and has been revised severaltimes since then. The title, "Standard Test Method for Plane Strain Fracture Toughnessof Metallic Materials," is somewhat misleading. Although plane strain is a necessarycondition for a valid Kjc test, it is not sufficient; a specimen must also behave in a linearelastic manner. The validity requirements in this standard are very stringent because evena relatively small amount of plastic deformation invalidates the assumptions of K theory(see Chapter 2).

Four specimen configurations are permitted by the current version of E 399: thecompact, SENB, arc-shaped, and disk-shaped specimens. Specimens for Kjc tests are usu-ally fabricated with the width, W, equal to twice the thickness, B. They are fatigue pre-cracked so that the crack length/width ratio (a/W) lies between 0.45 and 0.55. Thus thespecimen design is such that all the critical dimensions, a, B, and W-a, are approximatelyequal. This design results in efficient use of material, since each of these dimensionsmust be large compared to the plastic zone.

376 Chapter 7

Most standardized mechanical tests (fracture toughness and otherwise), lead to validresults as long as the technician follows all of the procedures outlined in the standard.The KIC test, however, often produces invalid results through no fault of the technician.If the plastic zone at fracture is too large, it is not possible to obtain a valid Kjc, regard-less of how skilled the technician is.

Because of the strict size requirements, ASTM E 399 recommends that the user per-form a preliminary validity check to determine the appropriate specimen dimensions. Thesize requirements for a valid Kjc are as follows:

QA5<a/W<0.55 (7.1)

In order to determine the required specimen dimensions, the user must make a rough esti-mate of the anticipated Kjc for the material. Such an estimate can come from data forsimilar materials. If such data are not available, the ASTM standard provides a table ofrecommended thicknesses for various strength levels. Although there is a tendency fortoughness to decrease with increasing strength, there is not a unique relationship betweenKIC and ays in metals. Thus the strength-thickness table in E 399 should be used onlywhen better data are not available.

During the initial stages of fatigue precracking, the peak value of stress intensity ina single cycle, Kmax, should be no larger than 0.8 KIC, according to ASTM E 399. Asthe crack approaches its final size, Kmax should be less than 0.6 Kjc. If the specimen isfatigued at one temperature (Tj) and tested at a different temperature (T2), the final Kmax

must be < Q.6(GYS(l)/OYS(2))Kic- The fatigue load requirements are less stringent atinitiation because the final crack tip is remote from any damaged material that is producedin the early part of precracking. The maximum stress intensity during fatigue must al-ways be less than Kjc, however, in order to avoid premature failure of the specimen.

Of course, one must know Kjc in order to determine the maximum allowable fatigueloads. The user must specify fatigue loads based on the anticipated toughness of the ma-terial. If he or she is conservative and selects low loads, precracking could take a verylong time. On the other hand, if precracking is conducted at high loads, the user risks aninvalid result, in which case the specimen and the technician's time are wasted.

When a precracked test specimen is loaded to failure, load and displacement are mon-itored. Three types of load-displacement curves are shown in Fig. 7.13. The critical load,PQ, is defined in one of several ways, depending on the type of curve. One must con-struct a 5% secant line (i.e. a line from the origin with a slope equal to 95% of the initialelastic loading slope) to determine Pj. In the case of Type I behavior, the load-displace-ment curve is smooth and it deviates slightly from linearity before ultimate failure atPmax- This nonlinearity can be caused by plasticity, subcritical crack growth, or both.For a Type I curve, PQ = Pj. With a Type II curve, a small amount of unstable crackgrowth (i.e. a pop-in) occurs before the curve deviates from linearity by 5%. In this casePQ is defined at the pop-in. A specimen that exhibits Type III behavior fails completelybefore achieving 5% nonlinearity. In such cases, PQ =


Prnax/

LOADPma

FIGURE 7.13 Three types of load-displacement behavior in a Kjc test.

The crack length must be measured from the fracture surface. Since there is a ten-dency for the crack depth to vary through the thickness, the crack length is defined as theaverage of three evenly spaced measurements. Once PQ and crack length are determined, aprovisional fracture toughness, KQ, is computed from the following relationship:

_ ê (7.2)

where f(a/W) is a dimensionless function of a/W. This function is given in polynomialform in the E 399 standard for the four specimen types. Individual values of f(a/W) arealso tabulated in ASTM E 399. (See Table 2.4 and Chapter 12 for K solutions for a vari-ety of configurations.)

The KQ value computed from Eq. (7.2) is a valid Kjc result only if all validity re-quirements in the standard are met, including

0.45<0/W<0.55 (7.3a)

B,a>2.5K

(7.3b)

Pmax<1.10Pe C7.3c)

378 Chapter 7

Additional validity requirements include the restrictions on fatigue load mentioned earlier,as well as limits on fatigue crack curvature. If the test meets all of the requirements ofASTM E 399, then KQ = Kjc.

Section 2.10 describes the limitations of stress intensity factor, and outlines thetheoretical reasons for the strict size requirements for K]c. Recall that Eqs. (7.3a) and(7.3b) ensure that the critical specimen dimensions, B, a, and (W-a), are at least -50 timeslarger than the plane strain plastic zone. The third requirement, Eq. (7.3c), is necessary tocorrect a loophole in the Kjc test procedure, as discussed below.

The deviation from linearity in a load-displacement curve can be caused by crackgrowth, plastic zone effects, or both. In the absence of plastic deformation, 5% deviationfrom the initial slope of the load-displacement curve corresponds to crack growth throughapproximately 2% of the ligament in test specimens with a/W~ 0.5; when a plastic zoneforms, a 5% deviation from linearity can be viewed as 2% apparent crack growth.(Recall Section 2.8, where crack tip plasticity was modeled by pretending that the crackwas slightly longer than the actual size.) If the nonlinearity in the load-displacementcurve is caused only by plasticity, a 5% deviation from linearity corresponds to a plasticzone size that is roughly 2% (i.e. 1/50) of the uncracked ligament. Thus the plastic zonesize at Pj in a Type I test is approximately equal to its maximum allowable size, as de-finedby Eq. (7.3b).

Consider a fracture toughness test that displays considerable plastic deformationprior to failure. Figure 7.14 schematically illustrates the load-displacement curve for sucha test. Since this is a Type I curve, PQ = Pj. A KQ value computed from PQ may justbarely satisfy the size requirements of Eq. (7.3b) for reasons described in the previousparagraph. Such a quantity, however, would have little relevance to the fracture tough-ness of the material, since the specimen fails well beyond PQ; the KQ value in this casewould grossly underestimate the true toughness of the material. Consequently the thirdvalidity requirement, Eq. (7.3c), is necessary to ensure that a Kjc value is indicative of thetrue toughness of the material.

LOAD

Pmax

FIGURE 7.14 Load-displacement curve foran invalid Kjc test, where ultimate failureoccurs well beyond PQ.

DISPLACEMENT


Because the size requirements of ASTM E 399 are very stringent, it is very difficultand sometimes impossible to measure a valid Kjc in most structural materials, asExamples 7.1 and 7.2 illustrate. A material must either be relatively brittle or the testspecimen must be very large for linear elastic fracture mechanics to be valid. In low- andmedium strength structural steels, valid Kfc tests are normally possible only on the lowershelf of toughness; in the ductile-brittle transition and the upper shelf, elastic-plastic pa-rameters such as the J integral and CTOD are required to characterize fracture.

Because of the strict validity requirements, the Kjc test is of limited value to struc-tural metals. The toughness and thickness of most materials precludes a valid KIC result.If, however, a valid Kjc test can be measured on a given material, it is probably too brit-tle for most structural applications.

EXAMPLE 7.1

Consider a structural steel with ays - 350 MPa (51 ksi). Estimate the specimen di-mensions required for a valid K[c test. Assume that this material is on the upper shelfof toughness, where typical Kjc values for initiation of microvoid coalescence in

these materials are around 200 MPa \ m.

Solution: Inserting the yield strength and estimated toughness into Eq. (7.1) gives

B, a = 2.5200 MPa

350 MPa J = 0.816m (32.1 in)

Since a/W - 0.5, W = 1.63 m (64.2 in)! Thus a very large specimen would be requiredfor a valid Kjc test. Materials are seldom available in such thicknesses. Even if a suf-

ficiently large section thickness were fabricated, testing such a large specimen wouldnot be practical; machining would be prohibitively expensive, and a special testingmachine with a high load capacity would be needed.

EXAMPLE 7.2

Suppose that the material in Example 7.1 is fabricated in 25 mm (1 in) thick plate.Estimate the largest valid KIC that can be measured on such a specimen.

380 Chapter 7

EXAMPLE 7.2 (cont.)

Solution: For the L-T or T-L orientation, a test specimen with a standard design couldbe no larger than B = a - 25 mm and W = 50 mm. Inserting these dimensions and theyield strength into Eq. (7.1) and solving for Kjc gives

= 350MPaV2!025 m= 35MPa

Figure 4.5 shows fracture toughness data for an A 572 Grade 50 steel. Note that thetoughness level computed above corresponds to the lower shelf in this material. Thusvalid KIC tests on this material would be possible only at low temperatures, where the

material is too brittle for most structural applications.

7.3 K-R CURVE TESTING

Some materials whose fracture behavior is predominantly linear elastic exhibit a rising Rcurve. The ASTM Standard E 561 [12] outlines a procedure for determining K versuscrack growth curves in such materials. Unlike ASTM E 399, the K-R standard does notcontain a minimum thickness requirement, and thus can be applied to thin sheets. Thisstandard, however, is appropriate only when the plastic zone is small compared to the in-plane dimensions of the test specimen. This test method is often applied to high strengthsheet materials, where the fracture behavior is plane stress linear elastic.

There is a common misconception about plane stress, plane strain, and R curves.A number of published articles and textbooks imply that a material in plane strain ex-hibits a single value of fracture toughness (K{c}, while the same material in plane stressdisplays a rising R curve. While this may occur in some cases, it is not a universal phe-nomenon. The shape of an R curve depends on the fracture mechanism as well as thestress state at the crack tip. Cleavage tends to exhibit a flat or falling R curve, while mi-crovoid coalescence can produce a rising R curve. The slope of an R curve tends to de-crease with increasing stress triaxiality, and the fracture mechanism (in steels) can changefrom ductile tearing to cleavage as the stress state ranges from plane stress to plane strain.This leads to the impression that plane stress conditions always produce a rising resis-tance curve while plane strain fracture can be described by a single toughness value (Kfc).It is possible, however, for cleavage, and thus a falling R curve, to occur in thin sheets.Similarly, rising R curves under plane strain conditions are common in ductile materials,as discussed in Section 7.4.

Figure 7.15 illustrates a typical K-R curve in a predominantly linear elastic mate-rial. The R curve is initially very steep, as little or no crack growth occurs with increas-ing Kj. As the crack begins to grow, încreases with crack growth until a steady state isreached, where the R curve becomes flat (see Section 3.5 and Appendix 3.5). It is possi-


ble to define a critical stress intensity, Kc, where the driving force is tangent to the Rcurve. This instability point is not a material property, however, because the point oftangency depends on the shape of the driving force curve, which is governed by the geom-etry of the cracked body. Thus Kc values obtained from laboratory specimens are notusually transferable to structures.

FIGURE 7.15 Schematic K-R curve; Kc occurs atthe point of tangency between the driving forceand R curve.

CRACK SIZE

7.3.1 Specimen Design

The ASTM standard for K-R curve testing [12] permits three configurations of test spec-imen: the middle tension (MT) geometry, the conventional compact specimen, and awedge loaded compact specimen. The latter configuration, which is similar to the com-pact crack arrest specimen discussed in Section 7.6, is the most stable of the three speci-men types, and thus is suitable for materials with relatively flat R curves.

Since this test method is often applied to thin sheets, specimens do not usually havethe conventional geometry, with the width equal to twice the thickness. The specimenthickness is normally fixed by the sheet thickness, and the width is governed by the antic-ipated toughness of the material, as well as the available test fixtures.

A modified nomenclature is applied to thin sheet compact specimens. For example,a specimen with W = 50 mm (2 in) is designated as a IT plan specimen, since the in-plane dimensions correspond to the conventional IT compact geometry. Standard fixturescan be used to test thin sheet compact specimens, provided the specimens are fitted withspacers, as illustrated in Fig. 7.16.

One problem with thin sheet fracture toughness testing is that the specimens aresubject to out-of-plane buckling, which leads to combined Mode I-Mode III loading of thecrack. Consequently, an antibuckling device should be fitted to the specimen. Figure7.16 illustrates a typical antibuckling fixture for thin sheet compact specimens. Plateson either side of the specimen prevent out-of-plane displacements. These plates shouldnot be bolted too tightly together, because loads applied by the test machine should becarried by the specimen rather than the antibuckling plates. Some type of lubricant (e.g.

382 Chapter 7

Teflon sheet) is usually required to allow the specimen to slide freely through the twoplates during the test.

7.3.2 Experimental Measurement of K-R Curves

The ASTM Standard E 561 outlines a number of alternative methods for computing bothKI and the crack extension in an R curve test; the most appropriate approach depends onthe relative size of the plastic zone. Let us first consider the special case of negligibleplasticity, which exhibits a load-displacement behavior that is illustrated in Figure 7.17.As the crack grows, the load-displacement curve deviates from its initial linear shape be-cause the compliance continuously changes. If the specimen were unloaded prior to frac-ture, the curve would return to the origin, as the dashed lines indicate. The compliance atany point during the test is equal to the displacement divided by the load. The instanta-neous crack length can be inferred from the compliance through relationships that aregiven in the ASTM standard. (See Chapter 12 for compliance-crack length equations for avariety of configurations.) The crack length can also be measured optically during testson thin sheets, where there is negligible through-thickness variation of crack length. Theinstantaneous stress intensity is related to the current values of load and crack length:

f ( a f W ) (7.4)

3iO

*=*SPECIMEN

ANTIBUCKLINGPLATES

FIGURE 7.16 Antibuckling fixtures for testing thin compact specimens


Consider now the case where a plastic zone forms ahead of the growing crack. Thenonlinearity in the load-displacement curve is caused by a combination of crack growthand plasticity, as Fig. 7.18 illustrates. If the specimen is unloaded prior to fracture, theload-displacement curve does not return to the origin; crack tip plasticity produces a finiteamount of permanent deformation in the specimen. The physical crack length can be de-termined optically or from unloading compliance, where the specimen is partially un-loaded, the elastic compliance is measured, the crack length is inferred from compliance.The stress intensity should be corrected for plasticity effects by determining an effectivecrack length. The ASTM standard suggests two alternative approaches for computingaeff. the Irwin plastic zone correction and the secant method. According to the Irwin ap-proach (Section 2.8.1), the effective crack length for plane stress is given by

aeff=a1

(7-5)

The secant method consists of determining an effective crack size from the effective com-pliance, which is equal to the total displacement divided by the load (Fig. 7.18). The ef-fective stress intensity factor for both methods is computed from the load and the effectivecrack length:

(7.6)

The Irwin correction requires an iterative calculation, where a first order estimate ofaeffisused to estimate Keff, which is inserted into Eq. (7.5) to obtain a new aeff, the process isrepeated until the Keff estimates converge.

LOAD

Crack Growth

DISPLACEMENT

FIGURE 7.17 Load-displacement curve for crack growth in the absence of plasticity.

384 Chapter 7

LOAD

PlasticDeformation

DISPLACEMENT

FIGURE 7.18 Load-displacement curve for crack growth with plasticity.

The choice of plasticity correction is left largely up to the user. When the plasticzone is small, ASTM E 561 suggests that the Irwin correction is acceptable, but recom-mends applying the secant approach when the crack tip plasticity is more extensive.Experimental data typically display less size dependence when the stress intensity is deter-mined by the secant method [13].

The ASTM K-R curve standard requires that the stress intensity be plotted againstthe effective crack extension (Aaejf). This practice is inconsistent with the J]c and J-Rcurve approaches (Section 7.4), where / is plotted against the physical crack extension.The estimate of the instability point (Kc} should not be sensitive to the way in whichcrack growth is quantified, particularly when both the driving force and resistance curvesare computed with a consistent definition of Aa.

The ASTM E 561 standard does not contain requirements on specimen size or themaximum allowable crack extension; thus there is no guarantee that a K-R curve producedaccording to this standard will be a geometry-independent material property. As discussedin Section 2.10, application of LEFM to thin sections is acceptable as long as the speci-men thickness matches the section thickness of the structure. The in-plane dimensions,however, must be large compared to the plastic zone in order for LEFM to be valid.Also, the growing crack must be remote from all external boundaries.

Unfortunately, the size dependence of R curves in high strength sheet materials hasyet to be quantified, so it is not possible to recommend specific size and crack growthlimits for this type of testing. The user must be aware of the potential for size depen-dence in K-R curves. Application of the secant approach reduces but does not eliminatethe size dependence. The user should test wide specimens whenever possible in order toensure mat the laboratory test is indicative of the structure under consideration.


7.4 J TESTING OF METALS

Two existing ASTM standards address J testing. The Jjc standard, E 813 [5], which wasfirst published in 1981 and revised in 1987, outlines a test method for estimating the crit-ical / near initiation of ductile crack growth. A J-R curve testing standard, E 1152 [6],was first published in 1987. An ASTM standard that combines E 813 and E 1152 is inthe final stages of balloting at this writing [14], and should appear in 1995. Thiscombined J standard not only covers Jjc and J-R curve measurement, but also addresses /testing of materials that experience unstable fracture (e.g., cleavage in ferritic steels). Thedraft ASTM common method [11] mentioned earlier includes provisions for J testing thatare nearly identical to the draft ASTM combined J standard [14]. Once Refs. [11] and [14]are officially adopted, E 813 and E 1152 will become obsolete and will eventuallydisappear from the ASTM Book of Standards.

Both the International Standards Organization (ISO) and the British StandardsInstitution (BSI) are currently drafting fracture toughness testing standards that are compa-rable in scope (though not in detail) to Refs. [11] and [14].

The ASTM standards E 813-87 and E 1152-87 both produce a J-R curve, a plot of/versus crack extension. The E 1152 standard applies to the entire J-R curve, while E 813is concerned only with Jjc, a single point on the R curve. The same test can be reportedin terms of both standards. This is analogous to a tensile test, where one can report eitherthe yield strength or the entire stress-strain curve. Both standards apply to compact andSENB specimens. The overlap between these two standards motivated the development ofthe combined J testing standard [14],

7.4.1 Jic Measurements

The R curve for Jjc measurements can be generated by either multiple specimen or singlespecimen techniques. With the multiple specimen technique, a series of nominally iden-tical specimens are loaded to various levels and then unloaded. Some stable crack growthoccurs in most specimens. This crack growth is marked by heat tinting or fatigue crack-ing after the test. Each specimen is then broken open and the crack extension is mea-sured.

The most common single specimen test technique is the unloading compliancemethod, which is illustrated in Fig. 7.19. The crack length is computed at regular inter-vals during the test by partially unloading the specimen and measuring the compliance.As the crack grows, the specimen becomes more compliant (less stiff). The various Jtesting standards provide polynomial expressions that relate a/W to compliance. Table12.4 lists these compliance equations for bend and compact specimens. The ASTM stan-dards require relatively deep cracks (0.50 < a/W < 0.70) because the unloading compliancetechnique is not sufficiently sensitive for a/W < 0.5. An alternative single specimen testmethod is the potential drop procedure (Fig. 7.10) in which crack growth is monitoredthrough the change in electrical resistance which accompanies a loss in cross sectionalarea. Both single specimen procedures are practical only in conjunction with a computerdata acquisition and analysis system.

386 Chapter 7

LOAD

LOAD LINE DISPLACEMENT

FIGURE 7.19. The unloading compliance method for monitoring crack growth.

Regardless of the method for monitoring crack growth, a corresponding J value mustbe computed for each point on the R curve. For estimation purposes, it is convenient todivide J into elastic and plastic components:

(7.7)

The elastic / is computed from the elastic stress intensity:

= K2(l- V2)d~ E

(7.8)

where K is inferred from load and crack size through Eq. (7.4). If, however, side groovedspecimens are used, the expression for K is modified:

(7.9)

where B is the gross thickness and Bpj is the net thickness (Fig 7.12). The ASTM 7/cstandard outlines a simplified method for computing Jpi from the plastic area under theload-displacement curve:2

2Since J is defined in terms of the energy absorbed divided by the net cross sectional area, Bjsj appears in the

denominator. For nonside grooved specimens BN = B.


BNb0

(7.10)

where 7] is a dimensionless constant, Api is the plastic area under the load-displacementcurve (see Fig. 7.20), and bo is the initial ligament length. For an SENB specimen,

77 = 2.0

and for a compact specimen,

(7.1 la)

(7. lib)

Recall from Section 3.2.5 that Eq. (7.10) was derived from the energy release rate defini-tion of /.

LOAD

FIGUREJ.20 Plastic energy absorbed by atest specimen during a Jjc test.

DISPLACEMENT

Note that Eqs. (7.10) and (7.1 Ib) do not correct J for crack growth, but are based onthe initial crack length. A more complicated procedure, in which / is computed incremen-tally with updated values of crack length and ligament length, can also be applied (seeSection 7.4.2). This more elaborate procedure is usually not necessary for TIC measure-ments, however, because crack growth is insignificant at the point on the R curve whereJlc is measured.. In the limit of a stationary crack, both formulas give identical results.Thus the measured initiation toughness is insensitive to the choice of J equation. Thesimplified method based on original ligament length is usually applied when the R curveis inferred from the multiple specimen technique, but the more complex procedure thatupdates the ligament length is often used in conjunction with single-specimen techniquessuch as unloading compliance and potential drop.

The ASTM procedure for computing JQ, a provisional Jjc, from the R curve is il-lustrated in Fig. 7.21. Exclusion lines are drawn at crack extension (Aa) values of 0.15

388 Chapter 7

and 1.5 mm. These lines have a slope of 2oy, where cry is the flow stress, defined asthe average of the yield and tensile strengths. The slope of the exclusion lines corre-sponds approximately to the component of crack extension that is due to crack blunting,as opposed to ductile tearing. A horizontal exclusion line is defined at a maximum valueof/:

(7.12)

All data that fall within the exclusion limits are fit to a power law expression:

(7.13)

The JQ is defined as the intersection between Eq. (7.13) and a 0.2 mm offset line. If allother validity criteria are met, JQ = Jjc as long the following size requirements are satis-fied:

O

(7-14)

i !

- Points used in regression analysis

O 400 |—

H2

0.25 0.50 0.75 1.0 1.25 1.50 1.75

CRACK EXTENSION, mm2.0 2.25

FIGURE 7.21 Determination of JQ from a J-R curve [5].


EXAMPLE 7.3

Estimate the specimen size requirements for a valid Jfc test on the material inExample 7. 1 . Assume OTS = 450 MPa and E = 207,000 MPa.

Solution: First we must convert the Kjc value in Example 7.1 to an equivalent Jjc:

KIc2 (1 - v2) (200 MPa^)2 (1 -0.32)

Jlc = £ = 207000 MPa = °-176Mpam

Substituting the above result into Eq. (7.14) gives,

B,bo ^ 7 ^ = 0.0110m = 11.0 mm (0.433 in)

which is nearly two orders of magnitude lower than the specimen dimension thatASTM E 399 requires for this material. Thus the Jjc size requirements are much morelenient than the Kjc requirements.

7.4.2 J-R Curve Testing

When the entire J-R curve is of interest, single specimen techniques should be used, andthe J should be corrected for crack growth. Test specimens should be side grooved in or-der to avoid tunneling and maintain a straight crack front.

There are a number of ways to compute J for a growing crack, as outlined in Section3.4.2. The ASTM procedure for J-R curve testing utilizes the deformation theory defini-tion of J, which corresponds to the rate of energy dissipation by the growing crack (i.e.,the energy release rate). Recall Fig. 3.22, which contrasts the actual loading path withthe "deformation" path. The deformation J is related to the area under the load-displace-ment curve for a stationary crack, rather than the area under the actual load-displacementcurve, where the crack length varies (see Eqs. (3.55) and (3.56)).

Since the crack length changes continuously during a J-R curve test, the J integralmust be calculated incrementally. For unloading compliance tests, the most logical timeto update the J value is at each unloading point, where the crack length is also updated.Consider an J test with n measuring points. For a given measuring point i, where

390 Chapter 7

l<i<n, the elastic and plastic components of Jean be estimated from the following ex-pressions (see Fig. 7.22)3:

(7.15a)

V/-X i-r,-, a, - fl,

(7.15b)

where 4Z'(W) is the plastic load line displacement, yt - 1.0 for SENB specimens and j[ -1 + 0.76bi/W for compact specimens; TH is as defined in Eq. (7.11), except that bo is re-placed by hi, the instantaneous ligament length. The instantaneous K is related to Pi andaj/W through Eq. (7.9).

Equation (7.15b) gives an approximation of the plastic component of the deforma-tion J. Appendix 7 explores the basis of this relationship, as well as its accuracy.

LOAD

LOAD LINE DISPLACEMENT

FIGURE 7.22 Schematic load-displacement curve for a J-R curve test.

Êquation (7.15b) contains different subscripts on b and y from the corresponding equation in E 1152, but it isconsistent with the ASTM combined J test standard [14]. E 1152 apparently contains a typographical error.The differences in the computed J values between the two equations is minimal, however.


Both ASTM E 1152 and the J testing standard [14] have the following limits onand crack extension relative to specimen size:

and

(7.16)

(7.17)

Figure 7.23 shows a typical J-R curve with the ASTM validity limits. The portion ofthe J-R curve that falls outside these limits is considered invalid.

The ASTM draft common method [11] currently allows crack growth to 30 percentof the initial ligament, as opposed to 10 percent in Eq. (7.17). This more relaxed require-ment may not survive the balloting process in ASTM, however.

2500

2000 -

oo

I§2 1500

iKg 1000

1— 1

500

n t

,

Jmax

— ®

1 ®®

©@®

~®ik., ' ! 1

Oo

o0

©

0

Ad. max

i i

O

A 710 STEEL2T Compact Specimens

• Valid Data

o Invalid Data

i i i i i 1 i i i

;

-j

— i

J

4 6 8

CRACK GROWTH, mm

10

FIGURE 7.23 J-R curve for A 710 steel [15]. In this case, the data exceed the maximum J (Eq. 7.16) be-fore the crack growth limit (Eq. 7.17).

392 Chapter 7

7.4.3 Critical J Values for Cleavage Fracture

Although E 813-87 and E 1152-87 apply only to ductile fracture, more recent standards[11,14] permit J testing of materials that fail by cleavage. In the ductile-brittle transitionregion of structural steels, cleavage is often preceded by significant plastic flow, and linearelastic parameters are not suitable to characterize toughness.

Many researchers and practitioners prefer to convert critical J values to equivalent Kvalues through the following relationship:

The Kjc values can be applied to structures that are elastically loaded, as discussed inChapter 9. This approach is valid if only if the critical J value is independent of speci-men size. The test specimen must be sufficiently large that further increases in size haveno effect on Jc.

Size requirements for /-controlled cleavage can be expressed in the form of Eqs.(7. 14) and (7. 16):

(7.19)

where M is a dimensionless constant. In 1991, Anderson and Dodds [16] recommended M= 200, based on their fracture toughness scaling model (Section 3.6.3) and plane strainelastic-plastic finite element analysis. More recent work based on 3-D finite elementanalysis [17] indicates that this requirement can be relaxed to M = 50. Both the ASTMcombined J and common methods require M = 200, based on the earlier recommendationof Anderson and Dodds. This requirement is conservative, and may be relaxed insubsequent revisions of these standards.

The recommended size limits of Eq. (7.19) only apply to cleavage without signifi-cant prior stable crack growth. In the upper transition region of steels, cleavage is usu-ally preceded by ductile tearing. Further research is necessary to understand upper transi-tion fracture better in order to develop suitable J testing procedures for this region.

7.5 CTOD TESTING

Because of the strict limits on plastic deformation, the Kfc test can be applied only on thelower shelf of toughness in structural steels and welds. The older ASTM Jjc and J-Rcurve test methods allow considerably more plastic deformation, but these tests are onlyvalid on the upper shelf. Until the newer standards [11,14] are published, the CTOD testis the only standardized method to measure fracture toughness in the ductile-brittle transi-tion region.


The first CTOD test standard was published in Great Britain in 1979 [18]. ASTMrecently published E 1290 [19], an American version of the CTOD standard. The BritishCTOD standard allows only the SENB specimen, while the ASTM standard provides forCTOD measurements on both the compact and SENB specimens. Both standards allowtwo configurations of SENB specimens: 1) a rectangular cross section with W = 2B, thestandard geometry for K]c and J\c tests; and 2) a square cross section with W— B. Therectangular specimen is most useful with L-T or T-L orientations (Fig. 7.5); the squaresection is generally applied to the L-S or T-S orientations.

Experimental CTOD estimates are made by separating the CTOD into elastic andplastic components, similar to the //c and J-R tests. The elastic CTOD is obtained fromthe elastic K:

K2(l-v2)2aYSE

(7.20)

The elastic K is related to applied load through Eq. (7.4). The above relationship assumesthat dn - 0.5 for linear elastic conditions (Eq. (3.48)). The plastic component of CTODis obtained by assuming that the test specimen rotates about a plastic hinge. This con-cept is illustrated in Fig. 7.24 for an SENB specimen. The plastic displacement at thecrack mouth, Vp, is related to the plastic CTOD through a similar triangles construction:

(7.21)rD(W-a)

where rp is the plastic rotational factor, a constant between 0 and 1 that defines the rela-tive position of the apparent hinge point. The mouth opening displacement is measuredwith a clip gage. In the case of an SENB specimen, knife edges must often be attached inorder to hold the clip gage. Thus Eq. (7.21) must take account of the knife edge height,z. The compact specimen can be designed so that z = 0. The plastic component of V isobtained from the load-displacement curve by constructing a line parallel to the elasticloading line, as illustrated in Fig. 3.6. According to ASTM E 1290, the plastic rota-tional factor is given by

rn=0.44 (7.22a)

for the SENB specimen and

rp =0.411af 0.5

1/2a^- + 0.5 (7.22b)

394 Chapter?

for the compact specimen. The original British standard for CTOD tests, BS 5762:1979applied only to SENB specimens and specified r^ = 0.40.

The crack mouth opening displacement, V", on an SENB specimen is not the sameas the load line displacement, A. The latter displacement measurement is required for /estimation because Api in Fig. 7.20 represents the plastic energy absorbed by the speci-men. The CTOD standard utilizes Vp because this displacement is easier to measure inSENB specimens. If rp is known, however, it is possible to infer / from a P-V curve orCTOD from a P-A curve [7,8]. The compact specimen simplifies matters somewhat be-cause V= A as long as z = 0.

FIGURE 7.24 Hinge model for plastic displacements in an SENB specimen.

The British ASTM CTOD standard test methods can be applied to ductile and brittlematerials, as well as steels in the ductile-brittle transition. These standards includes a no-tation for critical CTOD values that describes the fracture behavior of the specimen:

8C - Critical CTOD at the onset of unstable fracture with less than 0.2 mm of sta-ble crack growth. This corresponds to the lower shelf and lower transition region ofsteels where the fracture mechanism is pure cleavage.

Su - Critical CTOD at the onset of unstable fracture which has been preceded bymore than 0.2 mm of stable crack growth. In the case of ferritic steels, this correspondsto the "ductile thumbnail" observed in the upper transition region .

Si - CTOD near the initiation of stable crack growth.4 This measure of toughnessis analogous to Jjc.

The most recent revision of ASTM E 1290 dropped Si as a toughness parameter, but the draft commonmethod [11] retains Sj


8m - CTOD at the first attainment of a maximum load plateau. This occurs on ornear the upper shelf of steels.

Figure 7.25 is a series of schematic load-displacement curves that manifest each ofthe above failure scenarios. Curve (a) illustrates a test that results in a 8C value; cleavagefracture occurs at Pc. Figure 7.25(b) corresponds to a 8U result, where ductile tearing pre-cedes cleavage. The ductile crack growth initiates at P/. A test on or near the upper shelfproduces a load-displacement curve like Fig. 7.25(c); a maximum load plateau occurs atPm. The specimen is still stable after maximum load because the material has a rising Rcurve and the test is performed in displacement control. Three types of CTOD result, 8C,8U and Sm, are mutually exclusive; i.e, they cannot occur in the same test. It is possi-ble, however, to measure a 5f value in the same test as either a 8m or 8U result.

As Fig. 7.25 illustrates, there is usually no detectable change in the load-displace-ment curve at PI. The only deviation in the load-displacement behavior is the reducedrate of increase in load as the crack grows. The maximum load plateau (Fig. 7.25(c)) oc-curs when the rate of strain hardening is exactly balanced by the rate of decrease in thecross section. However, the initiation of crack growth cannot be detected from the load-displacement curve because the loss of cross section is gradual. Thus 5f must be deter-mined from an R curve.

The only specimen size requirement of the British and ASTM CTOD standards is arecommendation to test full section thicknesses. For example, if a structure is to be madeof 25 mm (1 in) thick plate, then B in the test specimens should be nominally 25 mm.If the specimen is notched from the surface (L-S or T-S orientations), a square sectionspecimen is required for B to equal the plate thickness. The British CTOD standard al-lows a/W ratios ranging from 0.15 to 0.70, while the ASTM standard restricts the per-missible a/W values to the range of 0.45 to 0.55. Shallow cracked specimens have cer-tain advantages, particularly for weldment tests (see Section 7.7), but critical CTOD val-ues from such tests are usually geometry dependent [16,20].

(0

MOUTH OPENING DISPLACEMENT

FIGURE 7.25 Various types of load-displacement curves from CTOD tests.

396 Chapter?

7.6 DYNAMIC AND CRACK ARREST TOUGHNESS

When a material is subject to a rapidly applied load or a rapidly propagating crack, the re-sponse of that material may be drastically different from the quasistatic case. When rapidloading or unstable crack propagation are likely to occur in practice, it is important to du-plicate these conditions when measuring material properties in the laboratory.

The dynamic fracture toughness and the crack arrest toughness are two importantmaterial properties for many applications. The dynamic fracture toughness is a measureof the resistance of a material to crack propagation under rapid loading, while the crack ar-rest toughness quantifies the ability of a material to stop a rapidly propagating crack. Inthe latter case, the crack may initiate under either dynamic or quasistatic conditions, butunstable propagation is generally a dynamic phenomenon.

Dynamic fracture problems are often complicated by inertia effects, material rate de-pendence, and reflected stress waves. One or more of these effects can be neglected insome cases, however. Refer to Chapter 4 for additional discussion on this subject.

7.6.1 Rapid Loading in Fracture Testing

Aside from an Annex to E 399 [2], there are currently no ASTM Standards for high ratefracture testing. This type of testing is more difficult than conventional fracture tough-ness measurements, and requires considerably more instrumentation.

High loading rates can be achieved in the laboratory by a number of means, includ-ing a drop tower, a high rate testing machine, and explosive loading. With a drop tower,the load is imparted to the specimen through the force of gravity; a cross head of with aknown weight is dropped onto the specimen from a specific height. A pendulum devicesuch as a Charpy testing machine is a variation of this principle. Some servohydraulicmachines are capable of high displacement rates. While conventional testing machinesare closed loop, where the hydraulic fluid circulates through the system, high rate ma-chines are open loop, where a single burst of hydraulic pressure is released over short timeinterval. For moderately high displacement rates, a closed-loop machine may be adequate.Explosive loading involves setting off a controlled charge which sends stress wavesthrough the specimen [21].

The dynamic loads resulting from impact are often inferred from an instrumentedtup. Alternatively, strain gages can be mounted directly on the specimen; the output canbe calibrated for load measurements, provided the gages are placed in a region of the spec-imen that remains elastic during the test. Cross head displacements can be measured di-rectly through an optical device mounted to the cross head. If this instrumentation is notavailable, a load-time curve can be converted to a load-displacement curve through mo-mentum transfer relationships.

Certain applications require more advanced optical techniques, such photoelasticity[22,23] and the method of caustics [24]. These procedures provide more detailed informa-tion about the deformation of the specimen, but are also more complicated than globalmeasurements of load and displacement.

Because high rate fracture tests typically last only a few microseconds, conventionaldata acquisition tools are inadequate. A storage oscilloscope has traditionally been re-


quired to capture data in a high rate test; when a computer data acquisition system wasused, the data were downloaded from the oscilloscope after the test. The newest genera-tion of data acquisition cards for microcomputers removes the need for this two-step pro-cess. These cards are capable of collecting data at high rates, and enable the computer tosimulate the functions of an oscilloscope.

Inertia effects can severely complicate measurement of the relevant fracture parame-ters. The stress intensity factor and J integral cannot be inferred from global loads anddisplacements when there is a significant kinetic energy component. Optical methodssuch as photoelasticity and caustics are necessary to measure J and K in such cases.

The transition time concept [25,26], which was introduced in Chapter 4, removesmuch of the complexity associated with / and K determination in high rate tests. Recallthat the transition time, t-t, is defined as the time at which the kinetic energy and deforma-tion energy are approximately equal. At times much less than t^, inertia effects dominate,while inertia is negligible at times significantly greater than t^. The latter case corre-sponds to essentially quasistatic conditions, where conventional equations for / and K ap-ply. According to Fig. 4.4, the quasistatic equation for J, based on the global load dis-placement curve, is accurate at times greater that 2 t^. Thus if the critical fracture eventoccurs after 2 tT, the toughness can be inferred from the conventional quasistatic relation-ships. For drop tower tests on ductile materials, the transition time requirement is rela-tively easy to meet [27,28]. For brittle materials (which fail sooner) or higher loadingrates, the transition time can be shortened through specimen design.

7.6.2 Kia Measurements

In order to measure arrest toughness in a laboratory specimen, one must create conditionsunder which a crack initiates, propagates in an unstable manner, and then arrests.Unstable propagation followed by arrest can be achieved either through a rising R curve ora falling driving force curve. In the former case, a temperature gradient across a steelspecimen produces the desired result; fracture can be initiated on the cold side of the spec-imen, where toughness is low, and propagate into warmer material where arrest is likely.A falling driving force can be obtained by loading the specimen in displacement control,as Example 2.3 illustrates.

The Robertson crack arrest test [29] was one of the earliest applications of the tem-perature gradient approach. This test is only qualitative, however, since the arrest temper-ature, rather than Kra, is determined from this test. The temperature at which a crack ar-rests in the Robertson specimen is only indicative of the relative arrest toughness of thematerial; designing above this temperature does not guarantee crack arrest under all load-ing conditions. The drop weight test developed by Pellini (see Section 7.9) is anotherqualitative arrest test that yields a critical temperature. In this case, however, arrest is ac-complished through a falling driving force.

While most crack arrest tests are performed on small laboratory specimens, a limitednumber of experiments have been performed on larger configurations in order to validatethe small scale data. An extreme example of large scale testing is the wide plate crack ar-

398 Chapter?

rest experiments conducted at the National Institute of Standards and Technology (NIST)5

in Gaithersburg, Maryland [30]. Figure 7.26 shows a photograph of the NIST testingmachine and one of the crack arrest specimens. This specimen, which is a single edgenotched tensile panel, is 10 m long by 1 m wide. A temperature gradient is appliedacross the width, such that the initial crack is at the cold end. The specimen is thenloaded until unstable cleavage occurs. These specimens are heavily instrumented, so thata variety of information can be inferred from each test. The crack arrest toughness valuesmeasured from these tests is in broad agreement with small scale specimen data.

In 1988, ASTM published a standard for crack arrest testing, E 1221 [31]. Thisstandard outlines a test procedure that is considerably more modest than the NIST experi-ments. A side grooved compact crack arrest specimen is wedge loaded until unstable frac-ture occurs. Because the specimen is held at a constant crack mouth opening displace-ment, the running crack experiences a falling K field. The crack arrest toughness, Kfo, isdetermined from the mouth opening displacement and the arrested crack length.

The test specimen and loading apparatus for K\a testing are illustrated in Figs. 7.27and 7.28. In most cases, a starter notch is placed in a brittle weld bead in order to facili-tate fracture initiation. A wedge is driven through a split pin that imparts a displacementto the specimen. A clip gage measures the displacement at the crack mouth (Fig. 7.28).

Since the load normal to the crack plane is not measured in these tests, the stress in-tensity must be inferred from the clip gage displacement. The estimation of K is compli-cated, however, by extraneous displacements, such as seating of the wedge/pin assembly.Also, local yielding can occur near the starter notch prior to fracture initiation. TheASTM standard outlines a cyclic loading procedure for identifying these displacements;Fig. 7.29 shows a schematic load-displacement curve that illustrates this method. Thespecimen is first loaded to a predetermined displacement and, assuming the crack has notinitiated, the specimen is unloaded. The displacement at zero load is assumed to representthe effects of fixture seating, and this component is subtracted from the total displacementwhen stress intensity is computed. The specimen is reloaded to a somewhat higher dis-placement and then unloaded; this process continues until fracture initiates. The zero loadoffset displacements that occur after the first cycle can be considered to be due to notch tipplasticity. The correct way to treat this displacement component in K calculations is un-clear at present. Once the crack propagates through the plastic zone, the plastic displace-ment is largely recovered (i.e., converted into an elastic displacement), and thus may con-tribute to the driving force. It is not known whether or not there is sufficient time forthis displacement component to exert an influence on the running crack. The ASTMstandard takes the middle ground on this question, and requires that half of the plastic off-set be included in the stress intensity calculations.

-'NIST was formerly known as the National Bureau of Standards (NBS), which explains the initials on eitherend of the specimen in Fig. 7.27.


FIGURE 7.26 Photograph of a wide plate crack arrest test performed at NIST. [30]. (Photograph pro-vided by J.G. Merkle.)

400 Chapter 7

\ x\\\\\\\\\\\\\\vFIGURE 7.27 Apparatus for Kja tests [31].

After the test, the specimen should be heat tinted at 250-350°C for 10 to 90 min tomark the crack propagation. When the specimen is broken open, the arrested crack lengthcan then be measured on the fracture surface. The critical stress intensity at initiation,K0, is computed from the initial crack size and the critical clip gage displacement. Theprovisional arrest toughness, Ka, is calculated from the, final crack size, assuming con-stant displacement. These calculations assume quasistatic conditions. As discussed inChapter 4, this assumption can lead to underestimates of arrest toughness. The ASTMstandard, however, cites experimental evidence [32,33] that implies that the errors intro-duced by a quasistatic assumption are small in this case.

r7 Fracture Toughness Testing of Metals 401

FIGURE 7.28 Side-grooved compact crack arrest specimen [31].

LOAD

FIGURE 7.29 Schematic load-displacementcurve for a Kja test [31], where Vj and V2 arezero load offset displacement. When computingVcrif, all of the first offset and half of the subse-quent offsets are subtracted from the total dis-placement.

DISPLACEMENT

In order for the test to be valid, the crack propagation and arrest should occur underpredominantly plane strain linear elastic conditions. The following validity requirementsin ASTM E 1221-88 are designed to ensure that the plastic zone is small compared tospecimen dimensions, and that the crack jump length is within acceptable limits:

W-aa>Q.l5W (7.25a)

(7.25b)

(7.25c)

402 Chapter?

<Z25d)

where aa is the arrested crack length, a0 is the initial crack length, and cryj is the assumeddynamic yield strength, which the ASTM standard specifies at 205 MPa (30 ksi) abovethe quasistatic value. Since unstable crack propagation results in very high strain rates,the recommended estimate of ayd is probably very conservative.

If the above validity requirements are satisfied and all other provisions of ASTM E1221 are followed, Ka = Kja.

7.7 FRACTURE TESTING OF WELDMENTS

All of the test methods discussed so far are suitable for specimens extracted from uniformsections of homogeneous material. Welded joints, however, have decidedly heterogeneousmicrostructures and, in many cases, irregular shapes. Weldments also contain complexresidual stress distributions. Existing fracture toughness testing standards do not addressthe special problems associated with weldment testing. The factors that make weldmenttesting difficult (i.e. heterogeneous microstructures, irregular shapes, and residual stresses)also tend to increase the risk of brittle fracture in welded structures. Thus, one cannotsimply evaluate the regions of a structure where ASTM testing standards apply and ignorethe fracture properties of weldments.

Although there are currently no fracture toughness testing standards for weldments, anumber of laboratories and industries have significant experience in this area. TheWelding Institute in Cambridge, England, which probably has the most expertise, has re-cently published detailed recommendations for weldment testing [34]. The InternationalInstitute of Welding (IIW) has produced a similar document [35], although not as detailed.The American Petroleum Institute (API) has published guidelines for heat affected zone(HAZ) testing as part of a weld procedure qualification approach [36]. Committees withinASTM and BSI are currently drafting weldment test methods, relying heavily on 20 yearsof practical experience as well as the aforementioned documents.

Some of the general considerations and current recommendations for weldment test-ing are outlined below, with emphasis on The Welding Institute procedure [34] because itis the most complete document to date. Early drafts of both the ASTM and BSI guide-lines incorporate many of the suggestions in The Welding Institute document.

When performing fracture toughness tests on weldments, a number of factors needspecial consideration. Specimen design and fabrication are more difficult because of theirregular shapes and curved surfaces associated with some welded joints. The heteroge-neous microstructure of typical weldments requires special attention to the location of thenotch in the test specimen. Residual stresses make fatigue precracking of weldment spec-imens more difficult. After the test, a weldment must often be sectioned and examinedmetallographically to determine whether or not the fatigue crack sampled the intended mi-crostructure.


7.7.1 Specimen Design and Fabrication

The underlying philosophy of the Welding Institute [34] guidelines on specimen designand fabrication is that the specimen thickness should be as close to the section thicknessas possible. Thicker specimens tend to produce more crack tip constraint, and hencelower toughness (See Chapters 2 and 3). Achieving nearly full thickness weldment speci-mens often requires sacrifices in other areas. For example if a specimen is to be extractedfrom a curved section such as a pipe, one can either produce a subsize rectangular speci-men which meets the tolerances of the existing ASTM standards, or a full thickness spec-imen that is curved. The Welding Institute recommends the latter.

If curvature or distortion of a weldment is excessive, the specimen can be straight-ened by bending on either side of the notch to produce a "gull wing" configuration, whichis illustrated in Fig. 7.30. The bending must be performed so that the three loadingpoints (in an SENB specimen) are aligned.

FIGURE 7.30 The gull-wing configuration for weldment specimens with excessive curvature [34].

Fabrication of either a compact or SENB weldment specimen is possible, but theSENB specimen is preferable in nearly every case. Although the compact specimen con-sumes less material (for a given B and W) in parent metal tests, it requires more weldmetal in a through-thickness orientation (L-jTor T-L) than an SENB specimen (Fig. 7.2).It is impractical to use a compact geometry for surface notched specimens (T-S orL-S);such a specimen would be greatly undersized with the standard B~K2B geometry.

The Welding Institute recommendations cover both the rectangular and square sec-tion SENB specimens. The appropriate choice of specimen type depends on the orienta-tion of the notch.

7.7.2 Notch Location and Orientation

Weldments have a highly heterogeneous microstructure. Fracture toughness can varyconsiderably over relatively short distances. Thus it is important to take great care in lo-cating the fatigue crack in the correct region. If the fracture toughness test is designed tosimulate an actual structural flaw, then the fatigue crack must sample the same mi-crostructure as the flaw. For a weld procedure qualification or a general assessment of aweldment's fracture toughness, location of the crack in the most brittle region may be de-sirable, but it is difficult to know in advance which region of the weld has the lowesttoughness. In typical C-Mn structural steels, low toughness is usually associated withthe coarse grained HAZ and the intercritically reheated HAZ. A microhardness survey can

404 Chapter 7

help identify low toughness regions because high hardness is often coincident with brittlebehavior. The safest approach is to perform fracture toughness tests on a variety of re-gions in a weldment.

Once the microstructure of interest is identified, a notch orientation must be se-lected. The two most common alternatives are a through-thickness notch and a surfacenotch, which are illustrated in Fig. 7.31. Since full thickness specimens are desired, thesurface notched specimen should be square section (BXB), while the through thicknessnotch will usually be in a rectangular (B X 25) specimen.

For weld metal testing, the through-thickness orientation is usually preferable be-cause a variety of regions in the weld are sampled. However, there may be cases wherethe surface notched specimen is the most suitable for testing the weld metal. For exam-ple, a surface notch can sample a particular region of the weld metal, such as the root orcap, or the notch can be located in a particular microstructure, such as unrefined weldmetal.

(a) Through-thickness notch. (b) Surface notch.

FIGURE 7.31 Notch orientation in weldment specimens [34].

Notch location in the HAZ often depends on the type of weldment. If welds are pro-duced solely for mechanical testing, for example as part of a weld procedure qualificationor a research program, the welded joint can be designed to facilitate HAZ testing. Figure7.32 illustrates the K and half-A" preparations, which simulate double-V and single-Vwelds, respectively. The plates should be tilted when these weldments are made, to havethe same angle of attack for the electrode as in an actual single- or double- V joint. Forfracture toughness testing, a through-thickness notch is placed in the straight side of the/sTorhalf-^HAZ.

In many instances, fracture toughness testing must be performed on an actual pro-duction weldment, where the joint geometry is governed by the structural design. In such


cases, a surface notch is often necessary for the crack to sample sufficient HAZ material.The measured toughness is sensitive to the volume of HAZ material sampled by the cracktip because of the weakest link nature of cleavage fracture (see Chapter 5).

Another application of the surface notched orientation is the simulation of structuralflaws. Figure 7.33 illustrates HAZ flaws in a structural weld and a surface notched frac-ture toughness specimen that models one of the flaws.

Figure 7.33 demonstrates the advantages of allowing a range of a/W ratios in surfacenotched specimens. A shallow notch is often required to locate a crack in the desired re-gion, but existing ASTM standards do not allow a/W ratios less than 0.45. Shallownotched fracture toughness specimens tend to have lower constraint than deeply crackedspecimens, as Figs 3.28, 3.44 and 3.45 illustrate. Thus there is a conflict between theneed to simulate a structural condition and the traditional fracture mechanics approach,where a toughness value is supposed to be a size independent material property. One wayto resolve this conflict is through constraint corrections, such as that applied to the datain Figs. 3.44 and 3.45.

FIGURE 7.32 Special weld joint designs for frac-ture toughness testing of the heat-affected zone(HAZ) [34].

(b) K weld ^H Notch Plane

(a) Weldment with a flaw in the HAZ. (b) Test specimen with simulated structural flaw.

FIGURE 733 Test specimen with notch orientation and depth that matches a flaw in a structure [34].

406 Chapter?

7.7.3 Fatigue Precracking

Weldments that have not been stress relieved typically contain complex residual stress dis-tributions that interfere with fatigue precracking of fracture toughness specimens. Tensileresidual stresses accelerate fatigue crack initiation and growth, but compressive stresses re-tard fatigue. Since residual stresses vary through the cross section, fatigue crack fronts inas-welded samples are typically very nonuniform.

Towers and Dawes [37] evaluated the various methods for producing straight fatiguecracks in welded specimens, including reverse bending, high R ratio, and local compres-sion.

The first method bends the specimen in the opposite direction to the normal loadingconfiguration to produce residual tensile stresses along the crack front that counterbalancethe compressive stresses. Although this technique gives some improvement, it does notproduce acceptable fatigue crack fronts.

The R ratio in fatigue cracking is the ratio of the minimum stress to the maximum.A high R ratio minimizes the effect of residual stresses on fatigue, but also tends to in-crease the apparent toughness of the specimen. In addition, fatigue precracking at a highR ratio takes much longer than precracking at R = 0.1, the recommended R ratio of thevarious ASTM fracture testing standards.

The only method evaluated that produced consistently straight fatigue cracks was lo-cal compression, where the ligament is compressed to produce nominally 1% plasticstrain through the thickness, mechanically relieving the residual stresses. However, localcompression can reduce the toughness slightly. Towers and Dawes concluded that thebenefits of local compression outweigh the disadvantages, particularly in the absence of aviable alternative.

7.7.4 Post-Test Analysis

Correct placement of a fatigue crack in weld metal is usually not difficult because this re-gion is relatively homogeneous. The microstructure in the HAZ, however, can changedramatically over very small distances. Correct placement of a fatigue crack in the HAZis often accomplished by trial and error. Because fatigue cracks are usually slightlybowed, the precise location of the crack tip in the center of a specimen cannot be inferredfrom observations on the surface of the specimen. Thus HAZ fracture toughness speci-mens must be examined metallographically after the test to determine the microstructurethat initiated fracture. In certain cases, post-test examination may be required in weldmetal specimens.

Figure 7.34 illustrates a procedure for sectioning surface notched and through-thick-ness notched specimens [34]. First, the origin of the fracture must be located by thechevron markings on the fracture surface. After marking the origin with a small spot ofpaint the specimen is sectioned perpendicular to the fracture surface and examined metal-lographically. The specimen should be sectioned slightly to one side of the origin andpolished down to the initiation site. The spot of paint appears on the polished specimenwhen the origin is reached.


The API document RP2Z [36] outlines a post-test analysis of HAZ specimenswhich is more detailed and cumbersome than the procedure outlined above. In addition tosectioning the specimen, the amount of coarse-grained material at the crack tip must bequantified. For the test to be valid, at least 15% of the crack front must be in the coarse-grained HAZ. The purpose of this procedure is to pre qualify steels with respect to HAZtoughness, identifying those that produce low HAZ toughness so that they can be rejectedbefore fabrication.

FIGURE 7.34 Post-test sectioning of a weldment fracture toughness specimen to identify the mi-crostructural that caused fracture.

7.8 TESTING AND ANALYSIS OF STEELS IN THE DUCTILE-BRITTLE TRANSITION REGION

Chapter 5 described the micromechanisms of cleavage fracture, and indicated that cleavagetoughness data tend to be highly scattered, especially in the transition region. Because ofthis substantial scatter, data should be treated statistically rather than deterministically.That is, a given steel does not have a single value of toughness at a particular temperaturein the transition region; rather, the material has a toughness distribution. Testing numer-ous specimens to obtain a statistical distribution can be expensive and time-consuming.Fortunately, a methodology has been developed that greatly simplifies this process forstructural steels. A draft ASTM standard [38] for the ductile-brittle transition region im-plements this methodology.

As discussed in Chapter 5, the cleavage fracture toughness distribution can be repre-sented by three-parameter Weibull distribution with a slope of 4 (Eq. 5.26). The draft

ASTM standard sets the threshold toughness, Kmin equal to 20 MPaVm, resulting in

408 Chapter?

\4

IT 1 ,^-20F = 1 - exp Jc- '• ' (7.23)®K-20)

where F is the cumulative probability, Kjc is the fracture toughness (inferred by convert-ing Jc to an equivalent K through Eq. (7.18)), and ©g is the 63rd percentile toughness.Note that two of the three parameters in the Weibull distribution are specified, in Eq.(7.23) leaving only one degree of freedom. A distribution that contains only one parame-ter can be fit with a relatively small sample size.

The draft ASTM standard accounts for temperature dependence of toughness through& fracture toughness master curve approach developed by Wallin [39]. Wallin observedthat a wide range of ferritic steels have a characteristic fracture toughness-temperaturecurve, and the only difference between different grades and heats of steel was the absoluteposition of the curve with respect to temperature. High toughness steels have a low tran-sition temperature and low toughness steels have a high transition temperature. The tem-perature dependence of the median fracture toughness in the transition region can be esti-mated from

*««*o = 30 + 70exp[0.019(r- T,)] (7.24)

where T0 is a reference transition temperature in °C and the units on Kjc are MPaVm. At

T= T0, the medial fracture toughness =100 MPavm. Once T0 is known for a givenmaterial, the fracture toughness distribution can be inferred as a function of temperaturethrough Eqs. (7.23) and (7.24).

The first step in determining T0 is to perform replicate fracture toughness tests at aconstant temperature. The draft ASTM standard recommends at least 6 tests. These dataare then fit to Eq. (7.23) to determine &K at the test temperature. The median toughnessat this temperature can be inferred by setting F = 0.5 and solving for Kjc in Eq. (7.23).Finally, T0 is computed by rearranging Eq. (7.24):

—Jc(median} - 1 (7.25)

The draft ASTM standard recommends applying this procedure at several test temperaturesin order to obtain more than one estimate of T0. Figure 7.35 schematically illustrates thefracture toughness master curve for a particular steel. By combining Eqs. (7.23) and(7.24), it is possible to infer median, upper-bound and lower-bound toughness as a func-tion of temperature.

The master curve approach works best the ductile-brittle transition region. It maynot fit data in the lower shelf very well, and it is totally unsuitable for the upper shelf.Equation (7.24) increases without bound with increasing temperature, and thus does notmodel the upper shelf.


K•Jc

ExperimentalData I °

TEMPERATURE

FIGURE 7.35 Fracture toughness master curve [38].

7.9 QUALITATIVE TOUGHNESS TESTS

Before the development of formal fracture mechanics methodology, engineers realized theimportance of material toughness in avoiding brittle fracture. In 1901 a French scientistnamed G. Charpy developed a pendulum test that measured the energy of separation innotched metallic specimens. This energy was believed to be indicative of the resistance ofthe material to brittle fracture. An investigation of the Liberty ship failures during WorldWar II revealed that fracture was much more likely in steels with Charpy energy less than20/(15ft-lb).

During the 1950s, when Irwin and his colleagues at the Naval Research Laboratory(NRL) were formulating the principles of linear elastic fracture mechanics, a metallurgistat NRL named W.S. Pellini developed the drop weight test, a qualitative measure of crackarrest toughness.

Both the Charpy test and the Pellini drop weight test are still widely applied todayto structural materials. ASTM has standardized the drop weight tests, as well as a numberof related approaches, including the Izod, drop weight tear and dynamic tear tests [40-43](see below). Although these tests lack the mathematical rigor and predictive capabilitiesof fracture mechanics methods, these approaches provide a qualitative indication of mate-rial toughness. The advantage of these qualitative methods is that they are cheaper andeasier to perform than fracture mechanics tests. These tests are suitable for materialscreening and quality control, but are not reliable indicators of structural integrity.

7.9.1 Charpy and Izod Impact Test

The ASTM Standard E 23-88 [40] covers Charpy and Izod testing. These tests both in-volve impacting a small notched bar with a pendulum and measuring the fracture energy.

410 Chapter 7

The Charpy specimen is a simple notched beam that is impacted in three-point bending,while the Izod specimen is a cantilever beam that is fixed at one end and impacted at theother. Figure 7.36 illustrates both types of test.

Charpy and Izod specimens are relatively small, and thus do not consume much ma-terial. The standard cross section of both specimens is 10 mm x 10 mm, and the lengthsare 55 and 75 mm for Charpy and Izod specimens, respectively.

The pendulum device provides a simple but elegant method for quantifying fractureenergy. As Fig. 7.37 illustrates, the pendulum is released from a height yy and swingsthrough the specimen to a height y2- Assuming negligible friction and aerodynamic drag,the energy absorbed by the specimen is equal to the height difference times the weight ofthe pendulum. A simple mechanical device on the Charpy machine converts the heightdifference to a direct read-out of absorbed energy.

