14 FatigueofMaterials 14.1INTRODUCTION Fatigueistheresponseofamaterialtocyclicloadingbytheinitiationand propagationofcracks.Fatiguehasbeenestimatedtoaccountforupto80– 90%ofmechanicalfailuresinengineeringstructuresandcomponents (Illstonetal.,1979).Itis,therefore,notsurprisingthataconsiderable amountofresearchhasbeencarriedouttoinvestigatetheinitiationand propagationofcracksbyfatigue.Asummaryofpriorworkonfatiguecan befoundinacomprehensivetextbySuresh(1999).Thischapterwill,there- fore,presentonlyageneraloverviewofthesubject. Theearliestworkonfatiguewascarriedoutinthemiddleofthe19th century,followingtheadventoftheindustrialrevolution.Albert(1838) conductedaseriesoftestsonminingcables,whichwereobservedtofail afterbeingsubjectedtoloadsthatwerebelowthedesignloads.However, Wo ¨ hler(1858–1871)wasthefirsttocarryoutsystematicinvestigationsof fatigue.Heshowedthatfatiguelifewasnotdeterminedbythemaximum load,butbytheloadrange.WohlerproposedtheuseofSÀNcurvesof stressamplitude,S a (Fig.14.1),orstressrange,ÁS(Fig.14.1),versusthe numberofcyclestofailure,N f ,fordesignagainstfatigue.Suchdataarestill obtainedfrommachinesofthetypeshowninFig.14.2.Healsoidentifieda Copyright © 2003 Marcel Dekker, Inc. ‘‘fatigue limit’’ below which smooth specimens appeared to have an infinite fatigue life. Rankine (1843) of mechanical engineering fame (the Rankine cycle) noted the characteristic ‘‘brittle’’ appearance of material broken under repeated loading, and suggested that this type of failure was due to recrys- FIGURE 14.1 Basic definitions of stress parameters that are used in the char- acterization of fatigue cycles. (From Callister, 2000—reprinted with permis- sion of John Wiley & Sons.) Copyright © 2003 Marcel Dekker, Inc. tallization.Thegeneralopinionsoondevelopedaroundthisconcept,andit wasgenerallyacceptedthatbecausethesefailuresappearedtooccursud- denlyinpartsthathadfunctionedsatisfactorilyoveraperiodoftime,the materialsimplybecame‘‘tired’’ofcarryingrepeatedloads,andsudden fractureoccurredduetorecrystallization.Hence,thewordfatiguewas coined(fromthelatinword‘‘fatigare’’whichmeanstotire)todescribe suchfailures. ThismisunderstandingofthenatureoffatiguepersisteduntilEwing andHumfrey(1903)identifiedthestagesoffatiguecrackinitiationand propagationbytheformationofslipbands.Thesethickentonucleate microcracksthatcanpropagateunderfatigueloading.However,Ewing andHumfreydidnothavethemodelingframeworkwithinwhichthey couldanalyzefatiguecrackinitiationandpropagation.Also,asaresult ofanumberofwell-publicizedfailuresduetobrittlefracture(Smith, 1984),thesignificanceofthepre-existenceofcracksinmostengineering structuresbecamewidelyrecognized.Thisprovidedtheimpetusforfurther researchintothecausesoffatiguecrackgrowth. AsdiscussedearlierinChapter11,Irwin(1957)proposedtheuseof the stress intensity factor (SIF) as a parameter for characterizing the stress and strain distributions at the crack tip. The SIF was obtained using a representation of the crack-tip stresses, proposed initially by Westergaard (1939) for stresses in the vicinity of the crack tip. It was developed for brittle fracture applications, and was motivated by the growing demands for devel- opments in aerospace, pressure vessels, welded structures, and in particular from the U.S. Space Program. This led to the rapid development of fracture mechanics, which has since been applied to fatigue crack growth problems. Paris et al. (1961) were the first to recognize the correlation between fatigue crack growth rate, da/dN, and the stress intensity range, ÁK. Although the work of Paris et al. (1961) was rejected initially by many of FIGURE 14.2 Schematic of rotating bending test machine. (From Keyser 1973—reprinted with permission