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11 FundamentalsofFractureMechanics 11.1INTRODUCTION Inthe17thcentury,thegreatscientistandpainter,LeonardodaVinci, performedsomestrengthmeasurementsonpianowireofdifferentlengths. Somewhatsurprisingly,hefoundthatthestrengthofpianowiredecreased withincreasinglengthofwire.Thislength-scaledependenceofstrengthwas notunderstooduntilthe20thcenturywhentheseriousstudyoffracturewas revisitedbyanumberofinvestigators.Duringthefirstquarterofthe20th century,Inglis(1913)showedthatnotchescanactasstressconcentrators. Griffith(1921)extendedtheworkofInglisbyderivinganexpressionforthe predictionofthebrittlestressinglass.Usingthermodynamicarguments, andtheconceptofnotchconcentrationfactorsfromInglis(1913),he obtainedaconditionforunstablecrackgrowthinbrittlematerialssuchas glass.However,Griffith’sworkneglectedthepotentiallysignificanteffects ofplasticity,whichwereconsideredinsubsequentworkbyOrowan(1950). AlthoughtheworkofGriffith(1921)andOrowan(1950)provided someinsightsintotheroleofcracksandplasticityinfracture,robustengi- neeringtoolsforthepredictionoffracturewereonlyproducedinthelate 1950sandearly1960safteranumberofwell-publicizedfailuresofshipsand aircraftinthe1940sandearlytomid-1950s.Someofthefailuresincluded thefractureoftheso-calledLibertyshipsinWorldWarII(Fig.11.1)and Copyright © 2003 Marcel Dekker, Inc. theCometaircraftdisasterinthe1950s.Theseledtosignificantresearchand developmenteffortsattheU.S.Navyandthemajoraircraftproducerssuch asBoeing. TheresearcheffortsattheU.S.NavalResearchlaboratorywereledby GeorgeIrwin,whomaybeconsideredasthefatheroffracturemechanics. Usingtheconceptsoflinearelasticity,hedevelopedacrackdrivingforce parameterthathecalledthestressintensityfactor(Irwin,1957).Ataround thesametime,Williams(1957)alsodevelopedmechanicssolutionsforthe crack-tipfieldsunderlinearelasticfracturemechanicsconditions.Workat theBoeingAircraftCompanywaspioneeredbyayounggraduatestudent, PaulParis,whowastomakeimportantfundamentalcontributionstothe subjectoffracturemechanicsandfatigue(Parisandcoworkers,1960,1961, 1963)thatwillbediscussedinChap.14. Following the early work on linear elastic fracture mechanics, it was recognized that further work was needed to develop fracture mechanics approaches for elastic–plastic and fully plastic conditions. This led to the development of the crack-tip opening displacement (Wells, 1961) and the J integral (Rice, 1968) as a parameter for the characterization of the crack driving force under elastic–plastic fracture mechanics conditions. Three- FIGURE 11.1 Fractured T-2 tanker, the S.S. Schenectady, which failed in1941. (Adapted from Parker, 1957—reprinted with permission from the National Academy of Sciences.) Copyright © 2003 Marcel Dekker, Inc. parameterfracturemechanicsapproacheshavealsobeenproposedby McClintocketal.(1995)forthecharacterizationofthecrackdriving forceunderfullyplasticconditions. Thesubjectoffracturemechanicsisintroducedinthischapter.The chapterbeginswithabriefdescriptionofGriffithfracturetheory,the Orowanplasticitycorrection,andtheconceptoftheenergyreleaserate. Thisisfollowed byaderivationofthestressintensity factor,K,andsome illustrationsoftheapplicationsandlimitationsofKinlinearelasticfracture mechanics.Elastic–plasticfracturemechanics conceptsarethen introduced alongwithtwo-parameterfractureconceptsfortheassessmentofcon- straint.Finally,therelativenewsubjectofinterfacialfracturemechanics is presented,alongwiththefundamentalsofdynamicfracturemechanics. 11.2FUNDAMENTALSOFFRACTUREMECHANICS It isnowgenerallyacceptedthatallengineeringstructuresandcomponents contain three-dimensional defects thatare known as cracks. However, as discussed in the introduction, our understanding of the significance of cracks has only been developed during the past few hundred years, with most of the basic understanding emerging during the last 50 years of the 20th century. 