A number of investigators [44-49] have attempted to correlate Charpy energy to frac-ture toughness parameters such as K/c. These empirical correlations seem to work rea-sonably well in some cases, but are unreliable in general. There are several important dif-ferences between the Charpy test and fracture mechanics tests that preclude simple rela-tionships between the qualitative and quantitative measures of toughness. The Charpytest contains a blunt notch, while fracture mechanics specimens have sharp fatigue cracks.The Charpy specimen is subsize, and thus has low constraint. In addition, the Charpyspecimen experiences impact loading, while most fracture toughness tests are conductedunder quasistatic conditions.

It is possible to obtain quantitative information from fatigue precracked Charpyspecimens, provided the tup (i.e. the striker) is instrumented [50,51]. Such an experimentis essentially a miniature dynamic fracture toughness test.

FIGURE 7.36 Charpy and Izod notched impact tests [40].


FIGURE 7.37 A pendulum device for impact testing. The energy absorbed by the specimen is equal tothe weight of the crosshead, times the difference in height before and after impact.

7.9.2 Drop Weight Test

The ASTM standard E 208-87 [41] outlines the procedure for performing the Pellini dropweight test. A plate specimen with a starter notch in a brittle weld bead is impacted inthree-point bending. A cleavage crack initiates in the weld bead and runs into the parentmetal. If the material is sufficiently tough, the crack arrests; otherwise the specimen frac-tures completely.

Figure 7.38 illustrates the drop weight specimen and the testing fixture. Thecrosshead drops onto the specimen, causing it to deflect a predetermined amount. The fix-ture is designed with a deflection stop, which limits the displacement in the specimen. Acrack initiates at the starter notch and either propagates or arrests, depending on the tem-perature and material properties. A "break" result is recorded when the running crackreaches at least one specimen edge. A "no-break" result is recorded if the crack arrests inthe parent metal. Figure 7.39 gives examples of break and no-break results.

A nil-ductility transition temperature (NDTT) is obtained by performing drop weighttests over a range of temperatures, in 5°C or 10°F increments. When a no-break result isrecorded, the temperature is decreased for the next test; test temperature is increased when aspecimen fails. When break and no-break results are obtained at adjoining temperatures, asecond test is performed at the no-break temperature. If this specimen fails, a test is per-formed at one temperature increment (5°C or 10°F) higher. The process is repeated untiltwo no-break results are obtained at one temperature. The NDTT is defined as 5°C or10°F below the lowest temperature where two no-breaks are recorded.

The nil-ductility transition temperature gives a qualitative estimate of the ability ofa material to arrest a running crack. Arrest in structures is more likely to occur if the ser-vice temperature is above NDTT, but structures above NDTT are not immune to brittlefracture.

412 Chapter 7

The ship building industry in the United States currently uses the drop weight testto qualify steels for ship hulls. The nuclear power industry relies primarily on quantita-tive fracture mechanics methodology, but uses the NDTT to index fracture toughness datafor different heats of steel (see Chapter 9).

FIGURE 7.38 Apparatus for drop weight testing according to ASTM E 208-87 [41]

No-Break Break Break

FIGURE 7.39 Examples of break and no-break behavior in drop weight tests. A break is recorded whenthe crack reaches at least one edge of the specimen.

7.9.3 Drop Weight Tear and Dynamic Tear Tests

Drop weight tear and dynamic tear tests are similar to the Charpy test, except that theformer are performed on large specimens. The ASTM standards E 604-83 [43] and E 436-74 [42] cover drop weight tear and dynamic tear tests, respectively. Both test methods


utilize three point bend specimens that are impacted in a drop tower or pendulum ma-chine.

Drop weight tear specimens are 41 mm (1.6 in) wide, 16 mm (0.625 in) thick, andare loaded over a span of 165 mm (6.5 in). These specimens contain a sharp machinednotch. A 0.13 mm (0.010 in) deep indentation is made at the tip of this notch. The frac-ture energy is measured in this test, much like the Charpy and Izod tests. Since dropweight specimens are significantly larger than Charpy specimens, the fracture energy ismuch greater, and the capacity of the testing machine must be scaled accordingly. If apendulum machine is used, the energy can be determined in the same manner as in theCharpy and Izod tests. A drop test must be instrumented, however; because only a por-tion of the potential energy is absorbed by the specimen; the remainder is transmittedthrough the foundation of the drop tower.

The dynamic tear test quantifies the toughness of steel through the appearance of thefracture surface. In the ductile-brittle transition region, a dynamic test produces a mixtureof cleavage fracture and microvoid coalescence; the relative amount of each depends on thetest temperature. The percent "shear" on the fracture surface is reported in dynamic teartests, where the so-called shear fracture is actually microvoid coalescence (Chapter 5).Dynamic tear specimens are 76 mm (3 in) wide, 305 mm (12 in) long, and are loaded overa span of 254 mm (10 in). The specimen thickness is equal to the thickness of the plateunder consideration. The notch is pressed into the specimen by indentation.

REFERENCES •

1. E 616-89, "Terminology Relating to Fracture Testing." American Society for Testingand Materials, Philadelphia, 1989.


3. Baker, A., "A DC Potential Drop Procedure for Crack Initiation and R CurveMeasurements During Ductile Fracture Tests." ASTM STP 856, American Society ofTesting and Materials, Philadelphia, 1985, pp. 394-410.

4. Schwalbe, K-H, Hellmann, D., Heerens, J., Knaack, J., Muller-Roos, J., "Measurementof Stable Crack Growth Including Detection of Initiation of Growth Using the DCPotential Drop and the Partial Unloading Methods." ASTM STP 856, American Societyof Testing and Materials, Philadelphia, 1985, pp. 338-362.

5. E 813-89, "Standard Test Method for JIG, a Measure of Fracture Toughness." AmericanSociety for Testing and Materials, Philadelphia, 1987.

6. E 1152-87 "Standard Test Method for Determining J-R Curves." American Society forTesting and Materials, Philadelphia, 1987.

7. Dawes, M.G. "Elastic-Plastic Fracture Toughness Based on the COD and J-ContourIntegral Concepts." ASTM STP 668, American Society of Testing and Materials,Philadelphia, 1979, pp. 306-333.

414 Chapter?

8. Anderson, T.L., McHenry, H.I. and Dawes, M.G., "Elastic-Plastic Fracture ToughnessTesting with Single Edge Notched Bend Specimens." ASTM STP 856, American Societyof Testing and Materials, Philadelphia, 1985. pp. 210-229.

9. Andrews, W.R. and Shih, C.F., "Thickness and Side-Groove Effects on J- and 8-Resistance Curves for A533-B Steel at 93°C." ASTM STP 668, American Society ofTesting and Materials, Philadelphia, 1979, pp. 426-450.

10. BS 5447:1974 "Methods of Testing for Plane Strain Fracture Toughness (Kic) ofMetallic Materials." British Standards Institution, London, 1974.

11. "Standard Method for Measurement of Fracture Toughness." (Draft), American Society ofTesting and Materials, Philadelphia, 1994.

12. E 561-92a, "Standard Practice for R-Curve Determination." American Society ofTesting and Materials, Philadelphia, 1986.

13. Stricklin, L.L., "Geometry Dependence of Crack Growth Resistance Curves in ThinSheet Aluminum Alloys." Master of Science Thesis, Texas A&M University, CollegeStation, TX, December 1988.

14. "Standard Test Method for J-Integral Characterization of Fracture Toughness." (Draft),American Society of Testing and Materials, Philadelphia, 1994.

15. Joyce, J.A. and Hackett, E.M., "Development of an Engineering Definition of theExtent of J Singularity Controlled Crack Growth." NUREG/CR-5238, U.S. NuclearRegulatory Commission, Washington, D.C., May 1989.

16. Anderson, T.L. and Dodds, R.H., Jr., "Specimen Size Requirements for FractureToughness Testing in the Ductile-Brittle Transition Region." Journal of Testing andEvaluation, Vol. 19, 1991, pp. 123-134.

17. Anderson, T.L. and Dodds, R.H., Jr., "An Experimental and Numerical Investigation ofSpecimen Size Requirements for Cleavage Fracture Toughness." NUREG/CR-6272,Nuclear Regulatory Commission, Washington DC, (in press).

18. BS 5762: 1979, "Methods for Crack Opening Displacement (COD) Testing." BritishStandards Institution, London, 1979.

19. E 1290-93 "Standard Test Method for Crack Tip Opening Displacement Testing."American Society for Testing and Materials, Philadelphia, 1989.

20. Sorem, W.A., "The Effect of Specimen Size and Crack Depth on the Elastic-PlasticFracture Toughness of a Low Strength High-Strain Hardening Steel." Ph.D.Dissertation, The University of Kansas, Lawrence, KS, May 1989.

21. Duffy, J. and Shih, C.F., "Dynamic Fracture Toughness Measurements for Brittle andDuctile Materials." Advances in Fracture Research: Seventh International Conferenceon Fracture., Pergamon Press, Oxford, 1989, pp. 633-642.

22. Sanford, R.J. and Dally, J.W., "A General Method for Determining Mixed-Mode StressIntensity Factors from Isochromatic Fringe Patterns." Engineering Fracture Mechanics,Vol. 11, 1979, pp. 621-633.


23. Chona, R., Irwin, G.R., and Shukla, A., "Two and Three Parameter Representation ofCrack Tip Stress Fields." Journal of Strain Analysis, Vol. 17, 1982, pp. 79-86.

24. Kalthoff, J.F., Beinart, J., Winkler, S., and Klemm, W., "Experimental Analysis ofDynamic Effects in Different Crack Arrest Test Specimens." ASTM STP 711, AmericanSociety for Testing and Materials, Philadelphia, 1980, pp. 109-127.

25. Nakamura, T., Shih, C.F. and Freund, L.B., "Analysis of a Dynamically Loaded Three-Point-Bend Ductile Fracture Specimen." Engineering Fracture Mechanics, Vol. 25,1986, pp. 323-339.

26. Nakamura, T., Shih, C.F. and Freund, L.B., "Three-Dimensional Transient Analysis of aDynamically Loaded Three-Point-Bend Ductile Fracture Specimen." ASTM STP 995,Vol. I, American Society of Testing and Materials, Philadelphia, 1989, pp. 217-241.

27. Joyce J.A. and Hacket, E.M., "Dynamic J-R Curve Testing of a High Strength SteelUsing the Multispecimen and Key Curve Techniques." ASTM STP 905, AmericanSociety of Testing and Materials, Philadelphia, 1984, pp. 741-774.

28. Joyce J.A. and Hacket, E.M., "An Advanced Procedure for J-R Curve Testing Using aDrop Tower." ASTM STP 995, American Society of Testing and Materials, Philadelphia,1989, 298-317.

29. Robertson, T.S., "Brittle Fracture of Mild Steel." Engineering, Vol. 172, 1951, pp.445-448.

30. Naus, D.J., Nanstad, R.K., Bass, B.R., Merkle, J.G., Pugh. C.E., Corwin, W.R., andRobinson G.C., "Crack-Arrest Behavior in SEN Wide Plates of Quenched and TemperedA 533 Grade B Steel Tested under Nonisothermal Conditions." NUREG/CR-4930, U.S.Nuclear Regulatory Commission, Washington, D.C., August 1987.

31. E 1221-88, "Standard Method for Determining Plane-Strain Crack-Arrest Toughness,Kja, of Ferritic Steels." American Society of Testing and Materials, Philadelphia, 1988.

32. Crosley, P.B., Fourney, W.L., Harm, G.T., Hoagland, R.G., Irwin, G.R., and Ripling,E.J., "Final Report on Cooperative Test Program on Crack Arrest ToughnessMeasurements." NUREG/CR-3261, U.S. Nuclear Regulatory Commission,Washington, D.C., April 1983.

33. Barker, D.B., Chona, R., Fourney, W.L., and Irwin, G.R., "A Report on the RoundRobin Program Conducted to Evaluate the Proposed ASTM Test Method of Determiningthe Crack Arrest Fracture Toughness, Kia, of Ferritic Materials." NUREG/CR-4996,January 1988.

34. Dawes, M.G., Pisarski, H.G. and Squirrell, H.G., "Fracture Mechanics Tests on WeldedJoints" ASTM STP 995, American Society of Testing and Materials, Philadelphia,1989, pp. 11-191 - 11-213.

35. Satok K. and Toyoda, M., "Guidelines for Fracture Mechanics Testing of WM/HAZ."Working Group on Fracture Mechanics Testing of Weld Metal/HAZ, InternationalInstitute of Welding, Commission X, IIW Document X-l 113-86.

416 Chapter 7

36. RP 2Z, "Recommended Practice for Preproduction Qualification of Steel Plates forOffshore Structures." American Petroleum Institute, 1987.

37. Towers O.L. and Dawes, M.G., "Welding Institute Research on the Fatigue Precrackingof Fracture Toughness Specimens." ASTM STP 856, American Society of Testing andMaterials, Philadelphia, 1985. pp. 23-46.

38. "Test Practice (Method) for Fracture Toughness in the Transition Range." (Draft)American Society of Testing and Materials, Philadelphia, 1994.

39. Wallin, K., "Fracture Toughness Transition Curve Shape for Ferritic Structural Steels."Proceedings of the Joint FEFG/ICF International Conference on Fracture of EngineeringMaterials, Singapore, August 6-8, 1991, pp. 83-88.

40. E 23-88, "Standard Test Methods for Notched Bar Impact Testing of Metallic Materials."American Society of Testing and Materials, Philadelphia, 1988.

41. E 208-87, "Standard Test Method for Conducting Drop-Weight Test to Determine Nil-Ductility Transition Temperature of Ferritic Steels." American Society of Testing andMaterials, Philadelphia, 1987.

42. E 436-74, "Standard Method for Drop-Weight Tear Tests of Ferritic Steels." AmericanSociety of Testing and Materials, Philadelphia, 1974.

43. E 604-83, "Standard Test Method for Dynamic Tear Testing of Metallic Materials."American Society of Testing and Materials, Philadelphia, 1983.

44. Marandet, B. and Sanz, G., "Evaluation of the Toughness of Thick Medium StrengthSteels by LEFM and Correlations Between Kjc and CVN." ASTM STP 631, AmericanSociety of Testing and Materials, Philadelphia, 1977, pp. 72-95.

45. Rolfe, S.T. and Novak, S.T., "Slow Bend KIc Testing of Medium Strength HighToughness Steels." ASTM STP 463, American Society of Testing and Materials,Philadelphia, 1970, pp. 124-159.

46. Barsom, J.M. and Rolfe, S.T., "Correlation Between KIC and Charpy V Notch TestResults in the Transition Temperature Range." ASTM STP 466, American Society ofTesting and Materials, Philadelphia, 1970, pp. 281-301.

47. Sailors, R.H. and Corten, H.T., "Relationship between Material Fracture ToughnessUsing Fracture Mechanics and Transition Temperature Tests." ASTM STP 514, AmericanSociety of Testing and Materials, Philadelphia, 1973, pp. 164-191.

48. Begley, J.A. and Logsdon, W.A., "Correlation of Fracture Toughness and CharpyProperties for Rotor Steels." Westinghouse Report, Scientific Paper 71-1E7, MSLRF-Pl-1971.

49. Ito, T., Tanaka, K. and Sato, M. "Study of Brittle Fracture Initiation from Surface Notchin Welded Fusion Line." IIW Document X-704-730, September 1973.

50. Wullaert, R.A., "Applications of the Instrumented Charpy Impact Test." ASTM STP 466,American Society of Testing and Materials, Philadelphia, 1970, pp. 148-164.


51. Turner, C.E., "Measurement of Fracture Toughness by Instrumented Impact Test."ASTM STP 466, American Society of Testing and Materials, Philadelphia, 1970, pp.93-114.

APPENDIX 7: EXPERIMENTALESTIMATES OF DEFORMATION J

This appendix presents a derivation of Eq. (7.15b), which estimates the plastic componentof the deformation theory J for a growing crack. Only the plastic component need beconsidered here, because the elastic J at any point in the test can be computed directlyfrom the current load and crack length.

Figure A7.1 shows a schematic load-plastic displacement curve for a growing crack.Consider two neighboring points on the curve, where the crack grows from aj to #2, andthe plastic component of the / integral varies from Jj to J2- Each of these J values canbe computed from the area under the appropriate "deformation theory" curve. For exam-ple, J'i is given by

1= „, (A7-D

where Apl(OABE) ^ me area defined by the corresponding points in Fig. A7.1. It isconvenient to separate /2 mto two components:

Jl =BNb2

= Ja + Jp (A7.2)

Assume that J] is known, and that we wish to compute /2- First, let us determine the re-lationship between Jj and Ja, the component of }% that corresponds to the displacementAj. This relationship can be approximated by

(A7.3)

The partial derivative, for the general case, can be solved as follows:

(dJPi]V 3a ^B,

418

Experimental Estimates of Deformation J 419

LOAD

A <C<

Pi

P2

"Deformation"Path

El G|

o Ai A2

PLASTIC DISPLACEMENT

FIGURE A7.1 Schematic load-plastic displacement curve for a specimen in which the crack grows froma/ to 02-

BNb| | PlBb2 Bb daN N

Recall the energy release rate definition of Ji:

(A7.4)

_~BN

(A7.5)

Substituting Eq. (A7.5) into Eq. (A7.4) leads to

(A7.6)

where

y= 77-1TlWd(b/W)J

(A7.7)

420 Appendix 7

Evaluating the derivative in Eq. (A7.3) at Jj gives

(A7.8)

For a deeply notched SENB specimen, 77 = 2; thus 7= 1, and Eq. (A7.8) reduces to

(A7.9)

For a deeply notched compact specimen, r] is approximately given by Eq. (7.lib).Substituting this result into Eq. (A7.7) gives

0.5226 IW

2 + 0.522b/W

which can be approximated by

(A7.10)

for 0.50 < a/W < 0.70.The second term in Eq. (A7.2), Jfi, can be estimated in a number of ways, including

_ 772P2(A2-A1)«/ /j """•'

P D K(A7.ll)

which is a good approximation, provided the load (on the deformation theory curve for &2)does not vary significantly between Aj and A2- The ASTM Standard E 1152-87 chosethe following estimate for Jfi.

1-

Summing /a (Eq. (A7.8)) and the ASTM estimate for Jfi (Eq. (A7.12) gives

J2 =

(A7.12)

(A7.13)

Experimental Estimates of Deformation J 421

which is essentially identical to Eq. (7.15b). The J-R curve can be computed by applyingEqs. (7.15a) and (7.15b) to successive increments of crack growth.

Figure A7.2 illustrates the numerical error that results from Eq. (A7.13). Note thatthis equation causes a slight underestimate of Jf$. There may also be small errors in theestimate of Jot, since a partial derivative is applied to a finite change in crack size (Eq.(A7.3)). Equation (A7.9), however, is rigorously correct for a deeply notched bend spec-imen. This can be readily shown by applying the dimensional argument that was invokedin Section 3.2.5.

LOAD

Pi

Ai AzPLASTIC DISPLACEMENT

FIGURE A7.2 Schematic illustration of the error in Jpi that results from Eq. (A7.13).

8. FRACTURE TESTING OFNONMETALS

The procedures for fracture toughness testing of metals, which are described in Chapter 7,are fairly well established. Fracture testing of plastics, composites and ceramics isrelatively new, however, and there are a number of unresolved issues.

Although many aspects of fracture toughness testing are similar for metals andnonmetals, there are several important differences. In some cases, metals fracture testingtechnology is inadequate on theoretical grounds. For example, the mechanical behavior ofplastics can be highly rate dependent, and composites often violate continuum as-sumptions (see Chapter 6). There are also more pragmatic differences between fracturetesting of metals and nonmetals. Ceramics, for instance, are typically very hard andbrittle, which makes specimen fabrication and testing more difficult.

This chapter briefly summarizes the current procedures for measuring fracturetoughness in plastics, fiber-reinforced composites, and ceramics. The reader should befamiliar with the material in Chapter 7, since much of the same methodology (e.g.,specimen design, instrumentation, fracture parameters) is currently being applied to non-metals.

8.1 FRACTURE TOUGHNESS MEASUREMENTS IN ENGINEERINGPLASTICS

Engineers and researchers who have attempted to measure fracture toughness of plasticshave relied almost exclusively on metals testing technology. Existing experimentalapproaches implicitly recognize the potential for time-dependent deformation, but do notspecifically address viscoelastic behavior in most instances. The recent work of Schapery[1,2], who developed a viscoelastic / integral (Chapter 4), has not seen widespreadapplication to laboratory testing.

The Mode I stress intensity factor, Kj, and the (conventional) J integral wereoriginally developed for time-independent materials, but may also be suitable forviscoelastic materials in certain cases. The restrictions on these parameters are exploredbelow, followed by a summary of procedures for K and J testing on plastics. Section8.1.5 briefly outlines possible approaches for taking account of viscoelastic behavior andtime-dependent yielding in fracture toughness measurements.

8.1.1 The Suitability of K and / for Polymers

A number of investigators [3-6] have reported K]c, J]c, and J-R curve data for plastics.They applied testing and data analysis procedures that are virtually identical to metalsapproaches (Chapter 7). The validity of K and / is not guaranteed, however, when amaterial exhibits rate dependent mechanical properties. For example, neither J nor K are

423

424 ChapterS

suitable for characterizing creep crack growth in metals (Section 4.2);l an alternateparameter, C*, is required to account for the time-dependent material behavior. Schapery[1,2] has proposed an analogous parameter, Jv, to characterize viscoelastic materials(Section 4.3).

Let us examine the basis for applying K and / to viscoelastic materials, as well asthe limitations on these parameters.

K-Controlled FractureIn linear viscoelastic materials, remote loads and local stresses obey the same

relationships as in the linear elastic case. Consequently, the stresses near the crack tip

exhibit a 7/Vrsingularity:

and KI is related to remote loads and geometry through the conventional linear elasticfracture mechanics (LEFM) equations introduced in Chapter 2. The strains anddisplacements depend on the viscoelastic properties, however. Therefore, the criticalstress intensity factor for a viscoelastic material can be rate dependent; a Kfc value from alaboratory specimen is transferable to a structure only if the local crack tip strain historiesof the two configurations are similar. Equation (8.1) only applies when yielding andnonlinear viscoelasticity are confined to a small region surrounding the crack tip.

Under plane strain linear viscoelastic conditions, Kj is related to the viscoelastic Jintegral, Jv, as follows [1]:

(8-2)

where ER is a reference modulus, which is sometimes defined as the short-time relaxationmodulus.

Figure 8.1 illustrates a growing crack at times t0 and to + /p.2 Linear viscoelasticmaterial surrounds a Dugdale strip yield zone, which is small compared to specimendimensions. Consider a point A, which is at the leading edge of the yield zone at to andis at the trailing edge at t0 + tp. The size of the yield zone and the crack tip openingdisplacement (CTOD) can be approximated as follows (see Chapters 2 & 3):

/T°cr J

(8.3)

The stress intensity factor is suitable for high temperature behavior in limited situations. At short times,when the creep zone is confined to a small region surrounding the crack tip, K uniquely characterizes cracktip conditions, while C* is appropriate for large scale creep,o

This derivation, which was adapted from Marshall, et al. [7], is only heuristic and approximate. Schapery[8] performed a more rigorous analysis that led to a result that differs slightly from Eq. (8.9).

Fracture Testing ofNonmetals 425

(a) Crack tip position at time t0.

and

(b) Crack tip position at time t0 + tp.

FIGURE 8.1 Crack growth at a constant CTOD in a linear viscoelastic material.

ICocrE(tp}

(8.4)

where ocr is the crazing stress. Assume that crack extension occurs at a constant CTOD.The time interval tp is given by

(8.5)

where a is the crack velocity. For many polymers, the time-dependence of the relaxationmodulus can be represented by a simple power law:

426 Chapters

E(t) = Efn (8.6)

where Ej and n are material constants that depend on temperature. If crazing is assumedto occur at a critical strain that is time-independent, the crazing stress is given by

Gcr = E(t}£cr (8.7)

Substituting Eqs. (8.5) to (8.7) into Eq. (8.4) leads to

(8.8)

Solving for pc and inserting the result in Eq. (8.8) gives

KIC =

Therefore, according to this analysis, fracture toughness is proportional to an, and crackvelocity varies as Kj^n. Several investigators have derived relationships similar to Eq.(8.9), including Marshall, et al. [7] and Schapery [8].

Figure 8.2 is a schematic plot of crack velocity versus Kj for various n values. In atime-independent material, n = 0; the crack remains stationary below K]c, and becomesunstable when Kf = Kjc. In such materials, Kjc is a unique material property. Mostmetals and ceramics are nearly time independent at ambient temperature. When n > 0,crack propagation can occur over a range of Kj values. If, however, n is small, the crack

•

velocity is highly sensitive to stress intensity, and the a - Kj curve exhibits a sharp knee.For example, if n = 0.1, the crack velocity is proportional to Kj'®. In typical polymersbelow Tg, «<0.1.

Consider a short-time Kjc test on a material with n < 0.1, where Kj increasesmonotonically until the specimen fails. At low Kj values (i.e., in the early portion ofthe test), the crack growth would be negligible. The crack velocity would accelerate

•rapidly when the specimen reached the knee in the a - K] curve. The specimen would thenfail at a critical Kjc that would be relatively insensitive to rate. Thus if the knee in thecrack velocity-stress intensity curve is sufficiently sharp, a short-time Kjc test can providea meaningful material property.

One must be careful in applying a Kfc value to a polymer structure, however.While a statically loaded structure made from a time-independent material will not fail aslong as Kj < KIC, slow crack growth below Kjc does occur in viscoelastic materials.Recall from Chapter 1 the example of the polyethylene pipe that failed by time-dependent


crack growth over a period of several years. The power law form of Eq. (8.9) enableslong-time behavior to be inferred from short-time tests, as Example 8.1 illustrates.

Equation (8.9) assumes that the critical CTOD for crack extension is rate-independent, which is a reasonable assumption for materials that are well below Tg. Formaterials near Tg, where E is highly sensitive to temperature and rate, the critical CTODoften exhibits a rate dependence [3].

n = 0(Time-Independent

Material)

FIGURE 8.2 Effect of applied Kj on crack velocity for a variety of material responses.

EXAMPLE 8.1

Short-time fracture toughness tests on a polymer specimen indicate a crack velocity

of 10 mm/s at Kjc = 5 MPa v m. If a pipe made from this material contains a flaw such

that Kj = 2.5 MPa v m, estimate the crack velocity, assuming n = 0.08.

Solution: Since the crack velocity is proportional to

MPa V m is given by

-^, the growth rate at 2.5

a = 10 mm/s2,5 MPa \12.5

5 MPa "V m= 0.0017 mm/s - 6.2 mm/hr.

J Controlled FractureSchapery [1,2] has introduced a viscoelastic J integral, Jv, that takes into account

various types of linear and nonlinear viscoelastic behavior. For any material that obeysthe assumed constitutive law, Schapery showed that Jv uniquely defines the crack tipconditions (Section 4.3.2). Thus Jv is a suitable fracture criteria for a wide range of time-dependent materials. Most practical applications of fracture mechanics to polymers,

428 Chapters

however, have considered only the conventional / integral, which does not account fortime-dependent deformation.

Conventional / tests on polymers can provide useful information, but is importantto recognize the limitations of such an approach. One way to assess the significance ofcritical / data for polymers is by evaluating the relationship between / and Jv. Thefollowing exercise considers a constant rate fracture test on a viscoelastic material.

Recall from Chapter 4 that strains and displacements in viscoelastic materials can berelated to pseudo elastic quantities through hereditary integrals. For example, the pseudoelastic displacement, Ae is given by

1 dAAe = El>l\E(t- T}^—di: (8.10)

0 ^

where A is the actual load line displacement and i is an integration variable. Equation(8.10) stems from the correspondence principle, and applies to linear viscoelasticmaterials for which Poisson's ratio is constant. This approach also applies to a widerange of nonlinear viscoelastic material behavior, although E(t) and ER have somewhatdifferent interpretations in the latter case.

For a constant displacement rate fracture test, Eq. (8.10) simplifies to

(8-11)R

a —where A is the displacement rate and E(t) is a time-average modulus, defined by

(8.12)

Figure 8.3 schematically illustrates load-displacement and load-pseudo displacementcurves for constant rate tests on viscoelastic materials. For a linear viscoelastic material(Fig. 8.3(a)), the P-Ae curve is linear, while the P-A curve is nonlinear due to timedependence. Evaluation of pseudo strains and displacements effectively removes the timedependence. When Ae is evaluated for a nonlinear viscoelastic material (Fig. 8.3(b)), thematerial nonlinearity can be de coupled from the time-dependent nonlinearity.

The viscoelastic J integral can be defined from the load-pseudo displacement curve:

Fracture Testing of Nonmetals 429

(a) Linear viscoelastic material.

(b) Nonlinear viscoelastic material.

FIGURE 8.3 Load-displacement and load-pseudo displacement curves for viscoelastic materials.

J PdA*0

(8.13)

where P is the applied load in a specimen of unit thickness. Assume that the P-Ae curveobeys a power law:

= M(Aef (8.14)

430 Chapters

where M and N are time-independent parameters; N is a material property, while Mdepends on both the material and geometry. For a linear viscoelastic material, N = 1, andM is the elastic stiffness. Inserting Eq. (8.14) into Eq. (8.13) leads to

N + l(8.15)

Solving for /v *n terms of physical displacement (Eq. (8.1 1)) gives

_(8.16)

Let us now evaluate / from the same constant rate test:

o(8.17)

The load can be expressed as a function of physical displacement by combining Eqs.(8.11) and (8.14):

(8.18)

Substituting Eq. (8.18) into Eq. (8.17) leads to

(8.19)

since A = At. Therefore,

where

(8.20)

(8.21)


Thus J and Jv are related through a dimensionless function of time in the case of aconstant rate test. For a linear viscoelastic material in plane strain, the relationshipbetween / and Kj is given by

, -/ = —L± - L (g 22)

The conventional / integral uniquely characterizes the crack tip conditions in aviscoelastic material for a given time. A critical J value from a laboratory test istransferable to a structure, provided the failure times in the two configurations are thesame.

A constant rate J test apparently provides a rational measure of fracture toughness inpolymers, but applying such data to structural components may be problematic. Manystructures are statically loaded at either a fixed load or remote displacement. Thus aconstant load creep test or a load relaxation test on a cracked specimen might be moreindicative of structural conditions than a constant displacement rate test. It is unlikelythat the J integral would uniquely characterize viscoelastic crack growth behavior under allloading conditions. For example, in the case of viscous creep in metals, plots of / versusda/dt fail to exhibit a single trend, but C* (which is a special case of /v) correlates crackgrowth data under different loading conditions (see Chapter 4).

Application of fracture mechanics to polymers presents additional problems forwhich both J and Jv may be inadequate. At sufficiently high stresses, polymericmaterials typically experience irreversible deformation, such as yielding, microcracking,and microcrazing. This nonlinear material behavior exhibits a different time dependencethan viscoelastic deformation; computing pseudo strains and displacements may notaccount for rate effects in such cases.

In certain instances, the J integral may be approximately applicable to polymers thatexhibit large scale yielding. Suppose that there exists a quantity Jy that accounts fortime-dependent yielding in polymers. A conventional / test will reflect material fracturebehavior if J and Jy are related through a separable function of time [9]:

Jy=J<})y(t) (8.23)

Section 8.1.5 outlines a procedure for determining Jy experimentally.In metals, the J integral ceases to provide a single parameter description of crack tip

conditions when the yielding is excessive. Critical / values become geometry dependentwhen the single parameter assumption is no longer valid (see Chapter 3). A similarsituation undoubtedly exists in polymers: the single parameter assumption becomesinvalid after sufficient irreversible deformation. Neither / nor Jy will give geometryindependent measures of fracture toughness in such cases. Specimen size requirements fora single parameter description of fracture behavior in polymers have yet to be established,although there has been some research in this area (see Sections 8.1.3 and 8.1.4).

Crack growth presents further complications when the plastic zone is large.Material near the crack tip experiences nonproportional loading and unloading when the

432 Chapters

crack grows, and the J integral is no longer path-independent. The appropriate definitionof J for a growing crack is unclear in metals (Section 3.4.2), and the problem iscomplicated further when the material is rate sensitive. The rate dependence of unloadingin polymers is often different from that of loading.

In summary, the / integral can provide a rational measure of toughness forviscoelastic materials, but the applicability of/data to structural components is suspect.When the specimen experiences significant time-dependent yielding prior to fracture, 7may give a reasonable characterization of fracture initiation from a stationary crack, aslong as the extent of yielding does not invalidate the single-parameter assumption. Crackgrowth in conjunction with time-dependent yielding is a formidable problem that requiresfurther study.

8.1.2 Precracking and Other Practical Matters

As with metals, fracture toughness tests on polymers require that the initial crack besharp. Precracks in plastic specimens can be introduced by a number of methodsincluding fatigue and razor notching.

Fatigue precracking in polymers can be very time consuming. The loadingfrequency must be kept low in order to minimize hysteresis heating, which can introduceresidual stresses at the crack tip.

Because polymers are soft relative to metals, plastic fracture toughness specimenscan be precracked by pressing a razor blade into a machined notch. Razor notching canproduce a sharp crack in a fraction of the time required to grow a fatigue crack, and themeasured toughness is not adversely affected if the notching is done properly [4].

Two types of razor notching are common: razor notch guillotine and razor sawing.In the former case, the razor blade is simply pressed into the material by a compressiveforce, while razor sawing entails a lateral slicing motion in conjunction with thecompressive force. Figures 8.4(a) and 8.4(b) are photographs of fixtures for the razornotch guillotine and razor sawing procedures, respectively.

In order to minimize material damage and residual stresses that result from razornotching, Cayard [4] recommends a three-step procedure: (1) fabrication of a conventionalmachined notch; (2) extension of the notch with a narrow slitting saw; and (3) finalsharpening with a razor blade (by either of the techniques described above). Cayard foundthat such an approach produced very sharp cracks with minimal residual stresses. Thenotch tip radius is typically much smaller than the radius of the razor blade, apparentlybecause a small pop-in propagates ahead of the razor notch.

While the relative softness of plastics aids the precracking process, it can causeproblems during testing. The crack opening force that a clip gage applies to a specimen(Fig. 7.8) is negligible for metal specimens, but this load can be significant in plasticspecimens. The conventional cantilever design may be too stiff for soft plasticspecimens; a ring-shaped clip gage may be more suitable.

One may choose to infer specimen displacement from the crosshead displacement.In such cases it is necessary to correct for extraneous displacements due to indentation ofthe specimen by the test fixture. A displacement calibration can be inferred from a load-displacement curve for an unnotched specimen. If the calibration curve is linear, thecorrection to displacement is relatively simple:


(a) Razor notched guillotine.

(b) Razor sawing.

FIGURE 8.4 Razor notching of polymer specimens. (Photographs provided by M. Cayard.)

434 Chapters

A=Atot-CtP (8.24)

where Atot is the measured displacement and Q is the compliance due to indentation.Since the deformation of the specimen is time-dependent, the crosshead rate in thecalibration experiment should match that in the actual fracture toughness tests.

8.1.3 Kic Testing

The American Society for Testing and Materials (ASTM) has published a number ofstandards for fracture testing of metals, which Chapter 7 describes. Committee D20within ASTM recently developed a standard method for Kjc testing of plastics [10].

The ASTM Kjc standard for plastics is very similar to E 399 [1 1], the ASTM Kjc

standard for metals. Both test methods define an apparent crack initiation load, PQ, by a5% secant construction (Fig. 7.13). This load must be greater than 1.1 times themaximum load in the test for the result to be valid. The provisional fracture toughness,KQ, must meet the following specimen size requirements:

B,a>2.5 —- (8.25a)

0.45< — <0.55 (8.25b)W

where B is the specimen thickness, a is the crack length, and W is the specimen width, asdefined in Fig. 7.1.

The yield strength, 073, is defined in a somewhat different manner for plastics.Figure 8.5 schematically illustrates a typical stress-strain curve for engineering plastics.When a polymer yields, it often experiences strain softening followed by strain hardening.The yield strength is defined at the peak stress prior to strain softening, as Fig. 8.5 illus-trates. Because the flow properties are rate dependent, the ASTM Kjc standard for plasticsrequires that the time to reach crys m a tensile test coincide with the time to failure inthe fracture test to within ± 20%.

The size requirements for metals (Eq. (8.25)) have been incorporated into the ASTMKjc standard for plastics, apparently without assessing the suitability of these criteria forpolymers. Recall from Chapters 2 and 7 the reasons for the Kfc size requirements:

• the plastic zone should be small compared to in-plane dimensions to ensurethe presence of an elastic singularity zone ahead of the crack tip.

• the plastic zone should be small compared to the thickness to ensurepredominantly plane strain conditions.


FIGURE 8.5 Typical stress-strain response ofengineering plastics

TRUE STRAIN

Because the yielding behavior of metals and plastics are different, one should not expectboth materials to exhibit the same size limits for a valid KIC. Even within a givenmaterial system, the sensitivity of toughness to specimen size is influenced by themicromechanism of fracture (see Appendix 5.1).

Cayard [4] has studied the size dependence of fracture toughness for a range ofengineering plastics. The results for two typical materials are described below.

Figure 8.6 shows the effect of specimen size on KQ values for a rigid poiyvinylchloride (PVC) and a polycarbonate (PC). In most cases, the specimens weregeometrically similar, with W=2B and a/W = 0.5. For specimen widths greater that 50mm in the PC, the thickness was fixed at 25 mm, which corresponds to the platethickness. Note that in the small specimens, KQ < Kjc, because PQ was defined from a5% secant; in small specimens, this deviation in linearity depends on flow propertiesrather than fracture properties, as discussed in Section 7.2. The ASTM E 399requirements for in-plane dimensions appear to be adequate for the PVC, but arenonconservative for the PC when the yield strength is defined by the peak stress (Fig.8.5).

The different size dependence for the two polymer systems can be partiallyattributed to strain softening effects. Figure 8.7 shows the stress-strain curves for thesetwo materials. Note that the PC exhibits significant strain softening, while the rigidPVC stress-strain curve is relatively flat after yielding. Significant strain softeningprobably increases the size of the yielded zone. If one defines ays as me lower flowstress plateau, the size requirements are more restrictive for materials that strain soften.Figure 8.6(b) shows the E 399 in-plane requirements corresponding to the lower yieldstrength in the polycarbonate. Although this latter requirement is still nonconservativefor this material, it represents a slight improvement over the approach in the ASTMstandard for plastics.

Cayard [4] also examined the effect of thickness at constant in-plane dimensions.Figures 8.8(a) and 8.8(b) are plots of fracture toughness versus thickness for the PVC andthe PC, respectively. Although all of the experimental data for the PVC are below the re-quired thickness (according to Eq. (8.25)), these data do not exhibit a thickness depen-

436 Chapter 8

dence; Fig. 8.8(a) indicates that the E 399 thickness requirement is too severe for thismaterial. In the case of the PC, most of the data are above the E 399 thicknessrequirement. These data also do not exhibit a thickness dependence, which implies thatthe E 399 requirement is at least adequate for this material. Further testing of thinnersections would be required to determine if the E 399 thickness requirement is overlyconservative for the PC.

a

i t 1 1 1 1

__Kic e __

8 ©^ '

8 |E 399 |

9©© Rigid PVC

QO Temperature = 25 °Ca/W = 0.5

© W = 2B(For W > 50 mm, B = 25 mm)

A — . . —©

1 | 1 I 1 1 I

0 -

80

(a) Rigid PVC.

3.5

3.0

a2.5

2.0

KlC0

ft— e_

PolycarbonateTemperature = 25 °C

a/ W = 0.5

1

0 10 20 30 40

SPECIMEN WIDTH, mm(b) Polycarbonate.

FIGURE 8.6 Effect of specimen size on KQ in two engineering plastics [4].

50

Fracture Testing ofNonmetals

140

437

Temperature 25 °CCrosshead Rate:

0.1 mm/ min

0.2 0.3 0.4

TRUE STRAIN

FIGURE 8.7 Stress-strain curves for the rigid PVC and polycarbonate [4].

The PVC material fails by crazing. Recall from Chapter 6 that a craze zone ahead ofa crack dp contains a high concentration of voids. The material inside of the craze zone issubject essentially to plane stress loading, regardless of the specimen thickness.Consequently, the fracture toughness of materials that craze may be relatively insensitiveto specimen thickness. Thickness effects are not necessarily absent from materials thatcraze, however. While the material in the craze zone is subject to plane stress, thesurrounding material may be experience plane strain or mixed conditions; the stress statein the surrounding material could influence the toughness by dictating the size and shapeof the craze zone.

Not all polymeric materials fracture by a crazing mechanism. For example, Cayardfound no evidence of cavitation (void formation) on the fracture surfaces of the PC. Thefracture toughness of this material may be more sensitive to specimen thickness thanmaterials that craze, such as the PVC. The development of rational thickness re-quirements for polymers requires further study.

One final observation regarding the ASTM Kjc standard for plastics is that theprocedure for estimating PQ ignores time effects. Recall from Section 7.2 thatnonlinearity in the load displacement curve from Kfc tests on metals can come from twosources: yielding and crack growth. In the case of polymers, viscoelasticity can alsocontribute to nonlinearity in the load-displacement curve. Consequently, at least aportion of the 5% deviation from linearity at PQ could result from a decrease in themodulus during the test. Linear elastic fracture mechanics (LEFM) is valid for linearviscoelastic deformation, even when the load-displacement curve is nonlinear. TheASTM standard could be unduly restrictive in defining the critical load by a 5% secant,irrespective of the source of the nonlinearity.

438 Chapter 8

FR

AC

TU

RE

TO

UG

HN

ES

S, M

Pa

m172

UJ

*.

Ul

CT\

vj

i i 1 1

ASTM E 399 ThicknessRequirement = 32 mm_

Rigid PVCTemperature = 25 °C

a/W = 0.5W= 50.8 mm

1 1 I 1) 5 10 15 20 25

SPECIMEN THICKNESS, mm

(a) Rigid PVC.

RA

CT

UR

E T

OU

GH

NE

SS

, MP

a m

17

2

K3

ls>

W

Wo

tn

O

in 1 1 1 1

0 9 8 ° o_Q n_— Q ^^ ^^ ^^ ^^ g ^^ -~~u "~~ o —

" ° ° o g 8 8 ° ° ~|E 399 |

Polycarbonate— Temperature = 25 °C —

a/ W = 0.5W = 50.8 mm

1 1 1 15 10 15 20

SPECIMEN THICKNESS, mm

25

(b) Polycarbonate.

FIGURE 8.8 Effect of specimen thickness on fracture toughness of plastics [4].

For most practical situations, however, viscoelastic effects are probably negligibleduring Kjc tests. In order to obtain a valid KIC result in most polymers, the testtemperature must be well below Tg, where rate effects are minimal at short times. The


duration of a typical KIC test is on the order of several minutes, and the elastic propertiesprobably will not change significantly prior to fracture. The rate sensitivity should bequantified, however, to evaluate the assumption that E does not change during the test.

8.1.4 J Testing

A number of researchers have applied J integral test methods to polymers [3-6], but astandard for measuring 7/c and J-R curves in plastics does not exist, as of this writing. Acommittee within ASTM is currently drafting a standard test method for J[c measurementsin plastics. The methodology that is being applied is very similar to that in E 813-87[11], the ASTM standard for Jjc testing of metals.

The current formulas for estimating J from plastic test specimens are identical tothose in E 813. For common test specimens, such as the compact and SENB geometries,/ is related to the area under the load versus load line displacement curve:

(8.26)

where 7] is a dimensionless parameter that depends on geometry.The most common approach for inferring the J-R curve for a polymer is the

multiple specimen method. A set of nominally identical specimens are loaded to variousdisplacements, unloaded, cooled to a low temperature, and then fractured. The initial cracklength and stable crack growth are measured optically from each specimen, resulting in aseries of data points on a J-Aa plot. The Jfc can then be inferred by fitting an equation,such as a power law or straight line, to the data. This latter exercise is described inSection 7.4.1 for Jjc testing of metals.

Single specimen techniques, such as unloading compliance, may also be applied tothe measurement of Jjc and the J-R curve [6]. Time-dependent material behavior cancomplicate unloading compliance measurements, however. Figure 8.9 schematicallyillustrates the unload-reload behavior of a viscoelastic material. If rate effects are signifi-cant during the time frame of the unload-reload, the resulting curve can exhibit ahysteresis effect. One possible approach to account for viscoelasticity in such cases is torelate instantaneous crack length to pseudo elastic displacements (see Section 8.1.5 ).

Critical / values for polymers exhibit less size dependence than KQ values. Figure8.10 compares KQ values for the polycarbonate with Kjc values, which were obtained byconverting critical / values at fracture to an equivalent critical K through the followingrelationship:

(8.27)

For sufficiently large specimens, where the global behavior is predominantly elastic, Kjc

= KQ = Kjc. Note that the Kjc values are independent of specimen size over the range ofavailable data.

440 Chapters

LOAD

£rt

fks

FIGURE 8.9 Schematic unloading behavior in apolymer. Hysteresis in the unload-reload curvecomplicates unloading compliance measurements.

DISPLACEMENT

3.5

3.0

2.5

2.0

° PolycarbonateTemperature = 25 °C

a/ W = 0.5W = 2Bo KQ« Kjc

10 20 30

SPECIMEN WIDTH, mm

40 50

FIGURE 8.10 Size dependence of KQ and ./-based fracture toughness for PC [4].

Crack growth resistance curves can be highly rate dependent. Figure 8.11 shows J-R curves for a polyethylene pipe material that was tested at three crosshead rates [3].Increasing the crosshead rate from 0.254 mm/min to 1.27 mm/min (0.01 and 0.05in/min, respectively) results in nearly a three-fold increase in Jjc in this case.

8.1.5 Experimental Estimates of Time-Dependent Fracture Parameters

While Jjc values may be indicative of a polymer's relative toughness, the existence of aunique correlation between / and crack growth rate is unlikely. Parameters such as Jv

may be more suitable for some viscoelastic materials. For polymers that experience largescale yielding, neither J nor Jv may characterize crack growth.

Fracture Testing ofNonmetals

12

441

0

0 0.200.04 0.08 0.12 0.16

CRACK EXTENSION, cm

FIGURE 8.11 Crack growth resistance curves for polyethylene pipe at three crosshead rates [3].

This section outlines a few suggestions for inferring crack tip parameters that takeinto account the time-dependent deformation of engineering plastics. Since most of theseapproaches have yet to be validated experimentally, much of what follows contains anelement of conjecture. These proposed methods, however, are certainly no worse thanconventional J integral approaches, and may be considerably better for many engineeringplastics.

The viscoelastic J integral, Jv, can be inferred by converting physical displacementsto pseudo displacements. For a constant rate test, Eq. (8.11) gives the relationshipbetween A and Ae. The viscoelastic J integral is given by Eq. (8.13); Jv can also beevaluated directly from the area under the P-Ae curve:

lv=±\ PdAe

b 0(8.28)

for a specimen with unit thickness. If the load-pseudo displacement is a power-law (Eq.(8.18)),Eq. (8.28) becomes

T]M(Aef+l

(8.29)

Comparing Eqs. (8.29) and (8.15) leads to

442 Chapters

(830)

Since M does not depend on time, the dimensionless 77 factor is the same for both J andJv.

Computing pseudo elastic displacements might also remove hysteresis effects inunloading compliance tests. If the unload-reload behavior is linear viscoelastic, the P-Ae

unloading curves would be linear, and crack length could be correlated to the pseudoelastic compliance.

Determining pseudo displacements from Eq. (8.11) or the more general expression(Eq. (8.10)) requires a knowledge of Eft). A separate experiment to infer E(t) would notbe particularly difficult, but such data would not be relevant if the material experiencedlarge scale yielding in a fracture test. An alternative approach to inferring crack tip pa-rameters that takes time effects into account is outlined below.

Schapery [9] has suggested evaluating a /-like parameter from isochronous (fixedtime) load-displacement curves. Consider a series of fracture tests that are performed overa range of crosshead rates (Fig. 8.12(a)). If one selects a fixed time and determines thevarious combinations of load and displacement that correspond to this time, the resultinglocus of points forms an isochronous load-displacement curve (Fig. 8.12(b)). Since theviscoelastic and yield properties are time-dependent, the isochronous curve represents theload displacement behavior for fixed material properties, as if time stood still while thetest was performed. A fixed-time / integral can be defined as follows:

(8.31)

inconstant

Suppose that the displacements at a given load are related by a separable function of time,such that it is possible to relate all displacements (at that particular load) to a referencedisplacement:

AR = Ay(t) (8.32)

The isochronous load-displacement curves could then be collapsed onto a single trend bymultiplying each curve by y(t), as Fig. 8.12(c) illustrates. It would also be possible todefine a reference J:

(8-33)

Note the similarity between Eqs. (8.23) and (8.33).The viscoelastic / is a special case of J^. For a constant rate test, comparing Eqs.

(8. 11) and (8.32) gives


A

(a)

A

(b)

AK = y(t) AFIGURE 8.12 Proposed method for removingtime dependence from load-displacement curves.First, a set of tests are performed over a range ofdisplacement rates (a). Next, isochronous loaddisplacement curves are inferred (b). Finally, thedisplacement axis of each curve is multiplied by afunction ~}(t), resulting in a single curve (c).

AR

(c)

Jv - Jf (8.34)

Thus for a linear viscoelastic material in plane strain,

E(t)(8.35)

Isochronous load-displacement curves would be linear for a linear viscoelastic material,since the modulus is constant at a fixed time.

444 Chapter 8

The parameter J^ is more general than Jv; the former may account for timedependence in cases where extensive yielding occurs in the specimen. The reference Jshould characterize crack initiation and growth in materials where Eq. (8.33) removes timedependence of displacement. Figure 8.13 schematically illustrates the postulated rela-

e

tionship between Jt, JR., and crack velocity. The Jf-a curves should be parallel on a log-

log plot, while a J^-a plot should yield a unique curve. Even if it is not possible to

produce a single fi-a curve for a material, the Jt parameter should still characterizefracture at a fixed time.

Although jR may characterize fracture initiation and the early stages of crack growthin a material that exhibits significant time-dependent yielding, this parameter wouldprobably not be capable of characterizing extensive crack growth, since unloading andnonproportional loading occur near the growing crack tip. (See Section 8.1.1 above.)

Log a Log a

LogJR

FIGURE 8.13 Postulated crack growth behavior in terms of Jt and J*.

8.1.6 Qualitative Fracture Tests on Plastics

The ASTM standard D 256-88 [13] describes impact testing of notched polymerspecimens. This test method is currently the most common technique for characterizingthe toughness of engineering plastics. The D 256 standard covers both Charpy and Izodtests (Fig. 7.36), but the plastics industry utilizes the Izod specimen in the vast majorityof cases.

The procedure for impact testing of plastics is very similar to the metals approach,which is outlined in ASTM E 23 [14] (see Section 7.9). A pendulum strikes a notchedspecimen, and the energy required to fracture the specimen is inferred from the initial andfinal heights of the pendulum (Fig. 7.37). In the case of the Izod test, the specimen is asimple cantilever beam that is restrained at one end and struck by the pendulum at theother end. One difference between the metals and plastics test methods is that theabsorbed energy is normalized by the net ligament area in plastics tests, while tests


according to ASTM E 23 report only the total energy. The normalized fracture energy inplastics is known as the impact strength.

The impact test for plastics is pervasive throughout the plastics industry because itis a simple and inexpensive measurement. Its most common application is as a materialscreening criterion. The value of impact strength measurements is questionable, however.

One problem with this test method is that the specimens contain blunt notches.Figure 8.14 [15] shows Izod impact strength values for several polymers as a function ofnotch radius. As one might expect, the fracture energy decreases as the notch becomessharper. The slope of the lines in Fig. 8.14 is a measure of the notch sensitivity of thematerial. Some materials are highly notch sensitive, while others are relativelyinsensitive to the radius of the notch. Note that the relative ordering of the materials'impact strengths in Fig. 8.14 changes with notch acuity. Thus a fracture energy for aparticular notch radius may not be an appropriate criterion for ranking material toughness.Moreover, the notch strength is often not a reliable indicator of how the material willbehave when it contains a sharp crack.

Since Izod and Charpy tests are performed under impact loading, the resultingfracture energy values are governed by the short-time material response. Many polymerstructures, however, are loaded quasistatically and must be resistant to slow, stable crackgrowth. The ability of a material to resist crack growth at long times is not necessarilyrelated to the fracture energy of a blunt-notched specimen in impact loading.

The British Standards Institution (BSI) specification for unplasticized polyvinylchloride (PVC-U) pipe, BS 3505:1986 [16], contains a procedure for fracture toughnesstesting. Although the toughness test in BS 3506 is primarily a qualitative screeningcriterion, it is much more relevant to structural performance than the Izod impact test.

Appendices C and D of BS 3506 outline a procedure for inferring toughness ofPVC-U pipe after exposure to an aggressive environment. A C-shaped section is removedfrom the pipe of interest and is submerged in dichloromethane liquid. After 15 min ofexposure, the specimen is removed from the liquid and the surface is inspected forbleaching or whitening. A sharp notch is placed on the inner surface of the specimenwhich is then dead-loaded for 15 min or until cracking or total fracture is observed.Figure 8.15 is a schematic drawing of the testing apparatus. The loading is such that thenotch region is subject to a bending moment. If the specimen cracks or fails completelyduring the test, the fracture toughness of the material can be computed from applied loadand notch depth by means of standard Kj formulae. If no cracking is observed during the15 min test, the toughness can be quantified by testing additional specimens at higherloads. The BS 3506 standard includes a semiempirical size correction for small pipes andhigh toughness materials that do not behave in an elastic manner.

8.2 INTERLAMINAR TOUGHNESS OF COMPOSITES

Chapter 6 outlined some of the difficulties in applying fracture mechanics to fiber-reinforced composites. The continuum assumption is often inappropriate, and cracks maynot grow in a self-similar manner. The lack of a rigorous framework to describe fracturein composites has led to a number of qualitative approaches to characterize toughness.

446 Chapter 8

FIGURE 8.14 Effect of notch radius onthe Izod impact strength of severalengineering plastics [15].

0.5 1 2 4 8 16 32NOTCH RADIUS, mm

= WL

FIGURE 8.15 Loading apparatus for evaluating the toughness of PVC-U pipe according to BS 3506 [16].


Interlaminar fracture is one of the few instances where fracture mechanics formalismis applicable to fiber-reinforced composites on a global scale. A zone of delamination canbe treated as a crack; the resistance of the material to the propagation of this crack is thefracture toughness. Since the crack typically is confined to the matrix material betweenplies, continuum theory is applicable, and the crack growth is self similar.

A standard for interlaminar fracture toughness does not exist as of this writing, butASTM, the European Group on Fracture (EOF) and Japanese Industrial Standards (JIS) arecurrently developing standardized test procedures for carbon/Epoxy and carbon/PEEKcomposites. The published literature contains a large amount of gfc and §i\c data forcomposites, but test methods differ widely between laboratories.

Figure 8.16 illustrates three common specimen configurations for interlaminarfracture toughness measurements. The double cantilever beam (DCB) specimen isprobably the most common configuration for this type of test. One advantage of thisspecimen geometry is that it permits measurements of Mode I, Mode II or mixed modefracture toughness. The end notched flexure (ENF) specimen has essentially the samegeometry as the DCB specimen, but the latter is loaded in three-point bending, whichimposes Mode II displacements of the crack faces. The edge delamination specimensimulates the conditions in an actual structure. Recall from Chapter 6 that tensilestresses normal to the ply are highest at the free edge (Fig. 6.16); thus delamination zonesoften initiate at the edges of a panel.

Procedures for measuring interlaminar toughness with DCB specimens are outlinedbelow; analogous methods can be applied to other specimen configurations. Theapproaches that follow are not definitive test methods, but are representative of currentpractice [17-20].

The initial flaw in a DCB specimen is normally introduced by placing a thin film(e.g. aluminum foil) between plies prior to molding. The film should be coated with arelease agent so that it can be removed prior to testing.

Figure 8.17 illustrates two common fixtures that facilitate loading the DCBspecimen. The blocks or hinges are normally adhesively bonded to the specimen. Thesefixtures must allow free rotation of the specimen ends with a minimum of stiffening.

The DCB specimen can be tested in Mode I, Mode II, or mixed-mode conditions, asFig. 8.18 illustrates. Recall from Chapter 2 that the energy release rate of this specimenconfiguration can be inferred from beam theory.

For pure Mode I loading (Fig. 8.18(a)), elastic beam theory leads to the followingexpression for energy release rate (see Example 2.2):

P}a2

Gr = -1 - (8.36)*! BEI

where

(8.37)

The corresponding relationship for Mode II (Fig 8.18(b)) is given by

448 Chapter 8

(a) Double cantilever beam specimen.

(b) End notched flexure specimen.

(c) Edge delamination specimen.

FIGURE 8.16 Common configurations for evaluating interlaminar fracture toughness.

(a) End blocks

(b) Piano hinges

FIGURE 8.17 Loading fixtures for DCB specimens.


Pi

(a) Mode I.

(b) Mode IIPu

(c) Mixed mode.

FIGURE 8.18 Mode I, II and mixed mode loading of DCB specimens.

4BEI(8.38)

assuming linear beam theory. Mixed loading conditions can be achieved by unequaltensile loading of the upper and lower portions of the specimens, as Fig. 8.18(c)illustrates. The applied loads can be resolved into Mode I and Mode II components asfollows:

450 Chapter 8

PI =\PL\

\PU\-\PL\2

(8.39a)

(8.39b)

where P\j and PL are the upper and lower loads, respectively. The components of ^"canbe computed by inserting P/ and P// into Eqs. (8.36) and (8.38). Recall from Chapter 2that Mode I and Mode II components of energy release rate are additive.

Linear beam theory may result in erroneous estimates of energy release rate,particularly when the specimen displacements are large. The area method [19-20] providesan alternative measure of energy release rate. Figure 8.19 schematically illustrates atypical load-displacement curve, where the specimen is periodically unloaded. The loadingportion of the curve is typically nonlinear, but the unloading curve is usually linear andpasses through the origin. The energy release rate can be estimated from the incrementalarea inside the load displacement curve, divided by the change in crack area:

AU_

BAa(8.40)

The Mode I and Mode II components of £ can be inferred from the Pj-A] and P//-.A//curves, respectively. The corresponding loads and displacements for Modes I and II aredefined in Fig. 8.18 and Eq. (8.39).

Figure 8.20 illustrates a typical delamination resistance curve for Mode I. Afterinitiation and a small amount of growth, delamination occurs at a steady-state Gjc value,provided the global behavior of the specimen is elastic.

FIGURE 8.19 Schematic load-displacementcurve for a delamination toughnessmeasurement.


CRACK EXTENSION

FIGURE 8.20 Schematic R curve inferred froma delamination experiment.

8.3 CERAMICS

Fracture toughness is usually the limiting property in ceramic materials. Ceramics tendto have excellent creep properties and wear resistance, but are excluded from many load-bearing applications because they are relatively brittle. The latest generation of ceramics(see Section 6.2) have enhanced toughness, but brittle fracture is still a primary area ofconcern in these materials.

Because toughness is a crucial property for ceramic materials, rational fracturetoughness measurements are absolutely essential. Unfortunately, fracture toughness testson ceramics can be very difficult and expensive. Specimen fabrication, for example,requires special grinding tools, since ordinary machining tools are inadequate. Precrackingby fatigue is extremely time-consuming; some investigators have reported precrackingtimes in excess of one week per specimen [21]. During testing, it is difficult to achievestable crack growth with most specimen configurations and testing machines.