of Prentice-Hall, Inc.) Copyright © 2003 Marcel Dekker, Inc. the leading researchers of the period, it was soon widely accepted by a global audience of scientists and engineers. Paris and Erdogan (1963) later showed that da/dN can be related to ÁK through a simple power law expression. This relationship is the most widely used expression for the modeling of fatigue crack growth. In general, however, the relationshp between da/dN and ÁK is also affected by stress ratio, R ¼ K min =K max . The effects of stress ratio are parti- cularly apparent in the so-called near-threshold regime, and also at high SIF ranges. The differences in the near-threshold regime have been attributed largely to crack closure (Suresh and Ritchie, 1984a, 1984b), which was first discovered by Elber (1970) as a graduate student in Australia. The high crack growth rates at high ÁK values have also been shown to be due to the additional contributions from monotonic or ‘‘static’’ fracture modes (Ritchie and Knott, 1973). Given the success of the application of the SIF to the correlation of the growth of essentially long cracks, it is not surprising that attempts have been made to apply it to short cracks, where the scale of local plasticity often violates the continuum assumptions of linear elastic fracture mechanics (LEFM) that were made in the derivation of K by Irwin (1957). In most cases, anomalous growth short cracks have been shown to occur below the so-called long-crack threshold. The anomalous behavior of short cracks has been reviewed extensively, e.g., by Miller (1987), and has been attributed largely to the combined effects of microstructure and microtexture localized plasticity (Ritchie and Lankford, 1986). Various parameters have been pro- posed to characterize the stress–strain fields associated with short cracks. These include the fatigue limit, Coffin–Manson type expressions for low cycle fatigue (Coffin, 1954; Manson, 1954), and elastic–plastic fracture mechanics criteria such as ÁJ and the crack opening displacement (Ritchie and Lankford, 1986). Considerable progress has also been made in the understanding of fatigue crack initiation and propagation mechanisms. Although Ewing and Humfrey observed the separate stages of crack initiation by slip-band formation and crack propagation as early as in 1903, it was not until about 50 years later that Zappfe and Worden (1951) reported fractographs of striations associated with fatigue crack propagation. However, they did not recognize the one-to-one correspondence of striations with the number of cycles. This was first reported by Forsyth (1961), a year before Laird and Smith (1962) proposed the most widely accepted model of crack propaga- tion. Since then, a great deal of research has been carried out to investigate various aspects of fatigue. A summary of the results obtained from well- established prior research on the fatigue of materials is presented in this chapter. Copyright © 2003 Marcel Dekker, Inc. 14.2MICROMECHANISMSOFFATIGUECRACK INITIATION Microcrackstendtoinitiateinregionsofhighstressconcentrationsuchas thosearoundnotchesandinclusions.Theymayalsoinitiateinthecentral regionsofgrains,orinthegrainboundaries,evenwhennomacroscopic stressraisersarepresent.Ingeneral,however,microcracksinitiateasaresult ofslipprocesses(Wood,1958)duetostressorplasticstraincycling. Dislocationseitheremergeatthesurfaceorpileupagainstobstaclessuch asgrainboundaries,inclusions,andoxidefilms,toformslipbands,which werefirstobservedbyEwingandHumfrey(1903).Thompsonetal.