Mathematical and Computational Analyses of Cracking Formation: Fracture Morphology and Its Evolution in Engineering Materials and Structures

285 Pages·2014·15.1 MB·English

by  Yoichi Sumi (auth.)

Mathematics for Industry 2
Yoichi Sumi
Mathematical and
Computational
Analyses of Cracking
Formation
Fracture Morphology and Its Evolution
in Engineering Materials and Structures
Mathematics for Industry
Volume 2
Forfurthervolumes:
http://www.springer.com/series/13254
EditorinChief
MasatoWakayama(KyushuUniversity,Japan)
Scientific Board Members
RobertS.Anderssen(CommonwealthScientificandIndustrialResearch
Organisation,Australia)
PhilipBroadbridge(LaTrobeUniversity,Australia)
JinCheng(FudanUniversity,China)
MoniqueChyba(UniversityofHawai‘iatMaNnoa,USA)
Georges-HenriCottet(JosephFourierUniversity,France)
JoséAlbertoCuminato(UniversityofSãoPaulo,Brazil)
Shin-IchiroEi(HokkaidoUniversity,Japan)
YasuhideFukumoto(KyushuUniversity,Japan)
JonathanR.M.Hosking(IBMT.J.WatsonResearchCenter,USA)
AlejandroJofré(UniversityofChile,Chile)
KerryLandman(TheUniversityofMelbourne,Australia)
RobertMcKibbin(MasseyUniversity,NewZealand)
GeoffMercer(AustralianNationalUniversity,Australia)(Deceased,2014)
JillPipher(BrownUniversity,USA)
KonradPolthier(FreeUniversityofBerlin,Germany)
W.H.A.Schilders(EindhovenUniversityofTechnology,TheNetherlands)
ZuoweiShen(NationalUniversityofSingapore,Singapore)
Kim-ChuanToh(NationalUniversityofSingapore,Singapore)
NakahiroYoshida(TheUniversityofTokyo,Japan)
Aims& Scope
The meaning of “Mathematics for Industry” (sometimes abbreviated as MI or MfI) is
differentfromthatof“MathematicsinIndustry”(orof“IndustrialMathematics”).Thelatter
is restrictive: it tends to be identified with the actual mathematics that specifically arises
in the daily management and operation of manufacturing. The former, however, denotes
a new research field in mathematics that may serve as a foundation for creating future
technologies. This concept was born from the integration and reorganization of pure and
appliedmathematicsinthepresentdayintoafluidandversatileformcapableofstimulating
awarenessoftheimportanceofmathematicsinindustry,aswellasrespondingtotheneeds
of industrial technologies. The history of this integration and reorganization indicates that
thisbasicideawillsomedayfindincreasingutility.Mathematicscanbeakeytechnologyin
modernsociety.
The series aims to promote this trend by (1) providing comprehensive content on
applicationsofmathematics,especiallytoindustrytechnologiesviavarioustypesofscientific
research,(2)introducingbasic,useful,necessaryandcrucialknowledgeforseveralapplica-
tionsthroughconcretesubjects,and(3)introducingnewresearchresultsanddevelopments
for applications of mathematics in the real world. These points may provide the basis for
opening a new mathematics-oriented technological world and even new research fields of
mathematics.
Yoichi Sumi
Mathematical and
Computational Analyses
of Cracking Formation
Fracture Morphology and Its Evolution in
Engineering Materials and Structures
123
YoichiSumi
SystemsDesignforOcean-Space
YokohamaNationalUniversity
Yokohama,Japan
ISSN2198-350X ISSN2198-3518(electronic)
ISBN978-4-431-54934-5 ISBN978-4-431-54935-2(eBook)
DOI10.1007/978-4-431-54935-2
SpringerTokyoHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014935235
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Preface
This book focuses attention on the pattern formation and the evolution of crack
propagation in engineering materials and structures, where efforts are made to
bridge mathematical analyses of cracks based on singular integral equations to
computationalsimulation of engineeringdesign. The aim of this book is not only
forstructuralengineerstounderstandthebasicbackgroundsofanalyses,butalsofor
thosemajoringinmathematicstoseehowsuchmathematicalsolutionsareevaluated
in industrial applications. Brittle crack propagation and fatigue crack growth are
consideredasthemechanismsofcrackpropagation,andtheyareinvestigatedbased
on the Griffith–Irwin theory for the former, while the latter is examined by the
repeated tension–compressionyielding behaviorincluding the contactmechanism
of plastic wake left behind a crack tip. This monograph consists of four parts:
elasticity, fracture, morphology, and design. The first two parts provide the basis
of the analysis for the discussions on the fracture morphology and its numerical
simulation which may lead to a simulation-based fracture control in engineering
structures.
