Sun01-fm-i-iv-97801238500102011/10/110:21Page1#1Fracture MechanicsSun01-fm-i-iv-97801238500102011/10/110:21Page3#3Fracture MechanicsC. T. SunSchool of Aeronautics and AstronauticsPurdue UniversityWest Lafayette, IndianaZ.-H. JinDepartment of Mechanical EngineeringThe University of MaineOrono, MaineAMSTERDAM BOSTON HEIDELBERG LONDONNEW YORK OXFORD PARIS SAN DIEGOSAN FRANCISCO SINGAPORE SYDNEY TOKYOAcademic Press is an imprint of ElsevierSun01-fm-i-iv-97801238500102011/10/110:24Page4#4Academic Press is an imprint of Elsevier225 Wyman Street, Waltham, MA 02451, USAThe Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UKc 2012 Elsevier Inc. All rights reserved.No part of this publication may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, recording, or any information storage and retrieval system,without permission in writing from the publisher. Details on how to seek permission, furtherinformation about the Publishers permissions policies and our arrangements with organizations suchas the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website:www.elsevier.com/permissions.This book and the individual contributions contained in it are protected under copyright by thePublisher (other than as may be noted herein).NoticesKnowledge and best practice in this eld are constantly changing. As new research and experiencebroaden our understanding, changes in research methods, professional practices, or medical treatmentmay become necessary.Practitioners and researchers must always rely on their own experience and knowledge in evaluatingand using any information, methods, compounds, or experiments described herein. In using suchinformation or methods they should be mindful of their own safety and the safety of others, includingparties for whom they have a professional responsibility.To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assumeany liability for any injury and/or damage to persons or property as a matter of products liability,negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideascontained in the material herein.Library of Congress Cataloging-in-Publication DataApplication submitted.British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.ISBN: 978-0-12-385001-0For information on all Academic Press publicationsvisit our Web site at www.elsevierdirect.comPrinted in the United States1112131415 10987654321Sun02-ded-v-vi-97801238500102011/10/110:34Pagev#1To my wife, Irisand my children, Edna, Clifford, and LeslieC. T. SunTo my wife, ZhenZ.-H. JinSun04-pre-xiii-xvi-97801238500102011/10/110:35Pagexiii #1PrefaceFracture mechanics is now considered a mature subject and has become an impor-tant course in engineering curricula at many universities. It has also become a usefulanalysis and design tool to mechanical, structural, and material engineers. Fracturemechanics, especially linear elastic fracture mechanics (LEFM), is a unique eld inthat its fundamental framework resides in the inverse square root type singular stresseld ahead of a crack. Almost all the fracture properties of a solid are characterizedusing a couple of parameters extracted from these near-tip stress and displacementelds. In view of this unique feature of fracture mechanics, we feel that it is essen-tial for the reader to fully grasp the mathematical details and their representation ofthe associated physics in these mathematical expressions because the rationale andlimitationsofthisseeminglysimpleapproachareembodiedinthesingularstresseld.There are already more than a dozen books dealing with fracture mechanics thatmaybeusedastextbooksfor teachingpurposes. Withdifferent emphases, thesebooks appeal to different readers and students from different backgrounds. This bookis based on the lecture notes that have been used at the School of Aeronautics andAstronautics, PurdueUniversity, formorethan30years. Itisintendedasabookforgraduatestudentsinaeronautical, civil, mechanical, andmaterialsengineeringwho areinterested in pickingup an in-depthunderstanding of how toutilize frac-ture mechanics for research, teaching, and engineering applications. As a textbook,our goal is to make it mathematically readable to rst-year graduate students with adecent elasticity background. To achieve this goal, almost all mathematical deriva-tionsareclearlypresentedandsuitableforclassroomteachingandforself-studyas well.In selecting and presenting the contents for this book, we use the aforementionedrationaleasaguide. InChapter2, theGrifththeoryoffractureandthesurfaceenergy concept are introduced. Chapter 3 presents the elastic stress and displacementelds near the crack tip and introduces Irwins stress intensity factor concept. Thechapter describes detailed derivations of the stress elds and stress intensity factor Kusing the complex potential method and Williams asymptotic expansion approach.Finally, the chapter introduces the fracture criterion based on the stress intensity fac-tor (K-criterion) and discusses the K-dominance concept to make the reader aware ofthe limitation of the K-criterion.Chapter 4 is totally devoted to energy release rate in conjunction with the path-independent J-integral. The energy release rate concept is rst introduced, and therelationship between the energy release rate G and stress intensity factor (GK rela-tion)isestablishedfollowedbythefracturecriterionbasedontheenergyreleaserate(G-criterion). TheJ-integral iswidelyacceptedbecauseitsvalueisequal toxiiiSun04-pre-xiii-xvi-97801238500102011/10/110:35Pagexiv#2xiv Prefacetheenergyreleaserateandit canbecalculatednumericallywithstressanddis-placement elds away from the singular stress at the crack tip. Another simple, yetefcient, crack-closuremethodhasbeenshowntobequiteaccurateinevaluatingenergyreleaserate. Therefore, acoupleof niteelement-basednumerical meth-odsforcalculationofenergyreleaserateandthestress-intensityfactorusingthecrack-closure method are included in this chapter.In most fracture mechanics books, the near-tip stress eld is presented in planeelasticity for Mode I and Mode II loadings and in generalized plane strain for ModeIII. In reality, none of these 2-D states exists. For instance, a thin plate containing acenter crack is usually treated as a 2-D plane stress problem. In fact, the plane stressassumption fails because of the presence of high stress gradients near the crack tip anda state-of-plane strain actually exists along most part of the crack front. The knowl-edgeofthe3-Dnatureofallthrough-the-thicknesscracksisimportantinLEFM.InChapter 4, asectionisdevotedtothethe3-Deffect onthevariationofstressintensity along the crack front.Under static Mode I loading, experimental results indicate that the direction ofcrack extension is self-similar. As a result, in determining Mode I fracture toughnessof a solid, the crack extension direction is not an issue. The situation is not as clearif the body is subjected to combined loads or dynamic loads. Of course, if the bodyis an anisotropic solid such as a berous composite, the answer to the question ofcracking direction is not as simple and is not readily available in general. In viewof this constraint, we only consider isotropic brittle solids in Chapter 5. The focusis on the prediction of crack extension direction. From a learning point of view, it isinteresting to follow a number of different paths of thinking taken by some earlierresearchers in the effort to predict the cracking direction.Chapters 6 and 7 present the result of the effort in extending the LEFM to treatfracture in elastic-plastic solids. In Chapter 6, plastic zones near the crack tip for thethree fracture modes are analyzed. Several popular and simple methods for estimat-ing the crack tip plastic zone size are covered. The initial effort in taking plasticityintoaccountinfractureswasproposedbyIrwinwhosuggestedusinganeffectivecrack length to account for the effect of plasticity. Later, the idea was extended tomodeling the so-called R-curve during stable crack growth. Another approach thatuses the J-integral derived based on deformation plasticity theory to model the cracktipstressandstrainelds(theHRR eld)alsohasmanyfollowers.InadditiontoIrwins adjusted crack length and the J-integral approach, crack growth modeled bycrack tip opening displacement (CTOD or CTOA) is also discussed in Chapter 7.Interfaces between dissimilar solids are common in modern materials and struc-tures. Interfaces are often the weak link of materials and structures and are the likelysites for crack initiation and propagation. Interfacial cracks have many unique phys-ical behaviors that are not found in homogeneous solids. However, surprisingly, thedevelopmentoffracturemechanicsforinterfacialcrackshasfollowedexactlythesame path as LEFM. In other words, fracture mechanics for interfacial cracks is allcentered on the crack tip stress eld. The only difference is in the violently oscilla-tory behavior of the crack tip stress eld of interfacial cracks. Chapter 8 presents aSun04-pre-xiii-xvi-97801238500102011/10/110:35Pagexv#3Preface xvthorough derivation of the crack tip stress and displacement elds. Attention is alsofocused on the signicance of stress oscillation at the crack tip and the nonconvergentnature of the energy release rates of the individual fracture modes.The cohesive zone model (CZM) has become a popular nite element-based toolformodelingfractureinsolids.CZMisoftenconsideredbysomeasamorereal-istic form of fracture mechanics because it does not employ the idealized singularstresses. Although there are fundamental differences between the two concepts, thepurposesofthetwoarethesame. Therefore, it isreasonabletoincludeCZMinthis book. In Chapter 9 we make an effort to present the basic formulation of CZM,especially the cohesive traction law. Instead of covering examples of applications ofthe cohesive zone model, we place greater emphasis on the logic in the formulationof CZM.Chapter 10 contains brief and condensed presentations of three additional topics,namely, anisotropic solids, nonhomogeneous solids, and dynamic fracture. The rea-son for including these three topics in this textbook is, perhaps, just for the sake ofcompleteness. For each topic, the coverage is quite brief and with a limited scope anddoes not warrant a full chapter.C. T. SunZ.-H. JinSun05-ata-xvii-xx-97801238500102011/10/112:16Pagexvii #1About the AuthorsC. T. Sun received his undergraduate education at National Taiwan University. Heobtained his M.S. in 1965 and Ph.D. in 1967 from Northwestern University. In 1968he joined Purdue University, where he is presently Neil A. Armstrong DistinguishedProfessorintheSchoolofAeronauticsandAstronautics.Hehasbeenengagedincomposites research for more than forty years. In addition to his work in composites,ProfessorSunhaspublishedextensivelyintheareasoffracturemechanics,smartmaterials, andnanomechanics. Hehasauthoredatextbookonaircraft structurespublished in 1998 with the second edition published in 2006.Z.-H. Jin is an Associate Professor in the Department of Mechanical Engineeringat the University of Maine. He obtained his Ph.D. in Engineering Mechanics fromTsinghua University in 1988. His research areas include fracture mechanics, thermalstresses, mechanical behavior of materials, and geodynamics. He has published morethan 70 refereed journal papers and three book chapters.xviiSun05-ata-xvii-xx-97801238500102011/10/112:16Pagexix#3Fracture MechanicsSun06-ch01-001-010-97801238500102011/10/317:11Page1#1CHAPTERIntroduction11.1FAILURE OF SOLIDSFailure of solids and structures can take various forms. A structure may fail withoutbreaking the material, such as in elastic buckling. However, failure of the material ina structure surely will lead to failure of the structure. Two general forms of failure insolids are excessive permanent (plastic) deformation and breakage. Plasticity can beviewed as an extension of elasticity for decribing the mechanical behavior of solidsbeyond yielding. The theory of plasticity has been studied for more than a centuryand has long been employed for structural designs. On the other hand, the latter formof failure is usually regarded as the strength of a solid, implying the total loss of load-bearing capability of the solid. For brittle solids, this form of failure often causes thebody under load to break into two or more separated parts.Unlike plasticity, the prediction of the strength of solid materials was all basedon phenomenological approaches before the inception of fracture mechanics. Manyphenomenological failure criteria in terms of stress or strain have been proposed andcalibrated against experimental results. In the commonly used failure criteria, suchas the maximum principal stress or strain criterion, a failure envelope in the stressor strain space is constructed based on limited experimental strength data. Failure isassumed to occur when the maximum normal stress at a point in the material exceedsthe strength envelope, that is,1fwhere1(> 0) is a principal stress andfis the tensile strength of the solid. Thefailure envelope has also been modied to distinguish the difference between tensileand compressive strengths and to account for the effects of stress interactions.In general, the classical phenomenological failure theories predict failure of engi-neering materials and structures with reasonable accuracy in applications where thestress eld is relatively uniform. These theories are often unreliable in the presence ofhigh-stress gradients resulting from cutouts. Moreover, there were many prematurestructural failures at stresses that were well below the critical values specied in theclassical failure theories.Fracture Mechanics. DOI: 10.1016/B978-0-12-385001-0.00001-8c 2012 Elsevier Inc. All rights reserved.1Sun06-ch01-001-010-97801238500102011/10/317:11Page2#22 CHAPTER 1IntroductionThe most frequently cited example is the failure of Liberty cargo ships built dur-ing World War II. Among roughly 2700 all-welded hull ships, more than 100 wereseriouslyfracturedandabout10werefracturedinhalf[1-1].Itwasdemonstrated[1-2]thatcrackswererstinitiatedatthestressconcentrationlocationsandthenpropagated in the hull, resulting in the catastrophic failure. Other signicant exam-ples include fuselage failure in Comet passenger jet airplanes from1953 to 1955 [1-3]and failure of heavy rotors in steam turbines from 1955 to 1956 [1-4].The aforementioned historical events led researchers to recognize that defects arethe original cause of failure and in strength predictions, materials cannot be alwaysassumed free of defects. Cracks and other forms of defects may be introduced duringmaterials manufacturing and processing, as well as during service. For instance, rapidquenching of cast irons results in microcracks in the material. Cyclic stresses inducecracks in the connections of the structural components. The stresses at the crack tipare much higher than the material strength, which is measured under a state of uni-formstress in laboratory condition. The high stresses near the crack tip drive the crackto extend, leading to the eventual catastrophic failure of the material. Failure causedby crack propagation is usually called fracture failure. The classical failure criteriaassumethatmaterialsarefreeofdefects, andhencearenotcapableofpredictingfracture failure, or failure of materials containing crack-like aws.1.2FRACTURE MECHANICS CONCEPTSFracture mechanics is a subject of engineering science that deals with failure of solidscaused by crack initiation and propagation. There are two basic approaches to estab-lish fracture criteria, or crack propagation criteria: crack tip stress eld (local) andenergybalance(global)approaches. Inthecracktipeldapproach, thecracktipstress and displacement states are rst analyzed and parameters governing the near-tip stress and displacement elds are identied. Linear elastic analysis of a crackedbody shows that stresses around the crack tip vary according to r1/2, where r is thedistance from the tip. It is clear that stresses become unbounded as r approaches thecrack tip. Such a singular stress eld makes the classical strength of materials failurecriteria inapplicable.A fundamental concept of fracture mechanics is to accept the theoretical stresssingularity at the crack tip but not use the stress directly to determine failure/crackextension. This is based on the fact that the tip stress is limited by the yield stressor the cohesive stress between atoms and singular stresses are the results of linearelasticity. It is also recognized that the singular stress eld is a convenient represen-tation of the actual nite stress eld if the discrepancy between the two lies in a smallregion near the crack tip. This notion is referred to as small-scale yielding.The stresses near the tip of a crack in linearly elastic solids have the followinguniversal form independent of applied loads and the geometry of the cracked bodySun06-ch01-001-010-97801238500102011/10/317:11Page3#31.2Fracture Mechanics Concepts 3(Chapter 3):xx=KI2r cos 12