(1956) latershowedthatiftheseslipbandsareremovedbyelectropolishing,they willreappearwhenfatiguingisrecommenced,andsotheyreferredtothem aspersistentslipbands(PSBs). Theresistancetotheinitiationofslipatthecentralportiondecreases withincreasinggrainsize,followingtheHall–Petchrelation(Hall,1951; Petch,1953).Theresistanceofthegrainboundaryregionscanalsobe weakenedinsoftprecipitatefreezones(PFZs)(MulvihillandBeevers, 1986)attheregionsofintersectionofgrainboundaries,e.g.,triplepoints (Miller,1987),byembrittlementduetograinboundarysegregation (Lewandowskietal.,1987),andalsobystresscorrosioneffects(Cottis, 1986).Hence,crackingcanoccurwithingrainsoratgrainboundaries. Theinitiationofmicrocracksmayalsobeinfluencedbyenvironment. LairdandSmith(1963)showedthatinitiationoffatiguecrackswasslower invacuumthaninair,andtheyattributedthislargelytotheeffectsofthe irreversibilityofslipinair. Fourmainstagesofcrackinitiationhavebeenidentified.They involve: 1.Localizedstrainhardeningorsofteningduetotheaccumulation ofslipstepsatthesurface.Thisoccursatsufficientlyhighalter- natingplanestrainamplitudes.AslipstepofoneBurgersvector iscreatedwhenadislocationemergesatthesurface.Sincedis- locationsemergeduringbothhalvesofeachfatiguecycle,slip stepscanaccumulateinalocalregion,andthisleadstosevere rougheningofthesurface. 2.Theformationofintrusionsandextrusions(Fig.14.3).Cottrell andHull(1957)havepostulatedthatthesecanbeformedwhen sequentialslipoccursontwointersectingslipplanes,asillu- stratedinFig.14.4.Slipoccursinthefirstslipsystemandthen in the second during the first half of the cycle, to give the inden- tation shown in Fig. 14.4(c). The slip systems may operate con- Copyright © 2003 Marcel Dekker, Inc. secutivelyorsimultaneouslyduringthereversecycletogiverise topairsofintrusionsandextrusions,asshowninFigs14.4(d) and14.4(e).Itisalsopossiblethatintrusionsandextrusionsmay formasaresultofadislocationavalanchealongparallelneigh- boringslipplanescontainingdislocationpile-upsofopposite signs,aspostulatedbyFineandRitchie(1979).Thisisillustrated inFig.14.5.Althoughitisunlikelythatintrusionsandextrusions form exactly by either of these mechanisms, they do illustrate the kind of slip processes that must be operative. 3. The formation of microcracks. This is often defined by the resol- ving power of the microscope or the resolution of the nondes- tructive inspection tool that is used. It is still not clear how intrusions and extrusions evolve into microcracks. These cracks often propagate initially along crystallographic planes of maxi- mum shear stress by Mode II (Forsyth Stage II) shear mechan- isms (Forsyth, 1961). Since the plasticity associated with the crack tips of these microcracks is often less than the controlling microstructural unit size, microstructural barriers, such as grain boundaries and dispersed precipitates may cause discontinuities in the crack growth. 4. The formation of macrocracks (usually larger than several grain sizes) as a result of microcrack coalescence or crack growth to a particular crack size where the crack begins to propagate by FIGURE 14.3 Formation of surface cracks by slip. Static slip forms unidirec- tional step: (a) optical microscope; (b) electron microscope. Fatigue slip by to- and-fro movements in slip band may form notch (c) or peak (d). (From Wood, 1958—reprinted with permission of Taylor & Francis Ltd.) Copyright © 2003 Marcel Dekker, Inc. Mode I (Forsyth Stage II) mechanisms (Forsyth, 1961), with the direction of crack propagation being perpendicular to the direc- tion of the principal axis. There is no universally accepted defini- tion of the transition from microcrack to macrocrack behavior, although a fatigue macrocrack is usually taken to be one that is sufficiently long to be characterized by LEFM. 14.3 MICROMECHANISMS OF FATIGUE CRACK PROPAGATION Various models of fatigue crack propagation