In Part I, the basic elasticity theory is summarized for two-dimensionalelastic
solids.Thedisplacementvectorandstraintensor,thestresstensorandAiry’sstress
function,and the relationbetweenstress and strain tensorsare first introducedfor
ahomogeneousisotropicelasticbody.Then,theelasticboundary-valueproblemis
defined,andsomefundamentalsolutionsaregiveninChapter1.Stressconcentra-
tion problemsare investigatedin Chapter 2, where the stress concentrationdue to
anexternalforceandtostructuraldiscontinuitiessuchascircularandellipticholes
aresolvedbyusingAiry’sstressfunctionandthecomplexpotentials.InChapter3,
crackproblemsare solvedby the complexpotentialmethod,and the elastic stress
singularitiesnearacracktipareidentified.Also,theanalyticalstructureofthestress
fieldisexaminedbyamethodofeigen-functionexpansionnearacracktip.
InPartII,analysesaremadeforbrittlefractureandfatiguecrackpropagation.In
Chapter4,brittlefractureisanalyzedbasedontheGriffith–Irwintheory,wherethe
effectsofplasticdeformationareexaminedbyconsideringthestripyieldingmodel,
J-integral,andstabilityofslowcrackgrowth.Fatiguecrackgrowthisdiscussedin
v
vi Preface
Chapter 5, in which Paris’ law is introduced. Then, the effective stress intensity
range proposed by Elber based on the crack closure concept and the repeated
tensile plasticity range proposed by Toyosada are discussed. The effects of stress
ratio and load sequence are analyzed based on a recently developedcrack growth
model.Fatigueandfractureareessentialforthestructuralintegrityofengineering
structures, so that the results in this part are relevant to the design codes for the
preventionofunstablefracturefollowingfatiguecrackgrowth.
InPartIIIweinvestigatemorphology,i.e.,theformationofsingleandmultiple
crack paths in brittle fracture and fatigue. In Chapter 6, morphologicalaspects of
crack propagation are discussed focusing attention on the pattern formation of a
systemofquasi-staticallygrowingstraightcracks,whichexhibitsstablebifurcation
phenomena, i.e., every other growing crack is arrested at a certain crack length,
resultinginaself-similarpatternformationofmultiplecracks.Then,inChapter7,
we discuss a kinked and curved crack by applying the first-order perturbation
method, in which crack path prediction in brittle solids is analyzed based on a
crack path criterion that includes some aspects of crack path stability. The first-
orderperturbationis then appliedto a system ofkinkedand curvedcracksforthe
simulationofcurvedorwavycrackpropagationinbrittle solids.InChapter8,the
perturbationmethodis extendedto the secondorderin examiningthe behaviorof
brittle fracture along butt-weld, where the residual stress and the degradation of
fracture toughness due to welding may play essential roles coupled with curved
crackpaths.InChapter9,fatiguecrackpathsareinvestigatedwithregardtocrack
pathcriteria,theeffectsofbiaxialstress,andwelding.