1 sin 12 sin 32

yy=KI2r cos 12

1 +sin 12 sin 32

(1.1)xy=KI2r sin 12 cos 12 cos 32where KIis the so-called stress intensity factor, which depends on the applied loadand crack geometry and (r, ) are the polar coordinates centered at the crack tip. Hereitisassumedthattheloadsandthegeometryaresymmetricaboutthecrackline.Equation (1.1) shows that KI is a measure of the stress intensity near the crack tip.Based on this obervation, Irwin [1-5] proposed a fracture criterion which statesthat crackgrowthoccurswhenthestressintensityfactorreachesacritical value,that is,KI=KIc(1.2)where KIc is called fracture toughness, a material constant determined by experiment.The preceding fracture criterion for cracked solids is fundamentally different fromtheclassical failure criteria based on stresses. It does not directly use stresses or strains,but a proportionality factor in the stress eld around the crack tip. KIis proportionalto the applied load but has a dimension of MPa m in the SI unit system and ksi in in the US customary unit system. KIc is a new material parameter introduced infracture mechanics that characterizes the resistance of a material to crack extension.ThecriterioninEq. (1.2)isbasedonlinearelasticitywithwhichtheinversesquare root singular stress eld exists and the stress intensity factor is well dened.The actual fracture process at the crack tip cannot be described using the linear elas-ticitytheory. Therationalityofthecriterionliesintheconditionthatthefractureprocess zone is sufciently small so that it is well contained inside the singular stresseld Eq. (1.1) characterized by the stress intensity factor KI.Thesecondapproachforestablishingafracturecriterionisbasedonthecon-siderationofglobal energybalanceduringcrackextension. Thepotential energyofacrackedsolidunderagivenloadisrst determinedanditsvariationwithavirtual crack extension is then examined. Consider a two-dimensional elastic bodywithacrackoflengtha. Thetotalpotentialenergyperunitthicknessofthesys-temisdenotedby=(a). Notethat thepotential energyisafunctionofthecrack length. For a small crack extension da, the decrease in the potential energy isd. Grifth [1-6] proposed that this energy decrease in the cracked body wouldbe absorbed into the surface energy of the newly created crack surface. Denote thesurface energy per unit area by, which can be calculated from solid state physics.The total surface energy of the new crack surface equals 2da. The Grifth energySun06-ch01-001-010-97801238500102011/10/317:11Page4#44 CHAPTER 1Introductionbalance equation becomesd=2da or dda =2The energy release rate G proposed by Irwin [1-7] is dened as the decrease inpotential energy per unit crack extension under constant load, that is,G =ddaThecrackgrowthorfailurecriterionusingtheenergybalanceapproachisestab-lished asG =Gc=2 (1.3)The fracture criterion given before is also fundamentally different from the clas-sical failure criteria. It involves the total energy of the cracked body as well as thesurface energy of the solid, which exists only in atomistic scale considerations. LikeKIc, GIc is also a new material constant introduced in fracture mechanics to measurethe resistance to fracture. GIc has a dimension of J/m2, or kJ/m2. The fracture criteriaEqs. (1.2) and (1.3) are actually equivalent (Chapter 4). However, the experimentallymeasuredcritical energyreleaserateforengineeringmaterials, especiallymetals,is signicantly larger than 2. This is because plastic deformations in the crack tipregion also contribute signicantly to the crack growth resistance. For perfectly brit-tle solids, it has been shown by MD simulations that GIc=2is valid in NaCl singlecrystal if the crack length is equal to or greater than 10 times the lattice constant [1-8].Fracturemechanicsintroducestwonovel concepts: stressintensityfactorandenergyreleaserate. Thesetwoquantitiesdistinguishfracturemechanicsfromtheclassical failure criteria. In using the stress intensity factor-based fracture criteriontopredict failureofamaterial orstructure, onerst needstocalculatethestressintensity factor for the given load and geometry. The second step is to measure thefracturetoughness. Oncethestressintensityfactorandthefracturetoughnessareknown, Eq. (1.2) can be used to determine the maximum allowable load that will notcause crack growth for a given crack length, or the maximum allowable crack lengththat will not propagate under the design load. The advantage of the stress intensityfactor approach is its ease in the calculation of stress intensity factors and the easymeasurement of fracture toughness. In contrast to the stress intensity factor approach,the energy release rate-based fracture criterion Eq. (1.3) is more naturally extendedto cases where nonlinear effects need to be accounted for because the energy conceptis universal.Stress intensity factor and energy release rate lay the foundation of linear elas-tic fracture mechanics (LEFM). In LEFM, the cracked solid is treated as a linearlyelastic medium and nonlinear effects are assumed to be minimal and can be ignored.Whileamodiedstressintensityfactorapproachmaybeusedtopredictfractureof a cracked solid when plastic deformations are small and conned in the near-tipregion, the approach, along with the energy release rate, would become futile whenSun06-ch01-001-010-97801238500102011/10/317:11Page5#51.3History of Fracture Mechanics 5the cracked solid undergoes large-scale plastic deformations. Several fracture param-eters have been proposed to predict fracture of solids under nonlinear deformationconditions, for example, the J-integral, the crack tip opening displacement (CTOD),and the crack tip opening angle (CTOA). Failure criteria based on these parameters,however,havenotbeenassuccessfulasthestressintensityfactorandtheenergyrelease rate critetia in LEFM.1.3HISTORY OF FRACTURE MECHANICSThis section briey describes the historical development of fracture mechanics fromGrifthspioneeringworkonbrittlefractureof glassin1920s, toIrwinsstressintensityfactorconceptandfracturecriterionin1950s,andtoelastic-plasticfrac-turemechanicsresearchin1960sandearly1970s. Abriefintroductionofrecentdevelopment of fracture mechanics research since 1990s is also included.1.3.1Grifth Theory of FractureTheadvent of fracturemechanics is usuallycreditedtothepoineeringworkofA. A. Grifthonbrittlefractureofglass[1-6]. ThispaperwasbasicallyhisPhDthesis work at Cambrige University under the guidance of G. I. Taylor. It had beenknown before Grifths work that the theoretical fracture strength of glass determinedbased on the breaking of atomic bonds exceeds the strength of laboratory specimensby one to two orders of magnitude. Grifth believed that this huge discrepancy couldbe due to microcracks in the glass and that these cracks could propagate under a loadlevel that is much smaller than the theoretical strength.Grifth adopted an energy balance approach to determine the strength of crackedsolids, that is, the work done during a crack extension must be equal to the surfaceenergy stored in the newly created surfaces. To calculate the strain energy in a crackedbody, he derived the stress eld in an innite plate with a through-thickness centralcrack under biaxial loading from Ingliss solution [1-9] for an elliptical hole in anelastic plate by reducing the minor axis to zero. Using this solution, Grifth was ableto calculate the total potential energies before and after crack extension. The differ-ence of the potential enegies of the two states were set equal to the correspondinggain in surface energy.It follows from the Grifth theory that the fracture strength (the remote appliedstress) of a solid with a crack is proportional to the square root of the surface energyand is inversely proportional to the square root of the crack size, that is,f