have been proposed (Forsyth and Ryder, 1961; Laird and Smith, 1962; Tomkins, 1968; Neumann, 1969, 1974; Pelloux, 1969, 1970; Tomkins and Biggs, 1969; Kuo and Liu, 1976). However, none of these models has been universally accepted. It is also unlikely that any single model of fatigue crack growth can fully explain the range of crack extension mechanisms that are possible in different mate- rials over the wide range of stress levels that are encountered in practice. Nevertheless, the above models to provide useful insights into the kinds of processes that can occur at the crack tips during crack propagation by FIGURE 14.4 Cottrell–Hull model for the formation of intrusions and extru- sions. (From Cottrell and Hull, 1957—reprinted with permission from the Royal Society.) Copyright © 2003 Marcel Dekker, Inc. fatigue.Manyofthemarebasedonthealternatingshearrupturemechan- ismwhichwasfirstproposedbyOrowan(1949),andmostofthemassume partialirreversibilityofslipduetothetanglingofdislocationsandthe chemisorptionofenvironmentalspeciesonfreshlyexposedsurfacesatthe cracktip. OneoftheearliestmodelswasproposedbyForsythandRyder(1961). Itwasbasedonobservationsoffatiguecrackgrowthinaluminumalloys. Theysuggestedthatfatiguecrackextensionoccursasaresultofburstsof brittleandductilefracture(Fig.14.6)andthattheproportionofbrittleand ductilefractureinasituationdependsontheductilityofthematerial.They alsoproposedthatcrackgrowthcouldoccurinsomecasesbyvoidlinkage. Thesevoidsareformedduringtheforwardcyclearoundparticlesthatfrac- tureduringthepreviousreversecycle.Crackingthenoccursbythenecking downofinterveningmaterialuntilthevoidlinksupwiththecrack,as showninFigs.14.7. FIGURE14.5Paireddislocationpile-upsagainstobstacleonmetalsurface grow with cyclic straining until they reach a critical size at which an avalanche occurs to form intrusions and extrusions. (From Fine and Ritchie, 1979— reprinted with permission of ASM International.) FIGURE 14.6 Bursts of brittle fracture (A) and ductile fracture (B) along stria- tion profile. (From Forsyth and Ryder, 1961—reprinted with permission from Cranfield College of Engineers.) Copyright © 2003 Marcel Dekker, Inc. Laird and Smith (1962) and Laird (1967) proposed an alternative model based on the repetitive blunting and sharpening of the crack tip due to plastic flow. In this model, localized slip occurs on planes of max- imum shear oriented at $ 708 to the crack tip (Irwin, 1957; Williams, 1957) on the application of a tensile load. As the crack opens during the forward cycle, the crack tip opens up, Figs 14.8(a) and 14.8(b). Further straining results in the formation of ears [Fig. 14.8(c)], which they observed clearly at the peak tensile strain, and the broadening of the slip bands, Fig. 14.8(c). The crack tip is also blunted progressively [Figs 14.8(b) and 14.8(c)] as a FIGURE 14.7 Forsyth and Ryder model of crack extrusion by void linkage. (From Forsyth and Ryder, 1961—reprinted with permission of Metallurgica.) FIGURE 14.8 Schematic representation of fatigue crack advance by Laird and Smith’s plastic blunting model. (From Laird and Smith, 1962—reprinted with permission of Taylor and Francis Ltd. Copyright © 2003 Marcel Dekker, Inc. resultofplasticflow,whichisreversedonunloading,Fig.14.8(d).Thecrack facesarebroughttogetherasthecrackcloses,buttheadsorptionofparticles intheenvironmentatthecracktipontothefreshlyexposedsurfacespre- ventscompleterewelding,andhenceperfectreversibilityofslip.Also,the newlycreatedsurfacesbuckleasthecrackextendsbyafractureofthecrack openingdisplacement,duringthereversehalfofthecycle,Figs14.8(e)and 14.8(f).Thecorrespondingcrack-tipgeometriesobtainedoncompressing