InPartIV,weconsidersimulation-basedfracturecontroldesign.Itiswellknown
that the simulation of the formation of brittle or fatigue cracks is essential for
the precise evaluation of fracture modes and fracture processes, so that several
numerical simulation systems have been developed based on the finite element
method, meshless method, and/or boundary element method. In Chapter 10, a
numerical simulation system is introduced for the path prediction of a system
of through-the-thickness cracks and their remaining life assessment for fatigue
crack propagation in three-dimensional plate structures. The method is based on
astep-by-stepfinite-elementanalysis.Crackpathsarepredictedbytheperturbation
methodapplyingthelocalsymmetrycriterion,whichgivesahigher-order(curved)
approximation of each incremental crack extension as described in Chapters 7,
8, and 9. The finite element re-zoning is automatically carried out, so that user
intervention is minimized to generate a very robust mesh during the entire crack
propagationprocess. In Chapter 11, several design concepts are discussed for the
prevention of fatigue and fracture in engineering structures, i.e., safe-life design,
fail-safe design, and damage-tolerant design. To further improve the structural
designagainstfracture,questionshavebeenraisedaboutwhetheronemaycontrol
thecrackpropagationphenomenato acertainextentbasedonfracturemechanics,
even though fatigue cracks initiate and then propagate. Attempts in this direction
aresoughtbypreciselyidentifyingpotentiallycriticallocationsandtheassociated
failure modes of structural details, predicting crack propagation lives considering
theeffectof retardationdueto loadsequenceandresidualstresses, andpredicting
Preface vii
crack paths and shapes during crack propagation. This kind of design concept is
sometimes called fracture control design, which is discussed in details based on
illustrativeapplicationstomarinestructures.
The results presented in Parts III and IV and some parts of Part II were based
on the work with my former advisers and my graduate students. I must express
myspecialthankstomyformeracademicadviserswhohavetaughtmemechanics
and worked together on crack problems: Professor Yoshiyuki Yamamoto, The
University of Tokyo; Professor Sia Nemat-Nasser, University of California, San
Diego; Professor Leon M. Keer, Northwestern University; and Professor Hiroshi
Itagaki,YokohamaNationalUniversity.Also,Iwishtoexpressmythankstoallthe
peoplewhohavehelpedmeduringthecourseofmyresearch:myformerstudents
Teppei Okawa, Yasunori Kagohashi, Chen Yang, Wang Zhinong, Mu Yang, Yuan
Kuilin,ShigetoshiHayashi,KazuhiroHirota,KazutoshiOhashi,SatoshiFunabashi,
andTakenobuInoue,aswellasmyfriendsYasumiKawamura,MasashiMohri,and
ShunichiMichiyama.
IgivemythankstoSaoriOgawaforherskillfultypinganddrawingsinadigital
environment and her secretarial work in various aspects for a couple of years.
I express my gratitude to my family: Katsuko, my wife, and Haruko, Naoko, and
Mariko,mydaughters,fortheirlong-timeunderstandingandpatiencewithme.
Finally, I acknowledgethe Ministry ofEducation,Culture, Sports,Science and
Technology(MEXT),JapanSocietyforthePromotionofScience(JSPS),andJapan
ShipTechnologyResearchAssociation(JSTRA)forthesupportofmyresearchin
theareaoffracturemechanicsofmarinestructures.
Yokohama YoichiSumi
December2013

Contents
PartI Elasticity
1 ElasticBoundary-ValueProblems........................................ 3
1.1 Displacement,Strain,andCompatibilityCondition................ 3
1.2 StressandEquilibriumConditions.................................. 5
1.3 Stress–StrainRelationship........................................... 6
1.4 StressFunction....................................................... 8
1.5 ElasticBoundary-ValueProblem.................................... 9
1.6 SolutionBasedonaPolarCoordinateSystem...................... 11
References.................................................................... 16
2 StressConcentrationProblems ........................................... 17
2.1 MechanismsandSolutionMethodsofStressConcentration....... 17
2.2 AConcentratedForceActingattheTipofaWedge ............... 18
2.3 AxisymmetricSolution .............................................. 20
2.4 StressConcentrationCausedbyaCircularHole ................... 22
2.5 StressConcentrationFactor.......................................... 24
2.6 EllipticHoleinTension.............................................. 25
References.................................................................... 30
3 AnalysisofTwo-DimensionalCracks .................................... 31
3.1 StressFieldAroundaCrack......................................... 31
3.2 Williams’ExpansionataCrackTip................................. 36
3.3 MethodofMuskhelishviliforaStraightCrack..................... 40
3.3.1 SolutionofaStraightCrackofLength2a ................ 40
3.3.2 SolutionofaSemi-InfiniteStraightCrack
SubjectedtoaForceCouple ............................... 43
References.................................................................... 44
ix