cEawherefis the applied failure stress,cis the specic surface energy, a is half thecrack length, and E is Youngs modulus.Sun06-ch01-001-010-97801238500102011/10/317:11Page6#66 CHAPTER 1IntroductionThe preceding relationship points out a specic functional form between the fail-urestressandthecracksize.TheGrifththeoryrepresentsabreakthroughinthestrength theory of solids. It successfully explains why there is an order of magnitudedifference between the theoretical strength and experimentally measured failure loadfor a solid. In particular, it provides a well-dened physical mechanism that controlsthe failure process, which is lacking in the classical phenomenological failure theo-ries. The original work of A. A. Grifth dealt with fracture of brittle glass. In metals,plastic deformations develop around the crack tip and the measured fracture strengthis much greater than that predicted by the Grifth theory. Orowan [1-10] and Irwin[1-11] suggested to add to 2the plastic work p associated with the creation of newcrack surfaces. For metals,p is much larger than the surface energy 2 , and hencethe modied Grifth theory by Orowan and Irwin explained the high fracture strengthof metals.1.3.2Fracture Mechanics as an Engineering ScienceAlthough the basic energy concept of fracture mechanics was presented byA. A. Grifthin1920, it wasonlyafter the1950sthat fracturemechanicswasaccepted as an engineering science with successful practical applications mainly as aresult of Irwins work ([1-5] and [1-7]). Irwin rst introduced the energy release rateto establish a fracture criterion as in Eq. (1.3). He then dened the stress intensityfactor K and derived the relationship between the energy release rate G and the stressintensity factor Kbased on Westergaards solutions for the stress and displacementelds in a cracked plate [1-12]. Because of the GK relationship, Irwin proposed touse the stress intensity factor as a fracture parameter, which is a more direct approachfor fracture mechanics applications as described by Eq. (1.2).At the same time, Williams [1-13] derived the asymptotic stress eld near a cracktip with the leading term exhibiting an inverse square root singularity under generalplanarloadingconditions.TheWilliamssolution,withbothsymmetricandasym-metric terms, gives a universal expression for the crack tip stress eld independent ofexternal loads and crack geometries. The load and crack geometry inuence the cracktip singular stresses through the stress intensity factors KI and KII, which govern theintensityofthesingularstresseld.Williamssolutionprovidesajusticationforadopting the stress intensity factors to establish fracture criteria.Thestressintensityfactorfracturecriterionassumesthatmaterialsbehaviorislinearly elastic, which is a good assumption for brittle materials such as glass andceramics. Forductilemetalsat roomandelevatedtemperatures, however, plasticyieldingoccursaroundthecracktipduetothestresssingularitypredictedintheelastic solution. For linear elastic fracture mechanics to be applicable to metals, theplastic deformation zone around the crack tip must be smaller than the dominancezone of the stress intensity factor. Irwin [1-14] estimated the size of plastic deforma-tion zone near the crack tip and found that the plastic zone size is proportional to thesquare of the stress intesnity factor to the yield strength ratio if the plastic zone issmall.Sun06-ch01-001-010-97801238500102011/10/317:11Page7#71.3History of Fracture Mechanics 7With the fracture criterion Eq. (1.2) in hand and the knowledge of the crack tipplastic zone size, the American Society for Testing and Materials (ASTM) formed aSpecial Technical Committee (ASMT STC, subsequently ASTM Committee E-24) todevelop the standard for measuring KIc, the plane strain fracture toughness (or simplyfracture toughness) for metallic materials. In the meantime, great efforts were madein 1960s and 1970s to develop analytical and numerical methods to compute stressintensityfactors. Most oftheapproachesandtechniquesareincludedinamulti-volume fracture mechanics monograph, Mechanics of Fracture, edited by G. C. Sihandhiscoworkers[1-15, 1-17]. Stressintensityfactorsforvariouscrackgeome-triesunderavarietyofloadingconditionsarecompiledinthehandbookbyTadaet al. [1-18].The LEFM based on stress intensity factor K and energy release rate G has beenverysuccessfulinpredictingfractureofmetalswhenthecracktipplasticzoneissmaller than the K-dominance zonealso termed small-scale yielding (SSY). Underlarge-scale yielding conditions, however, the LEFM generally becomes inadequateand fracture criteria based on plasticity of the cracked solids have to be used. Irwin[1-14] introduced an effective stress intensity factor concept to take the crack tip plas-ticity effect into account. The effective stress intensity factor is obtained by replacingthe crack length with an effective crack length that is equal to the original length plushalf the plastic zone size.Dugdale [1-19] presented a strip yielding zone model to determine the plastic zonesize in thin cracked sheets. Wells [1-20] and [1-21] proposed to use the crack openingdispalcement (COD) as a fracture parameter. The COD criterion is equivalent to theeffective stress intensity factor criterion under modest yielding conditions but can beextended to large-scale yielding when it is combined with the COD equation from theDugdale model. Rice [1-22] generalized the energy release rate concept to nonlinearelastic materials or elastic-plastic materials described by the deformation plasticityand found that the energy release rate can be represented by a line integral, the so-called path-independent J-integral.BegleyandLandes[1-23] later proposedtousetheJ-integral for predictingelastic-plasticcrackinitiationandexperimentallymeasuredthecriticalvalueofJat crack initiation. In 1968, Rice and Rosengren [1-24] and Hutchinson [1-25] pub-lished their work on the crack tip plastic stress eld (HRR eld) in the frameworkof deformation plasticity. The HRR eld shows that the J-integral characterizes theintensityofthesingularstresseldinasimilarwaytotheroleofstressintensityfactor in LEFM. Becasue the HRR eld is based on the deformation plasticity, theJ-integral in general may be used for crack initiation only. In other words, the HRReld disappears as the crack extends and unloading (a behavior that the deformationplasticity theory cannot model) takes place.1.3.3Recent Developments in Fracture Mechanics ResearchIn recent years, LEFM has found many new applications mostly dealing with newmaterials suchas nonhomogeneous andanisotropic ber-reinforcedcomposites.Sun06-ch01-001-010-97801238500102011/10/317:11Page8#88 CHAPTER 1IntroductionThe main issues that arise in these new applications include, for example, coupledthermal-mechanical loads in microelectronic packaging and multiscale issues in treat-ing composite materials as homogeneous solids. Because of the increasing interest innanotechnology, fracture of nanostructured materials has recently attracted the atten-tionofmanyresearchers.Moleculardynamics(MD)simulationsareemployedtomodel crack extension in atomistic systems. Researchers have attempted to answerthe question regarding the applicability of continuum theory-based LEFM in solids atnano scale ([1-8] and [1-26]). For instance, are the stress intensity factor and energyrelease rate introduced in LEFM still valid, and how can one evaluate their values?Other issues involve the denition of cracks that are equivalent to cracks adopted incontinuum LEFM.A new form of fracture model called cohesive zone model (CZM) has evolvedfromLEFMbut hastakenadifferent treatment ofthecracktipstressandstrainelds.ThemainmotivationinCZMwastoavoidtheseeminglyunrealisticstresssingularityatthecracktip.TheideaoftheCZMiscreditedtoBarenblatt[1-27],whoassumedthatfailurewouldoccurbydecohesionoftheupperandlowersur-faces of a volumeless cohesive zone ahead of the crack tip. In the cohesive zone theseparation displacement of the two surfaces that bound the cohesive zone follows acohesive traction law. Crack growth occurs when the opening displacement at the tailof the cohesive zone (physical crack tip) reaches a critical value at which the cohesivetraction vanishes. Clearly, the cohesive modeling approach does not involve stresssingularitiesandmaterialfailureiscontrolledbyquantitiessuchasdisplacementsand stresses, which are consistent with the usual strength of materials theory.Since Needleman [1-28] introduced the cohesive element technique in the niteelement frameworkforfracturestudies, CZMhasemergedasapopulartool forsimulatingfractureprocessesinmaterialsandstructuresduetothecomputationalconvenience. Although many researchers have reported successful results using theCZM approach, many issues remain to be resolved including the physics of the cohe-sive zone, a rational way to develop the cohesive traction law, and the uniqueness ofthe cohesive traction with respect to variation of loads and specimen geometry.References[1-1] H.P. Rossmanith, Thestrugglefor recognitionof engineeringfracturemechanics,in: H.P. Rossmanith (Ed.), Fracture Research in Retrospect, A.A. Balkema, Rotterdam,Netherlands, 1997, pp. 3794.[1-2] E. Hayes, Dr. Constance Tipper: testing her mettle in a materials world, Adv. Mater.Processes Vol. 153, 100. (1998).[1-3] A.A. Wells, The condition of fast fracture in aluminum alloys with particularly refer-ence to Comet failures, British Welding Research Association Report, NRB 129, April1955.[1-4] D.J. Winne, B.M. Wundt, Application of the Grifth-Irwin theory of crack propagationto the bursting behavior of disks, including analytical and experimental studies, Trans.ASME 80 (1958) 16431655.Sun06-ch01-001-010-97801238500102011/10/317:11Page9#9References 9[1-5] G.R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate,J. Appl. Mech. 24 (1957) 361364.[1-6] A.A. Grifth, The phenomena of rapture and ow in solids, Philos Trans R Soc LondA221 (1920) 163198.[1-7] G.R. Irwin, Relation of stresses near a crack to the crack extension force, in: Proceed-ings ofthe International Congresses of AppliedMechanics, Vol. VIII, UniversityofBrussels, 1957, pp. 245251.[1-8] A. Adnan, C.T. Sun, Evolution of nanoscale defects to planar cracks in a brittle solid,J. Mech. Phys. Sol. 58 (2010) 9831000.[1-9] C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners, Trans.Inst. Naval Architects 55 (1913) 219230.[1-10] E. Orowan, Notch brittleness and strength of metals, Trans. Inst. Engrs Shipbuilders inScotland 89 (1945) 165215.[1-11] G.R.Irwin, Fracturedynamics, in:FractureofMetals, ASM,Cleveland, OH,1948,pp.147166.[1-12] H.M. Westergaard, Bearing pressures and cracks, J. Appl. Mech. 6 (1939) 4953.[1-13] M.L. Williams, Onthestressdistribuionatthebaseofastationarycrack, J. Appl.Mech., 24 (1957) 109114.[1-14] G.R.Irwin,Plasticzonenearacrackandfracturetoughness,in:Proceedingsofthe7th Sagamore Ordnance Materials Conference, Syracuse University, 1960, pp. IV-63-IV-78.[1-15] G.C. Sih (Ed.), Mechanics of Fracture Vol. 1: Methods of Analysis and Solutions ofCrack Problems, Leyden, Noordhoff International Pub., 1973.[1-16] M.K. Kassir, G.C. Sih (Eds.), Mechanics of Fracture Vol. 2: Three-Dimensional CrackProblems, Leyden, Noordhoff International Pub., 1975.[1-17] G.C. Sih (Ed.), Mechanics of Fracture Vol. 3: Plates and Shells with Cracks, Leyden,Noordhoff International Pub., 1977.[1-18] H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, ASME Press,New York, 2000.[1-19] D.S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Sol. 8 (1960)100104.[1-20] A.A. Wells, Unstable crack propagation in metals: cleavage and fast fracture, in: Pro-ceedingsof theCrackPropagationSymposium, Vol. 1, Paper 84, Cracneld, UK,1961.[1-21] A.A. Wells, Applicationoffracturemechanicsat andbeyondgeneral yielding, Br.Weld. J. 11 (1963) 563570.[1-22] J.R. Rice, A path independent integral and the approximate analysis of strain concen-tration by notches and cracks, ASME J. Appl. Mech. 35 (1968) 379386.[1-23] J.A. Begley, J.D. Landes, The J-integral as a fracture criterion, in: ASTM STP 514,American Society for Testing and Materials, Philadelphia, 1972, pp. 120.[1-24] J.R.Rice,G.F.Rosengren,Planestraindeformationnearacracktipinapower-lawhardening material, J. Mech. Phys. Sol. 16 (1968) 112.[1-25] J.W. Hutchinson, Singular behavior at the end of a tensile crack in a hardening material,J. Mech. Phys. Sol. 16 (1968) 1331.Sun06-ch01-001-010-97801238500102011/10/317:11Page10#1010 CHAPTER 1Introduction[1-26] M.J. Buehler, H. Yao, B. Ji, H. Gao, Cracking and adhesion at small scales: atomisticand continuum studies of aw tolerant nanostructures, Model. Simul. Mater. Sci. Eng.14 (2006) 799816.[1-27] G.I. Barenblatt, Themathematical theoryof equilibriumcracks inbrittlefracture,in: Adv. Appl. Mech. 7 (1962) 55129.[1-28] A. Needleman, Acontinuummodel for voidnucleationbyinclusiondebonding,J. Appl. Mech. 54 (1987) 525531.Sun07-ch02-011-024-97801238500102011/10/317:17Page11#1CHAPTERGrifth Theory of Fracture22.1THEORETICAL STRENGTHThetheoreticalstrengthofasolidisusuallyunderstoodastheappliedstressthatfracturesaperfectcrystalofthematerialbybreakingtheatomicbondsalongthefractured surfaces. The theoretical strength may be estimated using the interatomicbonding force versus the atomic separation relation. This section gives two estimatesthat relate the theoretical strength to the Youngs modulus of the material based onthe atomic bonding strength and surface energy concept.2.1.1An Atomistic ModelIngeneral, failureof asolidis characterizedbyseparationof thebody. At theatomistic level the fracture strength of a perfect material depends on the strengthofitsatomicbonds. Considertwoarraysofatomsinaperfect crystal asshowninFigure2.1. Let a0betheequilibriumspacingbetweenatomicplanes intheabsenceofappliedstresses. Thestressrequiredtoseparatetheplanestoadis-tance a > a0increases until the theoretical strengthcis reached and the bonds area FIGURE 2.1Atomic planes in a perfect crystal.Fracture Mechanics. DOI: 10.1016/B978-0-12-385001-0.00002-Xc 2012 Elsevier Inc. All rights reserved.11Sun07-ch02-011-024-97801238500102011/10/317:17Page12#212 CHAPTER 2Grifth Theory of Fracturea0Resultant forceAttractive forceaRepulsive forceFIGURE 2.2Cohesive force versus separation between atoms.acEx =aa0a0/2FIGURE 2.3Cohesive force between atoms.broken. Furtherdisplacementsoftheatomscanoccurunderadecreasingappliedstress (see Figure 2.2). This stress-displacement curve can be approximated by a sinecurve (Figure 2.3) having wavelength as =csin