11.3 NOTCH CONCENTRATION FACTORS Inglis (1913) modeled the stress concentrations around notches with radii of curvature,,andnotchlength,a(Fig.11.2).Forelasticdeformation,hewas able to show that the notch stress concentration factor, K t , is given by K t ¼ maximumstressaroundnotchtip Remote stressawayfrom notch ¼ 1 þ2 ffiffiffi a r ð11:1Þ Hence,forcircularnotcheswitha¼,he wasable toshowthatK t 3. Thisratherlarge stressconcentrationfactorindicatesthatanapplied stress ofisamplifiedbya factorof3atthenotchtip.Failureis,therefore,likely toinitiatefrom thenotchtip,whentheapplied/remote stressesaresignifi- cantly belowthe fracturestrengthoftheun-notchedmaterial.Subsequent workbyNeuber (1945) extended the work of Inglis to include the effects of notch plasticity on stress concentration factors. This has resulted in the publication of handbooks of notch concentration factors for various notch geometries. Returning now to Eq. (11.1), it is easy to appreciate that the notch concentration factor will increase dramatically, as the notch-tip radius Copyright © 2003 Marcel Dekker, Inc. approachesthelimitingvaluecorrespondingtoasinglelatticespacing,b. Hence,foranatomisticallysharpcrack,therelativelyhighlevelsofstress concentrationarelikelytoresultindamagenucleationandpropagation fromthecracktip. 11.4GRIFFITHFRACTUREANALYSIS Theproblemofcrackgrowthfromasharpnotchinabrittlesolidwasfirst modeledseriouslybyGriffith(1921).Byconsideringthethermodynamic balancebetweentheenergyrequiredtocreatefreshnewcrackfaces,and thechangeininternal(strain)energyassociatedwiththedisplacementof specimenboundaries(Fig.11.3),hewasabletoobtainthefollowingenergy balanceequation: U T ¼ 2 a 2 B E 0 þ4a s Bð11:2Þ wherethefirsthalfontheright-handsidecorrespondstothestrainenergy andthesecondhalfoftheright-handsideisthesurfaceenergyduetothe upperandlowerfacesofthecrack,wihchhaveatotalsurfaceareaof4aB. Also,istheappliedstress,aisthecracklength,Bisthethicknessofthe specimen,E 0 ¼E=ð1 2 Þforplanestrain,andE 0 ¼Eforplanestress, FIGURE11.2Stressconcentrationaroundanotch.(AdaptedfromCallister, 1999—reprinted with permission from John Wiley.) Copyright © 2003 Marcel Dekker, Inc. where E is Young’s modulus, is Poisson’s ratio, and s is the surface energy associated with the creation of the crack faces. The critical condition at the onset of unstable equilibrium is deter- mined by equating the first derivative of Eq. 11.2 to zero, i.e., dU T =da ¼ 0. This gives dU T da ¼ 2 2 aB E 0 þ 4B ¼ 0 ð11:3Þ or c ¼ ffiffiffiffiffiffiffiffiffiffi 2E 0 a r ð11:4Þ where c is the Griffith fracture stress obtained by rearranging Eq. (11.3), and the other terms have their usual meaning. Equation (11.4) does not account for the plastic work that is done during the fracture of most mate- rials. It is, therefore, only applicable to very brittle materials in which no plastic work is done during crack extension. Equation (11.4) was modified by Orowan (1950) to account for plastic work in materials that undergo plastic deformation prior to catastrophic FIGURE 11.3 A center crack of length 2a in a large plate subjected to elastic deformations. (Adapted from Suresh, 1999—reprinted with permission from Cambridge University Press.) Copyright © 2003 Marcel Dekker, Inc. failure.Orowanproposedthefollowingexpressionforthecriticalfracture condition, c : c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð s þ p ÞE 0 a r ð11:5Þ where p isaplasticenergyterm,whichisgenerallydifficulttomeasure independently. Anotherimportantparameteristhestrainenergyreleaserate,G, whichwasfirstproposedbyIrwin(1964).Thisisgivenby: G¼ 1 B dðU L þU E Þ da ð11:6Þ whereU L isthepotentialenergyoftheloadingsystem,U E isthestrain energyofthebody,andBisthethicknessofthebody.Fractureshould initiatewhenGreachesacriticalvalue,G c ,whichisgivenby G C ¼2ð s þ p Þð11:7Þ 11.5ENERGYRELEASERATEANDCOMPLIANCE Thissectionpresentsthederivationofenergyreleaseratesandcompliance conceptsforprescribedloading[Fig.11.4(a)]andprescribeddisplacement [Fig.11.4(b)]scenarios.Thepossibleeffectsofmachinecomplianceare consideredattheendofthesection. 