thespecimenduringthereversecycleareshowninFigs14.8(g–i). TomkinsandBiggs(1969)andTomkins(1968)haveproposedamodel thatissimilartoLairdandSmith’splasticbuntingmodel.Theysuggestthat newcracksurfacesareformedbyplasticdecohesiononavailableshear planes,atthelimitoftensilestraining.ThismodelalsoappliestoStageI growthwheretheyhypothesizethatslipwillonlyoccurononeofthetwo availableslipplanes.CrackextensionbythismodelisillustratedinFig.14.9 forStageIIfatiguepropagation. Pelloux(1969,1970)hasformulatedadifferentmodelbasedonalter- natingshear.Thebehavorofthecracktipissimulatedusingfullyplastic specimenscontainingsharpnotches(Fig.14.10)—thiscanbejustifiedwhen theplasticzoneisseveraltimesthesizeofthestriationspacing.Pelloux’s modelisillustratedinFig.14.11.Crackextensionoccursonintersectingslip planes as a result of alternating slip, which takes place sequentially or FIGURE 14.9 Plastic flow model of crack advance proposed by Tomkins and Biggs (1969). (Reprinted with permission of Taylor & Francis Ltd.) Copyright © 2003 Marcel Dekker, Inc. [...]... the fatigue limits of aging steels are typically $ (UTS)/2 in steels Copyright © 2003 Marcel Dekker, Inc FIGURE 14.14 SÀN fatigue curve Curves of type A are typical of mild steel and alloys which strainage, and curves of type B are typical of nonaging alloys (From Knott, 1973—reprinted with permission from Butterworth-Heinemann.) A great deal of work has been carried out on the use of SÀN curves, and... low-cycle fatigue is dependent on the plastic strain range (Coffin 1954; Manson, 1954) The amount of plastic strain imposed per cycle can be found from the hysteresis loop in the plot of stress versus strain over one cycle, as shown in Fig 14 .15 The effect of plastic range, Á"p , on the number of cycles to failure, Nf , is expressed by the Coffin–Manson relationship: Á"p Á Nf 1 ¼ C1 ð14:1Þ where 1 ð%... permission of ASM International.) Copyright © 2003 Marcel Dekker, Inc FIGURE 14.12 Neumann’s coarse slip model of crack advance (From Neumann, 1974—reprinted with permission of AGARD.) although it requires crack extension to occur as a result of irreversibility of slip, it does not include crack blunting and sharpening stages The ‘‘unzipping’’ model by Kuo and Liu (1976) is a simple variant of Pelloux’s... propagation regions The number of cycles for fatigue failure, Nf , is then regarded as the sum of the number of cycles for fatigue crack initiation, Ni , and the number of cycles for fatigue crack propagation, Np The SÀN curves can, therefore, be regarded as the sum of two curves (SÀNi and SÀNp ), as shown in Fig 14.18 The extent to which either process contributes to the total number of cycles to failure depends... discontinuous shear bands are formed once every hundred of cycles There is, therefore, not a one-to-one correspondence between the number of shear bands and the number of fatigue cycles Further details on the mechanisms of crack growth in polymers can be found in texts by Hertzberg and Manson (1980) and Suresh (1999) 14.8 FATIGUE OF BRITTLE SOLIDS 14.8.1 Initiation of Cracks For highly brittle solids with strong... significance of the various components of closure is elucidated for nearthreshold fatigue crack propagation, and the main methods of determining closure loads are also assessed 14.9.2 Plasticity-induced Closure Plasticity-induced closure (Elber, 1970, 1971) occurs as a result of residual plastic deformation in the wake of a propagating crack Clamping stresses, due to the elastic constraint of surrounding... shown in Fig 14.30(b) Also, the irreversible nature of crack-tip deformation and the possibility of slip step oxidation in most environments can lead to mismatch of fracture surface asperities and higher levels of