2x

(2.1)where x =a a0 is the relative displacement between the atoms.At small displacement x we havesinx xSun07-ch02-011-024-97801238500102011/10/317:17Page13#32.1Theoretical Strength 13and, thus, c2x(2.2)The modulus of elasticity isE = stressstrain =x/a0 = Exao(2.3)Using Eqs. (2.2) and (2.3), we obtainExa0=c2xorc=E2a0(2.4)A reasonable value for is =ao, which yields the bond strengthc=E2(2.5)2.1.2The Energy ConsiderationTheoreticalstrengthmayalsobeestimatedusingthesimpleatomicmodelwithasurface energy concept. We now dene a quantity called the surface energy (energyper unit area) as the work done in creating new surface area by the breaking of atomicbonds. From the sine-curve approximation of the atomic force (see Figure 2.3), this issimply one-half the area under the stress-displacement curve since two new surfacesare created each time a bond is broken. Thus,2 =/2

0csin

2x

dx =cfrom whichc= 2 (2.6)However, from Eq. (2.4), = 2a0cE(2.7)Sun07-ch02-011-024-97801238500102011/10/317:17Page14#414 CHAPTER 2Grifth Theory of FractureWe obtain from substitution of Eq. (2.7) in Eq. (2.6)2c = 2 E2a0= Ea0Finally,c=

Ea0(2.8)Formanymaterials, isontheorderof0.01Ea0[2-1].Thus,anapproximateestimation of the theoretical strength is often given byc=E10(2.9)whichagreeswithEq.(2.5)intermsoforderofmagnitude.Formostmetals,thetheoreticalstrengthvariesbetween7GPa(1 106psi)and21GPa(3 106psi).However, bulk materials that are commercially produced for engineering applicationscommonly fracture at applied stress levels 10 to 100 times below these values, andthe theoretical strength is rarely obtained in engineering practice.The main reasons for the discrepancies are1. The existence of stress concentrators (aws such as cracks and notches)2. The existence of planes of weakness such as grain boundaries in polycrystallinematerials.Another type of fracture process (e.g., shear or rupture), which occurs by plasticdeformation, intervenes at a lower level of applied stress.2.2THE GRIFFITH THEORY OF FRACTUREAlan Grifths work [2-2] on brittle fracture of glass was motivated by the desire toexplain the discrepancy between the theoretical strength and actual stength of mate-rials. According to the preceding theoretical strength calculation, we may concludethat glass should be very strong. However, laboratory test results often indicate oth-erwise. Grifth argued that what we must account for is not the strength but ratherthe weakness, which is normally dominant in the failure process. One clue obviouslylies in the fact that actual glasses display a far more complex fracture behavior thanpredicted by our simple assumptions regarding the cohesive strength. In his pioneerpaper [2-2], Grifth postulated that all bulk glasses contain numerous minute awsin the form of microcracks that act as stress concentration generators. This new con-cept accompanied by the energy release approach that he introduced started the eraof modern fracture mechanics.Sun07-ch02-011-024-97801238500102011/10/317:17Page15#52.2The Grifth Theory of Fracture 15yxbaFIGURE 2.4An elliptic hole in an innite plate subjected to tension.The solution of an elliptic hole in a plate of innite extent under tension by Inglis[2-3] was the rst step toward relating observed failure stress to ultimate strength. Hesolved the problem as shown in Figure 2.4 and found that the greatest stress occursat the ends of the major axis:yy=

1 +2ab

(2.10)Ifa =b(acircularhole),thenyy=3,whichyieldsthewell-knownstresscon-centration factor near a circular hole. If b 0, then we have a line crack, and thestress yy increases without limit. If a stress-based failure criterion is used to predicttheextensionofsuchacrack,onewouldndtheunreasonableanswerthatanyamount of applied stress would cause the crack to grow.Grifth took an energy balance point of view and reasoned that the unstable prop-agation of a crack must result in a decrease in the strain energy of the system (for abody with a xed boundary where no work is done by external forces during the crackextension), and proposed that a crack would advance when the incremental releaseof energy dWassociated with a crack extension da in a body becomes greater thanthe incremental increase of surface energy dWSas new crack surfaces are created.That is,dW dWS(2.11)The equality indicates the critical point for crack propagation. In other words, if thesupply of energy from the cracked plate is equal to or greater than the energy requiredto create new crack surfaces, the crack can extend.Sun07-ch02-011-024-97801238500102011/10/317:17Page16#616 CHAPTER 2Grifth Theory of FractureIt is easy to calculate the surface energy for a crack (having two crack tips) withlength 2a, that is,WS=2(2a ) =4a (2.12)in whichis the surface energy density and the fact that two crack surfaces for acrack has been accounted for.GrifthusedInglissolutiontoobtainthetotal energyreleasedWduetothepresenceofacrackoflength2ainaninnitetwo-dimensionalbody.Hismethodfor calculating energy release was very complicated since he considered the energychange in the body as a whole. He obtained, for plane strain,W =a22(1 2)E(2.13)and for plane stress,W =a22E(2.14)Thus, the critical stresscrunder which the crack would start propagating may beobtained by substituting Eqs. (2.12) and (2.13) or Eq. (2.14) into Eq. (2.11):2a(1 2)2crEda =4 da (plane strain)From this equation, we have2cr=4 E2a(1 2)orcr=

2E(1 2)a(2.15)Similarly, for plane stress we havecr=

2Ea(2.16)Comparingthepreviouscritical stresscr, orthefracturestrengthofthein-nite plate with a crack of microscopic or macroscopic length 2a, and the theoreticalstrength c in Eq. (2.8), we note ccr if a ao. This explains qualitatively whyactual strengths of materials are much smaller than their theoretical strengths.The energy released dW for a crack extension of da can be expressed in terms ofthe strain energy release rate per crack tip G asdW =2Gda (2.17)Sun07-ch02-011-024-97801238500102011/10/317:17Page17#72.3A Relation among Energies 17Thus,G = 12dWda =a2(1 2)Efor plane strain (2.18)=a2Efor plane stress (2.19)The instability condition then readsG 2 (2.20)The value of G when equal to 2is denoted by Gc and is called the fracture tough-ness or the crack-resistant force of the material. This is, in fact, the Grifth energycriterion of brittle fracture. In theory, a crack would extend in a brittle material whentheloadproducesanenergyreleaserateGequalto2 .However,suchanenergyrelease rate turns out to be much smaller than the test data since most materals arenot perfectly brittle and plastic deformation occurs near the crack tip.To take this additional crack resistant force into account, Orowan [2-4] suggestedto add to 2the plastic workpassociated with the creation of the new crack sur-faces. For metals, 2is usually much smaller thanpand, thus, can be neglected.On the other hand, Irwin [2-5] took Gcas a new material constant to be measureddirectly from fracture tests. However, Eq. (2.15) points to a correct relation betweenthe failure stress and crack size.2.3A RELATION AMONG ENERGIESTheGrifththeoryforfractureofperfectlybrittleelasticsolidsisfoundedontheprinciple of energy conservation that is, energy added to and released from the bodymust be the same as that dissipated during crack extension. It states that, during crackextensionofda, theworkdonedWebyexternal forces, theincrement ofsurfaceenergy dWS, and the increment of elastic strain energy dU must satisfydWS+dU =dWe(2.21)For a conservative force eld, this condition can be expressed in the form(WS+U+V)/a =0 (2.22)whereWS=total crack surface energy associated with the entire crackU =total elastic strain energy of the cracked bodyV =total potential of the external forcesNote that a negative dV implies a positive work dWe done by external forces.Consider a single-edge-cracked elastic specimen subjected to a tensile load P ordisplacement as shown in Figure 2.5. The relationship between the applied tensileSun07-ch02-011-024-97801238500102011/10/317:17Page18#818 CHAPTER 2Grifth Theory of FractureaP, FIGURE 2.5A single-edge-cracked specimen.force P and the elastic extension, or displacement, , is =SP (2.23)whereSdenotes the elastic compliance of the specimen containing the crack. Thestrain energy stored in this specimen isU ==SP