11.5.1LoadControlorDeadweightLoading Letusstartbyconsideringthebasicmechanicsbehindthedefinitionofthe energyreleaserateofacracksubjectedtoremoteload,F,Fig.11.4(a).Also, uistheloadpointdisplacementthroughwhichloadFisapplied.Theenergy releaserate,G,isdefinedas G¼ 1 B @PE @a ð11:8Þ whereBisthethicknessofthespecimen,PEisthepotentialenergy,andais thecracklength.Thepotentialenergyforasystemwithprescribedload,F, isgivenby(Fig.11.5): ¼ PE ¼ SE WD ¼ 1 2 Fu Fu ¼ 1 2 Fu ð11:9Þ where SE is the strain energy and WD is the work done. By definition, the compliance, C, of the body is given simply by Copyright © 2003 Marcel Dekker, Inc. FIGURE 11.5 Schematic of load–displacement curve under prescribed load. (Adapted from Suresh, 1999—reprinted with permission from Cambridge University Press.) FIGURE 11.4 Schematic of notched specimens subject to (a) prescribed load- ing or (b) prescribed displacement. (Adapted from Suresh, 1999—reprinted with permission from Cambridge University Press.) Copyright © 2003 Marcel Dekker, Inc. C¼ u F ð11:10Þ whereCdependsongeometryandelasticconstantsEand.Hence,the potentialenergyisgivenby[Eqs(11.9)and(11.10)]: PE¼ 1 2 Fu¼ 1 2 FðCFÞ¼ 1 2 F 2 Cð11:11Þ andtheenergyreleaserateisobtainedbysubstitutingEq.(11.11)intoEq. (11.8)togive G¼ 1 B @PE @a ¼ 1 2B F 2 dC da ð11:12Þ 11.5.2Displacement-ControlledLoading Letusnowconsiderthecaseinwhichthedisplacementiscontrolled[Fig. 11.4(a)],andtheload,F,variesaccordingly(Fig.11.6).Whenthecrack advances by an amount Áa under a fixed displacement, u, the work done is zero and hence the change in potential energy is e qual to the strain energy. FIGURE 11.6 Schematic of load–displacement curve under prescribed dis- placement. (Adapted from Suresh, 1999—reprinted with permission from Cambridge University Press.) Copyright © 2003 Marcel Dekker, Inc. fixed u PE fixed u ¼SE¼GA¼GBað11:13Þ whereAisthechangeincrackarea,dA¼Ba.Hence,rearrangingEq. (11.13)nowgives G¼ 1 B @PE @a fixed u ð11:14Þ Thestrainenergy,SE,isgivensimplybytheshadedareainFigure11.6: SE ¼ 1 2 uF ¼ F 2 F ð11:15Þ Since the compliance is u/F, then from Equation 11.15, we have: G ¼ 1 B @PE @a fixed u ¼ u 2B @F @a fixed u ¼ F 2 2B dC da ð11:16Þ Hence, the expression for the energy release rate is the same for displace- ment control [Eq. (11.16)] and load control [Eq. (11.12)]. It is important to note the above equations for G are valid for both linear and non-linear elastic deformation. They are also independent of boundary conditions. 11.5.3 Influence of Machine Compliance Let us now consider the influence of machine compliance, C M , on the deformation of the cracked body shown in Fig. 11.7. The total displacement, FIGURE 11.7 Schematic of deformation in a compliant test machine. (Adapted from Hutchinson, 1979—with permission from the Technical University of Denmark.) Copyright © 2003 Marcel Dekker, Inc. Á T ,isnowthesumofthemachinedisplacement,Á M ,andthespecimen displacement,Á.Ifthetotaldisplacementisprescribed,thenwehave Á T ¼ÁþÁ M ¼ÁþFC M ð11:17Þ sinceC¼Á=F,wemayalsowrite: Á T ¼Áþ C M C Áð11:18Þ Thepotentialenergyisnowgivenby PE¼SEþ 1 2 C M F 2 ¼ 1 2B C 1 Á 2 þ 1 2B C 1 M ðÁ T ÁÞ 2 ð11:19Þ andtheenergyreleaserateis G¼ @PE @a Á T ¼ C 1 ÁC 1 M ðÁ T ÁÞ B "# @A @a Á T þ 1 2B C 2 Á 2 dC da ¼ 1 2B C 2 Á 2 dC da ¼ 1 2B F 2 dC da ð11:20Þ Hence,asbefore,theenergyreleaseratedoesnotdependonthenatureofthe loadingsystem.Also,themeasuredvalueofGdoesnotdependonthecom- plianceoftheloadingsystem.However,theexperimentaldeterminationofG isfrequentlydonewithrigidloadingsystemsthatcorrespondtoC M ¼0. 