roughness-induced closure (Suresh and Ritchie, 1982) In general, however, the level of roughness-induced closure depends on the height of the fracture surface asperities and the crack opening... strength materials where higher levels of plasticity-induced closure bring more asperities into contact to induce fretting associated with plasticity-induced and roughness-induced closure phenomena Evidence of oxide-induced closure in steels is often provided in the form of dark bands across the fracture surface These usually occur in the near-threshold regime, as shown in Fig 14.31 Measurements of the... clearly to the fracture toughness, KIc or Kc 14.7 FATIGUE OF POLYMERS A significant amount of work has been done on the fatigue behavior of plastics Most of the important results have been summarized in monograph by Hertzberg and Manson (1980), and the interested reader is referred to their book for further details Although the fatigue behavior of polymers exhibits several characteristics that are similar... the combined effects of mean and alternating stresses Gerber (1874) and Goodman (1899) proposed relationships of the form (Figure 14.17): Æ ¼ Æ0 1 À n ! m t ð14:3Þ where is the fatigue limit at a mean stress of m , 0 is the fatigue limit for m ¼ 0, t is the tensile strength of the material, and the exponent n ¼ 1 in the Goodman expression, and n ¼ 2 in Gerber’s versionof Eq (14.3) The resulting . 14 FatigueofMaterials 14.1INTRODUCTION Fatigueistheresponseofamaterialtocyclicloadingbytheinitiationand propagationofcracks.Fatiguehasbeenestimatedtoaccountforupto80– 90%ofmechanicalfailuresinengineeringstructuresandcomponents (Illstonetal.,1979).Itis,therefore,notsurprisingthataconsiderable amountofresearchhasbeencarriedouttoinvestigatetheinitiationand propagationofcracksbyfatigue.Asummaryofpriorworkonfatiguecan befoundinacomprehensivetextbySuresh(1999).Thischapterwill,there- fore,presentonlyageneraloverviewofthesubject. Theearliestworkonfatiguewascarriedoutinthemiddleofthe19th century,followingtheadventoftheindustrialrevolution.Albert(1838) conductedaseriesoftestsonminingcables,whichwereobservedtofail afterbeingsubjectedtoloadsthatwerebelowthedesignloads.However, Wo ¨ hler(1858–1871)wasthefirsttocarryoutsystematicinvestigationsof fatigue.Heshowedthatfatiguelifewasnotdeterminedbythemaximum load,butbytheloadrange.WohlerproposedtheuseofSÀNcurvesof stressamplitude,S a (Fig.14.1),orstressrange,ÁS(Fig.14.1),versusthe numberofcyclestofailure,N f ,fordesignagainstfatigue.Suchdataarestill obtainedfrommachinesofthetypeshowninFig.14.2.Healsoidentifieda Copyright. Inc. fatigue.Manyofthemarebasedonthealternatingshearrupturemechan- ismwhichwasfirstproposedbyOrowan(1949),andmostofthemassume partialirreversibilityofslipduetothetanglingofdislocationsandthe chemisorptionofenvironmentalspeciesonfreshlyexposedsurfacesatthe cracktip. OneoftheearliestmodelswasproposedbyForsythandRyder(1961). Itwasbasedonobservationsoffatiguecrackgrowthinaluminumalloys. Theysuggestedthatfatiguecrackextensionoccursasaresultofburstsof brittleandductilefracture(Fig.14.6)andthattheproportionofbrittleand ductilefractureinasituationdependsontheductilityofthematerial.They alsoproposedthatcrackgrowthcouldoccurinsomecasesbyvoidlinkage. Thesevoidsareformedduringtheforwardcyclearoundparticlesthatfrac- tureduringthepreviousreversecycle.Crackingthenoccursbythenecking downofinterveningmaterialuntilthevoidlinksupwiththecrack,as showninFigs.14.7. FIGURE14.5Paireddislocationpile-upsagainstobstacleonmetalsurface grow. success of the application of the SIF to the correlation of the growth of essentially long cracks, it is not surprising that attempts have been made to apply it to short cracks, where the scale of