Several test methods have been developed to overcome some of the difficultiesassociated with fracture toughness measurements in ceramics. The chevron-notchedspecimen [22-24] eliminates the need for precracking, while the bridge indentationapproach [21,25-29] is a novel method for introducing a crack without resorting to alengthy fatigue precracking process.

8.3.1 Chevron-Notched Specimens

A chevron notch has a V-shaped ligament, such that the notch depth varies through thethickness, with the minimum notch depth at the center. Figure 8.21 shows two commonconfigurations of chevron-notched specimens: the short bar and the short rod. Inaddition, single edge notched bend (SENB) and compact specimens (Fig. 7.1) are some-times fabricated with chevron notches. The chevron notch is often utilized inconventional fracture toughness tests on metals because this shape facilitates initiation ofthe fatigue precrack. For fracture toughness tests on brittle materials, the uniqueproperties of the chevron notch can eliminate the need for precracking altogether, asdiscussed below.

452 Chapters

(a) Short bar. (b) Short rod.

FIGURE 8.21 Two common designs of chevron notched specimens [22].

Figure 8.22 schematically compares the stress intensity factor versus crack lengthfor chevron and straight notch configurations. When the crack length = a0, the stressintensity factor in the chevron-notched specimen is very high, because a finite load isapplied over a very small net thickness. When a>aj, the Kj values for the two notchconfigurations are identical, since the chevron notch no longer has an effect. The Kj forthe chevron-notched specimen exhibits a minimum at a particular crack length, am, whichis between ao and aj.

The Kj v. crack length behavior of the chevron-notched specimen makes thisspecimen particularly suitable for measuring the toughness in brittle materials. Considera material in which the R curve reaches a steady-state plateau soon after the crack initiates(Fig. 8.23). The crack should initiate at the tip of the chevron upon application of asmall load, since the local Kj is high. The crack is stable at this point, because thedriving force decreases rapidly with crack advance; thus additional load is required to growthe crack further. The maximum load in the test, Pfrf, is achieved when the crack growsto am, the crack length corresponding to the minimum in the Kj-a curve. At this point,the specimen will be unstable if the test is conducted in load control, but stable crackgrowth may be possible beyond am if the specimen is subject to crosshead control. Thepoint of instability in the latter case depends on the compliance of the testing machine, asdiscussed in Section 2.5.

Since the maximum load occurs at am, and am is known a priori (from the Kj v.crack length relationship), it is necessary only to measure the maximum load in this test.The fracture toughness is given by

KIvM~B-JW

f(<*m/W) (8.41)


Ki

ao am ai

CRACK LENGTH

FIGURE 8.22 Comparison of stress intensity factors in specimens with chevron and straight notches.Note that Kj exhibits a minimum in the chevron notched specimens.

Ki

KlvM -

CRACK LENGTHDISPLACEMENT

FIGURE 8.23 Fracture toughness testing of a material with a flat R curve. The maximum load in thetest occurs when a = am.

where KJVM is the chevron-notched toughness defined at maximum load, and/fo/W) is thegeometry correction factor. Early researchers developed simple models to estimate/fo/W)for chevron-notched specimens, but more recent (and more accurate) estimates are based onthree-dimensional finite element and boundary element analysis of this configuration [23].

The maximum load technique for inferring toughness does not work as well whenthe material exhibits a rising R curve, as Fig. 8.24 schematically illustrates. The pointof tangency between the driving force and R curve may not occur at am in this case,resulting in an error in the stress intensity calculation. Moreover, the value of KR at thepoint of tangency is geometry dependent when the R curve is rising.

454 Chapters

FIGURE 8.24 Application of the chevron-notchedspecimen to a material with a rising R curve,

am

CRACK LENGTH

If both load and crack length are measured throughout the test, it is possible toconstruct the R curve for the material under consideration. Optical observation of thegrowing crack is not usually feasible for a chevron-notched specimen, but the crack lengthcan be inferred through an unloading compliance technique [22], in which the specimen isperiodically unloaded and the crack length is computed from the elastic compliance.

An ASTM standard for chevron-notched specimens, E 1304-90 [22], has recentlybeen published. This standard actually applies to brittle metals, such as high strengthaluminum alloys, but a corresponding standard for ceramics is currently underconsideration. The E 1304 standard includes both the maximum load and compliancemeasures of fracture toughness, which are designated KIVM and Kjv, respectively. Anumber of researchers have measured fracture toughness of chevron-notched ceramicspecimens with test techniques that are virtually identical to the provisions in ASTM E1304.

The chevron-notched specimen has proved to be very useful in characterizing thetoughness of brittle materials. The advantages of this test specimen include its compactgeometry, the simple instrumentation requirements (in the case of the K]VMmeasurement), and the fact that no precracking is required. One of the disadvantages ofthis specimen is its complicated design, which leads to higher machining costs. Also,this specimen is poorly suited to high temperature testing, and the KJVM measurementis inappropriate for material with rising R curves.

8.3.2 Bend Specimens Precracked by Bridge Indentation

A novel technique for precracking ceramic SENB specimens has recently been developedin Japan [25]. A number of researchers [21,25-29] have adopted this method, which hasbeen incorporated into an upcoming Japanese standard for fracture toughness testing ofceramics. Warren, et al. [26], who were among the first to apply this precrackingtechnique have termed it the "bridge indentation" method.


Figure 8.25 is a schematic drawing of the loading fixtures for the bridge indentationmethod of precracking. A starter notch is introduced into an SENB specimen by means ofa Vickers hardness indentation. The specimen is compressed between two anvils, as Fig.8.25 illustrates. The top anvil is flat, while the bottom anvil has a gap in the center.This arrangement induces a local tensile stress in the specimen, which leads to a pop-infracture. The fracture arrests because the propagating crack experiences a falling K field.

The bridge indentation technique is capable of producing highly uniform crack frontsin SENB specimens. After precracking, these specimens can be tested in three- or four-point bending with conventional fixtures. Nose and Fujii [21] showed that fracturetoughness values obtained from bridge precracked specimens compared favorably with datafrom conventional fatigue precracked specimens.

Bar-On, et al. [27] investigated the effect of precracking variables on the size of thecrack that is produced by this technique. Figure 8.26 shows that the length of the pop-inin alumina decreases with increasing Vickers indentation load. Also note that the pop-inload decreases with increasing indentation load. Large Vickers indentation loads producesignificant initial flaws and tensile residual stresses, which enable the pop-in to initiate ata lower load; the crack arrests sooner at lower loads because there is less elastic energyavailable for crack propagation. Thus it is possible to control the length of the precrackthough the Vickers indentation load.

The bridge indentation technique is obviously much more economical than fatigueprecracking of ceramic specimens. The SENB configuration is simple, and therefore lessexpensive to fabricate. Also, three- and four-point bend fixtures are suitable for hightemperature testing. One problem with the SENB specimen is that it consumes morematerial than the chevron notched specimens illustrated in Fig. 8.21; this is a majorshortcoming when evaluating new materials, where only small samples are available.Another disadvantage of the beam configuration is that it tends to be unstable; most testmachines are too compliant to achieve stable crack growth in brittle SENB specimens[28,29].

FIGURE 8.25 The bridge indentation method for precracking [21].

456 Chapter 8

0 100

14

600200 300 400

INDENT LOAD, N

FIGURE 8.26 Effect of bridge indentation load on the crack length after pop-in [27].

REFERENCES

1. Schapery, R.A., "Correspondence Principles and a Generalized J Integral for LargeDeformation and Fracture Analysis of Viscoelastic Media." International Journal ofFracture, Vol. 25, 1984, pp. 195-223.

2. Schapery, R.A., "Time-Dependent Fracture: Continuum Aspects of Crack Growth."Encyclopedia of Materials Science and Engineering, Pergamon Press, Oxford, 1986, pp.5043-5054.

3. Jones, R.E. and Bradley, W.L., "Fracture Toughness Testing of Polyethylene PipeMaterials." ASTM STP 995, Vol. 1, 1989, American Society for Testing and Materials,Philadelphia, PA, pp. 447-456.

4. Cayard, M. "Fracture Toughness Testing of Polymeric Materials." Ph.D. Dissertation,Texas A&M University, College Station, TX, September, 1990.

5. Williams, J.G. Fracture Mechanics of Polymers, , Halsted Press, John Wiley & Sons,New York, 1984.

6. Letton, A., "The Use of Specimen Compliance in Predicting Crack Growth." Submittedto Polymer Science and Engineering, 1991.

7. Marshall, G.P., Coutts, L.H., and Williams, J.G., "Temperature Effects in the Fracture ofPMMA." Journal of Materials Science, Vol. 13, 1974, pp. 1409-


8. Schapery, R.A. "A Theory of Crack Initiation and Growth in Viscoelastic Media--I.Theoretical Development." International Journal of Fracture, Vol 11, 1975, pp. 141-159.

9. Schapery, R.A., Private communication, 1990.

10. D 5045-91a "Standard Test Methods for Plane Strain Fracture Toughness and StrainEnergy Release Rate of Plastic Materials." American Society for Testing and Materials,Philadelphia, PA, 1991.

11. E 399-90, "Standard Test Method for Fracture Toughness of Metallic Materials."American Society for Testing and Materials, Philadelphia, PA, 1983.

12. E 813-89, "Standard Test Method for Jjc, a Measure of Fracture Toughness." AmericanSociety for Testing and Materials, Philadelphia, PA, 1987.

13. D 256-88, "Impact Resistance of Plastics and Electrical Insulating Materials."American Society for Testing and Materials, Philadelphia, PA, 1988.

14. E 23-88, "Standard Test Methods for Notched Bar Impact Testing of Metallic Materials."American Society of Testing and Materials, Philadelphia, PA, 1988.

15. Engineered Materials Handbook, Volume 2: Engineering Plastics. ASM International,Metals Park, OH, 1988.

16. BS 3505:1986, "British Standard Specification for Unplasticized Poly vinyl Chloride(PVC-U) Pressure Pipes for Cold Potable Water." British Standards Institution, London,1986.

17. Whitney, J.M., Browning, C.E., and Hoogsteden, W., "A Double Cantilever Beam Testfor Characterizing Mode I Delamination of Composite Materials." Journal ofReinforced Plastics and Composites, Vol. 1, 1982, pp. 297-313.

18. Prel, Y.J., Davies, P., Benzeggah, M.L., and de Charentenay, F.-X., "Mode I and ModeII Delamination of Thermosetting and Thermoplastic Composites." ASTM STP 1012,American Society for Testing and Materials, Philadelphia, PA, 1989, pp. 251-269.

19. Corleto, C.R. and Bradley, W.L., "Mode II Delamination Fracture Toughness ofUnidirectional Graphite/Epoxy Composites." ASTM STP 1012, American Society forTesting and Materials, Philadelphia, PA, 1989, pp. 201-221.

20. Hibbs, M.F., Tse, M.K., and Bradley, W.L., "Interlaminar Fracture Toughness and Real-Time Fracture Mechanism of Some Toughened Graphite/Epoxy Composites." ASTMSTP 937, American Society for Testing and Materials, Philadelphia, PA, 1987, pp. 115-130.

21. Nose, T. and Fujii, T., "Evaluation of Fracture Toughness for Ceramic Materials by aSingle-Edge-Precracked-Beam Method." Journal of the American Ceramic Society, Vol.71, 1988, pp. 328-333.

22. E 1304-89, "Standard Test Method for Plane-Strain (Chevron Notch) Fracture Toughnessof Metallic Materials." American Society for Testing and Materials, Philadelphia, PA,1989.

458 Chapters

23. Newman, J.C., "A Review of Chevron-Notched Fracture Specimens." ASTM STP 855,American Society for Testing and Materials, Philadelphia, PA, 1984, pp. 5-31.

24. Shannon, J.L., Jr. and Munz, D.G., "Specimen Size Effects on Fracture Toughness ofAluminum Oxide Measured with Short-Rod and Short Bar Chevron-Notched Specimens."ASTM STP 855, American Society for Testing and Materials, Philadelphia, PA, 1984,pp. 270-280.

25. Nunomura, S. and Jitsukawa, S. "Fracture Toughness for Bearing Steels by IndentationCracking under Multiaxial Stress." (In Japanese) Tetsu to Hagane, Vol. 64, 1978.

26. Warren, R. and Johannsen, B. "Creation of Stable Cracks in Hard Metals Using 'Bridge'Indentation." Powder Metallurgy, Vol. 27, 1984, pp. 25-29.

27. Bar-On, I., Beals, J.T., Leatherman, G.L., and Murray, C.M., "Fracture Toughness ofCeramic Precracked Bend Bars." Journal of the American Ceramic Society, Vol. 73,1990, pp. 2519-2522.

28. Barratta, F.I. and Dunlay, W.A., "Crack Stability in Simply Supported Four-Point andThree-Point Loaded Beams of Brittle Materials." Proceedings of the Army Symposiumon Solid Mechanics, 1989 - Mechanics of Engineered Materials and Applications, U.S.Army Materials Technology Laboratory, Watertown, MA, 1989, pp. 1-11.

29. Underwood, J.H., Barratta, F.I., and Zalinka, J.J., "Fracture Toughness Tests andDisplacement and Crack Stability Analyses of Round Bar Bend Specimens of Liquid-Phase Sintered Tungsten." Proceedings of the 1990 SEM Spring Conference onExperimental Mechanics, Albuquerque, NM, 1990, pp. 535-542.

9. APPLICATION TO STRUCTURES

Figure 9.1 illustrates the so-called fracture mechanics triangle. When designing a structureagainst fracture, there are three critical variables that must be considered: stress, flaw size,and toughness. Fracture mechanics provides a mathematical relationship between thesequantities. In most cases there are two degrees of freedom (i.e., one equation and threeunknowns); a knowledge of two quantities is required to compute the third. For example,if the stress is specified by the design and the material toughness is known, fracture me-chanics relationships can predict the critical flaw size in the structure.

A number of relationships are available that attempt to quantify the critical relation-ship between stress, flaw size, and toughness, but each of these approaches is only suit-able in limited situations. Linear elastic fracture models, for example, should not be ap-plied to structures that exhibit significant plastic flow.

APPLIEDSTRESS

FRACTUREMECHANICS

FLAWSIZE

FRACTURETOUGHNESS

FIGURE 9.1 The fracture mechanics triangle, which identifies the three critical variables in fracturedesign.

The fracture design methodology should be selected based on the available data, ma-terial properties, environment, and the loading on the structure. If KIC data are availableand the design stress is low, linear elastic fracture mechanics (LEFM) may be appropriate.If the behavior of the structure is linear elastic but the laboratory fracture toughness testsbehave in an elastic-plastic manner, it may be possible to convert a critical / value to theequivalent Kfc and analyze the structure with linear elastic relationships (Section 9.1.4).If the structure and test specimen both behave in an elastic-plastic fashion, /- and CTOD-based analyses are available (Section 9.3 to 9.7). Rapid loading may require special con-sideration, as will design for crack arrest. Time-dependent crack growth, such as fatigue,environmental assisted cracking, and creep crack growth may complicate the analysis fur-ther.

Most fracture analyses are deterministic. That is, the stress, flaw size, and tough-ness are assumed to be single-valued quantities. In practical situations, however, there is

459

460 Chapter 9

usually some degree of uncertainty associated with each of these variables. Consequently,it is usually not possible to predict the precise moment of failure. Probabilistic analyses(Section 9.9) can quantify the risk of failure, however.

This chapter focuses on fracture initiation and instability in structures made fromlinear elastic and elastic-plastic materials. A number of engineering approaches are dis-cussed; the basis of these approaches and their limitations are explored. This chapter onlycovers quasistatic methodologies, but such approaches can be applied to rapid loading andcrack arrest in certain circumstances (see Chapter 4). The analyses presented in this chap-ter do not address time-dependent crack growth. Chapter 10 considers fatigue crack growthin detail, and describes life predictions for all types of time-dependent crack growth.

9.1 LINEAR ELASTIC FRACTURE MECHANICS

Analyses based on LEFM apply to structures where crack tip plasticity is small. Chapter2 introduced many of the fundamental concepts of LEFM. The fracture behavior of a lin-ear elastic structure can be inferred by comparing the applied K (the driving force) to acritical K or a K-R curve (the fracture toughness). The elastic energy release rate, £, is analternative measure of driving force, and a critical value of Q quantifies the materialtoughness.

For Mode I loading (Fig. 2.14), the stress intensity factor can be expressed in thefollowing form:

Kj = Ya^/na (9.1)

where Fis a dimensionless geometry correction factor, <ris a characteristic stress, and a isa characteristic crack dimension. If the geometry factor is known, the applied Kj can becomputed for any combination of <J and a. The applied stress intensity can then be com-pared to the appropriate material property, which may be a Kjc value, a K-R curve, envi-ronmental assisted cracking data, or, in the case of cyclic loading, fatigue crack growthdata (see Chapter 10).

Fracture analysis of a linear elastic structure becomes relatively straightforward, oncea K solution is obtained for the geometry of interest. Stress intensity solutions can comefrom a number of sources, including handbooks, the published Jiterature, experiments, andnumerical analysis.

A large number of stress intensity solutions have been published over the past 35years. Several handbooks [1-3] contain compilations of solutions for a wide variety ofconfigurations. The published literature contains many more solutions. It is usuallypossible to find a K solution for a geometry that is similar to the structure of interest.

When a published K solution is not available, or the accuracy of such a solution isin doubt, one can obtain the solution experimentally or numerically. Deriving a closed-form solution is probably not a viable alternative, since this is only possible with simplegeometries, and nearly all such solutions have already been published. Experimentalmeasurement of K is possible through optical techniques, such as photoelasticity [4,5]and the method of caustics [6], or by determining § from the rate of change in compliance

Application to Structures 461

with crack length (Eq. (2.30)) and computing AT from g (Eq. (2.58)). Chapter 11 de-scribes a number of computational techniques for deriving stress intensity.

An alternative is to utilize the principle of elastic superposition, which enables newK solutions to be constructed from known cases. Section 2.6.4 outlined this approach,and applied the principle of superposition to a pressure loaded semicircular surface crack(Example 2.5). Influence coefficients [7], described below, are an application of the su-perposition principle. Section 2.6.5 introduced the concept of weight functions [8,9],from which K solutions can be obtained for arbitrary loading.

9.1.1 KI for Part-Through Cracks

Laboratory fracture toughness specimens usually contain idealized cracks, but naturallyoccurring flaws in structures are under no obligation to live up to these ideals. Structuralflaws are typically irregular in shape and are part-way through the section thickness.Moreover, severe stress gradients often arise in practical situations, while laboratory spec-imens experience relatively simple loading.

Newman and Raju [10] have published a series of Kj solutions for part-throughcracks. Figure 9.2 illustrates the assumed geometries. Newman and Raju approximateburied cracks, surface cracks and corner cracks as ellipses, half ellipses, and quarter el-lipses, respectively. These solutions apply to linear stress distributions, where the stressnormal to the flaw can be resolved into bending and membrane components, respectively(Fig. 9.3). If the stress distribution is not perfectly linear, equivalent membrane and bend-ing stresses can be inferred as follows: the equivalent membrane stress is equal to the in-tegrated average stress through the thickness, while the equivalent bending stress is in-ferred by computing a resultant moment (per unit width) and dividing by 6t2.

The Newman and Raju solutions for part- through flaws subject to membrane andbending stresses are expressed in the following form:

(9.2)

where Q is the flaw shape parameter, which is based on the solution of an elliptical inte-gral of the second kind (see Fig. 2.19), and F and H are geometry constants, whichNewman and Raju obtained from finite element analysis. The parameters F and H dependon a/c, a/t, and 0 (see Fig. 2.19), and plate width. Section 12.2 lists polynomial fits forF and H that correspond to each of the crack shapes in Fig. 9.2.

Equation (9.2) is reasonably flexible, since it can account for a range of stress gradi-ents, and includes pure tension and pure bending as special cases. This equation, how-ever, is actually a special case of the influence coefficient approach, an example of whichis described below.

462 Chapter 9

^ 2c fe>

t ^t

*

§§^ ;-(a) Semi-elliptical surface crack.

I

-ertl i flfrt-^ ~C ** "

' tt \. V \ V ' • • :>'>i

I <^S^\\^\\^^ * k 2stiff ^^•~^^\^vSx\\^^î"''i^t' ^y

A

td(b) Elliptical buried flaw.

m C flrlH C »|

'////SY///S/£&

w//^ 4///// a \6^ 4 t~ I

1

FIGURE 9.2 Part-through crack geometriesconsidered by Newman and Raju. See Section 12.2for the complete solutions.

(c) Quarter-elliptical corner crack.

^max LvX^^N-xx?vActual Stress >^^

Di&tribtiliQft ^**i^Ni*fc«11,

CJ +(T •®m 2

CW-<T '°b 2

_ > FIGURE 9.3 Approximating a nonuni-^min form stress distribution as linear, and re-

solving the stresses into membrane andbending components.

Suppose the normal stress acting on the crack plane (in the uncracked configuration)an be represented by a cubic equation:

<Tyy =A0+ AlX + A

3= IAJXJ

7=0

2,2

+ A3,3

(9.3)


Figure 9.4 schematically illustrates this stress distribution, and defines the coordinateaxes. The stress intensity factor that results from the above equation can be constructedby obtaining Kj solutions for power-law loading, where the exponent ranges from 0 to 3.The following dimensionless stress distributions can be applied to the crack face indepen-dently in a finite element model of the crack geometry:

for; = 0, 1,2, or 3 (9.4)

Figure 9.5 illustrates application of a power-law stress distribution to the crack faces.,Recall Figs. 2.25 and 2.26, which used the superposition principle to show that any setof boundary conditions can be represented by equivalent crack face tractions.

Raju and Newman [7] applied power law stress distributions to a wide range ofsemi-elliptical surface flaws in cylinders (Fig. 9.6). They considered t/R{ ratios of 0 (flatplate), 0.10, and 0.25. Their analysis included both internal flaws and external flaws.For the stress distribution in Eq. (9.4), the stress intensity factor can be expressed as fol-lows:

(9.5)

for j = 0, 1, 2, or 3, where Gj is an influence coefficient. Chapter 12 lists influence co-efficients for various flaw geometries. For a given flaw shape and j value, Gj is relativelyinsensitive to the t/fy ratio, which is indicative of the curvature at the free surface. Thusit is not critical to match the curvature of a structure exactly; the influence coefficients fora surface flaw in a flat plate (Table 12.23) should give reasonable estimates of K\ in moststructures.

FIGURE 9.4 Arbitrary stress distribution whichcan be fit to a four-term polynomial (Eq. (9.3)).

464 Chapter 9

FIGURE 9.5 Power-law stress distribution ap-plied to the crack face.

FIGURE 9.6 Internal and external axial surface flaws in a pressurized cylinder.

When an arbitrary stress distribution is approximated by Eq. (9.3), the contributionof each term in the polynomial can be summed to obtain the total Kj for the crack.Equations (9.4) and (9.5), however, must be scaled to the actual stress distribution.Equation (9.3) can be rewritten in the following form:


ayy =3

;=o

J(x_vaJ

(9-6)

By comparing Eqs. (9.4) and (9.6), we see that the scaling factor for each term in the

polynomial is Aj a!. Therefore, the stress intensity factor for a cubic polynomial stressdistribution is given by

(9.7)

Let us now consider the example of a pressurized cylinder with an internal axial sur-face flaw, as illustrated in Fig. 9.6. In the absence of the crack, the hoop stress in a thickwall pressure vessel is as follows [11]:

(9.8)RKo ~ Ki

where p is the internal pressure and the other terms are defined in Fig. 9.6. If we definethe origin at the inner wall (x = r - /?/) and perform a Taylor series expansion about x = 0,Eq. (9.8) becomes

(9.9)-R

where x is in the radial direction with the origin at Rf. The first four terms of thisexpansion give the desired cubic polynomial. An alternate approach would be to curve-fita cubic polynomial to the stress field. This latter method is necessary when the stressdistribution does not have a closed-form solution.

When computing Kj for the internal surface flaw, we must also take account of thepressure loading on the crack faces. Superimposing p on Eq. (9.9) and substituting theresulting coefficients (AJ) into Eq. (9.7) gives [7]:

Kr = 2G- (9.10)

Applying a similar approach to an external surface flaw leads to [7]:

466 Chapter 9

(9-11)

The origin in this case was defined at the outer surface of the cylinder, and a series expan-sion was performed as before. Thus Kj for a surface flaw in a pressurized cylinder can beobtained by substituting the appropriate influence coefficients into Eq. (9.10) or Eq.(9.11).

Of course one can also infer the stress intensity factor by performing a full finite el-ement analysis with the actual loading conditions. Figure 9.7 compares the K{ for an ex-ternal crack estimated from Eq. (9.11) with a solution published by Atluri and Kathiresan[12]. The full solution and the estimate from influence coefficients differ by less than10%. Raju and Newman [7] state that the discrepancies may be due to differences in nu-merical techniques, rather than inherent errors in the influence coefficient approach.

Influence coefficients are useful for inferring Kj values for cracks that emanate fromstress concentrations. Figure 9.8 schematically illustrates a surface crack at the toe of afillet weld. This geometry produces a local stress gradients that affect the K] of the crack.Performing a three-dimensional finite element analysis of this structural detail with crackwould be costly and time-consuming, and may be unnecessary. If the stress distributionin this detail is known for the uncracked case, these stresses can be fit to a cubic polyno-mial (Eq. 9.3), and Kj can be estimated by substituting the influence coefficients into Eq.(9.7). The uncracked stress distribution can be inferred from a two-dimensional elastic fi-nite element analysis with a relatively coarse mesh.

Ki 1.6

Q 1.2

0.8

0.4 -

External Axial Surface Flawa/c = 1.0 a/t = 0.8 t/R = 0.5

— - Atluri and Kathiresan [12]

0.2 0.4 0.6 0.8

FIGURE 9.7 Comparison of stress intensity solutions from the influence coefficient approach [7] withfinite element analysis of the actual geometry [12].


The previous example is only approximate, however. Since the influence coeffi-cients in Chapter 12 were not derived from the fillet weld geometry, there may be slighterrors if these Gj values are applied in this case. The influence coefficients for surfaceflaws depend on a/t and a/c, but are insensitive to the radius of curvature in the cross sec-tion. Thus as long as the crack shape and depth are taken into account, the Gj values inChapter 12 should be reasonably accurate. One can minimize errors by applying the Gjvalues for an internal flaw in a cylinder, since the concave shape comes closest to match-ing the profile of the fillet weld.

Since the flaw in Fig. 9.8 is near a weld, there is a possibility that weld residualstresses will be present. These stresses must be taken into account in order to obtain anaccurate estimate of Kj. Weld residual stresses are an example of secondary stresses, asdiscussed below.

1 fe»>•

e-ts»

Flat plate with a polynomial stress distributionFillet Weld

FIGURE 9.8 Application of the influence coefficient approach to a complex structural detail such as afillet weld.

9.1.2 Primary and Secondary Stresses

The loading in a structure can be divided into primary and secondary stresses. Primarystresses generally arise from externally applied loads and moments, while secondarystresses are localized and are self-equilibrating through the cross section. Primarystresses, if sufficiently large, are capable of leading to plastic collapse, but secondarystresses cannot cause collapse of the structure. The latter can, however, contribute to frac-ture if large tensile secondary stress occur near a crack. Examples of secondary stressesinclude weld residual stresses and thermal stresses. In some cases, however, thermal load-ing can produce primary stresses. A stress should be classified as primary when it is notclear which category is appropriate.

In linear elastic analyses, primary and secondary stresses are treated in an identicalfashion. The total stress intensity is simply the sum of the primary and secondary com-ponents:

(

468 Chapter 9

where the superscripts p and s denote primary and secondary quantities, respectively.The distinction between primary and secondary stresses is important only in elastic-

plastic and fully plastic analyses. Sections 9.3, 9.4, and 9.7 describe the treatment ofprimary and secondary stresses in such cases.

9.1.3 Plasticity Corrections

Section 2.8 describes approaches for incorporating small amounts of crack tip plasticityinto the estimation of the stress intensity. The Irwin approach [13] defines an effectivecrack length as the sum of the actual crack size, a, and a plastic zone correction, ry. Theeffective stress intensity factor is given by

effKe = Y{a + r}(7n(a + r) (9.13)

where Y{a + ry} denotes that the geometry correction factor is a function of the effectivecrack size (not Y times a + ry). The Irwin plastic zone corrections are as follows:

rv = - — — for plane stress (9.14a)y 2

and

rv = - — — for plane strain (9.14b)

This correction becomes significant at applied stresses greater than approximately half theyield strength, and is inaccurate above ~ 0.7 ffys (see Fig- 2.30).

The strip yield correction for a through crack in an infinite plate in plane stress isgiven by

Keff = -In sec ) (9.15)

Equation (9.15) does not result from adding a plastic zone correction to the crack size, butis based on an analysis by Burdekin and Stone [14] (see Appendix 3.1).1

Recall that the size of the strip yield zone was derived by requiring that the singularity vanish. Thus theeffective K cannot be defined in terms of a singularity amplitude. Instead, Burdekin and Stone derived theCTOD from Westergaard functions, and converted CTOD to an effective K through Eq. (3.7). A strip yieldequation for J can be derived by evaluating J contour integral along the strip yield zone (Eq. (3.43)).


The strip yield correction for a through crack in an infinite plate does not apply toother configurations. The strip yield model can be applied to other geometries, but eachconfiguration requires a separate analysis [15]. There is, however, an approximate methodof generalizing the strip yield model to a single equation that describes all cracked geome-tries (see Section 9.4).

Both the Irwin and strip yield plastic zone corrections have the effect of increasingKeff over the linear elastic value. Failure to apply an appropriate plasticity correctioncould, therefore, result in an underestimate of the crack driving force, which would lead toa nonconservative analysis.

9.1.4 KIC from Jcrit: Advantages and Pitfalls

For plane strain, small scale yielding conditions, Kjc and a critical J value (defined at thesame point on the load-displacement curve) are related as follows:

(9.16)

where Jcrit can either be a Jjc value, defined near the initiation of ductile crack growth, ora critical J for cleavage. When a test specimen behaves in a predominantly linear elasticmanner, either Kfc or Jcrit can b£ measured, but K ceases to be valid when the plasticzone becomes too large.

Recall from Chapter 7 that the size requirements for valid Jfc tests and JCrit valuesfor cleavage are much less strict than the requirements for a valid KIC test. Thus size-in-dependent fracture toughness values in terms of J can be obtained on much smaller speci-mens than are required for Kjc tests. A Jcrit value that meets the necessary size require-ments can be converted to an equivalent Kjc through Eq. (9.16). This quantity can beviewed as the Kjc that would be measured, given a sufficiently large specimen.

The equivalent Kjc value, which is usually given the designation Kjc, can be ap-plied to a structure that behaves in a linear elastic fashion. Quantifying the toughness interms of Kjc enables the designer to apply linear elastic relationships between stress, flawsize, and toughness. Linear elastic approaches are much simpler and more versatile than afracture design methodology based on the / integral (Section 9.5).

A conversion to Kjc is only appropriate when the critical / value is a size-indepen-dent measure of fracture toughness for the material. A //c value for ductile crack growthmust satisfy Eq. (7.14), while JCrit f°r cleavage must satisfy Eq. (7.19).

9.1.5 A Warning About LEFM

Performing a purely linear elastic fracture analysis and assuming that LEFM is valid ispotentially dangerous, because the analysis gives no warning when it becomes invalid.The user must rely on experience to know whether or not plasticity effects need to beconsidered. A general rule of thumb is that plasticity becomes important at around 50%of yield, but this is by no means a universal rule.

470 Chapter 9

The safest approach is to adopt an analysis that spans the entire range from linearelastic to fully plastic behavior. Such an analysis accounts for the two extremes of brittlefracture and plastic collapse. At low stresses, the analysis reduces to LEFM, but predictscollapse if the stresses are sufficiently high. At intermediate stresses, the analysis auto-matically applies a plasticity correction when necessary; the user does not have to decidewhether or not such a correction is needed.

Sections 9.4 to 9.7 give examples of fracture analyses that span the range of mate-rial behavior.

9.2 THE ASME REFERENCE CURVES

The American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Codeis a comprehensive guide for designers, fabricators, and operators of pressure vessels andrelated components. Section XI of the code, "Rules for Inservice Inspection of NuclearPower Plant Components [16]," contains guidelines for computing allowable flaw sizesbased on fracture mechanics principles.

Since fracture toughness data are not always available for a particular heat of steel,Section XI of the ASME code includes reference curves that give conservative estimatesof toughness versus temperature. These curves were generated by compiling Kjc, Kid,and Kia data for several heats of steel over a range of temperatures, and plotting these re-sults relative to a reference temperature, RTftDT-

As Fig. 9.9 schematically illustrates, different heats of pressure vessel steel typi-cally display ductile-brittle transitions at different temperatures; the reference temperatureis an attempt to collapse all data onto a single curve. The indexing temperature, RTj^Dj,is assigned through a combination of the drop weight nil-ductility transition temperature(NDTT) and Charpy properties; RTftDT'is defined as the higher of the following cases:

(1) The drop weight NDTT.

(2) 33°C (60°F) below the minimum temperature at which the lowest of threeCharpy results is at least 68 J (50 ft-lb).

in a typical pressure vessel steel occurs near the lower "knee" of the fracturetoughness transition curve.

Two reference toughness curves were originally developed: the KJC curve and theKlR curve. The former curve describes the lower envelope to a large set of Kjc data,while the latter is a lower envelope to a combined set of KIC, Kid, anc^ Kia data. Sincedynamic and crack arrest toughness values are generally lower than static initiation (Kjc]values, the KJR curve is the more conservative of the two. The KIC an(l %IR curves, inSI units, are given by

KIC = 36.5 + 3.084exp[0.036(7/ - RTNDT + 56)] (9.17a)


KIR = 29.5 + 1.344exp[0.026(r~ RTNDT + 89)] (9.17b)

where temperatures are in °C, and KIC and KIR are in MPa Vm. Figure 9.10 shows aplot of Eqs. (9.17a) and (9.17b), together with the experimental data that defined thecurves.

KIC eat 1 | |Hea2| [Heat3

TEMPERATURE X - RTNDT

FIGURE 9.9 The ASME Section XI [16] approach for indexing multiple heats of steel.

tiI-J

U

U

250

200 -

150 -

-250 -200 -150 -100 -50 0 50 100

100

FIGURE 9.10 The KIC an^ &1R curves, with the original data that was used to define the lower envelopecurves [17].

472 Chapter 9

According to the exponential curve fits for KJC and KIR, the toughness increaseswithout bound; these curves do not predict an upper shelf. In the late 1960s and early1970s, when these curves were defined, there was no way to quantify upper shelf tough-ness with a fracture mechanics test. A number of very large specimens, up to 305 mm(12 in) thick, were required to quantify Kjc in the transition region^; upper shelf Kjc

measurements were simply not possible. Because the upper shelf could not be quanti-

fied, a cut-off was imposed at 220 MPa Vm" (200 ksi Vin").Section XI of the ASME Code also gives guidelines for computing the applied Kj

in a pressure vessel. For a surface flaw, the stresses are linearized and divided into bend-ing and membrane components (Fig. 9.3), and stress intensity is estimated from Eq.(9.2).

9.3 THE CTOD DESIGN CURVE

The CTOD concept was applied to structural steels beginning in the late 1960s. TheBritish Welding Research Association (now known as The Welding Institute) and otherlaboratories performed CTOD tests on structural steels and welds. At that time there wasno way to apply these results to welded structures because CTOD driving force equationsdid not exist. Burdekin and Stone [14] developed the CTOD equivalent of the strip yieldmodel in 1966. Although their model provides a basis for a CTOD driving force relation-ship, they were unable to modify the strip yield model to account for residual stresses andstress concentrations. (These difficulties were later overcome when a strip yield approachbecame the basis of the R6 design method, as discussed in the next section)

In 1971, Burdekin and Dawes [18] developed the CTOD design curve, a semi-empir-ical driving force relationship, that was based on an idea that Wells [19] originally pro-posed. For linear elastic conditions, fracture mechanics theory was reasonably well devel-oped, but the theoretical framework required to estimate the driving force under elastic-plastic and fully plastic conditions did not exist until the late 1970s. Wells, however,suggested that global strain should scale linearly with CTOD under large scale yieldingconditions. Burdekin and Dawes based their elastic-plastic driving force relationship onWells' suggestion and an empirical correlation between small scale CTOD tests and widedouble-edge notched tension panels made from the same material. The wide plate speci-mens were loaded to failure, and the failure strain and crack size of a given large scalespecimen were correlated with the critical CTOD in the corresponding small scale test.

The correlation that resulted in the CTOD design curve is illustrated schematicallyin Fig. 9.11. The critical CTOD is nondimensionalized by the half crack length, a, of thewide plate and is shown on the ordinate of the graph. The nondimensional CTOD is plot-ted against the failure strain in the wide plate, normalized by the elastic yield strain, £y.Based on a plot similar to Fig. 9.11, Burdekin and Dawes [18,20] proposed the followingtwo-part relationship:

2^ original KJC curve is often referred to as the "Million Dollar Curve" which reflects the cost of testingsuch large specimens.

Application to Structures

< 0.5

and

271 ey a1.0

0.5

).5y -y

where <E> is the nondimensional CTOD. Equation (9.18a), which was derived from LEFMtheory, includes a safety factor of 2.0 on crack size. Equation (9.18b) represents an upperenvelope of the experimental data.

1.5

UNSAFE

I I0.5 1.0 1.5 2.0

FIGURE 9.11 The CTOD design curve.

The applied strain and flaw size in a structure, along with the critical CTOD for thematerial, can be plotted on Fig. 9.11. If the point lies above the design curve, the struc-ture is considered safe because all observed failures are below the design line. Equations(9.18a) and (9.18b) conform to the classical view of a fracture mechanics analysis, in re-lating stress (or strain in this case) to fracture toughness (8Crii) and flaw size (a). TheCTOD design curve is conservative, however, and does not relate critical combinations ofthese variables.

In 1980, the CTOD design curve approach was incorporated into the BritishStandards document PD 6493 [21]. This document addresses flaws of various shapes by

474 Chapter 9

relating them back to an equivalent through-thickness dimension, a. For example, if astructure contains a surface flaw of length 2c and depth a, the equivalent through-thicknessflaw produces the same stress intensity when loaded to the same stress as the structure

with the surface flaw. Thus a is a generalized measure of a flaw's severity. The CTOD

design curve can be applied to any flaw by replacing a with a in Eq. (9. 1 8).The original CTOD design curve was based on correlations with flat plates loaded in

tension. Real structures, however, often include complex shapes that result in stress con-centrations. In addition the structure may be subject to bending and residual stresses, aswell as tensile (membrane) stresses. The PD 6493:1980 approach accounts for complexstress distributions simply and conservatively by estimating the maximum total strain inthe cross section and assuming that this strain acts through the entire cross section. Themaximum strain can be estimated from the following equation:

(9.19)

where k{ is the elastic stress concentration factor, Pm is the primary membrane stress, Pîs the primary bending stress, and S is the secondary stress, which may include thermal orresidual stresses. Since the precise distribution of residual stresses is usually unknown, Sis often assumed to equal the yield strength in an as-welded weldment.

When Burdekin and Dawes developed the CTOD design curve, the CTOD and wideplate data were limited; the curve they constructed lay above all available data. In 1979,Kamath [22] reassessed the design curve approach with additional wide plate and CTODdata generated between 1971 and 1979. In most cases, there were three CTOD tests for agiven condition. Kamath used the lowest measured CTOD value to predict failure in thecorresponding wide plate specimen. When he plotted the results in the form of Fig. 9. 1 1,a few data points fell above the design curve, indicating Eq. (9.18) was nonconservative inthese instances. The CTOD design curve, however, was conservative in most cases.Kamath estimated the average safety factor on crack size to be 1.9, although individualsafety factors ranged from less than 1 to greater than 10. With this much scatter, the con-cept of a safety factor is of little value. A much more meaningful quantity is the confi-dence level. Kamath estimated that the CTOD design curve method corresponds to a97.5% confidence of survival. That is, the method in PD 6493:1980 is conservative ap-proximately 97.5% of the time.

9.4 FAILURE ASSESSMENT DIAGRAMS

Structures made from materials with sufficient toughness may not be susceptible to brit-tle fracture, but they can fail by plastic collapse if they are overloaded. The CTOD designcurve does not explicitly address collapse, and can be nonconservative if a separate col-lapse check is not applied.

Dowling and Townley [23] and Harrison, et al. [24] introduced the concept of a two-criteria failure assessment diagram (FAD) to describe the interaction between fracture and


collapse. The first FAD was derived from a modified version of the strip yield model, asdescribed below.

Equation (9.15) is the effective stress intensity factor for a through crack in an infi-nite plate, according to the strip yield model. As discussed earlier, this relationship isasymptotic to the yield strength. Equation (9.15) can be modified for real structures byreplacing GYS with the collapse stress, ffc, for the structure. This would ensure that thestrip yield model predicts failure as the applied stress approaches the collapse stress. Fora structure loaded in tension, collapse occurs when the stress on the net cross sectionreaches the flow stress of the material. Thus Gc depends on the tensile properties of thematerial and the flaw size relative to the total cross section of the structure. The next stepin deriving a failure assessment diagram from the strip yield model entails dividing the ef-fective stress intensity by the linear elastic K:

K.eff

a8 , [ n a

—«- In secn ( 2 or

(9.20)

This modification not only expresses the driving force in a dimensionless form but alsoeliminates the square root term that contains the half length of the through crack. ThusEq. (9.20) removes the geometry dependence of the strip yield modeP. This is analogousto the PD 6493 approach, where the driving force relationship was generalized by defining

an equivalent through thickness flaw, a. As a final step, we can define the stress ratio,Sr, and the K ratio, Kr, as follows:

V -A*. —K eff

and

Sr =

(9.21)

(9.22)

The failure assessment diagram is then obtained by inserting the above definitions intoEq. (9.20) and taking the reciprocal:

Kr = Sr71

-L(9.23)

•a-'This generalization of the strip yield model is not rigorously correct for all configurations, but it is a goodapproximation.

476 Chapter 9

1.2

Kr

0.8

0.6

0.4 -

0.2 -

0.2 0.4 0.6

s,0.8 1.2

FIGURE 9.12 The strip yield failure assessment diagram [23,24].

Equation (9.23) is plotted in Fig. 9.12. The curve represents the locus of predicted failurepoints. Fracture is predicted when K-eff- K!C- If me toughness is very large, the struc-ture fails by collapse when 5r= 1.0. A brittle material will fail when Kr - 1.0. In in-termediate cases, collapse and fracture interact, and both Kr and Sr are less than 1.0 atfailure. All points inside of the FAD are considered safe; points outside of the diagramare unsafe.

In order to assess the significance of a particular flaw in a structure, one must deter-mine the applied values of Kr and Sr, and plot the point on Fig. 9.12. The stress inten-sity ratio for the structure is given by

K _ KIr (structure) KIC

(9.24)

The applied stress ratio can be defined as the ratio of the applied stress to the collapsestress. Alternatively, the applied Sr can be defined in terms of axial forces or moments.If the applied conditions in the structure place it inside of the FAD, the structure is safe.


EXAMPLE 9.1

A middle tension (MT) panel (Fig. 7.1(e)) 1 m wide and 25 mm thick with a 200 mm

crack must carry a 7.00 MN load. For the material, Kjc = 200 MPa Vm, ays = 350MPa, and ojs - 450 MPa. Use the strip yield FAD to determine whether or not thispanel will fail.

Solution: We can take account of work hardening by assuming a flow stress that isthe average of yield and tensile strength. Thus Oflow = 400 MPa. The collapse loadis then defined when the stress on the remaining cross section reaches 400 MPa:

Pc = (400 MPa) (0.025 m) (1 m - 0.200m) = 8.00 MN

Therefore,

_Sr -

7.00 MN8.00MN

The applied stress intensity can be estimated from Eq. (2.46) (without the polynomialterm):

7.00 MN(0.025 m)(1.0 m)

^ /\/ Jt\ ( 0.100 m) sec

( n (0.100 m)\( — n^ - ) =\ LOO m /

Thus

= 0.805

The point (0.875, 0.805) is plotted in Fig. 9.12. Since this point falls outside of thefailure assessment diagram, the panel will fail before reaching 7 MN. Note that a col-lapse analysis or brittle fracture analysis alone would have predicted a "safe" condi-tion. Interaction of fracture and plastic collapse causes failure in this case.

In 1976, the Central Electricity Generating Board (CEGB) in Great Britainincorporated the strip yield failure assessment into a fracture analysis methodology, whichbecame known as the R6 approach [24]. A revised version of the R6 document, whichwas published in 1980 [25], offers practical advise on how to apply the strip yield FADto real structures. For example, it recommends that secondary stresses be taken intoaccount through a secondary stress intensity. The total stress intensity is obtained byadding the primary and secondary components:

478 Chapter 9

K r K f + K jKr=—1- = -J-—L (9.25)

Only the primary stresses are used to compute Sr, because secondary stresses, by defini-tion, do not contribute to collapse. Note that Kj is the LEFM stress intensity; it doesnot include a plastic zone correction. Plasticity effects are taken into account through theformulation of the failure assessment diagram (Eq. (9.23)).

The R6 procedure recommends that the fracture toughness input be obtained throughtesting the material according to ASTM E 399 or the equivalent British Standard (Chapter7). When it is not possible to obtain a valid Kjc value experimentally, one can measureJlc in the material and convert this toughness to an equivalent Kjc (or Kjc} by means ofEq. (9.16).

9.5 THE EPRI J ESTIMATION SCHEME

The R6 failure assessment diagram is based on a strip yield model. Since it as-sumes elastic-perfectly plastic material behavior, it is conservative when applied to strainhardening materials.

In 1976, Shih and Hutchinson [26] proposed a more advanced methodology forcomputing the fracture driving force that takes account of strain hardening. Their ap-proach was developed further and validated at the General Electric Corporation inSchenectady, New York in the late 1970s and early 1980s, and was published as an engi-neering handbook by the Electric Power Research Institute (EPRI) in 1981 [27].

The EPRI procedure provides a means for computing the applied J integral underelastic-plastic and fully plastic conditions. The elastic and plastic components of J arecomputed separately and added to obtain the total J:

Jtot = Jel + Jpl (9-26)

Figure 9.13 schematically illustrates a plot of /versus applied load. The plastic compo-nent of /is negligible at low loads, but dominates at high loads. The sum of elastic andplastic / values from the estimation scheme agrees well with an elastic-plastic finite ele-ment analysis.

9.5.1 Theoretical Background

Consider a cracked structure with a fully plastic ligament, where elastic strains are negli-gible. Assume that the material follows a power-law stress-strain curve:

(9.27)


J INTEGRAL

Elastic •+ Plastic(Estimation Scheme)

APPLIED LOAD (P)

FIGURE 9.13 The EPRIJ estimation scheme [27].

which is the second term in the Ramberg-Osgood model (Eq. (3.22)). The parameters a,n, BO, and Go are defined in Section 3.2.3. Close to the crack tip, under /-controlled con-ditions, the stresses are given by the HRR singularity:

\J

(9.28)

which is a restatement of Eq. (3.24a). Solving for / in the HRR equation gives

(9.29)

For J controlled conditions, the loading must be proportional. That is, the local stressesmust increase in proportion to the remote load, P. Therefore, Eq. (9.29) can be written interms of P:

480 Chapter 9

J = aen<jahL\ — (9.30). 0 )

where h is a dimensionless function of geometry and n, L is a characteristic length for thestructure, and Po is a reference load. Both I and P0 can be defined arbitrarily, and h canbe determined by numerical analysis of the configuration of interest.

It turns out that the assumptions of / dominance at the crack tip and proportionalloading are not necessary to show that / scales with Pn+' for a power-law material, butthese assumptions were useful for deriving the correct form of the J-P relationship.

Equation (9.30) is an estimate of the fully plastic J. Under linear elastic conditions,J must scale with P^. The EPRI J estimation procedure assumes that the total / is equalto the sum of the elastic and plastic components (Eq. (9.26) and Fig. 9. 13).

9.5.2 Estimation Equations

The fully plastic equations for J, crack mouth opening displacement (Vp\ and load linedisplacement (Ap) have the following form for most geometries:

( p r1Jpl = as^^hâ I W,n)\ — (9.31)

V = CC£0ah2(a I W» — (9.32)

Ap = a£0ah3(a I W,n)\ — (9.33)

where b is the uncracked ligament length, a is the crack length, and h], /Z2, and hj aredimensionless parameters that depend on geometry and hardening exponent. The h factorsfor various geometries and n values, for both plane stress and plane strain, are tabulated inseveral EPRI reports [28-30], as well as Chapter 12.

The reference load, P0, is usually defined by a limit load solution for the geometryof interest; Po normally corresponds to the load at which the net cross section yields.

The plastic load line displacement, Ap, defined in Eq. 9.33 is only that componentof plastic displacement that is due to the crack. Recall Section 3.2.5, where the dis-placement was divided into "crack" and "no crack" components; the latter is the displace-ment that would be measured if there were no crack, and the former is the additional dis-placement that results from the presence of the crack. The total displacement in a struc-ture is the sum of the elastic and plastic "crack" and "ho crack" components.


Several configurations have J expressions that are slightly different from Eq. (9.31).For example, the fully plastic / integral for a center cracked panel and a single edgenotched tension panel is given by

(9.34)

where, in the case of the center cracked panel, a is the half crack length and W is the halfwidth. This modification was made in order to reduce the sensitivity of hj to the cracklength/width ratio.

The elastic J is equal to ^(aeff), the energy release rate for an effective crack length,which is based on a modified Irwin plastic zone correction:

1(9-35)

where /? = 2 for plane stress and /? = 6 for plane strain conditions. Equation (9.35) is afirst-order correction, where aeffis computed from the elastic Kj, rather than Keff, thus it-eration is not necessary.

The plastic zone correction that is applied to Jei does not have a theoretical basis,but it was incorporated to provide a smooth transition from linear elastic to fully plasticbehavior. Estimated J values that include the plastic zone correction are closer to elastic-plastic finite element calculations than estimates of/ without this correction. Equation(9.35) has a relatively small effect on the computed / value (Example 9.3); the effect isnegligible at low loads, where the behavior is linear elastic, and at high loads, where thefully plastic term dominates.

The CTOD can be estimated from a computed J value as follows:

8 = dn - (9.36)

where dn is a dimensionless constant that depends on flow properties [31]. Figure 3.18shows plots of dn for both plane stress and plane strain. Equation (9.36) must be regardedas approximate in the elastic-plastic and fully plastic regimes, because the J-CTOD rela-tionship is geometry dependent in large scale yielding [31].

482 Chapter 9

EXAMPLE 9.2

Consider a single edge notched tensile panel with W = 1 m, B = 25 mm, and a = 125mm. Calculate J versus applied load assuming plane stress conditions. Neglect theplastic zone correction.Given: ao = 414 MPa; n=10; a= 1.0; E = 207,000 MPa eo = OQ/E = 0.002

Solution: From Table 12.13, the reference load for this configuration is given byv

P0 = 1.072 rjCfobBwhere

7] = V7 + (a/b)2 - a/b = 0.867 for a/b = 125/875 = 0.143

Solving for Po gives

P0 = 8.42 MN

For a/W= 0.125 and n = 10, /i; = 4.14 (from Table 12.13). Thus the fully plastic /isgiven by

77Jpl =. (1.0)(0.002)(414>000 kPa~)—: ~r~p.—: (4.14) I „ ._ ..... I

= 2.486 x 1Q-8 P11

where P is in MN and Jpi is in kJ/ mr. The elastic / is given by

Kf2 P2f^(a/W)Jel = ~~^~ = *)

E B2 W E

From the polynomial expression in Table 2A,f(a/W) = 0.770 for a/W = 0.125. Thus

1 WO P* ,0.770,* ^(0.025 m)2 (1.0 m) (207,000 MPa)

where P is in MN and Jei is in kJ/m^. The total / is the sum of ]e\ and }p\:

J = 4.584 P2 + 2.486 x W8 P1J

Figure 9.14 shows a plot of this equation. An analysis that includes the plastic zonecorrection (Eq. (9.35)) is also plotted for comparison.

rApplication to Structures 483

EXAMPLE 9.3

For the panel in Example 9.2, determine the effect of the plastic zone correction at P= Po-

Solution: From the previous problem, P0 = 8.42 MN. The elastic J without theIrwin correction is given by

Jel(a) = (4.584) (8.42 MN)2 = 325 kJ/m2

The plastic J is not influenced by the Irwin correction:

Jpl = (2.486 xW8) (8.42 MN)11 = 375 kJ/m2

The plastic zone correction is obtained by substituting the appropriate quantities intoEq. (9.35):

= 0.725m +1 + (l.O)2 2n

(8.42 MN)(0.770)

jfl.025 m) -v 1 m (414 MPa)

= 0.151m

For aejy/W = 0.151, f(aejy/W) = 0.874. Thus the corrected Jei is given by

= 418 kJ/m2= 325 kJ/m2

The total J without the plastic zone correction is as follows:

J(a) = 325kJ/m2 + 375 kJ/m2 = 700 kJ/m2

and J with the correction is given by

J(aeff) = 418kJ/m2 + 375 kJ/m2 = 793 kJ/m2

which is 13% higher than the estimate without the correction. This calculation repre-sents a worst-case situation. The relative effect of the plastic zone correction is sig-nificantly less at both lower and higher loads. Also, the correction is smaller inplane strain than in plane stress.

484 Chapter 9

1000

s5?

oB

800

600

400

200

EDGE CRACKED PANELW = 1.0 m B = 25 mm a = 125 mm

n = 10 ao = 414MPa eo = 0.002 a = 1.0

O Without PZ Correction

• With PZ Correction

2 4 6 8APPLIED LOAD, MN

FIGURE 9.14 Applied J versus applied load in an edge cracked panel.

10

9.5.3 Comparison with Experimental Estimates

Typical equations for estimating / from a laboratory specimen have the form

£LE

(9.37)

assuming unit thickness and a stationary crack. Equation (9.37) is convenient for exper-imental measurements because it relates / to the area under the load v. load line displace-ment curve, provided Ap does not contain a "no crack" component (see Section 3.2.5).

Since Eq. (9.33) gives an expression for the P-Ap curve for a stationary crack, it ispossible to compare Jpi estimates from Eqs. (9.31) and (9.34) with Eq. (9.37).According to Eq. (9.33), the P-Ap curve follows a power law, where the exponent is thesame as in a tensile test. The plastic energy absorbed by the specimen is as follows

71 + 1'•PA,


P0ae0ah3\ — I (9.38), 1 U U -JI TTtn + l ( P 0

Thus the plastic / is given by

Equating Eqs. (9.3 1) and (9.39) and solving for r\p gives

n + 1 Onb2h

7? = -- — 1 (9.40a)

Alternatively, if 7p/ is given by Eq. (9.34),

71 + 1 <7 J?2/iir}D = -- 2 - L (9.40b)* rt P0Wh3

Consider an SENB specimen in plane strain. The reference load, assuming unitthickness and the standard span of 4W, is given by

_ 0.364a0b2

p = - 2 - (9_41)0 wSubstituting Eq. (9.41) into Eq. (9.40a) gives

1 n + lWh?! - ---- L (9.42)

p 0.364 n a h3

Equation (9.42) is plotted in Fig. 9.15 for n = 5 and n = 10. According to the equationthat was derived in Section 3.2.5, f]p = 2. This derivation, however, is only valid fordeep cracks, since it assumes that the ligament length, b, is the only relevant length di-mension. Figure 9.15 indicates that Eq. (9.31) approaches the deep crack limit with in-creasing a/W. For n = 10, the deep crack formula appears to be reasonably accurate be-yond a/W ~ 0.3. Note that the rfp values computed from Eq. (9.42) for deep cracks fluc-tuate about an average of -1.9, rather than the theoretical value of 2.0. These fluctuationsmay be indicative of numerical errors in the h] and /zj values, while the average r\p

486 Chapter 9

slightly below 2.0 may indicate an a/W dependence that was not included in the dimen-sional analysis (Eq. (3.36)) in Section 3.2.5.

Equation (3.32) was derived for a double edge notched tension panel, but also appliesto a deeply notched center cracked panel. A comparison of Eq. (9.34) with the secondterm of Eq. (3.32) leads to the following relationship for a center cracked panel in planestress:

JEPRI n + l bJ

(9.43)DC

where JEPRI is the plastic / computed from Eq. (9.34) and JDC is the plastic / from thedeep crack formula. Figure 9.16 is a plot of Eq. (9.43). The deep crack formula underes-timates / at small a/W ratios, but coincides with J£PRI when a/W is sufficiently large.Note that the deep crack formula applied to a wider range of a/W for n = 10. The deepcrack formula assumes that all plasticity is confined to the ligament, a condition that iseasier to achieve in low-hardening materials.

SENS SPECIMENPlane Strain

1.4

1.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a / W

FIGURE 9.15 Comparison of the plastic r\ factor inferred from the EPRI Handbook with the deepcrack value of 2.0 derived in Chapter 3


3.5 hCENTER CRACKED PANEL

Plane Stress

0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a / W

FIGURE 9.16 Comparison of J estimates from the EPRI handbook with the deep crack formula for acenter cracked panel.

9.5.4 J-Based Failure Assessment Diagrams

The elastic-plastic driving force estimated from the EPRI procedure can also be expressedin terms of a failure assessment diagram, an idea first proposed by Bloom [32] and Shih,et al. [33]. The /ratio and stress ratio are defined as follows.

and

(9.44)

(9.45)

The equivalent Kr is equal to the square root of //-.Figure 9.17 shows the applied /for center cracked panels with a/W — 0.5 and n = 5,

10, and 20, plotted in terms of failure assessment diagrams. The strip yield diagram isincluded for comparison. Note that the shape of the FAD changes when strain hardeningis taken into account. The strip yield diagram is conservative in this case because it as-sumes collapse will occur when the net section stresses equal ff0\ a panel made from astrain hardening material can withstand somewhat greater stresses. Figure 9.18 illustrates

488 Chapter 9

that failure assessment diagrams derived from the EPRI procedure are geometry dependent,while the strip yield diagram is geometry independent. The shape of the EPRI failure as-sessment diagram depends not only on the geometry of the cracked body, but also on thestress state (plane stress or plane strain). This makes fracture analyses with the EPRI ap-proach more complicated, because a different FAD must be generated for each configura-tion analyzed.

9.5.5 Ductile Instability Analysis

Section 3.4.1 outlined the theory of stability of J controlled crack growth. Crack growthis stable as long as the rate of change in the driving force (J) is less than or equal to therate of change of the material resistance (Jp). Equations (3.49) and (3.50) defined thetearing modulus, which is a nondimensional representation of the derivatives of both thedriving force and the resistance:

Tapp~dJ}

da)A.and R

E dJR

cr? da(9.46)

where AT is the remote displacement:

AT = A + CM? (9.47)

1.2

0.4 -

0.2

CENTER CRACKED PANELa/W = 0.5

Strip Yield n = 20Model

0 0.2 0.4 0.6 0.8

Sr

1.2 1.4

FIGURE 9.17 Failure assessment diagrams for a center cracked panel with n = 5, 10, and 20 computedfrom the EPRI Handbook.