11.6LINEARELASTICFRACTUREMECHANICS Thefundamentalsoflinearelasticfracturemechanics(LEFM)arepresented inthissection.Followingthederivationofthecrack-tipfields,thephysical basisforthecrackdrivingforceparametersispresentedalongwiththe conditionsrequiredfortheapplicationofLEFM.TheequivalenceofG andtheLEFMcrackingdrivingforce(denotedbyK)isalsodemonstrated. 11.6.1DerivationofCrack-TipFields Beforepresentingthederivationofthecrack-tipfields,itisimportantto noteherethattherearethreemodesofcrackgrowth.Theseareillustrated schematicallyinFig.11.8.ModeI[Fig.11.8(a)]isgenerallyreferredtoas the crack opening mode. It is often the most damaging of all the loading modes. Mode II [Fig. 11.8(b)] is the in-plane shear mode, while Mode III [Fig. 11.8(c)] corresponds to the out-of-plane shear mode. Each of the Copyright © 2003 Marcel Dekker, Inc. [...]... the crack face that are intersected by 458 lines from the center of the crack (Fig 11.17) Values of the CTOD measured at the onset of fracture instability correspond to the fracture toughness of a material Hence, the CTOD is often used to represent the fracture toughness of materials that exhibit significant plasticity prior to the onset of fracture instability Guidelines for CTOD testing are given in... guidelines for the measurement of critical values of J One of the most widely used codes is the ASTM E-813 code for JIc testing developed by the American Society for Testing and Materials, Conshohocken, PA This code requires the use of deeply cracked SENB or C-T specimens with initial precrack length-to-specimen width ratios of 0.5 In general, such specimens provide a unique measure of the JIc when the remaining... model of the small-strain behavior of real elastic–plastic materials under the monotonic loads being considered 2 The regions in which finite strain effects are important and the region in which microscopic processes occur must be contained within the region of the small-strain solution dominated by the singularity fields The first condition is critical in any application of the deformation theory of plasticity... monotonic loading, the applications of uniaxial loads to stationary cracks do provide a good framework for the use of the deformation theory of plasticity The second condition is somewhat analogous to the so-called smallscale yielding condition for linear elastic fracture mechanics It also provides a physical basis for the determination of the inner radius of the annular region of J dominance This is illustrated... to be applied The size of the microstructural process zone may correspond to the mean void spacing in the case of ductile dimpled fracture, or the grain size in the case of cleavage or intergranular fracture Copyright © 2003 Marcel Dekker, Inc FIGURE 11.20 Schematic of annular region of J dominance (From Hutchinson, 1983—reprinted with permission from the Technical University of Denmark.) Under small-scale... corresponding to $ 20À25% of the plastic zone size However, under large-scale yielding conditions, the region of J dominance is highly dependent on specimen configuration In cases where the entire uncracked ligament is completely engulfed by the plastic zone, the size of the region of J dominance may be as low as 1% of the uncracked ligament for a center-cracked tension specimen or 7% of the uncracked ligament... in part 1 Solution 1 We may solve the problem by considering the in-plane deflection of two ‘‘cantilever’’ on either side of the crack For crack length, a, and deflection, Á=2, on either side of the crack, we may write the following expression from beam theory: Á Pa 3 ¼ 2 3EI ð11:71Þ where I is the second moment of area If the cantilever has a thickness, B, and a height, b, then the second moment of. .. Two-Parameter J–Q Under the condition of large-scale yielding, KÀT theory cannot adequately define the stress field ahead of the crack tip O’Dowd and Shih (1991) have proposed a family of crack-tip fields that are characterized