=0Pd ==SP

=0Sd=12S[2]SP0= 12SP2(2.24)The complianceSis a function of the crack length. The incremental strain energyunder the condition of varying a and P isdU = 12P2dS +SPdP (2.25)Case 2.1Suppose that the boundary is xed during the extension of the crack so that =SP =constantConsequently,d =SdP+PdS =0Sun07-ch02-011-024-97801238500102011/10/317:17Page19#92.3A Relation among Energies 19from which we obtainSdP =PdSSubstitution of the preceding equation into Eq. (2.25) yieldsdU| =12P2dS (2.26)Furthermore, dWe=0 in this case because d =0 and, thus, the external load does no work.Substituting Eq. (2.26) into Eq. (2.21) and using dWe=0, we havedWS= dU| =12P2dS (2.27)Thus,adecreaseinstrainenergyUiscompensatedbyanincreaseofthesameamountinthesurfaceenergy.Inotherwords,theenergyconsumedduringcrackextensionisentirelysupplied by the strain energy stored in the cracked body.Case 2.2Suppose that the applied force is kept constant during crack extension; thendP =0From Eq. (2.25) we havedU|P=12P2dS (2.28)Thus, there is a gain in strain energy during crack extension in this case. Moreover, we notethatdWe=Pd =P2dS (2.29)Substituting Eqs. (2.28) and (2.29) into Eq. (2.21), we again obtain Eq. (2.27), that is,dWS=12P2dSwhichishalfoftheworkdonebytheexternalforce.Itisinterestingtonotethattheworkdoneby theexternalforceis splitequallyintothesurfaceenergyandanincrease instrainenergy.For both boundary conditions discussed before, the energy released during crackextension isdW =dWedU =12P2dSSun07-ch02-011-024-97801238500102011/10/317:17Page20#1020 CHAPTER 2Grifth Theory of FractureThe corresponding energy release rate isG =dWda=12P2 dSda(2.30)Hence, the strain energy release rate is independent of the type of loading.The two loading cases can be illustrated graphically as in Figures 2.6a and 2.6b,respectively. In the gures, point B indicates the beginning of crack extension andpoint Ctheterminationof crackextension. TheareaOBCis thestrainenergyreleased, dW. It canbeshownrathereasilyfromthegraphicillustrationthat theenergies released in the two cases are equal.Under the xed load condition, we havedW =dU = 12dWeThus, the energy release rate can be obtained withG = dUda(2.31)in which the differentiation is performed assuming that the applied load is indepen-dent of a.Under the xed displacement condition, we have dW =dU, and henceG =dUda(2.32)In the previous equation, the applied load P should be considered as a function ofcrack length a in the differentiation. The result should be the same as that given byEq. (2.31). It is noted that the relation dW =dU =dWe/2 is not true for nonlinearsolids.O OC BP PPP +d aCBa+da a+daPdP(b) (a)aFIGURE 2.6Energy released during crack extension: (a) constant load, (b) constant displacement.Sun07-ch02-011-024-97801238500102011/10/317:17Page21#112.3A Relation among Energies 21Example 2.1The double cantilever beam (DCB) is often used for measuring fracture toughness of mate-rials.ConsiderthegeometryshowninFigure2.7wherebisthewidthofthebeam,andthe crack lengtha is much larger thanh and, thus, the simple beam theory is suitable formodeling the deection of the two split beams.Noting that the unsplit portion of the DCB is not subjected to any load and that in eachlegthebendingmomentisM =Px,wecalculatethetotalstrainenergystoredinthetwolegs of the DCB asUT =2a

0P2x22EIdx =P2a33EIwhereI =bh312The total strain energy per unit width isU =UT/bThe strain energy release rate is obtained asG =dUda =P2a2bEIIf the fracture toughnessGcof the material is known, then the load that could furthersplit the beam isPcr=bEIGcaahhPPbFIGURE 2.7A double cantilever beam subjected to concentrated forces.Sun07-ch02-011-024-97801238500102011/10/317:17Page22#1222 CHAPTER 2Grifth Theory of FractureReferences[2-1] A.H. Cottrell, Tewksbury Symposium on Fracture, University of Melbourne, 1963, p. 1.[2-2] A.A. Grifth, Thephenomenaof ruptureandowinsolids, Phil. Trans. Roy. Soc.(London) A221 (1920) 163198.[2-3] C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners, Trans.Inst. Naval Architects 55 (1913) 219230.[2-4] E.Orowan,Notchbrittlenessandstrengthofmetals,Trans.Inst.Engrs.ShipbuildersScot. 89 (1945) 165215.[2-5] G.R. Irwin, Fracturedynamics, in: Fractureof Metals, ASM, Cleveland, OH, 1948,pp. 147166.PROBLEMS2.1 Consider the cracked beam subjected to uniaxial tension shown in Figures 2.8and 2.9. Find the strain energy release rate (per crack tip). Consider both xed-end and constant force boundary conditions.LaP P hh/3h/3h/3FIGURE 2.8A cracked beam subjected to tension.Lahh/3h/3h/3P/2P/2PFIGURE 2.9A cracked beam subjected to tension and compression.2.2 Find the strain energy release rate Gfor the cracked beamshown inFigures2.10and2.11. Usesimplebeamtheorytomodel thecrackedanduncracked regions. The thickness of the beam is t.Sun07-ch02-011-024-97801238500102011/10/317:17Page23#13Problems 23hhaPLPFIGURE 2.10A cracked beam subjected to concentrated forces.hhaLPPFIGURE 2.11A cracked beam subjected to concentrated forces.2.3 A cracked beam is subjected to a pair of forces at the center of the crack (seeFigure 2.12). Find the minimum P that can split the beam. Assume E =70 GPaand Gc=200 Nm/m2.PP1 cm1 cm10 cm 10 cm0.5 cmFIGURE 2.12A center-cracked beam.2.4 Findthestrainenergyreleaseratefor theproblemshowninFigure2.13wherein a thin elastic lm of unit width is peeled from a rigid surface.Sun07-ch02-011-024-97801238500102011/10/317:17Page24#1424 CHAPTER 2Grifth Theory of FracturePLtFIGURE 2.13A thin lm peeled from the rigid substrate.2.5 Assume that the bending rigidity of the lm is negligible, that L is large, andthat the elastic constants of the lm are known. The thin lm is pulled parallelto the rigid surface as shown in Figure 2.14. Compare the strain energy releaserate and the strain energy gained by the lm during crack extension for bothproblems in Figures 2.13 and 2.14. For the problem of Figure 2.13, why is thestrain energy released not the same as the strain energy gained by the lm?LPFIGURE 2.14A thin lm pulled parallel to the surface of the rigid substrate.2.6 Show that the area OBC in Figure 2.6 does not depend on the loading conditionduring crack extension.Sun08-ch03-025-076-97801238500102011/10/317:15Page25#1CHAPTERThe Elastic Stress Fieldaround a Crack Tip3Brittle fracture in a solid in the form of crack growth is governed by the stress eldaround the crack tip and by parameters that describe the resistance of the material tocrack growth. Thus, the analysis of stresses near the crack tip constitutes an essen-tial part of fracture mechanics. For brittle materials exhibiting linear elastic behavior,methods of elasticity are used to obtain stresses and displacements in cracked bod-ies. These methods include analytical ones, such as the complex potential functionmethod and the integral transform method, and numerical ones, such as the nite ele-ment method. In this chapter, the complex potential function method is introducedand used to analyze the stresses and displacements around crack tips. The characteri-stics of the near-tip asymptotic stress and displacement elds and the crack growthcriterion based on the crack tip eld are discussed.3.1BASIC MODES OF FRACTURE AND STRESSINTENSITY FACTORA crack in a solid consists of disjoined upper and lower faces. The joint of the twocrack faces forms the crack front. The two crack faces are usually assumed to lie inthe same surface before deformation. When the cracked body is subjected to externalloads (remotely or at the crack surfaces), the two crack faces move with respect toeachotherandthesemovementsmaybedescribedbythedifferencesindisplace-ments ux, uy, and uz between the upper and lower crack surfaces, where(x, y, z) is alocal Cartesian coordinate system centered at the crack front with the x-axis perpen-dicular to the crack front, the y-axis perpendicular to the crack plane, and the z-axisalong the crack front.There are three independent movements corresponding to three fundamental frac-turemodesaspointedout byIrwin[3-1], whichareschematicallyillustratedinFigure 3.1. These basic fracture modes are usually called Mode I, Mode II, and ModeIII, respectively, and any fracture mode in a cracked body may be described by oneofthethreebasicmodes,ortheircombinations(seetheirdescriptionsonthenextpage).Fracture Mechanics. DOI: 10.1016/B978-0-12-385001-0.00003-1c 2012 Elsevier Inc. All rights reserved.25Sun08-ch03-025-076-97801238500102011/10/317:15Page26#226 CHAPTER 3The Elastic Stress Field around a Crack Tip(b) (c) (a)xyzxyzxyzFIGURE 3.1Schematic of the basic fracture modes: (a) Mode I (opening), (b) Mode II (sliding),(c) Mode III (tearing).1. Mode I (Opening Mode): The two crack surfaces experience a jump only in uy,that is, they move away symmetrically with respect to the undeformed crack plane(xz-plane).2. Mode II (Sliding Mode): The two crack surfaces experience a jump only in ux,that is, they slide against each other along directions perpendicular to the crackfront but in the same undeformed plane.3. Mode III (Tearing Mode): The two crack surfaces experience a jump only in uz,that is, they tear over each other in the directions parallel to the crack front but inthe same undeformed plane.The three basic modes of crack deformation can be more precisely dened by theassociated stresses ahead of the crack front, which may be considered as the cracktipintwo-dimensionalproblems.Itwillbeseeninthefollowingsectionsthatthenear-tip stresses in the crack plane (xz-plane) for these three modes can be expressedas ( y =0,x 0+)yy=KI2x+O(x), xy=yz=0xy=KII2x+O(x), yy=yz=0 (3.1)yz=KIII2x+O(x), yy=xy=0respectively, where the three parameters KI, KII, and KIIIare named stress intensityfactors corresponding to the opening, sliding, and tearing (anti-plane shearing) modesof fracture, respectively.These expressions show that the stresses have an inverse square root singularity atthe crack tip and the stress intensity factors KI, KII, and KIIImeasure the intensitiesof thesingular stresselds ofopening, in-planeshearing, andanti-plane shearing,respectively. The stress intensity factor is a new concept in mechanics of solids andSun08-ch03-025-076-97801238500102011/10/317:15Page27#33.2Method of Complex Potential for Plane Elasticity 27FIGURE 3.2Mode III deformation in a cracked cylinder under torsion.playsanessentialroleinthestudyoffracturestrengthofcrackedsolids. Variousmethods for determining stress intensity factors, including analytical, numerical, andexperimental approaches, have been developed in the past few decades.It is important to note that, except for Mode I deformation shown in Figure 3.1(a),the loading and specimen geometries shown in Figures 3.1(b) and (c) cannot be usedto produce pure Mode II and Mode III deformation, respectively. In fact, unless anadditional loading or boundary condition is specied, the cracked bodies cannot be inequilibrium. Other types of specimen are usually used. For instance, a long cylinderof a circular cross-section with a longitudinal slit under torsion (see Figure 3.2) canbe used to produce a pure Mode III crack deformation.3.2METHOD OF COMPLEX POTENTIAL FOR PLANEELASTICITY (THE KOLOSOV-MUSKHELISHVILIFORMULAS)Amongvarious mathematical methods inplaneelasticity, thecomplexpotentialfunction method by Kolosov and Muskhelishvili [3-2] are one of the powerful andconvenient methods to treat two-dimensional crack problems. In the complex poten-tial method, stresses and displacements are expressed in terms of analytic functionsof complex variables. The problem of obtaining stresses and displacements arounda crack tip is converted to nding some analytic functions subjected to appropriateboundary conditions. A brief introduction of the general formulation of the Kolosovand Muskhelishvili complex potentials is given in this section.3.2.1Basic Equations of Plane Elasticity and Airy Stress FunctionThe basic equations of elasticity consist of equilibrium equations of stresses, strain-displacementrelations, andHookeslawthatrelatesstressesandstrains. Inplaneelasticity (plane strain and plane stress), the equilibrium equations are (body forcesare absent)xxx+xyy= 0xyx+yyy= 0(3.2)Sun08-ch03-025-076-97801238500102011/10/317:15Page28#428 CHAPTER 3The Elastic Stress Field around a Crack Tipwhere xx, yy, and xy are stresses, and (x, y) are Cartesian coordinates. The strainsand the displacements are related byexx=uxx , eyy=uyy , exy= 12_uxy +uyx_(3.3)where exx, eyy, and exyare tensorial strain components, and uxand uyare displace-ments. The stressstrain relations are given byxx= _exx+eyy_+2exxyy= _exx+eyy_+2eyy(3.4)xy= 2exyor inverselyexx=12_xx2(+)_xx+yy__eyy=12_yy2(+)_xx+yy__(3.5)exy=12xywhere is the shear modulus and= 3 1in which =___3 4 for plane strain3 1 +for plane stress(3.6)In the previous relation, is Poissons ratio. The compatibility equation of strains canbe obtained from Eq. (3.3) by eliminating the displacements as follows:2exxy2+2eyyx2=22exyxy(3.7)By using the stressstrain relations Eq. (3.5) together with the equations of equilib-riumEq.(3.2),thecompatibilityconditionEq.(3.7)canbeexpressedintermsofstresses as2(xx+yy) =0 (3.8)Sun08-ch03-025-076-97801238500102011/10/317:15Page29#53.2Method of Complex Potential for Plane Elasticity 29where2=2x2 +2y2is the Laplace operator.The Airy stress function is dened throughxx=2y2 , xy=2xy, yy=2x2(3.9)Using these relations, the equilibrium equations in Eq. (3.2) are automatically satis-ed, and the compatibility Eq. (3.8) becomes4 =22 =0 (3.10)where4=22=4x4 +24x2y2 +4y4isthebiharmonicoperator.AnyfunctionsatisfyingEq.(3.10)iscalledabihar-monic function. A harmonic function f satises 2f =0. Thus, if f is harmonic, itis also biharmonic. However, the converse is not true. Once the Airy stress functionisknown,thestressescanbeobtainedbyEq.(3.9)andstrainsanddisplacementsobtained through Eqs. (3.5) and (3.3), respectively.3.2.2Analytic Functions and Cauchy-Riemann EquationsIn a Cartesian coordinate system (x, y), the complex variable z and its conjugate z aredened asz =x +iyand z =x iyrespectively, where i =1. Theycanalsobe expressedinpolar coordinates(r, ) asz =r(cos +i sin) =reiand z =r(cos i sin) =reirespectively.Sun08-ch03-025-076-97801238500102011/10/317:15Page30#630 CHAPTER 3The Elastic Stress Field around a Crack TipConsider a function of the complex variable z, f (z). The derivative of f (z) withrespect to z is by denitiondf (z)dz=limz0f (z +z) f (z)zIf f (z) has a derivative at point z0 and also at each point in some neighborhood of z0,then f (z) is said to be analytic at z0. The complex function f (z) can be expressed inthe formf (z) =u(x, y) +iv(x, y)where u and v are real functions. If f (z) is analytic, we havexf (z) = f