1.2

Kr

0.8

0.6 p

0.4

0.2

CENTER CRACKED PANELn = 10

Strip YieldModel

77777777/,

0 0.2 0.4 0.6 0.8 1.2 1.4

FIGURE 9.18 EPRI /-based failure assessment diagrams for a center cracked panel with various a/Wratios.

where CM is the system compliance (Fig. 2.12). Recall that the value of CM influencesthe relative stability of the structure; CM = °° corresponds to dead loading, which tends tobe unstable, while CM = 0 represents the other extreme of displacement control, which ismore stable. Crack growth is unstable when

T.app TR (9.48)

The rate of change in driving force at a fixed remote displacement is given by4

da (If)'

-1

(9.49)

Since the EPRI J estimation approach provides expressions for J and A as a functionof load and crack length, it is possible to evaluate the derivatives in Eq. (9.49) numeri-cally at any P and a in the structure of interest. The EPRI Handbook [27] recommends aforward difference approach for numerical differentiation:

The distinction between local and remote displacements (A and CM P, respectively) is arbitrary, as long asall displacements due to the crack are included in A. The local displacement, A, can contain any portion ofthe "no crack" elastic displacements without affecting the term in square brackets in Eq. (9.49).

490 Chapter 9

a + Aa,P)-J(a,P)(9.50a)

Aa

AP

Aa,P)-A(a,P)Aa

AP

(9.50b)

(9.50c)

(9.50d)

One must exercise extreme caution in computing these derivatives: / and A are highlynonlinear functions of load and crack size, particularly in the fully plastic regime. Theload and crack length increments, AP and Aa, should be chosen to minimize numerical er-rors. One possible approach is to chose progressively smaller increments until the nu-merical derivatives converge. Nonlinear differentiation is another alternative. For exam-ple, taking the logarithm of /, A, a, and P before differentiation may increase numericalaccuracy.

The EPRI Handbook outlines two approaches for assessing structural stability:crack driving force diagrams and stability assessment diagrams. The former is a plot of /and JR versus crack length, while a stability assessment diagram is a plot of tearing mod-ulus versus /. These diagrams are merely alternative methods for plotting the same in-formation.

Figure 3.20 shows a schematic driving force diagram for both load control and dis-placement control. In this example, the structure is unstable at Pj and A$ in load con-trol, but the structure is stable in displacement control. Figure 9.19 illustrates drivingforce curves for this same structure, but with fixed remote displacement, AT, and a finitesystem compliance, CM- The structure is unstable at AT(4) in this case.

Figure 9.20 illustrates the load-displacement curve for this hypothetical structure. Amaximum load plateau occurs at Pj and 43- and the load decreases with further displace-ment. In load control, the structure is unstable at Pj, because the load cannot increasefurther. The structure is always stable in pure displacement control (Qv/ = 0), but is un-stable at A4 (and A?(4) -A4 + CM PJ) for the finite compliance case.

Figure 9.21 is a schematic stability assessment diagram. The applied and materialtearing modulus are plotted against J and JR, respectively. Instability occurs when theTapp-J curve crosses the TR-}R curve. The latter curve is relatively easy to obtain, sinceJR depends only on the amount of crack growth:

JR = jR(a-a0) (9.51)


Thus there is a unique relationship between TR and //?, and the TR-JR curve can be de-fined unambiguously. Suppose, for example, that the J-R curve is Fit to a power law:

R

CRACK SIZE

FIGURE 9.19 Schematic driving force diagram for a fixed remote displacement. Refer to Fig. 3.20 forthe corresponding diagram for pure load control and displacement control.

LOAD

A2 AS A4

DISPLACEMENT

FIGURE 9.20 Schematic load-displacement curve for the material in Fig. 9.19.

492 Chapter 9

1 app» 1R

J INTEGRAL

FIGURE 921 Schematic stability assessment diagram for the material in the two previous figures.

(9.52)

The material tearing modulus is given by

— —^- Co Ci 2 /C2-l

/? Co (9.53)

The applied tearing modulus curve is less clearly defined, however. There are anumber of approaches for defining the Tapp-J curve, depending on the application. Figure9.21 illustrates two possible approaches, which are discussed below.

Suppose that the initial crack size, a0, is known, and one wishes to determine theloading conditions (P, A, and A7) at failure. In this case, the Tapn should be computedat various points on the R curve. Since J=JR during stable crack growth, the applied Jat a given crack size can be inferred from the J-R curve (Eq. (9.50). The remote displace-ment, AT, increases as the loading progresses up the J-R curve (see Figs. 9.19 and 9.21);instability occurs at AT(4)- The final load, local displacement, crack size, and stablecrack extension can be readily computed, once the critical point on the J-R curve has beenidentified.

The Tapp-J curve can also be constructed by fixing one of the loading conditions (P,A, or AJ), and determining the critical crack size at failure, as well as ao. For example,if we fix AT at AT(4) in the structure, we would predict the same failure point as the pre-vious analysis but the Tapp-J curve would follow a different path (Fig. 9.21). If, how-ever, we fix the remote displacement at a different value, we would predict failure at an-


other point on the TR-JR curve; the critical crack size, stable crack extension, and ao

would be different from the previous example.

9.5.6 Some Practical Considerations

If the material is sufficiently tough or if crack-like flaws in the structure are small, thestructure will not fail unless it is loaded into the fully plastic regime. When performingfracture analyses in this regime, there are a number of important considerations that manypractitioners overlook.

In the fully plastic regime, the /integral varies with pn+l; a slight increase in loadleads to a large increase in the applied J. The J versus crack length driving force curvesare also very steep in this regime. Consequently, the failure stress and critical crack sizeare insensitive to toughness in the fully plastic regime; rather, failure is governed by theflow properties of the material. The problem is reduced to a limit load situation, wherethe main effect of the crack is to reduce the net cross section of the structure.

Predicting failure stress or critical crack size under fully plastic conditions need notbe complicated. A detailed tearing instability analysis and a simple limit load analysisshould lead to similar estimates of failure conditions.

Problems arise, however, when one tries to compute the applied J at a given loadand crack size. Since J is very sensitive to load in the fully plastic regime, a slight errorin P produces a significant error in the estimated /. For example, a 10% overestimate inthe yield strength, cr0, will produce a corresponding error in Po, which will lead to an un-derestimate of /by a factor of 3.2 for n = 10. Since flow properties typically vary byseveral percent in different regions of a steel plate, and heat-to-heat variations can be muchlarger, accurate estimates of the applied J at a fixed load are virtually impossible.

If estimates of the applied / are required in the fully plastic regime, the displace-ment, not the load, should characterize conditions in the structure. While the plastic / is

proportional to Pn+1, Jpi scales with Ap /n, according to Eqs. (9.31) and (9.33).Thus a J-A plot is nearly linear in the fully plastic regime, and displacement is a muchmore sensitive indicator of the applied / in a structure. Figure 9.22 compares J-P and J-Aplots for a center cracked panel with three strain hardening exponents.

Recall Section 3.3, where the empirical correlation of CTOD and wide plate datathat resulted in the CTOD design curve was plotted in terms of strain (i.e., displacementover a fixed gage length) rather than stress [18,20], A correlation based on stress wouldnot have worked, because the failure stresses in the wide plate specimens were clusteredaround the flow stress of the material.

9.6 THE REFERENCE STRESS APPROACH

The EPRI equations for fully plastic /, Eqs. (9.31) and (9.34), assume that the material'sstress-plastic strain curve follows a simple power law. Many materials, however, haveflow behavior that deviates considerably from a power law. For example, most low car-bon steels exhibit a plateau in the flow curve immediately after yielding. Applying Eq.

494 Chapter 9

1000

« 800fi

i 1 1 r

1000

CENTER CRACKED PANELPlane Stress, a/W = 0.25

o n = 5o n = 10

= 20

0.5 1

LOAD, MN(a) / versus load.

CENTER CRACKED PANELPlane Stress, a/W = 0.25

o n = 5

n = 10

n = 20

1 2

LOAD LINE DISPLACEMENT, mm

(b) / versus load line displacement.

FIGURE 9.22 Comparison of J -load and /-displacement curves for a center cracked panel. (W = 50mm, B = 25 mm, L = 400 mm, aa = 420 MPa, e0 = 0.002, a = 1.0.)

TIB


(9.31) or (9.34) to such a material, results in significant errors. Ainsworth [34] modifiedthe EPRI relationships to reflect more closely the flow behavior of real materials. He de-fined a reference stress as follows:

Gref=(PIP0}G0 (9.54)

He further defined the reference strain as the total axial strain when the material is loadedto a uniaxial stress of oref. Substituting these definitions into Eq. (9.31) gives

reaJpl = VrefbhA eref -- J— (9.55)

For materials that obey a power law, Eq. (9.55) agrees precisely with Eq. (9.31), but theformer is more general, in that it is applicable to all types of stress-strain behavior.

Equation (9.55) still contains h^ the geometry factor which depends on the powerlaw hardening exponent n. Ainsworth proposed redefining Po for a given configuration toproduce another constant, h]', that is insensitive to n. He noticed, however, that evenwithout the modification of P0, h] was relatively insensitive to n except at high n values(low hardening materials). Ainsworth was primarily interested in developing a drivingforce procedure for high hardening materials such as austenitic stainless steels. The stripyield failure assessment diagram was considered suitable for low hardening materials. Heproposed the following approximation.

(9.56)

where hj(n) is the geometry constant for a material with a strain hardening exponent of nand h](l) is the corresponding constant for a linear material. By substituting hj(l) intoEq. (9.31) (or (9.34)), Ainsworth was able to relate the plastic J to the linear elasticstress intensity factor:

're/Jpl =

where fJL = 0.75 for plane strain and p. = 1.0 for plane stress.Ainsworth's work has important ramifications. When applying the EPRI approach,

one must obtain a stress intensity solution to compute the elastic J, and a separate solu-tion for HI in order to compute the plastic term. The hj constant is a plastic geometrycorrection factor. However, Eq. (9.57) makes it possible to estimate Jpl from an elasticgeometry correction factor. The original EPRI Handbook [27] and subsequent additions[28-30] contain h] solutions for a relatively small number of configurations, but there arehundreds of stress intensity solutions in handbooks and the literature. Thus Eq. (9.57) is

496 Chapter 9

not only simpler than Eq. (9.31), but also more widely applicable. The relative accuracyof Ainsworth's simplified equation is examined in Section 9.7.

Ainsworth made additional simplifications and modifications to the reference stressmodel in order to express it in terms of a failure assessment diagram. This FAD has beenincorporated into a revision of the R6 procedure (see Section 9.4). The new documentalso contains more accurate procedures for analyzing secondary stresses. The revised R6approach still permits application of the strip yield FAD to low hardening materials.

The reference stress FAD has also been included in the revised PD 6493 procedure,which was published in 1991. Both the revised R6 and PD 6493 approaches are broadlysimilar, and are discussed in Section 9.8.

9.7 COMPARISON OF DRIVING FORCE EQUATIONS

The primary advantage of Ainsworth's reference stress approach is in accounting for thegeometry of a cracked structure through a linear elastic stress intensity solution. The hifactor is replaced by an LEFM geometry factor. The other contribution of Ainsworth'sanalysis, the generalization to stress-strain laws other than power-law, is of secondaryimportance.

In most cases, the EPRI procedure and Ainsworth's simplified approach producenearly identical estimates of critical flaw size and failure stress. This section presents theresults of a parametric study of the accuracy of Ainsworth's approach relative to that ofthe EPRI procedure. The relative accuracy of the strip yield model was also evaluated. Apower law hardening material was assumed for all analyses, since the main purpose ofthis exercise was to evaluate the errors associated with the LEFM geometry correction fac-tor in the elastic-plastic regime.

For a power-law material, the Ainsworth model gives the following expression forthe total J:

E(9.58)!

if the Irwin plastic zone correction is neglected. Since stress intensity is proportional toload, this relationship has the form :

/ = CaP2 + CpPn+l \

The first term dominates under linear elastic conditions; the second term dominates underfully plastic conditions. The EPRI approach (Eqs. (9.31) and (9.34)) has the same formjThe only difference between the EPRI equation and Eq. (9.58) is the value of the constantCfi; the equations agree precisely in the linear elastic range. Thus any discrepancies be-;tween the two approaches are observed only when the plastic term is significant. !

Fnl


Since load in the fully plastic range is insensitive to the applied J (Section (9.5.6),the predicted failure stress is insensitive to the differences between the EPRI approach andthe Ainsworth model. The latter approach assumes that the geometry factor, h](n), isequal to the linear elastic value, h](J). Errors in / that result from applying Eq. (9.57)are proportional to the ratio hj(n)/hi(l), which is plotted against n in Fig. 9.23 for a cen-ter cracked panel in plane strain with a/W - 0.75. Note that the HI ratio in this configu-ration is sensitive to the hardening exponent. Thus Eq. (9.57) leads to significant errorsin J, particularly at high n values. However, when the hj ratio is raised to the power

/(n+l)> it *s insensitive to n. This latter ratio is indicative of the differences in the pre-dicted failure stress between the Ainsworth and EPRI approaches.

A design engineer often wishes to use a fracture mechanics analysis to estimate thecritical flaw size at a given applied stress. To determine the sensitivity of critical flawsize estimates to the driving force equation, the author [35] performed a series of calcula-tions with the EPRI, Ainsworth, and strip yield models on center cracked panels and edgecracked bend specimens. The material was assumed to follow perfectly the power law ex-pression (Eq. (9.27)) for stress versus plastic strain. The constants a and eo were fixed1.0 and 0.002, respectively, for all n values; thus a0 corresponds exactly to the 0.2% off-set yield strength.

Two separate strip yield analyses were performed for each case: one assumed thatthe collapse stress was equal to the yield strength; the other based collapse on the flowstress, defined as the average of yield and tensile strengths. For a material whose truestress-true strain curve follows a power law, the flow stress can be estimated from the fol-lowing expression:

1.2

O 0-8e

0.4

0.2

CENTER CRACKED PANELa/W = 0.75

6 8 10 12 14 16HARDENING EXPONENT (n)

18 20

FIGURE 9.23 The effect of hardening exponent on the hi factor for a center cracked panel with a/W =0.75. The hj ratio raised to the 1/n+l power is indicative of the ratio of predicted failure stresses fromthe EPRI and reference stress approaches.

498 Chapter 9

+ 0.002;exp(AO

(9.59)

where N = 1/n. Equation (9.59) was derived by solving for the tensile instability point inEq. (9.27) and converting true stress and strain to engineering values.

Figure 9.24 shows typical results from this analysis. The three driving force equa-tions are applied to a center cracked panel with n = 5 and n = 10. Critical crack size,normalized by W, is plotted against critical J, normalized by width and yield strength.The nominal stress (P/BW) is fixed at 2/3 yield. At low toughness levels, all predictionsagree because linear elastic conditions prevail. At high toughness levels, the curves arerelatively flat, indicating that critical crack size is insensitive to toughness. In this re-gion, failure is controlled primarily by plastic collapse of the remaining cross section.The EPRI and Ainsworth equations agree well at all hardening rates.

The strip yield model is nonconservative for the high hardening material (Fig.9.24(a)) when it is based on the flow stress. For n = 10, however, the strip yield model,with collapse defined at Gfioyv, gives a good approximation of the other two curves; theagreement is even better at high n values. When the strip yield model is based on collapseat <jys it is always conservative.

This analysis indicates that the Ainsworth model can predict either critical crack sizeor failure stress in the elastic, elastic-plastic, and fully plastic regimes. The strip yieldmodel gives reasonable results for low hardening materials.

9.8 THE PD 6493 METHOD

The original PD 6493 approach, published in 1980 by the British Standards Institution[21], was based on the CTOD design curve. This methodology suffers from a number ofshortcomings. For example, the driving force equation is mostly empirical and has avariable level of conservatism. Improved driving force equations became available withthe R6 and EPRI procedures, but the CTOD design curve had already been widely acceptedby the welding fabrication industry in the United Kingdom and elsewhere. Many;engineers were reluctant to discard PD 6493:1980 because structures analyzed with the oldapproach might have to be re-analyzed if the a method rendered the 1980 version obsolete.

The conflicting goals of improving PD 6493 and maintaining continuity with thepast have been largely satisfied in the 1991 edition of this procedure [36], which utilizes a;three-tier approach. The three tier philosophy assesses fracture problems at a level of;complexity and accuracy appropriate for the situation. All three levels of PD 6493 areexpressed as failure assessment diagrams. Level 1 is consistent with the CTOD design!curve approach; Level 2 utilizes a strip yield model and Level 3 is based on the reference;stress approach. 1


lcrit

W

0.7

0.6 -

0.5

0.4

0.3 -

0.2

0.1

CENTER CRACKED PANEL, n=5

= o.67

EPRI Equation

- Ainsworth Model

Strip Yield Model

0.005 0.01

Jcrit

W a<

0.015 0.02

(a) n = 5

CENTER CRACKED PANEL, n=10

<J/a0 = 0.67

— — Based on Flow Stress

Based on Yield Strength

EPRI Equation

— - Ainsworth Model

Strip Yield Model

0.02

(b) « = 10

FIGURE 9.24 Comparison of predicted critical crack lengths from three driving force equations.

500 Chapter 9

9.8.1 Level 1

Level 1 is consistent with the CTOD design curve in the 1980 version of PD 6493. Themain differences are that the equations are expressed in terms of a failure assessment dia-gram, and an explicit collapse analysis is included. Level 1, which is conservative, is in-tended as a screening tool.

If Kjc data are used (or equivalent K values from / data), the K ratio is defined by Eq.

(9.21). For CTOD data, Kr is replaced by V<5» defined as

(9.60)

where §j is the applied CTOD obtained from a modified form of the CTOD design curve:

2

_ for 0-j / 0YS < 0.5 (9.61a)crYSEK

—

where &i/E= e/, the maximum membrane strain defined in Eq. (9.19). Recall that ej(and thus ffj) takes residual stresses, bending stresses, and stress concentradons into ac-count by assuming that the maximum value of the total stress acts uniformly through thecross section. Unlike Eq. (9.18a), the above expression does not include a safety factor oftwo on crack size. In the revised approach, this safety factor is included in the formula-

tion of the FAD, which is a horizontal line at ^~5~r = 1/Y2 for (Jj/ayS ^ 0-5- TheLevel 1 failure assessment diagram is illustrated in Fig. 9.25. For higher stress levels,the assessment line is defined from the empirical portion of the CTOD design curve:

K2 ( cs YY n \Sj = -=I—\ -\\-^l — 0.25 for 0i / GYS > 0.5 (9.61b)

The influence of Eq. (9.6 Ib) on the FAD is illustrated in Fig. 9.25. The revised CTODdesign curve contains a conservative collapse check in the form of a maximum stress ra-tio, Sr. For Level 1, Sr is defined as

(9.62)

where <jn is the effective primary net section stress and <Jfl0w *s t"le ôw stress> definedas (<JYS + GTSyZ or 1 -^ GYS, whichever is less. As Fig. (9.25) indicates, the Level 1approach is restricted to 0.8 Sy because Eq. (9.61) can be nonconservative near limit load[37].


9.8.2 Level 2

Level 2 utilizes a strip yield failure assessment diagram. The assessment equation is iden-tical to the original R6 relationship (Eq. (9.23)), except that it allows CTOD based analy-ses:

•In sec —, (9.63)

Figure 9.25 compares the Level 1 and Level 2 failure assessment diagrams. Note that theLevel 1 FAD is always conservative compared to the Level 2 method.

The treatment of stress concentration effects and secondary stresses is more complexin the upper two levels. The procedure recommends that accurate stress intensity solu-tions be obtained for the actual primary and secondary stress distributions. If this is notfeasible, an approximate solution can be obtained by linearizing the stress distribution andseparating the stresses into bending and membrane components, as discussed in Section9.1.1 For example, consider a surface crack of depth a. If the primary and secondarystresses are resolved into bending and membrane components, the approximate stress in-tensity factors are computed from the following expressions:

0.2 0.4 0.6 0.8

FIGURE 9.25 Failure assessment diagram for Levels 1 and 2 of the PD 6493 method.

502 Chapter 9

(9.64a)

(9.64b)

where Q is the flaw shape parameter, Pm and Pfo are the primary membrane and bendingstresses, Sm and S^ are the secondary stresses, and F and H are constants obtained fromthe Newman and Raju stress intensity solutions [10], which are given in Section 12.2.If, as in many cases, the actual distribution of secondary stresses is unknown, one shouldassume that S acts uniformly across the section. The British Standards document recom-mends that S be assumed to equal the material's yield strength in the case of as- weldedcomponents. For thoroughly stress relieved weldments, the estimate of S can be reducedto 30% of yield parallel to the weld and 15% of yield transverse to the weld.

The total Kj is the sum of the primary and secondary contributions. For assess-ments based on CTOD, Sj is estimated from Kj by assuming plane stress conditions:

(9-65)

There is a plastic interaction between primary and secondary stresses that must be takeninto account in Level 2. This is achieved with the correction factor, p, based on the workof Ains worth [38]. The applied toughness ratios for the structure are given by

(9.66)

and

(9.67)

The procedure for determining p is as follows:

p— P\ —— < 0.8 (9.68a)


p = 4ft 1.05--S&- 0.8 <-3=-< 1.05 (9.68b)

-^->1.05 (9.68c)

where

pl = 0 ^ < 0 (9.68d)

ft = 0.1#a714 - 0.007£2 + 0.00003/5 0 < % < 5.2 (9.68e)

p!=0.25 ^>5.2 (9.681)

where

-^ (9.68g)

When a structure is loaded by primary stresses, a portion of the residual stresses arerelieved by plastic strain. A simple way to model this mechanical stress relief is to as-sume that the sum of the primary and residual stresses cannot exceed the flow stress. Foryield magnitude residual stresses in the unloaded state, the revised PD 6493 approachpermits the user to incorporate the benefits of mechanical stress relief as follows:

(9.69)

where &R is the residual stress which is used to compute Kf.The stress ratio, Sr, is defined as the ratio of the effective net section stress to the

flow stress, as in Level 1. Refer to Fig. 9.24, which shows that the strip yield model isnonconservative for high hardening materials; restricting the assumed flow stress in Sr to1.2 ays prevents the nonconservatism that is seen in Fig. 9.24(a).

When the point defined by Eqs (9.62), (9.66) and (9.67) falls inside of the Level 2assessment line (Eq. (9.63)), the structure is considered safe.

504 Chapter 9

9.8.3 Level 3

The Level 3 failure assessment diagram is based on Ainsworth' s reference stress approach[34]5. The FAD is related to the material's stress-strain behavior:

2

The above quantity is plotted against the load ratio, Lr, defined as

°YS

Note that the reference stress and the effective net section stress are equivalent. Since theload ratio is defined in terms of the yield strength rather than the flow stress, Lr can begreater than 1 . The load ratio cannot exceed Oflow/<JYS > where Oflow is defined as theaverage between yield and tensile strengths. For Level 3, the alternate definition of flowstress (<Jfl0w = 1-2 <Jys ) does not apply. For Lr > OflOw/GYS ,Kr=Q.

If the stress-strain curve for the material is not available, such as would be the casewhen analyzing a flaw in a weld heat affected zone, the following FAD equation can beapplied at Level 3:

(9.72)

This expression also has a cut-off at Lr = Oflow/OYS • This alternate FAD requires aknowledge of only the yield and tensile strengths of the material, but this relationship canbe excessively conservative. For many materials, Eq. (9.72) is more conservative thanthe Level 2 FAD. Figure 9.26 is a plot of Eq. (9.72). Note that the upper cut-off on Lr

depends on the hardening characteristics of the material.The Level 3 analysis of Kr (or 5r ) for the structure is identical to the Level 2

procedures (Eqs. (9.65) to (9.68)), but Level 3 includes guidelines for ductile instabilityand tearing analysis.

When the original reference stress equation is expressed in terms of a failure assessment diagram, theresulting diagram is geometry dependent, much like the EPRI approach (Fig. 9.18). Equation (9.70), which isgeometry independent, is apparently a semi-empirical approximation of a range of geometries.


Kr, "^r = (l - 0.14 L r2) [o.3 + Q.7 exp (-0.65 Li6)]

FIGURE 9.26 Optional Level 3 failure assessment diagram, for cases where a stress strain curve is notavailable.

9.9 THE R6 METHOD

The current version of the R6 approach [36], the fracture analysis of nuclear powerindustry in the United Kingdom, bears a slight resemblance to the revised PD 6493method. Both methods utilize failure assessment diagrams, and both have adopted thereference stress model as an option. Also, both approaches contain three levels ofassessment, although the details at each level differ for the two documents. Spaceconsiderations preclude describing the minute details of R6 assessments; a brief overviewis given below.

The R6 method contains three options, which are analogous to the three levels inPD 6493. The appropriate option depends on the available data and the desired accuracy.

Option 1 uses the lower-bound FAD defined by Eq. (9.72). This option is appropri-ate when the relevant stress-strain data are not available.

The Option 2 FAD is based on the reference stress model. Thus it is necessary tohave access to the stress-strain curve for the material in question. The failure assessmentdiagram for Option 2 is given by

_ EerefT for Lr < (9.73)

506 Chapter 9

Note that Eq. (9.73) differs from the reference stress FAD that is used in Level 3 of PD6493 (Eq. (9.70)).

Option 3 provides the most accurate analysis. The Option 3 FAD is inferred from aJ integral solution for the structure of interest. Normally, such a solution would requirean elastic-plastic finite element analysis that incorporates the stress-strain response of thematerial of interest.

Within each option, there are three categories of analysis:

Category 1: Fracture initiation.

Category 2: Limited stable crack growth.

Category 3: Tearing instability.

The appropriate category depends on the intent of the analysis. For example, designwould normally be based on avoiding fracture initiation (Category 1) under normal condi-tions, but analyses of potential accidents may consider stable and unstable crack growth(Categories 2 and 3).

9.10 PROBABILISTIC FRACTURE MECHANICS

Most fracture mechanics analyses are deterministic; i.e., a single value of fracture tough-ness is used to estimate failure stress or critical crack size. Much of what happens in thereal world, however, is not predictable. Since fracture toughness data in the ductile-brittletransition region are widely scattered, it is not appropriate to view fracture toughness as asingle-valued material constant. Other factors also introduce uncertainty into fractureanalyses. A structure may contain a number of flaws of various sizes, orientations andlocations. Extraordinary events such as hurricanes, tidal waves and accidents can result instresses significantly above the intended design level. Because of these complexities, frac-ture should be viewed probabilistically rather than deterministically.

Figure 9.27 is a schematic probabilistic fracture analysis for the case of a linearelastic structure. The curve on the left represents the distribution of driving force in thestructure, while the curve on the right is the toughness distribution. The former distribu-tion depends on the uncertainties in stress and flaw size. When the distributions ofapplied Kj and Kjc overlap, there is a finite probability of failure, indicated by the shadedarea. For example, suppose the cumulative distribution of the driving force is Fj(Kf) andthe cumulative toughness distribution is F2(Kic). The failure probability, Pf, is given by

KJ (9.74)

Time-dependent crack growth, such as fatigue and stress corrosion cracking, can be takeninto account by applying the appropriate growth law to the flaw distribution. Flaw


growth would cause the applied KI distribution to shift to the right with time, therebyincreasing failure probability.

The overlap of two probability distributions (Fig. 9.27) represents a fairly simplecase. In most practical situations, there is randomness or uncertainty associated withseveral variables, and a simple numerical integration to solve for Pf (Eq. 9.74)) is notpossible. Monte Carlo simulation can estimate failure probability when there are multiplerandom variables. Such an analysis is relatively easy to perform, since it merely involvesincorporating a random number generator into a deterministic model. Monte Carloanalysis is very inefficient, however, as numerous "trials" are required for convergence.First-order reliability methods [40] are much more efficient and yield more informationthat Monte Carlo analysis, but involve relatively complicated numerical algorithms.

The mathematics of probabilistic analysis is well established. Reliability en-gineering is currently applied in a variety of circumstances, ranging from quality controlin manufacturing to structural integrity. Probabilistic fracture analyses are rare, however,because the input data are usually not available. Scatter in fracture toughness data is oneof the largest uncertainties in a fracture analysis. A probabilistic analysis traditionallyrequires performing a large number of fracture toughness tests to define the toughnessdistribution, but the fracture toughness master curve described in Section 7.8 can greatlyreduce the amount of testing required.

FREQUENCY Distribution ofApplied K[

STRESS INTENSITY FACTOR

FIGURE 9.27 Schematic probabilistic fracture analysis.

9.11 LIMITATIONS OF EXISTING APPROACHES

Section 9.7 compared three driving force equations: the EPRI procedure, the referencestress model, and the strip yield model. Each equation reduces to LEFM in the limit ofsmall scale yielding, and each approaches a collapse limit under fully plastic conditions.

508 Chapter 9

Although the strip yield model and the reference stress model contain simplifying as-sumptions, they predict similar results to the more advanced EPRI approach that incorpo-rates a fully plastic J analysis. Any errors in the simpler models, relative to the EPRIapproach, are overshadowed by more serious shortcomings in all existing methods.

Although some of the fracture design analyses discussed in this chapter are complex,they still do not take into account all aspects of the problem. These analyses are two-di-mensional and assume that the material is homogeneous. In addition, all fracture designanalyses contain an inherent assumption that the computed driving force parameter (K, J,or CTOD) uniquely characterizes crack tip conditions.

The items discussed below do not constitute an exhaustive list of unresolved issues,but are key areas which need to be understood better before fracture analysis methods canbe improved.

It should be noted that current methods of fracture analyses are generally safe. Theeffects discussed below tend to make current predictions conservative. A better under-standing of these complexities would merely make analyses of critical conditions moreaccurate.

9.11.1 Driving Force in Weldments

A steel weld invariably has different flow properties than the parent metal. In most cases,the yield strength of the weld metal overmatches that of the parent metal, although un-dermatching sometimes occurs. Analyses such as the EPRI approach are unable to handlestructures whose flow properties are heterogeneous. If a crack occurs in or near a weld, itis impossible to determine the driving force accurately without performing an elastic-plas-tic finite element analysis of the component.

9.11.2 Residual Stresses

The assumptions for secondary stresses have a significant effect on predictions with eitherthe R6 method or PD 6493, but accurate information on the distribution of residualstresses is rarely available for the weld in question.

Conventional methods for measuring the through-thickness distribution of residualstress are destructive. Material from one side of the welded plate is typically removed bya milling machine while strain gage readings on the other side of the plate are recorded.Such an approach is obviously impractical for a structure in service. The center holedrilling technique does minimal damage to the structure, but it provides information onlyon the surface stresses. The only available nondestructive method for through-thicknessresidual stress measurement is neutron diffraction, a technique that is not portable.

A reliable, portable, nondestructive method for measuring residual stresses is desper-ately needed. Accurate finite element models that predict the residual stress distributionfrom the joint geometry and welding procedure are also desirable.


9.11.3 Three-Dimensional Effects

Existing elastic-plastic analyses do not account for the variation of the driving force alongthe crack front. Both J and CTOD vary considerably along the tip of a crack, however.This effect is particularly pronounced in semielliptical surface flaws. Thus the crack tipconditions cannot be uniquely characterized with a single value of J or CTOD.

These three-dimensional effects influence both cleavage and ductile tearing. Sincecleavage is statistical in nature (Chapter 5), good predictions will come only from sum-ming the incremental failure probabilities along the crack front. Such a calculation musttake account of the variation in the crack driving force with position. Since ductile crackgrowth occurs fastest where the driving force is highest, accurate tearing predictions arepossible only with a three-dimensional analysis.

9.11.4 Crack Tip Constraint

Constraint is related to the three-dimensional issue. Plane strain fracture analyses assumethat / or CTOD uniquely characterizes crack tip stresses and strains. If the entire crackfront is not in plane strain, however, there are regions where a single parameter does notcharacterize crack tip conditions. Similarly, the single parameter assumption breaks downunder large scale plasticity, which decreases the crack tip constraint. This constraint losscan occur at very low J (or CTOD) values in structures with shallow flaws loadedpredominantly in tension (see Section 3.6). In such cases, the structure has a higherapparent toughness than the small-scale fracture toughness tests, which contain deepcracks and are loaded predominantly in bending.

REFERENCES

1. Rooke, D.P. and Cartwright, D.J., Compendium of Stress Intensity Factors. HerMajesty's Stationary Office, London, 1976.

2. Tada, H., Paris, P.C., and Irwin, G.R. The Stress Analysis of Cracks Handbook. DelResearch Corporation, Hellertown, Pa, 1973.

3. Murakami, Y. Stress Intensity Factors Handbook. Pergamon Press, New York, 1987.

4. Sanford, R.J. and Dally, J.W., "A General Method for Determining Mixed-Mode StressIntensity Factors from Isochromatic Fringe Patterns." Engineering Fracture Mechanics,Vol. 11, 1979, pp. 621-633.

5. Chona, R. , Irwin, G.R., and Shukla, A., "Two and Three Parameter Representation ofCrack Tip Stress Fields." Journal of Strain Analysis, Vol. 17, 1982, pp. 79-86.

6. Kalthoff, J.F., Beinart, J., Winkler, S., and Klernm, W., "Experimental Analysis ofDynamic Effects in Different Crack Arrest Test Specimens." ASTM STP 711, AmericanSociety for Testing and Materials, Philadelphia, 1980, pp. 109-127.

ill

510 Chapter 9

I. Raju, I.S. and Newman J.C., Jr., "Stress-Intensity Factors for Internal and ExternalSurface Cracks in Cylindrical Vessels." Journal of Pressure Vessel Technology, Vol.104, 1982, pp. 293-298.

8. Rice, J.R., "Some Remarks on Elastic Crack-Tip Stress Fields." International Journalof Solids and Structures, Vol. 8, 1972, pp. 751-758.

9. Rice, J.R., "Weight Function Theory for Three-Dimensional Elastic Crack Analysis."ASTM STP 1020, American Society for Testing and Materials, Philadelphia, 1989, pp.29-57.

10. Newman, J.C. and Raju, I.S., "Stress-Intensity Factor Equations for Cracks in Three-Dimensional Finite Bodies Subjected to Tension and Bending Loads." NASA TechnicalMemorandum 85793, NASA Langley Research Center, Hampton, VA, April 1984.

I I . Timoshenko, S., Strength of Materials, Advanced Theory and Problems. D. VanNostrand Company, New York, 1956.

12. Atluri, S.N. and Kathiresan, K., "3-D Analyses of Surface Flaws in Thick-Walled ReactorPressure Vessels Using Displacement-Hybrid Finite Element Method." NuclearEngineering and Design, Vol. 51, 1979, pp. 163-176.


14. Burdekin, F.M. and Stone, D.E.W., "The Crack Opening Displacement Approach toFracture Mechanics in Yielding Materials." Journal of Strain Analysis, Vol. 1, 1966,pp. 144-153.

15. Hayes, D.J. and Williams, J.G., "A Practical Method for Determining Dugdale ModelSolutions for Cracked Bodies of Arbitrary Shape." International Journal of FractureMechanics, Vol. 8, 1972, pp. 239-256.

16. American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code.Section XI: Rules for Inservice Inspection of Nuclear Power Plant Components.American Society of Mechanical Engineers, New York.

17. Marsdon, T.U., ed., "Flaw Evaluation Procedures: Background and Application ofASME Section XI, Appendix A." EPRI NP-719-SR, Electric Power Research Institute,Palo Alto, CA, 1978.

18. Burdekin, F.M. and Dawes, M.G., "Practical Use of Linear Elastic and Yielding FractureMechanics with Particular Reference to Pressure Vessels." Proceedings of the Instituteof Mechanical Engineers Conference, London, May 1971, pp. 28-37.

19. Wells, A.A., "Application of Fracture Mechanics at and Beyond General Yielding."British Welding Journal, Vol 10, 1963, pp. 563-570.

20. Dawes, M.G., "Fracture Control in High Yield Strength Weldments." Welding Journal,Vol. 53, 1974, pp. 369-380.

21. PD 6493:1980, "Guidance on Some Methods for the Derivation of Acceptance Levelsfor Defects in Fusion Welded Joints." British Standards Institution, March 1980.


22. Karnath, M.S., "The COD Design Curve: An Assessment of Validity Using Wide PlateTests." The Welding Institute Report 71/1978/E, September 1978.

23. Dowling, A.R. and Townley, C.H.A., "The Effects of Defects on Structural Failure: ATwo-Criteria Approach." International Journal of Pressure Vessels and Piping, Vol 3,1975, pp. 77-137.

24. Harrison, R.P., Loosemore, K., and Milne, I., "Assessment of the Integrity of StructuresContaining Defects." CEGB Report R/H/R6, Central Electricity Generating Board,United Kingdom, 1976.

25. Harrison, R.P., Loosemore, K., Milne, I, and Dowling, A.R., "Assessment of theIntegrity of Structures Containing Defects." CEGB Report R/H/R6-Rev 2, CentralElectricity Generating Board, United Kingdom, 1980.

26. Shih, C.F. and Hutchinson, J.W., "Fully Plastic Solutions and Large-Scale YieldingEstimates for Plane Stress Crack Problems." Journal of Engineering Materials andTechnology, Vol. 98, 1976, pp. 289-295.

27. Kumar, V., German, M.D., and Shih, C.F., "An Engineering Approach for Elastic-Plastic Fracture Analysis." EPRI Report NP-1931, Electric Power Research Institute,Palo Alto, CA, 1981.

28. Kumar, V., German, M.D., Wilkening, W.W., Andrews, W.R., deLorenzi, H.G., andMowbray, D.F., "Advances in Elastic-Plastic Fracture Analysis." EPRI Report NP-3607,Electric Power Research Institute, Palo Alto, CA, 1984.

29. Kumar, V. and German, M.D., "Elastic-Plastic Fracture Analysis of Through-Wall andSurface Flaws in Cylinders." EPRI Report NP-5596, Electric Power Research Institute,Palo Alto, CA, 1988.

30. Zahoor, A. "Ductile Fracture Handbook, Volume 1: Circumferential ThroughwallCracks." EPRI Report NP-6301-D, Electric Power Research Institute, Palo Alto, CA,1989.

31. Shih, C.F. "Relationship between the J-Integral and the Crack Opening Displacementfor Stationary and Extending Cracks." Journal of the Mechanics and Physics of Solids,Vol 29, 1981, pp. 305-326.

32. Bloom, J.M., "Prediction of Ductile Tearing Using a Proposed Strain Hardening FailureAssessment Diagram." International Journal of Fracture, Vol. 6., 1980, pp. R73-R77.

33. Shih, C.F., German, M.D., and Kumar, V., "An Engineering Approach for ExaminingCrack Growth and Stability in Flawed Structures." International Journal of PressureVessels and Piping, Vol. 9, 1981, pp. 159-196.

34. Ainsworth, R.A., "The Assessment of Defects in Structures of Strain HardeningMaterials." Engineering Fracture Mechanics, Vol. 19, 1984, p. 633.

35. Anderson, T.L., "Application of Elastic-Plastic Fracture Mechanics to WeldedStructures: A Critical Review." Mechanics and Materials Center Report MM 6165-89-1, Texas A&M University, College Station, TX, August 1989.

512 Chapter 9

36 PD 6493:1991, "Guidance on Some Methods for the Derivation of Acceptance Levelsfor Defects in Fusion Welded Joints." British Standards Institution, August 1991.

37. Anderson, T.L., Leggatt, R.H., and Garwood, S.J., "The Use of CTOD Methods inFitness for Purpose Analysis." The Crack Tip Opening Displacement in Elastic-PlasticFracture Mechanics, Springer-Verlag, Berlin, 1986, pp. 281-313.

38. Ainsworth, R.A., "The Treatment of Thermal and Residual Stresses in FractureAssessments." Central Electricity Generating Board Report TPRD/0479/N84, 1984.

39. Milne, I., Ainsworth, R.A., Dowling, A.R., and Stewart, A.T., "Assessment of theIntegrity of Structures Containing Defects." Central Electricity Generating BoardReport R/H/R6-Rev 3, May 1986.

40. Jutla, T., "Probabilistic Fracture Mechanics and Reliability Analysis: An Overview."Proceedings of the 30th Canadian Institute of Metallurgists Conference, Ottawa,Canada, August 18-22, 1991.

10. FATIGUE CRACK PROPAGATION

Most of the material in the preceding chapters has dealt with static or monotonic loadingof cracked bodies. This chapter considers crack growth in the presence of cyclic stresses.The focus is on fatigue of metals, but many of the concepts presented in this chapter ap-ply to other materials as well.

In the early 1960s, Paris, et al. [1,2] demonstrated that fracture mechanics is a usefultool for characterizing crack growth by fatigue. Since that time, the application of fracturemechanics to fatigue problems has become almost routine. There are, however, a numberof controversial issues and unanswered questions in this field.

The procedures for analyzing constant amplitude fatigue1 under small scale yieldingconditions are fairly well established, although a number of uncertainties remain.Variable amplitude loading, large scale plasticity, and short cracks introduce additionalcomplications that are not fully understood.

This chapter summarizes the fundamental concepts and practical applications of thefracture mechanics approach to fatigue crack propagation. Section 10.1 outlines the simil-itude concept, which provides the theoretical justification for applying fracture mechanicsto fatigue problems. This is followed by a summary of the more common empirical andsemiempirical equations for characterizing fatigue crack growth. Subsequent sections dis-cuss crack closure, variable amplitude loading, retardation, and growth of short cracks interms of the validity (or lack of validity) of the similitude assumption. The micromecha-nisms of fatigue are also discussed briefly. The final two sections are geared to practicalapplications; Section 10.7 outlines procedures for experimental measurements of fatiguecrack growth, and Section 10.8 summarizes the damage tolerance approach to fatigue safedesign. Appendix 10 at the end of this chapter addresses the applicability of the J integralto cyclic loading.

10.1 SIMILITUDE IN FATIGUE

The concept of similitude, when it applies, provides the theoretical basis for fracture me-chanics. Similitude implies that the crack tip conditions are uniquely defined by a singleloading parameter such as the stress intensity factor. In the case of a stationary crack, twoconfigurations will fail at the same critical K value, provided an elastic singularity zoneexists at the crack tip (Section 2.10). Under certain conditions, fatigue crack growth canalso be characterized by the stress intensity factor, as discussed below.

Consider a growing crack in the presence of a constant amplitude cyclic stress inten-sity (Fig. 10.1). A cyclic plastic zone forms at the crack tip, and the growing crackleaves behind a plastic wake. If the plastic zone is sufficiently small that it is embeddedwithin an elastic singularity zone, the conditions at the crack tip are uniquely defined by

In this chapter, constant amplitude loading is defined as a constant stress intensity amplitude rather thanconstant stress amplitude.

513

514 Chapter 10

the current K value2, and the crack growth rate is characterized by Kmin and Kmax. It isconvenient to express the functional relationship for crack growth in the following form:

daUN (10.1)

where AK = (Kmax - Kmin), R & Kmin/Kmax, and da/dN is the crack growth per cycle.The influence of the plastic zone and plastic wake on crack growth is implicit in Eq.(10.1), since the size of the plastic zone depends only on Km{n and Kmax.

A number of expressions for f j have been proposed, most of which are empirical.Section 10.2 outlines some of the more common fatigue crack growth relationships.Equation (10.1) can be integrated to estimate fatigue life. The number of cycles requiredto propagate a crack from an initial length, a0, to a final length, of, is given by

da(10.2)

If Kmax or Kmin vary during cyclic loading, the crack growth in a given cycle maydepend on the loading history as well as the current values of Km[n and Kmax:

da~dN

(10.3)

Kmax

Kmin

TIME

FIGURE 10.1 Constant amplitude fatigue crack growth under small scale yielding conditions.

^The justification for the similitude assumption in fatigue is essentially identical to the dimensional argumentfor steady state crack growth (Section 3.5.2 and Appendix 3.5.2). If the tip of the growing crack issufficiently far from its initial position, and external boundaries are remote, the plastic zone size and width ofthe plastic wake will reach steady state values.

Fatigue Crack Propagation 515

where Vindicates the history dependence, which results from prior plastic deformation.Equation (10.3) violates the similitude assumption; two configurations cyclically loadedat the same AK and R will not exhibit the same crack growth rate unless both configura-tions are subject to the same prior history.

Figure 10.2 illustrates several examples where the similitude assumption is invalid.In each case, prior loading history influences the current conditions at the crack tip.Section 10.4 discusses the reasons for history-dependent fatigue, and gives an example ofa model that accounts for loading history.

Fatigue crack growth analyses become considerably more complicated when priorloading history is taken into account. Consequently, equations of the form of Eq. (10.1)are applied whenever possible. It must be recognized, however, that such analyses areonly approximate in the case of variable amplitude loading.

Excessive plasticity during fatigue can violate similitude, since K no longer charac-terizes the crack tip conditions in such cases. A number of researchers [3,4] have appliedthe J integral to fatigue accompanied by large scale yielding; they have assumed a growthlaw of the form

— = /3(AJ,/?) (10.4)dN 3

where A/ is a contour integral for cyclic loading, analogous to the J integral for mono-tonic loading (see Appendix 10). Equation (10.4) is valid in the case of constant ampli-tude fatigue in small scale yielding, because of the relationship between J and K underlinear elastic conditions^. The validity of Eq. (10.4) in the presence of significant plastic-ity is less clear, however.

Recall from Chapter 3 that deformation plasticity (i.e., nonlinear elasticity) is an es-sential component of / integral theory. When unloading occurs in an elastic-plastic mate-rial, deformation plasticity theory no longer models the actual material response (see Fig.3.7). Consequently, the ability of the J integral to characterize fatigue crack growth inthe presence of large scale cyclic plasticity is questionable, to say the least.

There is, however, some theoretical and experimental evidence in favor of Eq.(10.4). If certain assumptions are made with respect to the loading and unloadingbranches of a cyclic stress-strain curve, it can be shown that AJ is path independent, and ituniquely characterizes the change in stresses and strains in a given cycle [5,6]. Appendix10 summarizes this analysis. Experimental data [3,4] indicate that AJ correlates crackgrowth data reasonably well in certain cases. Several researchers have found that CTODmay also be a suitable parameter for fatigue under elastic-plastic conditions [7].

o 9J AJ = AK^/E' in the case of small-scale yielding. Thus AJ cannot be interpreted as the range of applied Jvalues. That is, AJ & Jmax ' ^min m general. See Appendix 10 for additional background on thedefinition of AJ.

516 Chapter 10

Ki

TIME

(a) K increasing.

Ki

TIME

(b) K decreasing

Ki

TIME(c) Random loading.

FIGURE 10.2. Examples of cyclic loading that violate similitude.


The validity of Eq. (10.4) has not been proven conclusively, but this approach ap-pears to be useful for many engineering problems. Of course, Eq. (10.4) is subject to thesame restrictions on prior history as Eq. (10.1). The crack growth rate may exhibit a his-tory effect if AJ or R vary during cyclic loading.

10.2 EMPIRICAL FATIGUE CRACK GROWTH EQUATIONS

Figure 10.3 is a schematic log-log plot of da/dN versus AK, which illustrates typical fa-tigue crack growth behavior in metals. The sigmoidal curve contains three distinct re-gions. At intermediate AK values, the curve is linear, but the crack growth rate deviatesfrom the linear trend at high and low AK levels. In the former case, the crack growth rateaccelerates as Kmax approaches Kcrit, the fracture toughness of the material. At the otherextreme, da/dN approaches zero at a threshold AK; Section 10.3 explores the causes ofthis threshold.

The linear region of the log-log plot in Fig. 10.3 can be described by a power law:

da _ . T,m— = CAjrm (10.5)dN

where C and m are material constants that are determined experimentally. According toEq. (10.5), the fatigue crack growth rate depends only on AK; da/dN is insensitive to theR ratio in Region II.

Paris and Erdogan [2] were apparently the first to discover the power law relation-ship for fatigue crack growth in Region II. They proposed an exponent of four, whichwas in line with their experimental data. Subsequent studies over the past three decades,however, have shown that rn is not necessarily four, but ranges from two to seven forvarious materials. Equation (10.5) has become widely known as the Paris Law.

A number of researchers have developed equations that model all or part of the sig-moidal da/dN - AK relationship. Many of these equations are empirical, although someare based on physical considerations. Forman [8] proposed the following relationship forRegions II and III:

da CAKm

= ) (10.6)dN (l-K)Kcrit-&K

This equation can be rewritten in the following form:

da CAKm~l

= -jp (10.7)dN ^crit i

518 Chapter 10

LOG

AKthLOG AK

FIGURE 10.3 Typical fatigue crack growth behavior in metals.

Thus the crack growth rate becomes infinite as Kmax approaches Kcrif Note that theabove relationship accounts for R ratio effects, while Eq. (10.5) assumes that da/dN de-pends only on AK. Another important point is that the material constants C and m inthe Fonnan equation do not have the same numerical values or units as in the Paris-Erdogan equation (Eq. (10.5)).

Weertman [9] proposed an alternative semiempirical equation for Regions n and III:

da CM:dN K . 2 - K 2cnt max

(10.8)

This equation can be made more general with a variable exponent, m, on AK. Again, thefitting parameters, C and m, do not necessarily have the same values or units in the vari-ous crack growth equations.

Both the Fonnan and Weertman equations are asymptotic to Kmax = Kcrit, but nei-ther predicts a threshold. Klesnil and Lukas [10] modified Eq. (10.5) to account for thethreshold:

daUN (10.9)

Donahue [11] suggested a similar equation, but with the exponent, m, applied to thequantity (AK - AKth). m both cases, the threshold is a fitting parameter to be determined


experimentally. One problem with these equations is that AKfo often depends on the Rratio (see Section 10.3).

A number of equations attempt to describe the entire crack growth curve, taking ac-count of both the threshold and KCrii- For example, Priddle proposed the following em-pirical relationship:

AK-AK Y*(10.10)

Kcrit-Kmax

McEvily [12] developed another equation that can be fit to the entire crack growth curve:

da _. . T^ . _., . iC(AK-AKth) 1 + - - — - (10.11)

dN V Kcrit ~ ^max )

Equation (10.1 1) is based on a simple physical model rather than a purely empirical fit.Equations (10.5) to (10.11) all have the form of Eq. (10.1). Each of these equations

can be integrated to infer fatigue life (Eq. (10.2)). The most general of these expressionscontain four material constants:'* C, m, Kcrit and AKfh- For a given material, the fa-tigue crack growth rate depends only on the loading parameters AK and R, at least accord-ing to the Eqs. (10.5) to (10.12). Thus all of the preceding expressions assume elasticsimilitude of the growing crack; none of these equations incorporate a history dependence,and thus are strictly valid only for constant (stress intensity) amplitude loading. Many ofthese formula, however, were developed with variable amplitude loading in mind.Although there are situations where similitude is approximately satisfied for variable am-plitude loading, one must always bear in mind the potential for history effects. SeeSections 10.3 and 10.4 for additional discussion of this issue.

Dowling and Begley [3] applied the /integral to fatigue crack growth under largescale yielding conditions where K is no longer valid. They fit the growth rate data to apower law expression in AJ:

da _ . Tm- = CAJm (10.12)T X T ^dN

Appendix 10 outlines the theoretical justification and limitations of /-based approaches.

^The threshold stress intensity range, AKtfo, is not a true material constant since it usually depends on the Rratio (Section 10.3).

520 Chapter 10

EXAMPLE 10.1

Derive an expression for the number of stress cycles required to grow a semicircularsurface crack from an initial radius ao to a final size af, assuming the Paris-Erdoganequation describes the growth rate. Assume that ay is small compared to plate dimen-sions, and that the stress amplitude, Ac, is constant.

Solution: The stress intensity amplitude for a semicircular surface crack in an infiniteplate (Fig. 2.19) can be approximated by

AK = -==.Aayjui = 0.663

If we neglect the 0 dependence of A^.. Substituting this expression into Eq. (10.5)gives

~ = c(o.6634o) m (K a)m/2

which can be integrated to determine fatigue life:

7N = — - —n - — la~m/2 daJ

a0*~""* - fl^-^/2

(j- lJ(o.(form #2)

Closed-form integration is possible in this case because the K expression is rela-tively simple. In most instances, numerical integration is required.

10.3 CRACK CLOSURE AND THE FATIGUE THRESHOLD

Soon after the so-called Paris law (Eq. (10.5)) gained wide acceptance as a predictor of fa-tigue crack growth, many researchers came to the realization that this simple expressionwas not universally applicable. As Fig 10.3 illustrates, a log-log plot of da/dN v. AK issigmoidal rather than linear when crack growth data are obtained over a sufficiently wide


range. Also, the fatigue crack growth rate exhibits a dependence on the R ratio, particu-larly at both extremes of the crack growth curve. As discussed in the previous section,the R ratio effects at the upper end of the curve can be explained in terms of the interac-tion between fatigue and ultimate failure at or near Kc. This section addresses the behav-ior at the lower end of the da/dN - AK curve.

A discovery by Elber [13] provided at least a partial explanation for both the fatiguethreshold and R ratio effects. He noticed an anomaly in the elastic compliance of severalfatigue specimens, which Fig. 10.4(a) schematically illustrates. At high loads, the com-pliance (dA/dP) agreed with standard formulas for fracture mechanics specimens (seeChapters 7 and 12), but at low loads, the compliance was close to that of an uncrackedspecimen. Elber believed that this change in compliance was due to the contact betweencrack surfaces (i.e., crack closure) at loads that were low but greater than zero.

Elber postulated that crack closure decreased the fatigue crack growth rate by reduc-ing the effective stress intensity range. Figure 10.4(b) illustrates the closure concept.When a specimen is cyclically loaded at Kmax and Kmin, the crack faces are in contactbelow KOp, the stress intensity at which the crack opens. Elber assumed that the portionof the cycle that is below K0p does not contribute to fatigue crack growth. He defined aneffective stress intensity range as follows:

LOAD

DISPLACEMENT

(a) Load-displacement behavior.

Kmax

FIGURE 10.4 Crack closure during fatigue crackgrowth. The crack faces contact at a positiveload (a), resulting in a reduced driving force forfatigue,

TIME

(b) Definition of effective stress intensity range.

522 Chapter 10

K0p (10.13)

He also introduced an effective stress intensity ratio:

Elber then proposed a modified Paris-Erdogan equation:

(10.15)dN

Equation (10.15) has been reasonably successful in correlating fatigue crack growth data atvarious R ratios.

Since Elber' s original study, numerous researchers have confirmed that crack closuredoes in fact occur during fatigue crack propagation. Suresh and Ritchie [ 14] identified fivemechanisms for fatigue crack closure, which are illustrated in Fig. 10.5

Plasticity-induced closure, Fig. 10.5(a), results from residual stresses in the plasticwake. Budiansky and Hutchinson [15] applied the Dugdale-Barenblatt strip yield model tothis problem and showed that residual stretch in the plastic wake causes the crack faces toclose at a positive remote stress. Although quantitative predictions from the Budianskyand Hutchinson model do not agree with experimental data [16], this model is useful fordemonstrating qualitatively the effect of plasticity on crack closure. Recently, several in-vestigators [17,18] have studied plasticity-induced closure with finite element analysis.

Roughness-induced closure, which is illustrated in Fig. 10.5(b), is influenced by themicrostructure. Although fatigue cracks propagate in pure Mode I conditions on a globalscale, crack deflections due to microstructural heterogeneity can lead to mixed mode condi-tions on the microscopic level. When the crack path deviates from the Mode I symmetryplane, the crack is subject to Mode II displacements, as Fig. 10.5(b) illustrates. Thesedisplacements cause mismatch between upper and lower crack faces, which in turn resultsin a positive closure load. Coarse-grained materials usually produce a higher degree of sur-face roughness in fatigue, and correspondingly higher closure loads [19]. Figure 10.6 il-lustrates the effect of grain size on fatigue crack propagation in 1018 steel. At the lowerR ratio, where closure effects are most pronounced (see below), the coarse grained materialhas a higher AKfh, due to a higher closure load that is caused by greater surface roughness(Fig. 10.6(b)). Note that grain size effects disappear when the data are characterized by

Fatigue Crack Propagation

(a) Plasticity-induced closure.

Mode IX Displacement

(b) Roughness-induced closure.

(c) Oxide-induced closure.

523

(d) Closure induced by a viscous fluid.

(e) Transformation-induced closure.

FIGURE 10.5 Fatigue crack closure mechanismsin metals [14].

524 Chapter 10

Oxide-induced closure, Fig. 10.5(c), is usually associated with an aggressive envi-ronment. Oxide debris or other corrosion products become wedged between crack faces.

Crack closure can also be introduced by a viscous fluid, as Fig. 10.5(d) illustrates.The fluid acts as a wedge between crack faces, somewhat like the oxide mechanism.

A stress induced martensitic transformation at the tip of the growing crack can resultin a process zone wake5, as Fig. 10.5(e) illustrates. Residual stresses in the transformedzone can lead to crack closure.

The relative importance of the various closure mechanisms depends on microstrac-ture, yield strength, and environment.

7M

in"510

10"6

> io"7

10-°10'9

10 -10

1 1 1 f 1 i -- AISI 1018 Steel -

n- A A Coarse Grain ~L— a n Fine Grain c^y A __

- T HR = 0.1 rcP *A I

_ 1 — R = 0 7 î- dSJ^1 ~" J^^ "r jp!? i

djrb ^

"f A -Ik D4 1 1 fe 1 i 1

Sfi 6p

ft. ChH if

o1 4

3

1 1 1 1 |

R = 0.1~ D Fine Grain ê

ACoarse Grain \.^"^^S>N*X^ A

"" ^Â —

1 IT-I 1 1 1

6 7 8 9 10 11 12

AK^MPam172

(b) Qosure measurements.

0)

U*

•7

4 6 8 10 20 40 60 5TO

AK^MPam172 ^ lo'10

1 1 • | - i

R-0.1

mFine Grain^Coarse Grain

i i i i 1 1 . 1

(a) Uncorrected data. 1/9AKeff, MPa m

(c) Corrected data.

FIGURE 10.6 Effect of grain size on fatigue crack growth in mild steel [19].

See Section 6.2.2 for a discussion of stress-induced martensitic transformations in ceramics.


10.3.1 A Simplistic View of Closure and A K t h

Crack closure reduces the fatigue crack growth rate and introduces a threshold. It is possi-ble to estimate the effects of fatigue crack closure by making two simplifying assump-tions:

(1) The opening stress intensity, K0p, is a material constant; i.e. it is independentof Kmin, Kmax, and prior history.

(2) There is no intrinsic AKth f°r me material; Eq. (10.15) applies as long asAKeff > 0.

Although neither assumption is valid for real materials (Section 10.3.2 and 10.3.3), theintent of the present exercise is merely to illustrate the consequences of closure, indepen-dent of other factors.

The Elber crack growth expression (Eq. (10.15)) can be written in the followingform:

(10.16)dN

where U is defined by Eq. (10.14) for Kmin < KOp. When Kmin >K0p, U = 1, and clo-sure does not influence the results; the crack growth law reduces to the Paris-Erdoganequation in this latter case.

Rewriting Eq. (10.14) in terms of AST and R gives

l-R AK

The threshold can be inferred by setting U = 0:

AKth=Kop(l-R) (10.18)

Equation (10.17) applies to the following range of AK values:

1

Figure 10.7 is a nondimensional plot of Eq. (10.16) for various R values, with U givenby Eq. (10.17). The curves in this figure exhibit typical Region I and Region II behav-ior. The threshold stress intensity range, AKth, decreases with increasing R ratio, but thepredicted fatigue crack growth rate in the Paris law regime is insensitive to R.