by a triaxility parameter The application of these fields has given rise to a two-parameter ðJÀQÞ theory for the characterization of the effects of constraint on cracktip fields As... is affected by the sizes of these process zones 11.6.5 Equivalence of G and K The relationship between G and K is derived in this section Consider the generic crack-tip stress profile for Mode I that is shown in Fig 11.14(a) The region of high stress concentration has strain energy stored over a distance ahead of the crack tip This strain energy is released when a small amount of crack advance occurs... crack-tip The shape and size of this zone depends strongly on the stress state (plane stress versus plane strain) and the stress intensity factor, K, under small-scale yielding conditions (Fig 11 .12) In general, however, the size of the plastic zone may be estimated from the boundary of the region in which the tensile yield stress, ys , is exceeded (Irwin, 1960) For the region ahead of the crack-tip in which . 11 FundamentalsofFractureMechanics 11.1INTRODUCTION Inthe17thcentury,thegreatscientistandpainter,LeonardodaVinci, performedsomestrengthmeasurementsonpianowireofdifferentlengths. Somewhatsurprisingly,hefoundthatthestrengthofpianowiredecreased withincreasinglengthofwire.Thislength-scaledependenceofstrengthwas notunderstooduntilthe20thcenturywhentheseriousstudyoffracturewas revisitedbyanumberofinvestigators.Duringthefirstquarterofthe20th century,Inglis(1913)showedthatnotchescanactasstressconcentrators. Griffith(1921)extendedtheworkofInglisbyderivinganexpressionforthe predictionofthebrittlestressinglass.Usingthermodynamicarguments, andtheconceptofnotchconcentrationfactorsfromInglis(1913),he obtainedaconditionforunstablecrackgrowthinbrittlematerialssuchas glass.However,Griffith’sworkneglectedthepotentiallysignificanteffects ofplasticity,whichwereconsideredinsubsequentworkbyOrowan(1950). AlthoughtheworkofGriffith(1921)andOrowan(1950)provided someinsightsintotheroleofcracksandplasticityinfracture,robustengi- neeringtoolsforthepredictionoffracturewereonlyproducedinthelate 1950sandearly1960safteranumberofwell-publicizedfailuresofshipsand aircraftinthe1940sandearlytomid-1950s.Someofthefailuresincluded thefractureoftheso-calledLibertyshipsinWorldWarII(Fig.11.1)and Copyright. 11 FundamentalsofFractureMechanics 11.1INTRODUCTION Inthe17thcentury,thegreatscientistandpainter,LeonardodaVinci, performedsomestrengthmeasurementsonpianowireofdifferentlengths. Somewhatsurprisingly,hefoundthatthestrengthofpianowiredecreased withincreasinglengthofwire.Thislength-scaledependenceofstrengthwas notunderstooduntilthe20thcenturywhentheseriousstudyoffracturewas revisitedbyanumberofinvestigators.Duringthefirstquarterofthe20th century,Inglis(1913)showedthatnotchescanactasstressconcentrators. Griffith(1921)extendedtheworkofInglisbyderivinganexpressionforthe predictionofthebrittlestressinglass.Usingthermodynamicarguments, andtheconceptofnotchconcentrationfactorsfromInglis(1913),he obtainedaconditionforunstablecrackgrowthinbrittlematerialssuchas glass.However,Griffith’sworkneglectedthepotentiallysignificanteffects ofplasticity,whichwereconsideredinsubsequentworkbyOrowan(1950). AlthoughtheworkofGriffith(1921)andOrowan(1950)provided someinsightsintotheroleofcracksandplasticityinfracture,robustengi- neeringtoolsforthepredictionoffracturewereonlyproducedinthelate 1950sandearly1960safteranumberofwell-publicizedfailuresofshipsand aircraftinthe1940sandearlytomid-1950s.Someofthefailuresincluded thefractureoftheso-calledLibertyshipsinWorldWarII(Fig.11.1)and Copyright. introduced alongwithtwo-parameterfractureconceptsfortheassessmentofcon- straint.Finally,therelativenewsubjectofinterfacialfracturemechanics is presented,alongwiththefundamentalsofdynamicfracturemechanics. 11.2FUNDAMENTALSOFFRACTUREMECHANICS It
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