(z)zx = f

(z)andyf (z) =f

(z)zy =if

(z)where a prime stands for differentiation with respect to z. Thus,xf (z) =iyf (z)orux +ivx =vyiuyFrom this equation, we obtain the Cauchy-Riemann equations:ux =vy,uy =vx(3.11)These equations can also be shown to be sufcient for f (z) to be analytic.From the Cauchy-Riemann equations it is easy to derive the following:2u =2v =0that is, the real and imaginary parts of an analytic function are harmonic.3.2.3Complex Potential Representation of the Airy Stress FunctionThe Airy stress function is biharmonic according to Eq. (3.10). Introduce a functionP by2 =P (3.12)Sun08-ch03-025-076-97801238500102011/10/317:15Page31#73.2Method of Complex Potential for Plane Elasticity 31then2P =22 =0This simply says that P is a harmonic function. Hence,P =Real part of f (z) Re{ f (z)}where f (z) is an analytic function and can be expressed asf (z) =P+iQLet(z) = 14_f (z)dz =p +iqthen is also analytic and its derivative is given by

(z) = 14f (z)According to the Cauchy-Riemann equations, we have

(z) =px +iqx =qy ipyA relation between P and p (or q) can then be obtained:P =4px =4qy(3.13)Consider the function (xp +yq). It can be shown that2[ (xp +yq)] =0Thus, (xp +yq) is harmonic and is a real (or imaginary) part of an analytic func-tion, say (z), that is, (xp +yq) = Re{(z)}Using the relationxp +yq = Re{ z(z)}we obtain the complex potential representation of the Airy stress function = Re{ z(z) +(z)}2(x, y) = z(z) +z(z) +(z) +(z)(3.14)Sun08-ch03-025-076-97801238500102011/10/317:15Page32#832 CHAPTER 3The Elastic Stress Field around a Crack Tip3.2.4Stress and DisplacementFrom the denition of the Airy stress function we obtainxx+ixy=2y2 i2xy =iy_x +iy_yyixy=2x2 +i2xy =x_x +iy_(3.15)Note that for an analytic function f (z) we havef (z)x= f

(z)zx =f

(z)f (z)x=_f (z)x_=f

(z)f (z)y= f

(z)zy =if

(z)f (z)y=_f (z)y_=if

(z)Using the preceding relations together with Eq. (3.14), we obtainx +iy =(z) +z

(z) +

(z) (3.16)Substitution of the relation in (3.16) in Eq. (3.15) leads toxx+ixy=

(z) +

(z) z

(z)

(z)yyixy=

(z) +

(z) +z

(z) +

(z)(3.17)Summing the two equations in Eq. (3.17), we havexx+yy=2[

(z) +

(z)] =4Re[

(z)] (3.18)Subtracting the rst equation from the second one in Eq. (3.17), we obtainyyxx2ixy=2z

(z) +2

(z)The equation above can be rewritten in the following form by taking the conjugate ofthe quantities on both sides:yyxx+2ixy=2[ z

(z) +

(z)] (3.19)Equations (3.18) and (3.19) are the convenient analytic function representations ofstresses.We now turn to the complex potential representation of displacements. Substitu-ting the strain-displacement relations Eq. (3.3) and the stresses in Eq. (3.9) into theSun08-ch03-025-076-97801238500102011/10/317:15Page33#93.2Method of Complex Potential for Plane Elasticity 33stressstrain relations Eq. (3.5) yields2uxx=2y2 2(+)22uyy=2x2 2(+)2 (3.20)_uxy +uyx_=2xyFrom Eqs. (3.12) and (3.13) we have2 =P =4px =4qySubstitution of this equation into the rst two equations in Eq. (3.20) yields2uxx= 2x2 +2(+2)+px2uyy= 2y2 +2(+2)+qyIntegrating the preceding equations, we obtain2ux= x +2(+2)+p +f1(y)2uy= y +2(+2)+q +f2(x)(3.21)Substituting these expressions in the third equation in Eq. (3.20), we can concludethatf1(y)andf2(x)representrigidbodydisplacementsandthuscanbeneglected.Rewrite Eq. (3.21) in complex form:2(ux+iuy) =_x +iy_+2(+2)+(z)UsingEq. (3.16) inthepreviousexpression, wearriveat thecomplexpotentialrepresentation of displacements:2(ux+iuy) =(z) z

(z)

(z) (3.22)In deriving Eq. (3.22), the relation =+3+is used. Equations (3.18), (3.19), and (3.22) are the Kolosov-Muskhelishviliformulas.Sun08-ch03-025-076-97801238500102011/10/317:15Page34#1034 CHAPTER 3The Elastic Stress Field around a Crack Tip3.3WESTERGAARD FUNCTION METHODThe Kolosov-Muskhelishvili formulas hold for general plane elasticity problems. Inapplications to crack problems, however, the three basic fracture modes discussed inSection 3.1 possess symmetry or antisymmetry properties. The Westergaard functionmethod [3-3, 3-4] is more convenient for discussing these basic crack problems. Wewill introduce the Westergaard functions using the general Kolosov-Muskhelishviliformulas, which are rewritten here for convenience:xx+yy= 4Re{

(z)} (3.23)yyxx+2ixy= 2{ z

(z) +

(z)} (3.24)2(ux+iuy) = (z) z

(z)

(z) (3.25)3.3.1Symmetric Problems (Mode I)Consider an innite plane with cracks along the x-axis. If the external loads are sym-metric with respect to the x-axis, then xy=0 along y =0. From Eq. (3.24), we haveIm{ z

(z) +

(z)} =0 at y =0 (3.26)The preceding equation can be satised if and only if

(z) +z

(z) +A =0 (3.27)in which A is a real constant.ProofIt is clear that Eq. (3.27) leads to Eq. (3.26) because z = z at y =0. For the converse case,consider

(z) +z

(z) =A(z) (3.28)WenowprovefromEq.(3.26)thatA(z)isareal constant.First, A(z)isanalyticsince

andz

areanalytic.Second,A(z)isboundedintheentireplaneaccordingtoEq.(3.24)(stressesareniteeverywhereexceptatcracktips)andthegeneral asymptopticsolutionsof(z) and(z) at crack tips (see Section 3.6).A(z) is thus a constant (=A) by Liouvillestheorem. Substituting Eq. (3.28) into the left side of Eq. (3.26), we haveIm{( z z)

(z) A} = Im{2iy

(z) A} = 0 at y =0orIm{A} =0Hence, we can conclude thatA = real constantSun08-ch03-025-076-97801238500102011/10/317:15Page35#113.3Westergaard Function Method 35Because (z) and (z) are related according to Eq. (3.27), stresses and displace-ments may be expressed by only one of the two analytic functions. From Eq. (3.27),we obtain

(z) =z

ASubstituting this equation into Eqs. (3.23) through (3.25) and solving the resultingequations, we havexx= 2Re{

} 2yIm{

} +Ayy= 2Re{

} +2yIm{

} Axy= 2yRe{

} (3.29)2ux= ( 1)Re{} 2yIm{

} +Ax2uy= ( +1)Im{} 2yRe{

} AyDene

= 12(ZI+A)Thus, =12(ZI+Az)

=12Z

Iwhere Z

IZI. The use of these two equations in Eq. (3.29) results inxx= Re{ZI} yIm{Z

I} +2Ayy= Re{ZI} +yIm{Z

I}xy= yRe{Z

I} (3.30)2ux=( 1)2Re{ZI} yIm{ZI} +12( +1)Ax2uy=( +1)2Im{ZI} yRe{ZI} +12( 3)AyZIis the so-called Westergaard function for Mode I problems. It is obvious that thestress eld associated with A is a uniform uniaxial stressxx=2A. This stress elddoes not add to the stress singularity at the crack tip.Sun08-ch03-025-076-97801238500102011/10/317:15Page36#1236 CHAPTER 3The Elastic Stress Field around a Crack TipFrom Eq. (3.30), we note thatxxyy=2A =constant at y =0For the case where uniform tension is applied in the y-direction; that is, xx=0 andyy=0, A =0/2. If the panel is subjected to biaxial tensions of equal magnitude,then A =0.TheAirystressfunctioncorrespondingtotheconstantstresseldxx=2A =2/y2is given by =Ay2It can be easily veried that the stresses and displacements of Eq. (3.30) are derivedfrom the stress function (see Problem 3.3) =Re{ZI} +yIm{ZI} +Ay2The function ZI is usually associated with Westergaard [3-3], who used it to solvethe contact pressure distribution resulting fromthe contact of many surfaces and somecrack problems. In its original form used by Westergaard, A =0. In 1957, Irwin [3-1]used Westergaards solutions to obtain the stress eld at the crack tip and related thatto the strain energy release rate.3.3.2Skew-Symmetric Problems (Mode II)For loads that are skew-symmetric with respect to the crack line (x-axis), the normalstress yy is zero along y =0. From Eqs. (3.23) and (3.24), this condition gives rise toRe{2