526 Chapter 10

100 E

0.0010.1

FIGURE 10.7 Nondimensional crack growth curves computed from Eqs. (10.16) and (10.17).

Figure 10.8 shows experimental data for a mild steel [7], which exhibit the same trends asFig. 10.7. Thus this simple analysis predicts behavior that is qualitatively consistentwith experiment.

10.3.2 Effects of Loading Variables on Closure

The stress intensity for crack closure is not actually a material constant, but depends on anumber of factors. Elber measured the closure stress intensity in 2023-T3 aluminum atvarious load levels and R ratios, and obtained the following empirical relationship:

= 0.5 +OAR (-0.1 </?<0.7) (10.19)

Subsequent researchers [20-22] inferred similar empirical expressions for other alloys.According to Eq. (10.19), U depends only on R. Shih and Wei [23,24], however,

argued that the Elber expression, as well as many of the subsequent equations, are over-simplified. Shih and Wei observed a dependence on Kmax for crack closure in a T1-6A1-4V titanium alloy. They also showed that experimental data of earlier researchers, whenreplotted, exhibits a definite Kmax dependence.

There has been a great deal of confusion and controversy about the Kmax dependenceof U. Hudak and Davidson [16] cited contradictory examples from the literature wherevarious researchers reported U to increase, decrease, or remain constant with increasing

Hudak and Davidson performed closure measurements on a 7091 aluminum alloy

IFatigue Crack Propagation 527

and 304 stainless steel over a wide range of loading variables. For both materials, theyinferred a closure relationship of the form:

TS

£7 = 1 - oJC• max

K0(l-R)iAK

(10.20)

10

10

10 "8

10 "9

10 -10

10

MILD STEEL

Open Symbols: AKFilled Symbols: AKeffHalf-filled symbols:

AK = AKeff

I Jin, A2 5 10 20

AK, AKeff

50

FIGURE 10.8. Fatigue crack growth data for mild steel at various R ratios [7],

528 Chapter 10

where Ko is a material constant. Hudak and Davidson concluded that the inconsistent re-sults from the literature could be attributed to a number of factors: (1) the range of AKvalues in the experiments was too narrow; (2) the AK values were not sufficiently closeto the threshold; and (3) the measurement techniques lacked the required sensitivity.

The effect of loading variables on KOp can be inferred by substituting the relation-ships for U into Eq. (10.14). For example, the Elber equation implies the following rela-tionship for K0p.

opKo = A £ - - 0 . 5 - 0 . 4 # (10.21)

Note that the ratio KOp/AK depends only on the R ratio. Equation (10.20), however,leads to a different expression for KOp.

„ „ m AKR= K0(l-R) + —- (10.22)

I — K

Thus Ko is the opening stress intensity for R = 0.Hudak and Davidson [16] attributed the confusion and controversy over the effect of

loading variables on closure to experimental factors. More recently, McClung [25] con-ducted an extensive review of experimental and analytical closure results and concludedthat there are three distinct regimes of crack closure. Near the threshold, closure levels de-crease with increasing stress intensity, while U is independent of Kmax at intermediate Klevels. At high AK values, the specimen experiences a loss in constraint, and U decreaseswith increasing stress intensity. McClung found that no single equation could describeclosure in all three regimes. Most of the seemingly contradictory data in the literature canbe reconciled by considering the regimes in which the data were collected.

Micro structural effects can also lead to differences in observed closure behavior invarious materials. Figure 10.6, for example, shows the effect of grain size on crack clo-sure in 1018 steel. Equation (10.20) contains one fitting parameter (K0) that must be ob-tained experimentally. It remains to be seen whether or not this empirical expression issufficiently flexible to account for the full range of fatigue behavior in various alloys.

10.3.3 The Fatigue Threshold

Crack closure influences the fatigue threshold, but closure is not the only cause of AKfh-The threshold stress intensity range can be resolved into intrinsic and extrinsic compo-nents:


where AKfh(eff) is the intrinsic fatigue threshold and AKfh(cl) is the contribution fromcrack closure. The precise mechanism for the intrinsic threshold has not been established,but several researchers have developed models for AKth(eff) based on dislocation emissionfrom the crack tip [26] or blockage of slip bands by grain boundaries [27] (see Section10.6.2).

Figure 10.8 shows fatigue crack growth data near the threshold for a mild steel [7].The data exhibit the expected R ratio effects when characterized by AK. Expressing thedata in terms of AKeff removes the R dependence, but the threshold remains. Thus theElber power law relationship (Eq. (10.15)) may not apply at low growth rates. The in-

trinsic threshold, however, is very small in this case (~ 3 MPa v m), and the crack growthrates must be less than 10"^ in/cycle for a noticeable deviation from Eq. (10.15).Therefore, the power law equation is acceptable in most practical circumstances.

Klesnil and Lukas [28] proposed the following empirical relationship betweenandfl:

(l-R) (10.24)

where AKtho 1S the threshold for R - 0 and /is a fitting parameter. When 7= 7, Eq.(10.24) is consistent with Eq. (10.18).

The effect of loading variables on the threshold can also be inferred from the crackclosure relationships described in Section 10.3.2. Threshold conditions are obtained bysetting AKeff = AKth(eff) and AK = AKth- For example, the Elber equation (Eq. (10.19))leads to the following expression

_th 0.5 + OAR

Solving for the threshold stress intensity from the Hudak and Davidson relationship (Eq.(10.20)) gives

AKth = &Kth(eff) + KO (1 - K} (10.26)

If AKth(eff) is small compared to AKfh, Eq. (10.26) coincides with Eq. (10.18), as wellas Eq. (10.24) for 7= 1. Recall that Eq. (10.18) was derived by assuming KOp is a mate-rial constant. According to Eq. (10.26), however, such an assumption is not necessary toobtain a linear relationship between AKfh and R.

Figure 10.9 shows a compilation of threshold values for a variety of steels [7].Aside from a high strength inartensitic steel, where AKth is apparently independent of R,the relationship between AKth and R is reasonably linear between R = 0 and R = 0.8.Above R = 0.8, the data appear to reach a plateau, which may be indicative of the intrin-sic threshold.

530 Chapter 10

rtPL,

CD</>OJ

*H4-1

CD

0.2 0.4 0.6

Stress Ratio (R)

FIGURE 10.9 Effect of R ratio on the threshold stress intensity range [7].

10.3.4 Pitfalls and Limitations of AK eff

The effective stress intensity range, AK-effi nas t>een fairly successful in characterizing fa-tigue crack growth. Seemingly random trends in data can often be rationalized by takingaccount of crack closure. This approach contains a number of pitfalls, however.

Since closure is influenced by microstructure and loading history, and distinctregimes of closure operate at various K levels [25], the empirical relationships for U andKOp described above are not reliable for estimating AKeff- Empirical fits to a given setof data only apply to a particular loading regime (e.g. near-threshold behavior) and shouldnot be extrapolated to other regimes or other materials.

The only reliable method for inferring closure loads is through direct measurement.Section 10.7 outlines several methods for measuring KOp. Such measurements, however,tend to be ambiguous. Crack opening and closure are progressive events that do not occurat a distinct load. Figure 10.3(a) schematically illustrates the typical load-displacementbehavior, where the compliance gradually changes from the closed to fully open case.


The definition of the closure load is somewhat arbitrary in such cases. The most appro-priate definition ofKOp is the subject of ongoing research.

Even without uncertainties in closure load measurements, the definition of AKeffiscomplicated by history effects. All of the equations in Section 10.3 implicitly assumesimilitude. That is, the fatigue crack growth rate is assumed to be a function only ofKmin and Kmax (Eq. (10.1)). This assumption is strictly valid only for constant ampli-tude fatigue; i.e., dK/da = 0.

The fatigue crack growth rate depends on dK/da [29,30]. Consequently, a rising Kfield and a falling K field result in different fatigue crack growth rates for a given Kmin

and Kmax. Most history effects can be explained in terms of crack closure: loading his-tory determines KOp* which in turn affects crack growth rate. Hertzberg, et al. [30]showed that fatigue tests at a constant R ratio but with a decreasing AK can result inoverestimates of AKfh', the negative dK/da produces large closure loads, which reduceAKeff. Applying such data to structures may be nonconservative, particularly if dK/da >0 in the structure.

One method to account for history effects is to measure K0p throughout the test toensure that AK^reflects the current closure behavior. Such an approach complicates fa-tigue experiments when KOp is determined intermittently, and a cycle-by-cycle evaluationof closure loads is totally impractical. Moreover, there is no guarantee that the instanta-neous AKeff uniquely characterizes crack growth rate when dK/da #0. Even if similitudeof AKeffvjeiQ valid in the general case, it would be of little practical use unless KOp wereknown for the loading spectrum in the structure of interest. History effects and variableamplitude loading are explored further below.

10.4 VARIABLE AMPLITUDE LOADING AND RETARDATION

Similitude of crack tip conditions, which implies a unique relationship between da/dN,AK, and R, is strictly valid only for constant amplitude loading (i.e., dK/da = 0). Realstructures, however, seldom conform to this ideal. A typical structure experiences a spec-trum of stresses over its lifetime. In such cases, the crack growth rate at any moment intime depends not only on the current loading conditions, but also the prior history.Equation (10.3) is a general mathematical representation of the dependence on past andpresent conditions.

10.4.1 Reverse Plasticity at the Crack Tip

History effects in fatigue are a direct result of the history dependence of plastic de-formation. Figure 10.10 schematically illustrates the cyclic stress-strain response of anelastic-plastic material which is loaded beyond yield in both tension and compression. Ifwe desire to know the stress at a particular strain, £*, it is not sufficient merely to specifythe strain. For the loading path in Fig. 10.10, there are three different stresses that corre-spond to £*; we must specify not only e*, but also the deformation history that precededthis strain.

532 Chapter 10

STRESS

FIGURE 10.10. Schematic stress-strain responseof a material that is yielded in both tension andcompression. The stress at a given strain, e*, de-pends on prior loading history.

Figure 10.11 illustrates the crack tip plastic deformation that results from a singlestress cycle. A plastic zone forms when the structure is loaded to Kmax. Upon unload-ing, material near the crack tip exhibits reverse plasticity, which results in a compressiveplastic zone. The compressive stress field at the crack tip influences subsequent deforma-tion and crack growth. Retardation of crack growth after an overload (Section 10.4.2) isan example of this effect.

Following an approach proposed by Rice [31], we can analyze reverse plasticity bymeans of the Dugdale-Barenblatt strip yield model. The advantage of this model is that itpermits superposition of loading and unloading stress fields. Refer to Fig. 10.11 (a),where the structure is loaded to Kmax. Assuming small scale yielding, the size of theplastic zone is given by

(10.27)

Let us now superimpose a compressive stress intensity, -AK. The effective yield stressfor reverse yielding is -2cys, since the material in the compressive plastic zone must bestressed to -<jys from an initial value of +(?YS- Figure 10.1 l(b) illustrates the super-imposed stress Field, and Fig. 10.11(c) shows the net stress field after unloading. The es-timated size of the compressive plastic zone is

p* = (10.28)


(a) Crack loaded to Kmax.

-2 GYS

(b) Superimposed stress field.

PreviouslyYielded Material

(c) Stress field after unloading.

FIGURE 10.11. Formation of a reverse plastic zone during cyclic loading.

534 Chapter 10

Because of the redistribution of stresses upon unloading, much of the material that waspreviously in the monotonic plastic zone is now in compression. According to Eq.(10.28), the compressive plastic zone is 1/4 the size of the monotonic zone. Finite ele-ment analysis, however, predicts a much smaller cyclic plastic zone [32].

A somewhat more complicated version of the above analysis forms the basis of theBudiansky and Hutchinson [15] crack closure model. They incorporated a plastic wakeinto the strip yield model, and showed that residual stretch in the wake results in positiveclosure loads.

10.4.2 The Effect of Overloads

Consider the fatigue loading history illustrated in Fig. 10.12. Constant amplitude load-ing is interrupted by a single overload, after which the K amplitude resumes its previousvalue. Prior to the overload, the plastic zone would have reached a steady state size, butthe overload cycle produces a significantly larger plastic zone. When the load drops tothe original Kmin and Kmax, the residual stresses that result from the overload plasticzone (Fig. 10.1 1) are likely to influence subsequent fatigue behavior.

Figure 10.13 shows actual experimental data [33], where a single overload is im-posed in an otherwise constant amplitude test. Immediately after application of the over-load, da/dN drops dramatically. The overload results in compressive residual stresses atthe crack tip, which retard fatigue crack growth. The growth rate resumes its earlier valueonce the crack has grown through the overload plastic zone. Figure 10.13 is an obviousexample where similitude is violated, as the instantaneous values of AK and R are notsufficient to define da/dN.

Retardation following an overload is a complicated phenomenon that has so fareluded rigorous mathematical description. There are a number of empirical and semiem-pirical models for retardation, which contain one or more fitting parameters that must beobtained experimentally. Some models assume that crack closure effects are responsiblefor retardation, while others consider the plastic zone in front of the crack tip. TheWheeler model [34], which takes the latter approach, is one of the simplest and mostwidely used retardation analyses. This model relates the crack growth rate to the overloadplastic zone size and the current plastic zone size (Fig. 10.14). The former quantity isgiven by

where K0 is the stress intensity at the peak overload, and ft = 2 for plane stress and /? = 6for plane strain. The plastic zone size that corresponds to the current Kmax is given by


Ki

TIME

FIGURE 10.12 A single overload during cyclic loading.

U ID"4

18

10 -5

0 1 2 3 4Distance from Overload, mm

FIGURE 10.13 Retardation of crack growth fol-lowing an overload [33].

Wheeler assumed that retardation effects persist as long as r-y(c) is contained within ry(o)(Fig. 10.14(a) and (b)), but the overload effects disappear when the current plastic zonetouches the outer boundary of ry(o), as Fig. 10.14(c) illustrates. For a crack that hasgrown Ao. since the overload, Wheeler defined a retardation factor as follows:

ry(c)ry(o)

(10.31)

536 Chapter 10

where /is a fitting parameter. The crack growth rate is reduced from a baseline value by

da(10.32)

The baseline crack growth rate is obtained from a growth law of the form of Eq. (10.1).Thus for a single overload, the number of cycles required to grow through the overloadplastic zone can be integrated as follows:

da(10.33)

where ao is the crack size at the application of the overload,// is the baseline growth law(Eq. (10.1)), and a* = ao + ry(0) - ry(c).

(a) Immediately following the overload. (b) After the crack propagates Aa.

FIGURE 10.14 The Wheeler model for fatigueretardation. The crack growth rate depends onthe size and position of the current plastic zonerelative to the overload plastic zone.

(c) Propagation through the overload plasticzone.


10.4.3 Analysis of Variable Amplitude Fatigue

Variable amplitude fatigue can involve either a regular pattern of cyclic stresses or a ran-dom sequence of loads. Figure 10.2 illustrates several examples of variable amplitudeloading. Similitude is not satisfied in such cases, and history effects can be quite pro-nounced.

In some instances, it may be appropriate to analyze variable amplitude fatigue witha growth law of the form of Eq. (10.1). In a rising or falling K field, for example, simili-tude may be approximately satisfied ifdK/da is small. Similitude may also be valid whenconstant amplitude loading follows a single overload, provided the crack has propagatedout of the overload plastic zone.

Simple fatigue growth laws that assume similitude are usually conservative whenapplied to variable amplitude loading. Retardation effects, which the simple equations donot consider, tend to extend the fatigue life of a structure.

Variable amplitude fatigue analyses that account for retardation can be very compli-cated and computationally intensive. Some models require a cycle-by-cycle integration ofcrack growth. For example, consider the Wheeler model as applied to a variable amplitudeloading case.

The Wheeler retardation model is implemented in much the same manner as foroverload cases, except that the overload plastic zone size, ry(o \, and the current plasticsize, ry(c), must be evaluated for each cycle in a variable amplitude problem. Figure10.15 illustrates a typical scenario, where a very high stress cycle occurred earlier in thehistory, and a moderately high stress occurred in the previous cycle. The overload plasticzone is chosen such that the retardation factor, (j)R, is minimized. In this case, the morerecent overload is considered, despite the fact that it produced a smaller plastic zone thanthe earlier peak stress.

Figure 10.16 shows a flow chart for computing fatigue crack growth with theWheeler model. Although the algorithm is relatively simple, the analysis can be verytime-consuming, since a cycle-by-cycle summation is required. The stress input consistsof two components: the spectrum and the sequence. The former is a statistical distribu-tion of stress amplitudes, which quantifies the relative frequency of low, medium and highstress cycles. The sequence, which defines the order of the various stress amplitudes, canbe either random or a regular pattern.

An important point about the Wheeler model is that the exponent, y, in Eq. (10.31)depends on material properties and stress spectrum. Therefore, this parameter must be ob-tained empirically from an experiment with a stress spectrum that has a similar characterto that of the structure. A variable amplitude loading analysis must first be performed onthe experiment to determine the /value that gives the best prediction of crack growth.The model can then be applied to structural predictions. If a structure with a differentstress spectrum is to be analyzed, the Wheeler model must be recalibrated with a new ex-periment. Empirical retardation models such as the Wheeler method contain adjustableparameters and must be calibrated experimentally.

Newman [35] has developed a crack closure model that can be applied to variableamplitude loading. One advantage of the Newman model is that it is capable of a prioripredictions, while empirical approaches such as the Wheeler model are merely able to cor-

538 Chapter 10

relate crack growth data after the fact. Newman's method is based on the Dugdale-Barenblatt strip yield model, and is an expansion of the superposition-reverse plasticityconcept illustrated in Fig. 10.11.

Figure 10.17 illustrates the Newman closure model. The plastic zone is divided intosegments, and the residual stress in each segment is computed for a given loading history.At the maximum far-field stress, Smax, the crack is fully open, and plastic zone isstressed to aoy, where a = 1 for plane stress and a - 3 for plane strain. At the minimumstress, Smin, the crack is closed. The residual stress distribution in the plastic wake de-termines the far-field opening stress, S0. The effective stress intensity, Keff, is then com-puted from the effective stress amplitude, Smax - S0.

Ki

TIME

y(c)

y(o)2

FIGURE 10.15 Analysis of variable amplitude fatigue with the Wheeler retardation model. The over-load plastic zone size, fy(o) >s chosen so as to minimize QR.


FIGURE 10.16 Flow chart for variable amplitude fatigue analysis with the Wheeler model.

10.5 GROWTH OF SHORT CRACKS

The fatigue behavior of short cracks is often very different from that of longer cracks.There is not a precise definition of what constitutes a "short" crack, but most experts con-sider cracks less than 1 mm long to be small.

Figure 10.17 compares short crack data with long crack data near the threshold [36].In this case, the short cracks were initiated from a blunt notch. Note that the short cracksexhibit finite growth rates well below AKth for long cracks. Also, the trend in da/dN isinconsistent with expected behavior; the crack growth rate actually decreases with AKwhen the stress range is 60 MPa, and the da/dN - AK curve exhibits a minimum at theother stress level.

1 Iill!

540 Chapter 10

FIGURE 10.17 The Newman closure model.

A number of factors can contribute to the anomalous behavior of small fatiguecracks. The fatigue mechanisms depend on whether the crack is microstructurally short ormechanically short, as described below.

10.5.1 Microstructurally Short Cracks

A microstructurally short crack has dimensions that are on the order of the grain size.Cracks less than 100 jam long are generally considered microstructurally short. The mate-rial no longer behaves as a homogeneous continuum at such length scales; the growth isstrongly influenced by microstructural features in such cases. The growth of microstruc-turally short cracks is often very sporadic; the crack may grow rapidly at certain intervals,and then virtually arrest when it encounters barriers such as grain boundaries and second-phase particles [7].


10

10

-7

-8

GJF—IU .0

10

io-10

10

10

-11

-12

I

Low Carbon SteelR = 0

0 Ao = SOMPa^ Ao = 76 MPa

— Long Crack Data

OQ

101/2

20

AK,MPam'

FIGURE 10.18 Growth of short cracks in a low carbon steel [36].

10.5.2 Mechanically Short Cracks

A crack that is between 100 (im and 1 mm in length is mechanically short. The size issufficient to apply continuum theory, but the mechanical behavior is not the same as inlonger cracks. Mechanically short cracks typically grow much faster than long cracks atthe same AK level, particularly near the threshold (Fig. 10.18).

Two factors have been identified as contributing to faster growth of short cracks:plastic zone size and crack closure.

When the plastic zone size is significant compared to the crack length, an elasticsingularity does not exist at the crack tip, and K is invalid. The effective driving forcecan be inferred by adding an Irwin plastic zone correction. El Haddad, et al. [37] intro-duced an "intrinsic crack length" which, when added to the physical crack size, bringsshort crack data in line with the corresponding long crack results. The intrinsic cracklength is merely a fitting parameter, however, and does not correspond to a physicallength scale in the material. Tanaka [7], among others, proposed adjusting the data forcrack tip plasticity by characterizing da/dN with AJ rather than AK.

According to the closure argument, short cracks exhibit different crack closure be-havior than long cracks, and data for different crack sizes can be rationalized throughAKeff. Figure 10.19(a) [36] shows KOp measurements for the short and long crack data

542 Chapter 10

in Fig. 10.18. The closure loads are significantly higher in the long cracks, particularlyat low AK levels. Figure 10.19(b) shows the small and large crack data lie on a commoncurve when da/dN is plotted against AKeff, thereby lending credibility to the closure the-ory of short crack behavior.

6

5

S 4s& 3%a 2

1

0

12*d


OAcr = 60MPa£kAa = 76MPaD Long Crack Data

10 15

(a) Crack closure data for short and long cracks.

20

10 "7 r"T

10"

IV"u -9^ 10

io-10

10-11

10-12


0 Aa = 60 MPa£>. Aa = 76 MPa

— Long Crack Data

3 5 1 0 2 01/2

AKeff,MPam

(b) Closure-corrected data.

FIGURE 10.19 Short crack fatigue crack growth data from Fig. 10.17, corrected for closure [36].


10.6 MICROMECHANISMS OF FATIGUE

Figure 10.20 summarizes the failure mechanisms for metals in the three regions of the fa-tigue crack growth curve. In Region II, where da/dN follows a power law, the crackgrowth rate is relatively insensitive to microstructure and tensile properties, while da/dNat either extreme of the curve is highly sensitive to these variables. The fatigue mecha-nisms in each region are described in more detail below.

10.6.1 Fatigue in Region II

In Region II, the fatigue crack growth rate is not a strong function of microstructure ormonotonic flow properties. Two aluminum alloys with vastly different mechanical prop-erties, for instance, are likely to have very similar fatigue crack growth characteristics.Steel and aluminum, however, exhibit significantly different fatigue behavior. Thusda/dN is not sensitive to microstructure and tensile properties within a given material sys-tem.

One explanation for the lack of sensitivity to metallurgical variables is that cyclicflow properties, rather than monotonic tensile properties, control fatigue crack propaga-tion. Figure 10.21 schematically compares monotonic and cyclic stress-strain behaviorfor two alloys of a given material. The low strength alloy tends to strain harden, whilethe strong alloy tends to strain soften with cyclic loading. In both cases, the cyclicstress-strain curve tends toward a steady-state hysteresis loop, which is relatively insensi-tive to the initial strength level.

LOGdN

II

Sensitive tomicrostrure

and flowproperties

Ill

Sensitive tomicrostrure

and flowproperties

Insensitive tomicrostructure and

flow properties

AKthLOG AK

FIGURE 10.20 Micromechanisms of fatigue in metals.

544 Chapter 10

STRESS

STRAIN

FIGURE 10.21 Schematic comparison of mono-tonic and cyclic stress-strain curves.

Propagating fatigue cracks often produce striations on the fracture surface.Striations are small ridges that are perpendicular to the direction of crack propagation.Figure 10.22 illustrates one proposed mechanism for striation formation during fatiguecrack growth [38]. The crack tip blunts as the load increases, and an increment of growthoccurs as a result of the formation of a stretch zone. Local slip is concentrated at ±45°from the crack plane. When the load decreases, the direction of slip reverses, and the cracktip folds inward. The process is repeated with subsequent cycles, and each cycle producesa striation on the upper and lower crack faces. The striation spacing, according to thismechanism, is equal to the crack growth per cycle (da/dN).

An alternative view of fatigue is the damage accumulation mechanism, which statesthat a number of cycles are required to produce a critical amount of damage, at which timethe crack grows a small increment [39]. This mechanism was supported by Lankford andDavidson [40], who observed that the striation spacing did not necessarily correspond thecrack growth after one cycle. Several cycles may be required to produce one striation, de-pending on the AK level; the number of cycles per striation apparently decreases with in-creasing AK, and striation spacing = da/dN at high AK values.

A number of researchers have attempted to relate the observed crack growth rate tothe micromechanism of fatigue, with limited success. The blunting mechanism, wherecrack advance occurs through the formation of a stretch zone, implies that the crackgrowth per cycle is proportional to ACTOD. This, in turn, implies that da/dN should beproportional to AK^. Actual Paris law exponents, however, are typically between threeand four for metals. One possible explanation for this discrepancy is that the bluntingmechanism is incorrect. An alternate explanation for exponents greater than two is thatthe shape of the blunted crack is not geometrically similar at high and low K values [7].Figure 10.23 [7] shows the crack opening profile for copper at two load levels; note thatthe shapes of the blunted cracks are different.


(a) (d)

(b) (e)

(c)

FIGURE 10.22 The crack blunting mechanismfor striation formation during fatigue crackgrowth [38].

ACTOD/ 2, (im

-60 -50 -40 -30 -20-10 0 10 20 30 40 50

X, fim

FIGURE 10.23 Crack opening profile in copper [7].

Jlil

546 Chapter 10

10.6.2 MicFomechanisms Near the Threshold

The fracture surface that results from fatigue near the threshold has a flat, faceted appear-ance that resembles cleavage [41]. The crack apparently follows specific crystallographicplanes, and changes directions when it encounters a barrier such as a grain boundary.

The fatigue crack growth rate in this region is sensitive to grain size, in part becausecoarse grained microstructures produce rough surfaces and roughness-induced closure (Fig.10.6). Grain size can also affect the intrinsic threshold in certain cases. One model forAKth(eff) [27] states that the threshold occurs when grain boundaries block slip bands andprevent them from propagating into the adjoining grain. This apparently happens whenthe plastic zone size is approximately equal to the average grain diameter, which suggeststhe following relationship between AKêff) and grain size:

where d is average grain diameter and A is a constant. Thus the intrinsic threshold in-creases with grain size, assuming oyS is constant. The Hall-Petch relationship, how-ever, predicts that yield strength decreases with grain coarsening:

_2 (10.35)

Consequently, the grain size dependence of yield strength offsets the tendency for the in-trinsic threshold to increase with grain coarsening.

10.6.3 Fatigue at High AK Values

In Region IE, da/dN accelerates due to an interaction between fatigue and fracture mecha-nisms. Fracture surfaces in this region typically include a mixture of fatigue striations,microvoid coalescence, and (depending on the material and the temperature) cleavagefacets. The overall growth rate can be estimated by summing the effects of the variousmechanisms:

da

dN total

da

~dNda

fatigue MVC

da_dN cleavage

(10.36)

The relative contribution of fatigue decreases with increasing Kmax. At Kc, crack growthis completely dominated by microvoid coalescence, cleavage, or both.


10.7 EXPERIMENTAL MEASUREMENT OF FATIGUE CRACKGROWTH

The American Society for Testing and Materials (ASTM) recently published Standard E647-93 [42], which outlines a test method for fatigue crack growth measurements. Theprocedure, which is summarized below, does not take crack closure into account. Acommittee within ASTM, however, is currently studying crack closure measurement andanalysis. Section 10.7.2 below describes some of the closure measurement techniquesthat are currently available.

10.7.1 ASTM Standard E 647

The Standard Test Method for Measurement of Fatigue Crack Growth Rates, ASTM E647-93 [42], describes how to determine da/dN as a function of AK from an experiment.The crack is grown by cyclic loading, and Km{n, Kmax, and crack length are monitoredthroughout the test.

The test fixtures and specimen design are essentially identical to those required forfracture toughness testing, which are described in Chapter 7. The E 647 document allowstests on compact specimens and middle tension panels (Fig. 7.1).

The ASTM standard for fatigue crack growth measurements requires that the behav-ior of the specimen be predominantly elastic during the tests. This standard specifies thefollowing requirement for the uncracked ligament of a compact specimen:

(1037)

There are no specific requirements on specimen thickness; this standard is often applied tothin sheet alloys for aerospace applications. The fatigue properties, however, can dependon thickness, much like fracture toughness is thickness dependent. Consequently, thethickness of the test specimen should match the section thickness of the structure of in-terest.

All specimens must be fatigue precracked prior to the actual test. The Kmax at theend of fatigue precracking should not exceed the initial Kmax in the fatigue test.Otherwise, retardation effects may influence the growth rate.

During the test, the crack length must be measured periodically. Crack length mea-surement techniques include optical, unloading compliance, and potential drop (seeChapter 7). Accurate optical crack length measurements require a traveling microscope.One disadvantage of this method is that it can only detect growth on the surface; in thickspecimens, the crack length measurements must be corrected for tunneling, which cannotbe done until the specimen is broken open after the test. Another disadvantage of the op-tical technique is that the crack length measurements are usually recorded manually",

"It may be possible to automate optical crack length measurements with image analysis hardware andsoftware, but most mechanical testing laboratories do not have this capability.

548 Chapter 10

while the other techniques can be automated. The unloading compliance technique re-quires that the test be interrupted for each crack length measurement. If the specimen isstatically loaded for a finite length of time, material in the plastic zone may creep. In anaggressive environment, long hold times may result in environmentally assisted crackingor the deposition of an oxide film on the crack faces. Consequently, the compliance mea-surements should be made as quickly as possible. The ASTM standard requires that holdtimes be limited to ten minutes; it should be possible to perform an unloading compli-ance measurement in less than one minute.

The ASTM standard E 647 outlines two types of fatigue tests: (1) constant loadamplitude tests where K increases, and (2) ^-decreasing tests. In the latter case, the loadamplitude decreases during the test to achieve a negative K gradient. The ^-increasingtest is suitable for crack growth rates greater that 10"° m/cycle, but is difficult to apply atlower rates because of fatigue precracking considerations (see above). The ^-decreasingprocedure is preferable when near-threshold data are required. Because of the potential forhistory effects when the K amplitude varies, ASTM E 647 requires that the normalized Kgradient be computed and reported:

1 dK I dAK I dKmin 1-- = -- = -- smL = -- (10.38)K da AK da K da ^ aa

The /if-decreasing test is more likely to produce history effects, because prior cycles pro-duce larger plastic zones, which can retard crack growth. Retardation in a rising K test isnot a significant problem, since the plastic zone produced by a given cycle is slightlylarger than that in the previous cycle. A ^-increasing test is not immune to history ef-fects, however; the width of the plastic wake increases with crack growth, which may re-sult in different closure behavior than in a constant K amplitude test.

The ASTM standard recommends that the algebraic value of G be greater than -0.08mm'* in a ^-decreasing test. If the test is computer controlled, the load can be pro-grammed to decrease continuously to give the desired K gradient. Otherwise, the loadamplitude can be decreased in steps, provided the step size is less than 10% of the currentAP, In either case, the load should be decreased until the desired crack growth rate isachieved. It is usually not practical to collect data below da/dN = 10" m/cycle.

The E 647 standard outlines a procedure for assessing whether or not history effectshave occurred in a ^-decreasing test. First, the test is performed at a negative G value un-til the crack growth rate reaches the intended value. Then the K gradient is reversed, andthe crack is grown until the growth rate is well out of the threshold region. The ^-de-creasing and AT-increasing portions of the test should yield the same da/dN - AK curve.This two-step procedure is time consuming, but it need only be performed once for agiven material and R ratio to ensure that the true threshold behavior is achieved by subse-quent /^-decreasing tests.

Figure 10.24 schematically illustrates typical crack length versus TV curves. Thesecurves must be differentiated to infer da/dN. The ASTM standard E 647 suggests two al-ternative numerical methods to compute the derivatives. A linear differentiation approachis the simplest, but it is subject to scatter. The derivative at a given point on the curve


can also be obtained by fitting several neighboring points to a quadratic polynomial (i.e.,a parabola).

The linear method computes the slope from two neighboring data points: (aj, Nj)

and (ai+i, NI+J). The crack growth rate for a = a is given by

— fl;- L. (10.39)

where a =The incremental polynomial approach involves fitting a quadratic equation to a local

region of the crack length versus TV curve, and solving for the derivative mathematically.A group of (2n + 1 ) neighboring points are selected, where n is typically 1, 2, 3, or 4,and (a/, TV/) is the middle value in the (2rc +1) points. The following equation is fitted tothe range a[.n <a <ai+n:

aj=b0+bi\ J \ + b2 \-+- (i-n<j<i + n) (10.40)

where bo, b], and b2 are the curve fitting coefficients, and aj is the fitted value of cracklength at Nj. The coefficients Cf = (Ni-n + Ni+n)/2 and C2 - (Ni-n - Ni+n)/2 scale thedata in order to avoid numerical difficulties. (Nj is often a large number.) The crackgrowth rate at a/ is determined by differentiating Eq. (10.40):

da\ b\ 2B2(Ni-G}=__L + ^—l- U- (10.41)vJ - TO r ov J a( V2 ^22

An appendix in ASTM E 647 lists a Fortran program which performs the curve fittingoperation and solves for da/dN.

10.7.2 Closure Measurements

A number of experimental techniques for measurement of closure loads in fatigue are cur-rently available. Allison [43] has reviewed the existing procedures. A brief summary ofthe more common techniques is given below.

550 Chapter 10

CRACKSIZE

CRACKSIZE

CYCLES CYCLES

FIGURE 10.24 Schematic fatigue crack growth curves. da/dN is inferred from numerical differentia-tion of these curves.

Most measurements of closure conditions are inferred from compliance. Figure10.25 schematically illustrates the load-displacement behavior of a specimen that exhibitscrack closure. The precise closure load is ill-defined, because there is often a significantrange of loads where the crack is partially closed. The closure load can be defined by a de-viation in linearity in either the fully closed or fully open case (P/ and Pj, respectively),or by extrapolating the fully closed and fully open load-displacement curve to the point ofintersection (P2)-

Figure 10.26 illustrates the required instrumentation for the three most commoncompliance techniques for closure measurements. The closure load can be inferred fromclip gage displacement at the crack mouth, back-face strain measurements, or laser inter-ferometry applied to surface indentations. Specimen alignment is critical when inferringclosure loads from compliance measurements.

Crack mouth opening displacement measurements with a clip gage are relativelysimple, but extra care is necessary when in attaching the clip gage. Nonlinearity or hys-teresis can result from improper gage attachment. Crack tip plasticity can also producehysteresis effects in clip gage measurements. Displacement measurements remote fromthe crack tip often lack sensitivity. A signal processing technique called differential com-pliance can enhance sensitivity of global displacement measurements. A baseline com-pliance is inferred from the fully open portion of the load-displacement curve, as Fig.10.25 illustrates. A differential clip gage displacement, AV, is defined as follows:

AV=k(V~CP) (10.42)

where C is the compliance when the crack is fully open and k is a gain factor. Figure10.27 shows a schematic load-differential displacement curve. As the specimen is un-loaded, the initially vertical line exhibits a finite slope when the crack closes. The sensi-tivity of this technique can be adjusted through the gain factor, k.

A back face strain gage has a relatively high degree of sensitivity. This measure-ment is not subject to hysteresis effects, provided the plastic zone is small.


T

LOAD

P3

P2

Pi

Fully Open

Fully ClosedFIGURE 10.25 Alternative definitions of the clo-sure load.

DISPLACEMENT

FIGURE 10.26 Instrumentation for the three most common closure measurement techniques.

LOAD

Pop

FIGURE 10.27 Schematic load versus differentialdisplacement curve. This technique enhances thesensitivity of closure measurements from clipgage displacement.

AV

552 Chapter 10

Interferometric techniques provide a local measurement of crack closure [44].Monochromatic light from a laser is scattered off of two indentations on either side of thecrack. The two scattered beams interfere constructively and destructively, resulting infringe patterns. The fringes change as the indentations move apart.

Crack closure is a three-dimensional phenomenon. The interior of a specimen ex-hibits different closure behavior than the surface. The clip cage and back-face strain gagemethods provide a thickness-average measure of closure, while laser interferometry isstrictly a surface measurement.

More elaborate experimental techniques are available to study three-dimensional ef-fects. For example optical interferometry [45] has been applied to transparent polymersto infer closure behavior through the thickness. Fleck [46] has developed a special gageto measure crack opening displacements at interior of a specimen.

10.8 DAMAGE TOLERANCE METHODOLOGY

The first six sections of this chapter described macroscopic and microscopic aspects of fa-tigue, and outlined various equations and analyses for characterizing crack growth.Section 10.7 addressed experimental measurements of fatigue behavior. This section de-scribes how to apply fatigue data and growth models to structures, as part of a damagetolerance design scheme.

The term damage tolerance has a variety of meanings, but normally refers to a de-sign methodology in which fracture mechanics analyses predict remaining life and quan-tify inspection intervals. This approach is usually applied to structures that are suscepti-ble to time-dependent flaw growth (e.g., fatigue, environmental-assisted cracking, creepcrack growth). As its name suggests, the damage tolerance philosophy allows flaws toremain in the structure, provided they are well below the critical size.

Fracture control procedures vary considerably among various industries; a detaileddescription of each available approach is beyond the scope of this book. This section out-lines a generic damage tolerance methodology and discusses some of the practical consid-erations. Although fatigue is the primary subject of this chapter, the approaches describedbelow can, in principle, be applied to all types of time-dependent crack growth.

One of the first tasks of a damage tolerance analysis is the estimation of the criticalflaw size, ac. Chapter 9 describes approaches for computing critical crack size.Depending on material properties, ultimate failure may be governed by fracture or plasticcollapse. Consequently, an elastic-plastic fracture mechanics analysis that includes theextremes of brittle fracture and collapse as special cases is preferable. (The possibility ofgeometric instabilities, such as buckling, should also be considered.)

Once the critical crack size has been estimated, a safety factor is normally applied todetermine the tolerable flaw size, a^ The safety factor is often chosen arbitrarily, but amore rational definition should be based on uncertainties in the input parameters (e.g..,stress and toughness) in the fracture analysis. Another consideration in specifying thetolerable flaw size is the crack growth rate; at should be chosen such that da/dt at this flawsize is relatively small, and a reasonable length of time is required to grow the flaw fromaj to ac.


Fracture mechanics analysis is closely tied to nondestructive evaluation (NDE) infracture control procedures. The NDE provides input to the fracture analysis, which inturn helps to define inspection intervals. A structure is inspected at the beginning of itslife to determine the size of initial flaws. If no significant flaws are detected, the initialflaw size is set at an assumed value, ao, which corresponds to the largest flaw that mightbe missed by NDE. This flaw size should not be confused with the NDE detectabilitylimit, which is the smallest flaw that can be detected by the NDE technique (on a goodday). In most cases, a0 is significantly larger than the detectability limit, due to the vari-ability in operating conditions and the skill of the operator.

Figure 10.28(a) illustrates the procedure for determining the first inspection intervalin the structure. The lower curve defines the "true" behavior of the worst flaw in thestructure, while the predicted curve assumes the initial flaw size is a0. The time requiredto grow the flaw from ao to at (the tolerable flaw size) is computed. The first inspectioninterval, //, should be less than this time, in order to preclude flaw growth beyond at be-fore the next inspection. If no flaws greater than ao are detected, the second inspection in-terval, /2, is equal to //, as Fig. 10.28(b) illustrates. Suppose that the next inspectionreveals a flaw of length ay, which is larger than a0. In this instance, a flaw growth analy-sis must be performed to estimate the time required to grow from aj to ap The next in-spection interval, Tj, might be shorter than /2, as Fig. 10.28(c) illustrates. Inspection in-tervals would then become progressively shorter as the structure approaches the end of itslife. The structure is repaired or taken out of service when the flaw size reaches the max-imum tolerable size, or when required inspections become too frequent to justify contin-ued operation.

In many applications, a variable inspection interval is not practical; inspectionsmust be often carried out at regular times that can be scheduled well in advance. In suchinstances a variation of the above approach is required. The main purpose of any damagetolerance assessment is to ensure that flaws will not grow to failure between inspections.The precise methods for achieving this goal depend on practical circumstances.

The schematic in Fig. 10.28(c) illustrates a flaw growth analysis that is conserva-tive. If retardation effects are not taken into account, the analysis will be considerablysimpler and will tend to overestimate growth rates. If a more detailed analysis is applied,a comparison of actual and predicted flaw sizes after each inspection interval can be usedto calibrate the analysis.

554Chapter 10

ac

FLAWSIZE

at

CriticalFlaw_Size_ — —

TIME

(a) Determination of first inspection interval,

FLAWSIZE

at I—

aol

TIME

(b) Determination of second inspection interval,

FLAWSIZE

TIME

(c) Determination of third inspection interval, 13.

FIGURE 10.28 Schematic damage tolerance analysis.


REFERENCES

1. Paris, P.C., Gomez, M.P., and Anderson, W.P., "A Rational Analytic Theory ofFatigue." The Trend in Engineering, Vol. 13, 1961, pp. 9-14.

2. Paris, P.C. and Erdogan, P., "A Critical Analysis of Crack Propagation Laws." Journalof Basic Engineering, Vol. 85, I960, pp. 528-534.

3. Dowling, N.E. and Begley, J.A., "Fatigue Crack Growth During Gross Plasticity and theJ Integral." ASTM STP 590, American Society for Testing and Materials, Philadelphia,1976, pp. 82-103.

4. Lambert, Y., Saillard, P., and Bathias, C, "Application of the J Concept to FatigueCrack Growth in Large-Scale Yielding." ASTM STP 969, American Society for Testingand Materials, Philadelphia, 1988, pp. 318-329.

5. Lamba, H.S., "The J-Integral Applied to Cyclic Loading." Engineering FractureMechanics, Vol. 7, 1975, pp. 693-703.

6. Wtithrich, C., "The Extension of the J-Integral Concept to Fatigue Cracks."International Journal of Fracture, Vol. 20, 1982, pp. R35-R37.

7. Tanaka, K., "Mechanics and Micromechanics of Fatigue Crack Propagation." ASTMSTP 1020, American Society for Testing and Materials, Philadelphia, 1989, pp. 151-183.

8. Foreman, R.G., Keary, V.E., and Engle, R.M., "Numerical Analysis of CrackPropagation in Cyclic-Loaded Structures." Journal of Basic Engineering, Vol. 89,1967, pp. 459-464.

9. Weertman, J., "Rate of Growth of Fatigue Cracks Calculated from the Theory ofInfinitesimal Dislocations Distributed on a Plane." International Journal of FractureMechanics, Vol. 2, 1966, pp. 460-467.

10. Klesnil, M. and Lukas, P., "Influence of Strength and Stress History on Growth andStabilisation of Fatigue Cracks." Engineering Fracture Mechanics, Vol. 4, 1972, pp.77-92.

11. Donahue, R.J., Clark, H.M., Atanmo, P., Kumble, R., and McEvily, A.J., "CrackOpening Displacement and the Rate of Fatigue Crack Growth." International Journal ofFracture Mechanics, Vol 8, 1972, pp. 209-219.

12. McEvily, A.J., "On Closure in Fatigue Crack Growth." ASTM STP 982, AmericanSociety for Testing and Materials, Philadelphia, 1988, pp. 35-43.

13. Elber, W., "Fatigue Crack Closure Under Cyclic Tension." Engineering FractureMechanics, Vol. 2, 1970, pp. 37-45.

14. Suresh, S. and Ritchie, R.O., "Propagation of Short Fatigue Cracks." InternationalMetallurgical Reviews, Vol. 29, pp. 445-476.

15. Budiansky, B. and Hutchinson, J.W., "Analysis of Closure in Fatigue Crack Growth."Journal of Applied Mechanics, Vol. 45, 1978, pp. 267-276.

556 Chapter 10

16. Hudak, S.J., Jr. and Davidson, D.L., "The Dependence of Crack Closure on FatigueLoading Variables." ASTM STP 982, American Society for Testing and Materials,Philadelphia, 1988, pp. 121-138.

17. Newman, J.C., "A Finite Element Analysis of Fatigue Crack Closure." ASTM STP 590,American Society for Testing and Materials, Philadelphia, 1976, pp. 281-301.

18. McClung, R.C. and Sehitoglu, H., "On the Finite Element Analysis of Fatigue CrackClosure - 1. Basic Modeling Issues." Engineering Fracture Mechanics, Vol. 33, 1989,pp. 237-252.

19. Gray, G.T., Williams, J.C., and Thompson, A.W., "Roughness Induced Crack Closure:An Explanation for Microstructurally Sensitive Fatigue Crack Growth." MetallurgicalTransactions, Vol. 14A, 1983, pp. 421-433.

20. Schijve, J., "Some Formulas for the Crack Opening Stress Level." Engineering FractureMechanics, Vol. 14, 1981, pp. 461-465.

21. Gomez, M.P., Ernst, H., and Vazquez, J., "On the Validity of Elber's Results on FatigueCrack Closure for 2024-T3 Aluminum." International Journal of Fracture, Vol. 12,1976, pp. 178-180.

22. Clerivet, A. and Bathias, C, "Study of Crack Tip Opening under Cyclic Loading Takinginto Account the Environment and R Ratio." Engineering Fracture Mechanics, Vol 12,1979, pp. 599-611.

23. Shih, T.T. and Wei, R.P., "A Study of Crack Closure in Fatigue." Engineering FractureMechanics, Vol. 6, 1974, pp. 19-32.

24. Shih, T.T. and Wei, R.P., "Discussion." International Journal of Fracture, Vol. 13,1977, pp. 105-106.

25. McClung, R.C., "The Influence of Applied Stress, Crack Length, and Stress IntensityFactor on Crack Closure." Metallurgical Transactions, Vol. 22A, 1991, pp. 1559-1571.

26. Yokobori, T., Yokobori, A.T., Jr., and Kamei, A., "Dislocation Dynamic Theory forFatigue Crack Growth." International Journal of Fracture, Vol. 11, 1975, pp. 781-788.

27. Tanaka, K., Akiniwa, Y., and Yamashita, M, "Fatigue Growth Threshold of SmallCracks." International Journal of Fracture, Vol. 17, 1981, pp. 519-533.

28. Klesnil, M. and Lucas. P., "Effect of Stress Cycle Asymmetry on Fatigue CrackGrowth." Materials Science and Engineering, Vol. 9, 1972, pp. 231-240.

29. Schijve, J., "Fatigue Crack Closure: Observations and Technical Significance." ASTMSTP 982, American Society for Testing and Materials, Philadelphia, 1988, pp. 5-34.

30. Hertzberg, R.W., Newton, C.H., and Jaccard, R., "Crack Closure: Correlation andConfusion." ASTM STP 982, American Society for Testing and Materials,Philadelphia, 1988, pp, 139-148.


31. Rice, J.R., "Mechanics of Crack-Tip Deformation and Extension by Fatigue." ASTMSTP 415, American Society for Testing and Materials, Philadelphia, 1967, pp. 247-309.

32. McClung, R.C., "Crack Closure and Plastic Zone sizes in Fatigue." Fatigue and Fractureof Engineering Materials and Structures, Vol. 14, 1991, pp. 455-468.

33. von Euw, E.F.J., Hertzberg, R.W., and Roberts, R., "Delay Effects in Fatigue-CrackPropagation." ASTM STP 513, American Society for Testing and Materials,Philadelphia, 1972, pp. 230-259.

34. Wheeler, O.E., "Spectrum Loading and Crack Growth." Journal of Basic Engineering,Vol 94, 1972, pp. 181-186.

35. Newman, J.C., "Prediction of Fatigue Crack Growth under Variable Amplitude andSpectrum Loading Using a Closure Model." ASTM STP 761, American Society forTesting and Materials, Philadelphia, 1982, pp. 255-277.

36. Tanaka, K. and Nakai, Y., "Propagation and Non-Propagation of Short Fatigue Cracks ata Sharp Notch." Fatigue of Engineering Materials and Structures, Vol. 6., 1983, pp.315-327.

37. El Haddad, M.H., Topper, T.H., Smith, K.N., "Prediction of Non-Propagating Cracks."Engineering Fracture Mechanics, Vol. 11, 1979, pp. 573-584.

3 8. Laird, C., "Mechanisms and Theories of Fatigue." Fatigue and Microstructure, AmericanSociety for Metals, Metals Park, OH, 1979, pp. 149-203.

39. Starke, E.A. and Williams, J.C., "Microstructure and the Fracture Mechanics of FatigueCrack Propagation." ASTM STP 1020, American Society for Testing and Materials,Philadelphia, 1989, pp. 184-205.

40. Lankford, J. and Davidson, D.L., "Fatigue Crack Micromechanisms in Ingot and PowderMetallurgy 7XXX Aluminum Alloys in Air and Vacuum." Acta Metallurgica, Vol 31,1983, pp. 1273-1284.

41. Hertzberg, R.W., Deformation and Fracture of Engineering Materials, John Wiley andSons, New York, 1989.

42. E 647-93 "Standard Method for Measurement of Fatigue Crack Growth Rates."American Society for Testing and Materials, Philadelphia, 1993.

43. Allison, J.E., "The Measurement of Crack Closure During Fatigue Crack Growth."ASTM STP 945, American Society for Testing and Materials, Philadelphia, 1988, pp.913-933.

44. Sharpe, W.N., and Grandt, A.F., ASTM STP 590, American Society for Testing andMaterials, Philadelphia, 1976, pp. 302-320.

45. Pitoniak, F.J., Grandt, A.F., Jr., Montulli, L.T., and Packman, P.P., "Fatigue CrackRetardation and Closure in Polymethylmethacrylate." Engineering Fracture Mechanics,Vol. 6, 1974, pp. 663-670.

558 Chapter 10

46 Fleck N A. and Smith, R.A., "Crack Closure - Is it Just a Surface Phenomenon?"International Journal of Fatigue, Vol. 4, 1982, pp. 157-160.

10: APPLICATION OF THE JINTEGRAL TO CYCLIC

LOADING

A10.1 DEFINITION OF AJ

Material ahead of a growing fatigue crack experiences cyclic elastic-plastic loading, asFig. A 10.1 illustrates. The material deformation can be characterized by the stress range,Acty, and the strain range, Ae^, in a given cycle.

Consider the loading branch of the stress-strain curve, where the stresses and strainshave initial values (Jif^ and £f/^, and increase to o^) and e^2^. It is possible to de-fine a 7-like integral as follows [3-6]:

AJ = f |i\

- AT; (AlO.l)

where F defines the integration path around the crack tip, and AT} and Aui are the changesin traction and displacement between points (1) and (2). The quantity i//is analogous tothe strain energy density:

'ij M?.

FIGURE AlO.l Schematic cyclic stress-strainbehavior ahead of a growing fatigue crack.

559

560 Chapter 10

(2)kl

(A10.2)

Note that y/" represents the stress work per unit volume performed during loading, ratherthan the stress work in a complete cycle. The latter corresponds to the area inside thehysteresis loop (Fig. A10.1). For the special case where <7//-^ = e^1) = 0, AJ = J.Thus AJ is merely a generalization of the / integral, in which the origin is not necessarilyat zero stress and strain.

Although AJ is normally defined from the loading branch of the cyclic stress-straincurve, it is also possible to define a AJ from the unloading branch. The two definitionscoincide if the cyclic stress-strain curve forms a closed loop, and the loading and unload-ing branches are symmetric.

Just as it is possible to estimate / experimentally from a load-displacement curve(Chapters 3 and 7), AJ can be inferred from the cyclic load-displacement behavior.Consider a specimen with thickness B and uncracked ligament length b, that is cyclicallyloaded between the loads Pmin and Pmax and the load line displacements Vmin andVmax, as Fig. A10.2(a) illustrates.7 The AJ can be computed from an equation of theform

AVAJ = -

Bb

ri VTI l / i j „ ^,, (A10.3)Bb

'min

where the dimensionless constant rj has the same value as for monotonic loading. Forexample, 77 = 2.0 for a deeply notched bend specimen.

Because the AJ parameter is often applied to crack growth under large scale yieldingconditions, plasticity induced closure often has a significant effect on the results. If thecrack is closed below Pc\ and Vc\ (Fig. 10.2(b)), Eq. (A 10.3) can be modified as fol-lows.

The convention of previous chapters, where A represents the load line displacement, is suspended here toavoid confusion with the present use of this symbol.o°The global displacement at closure, Vcj, is not necessarily zero. The crack tip region may be closed whilethe crack mouth is open. Thus, Vc\ is often positive.


LOAD

(Pmax, Vmax)

(Pmin, Vmin)

DISPLACEMENT

(a) No closure.

LOAD

(Pel, Vcl)

(Pmax, Vmax)

DISPLACEMENT

(b) With crack closure.

FIGURE A10.2 Cyclic load displacement behavior for fatigue under large scale yielding conditions.

f\ (A 10.4)

A10.2 PATH INDEPENDENCE OF AJ

If i//" exhibits the properties of a potential, the stresses can be derived by differentiating if/with respect to the strains:

AACT;; =

17(A10.5)

562 Chapter 10

The validity of Eq. (A10.5) is both necessary and sufficient for path independence of AT.The proof of path independence is essentially identical to the analysis in Appendix 3.2,except that stresses, strains and displacements are replaced by the changes in these quanti-ties from states (1) to (2). Evaluating AJ along a closed contour, F*, (Fig. A3.2) and in-voking Green's theorem gives

AJ*=J f- A C 7 < / 3A* - ^/l ^ )

dxdy (A10.6)

where A* is the area enclosed by F*. By assuming y displays the properties of a poten-tial (Eq. A 10.5), the first term in the integrand can be written as

= AeT;y . (A10.7)

By invoking the strain-displacement relationships for small strains, it can be shown thatEq. (A 10.7) is equal to the absolute value of the second term in the integrand in Eq.(A10.6). (See Eqs. (A3.16) to (A3.18) for the mathematical details.) Thus AJ* = 0 forany closed contour. Path independence of AJ evaluated along a crack tip integral can thusbe readily demonstrated by considering the contour illustrated in Fig. A3.3, and notingthat/7 = -/2-

The validity of Eq. (A10.5) is crucial in demonstrating path independence of AJ.This relationship is automatically satisfied when there is proportional loading on eachbranch of the cyclic stress-strain curves. That is, Actij must increase (or decrease) in pro-portion to Acrid, and the shapes of the AOy-Ae/y hysteresis loops must be similar to oneanother.

Proportional loading also implies a single-parameter characterization of crack tipconditions. Consequently, AJ uniquely defines the changes in stress and strain near thecrack tip when there is proportional loading in this region.

In the case of monotonic loading, the / integral ceases to provide a single parameterdescription of crack tip conditions when there is excessive plastic flow or crack growth(Section 3.6). Similarly, one would not expect AJ to characterize fatigue crack growthbeyond a certain level of plastic deformation. The limitations of AJ have yet to be estab-lished.

A10.3 SMALL SCALE YIELDING LIMIT

When the cyclic plastic zone is small compared to specimen dimensions, AJ should char-acterize fatigue crack growth, since it is related to AK. The precise relationship betweenAK and AJ under small scale yielding conditions can be inferred by evaluating Eq.


(A 10.1) along a contour in the elastic singularity dominated zone. For a given AKj, thechanges in the stresses, strains and displacements are given by

(0) (AlO.Sa)

(A10.8b)

(A10.8c)

where //y and h{j are given in Tables 2. 1 and 2.2, and gij can be inferred from Hooke's lawor the strain-displacement relationships.

Inserting Eqs (AlO.Sa) to (A10.Sc) into Eq. (A10.1) and evaluating J along a circu-lar contour of radius r leads to

A T ,A/ = - '— (A10.9)

where E' = E for plane stress conditions and £" = £7f7 - v^J for plane strain. Note that al-though AK = (Kmax - Kmin), AJ*(Jmax - Jmfn) since

11. COMPUTATIONAL FRACTUREMECHANICS

Computers have had an enormous influence in virtually all branches of engineering, andfracture mechanics is no exception. Numerical modeling has become an indispensabletool in fracture analysis, since relatively few practical problems have closed-formanalytical solutions.

Stress intensity solutions for literally hundreds of configurations have beenpublished, the majority of which were inferred from numerical models. Elastic-plasticanalyses to compute the J integral and crack tip opening displacement (CTOD) are alsobecoming relatively common. In addition, researchers are applying advanced numericaltechniques to special problems, such as fracture at interfaces, dynamic fracture, and ductilecrack growth.

Rapid advances in computer technology are primarily responsible for the exponentialgrowth in applications of computational fracture mechanics. The personal computers thatmost engineers have on their desks are more powerful than mainframe computers of 20years ago. The latest supercomputers require only a few minutes to solve problems thatwould take months or even years on older machines.

Hardware does not deserve all of the credit for the success of computational fracturemechanics, however. More efficient numerical algorithms have greatly reduced solutiontimes in fracture problems. For example, the domain integral approach (Section 11.3)enables one to generate K and J solutions from finite element models with surprisinglycoarse meshes. Commercial numerical analysis codes have become relatively userfriendly, and many codes have incorporated fracture mechanics routines.

This chapter will not turn the reader into an expert on computational fracturemechanics, but it should serve as an introduction to the subject. The sections that followdescribe some of the traditional approaches in numerical analysis of fracture problems, aswell as some recent innovations.

The format of this chapter differs from earlier chapters, in that the main body of thischapter contains several relatively complicated mathematical derivations; previouschapters confined such material to appendices. This information is unavoidable whenexplaining the basis of the common numerical techniques. Readers who are intimidatedby the mathematical details should at least skim this material and attempt to understandits significance.

11.1 OVERVIEW OF NUMERICAL METHODS

It is often necessary to determine the distribution of stresses and strains in a body that issubject to external loads or displacements. In limited cases, it is possible to obtain aclosed-form analytical solution for the stresses and strains. If, for example, the body issubject to either plane stress or plane strain loading and it is composed of an isotropiclinear elastic material, it may be possible to find a stress function that leads to the desired

565

566 Chapter 11

solution. Westergaard [1] and Williams [2] used such an approach to derive solutions forthe stresses and strains near the tip of a sharp crack in an elastic material (see Appendix2). In most instances, however, closed-form solutions are not possible, and the stressesin the body must be estimated numerically.1

A variety of numerical techniques have been applied to problems in solid mechanics,including finite difference [3], finite element [4] and boundary integral equation methods[5]. In recent years, the latter two numerical methods have been applied almostexclusively. The vast majority of analyses of cracked bodies utilize finite elements,although the boundary integral method may be useful in limited circumstances.

11.1.1 The Finite Element Method

In the finite element method, the structure of interest is subdivided into discrete shapescalled elements. Element types include one-dimensional beams, two-dimensional planestress or plane strain elements, and three-dimensional bricks. The elements are connectedat node points where continuity of the displacement fields is enforced. Thedimensionality of the structure need not correspond to the element dimension. Forexample, a three-dimensional truss can be constructed from beam elements.

The stiffness finite element method [4] is usually applied to stress analysisproblems. This approach is outlined below for the two-dimensional case.

Figure 1 1.1 shows an isoparametric continuum element for two-dimensional planestress or plane strain problems, together with local and global coordinate axes. The localcoordinates, which are also called parametric coordinates, vary from -1 to +1 over theelement area; the node at the lower left-hand corner has parametric coordinates (-1, -1)while upper right-hand corner is at (+1, +1) in the local system. Note that the parametriccoordinate system is not necessarily orthogonal. Consider a point on the element at (£77). The global coordinates of this point are given by

/=!(11.1)

1=1where n is the number of nodes in the element and NI are the shape functionscorresponding to the node i, whose coordinates are (xi, >'/) in the global system and (&,77,') in the parametric system.

Experimental stress analysis methods, such as photoelasticity, Moire' interferometry, and caustics areavailable, but even these techniques often require numerical analysis to interpret experimental observations.

Computational Fracture Mechanics 567

(1,1)

(-1, -l)

FIGURE 11.1 Local and global coordinates for a two-dimensional element.

The shape functions are polynomials that interpolate field quantities within theelement. The degree of the polynomial depends on the number of nodes in the element.If, for example, the element contains nodes only at the corners, N; are linear. Figure 11.1illustrates a four-sided, eight-node element, which requires a quadratic interpolation.Appendix 11 gives the shape functions for the latter case.

The displacements within an element are interpolated as follows:

(11.2)

i=\

where (ui, v{) are the nodal displacements in the x and y directions, respectively. Thestrain matrix at (x, y) is given by

where

U;(11.3)

568 Chapter 11

3JV- 0

3?

a*

(11.4)

and

3tf;

01.5)

I 3 y jwhere [J] is the Jacobian matrix, which is given by

• • ' Bx '

a?? a?]_

The stress matrix is computed as follows:

dy

(11.6)

where [D] is the stress-strain constitutive matrix. For problems that incorporateincremental plasticity, stress and strain are computed incrementally and [D] is updated ateach load step:

(ii.Tb)

Thus the stress and strain distribution throughout the body can be inferred from thenodal displacements and the constitutive law. The stresses and strains are usuallyevaluated at several Gauss points or integration points within each element. For 2-Delements, 2x2 Gaussian integration is typical, where there are four integration points oneach element.

The displacements at the nodes depend on the element stiffness and the nodal forces.The elemental stiffness matrix is given by:

TV


1 1[k]= I l[B]T[D][B]dQt\J\d^d7] (11.8)

where the subscript T denotes the transpose of the matrix. Equation (1 1.8) can be derivedfrom the principle of minimum potential energy [4].

The elemental stiffness matrices are assembled to give the global stiffness matrix,[3Q. The global force, displacement, and stiffness matrices are related as follows:

11.1.2 The Boundary Integral Equation Method

Most problems in nature cannot be solved mathematically without specifying appropriateboundary conditions. In solid mechanics, for example, a well posed problem is one inwhich either the tractions or the displacements (but not both) are specified over the entiresurface. In the general case, the surface of a body can be divided into two regions: Su,where displacements are specified, and ST, where tractions are specified. (One cannotspecify both traction and displacement on the same area, since one quantity depends on theother.) Given these boundary conditions, it is theoretically possible to solve for thetractions on Su and the displacements on Sj1, as well as the stresses, strains, and dis-placements within the body.

The boundary integral equation (BIE) method [5-9] is a very powerful technique forsolving for unknown tractions and displacements on the surface. This approach can alsoprovide solutions for internal field quantities, but finite element analysis is more efficientfor this purpose.

The BIE method stems from Betti's reciprocal theorem, which relates work done bytwo different loadings on the same body. In the absence of body forces, Betti's theoremcan be stated as follows:

(11.10)

where TI and ui are components of the traction and displacement vectors, respectively, andthe superscripts denote loadings (1) and (2). The standard convention is followed in thischapter, where repeated indices imply summation. Equation (1 1.10) can be derived fromthe principle of virtual work, together with the fact that ay^£y^) = Oy^fy-C-O for alinear elastic material.

Let us assume that (1) is the loading of interest and (2) is a reference loading with aknown solution. Figure 11.2 illustrates the conventional reference boundary conditionsfor BIE problems. A unit force is applied at an interior point p in each of the threecoordinate directions, jc/, resulting in displacements and tractions at surface point Q in the

570 Chapter 11

xj direction2. For example, a unit force in the ^7 direction may produce displacementsand tractions at Q in all three coordinate directions. Consequently, the resultingdisplacements and tractions at Q, ufj and Ty, are second-order tensors. The quantitiesuij(p.Q) and Tij(p,Q) have closed-form solutions for several cases, including a point forceon the surface of a semi-infinite elastic body [5].

Applying the Betti reciprocal theorem to the boundary conditions described aboveleads to [5]:

where utfp) is the displacement vector at the interior point p; Uj(Q) and Tj(Q) are thereference displacement and traction vectors at the boundary point Q. Note that utfp),uj(Q), and Tj(Q) correspond to the loading of interest; i.e., loading (1). At a given pointQ on the boundary, either traction or displacement is known a priori, and it is necessaryto solve for the other quantity. If we let p -* P, where P is a point on the surface, Eq.(11. 11) becomes [5]:

(11.12)

assuming the surface is smooth. (This relationship is modified slightly when P is near acorner or other discontinuity.) Equation (11.12) represents a set of integral constraintequations that relate surface displacements to surface tractions. In order to solve for theunknown boundary data, the surface must be subdivided into segments (i.e., elements),and Eq. (1 1.12) approximated by a system of algebraic equations. If it is assumed that u\and TI vary linearly between discrete nodal points on the surface, Eq. (11.12) can bewritten as

(1U3)

where <5y is the Kronecker delta. Equation (1 1.13) represents a set of 3n equations for athree-dimensional problem, where n is the number of nodes. Once all of the boundaryquantities are known, displacements at internal points can be inferred from Eq. (11.11).

^For the remainder of this chapter, we will adopt the xi-X2-*3 coordinate system, rather than x-y-z. Theformer notation is more convenient when manipulating tensor quantities.


T2j(p,Q)U2j(p,Q)

Tij(p,Q)FIGURE 11.2 Reference boundary conditions fora boundary integral element problem. Unitforces are applied in each of the coordinatedirections at point p, resulting in tractions anddisplacements on the surface point Q.

Unit Forces

The boundary elements have one less dimension than the body being analyzed. Thatis, the boundary of a two-dimensional problem is surrounded by one-dimensionalelements, while the surface of a three-dimensional solid is paved with two-dimensionalelements. Consequently, boundary element analysis can be very efficient, particularlywhen the boundary quantities are of primary interest. This method tends to be inefficient,however, when solving for internal field quantities.

The boundary integral equation method is usually applied to linear elastic problems,but this technique can also be utilized for elastic-plastic analysis [6,9]. As with the finiteelement technique, nonlinear BIE analyses are typically performed incrementally, and thestress-strain relationship is assumed to be linear within each increment.

11.2 TRADITIONAL METHODS IN COMPUTATIONAL FRACTUREMECHANICS

This section describes several of the earlier approaches for inferring fracture mechanicsparameters from numerical analysis. Most of these methods have been made obsolete bymore recent techniques that are significantly more accurate and efficient (Section 11.3).

The approaches outlined below can be divided into two categories: point matchingand energy methods. The former technique entails inferring the stress intensity factorfrom the stress or displacement fields in the body, while energy methods compute theenergy release rate in the body and relate Q to stress intensity. One advantage of energymethods is that they can be applied to nonlinear material behavior; a disadvantage is thatit is often very difficult to separate energy release rate into mixed-mode K components.

Most of the techniques described below can be implemented with either finiteelement or boundary element methods. The stiffness derivative approach (Section11.2.4), however, was formulated in terms of the finite element stiffness matrix, and thusis not compatible with boundary element analysis.

572 Chapter 11

11.2.1 Stress and Displacement Matching

Consider a cracked body subject to pure Mode I loading. On the crack plane (6 = 0),related to the stress in the X2 direction as follows:

Kf = lim[<722V27Tr] (0 = 0) (11.14)

The stress intensity factor can be inferred by plotting the quantity in square bracketsagainst distance from the crack tip, and extrapolating to r = 0. Alternatively, Kj can beestimated from a similar extrapolation of crack opening displacement:

(11.15)

Equation (11.15) tends to give more accurate estimates of K than Eq. (11.14) becausenodal displacements can be inferred with a higher degree of precision than stresses. Bothmethods, however, are vastly inferior to current approaches. These extrapolationapproaches require a high degree of mesh refinement for reasonable accuracy. Forexample, with a two-dimensional finite element mesh with 2000 degrees of freedom, theextrapolation methods typically give errors in KI of around 5% [10]; present-day energymethods (Section 11.3) provide much better accuracy and do not require such fine meshes.

The boundary collocation method [11,12] is an alternative point matching techniquefor computing stress intensity factors. This approach entails finding stress functions thatsatisfy the boundary conditions at various nodes, and inferring the stress intensity factorfrom these functions. For plane stress or plane strain problems, the Airy stress function(Appendix 2) can be expressed in terms of two complex analytic functions, which can berepresented as polynomials in the complex variable z (= xj + 1x2). In a boundarycollocation analysis, the coefficients of the polynomials are inferred from nodal quantities.The minimum number of nodes utilized in the analysis corresponds to the number ofunknown coefficients in the polynomials. The results can be improved by analyzingmore than the minimum number of nodes and solving for the unknowns by least squares.This approach can be highly cumbersome; energy methods are preferable in mostinstances.

Early researchers in computational fracture mechanics attempted to reduce the meshsize requirements for point matching analyses by introducing special elements at the crack

tip that exhibit the 7/Vr singularity [13]. Barsoum [14] later showed that this same effectcould be achieved by a slight modification to conventional isoparametric elements (seeSection 11.4 and Appendix 11).

11.2.2 Elemental Crack Advance

Recall from Chapter 2 that the energy release rate can be inferred from the rate of changein global potential energy with crack growth. If two separate numerical analyses of a

Computational Fracture Mechanics

given geometry are performed, one with crack length a, and the other with crack length a+ Aa, the energy release rate is given by

V î# J fixed boundary conditions

assuming a two-dimensional body with unit thickness.This technique requires minimal post-processing, since total strain energy is output

by many commercial analysis codes. This technique is also more efficient than the pointmatching methods, since global energy estimates do not require refined meshes.

One disadvantage of the elemental crack advance method is that multiple solutionsare required in this case, while other methods infer the desired crack tip'parameter from asingle analysis. This may not be a serious shortcoming if the intention is to compute Cj(or K) as a function of crack size. The numerical differentiation in Eq. (1 1.16), however,can result in significant errors unless the crack length intervals (Aa) are small.

11.2.3 Contour Integration

The / integral can be evaluated numerically along a contour surrounding the crack tip.The advantages of this method are that it can be applied both to linear and nonlinearproblems, and path independence (in elastic materials) enables the user to evaluate / at aremote contour, where numerical accuracy is greater. For problems that include path-dependent plastic deformation or thermal strains, it is still possible to compute J at a re-mote contour, provided an appropriate correction term (i.e., an area integral) is applied[15,16].

For three-dimensional problems, however, the contour integral becomes a surfaceintegral, which is extremely difficult to evaluate numerically.

More recent formulations of 7 apply an area integration for two-dimensionalproblems and a volume integration for three-dimensional problems. Area and volumeintegrals provide much better accuracy than contour and surface integrals, and are mucheasier to implement numerically. The first such approach was the stiffness derivativeformulation of the virtual crack extension method, which is described below. Thisapproach has since been improved and made more general, as Sections 11.2.4 and 1 1.3discuss.

11.2.4 Virtual Crack Extension: Stiffness Derivative Formulation

In 1974, Parks [10] and Hellen [17] independently proposed the following finite elementmethod for inferring energy release rate in elastic bodies. Several years later Parks [18]extended this method to nonlinear behavior and large deformation at the crack tip.Although the stiffness derivative method is now outdated, it was the precursor to themodern approaches described in Sections 1 1.2.5 and 1 1.3.

574 Chapter 11

Consider a two-dimensional cracked body with unit thickness, subject to Mode Iloading. The potential energy of the body, in terms of the finite element solution, isgiven by

(11.17)

where /I is the potential energy, and the other quantities are as defined in Section 11.1.1.Recall from Chapter 2 that the energy release rate is the derivative of IT with respect tocrack area, for both fixed load and fixed displacement conditions. It is convenient in thisinstance to evaluate Bunder fixed load conditions:

load

da 2 da da(lug)

Comparing Eq. (1 1.9) to the above result, we see that the first term in Eq. (11.18) mustbe zero. In the absence of tractions on the crack face, the third term must also vanish,since loads are held constant. Thus the energy release rate is given by

.E 2 da

Thus the energy release rate is proportional to the derivative of the stiffness matrix withrespect to crack length.

Suppose that we have generated a finite element mesh for a body with crack length aand we wish to extend the crack by Aa. It would not be necessary to change all of theelements in the mesh; we could accommodate the crack growth by moving elements nearthe crack tip and leaving the rest of the mesh intact. Figure 11.3 illustrates such a pro-cess, where elements inside the contour Fo are shifted by Aa, and elements outside of thecontour Fj are unaffected. Each of the elements between Fo and Fj is distorted, such thatits stiffness changes. The energy release rate is related to this change in element stiffness:

where [kj] are the elemental stiffness matrices and Nc is the number of elements betweenthe contours F0 and Fj. Parks [10] demonstrated that Eq. (1 1.20) is equivalent to the J


integral. The value of £ (or /) is independent of the choice of the inner and outercontours.

It is important to note that in a virtual crack extension analysis, it is not necessaryto generate a second mesh with a slightly longer crack. It is sufficient merely to calculatethe change in elemental stiffness matrices corresponding to shifts in the nodal coordinates.

One problem with the stiffness derivative approach is that it involves cumbersomenumerical differencing. Also, this formulation is poorly suited to problems that includethermal strain. A more recent formulation of the virtual crack extension methodovercomes these difficulties, as discussed below.

n

(a) Initial conditions. (b) After virtual crack advance.

FIGURE 11.3 Virtual crack extension in a finite element model [10,17]. Elements between Fj and F0are distorted to accommodate a crack advance.

11.2.5 Virtual Crack Extension: Continuum Approach

Parks [10] and Hellen [17] formulated the virtual crack extension approach in terms offinite element stiffness and displacement matrices. deLorenzi [19,20] improved the virtualcrack extension method by considering the energy release rate of a continuum. The mainadvantages of the continuum approach are two-fold: first, the methodology is notrestricted to the finite element method; and second, deLorenzi's approach does not requirenumerical differencing.

Figure 11.4 illustrates a virtual crack advance in a two-dimensional continuum.Material points inside F0 experience rigid body translation a distance Aa in the x jdirection, while points outside of Fj remain fixed. In the region between contours,virtual crack extension causes material points to translate by Ax]. For an elasticmaterial, or one that obeys deformation plasticity theory, deLorenzi showed that energyrelease rate is given by

576 Chapter 11

(11.21)

where w is the strain energy density. Equation (11.21) assumes unit thickness, crackgrowth in the xj direction, no body forces within F/, and no tractions on the crack faces.Note that dAxj/dxi = 0 outside of F/ and within F0; thus the integration need only beperformed over the annular region between Fo and F/.

deLorenzi actually derived a more general expression that considers a three-dimensional body, tractions on the crack surface, and body forces:

1^cy

dV

1AAC

fc •idS (11.22)

where AAC is the increase in crack area generated by the virtual crack advance, V is thevolume of the body, and FI are the body forces. In this instance, two surfaces enclose thecrack front. Material points within the inner surface, So, are displaced by Aa/, while thematerial outside of the outer surface, S j , remains fixed. The displacement vector betweenS0 and Sj is Axf, which ranges from 0 to Aa{. Equation (11.22) assumes a fixedcoordinate system; consequently, the virtual crack advance, Aai, is not necessarily in thexj direction when the crack front is curved. The above expression, however, only appliesto virtual crack advance normal to the crack front, in the plane of the crack.

(a) Initial state (b) After virtual crack advance.

FIGURE 11.4 Virtual crack extension in a two-dimensional elastic continuum.


In a three-dimensional problem, Q may vary along the crack front. In computing Q,one can consider a uniform virtual crack advance over the entire crack front or a crackadvance over a small increment, as Fig. 11.5 illustrates. In the former case, AAC = Aa L,and the computed energy release rate would be a weighted average. Defining Ac

incrementally along the crack front would result in a local measure of §.For two-dimensional problems, the virtual crack extension formulation of £ requires

an area integration, while three-dimensional problems require a volume integration. Suchan approach is easier to implement numerically and is more accurate than contour andsurface integrations for two- and three-dimensional problems, respectively.

Numerical implementation of the virtual crack extension method entails applying avirtual displacement to nodes within a specified contour. Since the domain integralformulation is very similar to the above method, further discussion on numericalimplementation is deferred to Section 11.3.3.

(a) Uniform crack advance. (b) Advance over an increment of crack front.

FIGURE 11.5 Virtual crack extension along a three-dimensional crack front.

11.3 THE ENERGY DOMAIN INTEGRAL

Shin, et. al. [21,22] have recently formulated the energy domain integral methodology,which is a general framework for numerical analysis of the / integral. This approach isextremely versatile, as it can be applied to both quasistatic and dynamic problems withelastic, plastic, or viscoplastic material response, as well as thermal loading. Moreover,the domain integral formulation is relatively simple to implement numerically, and it isvery efficient. This approach is very similar to the virtual crack extension method.

578 Chapter 11

11.3.1 Theoretical Background

Appendix 4.2 presents a derivation of a general expression for the / integral that includesthe effects of inertia as well as inelastic material behavior. The generalized definition of Jrequires that the contour surrounding the crack tip be vanishingly small:

(11.23)

where T is the kinetic energy density. Various material behavior can be taken intoaccount through the definition of w, the stress work.

Consider an elastic-plastic material loaded under quasistatic conditions (7 = 0). Ifthermal strains are present, the total strain is given by

_ ~ , p ,~~

_ ~ ,~ fc ~r (11.24)

where a is the coefficient of thermal expansion and 0 is the temperature, relative toambient. The superscripts e, p, m, and t denote elastic, plastic, mechanical, and thermalstrains, respectively. The mechanical strain is equal to the sum of elastic and plasticcomponents. The stress work is given by

cm£klJ0

(11.25)

The form of Eq. (11.23) is not suitable for numerical analysis, since it is notfeasible to evaluate stresses and strains along a vanishingly small contour. Let usconstruct a closed contour by connecting inner and outer contours, as Fig. 11.6illustrates. The outer contour, F/, is finite, while Fo is vanishingly small. For a linearor nonlinear elastic material under quasistatic conditions, J could be evaluated along eitherF] or Fo, but only the inner contour gives the correct value of / in the general case. Forquasistatic conditions, where T= 0, Eq. (11.23) can be written in terms of the followingintegral around the closed contour, F* = F] + F+ + F. - F0 [21,22]:

'= /r*

_ f (11.26)cl

where F~*~ and JT are the upper and lower crack faces, respectively, m/ is the outwardnormal on /""*, and q is an arbitrary but smooth function that is equal to unity on F0 and


zero on F/. Note that mi = - n{ on To\ also, mi - 0 and m-2 = ir7 on r4" and F". In theabsence of crack face tractions, the second integral in Eq. (11.26) vanishes.

X2

XI

FIGURE 11.6 Inner and outer contours, which form a closed contour around the crack tip whenconnected by F+ and r_.

For the moment, assume that the crack faces are traction free. Applying thedivergence theorem to Eq. (11.26) gives

qdA/= I -—-A* i

= \A*

- ^v8^; £•*+/A*

IJ qdA (11.27)

where A* is the area enclosed by F*. Referring to Appendix 3.2, we see that

3 ( dw/1 3w rt

H^^rr~=0oxi ^ OJCj j dî

when there are no body forces and w exhibits the properties of an elastic potential:

(11.29)

580 Chapter 11

It is convenient at this point to divide w into elastic and plastic components:

oe£klj

0(11.30)

0

where Sn is the deviatoric stress, defined in Eq. (A3.62). While the elastic componentsof w and EIJ satisfy Eq. (11.29), plastic deformation does not, in general, exhibit theproperties of a potential. (Equation (11.29) may be approximately valid for plasticdeformation when there is no unloading.) Moreover, thermal strains would cause the leftside of Eq. (11.28) to be nonzero. Thus the second integrand in Eq. (11.27) vanishes inlimited circumstances, but not in general. Taking account of plastic strain, thermalstrain, body forces, and crack face tractions leads to the following general expression for /in two dimensions:

A*CF; • + CF,

(11.31)r++r_

where the body force contribution is inferred from the equilibrium equations, and thecontribution from thermal loading is obtained by substituting Eqs. (11.24) and (11.30)into Eq. (11.27). Inertia can be taken into account by incorporating T, the kinetic energydensity, into the group of terms that are multiplied by q. For a linear or nonlinear elasticmaterial under quasistatic conditions, in the absence of body forces, thermal strains, andcrack face tractions, Eq. (11.31) reduces to

'=/A*

•dA (11.32)

Equation (11.32) is equivalent to Rice's path-independent / integral (Chapter 3). Whensum of the additional terms in the more general expression (Eq. (11.31)) is nonzero, /ispath dependent.

Comparing Eqs. (11.21) and (11,32) we see that the two expressions are identical ifq = Axj/Aa. Thus q can be interpreted as a normalized virtual displacement, although theabove derivation does not require such an interpretation. The q function is merely amathematical device that enables the generation of an area integral, which is better suitedto numerical calculations. Section 11.3.3 provides guidelines for defining q.


11.3.2 Generalization to Three Dimensions

Equation (11.23) defines the J integral in both two and three dimensions, but the form ofthis equation is poorly suited to numerical analysis. In the previous section, / wasexpressed in terms of an area integral in order to facilitate numerical evaluation. Forthree-dimensional problems, it is necessary to convert Eq. (11.23) into a volume integral.

Figure 11.7 illustrates a planar crack in a three-dimensional body; 77 corresponds tothe position along the crack front. Suppose that we wish to evaluate / at a particular 77on the crack front. It is convenient to define a local coordinate system at 77, with xjnormal to the crack front, X2 normal to the crack plane, and x$ tangent to the crack front.The J integral at 77 is defined by Eq. (11.23), where the contour F0 lies in the xj-x2plane.

Let us now construct a tube of length AL and radius ro that surrounds a segment ofthe crack front, as Fig. 11.7 illustrates. Assuming quasistatic conditions, we can define aweighted average J over the crack front segment AL as follows:

JAL= jAL

= lim J qn(dS (11.33)

where J(r\) is the point-wise value of J, So is the surface area of the tube in Fig. 11.7,and q is a weighting function that was introduced in the previous section. Note that theintegrand in Eq. (11.33) is evaluated in terms of the local coordinate system, where xj istangent to the crack front at each point along AL.

FIGURE 11.7 Surface enclosing an increment of a three-dimensional crack front.

582 Chapter 11

Recall from the previous section that q can be interpreted as a virtual crack advance.For example, Fig. 11.8 illustrates an incremental crack advance over AL, where q isdefined by

(11.34)

and the incremental area of the virtual crack advance is given by

JAL

(11.35)

The q function need not be defined in terms of a virtual crack extension, but attaching aphysical significance to this parameter may aid in understanding.

FIGURE 11.8 Interpretation of q in terms of avirtual crack advance along AL,

If we construct a second tube of radius r] around the crack front (Fig. 11.9), it ispossible to define the weighted average J in terms of a closed surface, analogous to thetwo-dimensional case (Fig. 11.6 and Eq. (11.26)):

JAL= js*

\s++s_

(11.36)

where the closed surface S* = S] + 5"+ + S. - So> and S+ and S. are the upper and lowercrack faces, respectively, that are enclosed by Sj. From this point, the derivation of thedomain integral formulation is essentially identical to the two-dimensional case, exceptthat Eq. (11.34) becomes a volume integral:

• + q\dV

- JS++S_

(11.37)


Si

FIGURE 11.9 Inner and outer surfaces, S0 and Sj, which enclose V*.

Equation (11.37) requires that q = 0 at either end of AL; otherwise, there may be acontribution to / from the end surfaces of the cylinder. The virtual crack advanceinterpretation of q (Fig. 1 1.8) fulfills this requirement.

If the point-wise value of the J integral does not vary appreciably over AL, to a firstapproximation, J(r\) is given by

JAL(11.38)

AL

Equation (1 1.38) is a reasonable approximation if the q gradient along the crack front issteep relative to the variation in J(rf).

Recall that Eq. (1 1.22) was defined in terms of a fixed coordinate system, while Eq.(11.37) assumes a local coordinate system. The domain integral formulation can beexpressed in terms of a fixed coordinate system by replacing q with a vector quantity, qi,and evaluating the partial derivatives in the integrand with respect to */ rather than xj,where the vectors qi and jt/ are parallel to the direction of crack growth. Severalcommercial codes that incorporate the domain integral definition of J require that the qfunction be defined with respect to a fixed origin.

11.3.3 Finite Element Implementation

Shih, et al. [21] and Dodds and Vargas [23] give detailed instructions for implementingthe domain integral approach. Their recommendations are summarized briefly below.

In two-dimensional problems, one must define the area over which the integration isto be performed. The inner contour, Fo is often taken as the crack tip, in which case A *corresponds to the area inside of Fj . The boundary of Fj should coincide with element

584 Chapter 11

boundaries. An analogous situation applies in three dimensions, where it is necessary todefine the volume of integration. The latter situation is somewhat more complicated,however, since J(77) is usually evaluated at a number of locations along the crack front.

The q function must be specified at all nodes within the area or volume ofintegration. The shape of the q function is arbitrary, as long as q has the correct valueson the domain boundaries. In a plane stress or plane strain problem, for example, q = 1 atr0 (which is usually the crack tip) and q = 0 at the outer boundary. Figure 11.10illustrates two common examples of q functions for two-dimensional problems, with thecorresponding virtual nodal displacements. This example shows 4-node square elementsand rectangular domains for the sake of simplicity. The pyramid function (Fig. 11.10(a))is equal to 1 at the crack tip but varies linearly to zero in all directions, while the plateaufunction (Fig. 11.10(b)) equals 1 in all regions except the outer ring of elements. Shih,et al. [21] have shown that the computed value of / is insensitive to the assumed shape ofthe q function.

Figure 11.11 illustrates the pyramid function along a three-dimensional crack front,where the crack tip node of interest is displaced a unit amount, and all other nodes arefixed. If desired, J(r\) can be evaluated at each node along the crack front.

X2

(a) The pyramid function.

X2\

v~

» / • s ' /

\.

nc

* <

^™.

» «

i

>— e

-<

—<

^ >

\

_ ,

r >

\

r . /\

I

-4

(b) The plateau function.

FIGURE 11.10 Examples of q functions in two dimensions, with the corresponding virtual nodaldisplacements [21].


FIGURE 11.11 Definition of q in terms of avirtual nodal displacement along a three-dimensional crack front.

The value of q within an element can be interpolated as follows:

q(xt)= (11.39)1=1

where n is the number of nodes per element, qj are the nodal values of q, and Nj are theelement shape functions, which were introduced in Section 11.1.1.

The spatial derivatives of q are given by

(11.40)7=1 k=i

where £/ are the parametric coordinates for the element.In the absence of thermal strains, path-dependent plastic strains, and body forces

within the integration volume or area, the discretized form of the domain integral is asfollows:

Y \- i.~ L \V2j 3

crack faces V °*l(11.41)

where m is the number of Gaussian points per element, and Wp and w are weightingfactors. The quantities within { }p are evaluated at the Gaussian points. Note that theintegration over crack faces is necessary only when there are nonzero tractions.

586 Chapter 11

11.4 MESH DESIGN

The design of a finite element mesh is as much an art form as it is a science. Althoughmany commercial codes have automatic mesh generation capabilities, construction of aproperly designed finite element model invariably requires some human intervention.Crack problems, in particular, require a certain amount of judgement on the part of theuser.

This section gives a brief overview of some of the considerations that should governthe construction of a mesh for analysis of crack problems. It is not possible to address ina few pages all of the situations that may arise. Readers with limited experience in thisarea should consult the published literature, which contains numerous examples of finiteelement meshes for crack problems.

Figure 11.12 illustrates several common element types for crack problems. Shih, etal. [21] recommend 9-node biquadratic Lagrangian elements for two-dimensional problemsand 27-node biquadratic Lagrangian elements in three dimensions. The 8- and 20-node 2-D and brick elements are also common in crack problems.

At the crack tip, four-sided elements (in 2-D problems) are often degenerated down totriangles, as Fig. 11.13 illustrates. Note that three nodes occupy the same point in space.Figure 11.14 shows the analogous situation for three dimensions, where a brick elementis degenerated to a wedge.

In elastic problems, the nodes at the crack tip are normally tied, and the mid-side

nodes moved to the */4 points (Fig. 11.15(a)). Such a modification results in a Jf/Vrstrain singularity in the element, which enhances numerical accuracy.^ A similar resultcan be achieved by moving the mid-side nodes to */4 points in 4-sided elements, but thesingularity would only exist on the element edges [14,24]; triangular elements arepreferable in this case because the singularity exists within the element as well as on theedges. Appendix 11 presents a mathematical derivation that explains why moving themid-side nodes results in the desired singularity for elastic problems.

When a plastic zone forms, the 7/Vr singularity no longer exists at the crack tip.Consequently, elastic singular elements are not appropriate for elastic-plastic analyses.Figure 11.15(b) shows an element that exhibits the desired strain singularity under fullyplastic conditions. The element is degenerated to a triangle as before, but the crack tipnodes are untied and the location of the mid-side nodes is unchanged. This elementgeometry produces a 1/r strain singularity, which corresponds to the actual crack tip strainfield for fully plastic, nonhardening materials.

One side benefit of the plastic singular element design is that it allows the crack tipopening displacement (CTOD) to be computed from the deformed mesh, as Fig. 11.16illustrates. The untied nodes initially occupy the same point in space, but move apart asthe elements deform. The CTOD can be inferred from the deformed crack profile bymeans of the 90° intercept method (See Fig. 3.4).

•3Jln principle, the desired singularity can be achieved with ordinary elements, but a great deal of meshrefinement would be required to capture the crack tip fields. Moving the mid-side nodes forces the elementto exhibit the intended behavior,


(a) S-node 2-D element. (b) 9-node 2-D element.

(b) 20 node brick element. (d) 27-node brick element.

FIGURE 11.12 Isoparametric elements that are commonly used in two- and three-dimensional crackproblems.

FIGURE 11.13 Degeneration of a quadrilateral element into a triangle at the crack tip.

588 Chapter 11

(a) Elastic singularity element

FIGURE 11.14 Degeneration of a brick elementinto a wedge.

(b) Plastic singularity element.

FIGURE 11.15 Crack tip elements for elastic and elastic-plastic analyses. Element (a) produces a Inrstrain singularity, while (b) exhibits a 1/r strain singularity.

FIGURE 11.16 Deformed shape of plastic singularity elements (Fig. 11.15(b)). The crack tip elementsmodel blunting, and it is possible to measure CTOD.


For most problems, the most efficient mesh design for the crack tip region hasproven to be the "spider web" configuration, which consists of concentric rings of foursided elements that are focused toward the crack tip. The inner-most ring of elements aredegenerated to triangles, as described above. Since the crack tip region contains steepstress and strain gradients, the mesh refinement should be greatest at the crack tip. Thespider web design facilitates a smooth transition from a fine mesh at the tip to a coarsermesh remote to the tip. Figure 11.17 shows a half-symmetric model of a simple crackedbody, in which a spider-web mesh transitions to coarse rectangular elements.

The appropriate level of mesh refinement depends on the purpose of analysis.Elastic analyses of stress intensity or energy release can be accomplished with relativelycoarse meshes since modern methods, such as the domain integral approach, eliminate theneed to resolve local crack tip fields accurately. The area and volume integrations in thenewer approaches are relatively insensitive to mesh size for elastic problems. The meshshould include singularity elements at the crack tip, however, when the domain is definedas a small region near the crack tip. If the domain is defined over a large portion of themesh, singularity elements are unnecessary, because the crack tip elements contributelittle to J. The relative contribution of the crack tip elements can be adjusted through thedefinition of the q function. For example, in elastic problems, the crack tip elements donot contribute to / when the plateau function (Fig. 11.10(b)) is adopted, since dq/dxj = 0at the crack tip.

FIGURE 11.17 Half-symmetric model of acracked panel.

590 Chapter 11

FIGURE 11.18 Crack tip region of a mesh for large strain analysis. Note that the initial crack tip radiusis finite and the crack tip elements are not degenerated.

Elastic-plastic problems require more mesh refinement in regions of the body whereyielding occurs. When a body experiences net section yielding, narrow deformation bandsoften propagate across the specimen (Fig. 3.26). The high level of plastic strain in thesebands will make a significant contribution to the J integral; the finite element mesh mustbe sufficiently refined in these regions to capture this deformation accurately.

When the purpose of the analysis is to analyze crack tip stresses and strains, a veryhigh level of mesh refinement is required [25,26]. As a general rule, it is desirable tohave at least 10 elements on a radial line in the region of interest. In addition, if it isnecessary to infer crack tip fields at distances less than twice the CTOD from the cracktip, the analysis code must incorporate large strain theory. McMeeking and Parks [26]were among the first to apply a large strain analysis to the crack tip region. Figure 3.13is a plot of some of their results.

In a large strain analysis, it is advisable to begin with a finite radius at the crack tip,as Fig. 11.18 illustrates. Note that the crack tip elements are not collapsed to triangles inthis case. Provided the CTOD after deformation is at least 5 times the initial value, theresults should be not be affected by the initial blunt notch [26].

We will not address boundary conditions in detail, but it is worth mentioning acommon pitfall. Many problems require forces to be applied at the boundaries of thebody. For example, a single edge notched bend specimen is loaded in three point bending,with a load applied at mid-span, and appropriate restraints at each end. In elastic-plasticproblems, the manner in which the load is applied can be very important. Figure 11.19shows both acceptable and unacceptable ways of applying this boundary condition. If theload is applied to a single node (Fig. 11.19(a)), a local stress and strain concentration willoccur, and the element connected to this node will yield almost immediately. Theanalysis code will spend an inordinate amount of time solving a punch indentationproblem at this node, while the events of interest may be remote from the boundary. Abetter way of applying this boundary condition might be to distribute the load overseveral nodes, and specifying that the elements on which the load acts remain elastic (Fig.11.19(b)); the load will then be transferred to the body without wasting computer timesolving a local indentation problem. If, however, the local indentation is of interest,


(e.g., if one wants to simulate the effects of the loading fixture) the load can be applied bya rigid or elastic indenter with a finite radius, as Fig. 11.19(c) illustrates. Note, however,that greater mesh refinement is required to resolve the plastic deformation at the indenter.

ft dfin ir *

J. •<

§ -,,,-m t• w '

i- «

1§ 11 M.liijl 1

» J

lUinniBfftll ... ,r"""P' 1

1 <

F

ri™«Jft i

fe 1

j»». jr ""V"""S

F 1

1- 4

i 4

a i

) 1

I <

1, .,„„,,„,*, it

» 1

1- 1

''*—.-A>-~.,^

1 j% (j ;

1 <

I" <

fe <g> u

F/7

1> «

: 4

,—»-» K

ft ft fl

* <

> ' <

(a) Point force applied to a single node. NOT (b) Distributed force applied to elastic elements.RECOMMENDED.

Linear Elastic Elements

Elastic-Plastic Elements

FIGURE 11.19 Examples of improper (a) andproper (b & c) methods for applying a force to aboundary.

(c) Finite radius indenter. (Nodes omitted forclarity.)

11.5 LIMITATIONS OF NUMERICAL FRACTURE ANALYSIS

Although computational methods are very useful in fracture mechanics, they cannotreplace experiments. A numerical fracture simulation of a cracked body can computecrack tip parameters, but such an analysis alone cannot predict when fracture will occur.Techniques such as finite element analysis and boundary integral elements rely on contin-uum theory. A continuum does not contain voids, microcracks, second-phase particles,grain boundaries, dislocations, atoms, or any of the other microscopic or submicroscopicfeatures that control fracture behavior in engineering materials (see Chapters 5 and 6).

592 Chapter 11

A numerical analysis of a cracked body can provide information on local stresses andstrains at the crack tip, as well as global fracture parameters. Existing analyses,however, model only the deformation of the material. Fracture can be modeled, but aseparate failure criterion is required. For example, one might model cleavage fracture byimposing a stressed-based failure criterion, in which the analysis would predict failurewhen a user-specified stress is reached at a particular point ahead of the crack tip.Predictions of fracture could not be made a priori in such cases, but would require one ormore experiments to infer material-dependent parameters in the local fracture model.

Several researchers have attempted to combine flow and fracture behavior into asingle constitutive model, and have incorporated such approaches into finite elementanalyses. The Gurson model, for example (Chapter 5), was intended to model bothplastic flow and ductile fracture in metals. Because this approach is a continuum modeland does not include voids, however, it does not capture the important microscopic eventsthat lead to fracture, and it is unable to predict failure in real materials. A number ofadjustable parameters have recently been added to this model in order to bring predictionsin line with experimental data, but such parameters are based on curve fitting rather thansound physics.

Numerical analysis will undoubtedly play a major role in developingmicromechanical models for fracture. Computer simulation of processes such asmicrocrack nucleation, void growth, and interface fracture should lead to new insights intofracture and damage mechanisms. Such research may then lead to rational failure criteriathat can be incorporated into global continuum models of cracked bodies.

To reiterate the statement at the beginning of this section: computer modelingcannot replace experimentation. Any mathematical model, regardless of how sophisticatedit is, must omit much of the real world in its formulation. Models often leave out thevery feature that controls the physical process. Unlike a mathematical model, anexperiment is obliged to obey all laws of nature, down to the quantum level. Thus anexperiment often conveys important information that a simulation overlooks.

REFERENCES



3. Lapidus, L. and Finder, G.F., Numerical Solution of Partial Differential Equations inScience and Engineering. John Wiley and Sons, New York, 1982.

4. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method. (Fourth Edition)McGraw-Hill, New York, 1989.

5. Rizzo, F.J., "An Integral Equation Approach to Boundary Value Problems of ClassicalElastostatics." Quarterly of Applied Mathematics, Vol. 25, 1967, pp. 83-95.


6. Cruse, T.A., Boundary Element Analysis in Computational Fracture Mechanics. KluwerAcademic Publishers, Dordrect, Netherlands, 1988.

7. Blandford, G.E. and Ingraffea, A.R., "Two-Dimensional Stress Intensity FactorComputations Using the Boundary Element Method." International Journal forNumerical Methods in Engineering, Vol 17, 1981, pp. 387-404.

8. Cruse, T.A., "An Improved Boundary-Integral Equation for Three Dimensional ElasticStress Analysis." Computers and Structures, Vol. 4, 1974, pp. 741-754.

9. Mendelson, A. and Albers, L.U., "Application of Boundary Integral Equations toElastoplastic Problems." Boundary Integral Equation Method: ComputationalApplications in Applied Mechanics, AMD-Vol 11, American Society of MechanicalEngineers, New York, 1975, pp. 47-84.

10. Parks, D.M., "A Stiffness Derivative Finite Element Technique for Determination ofCrack Tip Stress Intensity Factors." International Journal of Fracture, Vol. 10, 1974,pp. 487-502.

11. Kobayashi, A.S., Cherepy, R.B., and Kinsel, W.C., "A Numerical Procedure forEstimating the Stress Intensity Factor of a Crack in a Finite Plate." Journal of BasicEngineering, Vol. 86, 1964, pp. 681-684.

12. Gross, B. and Srawley, J.E., "Stress Intensity Factors of Three Point Bend Specimensby Boundary Collocation." NASA Technical Note D-2603, 1965.

13. Tracey, D.M., "Finite Element Methods for Determination of Crack Tip Elastic StressIntensity Factors." Engineering Fracture Mechanics, Vol. 3, 1971, pp. 255-266.

14. Barsoum, R.S, "On the Use of Isoparametric Finite Elements in Linear FractureMechanics." International Journal for Numerical Methods in Engineering, Vol. 10,1976, pp. 25-37.

15. Budiansky, B. and Rice, J.R., "Conservation Laws and Energy Release Rates." Journalof Applied Mechanics, Vol. 40, 1973, pp. 201-203.

16. Carpenter, W.C., Read, D.T., and Dodds, R.H. Jr., "Comparison of Several PathIndependent Integrals Including Plasticity Effects." International Journal of Fracture,Vol. 31, 1986, pp. 303-323.

17. Hellen, T.K., "On the Method of Virtual Crack Extensions." International Journal forNumerical Methods in Engineering, Vol 9, 1975, pp. 187-207.

18. Parks, D.M.. "The Virtual Crack Extension Method for Nonlinear Material Behavior."Computer Methods in Applied Mechanics and Engineering, Vol. 12, 1977, pp. 353-364.

19. deLorenzi, H.G., "On the Energy Release Rate and the J-Integral of 3-D CrackConfigurations." International Journal of Fracture, Vol. 19, 1982, pp. 183-193.

20. deLorenzi, H.G., "Energy Release Rate Calculations by the Finite Element Method,"Engineering Fracture Mechanics, Vol. 21, 1985, pp. 129-143.

594 Chapter 11

21. Shih, C.F., Moran, B., and Nakamura, T., "Energy Release Rate Along a Three-Dimensional Crack Front in a Thermally Stressed Body." International Journal ofFracture, Vol. 30, 1986, pp. 79-102.

22. Moran, B. and Shih, C.F., "A General Treatment of Crack Tip Contour Integrals."International Journal of Fracture, Vol. 35, 1987, pp. 295-310.

23. Dodds, R.H., Jr. and Vargas, P.M., "Numerical Evaluation of Domain and ContourIntegrals for Nonlinear Fracture Mechanics." Report UILU-ENG-88-2006, University ofIllinois, Urbana, IL, August 1988.

24. Henshell, R.D. and Shaw, K.G., "Crack Tip Finite Elements are Unnecessary."International Journal for Numerical Methods in Engineering, Vol. 9, 1975, pp. 495-507.

25. Dodds, R.H. Jr., Anderson T.L., and Kirk, M.T. "A Framework to Correlate aAV Effectson Elastic-Plastic Fracture Toughness (Jc)." to be published in International Journal offracture.

26. McMeeking, R.M. and Parks, D.M., "On Criteria for J-Dominance of Crack Tip Fieldsin Large-Scale Yielding." ASTM STP 668, American Society for Testing and Materials,Philadelphia, 1979, pp. 175-194.

APPENDIX 11: PROPERTIES OFSINGULARITY ELEMENTS

Certain element/node configurations produce strain singularities. While such behavior isundesirable for most analyses, it is ideal for elastic crack problems. Forcing the elements

at the crack tip to exhibit a 7/v r strain singularity greatly improves accuracy and reducesthe need for a high degree of mesh refinement at the crack tip.

The derivations that follow show that the desired singularity can be produced inquadratic isoparametric elements by moving the mid-side nodes to the V4 points. Thisbehavior was first noted by Barsoum [14] and Henshell and Shaw [24].

From Eqs. (11.3) and (11.4), the strain matrix for a two-dimensional element can bewritten in the following form:

(All.l)

where

[B*] =

0

o37]

377

(All.2)

where (£, 77) are the parametric coordinates of a point on the element. Since the nodaldisplacements, (w/, v/}, are bounded, the strain matrix can only be singular if either [B*]or [J]~l is singular.

Consider an 8-noded quadratic isoparametric 2-D element (Fig 1 1. 12(a)). The shapefunctions for this element are as follows [4]:

t =[(1 7777,-) -(1-

(A11.3)

where (£, rj) are the parametric coordinates of a point in the element and (^j, 77;) are thecoordinates of the ith node.

595

- 0 y- Appendix 11

In general the shape functions are polynomials. Equation (All.3), for example, isa quadratic equation. Thus tf/, dNj/d& or dNi/dr] are all nonsingular, and [J] must be the

cause of the singularity.A strain singularity can arise if the determinant of the Jacobian matrix vanishes at

the crack tip:

All.l QUADRILATERAL ELEMENT

Consider an 8-noded quadrilateral element with the mid-side nodes at the 1/4 point, as Fig.All.l illustrates. For convenience, the origin of the x-y global coordinate system isplaced at node 1 . Let us evaluate the element boundary between nodes 1 and 2. From Eq.(Al 1.3), the shape functions along this line at nodes 1, 2, and 5 are given by

(A11.5)

Inserting these results into Eq. (Al 1.1) gives

Setting xi - 0, x2 = L, and *5 = /4 results in

where L is the length of the element between nodes 1 and 2. Solving for % gives

The relevant term of the Jacobian is given by

Properties of Singularity Elements 597

3L/4

FIGURE All.l Quadrilateral isoparametricelement with mid-side nodes moved to thequarter points.

OX L- = - (All.9)

which vanishes at x = 0; thus the strain must be singular at this point. Considering onlythe displacements of points 1, 2, and 5, the displacements along the element edge are asfollows:

(All. 10)

Substituting Eq. (All.8) into Eq. (All.10) gives

2 _ 2 £LU

r if r- -_, i A- -^ I -A- ,

-1 + 2, — 2J— MO + 4 J (All.11)

Solving for the strain in the * direction leads to

598 Appendix 11

U5 (All. 12)

Therefore, the strain exhibits a In r singularity along the element boundary.

A11.2 TRIANGULAR ELEMENT

Let us now construct a triangular element by collapsing nodes 1, 4, and 8 (Fig. A11.2).

Nodes 5 and 7 are moved to the quarter points in this case. The 7/V r strain singularityexists along the 1-5-2 and 4-7-3 edges, as with the quadrilateral element. In this instance,however, the singularity also exists within the element.

Consider the x axis, where r\ = 0. The relationship between x and £ is given by

2 2t n_A4

where LI is the length of the element in the x direction. Solving for <fj gives

(All.13)

(All. 14)

which is identical to Eq. (All.8). Therefore, the strain is singular along the x axis inthis element. By solving for strain as before (Eqs. (A 11.10) to (A 11.12)) it can easily be

shown that the singularity is the desired 7/V r type.

FIGURE A11.2 Degenerated isoparametricelement, with mid-side nodes at the quarterpoints.

PART V: REFERENCE MATERIAL

12. COMPILATION OF K, J,COMPLIANCE AND LIMIT LOAD

SOLUTIONS

This chapter lists stress intensity, compliance, limit load, plastic / and plasticdisplacement solutions for selected configurations. Table 12.1 summarizes thegeometries and solutions that are provided. The geometries are divided into threecategories: (1) through-thickness cracks in flat plates, (2) part-through cracks in flatplates, and (3) flawed cylinders. Note that Kj solutions are given in all cases but thatfully plastic J solutions are available for only a limited number of configurations.

12.1 THROUGH-THICKNESS CRACKS-FLAT PLATES

Figure 12.1 illustrates eight flat plate/through thickness crack configurations. Note thatthis geometry category includes most of the common test specimens.

12.1.1 Stress Intensity and Elastic Compliance

Table 12.2 gives polynomial Kj expressions for the eight configurations shown in Fig.12. 1 . The load line compliance solutions for some of these geometries are listed in Table12.3. It is important to note that the compliance values in Table 12.3 correspond to thetotal load line displacement. Recall from Chapter 3 that the load line displacement can bedivided into crack and no crack components:

where Anc is the displacement (at a given load) that would be measured in the absence ofa crack, and Ac is the additional displacement that is due to the crack. The no crackcompliance for each of the configurations in Table 12.3 can be readily inferred by setting

601

602 Chapter 12

TABLE 12.1Summary of solutions in Chapter 12.

Geometry

(a) ThroughCracks, Flat

Plates.Edge crack

"»"

Center crackDouble edge notch

Compact spec.Disk compactArc specimen

(b) Part-ThroughCracks, Flat

Plates.Surface crack

I!

Ellip. buried crackCorner crack3. FlawedCylinders.

Thr-wall, circum.ti11

Long, circum.Part-thr., circum.

Thr-wall, axialLong, axial

Part-thr., axial11

Loading

Tension3 Pt. BendPure Bend

T + BTensionTensionTensionTensionTension

T + BPolynom.Tension

T+B

TensionBendingT + B

TensionTensionPressurePressurePressure

Polynom.

KI

VVVVVVVVV

VVVV

VVVVVVVVV

Compl

VV

VVV

V

V

PL

V- V

VVVV

V

VVVV

V

Jpl

VV

VVVV

VVVV

V

Ap, Vp

A/

VVVVV

V

V

- Mode I stress intensity.Comp. - Elastic compliance.PL - Limit load.Jpl - Plastic J integral.Ap, Vp - Plastic displacements.

K, J, Compliance, and Limit Load Solutions 603

1.25 W

(a) Compact specimen.

P/2

(b) Disk-shaped compact specimen.

IP/2

W

P,A(c) Single edge notched bend (SENS) specimen.

P,Af tP,A

•2W

•2a

t t t t(d) Middle tension (MT) panel. (e) Arc-shaped specimen.

FIGURE 12.1 Flat plates with through-thickness cracks.

604 Chapter 12

SJ

W

P,A

(f) Single edge notched tension (SENT) panel.

2W

IDouble edge notched tension (DENT) panel.

P,A

(g)

*w

(h) Edge cracked plate in pure bending.

M,

W

(i) Edge cracked plate in combined bending and tension.

FIGURE 12.1 (Cont.)


TABLE 12.2Nondimensional Kj solutions for through-thickness cracks in flat plates [1,2]. See Fig. 12.1 for a definitionof dimensions for each configuration.

^J w p(a) Compact specimen.

W

1-W

\w w w w

(b) Disk shaped compact specimen.

W

W

0.76 + 4.8 — -11., ,WJ {W W W

(c) Single edge notched bend specimen loaded in three-point bending.

-V K V W

w w

(d) Middle tension (MT) panel.

Tta Jtasec

4W 2W1-0.025 —

W {w

(e) Arc shaped specimen.

—+ 1.9 + 1.1— 1 + 0.25 1-—W WJ\ { W

where

1- —W

w w w

606 Chapter 12

TABLE 12.2 (Cent.)

(f) Single edge notched tension (SENT) panel.

2 tanno.

2W

cos-na

2W

(g) Double edge notched tension (DENT) panel.

II2W

(h) Edge cracked plate subject to pure bending.

.3/2 6j2tanna

M ( na\cos

(2W

s •^

1-sin - -UwJ

(i) Edge cracked plate subject to combined bending and tension.

W

where ft and/£ are given above in (f) and (h), respectively.


TABLE 123Nondimensional load line compliance solutions for through-thickness cracks in flat plates [3]*. See Fig.12.1 for a definition of dimensions for each configuration.

Nondimensional Compliance:ABE'

where A = Ac + Afl


W

\ YY yW

2.163 +12.21 g( — ) - 20.065f—} - 0.992sf—(w) (w) (w,

+20.609f—l -9.9314^—Y\wj (w)

(b) Single edge notched bend (SENS) specimen loaded in three-point bending.

_ ^2

53(l-v2)

W'I

W— | (1 + V)O

+ 1.5 — W 5.58-19.57 —W

.82W2 -34.94f-^f +12.77Wl^J IwJ Iw^J

(c) Middle tension (MT) panel.

2W-1.071 +0.25o

w(d) Single edge notched tension (SENT) specimen.

,29 4l'

L(1-V2) +

2WW

w .w w

(e) Double edge notched tension (DENT) panel.

2W0.0629-0.0610| cos - - 1 -0.0019[ cos^- ) +ln|sec-^-]

2WJ 2W 2WJ

*For side-grooved specimens, B should be replaced by an effective thickness:

Be=B- N where Btf is the net thickness.B

608 Chapter 12

TABLE 12.4Crack length-compliance relationships for compact and three-point bend specimens [4].*

Compact Specimen.

— = 1.00196 - 4.06319ULL +11.24217^ -106.043t/|, + 464.335f/fr - 650.677 U5T

W

where

'LL

7ZLL =BEA

Single Edge Notched Bend Specimen Loaded in Three-Point Bending.

a~W

where

UV = arid Zy =BEV

+ 1

*For side-grooved specimens, B should be replaced by an effective thickness:

(B - BN )Bo = B -- where BN is the net thickness.

6 B

For large rotations, a compliance correction is required for the compact specimen. (See ASTM E 813

[4].)

SENB Specimen. Compact specimen.


12.1.2 Limit Load

TABLE 12.5Limit load solutions for through-thickness cracks in flat plates [5,6].* See Fig. 12.1 for a definition ofdimensions for each configuration.


where

Pi = 1.455 T\Bb(JY (Plane Strain)

(Plane Stress)

(2aY 4a . (2a ,= 1"~T + —+ 2- —+1b ) b V b

(b) Single edge notched bend (SENS) specimen loaded in three-point bending.

p, -L

™ o - v(Plane Strain)

(c) Middle tension (MT) panel.

PL = —= Bbo-y (Plane Strain)

= 2BbOy (plane Stress)

(d) Single edge notched tension (SENT) specimen.

PL = \. 455 T]BbOy (Plane Strain)

Pi = 1.072 rjBbCfy (Plane Stress)where

(e) Double edge notched tension (DENT) panel.

Pr =10.72 + 1.82—Way (Plane Strain)L ( WJ Y

"•'* (plane Stress)

*The flow stress, ay, is normally taken as the average of 0^5 and

610 Chapter 12

TABLE 12.5 (Cont.)*

Edge Crack Subject to Combined Bending and Tension.

PL~ V3 w (Plane Strain)

where

PL = BbaY

M

PW

(Plane Stress)

*The flow stress, cry, is normally taken as the average of ays an^

12.1.3 Fully Plastic J and Displacement

The Electric Power Research Institute (EPRI) / estimation scheme [5] provides a meansfor computing the / integral in a variety of configurations and materials. A fully plasticsolution is combined with the stress intensity solution to obtain an estimate of theelastic-plastic /. Section 9.5 describes the theoretical background and applications of thisapproach.

The fully plastic J integral in the EPRI / estimation scheme is normally expressedin the following form:

Jpl = ae0o0bhl\ (12.2)

where b is a characteristic length dimension, hj is a geometry factor, P is a characteristicload, and P0 is a reference load, which is usually inferred from the limit load solution forthe geometry of interest. The other parameters in Eq. (12.2) are flow properties that aredefined by a Ramberg-Osgood fit to the material stress-strain curve:

a (J(12.3)


where cr and £ are uniaxial stress and strain, respectively, o0 is a reference stress (usuallydefined at yield), eo = (?o/E, a is a fitting constant, and « is a hardening exponent.

The geometry factor, h], has been tabulated for a variety of configurations andhardening exponents (Tables 12.6-12.21, 12.34-12.51) [5-7].

The total J is given by the sum of the fully plastic value and an effective elastic J:

(12-4)

where Jel is computed from the elastic stress intensity factor of an effective crack size:

oK

02.5)

where E' = E for plane stress and E' = E/(l - v*) for plane strain conditions. Theparentheses in Eq. (12.5) indicates that Kj is a function of aeff, rather than amultiplication product. The effective crack size is inferred from a first order Irwincorrection:

0

1 (n-l\Kr(a)}^ } (12.6)

V '

where^ = 2 for plane stress and /? = 6 for plane strain conditions.Elastic-plastic displacements can also be computed from the EPRI approach by

adding the elastic displacement, inferred from the compliance solution with crack size =aeff, to the plastic displacement. The fully plastic crack mouth opening displacement,Vp, and plastic load line displacement, Ap are normally expressed in the following form:

Vp = ae0ah2(a I WM — (12.7)

Ap = ae0ah3(a I Wtri)\ — (12.8)\"o J

where h.2 and /zj are geometry factors.