(z) + z

(z) +

} =0 at y =0 (3.31)Following the same procedure described for the symmetric problem, we obtain

(z) +2

(z) +z

(z) +iB =0inwhichBis a real constant. Usingthis equation, (z) canbe eliminatedinEqs. (3.23) through (3.25) and we havexx= 4Re{

(z)} 2yIm{

(z)}yy= 2yIm{

(z)}xy= 2Im{

(z)} 2yRe{

(z)} B (3.32)2ux= ( +1)Re{(z)} 2yIm{

(z)} By2uy= ( 1)Im{(z)} 2yRe{

(z)} BxSun08-ch03-025-076-97801238500102011/10/317:15Page37#133.3Westergaard Function Method 37Dene an analytic function (z) by

(z) =

(z) +i2BThen(z) = (z) +i2Bz

(z) =

(z)Substituting these denitions into Eq. (3.32), we obtainxx= 4Re{

(z)} 2yIm{

(z)}yy= 2yIm{

}xy= 2Im{

(z)} 2yRe{

(z)} (3.33)2ux= ( +1)Re{(z)} 2yIm{

(z)} + +12By2ux= ( 1)Im{(z)} 2yRe{

(z)} +12BxThelast terminthedisplacement componentsuxanduyrepresentsarigidbodyrotation. Dene Westergaard function ZII asZII=2i

(z)Then Eq. (3.33) becomesxx= 2Im{ZII} +yRe{Z

II}yy= yRe{Z

II}xy= Re{ZII} yIm{Z

II} (3.34)2ux=12( +1)Im{ZII} +yRe{ZII} + +12By2uy= 12( 1)Re{ZII} yIm{ZII} +12BxThus, theWestergaardfunctionZII(z)providesthegeneralsolutionfortheskew-symmetric problems.We conclude that any plane elasticity problem involving collinear straight cracksin an innite plane can be completely solved by the stress function = Re{ZI} +yIm{ZI} yRe{ZII} +Ay2ThisisanalternativeformtoEq.(3.14).TheadvantageofusingtheWestergaardfunctions is that the two modes of fracture are represented separately by two analyticfunctions.Sun08-ch03-025-076-97801238500102011/10/317:15Page38#1438 CHAPTER 3The Elastic Stress Field around a Crack Tip3.4SOLUTIONS BY THE WESTERGAARD FUNCTION METHODA center crack in an innite plate under uniform remote loading is perhaps the bestexample to introduce the basic concepts of stress intensity factor and near-tip stressand deformation elds. In this section, the Westergaard function method is used to ndthe elasticity solutions for an innite plane with a center crack under uniform biaxialtension, in-plane shear, and antiplane shear loading, respectively. We will see that thenear-tip singular stress elds can be easily extracted from the complete solutions andthestressintensityfactorscanbeobtainedfromthesolutionsusingthedenitioninEq. (3.1). Furthermore, stressintensityfactorsmayalsobeconvenientlydeter-mined directly from the general Kolosov-Muskhelishvili potentials or Westergaardfunctions.3.4.1Mode I CrackOneofthemosttypicalcrackproblemsinfracturemechanicsisaninniteplanewith a line crack of length 2a subjected to biaxial stress0at innity, as shown inFigure 3.3. In practice, if the crack length is much smaller than any in-plane size ofthe concerned elastic body, the region may be mathematically treated as an inniteplane with a nite crack. The problem is Mode I since the loads are symmetric withrespect to the crack line.xya0000aFIGURE 3.3A crack in an innite elastic plane subjected to biaxial tension.Sun08-ch03-025-076-97801238500102011/10/317:15Page39#153.4Solutions by the Westergaard Function Method 39Solution of StressesThe boundary conditions of the crack problem arexy= yy=0 at |x| aand y =0 (crack surfaces)xx= yy=0, xy=0 at x2+y2(3.35)ThesolutiontotheprecedingboundaryvalueproblemwasgivenbyWestergaard[3-3] withZI(z) =0zz2a2, A =0 (3.36)Since all the equations of plane elasticity are automatically satised, we only need toverify that the stresses obtained by Eq. (3.30) with ZIgiven in Eq. (3.36) satisfy theboundary conditions Eq. (3.35).To nd the explicit expressions of the stress eld, it is more convenient to use thepolar coordinates shown in Figure 3.4. The following relations are obvious:z = reiz a = r1ei1(3.37)z +a = r2ei2In terms of these polar coordinates, the function ZI becomesZI=0zz +az a=0rr1r2expi_ 121122_(3.38)a ayxr12r1r2FIGURE 3.4Polar coordinate systems.Sun08-ch03-025-076-97801238500102011/10/317:15Page40#1640 CHAPTER 3The Elastic Stress Field around a Crack TipThe derivative of ZI is obtained asZ

I=0z2a2 0z2(z2a2)3/2 =0a2(z2a2)3/2=0a2(r1r2)3/2 exp_i32(1+2)_(3.39)It follows from Eqs. (3.38) and (3.39) thatRe{ZI} =0rr1r2cos_ 121122_Re{Z

I} =0a2(r1r2)3/2 cos 32(1+2)Im{Z

I} =0a2(r1r2)3/2 sin 32(1+2)Using these expressions, the normal stress xx is obtained asxx= Re{ZI} yIm{Z

I}=0rr1r2cos_ 121122_0a2(r1r2)3/2r sin sin 32(1+2)=0rr1r2_cos_ 121122_a2r1r2sin sin 32(1+2)_(3.40)Similarly, the other two stress components can be obtained:yy=0rr1r2_cos_ 121122_+a2r1r2sin sin 32(1+2)_(3.41)xy=0rr1r2_a2r1r2sin cos 32(1+2)_(3.42)Using these stress expressions, it is easy to show that the tractions (yyandxy) onthe crack surface (1=, 2=0, =0, ) vanish completely.At large distances (r ) from the crack it is easily seen thatr1 r2 r 1 2 and consequently thatxx=0, yy=0, xy=0Thus, theboundaryconditionsaresatisedeverywhere. Equations(3.40)through(3.42) are the complete solution of the stress eld in the entire cracked plane.Sun08-ch03-025-076-97801238500102011/10/317:15Page41#173.4Solutions by the Westergaard Function Method 41The Near-Tip SolutionIn fracture mechanics, crack growth is controlled by the stresses and deformationsaround the crack tip. We thus study the near-tip asymptotic stress eld. In the vicinityof the crack tip (say the right tip), we haver1a 1, 0, 20r a, r22asin r1asin1sin 32(1+2) sin 321cos_ 121122_cos 121cos 32(1+2) cos 321By using the preceding asymptotic expressions, Eq. (3.40) reduces toxx=0a2ar1_cos 121a22ar1r1asin1sin 321_=0a2r1_cos 12112 sin1sin 321_=0a2r1cos 121_1 sin 121sin 321_Similarly,yy=0a2r1cos 121_1 +sin 121sin 321_xy=0a2r1sin 121cos 121cos 321Along the crack extended line ( =1=2=0), these near-tip stresses areyy=0a2r1xy= 0Sun08-ch03-025-076-97801238500102011/10/317:15Page42#1842 CHAPTER 3The Elastic Stress Field around a Crack TipComparing the previous stresses with Eq. (3.1) in Section 3.1, we have the Mode Istress intensity factor KI for the crack problem asKI=0a (3.43)and the Mode II stress intensity factor KII=0.If the origin of the coordinate system(r, ) is located at the crack tip, then thestresseldnearthecracktipcanbewrittenintermsofthestressintensityfactorKI asxx=KI2r cos 12_1 sin 12 sin 32_yy=KI2r cos 12_1 +sin 12 sin 32_(3.44)xy=KI2r sin 12 cos 12 cos 32Following the same procedures, the near-tip displacements are obtained asux=KI82r_(2 1)cos 2cos 32_uy=KI82r_(2 +1)sin 2sin 32_(3.45)Equation(3.44)showsthatstresseshaveaninversesquarerootsingularityatthecracktipandtheintensityof thesingular stresseldisdescribedbythestressintensity factor.Equations (3.44) and (3.45) are derived for a crack in an innite plane subjectedto remote biaxial tension. It will be seen in Section 3.7 that these expressions for thenear-tipstressesanddisplacementsholdforanycrackedbodyundergoingModeIdeformations. The difference is only the value of the stress intensity factor. Hence,once the stress yy along the crack extended line is known, the Mode I stress intensityfactor can be obtained fromKI= limr02r yy( =0) (3.46)Crack Surface DisplacementBesides the near-tip stress eld, the displacements of crack faces are also relaventtocrackgrowth. Under ModeI deformationconditions, thecracksurfacesopenup, which is quantied by the vertical displacement component uy. For the problemshown in Figure 3.3, we already haveZI=0zz2a2Sun08-ch03-025-076-97801238500102011/10/317:15Page43#193.4Solutions by the Westergaard Function Method 43andZI=0_z2a2In terms of the polar coordinates dened in Eq. (3.37) and Figure 3.4, the function ZIcan be expressed asZI=0r1r2e12i(1+2)The corresponding vertical displacement is obtained fromthe last equation ofEq. (3.30):4uy= ( +1)ImZI= ( +1)0r1r2sin 12(1+2)The upper crack surface corresponds to 1=, 2=0 and the lower surface is 1=, 2=0. The displacement of the upper crack surface is thus given byuy= +140r1r2= +140_a2x2(3.47)Near the crack tip, r1a, r22a, and Eq. (3.47) reduces touy= +140_2ar1= +14KI_2r1which is consistent with Eq. (3.45).3.4.2Mode II CrackConsider a cracked plate of innite extent that is subjected to uniform shear stress 0at innity as shown in Figure 3.5. This is a basic Mode II problem with the followingskew-symetric boundary conditions:xy= 0, yy=0 at |x| aand y =0 (crack surfaces)xx= yy=0, xy=0at x2+y2(3.48)It can be shown that the following Westergaard function,ZII(z) =0zz2a2(3.49)yields stresses that satisfy the boundary conditions Eq. (3.48) and, hence, yields thesolution for the problem. The stress eld can be computed according to Eqs. (3.34)Sun08-ch03-025-076-97801238500102011/10/317:15Page44#2044 CHAPTER 3The Elastic Stress Field around a Crack Tipxya0000aFIGURE 3.5A crack in an innite elastic plane subjected to pure in-plane shear.and (3.49). Following the same procedure described for the Mode I crack and usingthepolarcoordinatesdenedinEq. (3.37)andFigure3.4, wehavethecompletestress eld,xx=0rr1r2_2sin_ 121122_a2r1r2sin cos 32(1+2)_yy=0a2r(r1r2)3/2 sin cos 32(1+2) (3.50)xy=0rr1r2_cos_ 121122_a2r1r2sin sin 32(1+2)_Along the crack extended line (1=2= =0) and near the crack tip (r1/a 1),the stresses areyy= 0xy=0a2r1Comparing these stresses with Eq. (3.1) in Section 3.1, we have the Mode II stressintensity factor KII for the crack problem asKII=0a (3.51)Sun08-ch03-025-076-97801238500102011/10/317:15Page45#213.4Solutions by the Westergaard Function Method 45and the Mode I stress intensity factor KI=0. In general, the stresses in the vicinityof the right crack tip are derived asxx= KII2r sin 12_2 +cos 2 cos 32_yy=KII2r sin 2 cos 2 cos 32 (3.52)xy=KII2r cos 12_1 sin 12 sin 32_and the near-tip displacements areux=KII82r_(2 +3)sin 2+sin 32_uy= KII82r_(2 3)cos 2+cos 32_(3.53)where the origin of the (r, ) system has been shifted to the right crack tip.Equation (3.52) shows that stresses also have an inverse square root singularityat the crack tip and the intensity of the singular stress eld is described by the ModeII stress intensity factor. It will be seen again in Section 3.7 that the near-tip stressesEq. (3.52) and displacements Eq. (3.53) hold for any cracked body under Mode IIdeformationconditionswithdifferencesonlyinthevalueofKII.Hence,oncethestress xy along the crack extended line is known, the Mode II stress intensity factorcan be obtained fromKII= limr02r xy( =0) (3.54)The crack surface displacment may be obtained by setting y =0 in Eq. (3.34) (Bis ignored as it represents rigid displacements):4ux= ( +1)ImZII4uy= (1 )ReZIISinceZII=0_z2a2we haveZII=0_x2a2at y =0The crack surfaces lie in the region |x| < a. Thus,ZII=i0_a2x2Sun08-ch03-025-076-97801238500102011/10/317:15Page46#2246 CHAPTER 3The Elastic Stress Field around a Crack TipIt is then obvious that the displacements of the upper crack surface areuy= 0ux= +140_a2x2(3.55)The preceding expressions show that the two crack surfaces slide with each other.3.4.3Mode III CrackTheModeIIIfractureisassociatedwiththeanti-planedeformationforwhichthedisplacements are given byux=0, uy=0, uz=w(x, y) (3.56)The nonvanishing strains are thus given byexz= 12wx , eyz= 12wy(3.57)and the corresponding stresses follow Hookes law:xz=2exz, yz=2eyz(3.58)The equations of equilibrium reduce toxzx+yzy=0 (3.59)which can be written in terms of displacement w by using Eqs. (3.57) and (3.58) asfollows:2w =0 (3.60)Thus, w must be a harmonic function. Letw =1Im{ZIII(z)} (3.61)where ZIII(z) is an analytic function. Substituting Eq. (3.61) in Eq. (3.57) and thenEq. (3.58), the stresses can be represented by ZIII(z) as follows:xziyz=iZ