612 Chapter 12

TABLE 12.6Fully plastic / and displacement for a compact specimen in plane strain [5].

a/W:

0.250

0.375

0.500

0.625

0.750

-» 1

hjfa3

hifa2

h3

hi2

h3

hi2

h3

h jh2

h3

hjh2

h3

n-l

2.2317.99.852.1512.67.941.949.336.411.767.615.521.716.374.861.575.394.31

n = 2

2.0512.58.511.728.185.761.515.854.271.454.573.431.423.953.051.453.742.99

n = 3

1.7811.78.171.396.524.641.244.303.161.243.422.581.263.182.461.353.092.47

n = 5

1.4810.87.770.9704.323.100.9192.752.020.9742.361.791.0332.341.811.182.431.95

n = 7

1.3310.57.710.6932.972.14

0.6851.911.41

0.7521.811.37

0.8641.881.451.082.121.79

n = 10

1.2610.77.92

0.4431.791.29

0.4611.20

0.8880.6021.321.00

0.7171.441.11

0.9501.801.44

n = 13

1.2511.58.52

0.2761.10

0.7930.3140.7880.5850.4590.9830.7460.5751.12

0.8690.8501.571.26

n=16

1.3212.69.31

0.1760.6860.4940.2160.5300.3930.3470.7490.5680.4480.8870.6860.7301.331.07

n = 20

1.5714.610.9

0.0980.3700.2660.1320.3170.2360.2480.4850.3680.3450.6650.5140.6301.14

0.909

Jpl = ae0o0bhi(alW,n}\ —

Ap = a£0ah3(a/W,n)\ —

P0= 1.455 T]Bb<J0

where

\ 4a „ (la+ — + 2- —

) b (b

Thickness = B


TABLE 12.7Fully plastic J and displacement for a compact specimen in plane stress [5].

a/W:

0.250

0.375

0.500

0.625

0.750

-

hi

^2

fa!

h3

hifa2

fa3

hjh2

hi

fa2

h3

h t

h2

h3

n = 1

1.6117.69.67

1.5512.47.801.409.166.291.277.475.42

1.236.254.77

1.135.294.23

n=2

1.4612.08.00

1.258.205.731.085.674.151.034.483.38

0.9773.782.92

1.013.542.83

n = 3

1.2810.77.21

1.056.544.62

0.9014.213.110.8753.352.540.8332.892.24

0.7752.411.93

n = 5

1.068.745.94

0.8014.563.250.6862.802.090.6952.371.80

0.6832.141.66

0.6801.911.52

n = 7

0.9037.325.00

0.6473.452.48

0.5582.121.59

0.5931.921.47

0.5981.781.38

0.6501.731.39

n=10

0.7295.743.95

0.4842.441.77

0.4361.571.18

0.4941.541.18

0.5061.441.12

0.6201.591.27

n = 13

0.6014.633.19

0.3771.831.33

0.3561.25

0.9380.4231.29

0.9880.4311.20

0.9360.4901.23

0.985

n = 16

0.5113.752.59

0.2841.36

0.9900.2981.03

0.7740.3701.12

0.8530.3731.03

0.8000.4701.17

0.933

n = 20

0.3952.92

2.0230.2201.02

0.7460.2380.8140.6140.3100.9280.7100.3140.8570.6660.4201.03

0.824

Jpl = ae0G0bhi (a 1 W, n)\ —

Ap = ae0ah^(a/W,n)

P0=l.Q12r]Bb(70

where

b b

Thickness = B

614 Chapter 12

FuHj piastic J and displace*,,,, for a singie ed^nSed bend (SENB, specimen in piane strain subject

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hifa2

h3hj

h?,h3hih2h3hih?,h3

hih?,h3

hih2

h3

hih2h3

n = l

0.9366.973.001.205.804.081.335.184.511.414.874.691.464.644.711.484.474.491.504.364.15

n=2

0.8696.7722.11.0344.679.721.153.936.011.093.284.331.072.863.491.152.753.141.352.903.08

n = 3

0.8056.2920.00.9304.018.361.023.205.030.9222.533.490.8962.162.700.9742.102.401.202.312.45

n = 5

0.6875.2915.0

0.7623,085.860.0842.383.740.6751.692.350.6311.371.72

0.6931.361.561.021.701.81

n = 7

0.5804.3811.7

0.6332.454.470.6951.933.020.4951.191.66

0.4360.9071.14

0.5000.9361.07

0.8551.331.41

n = 10

0.4373.248.39

0.5231.933.42

0.5561.472.300.3310.7731.08

0.2550.5180.6520.3480.6180.7040.6901.001.06

n = 13

0.3292.406.14

0.3961.452.54

0.4421.151.80

0.2110.4800.6690.1420.2870.3610.2230.3880.4410.5510.7820.828

n = 16

0.2451.784.54

0.3031.091.90

0.3600.9281.45

0.1350.3040.4240.0840.1660.2090.1400.2390,2720.4400.6130.646

n = 20

0.1651.193.01

0.2150.7581.32

0.2650.6841.07

0.07410.1650.2300.04110.08060.1020.07450.1270.1440.3210.4590.486

Ap =

\A55Bb2a',

n+1

Thickness = B


TABLE 12.9Fully plastic / and displacement for a single edge notched bend (SENS) specimen in plane stress subjectto three-i

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

>omt bending [5].

hih2

h3hih2

h3

hih2

h3

hih2

h3

hih2

h3

hih2

h3

hih2

h3

n = l

0.6766.842.950.8695.694.010.9635.094.421.024.774.601.054.554.621.074.394.391.0864.284.07

n = 2

0.6006.3020.10.7314.508.810.7973.735.530.7673.124.090.7862.833.430.7862.663.010.9282.762.93

n = 3

0.5485.6614.6

0.6293.687.190.6802.934.480.6212.323.090.6492.122.600.6431.972.240.8102.162.29

n = 5

0.4594.5312.2

0.4792.614.730.5272.073.170.4531.552.080.4941.461.79

0.4741.331.516.461.561.65

n = 7

0.3833.649.120.3701.953.390.4181.582.410.3241.081.44

0.3571.021.26

0.3430.9281.05

0.5381.231.30

n = 10

0.2972.726.75

0.2461.292.20

0.3071.131.73

0.2020.6550.8740.2350.6560.803

'0.2300.6010.6800.4230.9220.975

n=13

0.2382.125.20

0.1740.8971.52

0.2320.8411.28

0.1280.4100.5450.1730.4720.5770.1670.4270.4830.3320.7020.742

n = 16

0.1921.674.09

0.1170.6031.01

0.1740.6260.9480.08130.2590.3440.1050.2860.3490.1100.2800.3160.2420.5610.592

n = 20

0.1481.263.07

0.05930.3070.5080.1050.3810.5750.02980.09740.1290.04710.1300.1580.04420.1140.1290.2050.4280.452

Jpl = ae0<J0bhi (a I W, n)

Ap=as0ah3(a/W,n)\—-

_ .in —

n+1

Thickness = B

616 Chapter 12

TABLE 12.10Fully plastic J and displacement for a middle tension (MT) specimen in plane strain [5].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih2h*hih2

h3h!h2

h3hih2

h3

hih2h3

hih2

h3hih2h3

n = l

2.803.050.3032.542.680.5362.342.350.6992.212.030.8032.121.71

0.8442.071.35

0.8052.080.8890.632

n = 2

3.613.620.5743.012.990.9112.622.391.062.291.861.071.961.32

0.9371.73

0.8570.7001.64

0.4280.400

n = 3

4.063.910.8403.213.011.222.652.231.282.201.601.161.761.04

0.8791.47

0.5960.5551.40

0.2870.291

n = 5

4.354.061.303.292.851.642.511.881.441.971.231.101.43

0.7070.7011.11

0.3610.3591.14

0.1810.182

n = 7

4.333.931.633.182.611.842.281.581.401.761.00

0.9681.17

0.5240.5220.8950.2540.2540.9870.1390.140

n = 10

4.023.541.952.922.301.851.971.281.231.52

0.7990.7960.8630.3580.3610.6420.1670.1680.8140.1050.106

n = 13

3.563.072.032.631.971.801.711.071.051.32

0.6640.6650.6280.2500.2510.4610.1140.1140.6880.08370.0839

n = 16

3.062.601.962.341.711.641.46

0.8900.8881.16

0.5640.5650.4580.1780.1780.3370.08100.08130.5730.06820.0683

n = 20

2.462.061.772.031.451.431.19

0.7150.7190.9780.4660.4690.3000.1140.1150.2160.05110.05160.4610.05330.0535

(Jpl = ae0a0~hi(alW,n)\ —

yy v ff\

~ 0

V C


TABLE 12.11Fully plastic / and displacement for a middle tension (MT) specimen in plane stress [5].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih2

fa3

hih2

h3hih2

h3hih2h3hih2h3hih2

h3hih2

h3

n = l

2.803.530.3502.543.100.6192.342.710.8072.212.340.9272.121.97

0.9752.071.55

0.9292.081.03

0.730

n = 2

3.574.090.6612.973.291.012.532.621.202.202.011.191.911.461.051.71

0.9700.8021.57

0.4850.452

n = 3

4.014.430.9973.143.301.352.522.411.432.061.701.261.691.13

0.9701.46

0.6850.6421.31

0.3100.313

n = 5

4.474.741.553.203.151.832.352.031.591.811.301.181.41

0.7850.7631.21

0.4520.4501.08

0.1960.198

n = 7

4.654.792.053.112.932.082.171.751.571.631.071.041.22

0.6170.6201.08

0.3610.3610.9720.1570.157

n = 10

4.624.632.562.862.562.191.951.471.431.43

0.8710.8671.01

0.4740.4780.8670.2620.2630.8620.1270.127

n = 13

4.414.332.832.652.292.121.771.281.271.30

0.7570.7580.8530.3830.3860.7450.2160.2160.7780.1090.109

n = 16

4.134.002.952.472.082.011.611.131.131.17

0.6660.6680.7120.3130.3180.6460.1830.1830.7150.09710.0973

n = 20

3.723.552.922.201.811.791.43

0.9880.9941.00

0.5570.5600.5730.2560.2730.5320.1480.1490.6300.08420.0842

Ap(c)=a£0ahi(aIW,n)\ —

P0 = 2Bba0

^p(nc) = a£oL\

fK-2a.

618 Chapter 12

TABLE 12.12Fully plastic J and displacement for a single edge notched tension (SENT) specimen in plane strain [5].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih3hih2

hl

h3hihh3hl

h3

h1

h

he,hlh2h^

n= l

4.955.2526.64.344.7610.33.884.545.143.404.453.152.864.372.312.344.322.021.914.292.01

n=2

6.936.4725.8

4.774.567.64

3.253.492.99

2.302.771.54

1.802.441.08

1.612.521.10

1.572.751.27

n = 3

8.577.5625.24.644.285.872.632.671.901.691.89

0.9121.301.62

0.6811.251.79

0.7651.372.14

0.988

n = 5

11.59.4624.23.823.393.701.681.57

0.9230.9280.9540.4170.6970.0810.3290.7691.03

0.4351.101.55

0.713

n = 7

13.511.123.63.062.642.481.06

0.9460.5150.5140.5070.2150.3780.4230.1710.4770.6190.262

1 0.9251.23

0.564

/• \n-fl

J pi ~ &£ba

°°°W*,(«/«/ n\\

' } P I\ro J

[\n

P }—P 1ro J

Ap(c) - 0(.£0ah^(al W,n)V o )

where

( ' a\ a

\ \b ) b

n = 10

fix!12.923.2

2.171.811.50

0.5390.4580.2400.20.2040.0850.1530.1670.0670.2330.2960.1250.7020.9210.424

n=13

18.114.423.21.551.25

0.9700.2760.2290.119

0.09020.08540.03580.06250.06710.02680.1160.1460.0617

n = 16

19.915.723.51.11

0.8750.6540.1420.1160.0600.03850.03560.01470.02560.02720.01080.0590.07350.0312

n = 20

21.216.823.7

0.7120.5520.4040.05950.0480.02460.01190.01100.00450.00780.00820.00330.02150.02670.0113

nr / /— \n

^p(nc)

iV

V-3 r V3F— rtr T \

2 ° {4BW(J0)A AA Jp JaP

j

4 1 M M M . W , , , - f e t o r"' '" W "" ''""•"pw

4+ - iJMlUlr

r *

^J T»v\\\.\

|

1L

i

i

1


TABLE 12.13Fully plastic / and displacement for a single edge notched tension (SENT) specimen in plane stress [5].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

h,n2"3hln2h3h1h2h3hlh2h3hlh2"3nlh2D3h1n2h3

n = 1

3.585.1526.13.144.6710.12.884.475.052.464.373.102.074.302.271.704.241.981.384.221.97

n = 2

4.555.4321.63.264.306.492.373.432.651.672.731.431.412.551.131.142.471.091.112.681.25

n = 3

5.066.0518.02.923.704.361.942.631.601.251.91

0.8711.1051.84

0.7710.9101.81

0.7840.9622.08

0.969

n = 5

5.306.0112.72.122.532.191.371.69

0.8120.7761.09

0.4610.7551.16

0.4780.6241.15

0.4940.7921.54

0.716

f \n+lba A P\

P ° ° W ( Pn }V o j

( \ nP }

™o )

^.-^(./^J1

P0.1.072i,«fro-0

1 X S?

ll + l a \ a

^ \ (b) b

n = 7

4.965.479.241.531.761.241.011.18

0.5250.5100.6940.2860.5510.8160.3360.4470.7980.3440.6771.27

0.591

n=10

4.144.465.98

0.9601.05

0.6300.6770.7620.3280.2860.3800.1550.3630.5230.2150.2800.4900.2110.5741.04

0.483

n=13

3.293.483.94

0.6150.6560.3620.4740.5240.2230.1640.2160.0880.2480.3530.1460.1810.3140.136

n = 16

2.602.742.720.4000.4190.2240.3420.3720.157

0.09560.124

0.05060.1722.42

0.1000.1180.2030.0581

n = 20

1.922.022.0

0.2300.2370.1230.2260.2440.1020.04690.06070.02470.1070.1500.06160.06700.1150.0496

f \nA ac L( P }^^ 0(*BW*o)

A4^p

±

1

nf lA , , , , , , , , , , , , , , ,,tA7 , n , „ , „ , „„ , . ifelir-

4K

«JL*.* J&a

w^_.V ^^>

T

— i_L_JfiSte.

J

/1»\

A

T

L

1 T

^ , v

620Chapter 12

TABLE 12.14Fully plastic J and displacement for a double edge notched tension (DENT) specimen in plane strain [5].

••—II •— •—

a/W:•••••••i »••

0.125

•mi •• —

0.250

_

0.375

••PBHHBIBBMIBB*-""-""1!

0.500

^ •*»••••• •

0.625

0.750

0 875

••*»»«•»••»»•»•«••

••.••BMPBBIBJMM

hi"1^2h-a"3hi"1h2h-a"3 _hi1h2h-."3hih2hi11 3hi1 1h2h3

hi1h2h3

»1h-i"2h3

•n.i i"

n-I

0.5720.7320 063

••••••—•»•••«••'

1.101.56

0 267i i '

1.612.510 637

"2.223.731.26

— "3.165.572.365.249.104.7314.220.112.7

•• i '--

is=2

"0.7720.8520.126TsT

1.630.4791.832.411.05

"2.433.401.923.384.763.296.297.766.2624.819.418.2

!•• 1'"

n = 3

0.9220.9610.200

•"—•

1.381.700.6981.922.351.402.483.152.37

«™»^— "•«"—3.454.233.747.177.147.0339.022.724.1

NWMMMH>^">-

n = 5

•••••••a 'i N ""

1.131.14

0.372•— •— ••P^™—1.65

1.781.111.922.151.87

••••IBB-™— «••

2.432.712.793.423.463.908.446.647.6378.436.140.4

•M— «!—»"•—

n = 7

.HIM ••I-"

1.351.29

0.5711.751.801.471.841.942.112.322.372.853.282.973.689.466.838.14140.058.965.8

••— «——•••••

n = 10

1.611.50

0.9111.821.811.921.681.682.202.122.012.683.002.483.2310.97.489.04

341.0133.0149.0

n = 13

1.861.701.301.861.792.251.491.442.091.911.722.402.542.022.6611.97.799.40

294.0327.0

n = 16

2.081.941.741.891.782.491.321.251.921.601.401.992.361.822.401 1.37.148.58

1570.0585.0650.0

n = 20

2.442.172.291.921.762.731.121.051.671.511.381.942.271.662.19

11.113.5

1400.01560.0

( P Y+1

I r - ap cr —hi(alWnn — \w V/o J

( P VVp = a£0ah2(a/W,n)\ — \p 01 | J

OYAp(c) = cce0ahi(aIW,n) — -voy

P0=( 0.72 + 1.82AV,cr0\ W J

V3 T( V3PAp(nc) ~ 2 <**oJ\ 4BWaf

iV

T

4 4J'F

llliill || 11 111111

iilllllil l;i 11; iiiill?Wff^^Wf ****$*

rssiiiilr•tff&xZyfffttf ¥•:;: SSI: !:;:•:•: ggSS?

Illllllll I If 11 1111

i

.L

^

k


TABLE 12.15Fully plastic / and displacement for a double edge notched tension (DENT) specimen in plane stress [5].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih2h3

hlh2h3hih2

h3

h ]_h2h3hih2h3

hih2h3

hih2h3

n = 1

0.5830.8530.072

91.011.73

0.2961.292.590.6581.483.511.181.594.561.931.655.903.061.698.025.07

n=2

0.8251.05

0.159

1.231.82

0.5371.422.391.041.472.821.581.453.152.14

1.433.372.671.433.513.18

n = 3

1.021.230.26

1.361.89

0.7701.432.221.301.382.341.691.292.321.951.222.222.061.222.142.16

n = 5

1.371.55

0.504

1.481.921.171.341.861.521.171.671.561.041.451.44

0.9791.301.31

0.9791.271.30

n = 7

1.711.87

0.821

1.541.911.491.241.591.551.011.281.32

0.8821.061.09

0.8340.9660.9780.8450.9710.980

n = 10

2.242.381.41

1.581.851.821.091.281.41

0.8450.9441.01

0.7370.7900.8090.7010.7410.7470.7380.7750.779

n = 13

2.842.962.18

1.591.802.02

0.9701.071.23

0.7320.7620.8090.6490.6570.6650.6300.6360.6380.6640.6630.665

n — 16

3.543.653.16

1.591.752.12

0.8730.9221.07

0.6250.6300.6620.4660.4730.4870.2970.3120.3180.6140.5960.597

n = 20

4.624.704.73

1.591.702.20

0.6740.7090.8300.2080.2320.2660.02020.02770.0317

0.5620.5350.538

3 pl = ~: (a/ W, «) —

= CCE0ah3(a/W,n) —

T PAp(nc)=<X£oL\

±V

T

622 Chapter 12

TABLE 12.16Fully plastic J, displacement, and rotation for an edge cracked plate in plane strain subject to combinedtension and bending [6].

( \n+l

Jpl = 0£0a0-£hi(al W,n,A) —W {P0j

0 V3- 2A + —

W

W

M

Tables 12.17 to 12.21 list h^, /£, hj, and /z^ values for various A.

- **'• ii


TABLE 12.17h factors for an edge cracked plate in plane strain subject to combined bending and tension with A =0.125 [6].

a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih2

h3

h4

hih2

h3h4

hih2h3h4

hih2

h3h4

hih2

h3h4

hih2

h3h4

hih2h3h4

n = 1

4.7615.2750.3940.3283.5684.5100.7450.8892.7424.0270.9611.5362.2003.7671.1472.1161.7383.5181.2752.5601.4013.3731.3902.8811.1683.3481.5323.171

n '= 2

4.5444.9880.7790.3092.5363.1930.8680.6821.6572.3160.7870.9521.3051.9770.7241.1921.0561.7810.7071.3410.9011.7270.7411.5130.7851.7510.8081.664

n = 3

3.8819.3140.9250.3241.7732.1950.7650.5481.0161.3570.5390.6120.8041.1310.4650.7140.6781.0550.4410.8300.6091.0730.4620.9460.5511.1130.5141.059

n = 5

2.6322.8720.8980.2890.8431.0060.4430.3210.3730.4680.2040.2370.3100.3990.1570.2590.2900.4110.1620.3200.2930.4590.1930.4020.2830.5120.2350.486

n = 7

1.7341.8730.7480.2400.3920.4450.2270.1650.1360.1630.0730.0850.1200.1490.0570.0970.1290.1740.0670.1350.1420.2140.0900.1880.1530.2600.1190.247

n = 10

0.9050.9620.4880.1580.1190.1380.0750.0540.0290.0340.0150.0180.0300.0360.0130.0230.0390.0510.0190.0400.0500.0730.0300.0640.0600.1000.0460.095

624 Chapter 12


a/W:

0.125

0.250

0.375

0.500

0.625

0.750

0.875

hih2

h3h4hih2h3h4hih2

h3h4

hih2h3h4hih2h3h4hih2h3h4

hih2

h3h4

n = 1

1.9103.1110.2790.1812.0143.0170.5160.6212.0533.0690.7661.2001.9853.1841.0021.8281.7373.2091.1882.3541.4433.2171.3412.7541.1923.2841.5093.107

n = 2

1.7812.4370.3530.2531.9432.4240.6180.7121.6352.1190.7380.9461.2321.7420.7071.0061.0121.5890.6731.1760.8861.6070.7101.3700.7841.6970.7891.602

n = 3

1.4941.8230.3640.2521.7141.9520.6012.4741.1451.3870.5810.6040.7510.9830.4400.5490.6130.8800.3790.6430.5720.9400.4080.8040.5381.0440.4820.987

o = 5

1.1011.0980.2940.1391.2531.2990.5030.3850.5300.5920.3050.2150.2770.3310.1570.1750.2320.3050.1280.2220.2500.3760.1610.3240.2680.4700.2170.445

n = 7

0.8650.6830.3100.0070.7820.8640.3980.2350.2420.2570.1470.0830.1560.1700.0790.0910.0890.1130.0470.0830.1130.1630.0690.1410.1360.2290.1050.217

n = 10

0.6920.3440.428-0.0630.4710.4480.2510.1010.0740.0760.0460.0220.0230.0250.0120.0130.0220.0270.0110.0200.0350.0490.0210.043


TABLE 12.19h factors for an edge cracked plate in plane strain subject to combined bending and tension with A = -0.125 [6].

a/W:

0.375

0.500

0.625

0.750

0.875

hih2

fa3h4

hih2h3fa4

hih2

h3

fa4

hih2

fash4

hih2

h3

h4

n = 1

1.4902.2960.5980.9281.7232.6850.8691.5741.6822.9491.1112.1781.4563.1011.3032.6591.2173.2591.5003.081

n = 2

1.3641.6460.5650.7981.2351.6460.6890.9850.9881.4910.6711.0930.8741.5560.7241.2790.7841.6940.7941.585

n = 3

1.1801.2760.5070.6280.8161.0010.4690.5470.5940.8250.3950.5710.5530.8990.4120.7400.5321.0320.4810.971

n = 5

0.8040.7860.3740.3270.3460.3860.2100.1790.2170.2790.1360.1850.2310.3450.1540.2890.2590.4540.2100.429

n = 7

0.5090.4720.2600.1660.1470.1550.0910.0660.0800.1000.0480.0670.0990.1430.0630.1200.1300.2190.1010.207

e = 10

0.2410.2130.1370.0570.0420.0430.0290.0160.0190.0240.0130.0150.0290.0420.0190.0350.0650.1040.0480.099

626 Chapter 12

TABLE 12.20h factors for an edge cracked plate in plane strain subject to combined bending and tension with A = -0.1875 [6].

a/W:

0.500

0.625

€.750

0.875

hih2

h3h4

hih2h3h4

hih2

h3h4

hih2

h3h4

n = 1

1.2461.8120.636-1.2031.5262.4710.962-2.7441.4562.8991.2340.7581.2233.1811.4692.671

n = 2

1.1481.2890.521-5.1530.9761.3520.612-0.6010.8551.4400.6850.9290.7811.6450.7761.516

n = 3

0.9730.9880.438-2.8950.6090.7770.3790.0450.5250.8140.3870.6170.5220.9910.4640.926

n = 5

0.5980.5470.285-0.8620.2340.2720.1360.1320.2040.2870.1290.2380.2450.4200.1940.396

n = 7

0.3500.3000.179-0.2380.0900.1000.0500.0590.0810.1100.0490.0920.1190.1960.0900.185

n = 10

0.1570.1270.084-0.0220.0440.0230.0110.0140.0210.0280.0120.0230.0420.0660.0310.063


a/W:

0.625

0.750

0.875

hih2h3h4

hih2h3h4

hih2h3h4

n = 1

1.1481.5210.659-7.0601.4052.4911.091-1.0081.4453.3841.5722.533

n = 2

0.9891.0740.516-3.1180.8281.2640.6120.4190.9771.8950.9021.693

n = 3

0.7370.7360.385-1.2460.4970.7050.3520.4260.7031.2280.5801.126

n = 5

0.3820.3480.201-0.1330.1820.2360.1120.1820.3750.5950.2770.555

n = 7

0.1940.1680.1020.0240.0670.0840.0390.0670.2070.3160.1470.296

n = 10

0.0710.0590.0360.0230.0150.0190.0090.0150.0870.1300.0600.122


12.2 PART-THROUGH CRACKS-FLAT PLATES

TABLE 12.22Stress intensity solution for a semieiliptical surface flaw in a flat plate for a<c [8],

am - Membrane (tensile) stress.

CTb - Bending stress

= —£Tfc~ 21

where

/ =Wr

a a

where

r*

MI =1.13 -0.09| -U

M2=_0.54 + -°-89

f(j>fwS

0.2-f-^

M3 =0.5 —— + 14(1.0--0.65 + - ^ c

cos 0 + sin'c

24

Jw ~ secTTC a2W Vt

/2

0.1 + 0.351-t

- + 0.6-

= -1.22-0.121-c

/ xO.75 / Nl.5G2 =0.55-1.05 - +0.47 -

\cj \cj

628 Chapter 12

TABLE 12.23Stress intensity solution for a semielliptical surface flaw in a flat plate for oa [8].

s

-2W Y


c?b - Bending stress

Mt

where

/ =Wt

Q U V wwhere

( ~\1'65

0 = 1 + 1.464 -

F = T l +M3|j

M2=0.2|-

=-0.111-.a

a)\l

sin2 <b + cos2

fw -

+ 0.6a U

Gn=-0.04-0.4lf-]\a)

Go =0.55-1.93 - +1.38 -12

G21 =-2.11-0.771-

G,, =0.55-0.721-1 +0.141-a22


TABLE 12.24Stress intensity solution for an elliptical buried flaw in a flat plate [8].

4444


P- _,,. J7~ ~where

+ M2 A2 + M3

a + d

0-05'2 -

M3 =0-29

0.23 +

,~

1 + 4A

For a/c <1:

= 1 + 1/

.464 -

2 2~~ cos + sin

1/

Fora/c> 1:

Q = 1 + 1.464|-'65

630 Chapter 12

TABLE 1125Stress intensily solution for a quarter-ell ptical corner crack in a flat plate for a<c [8].

<i—^

•w


% - Bending stress

27where

7 =12

t

4® c ^

, flV fa x 4

F= M,+M,|- +M3 -t) \t

H=Hl+(H2-Hl)(sm<t>)p

fi = l + L464fÛ;

Af, =1.08-0. Osf-u1.06M, =-0.44 + -

0.3 + -c

M = -0.5-0. 25 - 1c j

0.08 + 0.41-t

0.08 + 0.15 -U

a\2 9 9— cos 0 + sin

(l-COS(/»)3

sec2W V r

0.1 + 0.35 - (1 — sin i

=-1.22-0.12

/ \0.75 /Â1

G2 =0.64-1.05 - +0.47 -\cj \.cj


TABLE 12.26Stress intensity solution for a quarter-elliptical corner crack in a flat plate for a>c [8].

/ ^ r>


G\) - Bending stress

Mt

where

12

Tt

1 a

i IK-'

F = Mj+M 2 -

rc. 464 -

ft =,|-l 1.08 + 0.03f-

M9 = 0.3751-1 Mo = -0.2s(-

S 2 = l +

0.08 + 0.4 -

0.08 + 0.15 -

(l-sin0)3

(1-cos^))3

— sin2 0 + cos2

//,=

Gn =-0.04-0.4l[-

G12= 0.55-1.93 -I +1.381-

21

/- \0.75 x \= 0.64-0.72-1 +0.14 -

632 Chapter 12

TABLE 12.27Influence coefficients for a semi-elliptical surface crack in a flat plate, where the remote opening modestress is fit to a cubic polynomial over the range 0 < x < a [9], See Section 9.1.1 for an example of thisapproach.

GI:

G0

GI

G2

G3

3^t''

2$K "

00.250.5

0.751

'00.250.5

0.7510

0.250.5

0.7510

0.250.5

0.751

* = 0.2c0 .2

0.6110.7480.9581.0901.1340.0800.2080.4260.6090.6800.0230.0760.2390.4320.5180.0100.0320.1470.3340.431

0.5

0.8160.9671.2401.4321.4980.1450.2780.5190.7260.8070.0550.1100.2850.4910.5830.0290.0520.1730.3690.470

0.8

1.2621.3821.6701.8401.8610.2750.4000.6460.8660.9480.1130.1650.3420.5610.6620.0600.0820.2050.4110.522

* = 0.4c0.2

0.7840.8180.9511.0511.0860.1270.2480.4450.6120.6760.0440.0980.2580.4430.5260.0220.0450.1620.3480.442

0.5

0.9650.9791.1121.2201.2580.1850.3010.4980.6700.7360.0730.1240.2840.4720.5560.0380.0600.1770.3640.460

0.8

1.2831.2221.2871.3721.3880.2750.3710.5490.7280.8000.1120.1550.3060.5040.5960.0590.0770.1880.3840.488

* « 1.0c0 .2

1.1501.0761.0391.0251.0210.2000.3620.5430.6710.7170.0750.1540.3340.5140.5890.0380.0760.2190.4170.513

0.5

1.2471.1481.0901.0681.0620.2290.3840.5590.6860.7330.0890.1650.3410.5220.5970.0460.0820.2230.4210.516

0.8

1.4001.2331.1061.0901.0860.2680.4060.5580.7010.7560.1040.1740.3380.5310.6150.0540.0860.2190.4270.530

= A0 + A\x + A2J(

for 0 < x < a

frmYirrfn/

1 + 1.464 -\c

l.65

nc asec

See Table 12.22 for definitions of c, W, and


TABLE 12.28Limit load solution for a semielliptical surface crack in a flat plate subject to combined bending andtension [10].

A

*—*^^&

-2W

3<ry(l-«)

where

for W>(c + t)

Q.Ca- for W<(c + t)tW

GYS +

12.3 FLAWED CYLINDERS

This section contains AT/, J, and displacement solutions for flawed cylinders. Part-throughand through-wall flaw geometries are included, in both axial and circumferentialorientations. Loading cases include uniform tension, bending, and internal pressure.Influence coefficients for part-through axial cracks are included, which enable the user todetermine Kj for a wide range of loading.

634 Chapter 12

12.3.1 Stress Intensity Factor

TABLE 12.29Stress intensity solutions for circumferential through-wail flaws in cylinders [7],

R - mean radius:

9 -crack half angle in radians

Axial Tension.

0V =-where

F=

2jtRt

5.33031 — 1 + 18.7731 —n

A =0.125-- 0.25t

,4= 0.4 — -3.0t

for 5< — <10t

for 10 < — <20r

Bending Moment.

where

Fb=

M«

TlR t

4.59671.5 / / JN4.24

+2.6422-

where A is as defined above for the pure tension case.

Internal Pressure.

where <Jm =mpR

Fm= 1 + 0.1501 r for 7<2

Fm= 0.8875 + 0.2625 7 for 2 < 7 < 5


TABLE 1230Stress intensity solutions for part-through internal circumferential flaws in cylinders subject to uniformtension [11]

at

Ro

Axisymmetric Flaw.

Kj = <

Ft=l. 1.948Û

1.5

v4.2

*

0-25

A = \ 0.125-±--0.25 for 5<-^t ) t

0.25= |0.4^-3.0| for 10<^-

t

Semielliptical Flaw. (A'/ at deepest point.)

-,

(R \Q'1'+0.0035(1 + 0.7^) --5

<3 =

2c

636 Chapter 12

TABLE 12.31Stress intensity factors for axial flaws in cylinders subject to internal pressure [1,11]-

Finite length through wall flaw. Valid forR»t.

K =

pR°h=t

1 + Q.52x+L29x -0.074^

Long Part-Through Internal Flaw. Valid for5 <R/t< 20 and ah < 0.750.

2pR a Ri\K}= 7 %V7TQF -,—

Rf-R U t )

4.95lf-l +1.092f-\t J U

0.25 DA = |0.125^--0.25] for5<^-<10

t

t

0.25

Finite Length Part-Through Internal Flaw. Validfor 5 < RA < 20, 2c/a < 12, and alt < 0.80. (Jfj atdeepest point.)

,2c t

F = 1.12 + 0.053^+ 0.0055^'

•2c 20- *

2c

t= l + 1.464-

1400

1.65


TABLE 1232Influence coefficients for a semi-elliptical surface crack on the inside of a cylinder with t/Rf = O.IO [9].See Section 9.1.1 for an example of this approach.

GJ:

Go

GI

G2

G3

§..t:

2i.71 *

00.250.5

0.7510

0.250.5

0.7510

0.250.5

0.7510

0.250.5

0.75I

- = 0.2c

0.2

0.6070.7400.9451.0731.1150.0790.2060.4220.6030.6730.0230.0750.2370.4290.5140.0100.0320.1460.3320.438

0.5

0.7910.9321.1881.3661.4270.1380.2680.5030.7050.7830.0520.1050.2770.4800.5710.0270.0490.1690.3630.462

0.8

1.1791.2841.5681.7981.8720.2530.3740.6190.8590.9600.1040.1540.3310.5600.6710.0560.0770.1990.4120.529

* = 0.4c0.2

0.7770.8100.9401.0381.0720.1250.2460.4420.6080.6720.0430.0970.2560.4410.5230.0210.0440.1610.3460.441

0.5

0.9360.9481.0761.1801.2170.1760.2910.4870.6570.7230.0690.1190.2790.4660.5490.0360.0580.1740.3600.456

0.8

1.2191.1641.2431.3571.3930.2590.3560.5380.7270.8060.1060.1490.3020.5050.6010.0560.0740.1870.3850.493

* = 1.0c0.2

1.1401.0681.0331.0191.0150.1970.3590.5410.6690.7150.0740.1530.3330.5140.5880.0380.0750.2180.4170.512

0.5

1.2191.1261.0741.0551.0500.2210.3770.5540.6830.7290.0850.1620.3390.5200.5960.0440.0800.2220.4200.515

0.8

1.3481.2001.0911.0901.0900.2550.3970.5550.7030.7600.0990.1700.3370.5330.6180.0510.0850.2190.4290.532

G(x) = A0 + A[X + A2XJ

for 0<x<a

+ 63 A-^

/ xl.65(2 = 1 + 1.464 -Vc;

secnc a

See Table 12.31 for definitions of c, /?/ and 0.

638 Chapter 12

TABLE 1233Influence coefficients for a semi-elliptical surface crack on the inside of a cylinder with t/Ri = 0.25 [9].

G|:

Go

Gi

G2

G3

§.t :

2171 *

00.250.5

0.7510

0.250.5

0.7510

0.250.5

0.7510

0.250.5

0.751

* = 0.2c

0.2

0.6060.7360.9351.0571.0970.0790.2050.4190.5980.6660.0230.0750.2360.4260.5110.0100.0320.1450.3300.426

0.5

0.7970.9251.1701.3431.4050.1410.2680.4980.6980.7760.0540.1060.2750.4770.5670.0280.0500.1680.3610.460

0.8

1.2011.2701.5491.8381.9590.2620.3720.6150.8760.9960.1080.1540.3300.5710.6920.0590.0770.1990.4190.542

* = 0.4c

0.2

0.7700.8010.9281.0241.0570.1230.2430.4380.6030.6660.0420.0960.2540.4390.5200.0210.0440.1600.3450.439

0.5

0.9240.9321.0561.1571.1930.1740.2870.4810.6500.7150.0680.1180.2760.4620.5450.0360.0570.1730.3580.454

0.8

1.2191.1541.2411.3851.4430.2630.3560.5400.7400.8280.1090.1500.3040.5130.6140.0590.0750.1880.3910.509

§ = 1.0c0.2

1.1281.0581.0251.0131.0090.1940.3560.5380.6670.7130.0720.1520.3320.5120.5880.0370.0750.2170.4160.511

0.5

1.1911.1051.0601.0451.0410.2140.3710.5500.6800.7260.0820.1590.3380.5190.5940.0430.0790.2210.4190.515

0.8

1.3161.1801.0881.0991.1050.2480.3930.5560.7080.7680.0970.1690.3390.5370.6230.0500.0850.2200.4310.536

<J(x} = A0

for 0<x<a

= 1 + 1.464|-\c

Jw ~ secnc a2W\ t

I/-

See Table 12.31 for definitions of c, RI and <j>.


TABLE 1234Influence coefficients for a semi-elliptical surface crack on the outside of a cylinder with tfRi = 0.10 [9].See Section 9.1.1 for an example of this approach.

G|:

Go

Gl

G2

G3

a.t :

2&.n '

00.250.5

0.7510

0.250.5

0.7510

0.250.5

0.7510

0.250.5

0.751

fi = 0.2c0.2

0.6120.7500.9651.1021.1470.0800.2080.4280.6140.6850.0230.0760.2400.4340.5210.0100.0320.1470.3350.432

0.5

0.8060.9681.2721.5021.5840.1420.2790.5300.7520.8390.0530.1100.2900.5040.6000.0280.0520.1770.3780.480

0.8

1.2621.4321.8672.2082.2980.2770.4190.7150.9931.0990.1140.1750.3770.6260.7390.0620.0880.2260.4500.568

* = 0.4c0.2

0.7880.8230.9581.0611.0960.1280.2500.4480.6160.6800.0450.0990.2590.4450.5280.0220.0460.1630.3490.444

0.5

0.9841.0021.1471.2671.3100.1920.3090.5110.6870.7550.0760.1280.2900.4810.5650.0400.0630.1810.3700.466

0.8

1.3781.3251.4251.5411.5650.3090.4060.5950.7840.8580.1290.1730.3290.5310.6250.0700.0880.2020.4000.505

* = 1.0c0.2

1.1561.0821.0441.0291.0250.2020.3630.5440.6730.7180.0760.1550.3350.5150.5900.0390.0770.2190.4180.513

0.5

1.2661.1651.1061.0831.0780.2360.3900.5650.6920.7380.0920.1680.3440.5240.6000.0480.0840.2240.4220.518

0.8

1.4531.2781.1441.1251.1180.2860.4210.5700.7120.7650.1130.1810.3440.5360.6190.0590.0910.2220.4300.533

cr(;t) = A0 + AIX + A2-*2 + A3*3

for 0 < x < a

ncsec

See Table 12.31 for definitions of c, /?/ and 0.

640 Chapter 12

TABLE 1235Influence coefficients for a semi-elliptical surface crack on the outside of a cylinder with t/Rf = 0.25 [9].See Section 9.1.1 for an example of this approach.

Gi:

G0

GI

G2

G3

§.t:

li.71 '

00.250.5

0.7510

0.250.5

0.7510

0.250.5

0.7510

0.250.5

0.751

* = 0.2c

0.2

0.6120.7520.9721.1141.1620.0800.2090.4300.6180.6910.0230.0760.2410.4370.5240.0100.0320.1480.3370.434

0.5

0.7860.9521.2781.5411.6400.1340.2720.5320.7670.8610.0490.1060.2910.5130.6130.0250.0500.1770.3830.488

0.8

1.1601.3461.8602.3442.5100.2420.3890.7131.0441.1780.0970.1590.3760.6540.7820.0510.0790.2250.4680.596

* = 0.4c

0.2

0.7930.8280.9671.0721.1090.1300.2520.4510.6200.6850.0450.1000.2610.4470.5300.0220.0460.1640.3500.445

0.5

0.9941.0161.1751.3111.3600.1950.3150.5210.7020.7730.0780.1300.2950.4890.5750.0410.0640.1840.3750.472

0.8

1.4001.3651.5131.6821.7270.3180.4210.6260.8330.9140.1340.1800.3450.5560.6530.0730.0930.2120.4160.523

* = 1.0c0.2

1.1631.0881.0491.0341.0300.2040.3650.5460.6740.7200.0770.1560.3360.5160.5910.0400.0770.2200.4180.513

0.5

1.2861.1841.1231.1001.0940.2430.3960.5700.6980.7430.0960.1710.3470.5270.6030.0510.0860.2260.4240.520

0.8

1.4981.3201.1831.1631.1560.3020.4350.5830.7240.7770.1220.1880.3500.5420.6250.0640.0950.2260.4330.536

a(x) = A0 + A\x + A2x'

for 0<x<a

rrffrrmfTl0 = 1 + 1.464 -

fw = secnc a

[2W

I/-

See Table 12.27 for definitions of c, Rj and


12.3.2 Limit Load

TABLE 1236Limit load solutions for circumferential through- wall flaws in cylinders [7].

Axial Tension.

PL = <JY (2R0t -12 )[2 cos"1 (0.5 sin 0) - 6]

a _Y

Bending Moment.

— (cos co -0.5 sin 9)

R

Corabhied Tension and Bending.

l-£ + — (cos 0) -0.5 sin 9)

e= .(1-1X1-0.55)

12.3.3 Fully Plastic J and Displacement

The pages that follow give fully plastic / and displacement solutions for severalconfigurations of flawed cylinders. See Sections 9.5 and 12.1.3 for the appropriateelastic-plastic estimation formulae.

642 Chapter 12

TABLE 1237(a)Fully plastic J and crack mouth opening displacement for an axially cracked cylinder under internalpressure [5].

n+1

[\n

~\Fo )

p = 2bo00

h factors for this geometry are listed in Tables 12.38 to 12.40.

TABLE 1237(b)Fully plastic / and displacement for a circumferentially cracked cylinder in tension [5].

i

L

1i

i

[\

<:

i

P,A/2'

',A/2^

k

RiPT

1

"T

t

T*Ko

^n*.

if,,,,,, ,D-

, f n \n+lba P }

Jjji = oce0G0 — h] (a 1 t,n, RI 1 1)\ —r (Po)

f ^ \n

I P 1V — ocEndh'y (a / 1 n R~ 1 1 )

( D \n

2 2 2V 3

/• \/zrf ^

&p(nc) - <XC0L 2 2

h factors for this geometry are listed in Tables 12.41 to 12.43.


TABLE 12 38h factors for an axially cracked cylinder under internal pressure with t/Rf = 0.20 [5].

a/t:

0.125

0.250

0.500

0.750

hih2

hih2

hih2

hih2

n = 1

6.325.837.005.929.797.0511.007.35

n = 2

7.937.018.348.7210.376.975.543.86

n = 3

9.327.969.037.079.076.012.840.186

n = 5

11.59.499.597.265.613.701.24

0.556

n = 7

13.1210.679.717.143.522.280.830.261

n =10

14.9411.969.456.712.111.25

0.4930.129

TABLE 1239h factors for an axially cracked cylinder under internal pressure with t/Rj = 0.10. [5]

a/t:

0.125

0.250

0.500

0.750

hih2

hih2

hih2

hih2

n = 1

5.225.316.165.5610.57.4816.19.57

n = 2

6.646.257.496.3111.67.728.195.40

n = 3

7.596.887.966.5210.77.013.872.57

n = 5

8.767.658.086.406.474.291.46

0.706

n = 7

9.348.027.786.013.952.581.05

0.370

n = 10

9.558..096.985.272.271.37

0.7870.232

644 Chapter 12

TABLE 12.40h factors for an axially cracked cylinder under internal pressure with t/Rf = 0.05 [5].

a/t:

0.125

0.250

0.500

0.750

h2

fa!

**2h!

fa2

h!

H2

n = 1

4.504.96

5.575.2910.87.6623.112.1

n = 2

5.795.716.915.9812.88.33

13.17.88

n = 3

6.626.20

7.376.16

12.88.135.873.84

n = 5

7.656.82

7.476.018.165.33

1.901.01

n = 7

8.077.02

7.215.63

4.883.20

1.230.454

n = 10

7.756.66

6.534.93

2.621.65

0.8830.240

TABLE 12.41h factors for a circumferentially cracked cylinder in tension with t/Rf = 0.20 [5],

a/t:

0.125

0.250

0.500

0.750

hifa2

fashlIlTj

ii3hjSi2hsfa |

fa2

h3

n = 1

3.784.560.3693.884.400.6734.404.361.334.123.461.54

n = 2

5.005.550.7004.955.121.254.784.301.933.032.191.39

n = 3

5.946.371.075.645.571.794.593.912.212.231.361.04

n = 5

7.547.791.966.496.072.793.793.002.231.5460.6380.686

n = 7

8.999.103.046.946.283.613.072.261.941.30

0.4360.508

n = 10

11.111.04.947.226.304.522.341.551.461.11

0.3250.366


TABLE 12.42h factors for a circumferentiaily cracked cylinder in tension with t/Rj = 0.10 [5].

a/t:

0.125

0.250

0.500

0.750

fa!h2

h3

hi1*21*3hi1*2fa3

hih2h3

n = 1

4.004.710.5484.174.580.7575.404.991.5555.184.221.86

n = 2

5.135.630.7335.355.361.355.905.012.263.782.791.73

n = 3

6.096.451.136.095.841.935.634.592.592.571.671.26

n = 5

7.697.852.076.936.312.964.513.482.571.59

0.7250.775

n = 7

9.099.093.167.306.443.783.492.562.181.310.480.561

n = 10

11.110.95.077.416.314.602.471.671.561.10

0.3000.360

TABLE 12.43h factors for a circumferentially cracked cylinder in tension with t/Ri = 0.05 [5].

a/t:

0.125

0.250

0.500

0.7.50

h!h2

h3

hifa2

h3hih2

h3hih2

h3

E = 1

4.044.820.6804.384.710.8186.555.671.806.645.182.36

n = 2

5.235.690.7595.685.561.437.175.772.594.873.572.18

n = 3

6.226.521.176.456.052.036.895.362.993.082.071.53

n = 5

7.827.902.137.296.513.105.464.082.981.68

0.8080.772

n = 7

9.199.113.237.626.593.914.132.972.501.30

0.4720.494

n = 10

11.110.85.127.656.394.692.771.881.741.07

0.3160.330

646 Chapter 12

TABLE 12.44Fully plastic / integral for circumferential through-wall flaws in cylinders [7].

(a) Axial tension.

Jpl=n+l

IP0=2a0Rt[x-9-2sin i(0.5sin9)]

(b) Bending moment.

n+1

cos — | - 0.5 sin 9u(c) Combined tension and bending.

Jpl=cce0<j0R(7i:-6)-hif ~\n+l

Soj

P0=0.5Mn

^2 0.5

A = —" PR

P0 and M0 are given in (a) and (b), respectively.

hj factors for this geometry are listed in Tables 12.45 to 12.56.


TABLE 12.45factors for through-wall flaws in cylinders in uniform tension, R/t = 5 [7],

e/7c

0.0000.0630.1000.1250.1500.1750.2000.2250.2500.2750.3000.3250.3500.3750.4000.4250.4500.4750.500

n = 1

0.0000.1770.2800.3520.4270.5040.5780.6520.7310.7930.8580.9180.9761.0251.0751.1171.1541.1841.224

n=2

0.0000.2300.3550.4370.5200.6000.6740.7400.8040.8400.8500.8530.8560.8590.8620.8650.8680.8730.879

H = 3

0.0000.2650.4000.4790.5600.6300.6950.7420.7650.7860.7950.7930.7860.7780.7600.7430.7200.6960.669

n = 5

0.0000.3070.4330.5000.5540.5800.5950.5960.5960.5950.5900.5800.5620.5450.5250.5060.4800.4540.426

n = 7

0.0000.3260.4250.4780.4940.4920.4850.4770.4640.4550.4400.4210.4040.3870.3700.3480.3300.3080.288

TABLE 12.46hj factors for through-wall flaws in cylinders in uniform tension, R/t = 10 [7].

e/7u

0.0000.0630.1000.1250.1500.1750.2000.2250.2500.2750.3000.3250.3500.3750.4000.4250.4500.4750.500

n = 1

0.0000.1860.3300.4030.5300.6260.7230.8240.9191.0051.0841.1631.2351.3001.3601.4201.4601.5051.546

n = 2

0.0000.2480.4150.5200.6200.7200.8220.9301.0401.0731.0941.1001.1051.1071.1101.1101.1101.1101.110

n = 3

0.0000.2910.4720.5890.6830.7900.8840.9701.0081.0301.0371.0351.0211.0070.9820.9590.9250.8900.857

n = 5

0.0000.3230.5180.6450.7300.7770.8000.8080.8100.8040.7940.7780.7590.7340.7040.6740.6400.6030.568

n = 7

0.0000.3820.5950.6380.6600.6660.6700.6610.6520.6350.6150.5970.5730.5500.5240.4960.4650.4390.408

648 Chapter 12

TABLE 12.47kj factors for through-wall flaws in cylinders in uniform tension, R/t = 20 [7].

e/jt

0.0000.0630.1000.1250.1500.1750.2000.2250.2750.3000.3250.3500.3750.4000.4250.4500.4750.500

n = 1

0.0000.2040.3680.4890.6300.7700.9201.0651.3401.4481.5481.6401.7281.8001.8681.9301.9702.020

n = 2

0.0000.2810.5040.6570.8000.9451.0851.2401.4601.4981.4961.4881.4791.4701.4641.4581.4521.445

n = 3

0.0000.3390.5840.7670.9281.1001.2411.3101.3701.3751.3641.3501.3281.2941.2601.2201.1761.121

n = 5

0.0000.4260.6840.8731.0001.0501.0801.0991.0881.0751.0491.0190.9840.9450.9040.8550.8040.758

n = 7

0.0000.4800.7100.8010.8800.9000.9070.9020.8750.8550.8240.8000.7600.7200.6840.6500.6060.564

TABLE 12.48hj factors for through-wall flaws in cylinders in bending, R/t = 5 [7],

Q/K

0.0000.0630.1000.1250.1500.1750.2000.2500.3000.3500.4000.4500.500

n = 1

0.0000.3130.5200.6640.8230.9831.1421.4611.7602.0482.3202.5762.795

n = 2

0.0000.3690.5600.7080.8360.9601.0801.3331.5261.7201.8721.9902.059

o = 3

0.0000.4030.5590.7130.8480.9601.0561.1981.3281.4201.4881.5561.598

n = 5

0.0000.4330.5580.6750.7600.8020.8800.9410.9901.0161.0361.0401.048

n = 7

0.0000.4320.5570.6110.6380.6640.6910.7440.7540.7600.7740.7800.794

n = 10

0.0000.4310.5360.5770.5820.5860.5910.6000.6040.6080.6120.6160.620

n = 20

0.0000.4310.4960.5290.5100.4900.4710.4320.4280.4250.4210.4180.414


TABLE 12.49factors for through-wall flaws in cylinders in bending, R/t - 10 [7].

e/7c

0.0000.0630.1000.1250.1500.1750.2000.2500.3000.3500.4000.4500.500

n = 1

0.0000.3330.6000.7660.9871.2091.4301.8732.2802.6803.0403.3603.646

n = 2

0.0000.4010.7600.8551.0401.2021.4001.7712.0032.2602.4402.5722.682

n = 3

0.0000.4500.7700.8971.0801.2401.4001.6291.7601.9701.8801.9702.105

n = 5

0.0000.5080.7880.9021.0721.1601.2441.3231.3431.3641.3831.4001.424

n = 7

0.0000.5310.7500.8570.9300.9881.0201.0801.1001.1001.0801.0521.035

n = 10

0.0000.5490.7100.8160.8340.8520.8760.9070.8890.8800.8400.8000.760

n = 20

0.0000.5700.7150.7770.7590.7420.7240.6880.6400.5920.5240.4720.392

TABLE 12.50hi factors for through-wall flaws in cylinders in bending, R/t = 20 [7],

e/7c

0.0000.0630.1000.1250.1500.1750.2000.2500.3000.3500.4000.4500.500

n = 1

0.00003650.6500.9331.2421.5521.8612.4802.9603.4503.9504.4004.859

n = 2

0.0000.4560.8501.0911.3501.6001.8632.3912.7003.0203.2603.4453.571

n = 3

0.0000.5270.9381.1871.4151.6501.8652.2242.4352.5752.6702.7502.821

n = 5

0.0000.6310.9851.2411.4631.6101.7081.8341.8571.8861.9041.9251.950

n = 7

0.0000.6961.0251.2081.2711.3341.3971.5241.5391.5601.5701.5951.600

n = 10

0.0000.7571.0101.1771.1951.2131.2311.2671.2771.2851.2991.3031.320

n = 20

0.0000.8641.0101.1431.1141.0851.0561.0101.0051.0031.0021.0011.000

650 Chapter 12

TABLE 12.51factors for through-wall flaws in cylinders in combined tension and bending, R/t = 10, 6/it - 0.0625

m.

X/(1+X)

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

n = 2

3.9674.3134.7365.1255.6146.0006.4386.7897.1407.5007.9018.0948.2878.3448.2578.1257.8117.5007.0636.5636.018

n = 5

5.5676.5007.3758.2509.0809.75010.50111.00011.45711.87512.15012.31312.23611.93811.64211.12510.6179.8759.1908.5007.620

n = 7

6.1046.5007.0807.8758.7879.87511.07812.12513.18814.00014.61015.00015.13014.87514.40813.62512.72911.68810.4479.2508.160

n = 10

6.5107.9699.72111.25012.93714.25015.46316.37517.06317.50017.83917.55017.24116.37515.50014.50013.36612.12510.7389.3137.928


m.TABLE 12.52

factors for through-wall flaws in cylinders in combined tension and bending, R/t = 10, 6/K - 0.125

W(1*X,

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

n = 2

4.1574.6255.0855.5315.9566.3756.7507.1257.5007.7197.9308.0008.0648.0637.9267.7507.4357.1256.7726.3755.987

n = 5

5.1635.7506.3316.8757.3237.8758.3098.6889.0089.3139.5009.6259.5669.4069.1658.8138.3907.9387.3916.8756.311

n = 7

5.1025.7196.3316.9067.5188.0008.5009.0009.3819.6309.87510.00010.0119.8139.4959.0008.5357.8757.3136.6255.996

n = 10

4.7505.2505.9826.8137.6888.5639.41610.00010.54310.81310.93510.84410.56710.2199.7029.1258.5787.8447.1586.3755.688

652 Chapter 12

TABLE 12.53hj factors for through-wall flaws in cylinders in combined tension and bending, R/t = 10, BJn = 0.250[7].

x/(l+>.)

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

n = 2

4.1594.5224.8855.2505.6146.0006.3476.6506.9007.1507.3637.5007.5087.4507.3077.1006.8856.5196.1525.7005.312

n = 5

3.2383.5503.9884.4004.8825.4005.9086.3506.8007.1007.3337.4507.3977.2507.0186.6196.2205.7005.1084.5503.969

n = 7

2.6053.0003.4904.0504.4915.2005.8166.3506.8007.1507.3757.5007.3977.0756.6886.1505.6415.0004.3723.8003.240

n = 10

3.0003.2253.5894.0004.5405.0505.7006.4007.2647.7078.1508.1507.9007.5007.0186.3005.6704.9504.3533.7003.125


TABLE 12,54factors for through-wall flaws in cylinders in combined tension and bending, R/t = 10, 0/n = 0.375

[7].

X/(1+X)

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

n = 2

2.8923.3153.7394.1644.5894.9575.3255.6515.9776.2376.4976.5866.6746.5366.3996.0935.7865.3604.9344.4593.984

n = 5

1.9922.0002.0442.2132.4412.6512.8623.0913.2333.2853.3363.3003.2003.0452.8902.6462.4012.1201.8381.6031.368

n = 7

1.4961.7001.8942.0942.2942.4842.6752.7912.9062.9452.9852.8832.7812.6292.4772.2512.0251.7671.5091.3011.092

e = 10

2.0002.1002.2432.3422.4412.5582.6752.7082.7402.7502.7392.6502.4472.2602.0641.8561.6491.4441.2381.0490.860

654 Chapter 12

TABLE 12.55factors for through-wall flaws in cylinders in combined tension and bending, R/t = 10, Q/it = 0.500

[7J.

X/(1+X)

0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

n = 2

2.2202.3002.5252.8503.3203.7844.2474.8005.2305.6005.8305.9005.8505.7005.3675.0004.5424.1003.6003.1002.682

n = 5

1.1371.3001.4961.7001.9532.2002.4752.7002.9003.1003.1603.2003.1703.0502.8902.7002.4302.1831.9351.7001.424

n = 7

0.8160.9501.1221.3501.6111.8002.0502.3002.4382.5752.5502.5252.4472.2972.1472.0001.8231.5891.3541.2001.008

n = 10

1.5001.5251.5701.6501.7571.8001.9002.0002.0502.1002.1072.1002.0001.9001.7341.6001.4461.2001.0830.9000.760


REFERENCES

1. Tada, H., Paris, P.C., and Irwin, G.R. The Stress Analysis of Cracks Handbook. (2nd Ed.)Paris Productions, Inc., St. Louis, 1985.


3. Towers, O.L., "Stress Intensity Factors, Compliances, and Elastic T) Factors for Six TestGeometries." Report 136/1981, The Welding Institute, Abington, UK, 1981.

4. E 813-87, "Standard Test Method for Jjc, a Measure of Fracture Toughness." AmericanSociety for Testing and Materials, Philadelphia, 1987.

5. Kumar, V., German, M.D., and Shih, C.F.,"An Engineering Approach for Elastic-PlasticFracture Analysis." EPRI Report NP-1931, Electric Power Research Institute, Palo Alto,CA, 1981.

6. Kumar, V., German, M.D., Wilkening, W.W., Andrews, W.R., deLorenzi, H.G., andMowbray, D.F., "Advances in Elastic-Plastic Fracture Analysis." EPRI Report NP-3607,Electric Power Research Institute, Palo Alto, CA, 1984.

7. Zahoor, A. "Ductile Fracture Handbook, Volume 1: Circumferential Throughwall Cracks."EPRI Report NP-6301-D, Electric Power Research Institute, Palo Alto, CA, 1989.

8. Newman, J.C. and Raju, I.S., "Stress-Intensity Factor Equations for Cracks in Three-Dimensional Finite Bodies Subjected to Tension and Bending Loads." NASA TechnicalMemorandum 85793, NASA Langley Research Center, Hampton, VA, April 1984.

9. Raju, I.S. and Newman J.C., Jr., "Stress-Intensity Factors for Internal and ExternalSurface Cracks in Cylindrical Vessels." Journal of Pressure Vessel Technology, Vol. 104,1982, pp. 293-298.

10. Miller, A.G., "Review of Limit Loads of Structures Containing Defects." InternationalJournal of Pressure Vessels and Piping, Vol. 32, 1988, pp. 197-327

11. Zahoor, A., "Closed Form Expressions for Fracture Mechanics Analysis of CrackedPipes." Journal of Pressure Vessel Technology, Vol. 107, 1985, pp. 203-205.

13. PRACTICE PROBLEMS

This chapter contains practice problems that correspond to material in Chapters 1 to 1 1 .Some of the problems for Chapters 7 to 10 require a computer program or spreadsheetmacro. This level of complexity was necessary in order to make the application-orientedproblems realistic.

All quantitative data are given in SI units, although the corresponding values inEnglish units are also provided in many cases.

13.1 CHAPTER 1

1 . 1 Compile a list of five mechanical or structural failures that have occurred withinthe last 20 years. Describe the factors that led to each failure and identify thefailures that resulted from misapplication of existing knowledge (Type 1) andthose that involved new technology or a significant design modification (Type 2).

1.2 A flat plate with a through-thickness crack (Fig. 1.8) is subject to a 100 MPa

(14.5 ksi) tensile stress and has a fracture toughness (£/c) of 50.0 MPa \m (45.5

ksi v in). Determine the critical crack length for this plate, assuming the materialis linear elastic.

1.3 Compute the critical energy release rate (£c) of the material in the previousproblem for E = 207,000 MPa (30,000 ksi).

1.4 Suppose that you plan to drop a bomb out of an airplane and that you areinterested in the time of flight before it bits the ground, but you cannot rememberthe appropriate equation from your undergraduate physics course. Your decide toinfer a relationship for time of flight of a falling object by experimentation. Youreason that the time of flight, t, must depend on the height above the ground, h,and the weight of the object, mg, where m is the mass and g is the gravitationalacceleration. Therefore, neglecting aerodynamic drag, the time of flight is givenby the following function:

Apply dimensional analysis to this equation and determine how many experimentswould be required to determine / to a reasonable approximation, assuming youknow the numerical value of g. Does the time of flight depend on the mass of theobject?

657

658 Chapter 13

13.2 CHAPTER 2

2.1 According to Eq. (2.25), the energy required to increase the crack area a unitamount is equal to twice the fracture work per unit surface area, wf. Why is thefactor of 2 in this equation necessary?

2.2 Derive Eq. (2.30) for both load control and displacement control by substitutingEq. (2.29) into Eqs. (2.27) and (2.28), respectively.

2.3 Figure 2.10 illustrates that the driving force is linear for a through-thickness crackin an infinite plate when the stress is fixed. Suppose that a remote displacement(rather than load) were fixed in this configuration. Would the driving force curvesbe altered? Explain. (Hint: see Section 2.5.3).

2.4 A plate 2W wide contains a centrally located crack 2a long and is subject to atensile load, P. Beginning with Eq. (2.24), derive an expression for the elasticcompliance, C (= A/P) in terms of the plate dimensions and elastic modulus, E.The stress in Eq. (2.24) is the nominal value; i.e., a = P/2BW'm this problem.(Note: Eq. (2.24) only applies when a « W; the expression you derive is onlyapproximate for a finite width plate.)

2.5 A material exhibits the following crack growth resistance behavior:

R = 6.95(a-a0)°-5

where a0 is the initial crack size. R has units of kJ/m^ and crack size is inmillimeters. Alternatively,

/? = 200(0-tf0)°-5

where R has units of in-lb/in^ and crack size is in inches. The elastic modulus ofthis material = 207,000 MPa (30,000 ksi). Consider a wide plate with a throughcrack (a « W) that is made from this material.

(a) If this plate fractures at 138 MPa (20.0 ksi), compute the following:(i) The half crack size at failure (ac).(ii) The amount of stable crack growth (at each crack tip) that precedesfailure (ac - a0}.

(b) If this plate has an initial crack length (2ao) of 50.8 mm (2.0 in)and the plate is loaded to failure, compute the following:

(i) The stress at failure.(ii) The half crack size at failure.(iii) The stable crack growth at each crack tip.