III(z) (3.62)Now consider an innite cracked body under anti-plane shear stress S shown inFigure 3.6. The boundary conditions of the crack problem are given byyz= 0 at |x| a and y =0yz= S at |y| (3.63)Sun08-ch03-025-076-97801238500102011/10/317:15Page47#233.4Solutions by the Westergaard Function Method 47xya aSSFIGURE 3.6A crack in an innite elastic body subjected to anti-plane shear.ChooseZIII=S_z2a2(3.64)We can easily show that the stresses calculated from this function satisfy the bound-ary conditions Eq. (3.63). Substituting Eq. (3.64) in Eq. (3.62) and using the polarcoordinates dened in Eq. (3.37), the stress components can be obtained asyz= Re{Z

III} =Srr1r2cos_ 121122_xz= Im{Z

III} =Srr1r2sin_ 121122_(3.65)Again consider the stress along the crack extended line (1=2= =0) and nearthe crack tip (r1/a 1). It is obtained from Eq. (3.65) asyz=Sa2r1Comparing these stresses with Eq. (3.1) in Section 3.1, we have the Mode III stressintensity factor KIII for the crack problem asKIII=Sa (3.66)Sun08-ch03-025-076-97801238500102011/10/317:15Page48#2448 CHAPTER 3The Elastic Stress Field around a Crack TipIn general, the near-tip stresses are obtained asyz=KIII2r1cos 121xz= KIII2r1sin 121(3.67)and the anti-plane displacement isw =_2KIIIr1sin121(3.68)The forms of Eqs. (3.67) and (3.68) hold for general Mode III cracks and the ModeIII stress intensity factor is calculated fromKIII=limr10_2r1yz( =0) (3.69)The complete displacement eld is obtained using Eqs. (3.64) and (3.61) asw =uz=Sr1r2sin 12(1+2)On the upper crack surface (y =0+, |x| < a, or 1=, 2=0), the displacement isuz=Sr1r2=S_a2x2(3.70)3.4.4Complex Representation of Stress Intensity FactorStress intensity factor is a key concept in linear elastic fracture mechanics. It will beseen in Section 3.7 that the asymptotic stress and displacement elds near a crack tiphave universal forms as described in Eqs. (3.44), (3.45), (3.52), and (3.53). Solvingfor the stresses and displacements around a crack tip thus reduces to nding the stressintensity factors, which may be directly calculated from the solutions of the complexpotential functions. It follows from the near-tip stress eld Eqs. (3.44) and (3.52) that(now use (r1,1) at the crack tip z =a)xx+yy=2KI2r1cos 121for Mode Ixx+yy=2KII2r1sin 121for Mode IIFor combined loading, we havexx+yy=2KI2r1cos 1212KII2r1sin 121Sun08-ch03-025-076-97801238500102011/10/317:15Page49#253.4Solutions by the Westergaard Function Method 49Dening the complex stress intensity factor K,K =KIiKII(3.71)and recalling the denition of polar coordinates (r1, 1),z a =r1ei1one can show thatxx+yy=2Re_K2(z a)_z aSincexx+yy=4Re{

(z)}we obtainRe_K2(z a)_=2Re{

(z)} z aThe complex stress intensity factor at the crack tip z = a follows fromthisequation:K =22 limza{(z a)

(z)} (3.72)In the last step of deriving the expression, we have used the relationRe{ f (z)} = Re{g(z)} f (z) =g(z) +iCwhere f (z) and g(z) are analytic functions and C is a real constant.ProofLeth(z) =f (z) g(z) =U3+iV3Thus, h(z) is also analytic. Since Re{h(z)} =0, i.e.,U3=0we have from the Cauchy-Riemann equation (3.11)V3x= U3y=0V3y=U3x=0Sun08-ch03-025-076-97801238500102011/10/317:15Page50#2650 CHAPTER 3The Elastic Stress Field around a Crack TipThus,V3=real constant =Ci.e.,f (z) =g(z) +iCIf the Westergaard functions are used, we can obtain the following expression forthe complex stress intensity factor at the tip z =a:K =2 limza_z a(ZIiZII)_(3.73)For Mode III cracks, it follows from Eqs. (3.62) and (3.67) thatKIII=2 limza_z aZ

III(z)_(3.74)3.5FUNDAMENTAL SOLUTIONS OF STRESS INTENSITYFACTORIntheprevioussection,thestressintensityfactorsforacrackinaninniteplanewerecalculatedunder uniformloadingconditions. Inengineeringapplications, acracked body is generally subjected to nonuniformly distributed loads. Stress inten-sityfactorsofacrackedbodyunderarbitraryloadingconditionsmaybeobtainedusing the superposition method and the fundamental solutions, which are the stressintensityfactorsforacrackedbodysubjectedtoconcentratedforcesonthecrackfaces.Consider a cracked body subjected to arbitrarily loads, as shown in Figure 3.7(a).The crack surfaces are assumed in a traction free state without loss of generality. Thestresses and displacements in the cracked body can be obtained by superposing thecorresponding solutions of the following two problems, as shown in Figures 3.7(b)and(c).Therstproblemisthesameelasticbodywithoutthecracksubjectedtothesameexternalloads.Thesolutionstothisrstproblemcanbeobtainedusingconventional methods in the theory of elasticity because no cracks are present. Thesecond problem consists of the same cracked body subjected to the crack face trac-tions, which are equal only in magnitude but opposite in sign to the tractions obtainedat the same crack location in the rst problem.It is evident that the superposition of the solutions of the two problems satisesalltheboundaryconditionsandthetractionfreeconditionsatthecracksurfaces.Inlinear elasticfracturemechanics, thesolutiontotherst problemisassumedSun08-ch03-025-076-97801238500102011/10/317:15Page51#273.5Fundamental Solutions of Stress Intensity Factor 51(a)Crack(b)= +(c)FIGURE 3.7Superposition method of linear elastic crack problems (a) the original crack problem,(b) the rst problem, (c) the second problem.abPPQQayxFIGURE 3.8A crack in an innite plate subjected to concentrated forces on the crack faces.tobeknownandthestressesareniteatthelocationofthecracktip. Thestressintensity factors therefore can be calculated from the solution to the second problemand may be obtained using the fundamental solutions. In the following sections, twofundamental solutions for a crack in an innite plane are introduced.3.5.1A Finite Crack in an Innite PlateConsider an innite plate containing a crack of length 2a subjected to a pair of com-pressive forces P per unit thickness at x =b, as shown in Figure 3.8 (the shear forceQ is assumed to be absent for now). The boundary conditions of the crack problemSun08-ch03-025-076-97801238500102011/10/317:15Page52#2852 CHAPTER 3The Elastic Stress Field around a Crack Tipare formulated as follows:yy= 0 at |x| a, x =band y =0a_ayydx = P at y =0+and y =0xy= 0 at |x| aandy =0xx,yy,xy0 at x2+y2(3.75)It can be shown that the Westergaard functionZI=P(z b)_a2b2z2a2givesthestressesthatsatisfytheboundaryconditionsEq. (3.75). SubstitutingthepreviousfunctionintoEq. (3.73)yieldsthecomplexstressintensityfactorat theright crack tip (z =a):KIiKII=2 limza___z aP(z b)_a2b2z2a2___=Pa_a +ba bSimilarly,wecangetthecomplexstressintensityfactorattheleftcracktip(notethat z a in Eq. (3.73) should be replaced by (z +a) when evaluating the stressintensity factor at the tip z =a). Finally, we have the stress intensity factors at thetwo crack tips:KI=Pa_a +ba b, at the right crack tip (x =a)=Pa_a ba +b, at the left crack tip (x =a) (3.76)When the crack faces are subjected to a pair of shearing forces Qper unit thicknessat x =b, as shown in Figure 3.8 (now the compressive force P is assumed absent),the Westergaard functionZII=Q(z b)_a2b2z2a2Sun08-ch03-025-076-97801238500102011/10/317:15Page53#293.5Fundamental Solutions of Stress Intensity Factor 53gives the stresses that satisfy the following boundary conditions of the Mode II crackproblem, that is,xy=0 at |x| a, x =band y =0a_axydx =Q at y =0+and y =0yy=0 at |x| aandy =0xx,yy,xy0 at x2+y2(3.77)SubstitutingtheWestergaardfunctionintoEq.(3.73),wehavetheModeIIstressintensity factors:KII=Qa_a +ba b, at the right crack tip (x =a)=Qa_a ba +b, at the left crack tip (x =a) (3.78)3.5.2Stress Intensity Factors for a Crack Subjected to ArbitraryCrack Face LoadsIn linear elastcity, it has been known that the stress (and displacement) in an elasticbody subjected to a number of external loads is the sum of the stresses correspondingto each individual load. This is the so-called superposition principle of linear elasticsystems. We now apply this principle to problems of cracks in linear elastic solids.The fundamental solutions Eqs. (3.76) and (3.78) can thus be used to