Practice Problems 659

2.6 Suppose that a double cantilever beam specimen (Fig. 2.9) is fabricated from thesame material considered in Problem 2.5. Calculate the load at failure and theamount of stable crack growth. The specimen dimensions are as follows:

B = 25.4 mm (1 in) h = 12.7 mm (0.5 in) s^ = 152 mm (6 in)

2.7 Consider a nominally linear elastic material with a rising R curve (e.g., Problems2.5 and 2.6). Suppose that one test is performed on wide plate with a throughcrack (Fig. 2.3) and a second test on the same material is performed on a DCBspecimen (Fig. 2.9). If both tests are conducted in load control, would the gc

values at instability be the same? If not, which geometry would result in a higher£c! Explain.

2.8 Example 2.3 showed that the energy release rate, Q, of the double cantilever beam(DCB) specimen increases with crack growth when the specimen is held at aconstant load. Describe (qualitatively) how you could alter the design of the DCBspecimen such that a growing crack in load control would experience a constant §.

2.9 Beginning with Eq. (2.20), derive an expression for the potential energy of a platesubject to a tensile stress <J with a penny-shaped flaw of radius a. Assume that a« plate dimensions.

2.10 Beginning with Eq. (2.20), derive expressions for the energy release rate and ModeI stress intensity factor of a penny-shaped flaw subject to a remote tensile stress.(Your KI expression should be identical to Eq. (2.44).)

2.11 Calculate KI for a rectangular bar containing an edge crack loaded in three pointbending.

P = 35.0 kN(78701b); W = 50.8 mm (2.0 in); B = 25 mm (1.0 in); a/W =0.2; S = 203 mm (8.0 in).

2.12 Consider a material where KJC = 35 MPaVm" (31.8 ksiVinj. Each of the fivespecimens in Table 2.4 and Fig. 2.23 have been fabricated from this material. Ineach case, B - 25.4 mm (1 in), W= 50.8 mm (2 in), and a/W - 0.5. Estimate thefailure load for each specimen. Which specimen has the highest failure load?Which has the lowest?

2.13 A large block of steel is loaded to a stress of 345 MPa (50 ksi). If the fracture

toughness (Kjc) is 44 MPa Vm (40 ksi "V in), determine the critical radius of apenny-shaped crack.

660

2.14 A semicircular surface crack in a pressure vessel is 10 mm (0.394 in) deep. Thecrack is on the inner wall of the pressure vessel and is oriented such that the hoopstress is perpendicular to the crack plane. Calculate Kj if the local hoop stress =200 MPa (29.0 ksi) and the internal pressure = 20 MPa (2900 psi). Assume thatthe wall thickness » 10 mm.

2.15 Calculate Kj for a semielliptical surface flaw at 0 = 0°, 30°, 60°, 90°.

a= 150 MPa (21.8 ksi); a = 8.00 mm (0.315 in); 2c = 40 mm (1.57 in).

2.16 Consider a plate subject to biaxial tension with a through crack of length 2a,oriented at an angle ft from the 02 axis (Fig. 13.1). Derive expressions for Kj andKH for this configuration. What happens to each K expression when 07 = 03?

1

FIGURE 13.1 Through-thickness crack in a

biaxially loaded plate (Problem 2.16).

2.17 A wide flat plate with a through-thickness crack experiences a nonuniform normalstress which can be represented by the following crack face traction:

where p0 = 300 MPa and {3 = 25 mm. The origin (x = 0) is at the left crack tip, asillustrated in Fig. 2.27. Using the weight function derived in Example 2.6,calulate Kj at each crack tip for 2a = 25, 50, and 100 mm. You will need tointegrate the weight function numerically,

2.18 Calculate Keff (Irwin correction) for a through crack in a plate of width 2 W (Fig.2.20(b)). Assume plane stress conditions and the following stress intensityrelationship:


nasec eff

a = 250 MPa (36.3 ksi);2a - 50.8 mm (2.0 in).

{ 2W

ays = 350 MPa (50.8 ksi); 2W = 203 mm (8.0 in);

2.19 For an infinite plate with a through crack 50.8 mm (2.0 in) long, compute andtabulate Keffv. stress using the three methods indicated below. Assume250 MPa (36.3 ksi).

Stress, MPa (ksi)25 (3.63)50 (7.25)100 (14.5)150(21.8)200 (29.0)225 (32.6)249(36.1)250 (36.3)

Keff, MPa Vm or ksi viriLEFM Irwin Correction Strip Yield Model

2.20 A material has a yield strength of 345 MPa (50 ksi) and a plane strain lineari— n—elastic fracture toughness of 110 MPa vm (100 ksi "V in). Determine the

minimum specimen dimensions (B, a, W) required to perform a valid K]c test onthis material. Comment on the feasibility of testing a specimen of this size.

2.21 You have been given a set of fracture mechanics test specimens, all of the samesize and geometry. These specimens have been fatigue precracked to various cracklengths. The stress intensity of this specimen configuration can be expressed asfollows:

f(a/W)

where P is load, B is thickness, W is width, a is crack length, and f(a/W) is adimensionless geometry correction factor.

662

Describe a set of experiments you could perform to determine f(a/W) for thisspecimen configuration. Hint: you may want to take advantage of therelationship between Kj and energy release rate for linear elastic materials.

2.22 Derive the Griffith-Inglis result for the potential energy of a through crack in aninfinite plate subject to a remote tensile stress (Eq. (2.16)). Hint: solve for thework required to close the crack faces; Eq. (A2.43b) gives the crack openingdisplacement for this configuration.

2.23 Using the Westergaard stress function approach, derive the stress intensity factorrelationship for an infinite array of collinear cracks in a plate subject to biaxialtension (Fig. 2.21).

13.3 CHAPTER 3

3.1 Repeat the derivation of Eqs. (3.1) to (3.3) for the plane strain case.

3.2 A CTOD test is performed on a three point bend specimen. Figure 13.3 showsthe deformed specimen after it has been unloaded. That is, the displacementsshown are the plastic components.

(a) Derive an expression for plastic CTOD (8p) in terms of Ap and specimendimensions.

(b) Suppose that Vp and Ap are measured on the same specimen, but that theplastic rotational factor, rp, is unknown. Derive an expression for rp in terms ofAp, Vp and specimen dimensions, assuming the angle of rotation is small.

/

FIGURE 13.2 Three-point bend specimen rotating about a plastic hinge (Problem 3.2).


3.3 Fill in the missing steps between Eqs. (3.36) and (3.37)

3.4 Derive an expression for the / integral for a -deeply notched three-point bendspecimen, loaded over a span 5", in terms of the area under the load-displacementcurve and ligament length, b. Figure 13.2 illustrates two displacementmeasurements on a bend specimen: the load line displacement (A) and the crackmouth opening displacement (V). Which of these two displacementmeasurements is more appropriate for inferring the J integral? Explain.

3.5 Derive an expression for the / integral for an axisymmetrically notched bar intension (Fig. 13.3), where the notch depth is sufficient to confine plasticdeformation to the ligament.

3.6 Derive an expression for the J integral in a deeply notched three-point bendspecimen in terms of the area under the load-crack mouth opening displacementcurve. Begin with the corresponding formula for the P-A curve (given below),and assume rotation about a plastic hinge (Fig. 13.3).

K2 9 ?= — + - f

E b J

for a specimen with unit thickness.

3.7 A wide plate with a through thickness crack fails at 30% of yield. Estimate theelevation in toughness (JCri/Jo) resulting from constraint loss. Assume n = 10and the J-Q locus for this material is given by Fig. 3.39 with 7= 1. Thebiaxiality ratio, ft, = 1 for this geometry. What do your results say about thevalidity of single-parameter fracture mechanics for this geometry?

FIGURE 13.3 Axisymmetrically notched bar

loaded in tension (Problem 3.5).

664

13.4 CHAPTER 4

4.1 A high rate fracture toughness test is to be performed on a high strength steel

with Kid = HO MPa Vm (100 ksi Vin). A three-point bend specimen will beused, with W = 50.8 mm (2.0 in), a/W - 0.5, B = W/2, and span = 4W. Also, cj= 5940 m/sec (19,500 ft/sec) for steel. Estimate the maximum loading rate atwhich the quasistatic formula for estimating Kjd is approximately valid.

4.2 Unstable fracture initiates in a steel specimen and arrests after the crack propagates8.0 mm (0.32 in). The total propagation time was 7.52 x 10~6 sec. The initialligament length in the specimen was 30.0 mm (1.18 in) and c] for steel = 5940m/sec (19,500 ft/sec). Determine whether or not reflected stress waves influencedthe propagating crack.

4.3 Fracture initiates at an edge crack in a 2.0 m (78.7 in) wide steel plate and rapidlypropagates through the material. The stress in the plate is fixed at 300 MPa (43.5ksi). Plot the crack speed versus crack size for crack lengths ranging from 10 to60 mm (or 0.4 to 2.4 in). The dynamic fracture toughness of the material isgiven by

KID~KIA

1-v

where KM = 55 MPa Vrn^ (50 ksi Vki) and Vf= 1500 m/sec (4920 ft/sec) Usethe Rose approximation (Eqs. (4.17) and (4.18)) for the driving force. The elasticwave speeds for steel are given below.

ci

C2

cr

5940 m/sec

3220 m/sec

2980 m/sec

19,500 ft/sec

10,600 ft/sec

9780 ft/sec

4.4 Derive an expression for C* in a double edge notched tension panel in terms ofspecimen dimensions, creep exponent, load, and displacement rate. See Section3.2.5 for the corresponding J expression.


4.5 A three-point bend specimen is tested in displacement control at an elevatedtemperature. The displacement rate is increased in steps as the test progresses.The load, load line displacement rate, a/W, and crack velocity are tabulated below.Compute C*, and construct a log-log plot of crack velocity versus C*. Thespecimen thickness and width are 25 mm and 50 mm, respectively. The creepexponent = 5.0 for the material.

©

A, m/s

l .Ox 10'7

5.0 x 10'7

l .Ox lO'6

5.0 x 10'6

l .Ox ID'5

5.0 x lO'5

Load, kN

10.813.814.919.020.424.0

a/W

0.520.540.560.580.600.65

o

a, m/s

3.67 x lO'9

1.79x 10-8

3.49 x lO'8

1.71 x lO'7

3.37 x 10- 7

1.65x 10-6

4.6 In a linear viscoelastic material, the pseudo elastic displacement and the physicaldisplacement are related through a hereditary integral:

4.7

Ae = {EdA}

Simplify this expression for the case of a constant displacement rate.

Consider a fracture toughness test on a nonlinear viscoelastic material at aconstant displacement rate. Assume that the load is related to the pseudo elasticdisplacement by a power law:

where M and N are constants that do not vary with time. Show that theviscoelastic / integral and the conventional J integral are related as follows:

Jv =

4.8

where 0 is a function of time. Derive an expression for <p(t). Hint: begin withEqs. (3.17) and (4.75). Also, the result from the previous problem may beuseful.

A fracture toughness test on a linear viscoelastic material results in a nonlinearload-displacement curve in a constant rate test. Yielding is restricted to a verysmall region near the crack tip. Why is the curve nonlinear? Does the stress

666 Chapter

intensity factor characterize the crack tip conditions in this case? Explain. Whatis the relationship between / and Kj for a linear viscoelastic material? Hint: referto the second equation in the previous problem.

13.5 CHAPTER 5

5.1 A body-centered cubic (BCC) material contains second phase particles. The sizeof these particles can be controlled through thermal treatment. Discuss theanticipated effect of particle size on the material's resistance to both cleavagefracture and microvoid coalescence, assuming the volume fraction of the secondphase remains constant.

5.2 An aluminum alloy fails by microvoid coalescence when the average void sizereaches ten times the initial value. If the voids grow according to Eq. (5.11), withays replaced by (Je, plot the equivalent plastic strain (eeq) at failure versus€>m/ffe f°r Gm/Ge ranging from 0 to 2.5. Assume the triaxiality ratio remainsconstant during deformation of a given sample; i.e.,

f n\ (}

In — =0.283exp -\Ro) \

5.3 The critical microstructural feature for cleavage initiation in a steel sample is a6.67 \im diameter spherical carbide; failure occurs when this particle forms amicrocrack that satisfies the Griffith criterion (Eq. (5.18)), where Yp = 14 J/m^, E= 207,000 MPa, and v = 0.30 for the material. Assuming Fig. 5.14 describes thestress distribution ahead of the macroscopic crack, where ao = 350 MPa, estimatethe critical J value of the sample if the particle is located 0.1 mm ahead of thecrack tip, on the crack plane. Repeat this calculation for the case where thecritical particle is 0.4 mm ahead of the crack tip.

5.4 Cleavage initiates in a ferritic steel at 3.0 jim diameter spherical particles. Thefracture energy on a single grain, Yp, is 14 J/m^ and the fracture energy requiredfor propagation across grain boundaries, Ygb> *s 50 J/m^. At what grain size doespropagation across grain boundaries become the con treeing step for cleavage frac-ture?

5.5 Compute the relative size of the 90% confidence band of Kjc data (as in Example5.1), assuming Eq. (5.24) describes the toughness distribution. Compute theconfidence band width for KQ/&K = 0, 0-5> 1-0, 2.0, and 5.0. What is the effectof the threshold toughness, K0, on the relative scatter? What is the physicalsignificance of O% in this case?

13 * Practice Problems 667

5.6 Compute the relative size of the 90% confidence band of Kjc data (as in Example5.1), assuming Eq. (5.26) describes the toughness distribution. Compute theconfidence band width for Ko/0K=Q, 0.5, 1.0, 2.0, and 5.0. What is the effectof the threshold toughness, K0, on the relative scatter?

13.6 CHAPTER 6

6.1 For the Maxwell spring and dashpot model (Fig. 6.6) derive an expression for therelaxation modulus.

6.2 Fill in the missing steps in the derivation of Eq. (6.14).

6.3 At room temperature, tensile specimens of polycarbonate show 60% elongationand no stress whitening, while thick compact specimens used in fracturetoughness testing show stress whitening at the crack tip. Explain theseobservations. Polycarbonate is an amorphous glassy polymer at roomtemperature.

6.4 A wide and thin specimen of PMMA has a 15 mm (0.59 in) long through crackwith a 1.5 mm (0.059 in) long craze at each crack tip. If the applied stress is 3.5MPa (508 psi), calculate the crazing stress in this material.

6.5 When a macroscopic crack grows in a ceramic specimen, a process zone 0.2 mmwide forms. This process zone contains 10,000 penny-shaped microcracks/mnPwith an average radius of 10 |im. Estimate the increase in toughness due to therelease of strain energy by these microcracks. The surface energy of the material =25 J/m2.

13.7 CHAPTER 7

7.1 A fracture toughness test is performed on a compact specimen. Calculate KQ anddetermine whether or not KQ = KIC.

B = 25.4mm (1.0 in); W = 50.8 mm (2.0 in); a = 27.7mm (1.09 in)PQ = 42.3 kN (9.52 kip); Pmax = 46.3 kN (10.4 kip); GyS = 759 MPa(HOksi)

7.2 You have been asked to perform a Kjc test on a material with <JYS - 690 MPa

(100 ksi). The toughness of this material is expected to lie between 40 MPa Vm

and 60 MPa Vm (1 ksi \ in = 1.10 MPa V m). Design an experiment tomeasure Kjc in this material using a compact specimen. Specify the following

668 Chapter 13

7.3

7.4

7.5

7.6

quantities: (a) specimen dimensions, (b) precracking loads, and (c) required loadcapacity of the test machine.

A titanium alloy is supplied in 15.9 mm (0.625 in) thick plate. If ays = 807MPa (117 ksi), calculate the maximum valid Kjc that can be measured in thismaterial.

Recall Problem 2.20, where a material with Kjc =110 MPa Vm (100 ksi Vin)required a 254 mm (10.0 in) thick specimen for a valid Kjc test. Suppose that acompact specimen of the appropriate dimensions has been fabricated. Estimatethe required load capacity of the test machine for such a test.

A 25.4 mm (1 in) thick steel plate has material properties which are tabulatedbelow. Determine the highest temperature at which it is possible to perform avalid KIC test.

Temperature, °C

-10-50510152025

Yield Strength, MPa

760725690655620586550515

Kjc, MPaVin~343642506285110175

A fracture toughness test is performed on a compact specimen fabricated from a 5mm thick sheet aluminum alloy. The specimen width (W) = 50.0 mm and B = 5mm (the sheet thickness). The initial crack length is 26.0 mm. Young'smodulus = 70,000 MPa. Compute the K-R curve from the load-displacement datatabulated below. Assume that all nonlinearity in the P-A curve is due to crackgrowth. (See Chapter 12 for the appropriate compliance and stress intensityrelationships.)


Load, kN

00.54331.0871.6302.1612.3612.5412.699

Load LineDisplacement, mm

00.06350.12700.19060.25520.28170.30960.3392

Load, kN

2.8512.9132.9032.8502.7492.6522.5532.457

Load LineDisplacement, mm

0.36980.38600.39710.41130.41910.42740.43550.4443

1 kN = 224.8 Ib 25.4 mm = 1 in 1 MPa = 0.145 ksi

7.7

7.9

A number of fracture toughness specimens have been loaded to various points andthen unloaded. Values of J and crack growth were measured in each specimen andare tabulated below. Plot the R curve for this material and determine JQ and, ifpossible, J/c.

= 350 MPa; 073 = 450 MPa; B = 25 mm; bo - 22 mm;

Specimen

123456

/, kJ/m2

100175185225250300

Crack Extension,mm

0.300.400.801.201.601.70

25.4 mm = 1 in 1 MPa = 0.145 ksi 1 kJ/m2 = 5.71 in-lb/in2

7.8 Recall Problem 2.20, where a material with Kfc =110 MPa Vrn (100 ksi Vin)and &YS = 345 MPa (50 ksi) required a specimen 254 mm (10 in) thick for avalid Kjc test. Estimate the thickness required for a valid //c test on this material.

= 483 MPa (70 ksi); E = 207,000 MPa (30,000 ksi); v = 0.3.

An unloading compliance test has been performed on a 3-point bend specimen.The data obtained at each unloading point is tabulated below.

(a) Compute and plot the J resistance curve according to ASTM E 1152.

670 Chapter 13

(b) Determine J/c according to ASTM E 813.

B = 25.0 mm; W = 50.0 mm; ao = 26.1 mm; E = 210,000 MPa, v = 0.3

ays = 345 MPa (50 ksi); &TS = 483 MPa (70 ksi)

(c) Plot and compare the J-R curves obtained from the simplified expression (Eq.(7.10)) and the incremental approach that takes account of crack growth (Eq.(7.15)). At what point does the crack growth correction become significant?

LOAD,kN

20.831.235.437.441.643.745.747.649.951.653.555.356.656.756.555.854.753.752.550.144.440.036.630.926.8

PlasticDisplacement,

mm

00.00320.0110.0200.0560.0920.1460.2280.3490.5250.7771.131.632.322.663.253.964.515.136.208.4310.0911.3713.5415.19

Crack Extension,mm

0.0130.0200.0230.0250.0310.0360.0440.0550.0710.0910.1280.1830.3210.7230.9281.291.742.082.483.174.675.816.708.239.41

1 kN = 224.8 Ib 25.4 mm = 1 in 1 MPa = 0.145 ksi

7.9 A CTOD test was performed on a three point bend specimen with B = W = 25.4mm (1.0 in). The crack depth, a, was 12.3 mm (0.484 in). Examination of the


fracture surface revealed that the specimen failed by cleavage with no prior stablecrack growth. Compute the critical CTOD in this test. Be sure to use the appro-priate notation (i.e., Sc, Su, 8}, or Sm).

Vp= 1.05 mm (0.0413 in); PCritical = 24.6 kN (5.53 kip); E = 207,000 MPa(30,000 ksi); crys = 400 MPa (58.0 ksi); v = 0.3.

7.10 A crack arrest test has been performed in accordance with ASTM E 1221. Theside-grooved compact crack arrest specimen has the following dimensions: W =100 mm (3.94 in), B - 25.4 mm (1.0 in), and BN = 19.1 mm (0.75 in). Theinitial crack length = 46.0 mm (1.81 in) and the crack length at arrest = 63.0 mm(2.48 in). The corrected clip gage displacements at initiation and arrest are V0 =0.582 mm (0.0229 in) and Va = 0.547 (0.0215 in), respectively. E = 207,000MPa (30,000 ksi) and ays(static) = 483 Mpa (70 ksi)- Calculate the stressintensity at initiation, K0, and the arrest toughness, Ka. Determine whether ornot this test satisfies the validity criteria in Eq. (7.25). The stress intensitysolution for the compact crack arrest specimen is given below.

!Wwhere

2.24(1. 72 -

9.85- 0.17* +11*2

13.8 CHAPTER 8

8.1 A 25.4 mm (1.0 in) thick plate of PVC has a yield strength of 60 MPa (8.70

ksi). The anticipated fracture toughness (Kfc) of this material is 5 MPa "Vm (4.5

ksi V in) Design an experiment to measure Kfc of a compact specimen machinedfrom this material. Determine the appropriate specimen dimensions (B, W, a) andestimate the required load capacity of the test machine.

8.2 A 15.9 mm (0.625 in) thick plastic plate has a yield strength of 50 MPa (7.25ksi). Determine the largest valid Kjc value that can be measured on this material.

8.3 A KIC test is to be performed on a polymer with a time-dependent relaxationmodulus which has been fit to the following equation:

£(?) = [0.417 + 0.0037f°<35r1

672 Chapter 13

where E is in GPa and t is in seconds. Assuming PQ is determined from a 5%secant construction, estimate the test duration (i.e. the time to reach PQ) at which90% of the nonlinearity in the load-displacement curve at PQ is due toviscoelastic effects. Does the 5% secant load give an appropriate indication ofmaterial toughness in this case? Explain.

8 . 4 Derive a relationship between the conventional J integral and the isochronous Jintegral, Jf, in a constant displacement rate test on a viscoelastic material forwhich Eqs. (8. 10) and (8.15) describe the load-displacement behavior.

8.5 A 500 mm wide plastic plate contains a through-thickness center crack that isinitially 50 mm long. The crack velocity in this material is given by

a =

i — * r~ — i —where K is in kPa v m anda is in mm/sec (1 psi \ in = 1.1 kPa vm, 1 in = 25.4mm). Calculate the time to failure in this plate assuming remote tensile stressesof 5 MPa and 10 MPa (1 ksi = 6.897 MPa). Comment on the sensitivity of thetime to failure on the applied stress. (As a first approximation, neglect the finitewidth correction on K. For an optional exercise, repeat the calculations with thiscorrection to assess its effect on the computed failure times.)

8.6 A composite double cantilever beam (DCB) specimen is loaded to 445 N (100 Ib)at which time crack growth begins. Calculate Gjc for this material assuminglinear beam theory.

E = 124,000 MPa (18,000 ksi); a = 76.2 mm (3.0 in); h = 2.54 mm (0.10in); B = 25.4mm (1.0 in).

8 . 7 One of the problems with testing brittle materials is that crack growth tends to beunstable in conventional test specimens and test machines. Consider, forexample, a single edge notched bend (SENB) specimen loaded in three pointbending. The influence of the test machine can be represented by a spring inseries, as Fig. 13.4 illustrates. Show that the stress intensity factor for this spec-imen can be expressed as a function of crosshead displacement and compliance asfollows:

K =

where At is the crosshead displacement, C is the specimen compliance, Cm is themachine compliance, and f(a/W) is defined in Table 12.2. Construct a


nondimensional plot of Kj versus crack size for a fixed crosshead displacement anda/W ranging from 0.25 to 0.75. Develop a family of these curves for a range ofmachine compliance. (You will have to express Cm in an appropriatenondimensional form.) What is the effect of machine compliance on the relativestability of the specimen? At what machine compliance would a growing crackexperience a relatively constant Kj between a/W = 0.5 and a/W= 0.6 ?

1 I

'I 'k\\\N^\\NI 1

tAT

r

SPAN = 4 W-

FIGURE 13.4 Single edge notch bend specimen loaded in crosshead control (Problem 8.7). The effect ofmachine compliance is schematically represented by a spring in series.

13.9 CHAPTER 9

9.1 Develop a computer program or spreadsheet macro to calculate stress intensityfactors for semielliptical surface flaws in flat plates subject to linear stressdistributions (Table 12.22). Plot a family of curves for F and H as a function ofa/t for a/c = 0, 0.2, 0.4, 0.6, 0.8, and 1.0, where 0 = 90° and c « W.

9.2 Calculate the stress intensity factor at the deepest point (0 = 90°) of an axial flawin a pressure vessel using both Eq. 9.10 and Table 12.31.

674 Chapter 13

p = 17.2 MPa (2500 psi); RI = 1.00 m; t/Ri = 0.10; a/t = 0.20; a/c = 0.40.

9.3 A nuclear reactor pressure vessel operates at an internal pressure of 17.2 MPa(2500 psi) and a temperature of 200°C (392°F). The steel in this pressure vesselhas an RT^DT of 100°C (212°F), and thus is relatively brittle at roomtemperature. Consequently, the full design pressure is not applied when thereactor is cold. Upon start-up, the temperature and pressure must be increased intandem in order to avoid brittle fracture.

(a) Determine the maximum allowable pressure-temperature curve, ranging fromambient to the design temperature. As a worst case, assume the vessel containsan internal axial surface flaw with a/t = 0.25 and a/c = 0.50, and that the fracturetoughness is given by the KIR curve (Eq. (9.17b)). Assume linear elastic con-ditions. The vessel dimensions are given below.

Ri = 2.16m (85.0 in); t = 216 mm (8.50 in)

(b) As the reactor operates over a period of several years, the steel becomesembrittled due to radiation damage, and the RTpjDT increases with time.Estimate the RTftDT at which it is no longer safe to start up the reactor.

(c) The pressure vessel is made from A 533 Grade B steel, which has a yieldstrength of 460 MPa (66.7 ksi). Was the assumption of linear elastic conditionsacceptable in this case?

9.4 A structure contains a through-thickness crack 20 mm long. Strain gages indicatean applied normal strain of 0.0042 when the structure is loaded to its design limit.The structure is made of a steel with £y = 0.0020 and 8crit =0.15 mm. Is thisstructure safe, according to the CTOD design curve?

9.5 A welded structure is loaded in combined bending and tension, with Pm = 200MPa and Pfr = 150 MPa. The structure is in the as-welded condition; the preciseresidual stress distribution in the weldrnent is unknown. Determine the

maximum allowable flaw size, a, according to the 1980 version of the PD 6493approach (Eqs. (9.18) and (9.19)).

0-75 = 400 MPa; E = 207,000 MPa; 8crit = 0.23 mm

9.6 A flat plate 1.0 m (39.4 in) wide and 50 mm (2.0 in) thick which contains asemi-elliptical surface flaw is loaded in uniaxial tension to 0.75 cryS- Assumingthe ratio a/2c - constant = 0.3, plot Kr and Sr values on a strip yield failureassessment diagram for various flaw sizes. Estimate the critical flaw size forfailure. (See Tables 12.22 and 12.28 for KI and limit load solutions.)


ays = 345 MPa (50 ksi); OTS = 448 MPa (65 ksi); E = 207,000 MPa (30,000

ksi); KIC = 110 MPa Vm" (100 ksi Vhi)

9.7 A pipe with 1.10 m (43.3 in) outside diameter and 50 mm (1.97 in) thick wallcontains a long internal axial flaw 10 mm (0.394 in) deep. The material flowproperties have been fit to a Ramberg-Osgood equation:

<J0 = 450 MPa (65.3 ksi); eo = (?o/E; a= 1.25; n = 9.72E = 207,000 MPa (30,000 ksi)

(a) Plot the applied J versus internal pressure.

(b) If J]c for this material is 300 kJ/m^ (1.71 in-kip/in^), determine the pressurerequired to initiate ductile crack growth.

(Disregard the Irwin plastic zone correction for all calculations.)

9.8 Suppose the edge cracked plate in Examples 9.2 and 9.3 is subject to a 5 MNtensile load.

(a) Calculate the applied / integral, both with and without the Irwin plastic zonecorrection.

(b) Calculate the load line displacement over a 5 m gage length.

(c) Calculate the load line displacement over a 50 mm gage length.

9 . 9 For the plate in the previous problem, estimate the following:

(a) dJ/da for fixed load (5 MN)

(b) dJ/da for fixed displacement at 5 m gage length. (P = 5 MN when a - 225mm.)

(c) dJ/da for fixed displacement at 50 mm gage length. (P = 5 MN when a = 225mm.)

9.10 When a single edge notched bend specimen is loaded in the fully plastic range, thedeformation can be described by a simple hinge model (Fig. 13.3). The plasticrotational factor can be estimated from load line displacement and crack mouthopening displacement as follows:

676 Chapter 13

rn =1

W-a

lwvpA n

-a

9.11

9.12

assuming a small angle of rotation. Beginning with Eqs (9.32) and (9.33) solvefor rn in terms of /z2> h$, and specimen dimensions. Use the resulting expressionto compute rp for n = 10 and a/W = 0.250, 0.375, 0.500, 0.625, and 0.750.Repeat for n = 3 and the same a/W values. Assume plane strain for allcalculations. How do the rn values estimated from the EPRI Handbook comparewith the assumed value of 0.44 in ASTM E 1290-89?

A welded panel 5 m wide and 50 mm thick contains a semielliptical surface crack(in the weld metal) with a = 10 mm and 2c = 54 mm. The primary membranestress is 260 MPa and the primary bending stress is 60 MPa. The panel is in theas-welded condition; the residual stress distribution is unknown. Perform a PD6493 Level 2 assessment on the weld flaw to determine whether or not it isacceptable. The material properties are as follows:

<JYS = 480 MPa; ajS = 610 MPa; E = 207,000 MPa; 8crit = 0.15 mm

Suppose that the panel in Problem 9. 1 1 is thermally stress relieved. Repeat theLevel 2 assessment, assuming the residual stresses after the heat treatment areequal to 30% of the yield strength.

9.13 Evaluate the structure from Problem 9.5 with Level 2 of the 1991 version of PD6493. Estimate the limiting flaw dimensions assuming a surface crack with a/2c= 0.25. GTS = 500 MPa; plate thickness = 25 mm.

13.10 CHAPTER 10

10.1 Using the Paris-Erdogan equation for fatigue crack propagation, calculate thenumber of fatigue cycles corresponding to the combinations of initial and finalcrack radius for a semicircular surface flaw tabulated below. Assume that thecrack radius is small compared to the cross section of the structure.

— = 6.87 x 10-12 (AK)3 , where da/dN is in m/cycle and AK is in MPa Vm".

Also, AG = 200 MPa.


Initial Crack Radius

1 mm1 mm2 mm2 mm

Final Crack Radius

10 mm20mm10 mm20mm

1 .1 MPaVin= 1 ksi Vin 25.4 mm = 1 in 1 MPa

ksi

Discuss the relative sensitivity of NfOt to:° initial crack size.9 final crack size.

10.2 A structural component made from a high strength steel is subject to cyclicloading, with (Jmax = 210 MPa and Gmin = 70 MPa. This componentexperiences 100 stress cycles per day. Prior to going into service, the componentwas inspected by nondestructive evaluation (NDE), and no flaws were found. The

material has the following properties: 0*75 = 1000 MPa, K\c = 2 5 M P a \ m .The fatigue crack growth rate in this material is the same as in Problem 10.1.

(a) The NDE technique can find flaws > 2 mm deep. Estimate the maximum safedesign life of this component, assuming that subsequent in-service inspectionswill not be performed. Assume that any flaws that may be present aresemicircular surface cracks and that they are small relative to the cross section ofthe component.

(b) Repeat part (a), assuming an NDE detectability limit of 10 mm.

10.3 Fatigue tests are performed on two samples of an alloy for aerospace applications.In the first experiment, R = 0, while R = 0.8 in the second experiment. Sketchthe expected trends in the data for the two experiments on a schematic log(da/dN)v. \og(AK] plot. Assume that the experiments cover a wide range of AK values.Briefly explain the trends in the curves.

10.4 Write a program or spreadsheet macro to compute fatigue crack growth behaviorin a compact specimen, assuming the fatigue crack growth is governed by theParis-Erdogan equation.

Consider a IT compact specimen (see Section 7.1.1) that is loaded cyclically at aconstant load amplitude with Pmax = 18 kN and Pmin = 5 kN. Using the fatiguecrack growth data in Problem 10.1, calculate the number of cycles required to

678 Chapter 13

grow the crack from a/W= 0.35 to a/W= 0.60. Plot crack size versus cumulativecycles for this range of a/W.

10.5 Write a program or spreadsheet macro to compute the fatigue crack growthbehavior in a flat plate that contains a semielliptical surface flaw and is subject toa cyclic membrane (tensile) stress. Assume that the flaw remains semielliptical,but take account of the difference in K at $ = 0° and 0 = 90°. Also, assume that c« W, but that aft is finite. Use the Paris-Erdogan equation to compute the crackgrowth rate.

Consider a 25.4 mm (1.0 in) thick plate that is loaded cyclically at a constantstress amplitude of 200 MPa (29 ksi). Given an initial flaw with a/t = 0.1 anda/2c = 0.1, calculate the number of cycles required to grow the crack to a/t = 0.8,using the fatigue crack growth data in Problem 10.1. Construct a contour plotthat shows the crack size and shape at a/t = 0.1, 0.2, 0.4, 0.6, and 0.8. Whathappens to the a/2c ratio as the crack grows?

10.6 Estimate U and K0p as a function of R and AK for the data in Fig. 10.8. DoesEq. (10.19) fit the data adequately or does U depend on Kmaxl Does Eq. (10.20)adequately describe the data? If so, determine the parameter Ko.

10.7 Suppose that the IT compact specimen in Problem 10.4 experiences a singleoverload of 36 kN when a/W'= 0.45. During all other cycles the load amplitudeis constant, with Pmax = 18 kN and Pmin = 5 kN. Using the Wheelerretardation model with 7= 1.5, estimate the number of cycles required to grow thecrack from a/W= 0.35 to a/W= 0.60. Plot crack size versus cumulative cycles,comparing the present case to the constant load amplitude case of Problem 10.4.Assume plane strain conditions at the crack tip and ays = 250 MPa.

10.8 You have been asked to perform ^"-decreasing tests on a material to determine thenear-threshold behavior at R = 0.1. Your laboratory has a computer-controlledtest machine that can be programmed to vary Pmax and Pmin on a cycle-by-cyclebasis.

(a) Compute and plot Pmax and fmin versus crack length for the range 0.5 <a/W < 0.75 corresponding to a normalized K gradient of - 0.07 mirr* in a ITcompact specimen.

(b) Suppose that the material exhibits the following crack growth behavior nearthe threshold:

— = 4.63jd(T12(A£3 -dN


where da/dN is in m/cycle and AK is in MPa Vm. For R = 0.1, zi^ = 8.50

MPa Vm". When the test begins, a/W = 0.520 and da/dN = 1.73 x 10~8 m/cycle.As the test continues in accordance with the loading history determined in part (a),the crack growth rate decreases. You stop the test when da/dN reaches 10"^m/cycle. Calculate the following:

(i) The number of cycles required to complete the test.(ii) The final crack length.(iii) The final AK.

13.11 CHAPTER 11

11.1 A series of finite element meshes have been generated that model compactspecimens with various crack lengths. Plane stress linear elastic analyses havebeen performed on these models. Nondimensional compliance values as afunction of a/W are tabulated below. Estimate the nondimensional stressintensity for the compact specimen from these data and compare your estimates tothe polynomial solution in Table 12.2.

aW

0.200.250.300.350.40

A B EP

8.6111.214.318.122.9

aW

0.450.500.550.600.65

A B EP

29.037.047.963.386.3

aW

0.700.750.800.850.90

A B EP

1231863065771390

11.2 A finite element analysis is performed on a through crack in a wide plate (Fig.2.3). The remote stress is 100 MPa, and the half crack length = 25 mm. Thestress normal to the crack plane (022) at 0 = 0 is determined at node points nearthe crack tip and is tabulated below. Estimate K] by means of the stress matchingapproach (Eq. 11.14) and compare your estimate to the exact solution for thisgeometry. Is the mesh refinement sufficient to obtain an accurate solution in thiscase?

r (Q f]\a (8 0)

0.0050.0100.0200.0400.060

Q"22o°°

11.08.076.004.543.89

fft ma

0.0800.1000.1500.2000.250

<j22o°°

3.503.242.832.582.41

680 Chapter 13

11.3 Displacements at nodes along the upper crack face («2 at 0 = TT) in the previousproblem are tabulated below. The elastic constants are as follows: fj. = 80,000MPa and K= 1.80. Estimate Kjby means of the displacement matching approach(Eq. 11.15) and compare your estimate to the exact solution for this geometry. Isthe mesh refinement sufficient to obtain an accurate solution in this case?

I (e = n)

0.0050.0100.0200.0400.060

HIa

9.99 x lO'5

1.41 x lO'4

1.99 x lO"4

2.80 x 10-4

3.41 x 10'4

I (9 = 10

0.0800.1000.1500.2000.250

HIa

3.92 x ID"4

4.36 x lO'4

5.27 x lO'4

6.00 x ID'4

6.61 x 10-4

11.4 Figure 13.6 illustrates a one-dimensional element with three nodes. Consider twocases: (1) Node 2 at x = 0.50L and (2) Node 2 at x = 0.25L.

(a) Determine the relationship between the global and parametric coordinates,x(%), in each case.

(b) Compute the axial strain, e(%) for each case in terms of the nodaldisplacements and parametric coordinate.

(c) Show that X2 = 0.25L leads to a 7/V x singularity in the axial strain.

2

£ = 0(a) Parametric coordinates.

XI = 0 X2 = L/2 X3 = L

(b) Global coordinates, Case (1).

2

XI = 0 X2 = 1/4 X3 = L FIGURE 13.6 One-dimensional element with 3

(c) Global coordinates, Case (2). nodes (Problem 11.4).

INDEX

INDEX

Advanced Turbine Technology Applications Program(ATTAP), 343

American Petroleum Institute (API), 402American Society for Testing and Materials (ASTM),

402, 407Standards. See Standards

American Society of Mechanical Engineers (ASME),470

Angled crack, 91-93, 95-96Applied tearing modulus, 143-144, 158-159Arc-shaped specimen, 8, 366-367, 602-603Arrest toughness, 213, 397-402, 664ASME reference curves, 470-472ASME standards. See StandardsAsymptotic analyses, 105-109, 176-177. See also

Deformation plasticityAtomic theory of fracture, 31-33, 38

B

Biaxiality ratio, 161-162Biaxial loading, 95-96Boundary collocation method, 572Boundary integral equation (BIE) method, 569-571Bridge design, 8-9Bridge indentation, 454-456Bridging toughening mechanism, 343, 352British Standards Institution (BSD, 365, 385, 402, 498British Welding Research Association, 12-13Buckingham II-theorem, 21-22Buried cracks, stress intensity solutions, 629

C* integral, 226-229, 664-665Carbon/epoxy composite materials, 447Carbon/PEEK composite materials, 447Center cracked tension (CCT), 155, 157Central Electricity Generating Board (CEGB), 477Ceramics, 313, 343-353

bridging toughening, 343, 352crack deflection toughening, 343ductile phase toughening, 343, 349-351fracture testing, 451-456microcrack toughening, 343-344, 347-348process zone toughening, 343-346, 667transformation toughening, 343-344, 348-350whisker toughening, 344, 350-353

Chain disentanglement, 323Chain scission, 323Challenger Space Shuttle, 7Charpy impact tests, 11, 206, 409-410, 444-445Chevron-notched specimen, 451-454Clamping, pinch, 4, 6, 313Cleavage fracture, 265-266, 282-297, 666

critical J values for, 392

initiation, 285-289models of fracture toughness, 289-297statistical modeling of, 307-312

Cleavage scaling model, 169-173Closure stresses, 70-72, 76-77, 355, 520-531Cohesive zone model, 354—357Comet jet aircraft, 12Compact specimen, 366-368, 667-668, 671-678Composite materials, 313. See also Polymers

compressive failure, 335-338fiber-reinforced plastics, 329-342fiber waviness, 337-338fracture mechanics applicable to, 330-331fracture testing, 445-451interlaminar toughness, 445-451notch strength, 339-341

Compressively loaded structures, 8Computer analysis of fracture mechanics, 565-592Concrete, fracture of, 354—357Corner cracks, stress intensity solutions, 630-631Cost of fracture, 3Crack arrest, 211, 213, 220-222, 291, 295

testing, 398, 670-671toughness, 396-402

Crack deflection, 343Crack driving force, 41, 658. See also Energy release

rateCrack extension, 41, 43, 211-222. See also Crack

growth; Crack growth resistance curves;Driving force curve; Energy release rate;Fracture toughness, CTOD testing; Fracturetoughness,KIc. J testing of metals

numerical analyses, 573-577, 591-592virtual, 573-577

Crack growth, 4, 18. See also Creep crack growth;J-controlled fractures

ductile, 279-281, 283, 488-493, 666, 675fatigue equations, 517-520, 661instability analysis, 104—105J-controlled, 152-155rate of, 18, 117,205-245steady state, 198-201stick/slip, 326-327

Crack growth resistance curves, 142-148, 658definition, 46-51J resistance curves, 146-148, 212shape of R curve, 48, 142, 154, 365, 451, 453-454stable crack growth, 47, 143-146, 198-201unstable crack growth, 47, 49, 143-146, 211, 213

Crack mouth opening displacement (CMOD),372-374, 394-395

Crack propagation, 42, 91-93, 95-96. Sec alsoFatigue conditional probability model,309-312, 506-507

Crack shape, 59, 115-116, 461, 673Crack speed, 214-216, 219-220Crack surface area, definition, 37-38Crack tip, 17, 40

683

Fracture Mechanics: Fundamentals and Applications

atomic spacing, 38blunting, 129-131, 153-154, 281, 545characterized by J, 152-155closure stresses, 70-72, 520-531constraint, 509ductile crack growth, 280elastodynamic parameters, 216-218, 251-255plasticity, 72-81, 84-86, 121-122, 148-155

corrections, 468-469, 662effect on stress fields, 149-150under large-scale yielding, 155-159, 197-198under small-scale yielding, 194-197, 258

polymers, 326-327reverse plasticity, 531-534shielding, 348-349stress analysis of, 51-55, 105-109. See also Finite

element methodstress fields in nonlinear materials, 13three-dimensional effects, 509viscous material. See Creep crack growth, C* integralyielding zone. See Crack tip, blunting; Irwin

plastic zone correction; Strip yield modelCrack tip finite element analyses, 141-142. See also

HRR singularityCrack tip opening displacement (CTOD), 13-14,

117-122, 586,590,670CTOD design curve, 13, 472-474, 498-500CTOD testing for fracture toughness, 392-395definition, 120hinge model analysis of, 120-12290 degree intercept, 140-141polymers, 326-327relationship with G, 120, 138relationship with J, 138-142, 148-155, 185-186rotational factor, 393, 675-676standards. See Standards, ASTM E 1290strip yield model analysis of, 119-120, 139,

183-186,241,501-503,658Craze zone, 324-326, 667Crazing stress, 324-327, 667Creep crack growth, 205, 225-233. See also Crack

growth; Crack growth resistance curvesC* integral, 226-229, 664-665

Critical stress intensity factor, 16-20, 87Crosshead displacement, 432, 434, 672CTOD. See Crack tip opening displacement (CTOD)C, parameter, 231-233C(t) parameter, 230-231Cup and cone fracture, 271-272, 274-275Cyclic stresses. See FatigueCylinders. See Flawed cylinders

Damage tolerance methodology, 552-554Deformation J, 419-422Deformation plasticity, 122-123, 201-204Degree of polymerization, 314Design, approaches to, 14-18Dimensional analysis, 21-24Disk-shaped compact specimen, 366-367, 602-603

Displacement control, 43-45, 48-51, 143-144, 658.See also Load control

Double cantilever beam (DCB) specimenapplied tearing modulus, 145crack arrest on, 222energy release rate for, 45-46, 659, 672in interlaminar toughness tests, 447-450

Double edge notched tension (DENT) panel, 133,155-156, 158-159

limit load solutions, 602, 604, 606-607, 609,620-621

Driving force curve, 46, 658Driving force equations, 478-498, 504-505. See also

Strip yield modelDrop weight tear test, 412-413Drop weight test, 397. 409, 411-412Ductile-brittle transition, 297-299, 407-408Ductile crack growth. See Crack growth, ductileDuctile fracture, 265-266, 269-270, 297-299Dynamic fracture mechanics, 19, 205-245

mathematical derivation of, 251-259Dynamic material resistance, 216, 218-220Dynamic tear tests, 412-413

Edge cracks, 23-24, 56-57, 120, 664, 675. See alsoArc-shaped specimen; Compact specimen;Disk-shaped compact specimen; Double edgenotched tension (DENT) panel; Hinge model;Single edge notched bend (SENB) panel;Single edge notched tension (SENT) panel

ductile crack growth of, 282plate in pure bending, 135-136, 603

Elastic-plastic fracture mechanics, 19-20, 23, 84,117-178. See also Crack tip openingdisplacement; J contour integral

mathematical derivation of, 183-204Elastic T stress, 160-163Elastodynamic fracture mechanics, 205-206, 216-218

mathematical derivation of, 251-255Electric Power Research Institute (EPRI), 13,

478-493, 496^498Element crack advance, 572-573Elliptical integral of the second kind, 115-116End notched flexure (ENF), 447-448Energy, surface, 15Energy domain integral methodology, 577-585Energy release rate, 11, 41-46, 657

definition, 15-16, 124mathematical derivation of, 255-259, 573of nonlinear materials, 13, 123-126, 188-190

Engineering plastics, 314-342. See also PolymersEPRI, 13, 478EPRI J estimation scheme, 478-493, 495-498European Group on Fracture (EOF), 447

Failure assessment diagrams, 474-478, 487-488,500-505

Index 685

Failure criterion, 84-86, 168-169Fatigue

crack closure, 520-531, 549-552crack growth, 12, 517-520, 677-678damage tolerance, 552—554experimental measurments, 547-552J contour integral, 559-564micromechanisms, 543-546polymers, 328-329, 341-342Region II, 543-545Region III, 546retardation, 531, 534-536short cracks, 539-542similitude, 16-17, 513-517threshold stress intensity, 528-530, 546, 678variable amplitude loading, 531, 537-539

Fatigue precracking, 370-371, 406Fiber failure, in composite materials. See Composite

materials, individual types of failureFiber waviness, 337-338Fictitious crack model, 354-357Finite element rnesh, 586-591, 679Finite element method, 566-569, 679First Law of Thermodynamics, 10, 36, 211Flawed cylinders, 633, 659, 673-675

fully plastic J and displacement, 641-654influence coefficients, 637-640limit load solutions, 641stress intensity solutions, 634—636, 673

Flaw size, 10, 58-63, 674Fractography, 283-285Fracture mechanisms in metals. See Cleavage fracture;

Ductile fracture; Intergranular fractureFracture testing

ceramics, 451-456composite materials, 445-451metals. See Fracture toughness,CTOD; Fracture

toughness,J-R curve testing; Fracturetoughness,KIa testing ; K-R curve testing

polymers, 423-445Fracture toughness, 84, 459, 664

cleavage, 289-297CTOD testing, 392-395ductile-brittle transition, 297-299, 407-408dynamic, 396-402effect of loading mode, 52-55, 87effect of loading rate, 211-214J contour integral. See J contour integral

J,c

definition, 143fracture testing of polymers, 423, 427-432,

439-440measurements for testing metals, 385-389, 675

J-Q locus, 163-169J-R curve testing, 389-391, 423-432, 439Kc 16-20, 87K!a'testing, 213, 397-402, 664Klc 87-89, 211-212, 371, 375-380

crack tip opening displacement, 117-119critical J value, 469fracture testing of polymers, 423-427, 434—439

practice problems, 659, 661, 666,-668size requirements for validity. See Standards,

ASTM E 399K,D 216, 218-220plane strain. See Fracture toughness, KIc

recent research in, 14specimen size requirements, 434—436, 438theory, 14-20

Fully plastic J and displacementflawed cylinders, 641 -654through-thickness cracks, flat plates, 610-626

Fundamental dimensional units (fdu), 21

G. See Energy release rateGc 16, 18,657General Electric Corporation, 12, 478Griffith model of energy balance, 10, 36-41, 258,

661,666modified for non-brittle solids, 39^1

Gull-wing configuration, 403Gurson model, 277-278

H

Hall-Petch equation, 287Hazard function, 309-310HAZ testing, 402-405, 407H factors for flawed cylinders, 623-626, 643-645Hillerborg model, 354-357Hinge model, estimation of CTOD, 120-122History of fracture mechanics, 3-14Hooke's law, 31HRR singularity, 127-131, 139-140

mathematical derivation of, 190-194non-requirement in J-dominated zone, 151

I

Impact loading, 206-211, 396-397Inertia forces, 205Influence coefficients

flawed cylinder, 637-640part-through cracks, flat plates, 633surface cracks, 632

Inglis stress analysis, 38-39, 661Instrumentation for testing of metals, 372-373Intergranular fracture, 2, 265-266, 299-301, 666Interlaminar toughness, 445-451International Institute of Standards (ISO), 365, 385International Institute of Welding, 402Irwin plastic zone correction, 12, 72-75, 78—79, 89,

660. See also Strip yield modelzone shape, 78-81,371

Izod impact test, 409-410, 444—446

Japanese Industrial Standards (US), 447Japan Society of Mechanical Engineers (JSME), 354

686 Fracture Mechanics: Fundamentals and Applications

J contour integral, 122-142, 663, 671. See alsoEnergy release rate, of nonlinear materials;EPRI J estimation scheme

application to cyclic loading, 559-564, 677contour integration numerical analysis, 573dynamic, 222-225energy domain integral methodology, 577-585history of research, 13laboratory measurement of, 131-138mathematical derivation of, 186-188newer approaches to, 18, 117, 205-245path independence, 126-127relationships with CTOD, 138-142, 148-155, 185-186viscoelasticity, 238-242, 478-493, 665

J-controlled fractures, 148-155J-dominated zone, 149-151J estimation scheme, 478^93, 496-498J0 parameter, 169-171J-Q theory, 163-169J resistance curves, 146-148, 212J testing of metals, 13-14. See also Fracture toughness

K

K. See Stress intensity factorK-dominated zone, 149-151Klc See Fracture toughness, KIC

Kink bands, 336, 338Kinked crack, 93-96K, solutions, 461-467, 659K-R curve testing, 380-384, 668Kurdistan oil tanker, 4-5

Load displacement curves, 491, 494, 675Loading mode. See Fracture toughness, effect of

loading mode; Mixed-mode fracture

Material resistance, 15Material toughness, 13, 15-17Mathematical derivations

dynamic fracture mechanics, 251-259elastic-plastic fracture mechanics, 183-204elastodynamic fracture mechanics, 251-255energy release rate, 255-259, 573HRR singularity, 190-194J contour integral, 186-188linear elastic fracture mechanics (LEFM), 101-116

Maxwell model, 319-322, 667Mere, 314Mesh design, 586-591, 679Metals, fracture mechanisms in. See Cleavage

fracture; Ductile fracture; Intergranularfracture

Microvoid coalescence, 268, 270, 273, 280-282, 666.See also Crack growth, Ductile

Middle tension (MT) specimen, 366-367, 603, 609,616-617

Mixed-mode fracture, 53, 69, 91-96Models to scale, 21-24Modified boundary layer analysis, 160Monte Carlo analysis, 507

N

LEFM. See Linear elastic fracture mechanics (LEFM)Liberty ships, 3-4, 10-11, 409Limit load solutions, 601, 609-610

flawed cylinders, 641part-through cracks, flat plates, 632surface cracks, 633through-thickness cracks, flat plates, 609-610

Linear elastic fracture mechanics (LEFM), 12-13,19-20, 31-96,679

applicability to composite materials, 340-341application to structures, 459-470fatigue crack propagation, 513-554inapplicability to concrete, 354laboratory testing, 375-384

metals, 385-413nonmetals, 423-456

limitations of, 117, 469-470, 507-508mathematical derivation of, 101-116nonlinear behavior, 19, 205-245, 251-259stress intensity factor. See Stress intensity factorvalidity limits, 89-90without nonlinear behavior, 205-206, 216-218

Linear variable differential transformer (LVDT),372-374

Load control, 43-45, 48-51, 143-144, 658. See alsoDisplacement control; Rapid loading

National Aerospace Plane, 343National Institute of Standards and Technology

(NIST), 398-399Natural gas piping, 4Naval Research Laboratory, 11, 409Nil-ductility transition temperature (NDTT), 411-412Nondestructive evaluation (NDE), 553, 677Nonlinear materials. See Crack tip, stress fields in

nonlinear materials; Elastic-plastic fracturemechanics; Energy release rate, of nonlinearmaterials

Notch location and orientation, 403-405Notch strength, 339-341Nuclear power, 13, 470Numerical fracture analysis, 565-571, 591-592

Overview of fracture mechanics, 3-7, 13, 15-17

Paris law, 517, 520, 676-677Part-through cracks, flat plates

influence coefficients, 632limit load solutions, 633stress intensity solutions, 627-631

Index 687

PD 6493 method, 498-505PE (polyethylene), 4, 6. See also PolymersPinch clamping, 4, 6, 313Plane strain, 82-84, 87-89

requirement for valid K!c See Standards, ASTM E399

Plane stress, 82-84Plasticity of crack tip. See Crack tip, plasticityPlastic work, 15Plastic zone. See Irwin plastic zone correctionPolyethylene (PE), 4, 6. See also PolymersPolymers, 4, 313. See also Composite materials;

Polyethyleneamorphous, 316-317, 667chain disentanglement, 323chain scission, 323crack tips, 326-327craze zone, 324-326, 667crazing stress, 324-327, 667crystalline, 316-317fatigue damage, 328-329, 341-342fracture testing, 423-445fracture toughness, 322-323, 334, 671glass transition temperature, 205Maxwell model, 319-322, 667molecular structure, 315-316molecular weight, 314-315rubber toughening, 328shear yielding, 323, 325-326stress-strain response, 434-435stress-whitened region, 324, 327, 667thermoset, 315viscoelastic behavior, 317-319, 440-444, 665Voigt model, 319-322

Potential drop technique, 372, 374Precracking

of ceramics, 454-456of polymers, 432-434

Primary stresses, 467Principle of superposition, 64-66, 131Probabilistic fracture mechanics, 506-507Process zone toughening, 343-346, 667

QQ stress, 163-169Quadrilateral element, 596-598

Ramberg-Osgood power law, 151, 675Rapid loading, 206-211, 396-397Razor notching of polymer, 432—433, 444-445R curve. See Crack growth resistance curves; K-R

curve testingReference stress approach, 493—496, 504-505

EPRI J estimation scheme, 496-498Reflected stress waves, 205, 217Research in fracture mechanics, 3-14Residual stresses, 467, 508Resistance curves. See Crack growth resistance curves

Reverse plasticity of crack tip, 531-534Rice-Drugan-Sham (RDS) analysis, 194-198Ritchie-Knott-Rice (RKR) model, 167, 290, 292, 339River patterns, 286R6 method, 505-506Robertson crack arrest test, 397Rock, fracture of, 354-357RP2Z, 407Rubber toughening, 328

Scale models, 21-24Secondary stresses, 467, 508Shear lips, 281Shear yielding, 323, 325-326Ships, fractures in, 3-5, 10-11, 409Short cracks, 539-542Side grooves, 373-374Similitude, 16-17,513-517Single edge notched bend (SENB) panel, 155,

157-159, 162, 366-367, 393, 454-456limit load solutions, 603, 609, 614-615practice problems, 608, 672, 675

Single edge notched tension (SENT) panel, 604, 609,618-619

Singularity dominated zone, 53, 55Singularity elements, 595-598Specimen configurations, 366-376Standards

ASTM D 256, 444-445ASTM D 5045, 434-439ASTM E 23, 409, 444-445ASTM £208,411-412ASTM E 399, 89, 375-380, 434-436ASTM E 436, 412-413ASTM £561,380-384ASTM E 604, 412^413ASTM £616,368-369ASTM E 647, 547-549ASTM £813, 385-392ASTME 1152,385-392ASTM E 1221,398,401ASTME 1290, 121,393ASTME 1304,451,454BS 3505, 445BS 3506, 445BS 5447, 375BS 5762, 394fracture toughness of metals, 365-366

Stress and displacement matching, 572Stress concentration factor (SCF), 339Stress intensity factor, 11, 17-18, 52-55, 64, 77-78,

466, 659chevron-notched specimen, 452-453computational method, 572effect of loading mode, 52-55, 87, 93-96elastic compliance solutions, 601-602, 605-608failure criterion, 84-86, 168-169J as a stress intensity parameter, 127-129relationship between K and G, 69-72

688 Fracture Mechanics: Fundamentals and Applications

relationship between K and global behavior, 55-58superposition, 64-66weight functions, 67-69

Stress intensity solutions, 601-608, 627-631,634-636, 679

Striations, 544-545Strip yield model, 72, 75-78, 326

to analyze CTOD, 119-120for failure assessment, 477-478

Superposition, 64-66, 95, 131, 234, 461-464Surface cracks

influence coefficients, 632, 637-640limit load solutions, 633stress intensity solutions, 627-628

Surface energy, 15

Tearing modulus, 143Tensile strength, 8-9Thermal stress, 4Thermoplastics, 315Thermoset polymer, 315Thickness. See Dimensional analysis; Fracture

toughness,K[c; Plane strainThree-point bend specimen, 207-209, 659, 662-664,

669-670, 672Through-thickness cracks, flat plates

configurations, 603-604fully plastic J and displacement, 610-626limit load solutions, 609-610stress intensity and elastic compliance, 601-608,

657, 659, 672Time-dependent fracture mechanics, 18, 117,

205-245. See also Dynamic fracturemechanics; Elastic-plastic fracture mechanics;Viscoelasticity; Viscoplasticity

Crack growth. See Crack growthTower Bridge in London, 8-9Transition time, 207-209Triangular element, 598T stress, 160-163Tunneling effect, 281Two-parameter fracture mechanics, 160-178. See aIso

Cleavage scaling model; J-Q theory; T stress

Unloading compliance technique, 372, 669

Van der Waals bonds, 318, 322Virtual crack extension, 573-577Viscoelasticity, 233-234

J contour integral, 238-242, 478-493, 665linear viscoelastic material, 234-237,

665-666nonlinear viscoelastic material, 242-245,

665polymers, 317-319, 440-444, 665

Viscoplasticity, 14, 19-20Void growth, 268-273, 276-282void nucleation, 267-268, 269, 270, 272Voigt model, 319-322Von Mises yield criterion, 78-79. See also Irwin

plastic zone correction

wWeakest link model, 307-309Weibull distribution, 296-297Weight functions, 67-69Welding Institute, 402-403Weldment testing, 402, 674, 676

API document RP2Z, 407driving force, 508fatigue precracking, 406gull-wing configuration, 403HAZ testing, 402-405, 407notch location and orientation, 403^1-05post-test analysis, 406-407residual stresses, 402, 676specimen design and fabrication, 402-403

Westergaard stress function, 11, 109-115, 661Whisker reinforced ceramics, 344, 350-353

Young's modulus, 16, 33, 122

Documents

Fracture Mechanics(Anderson)