A Microstructure-Based Fatigue-Crack-Initiation Model.pdf

A Microstructure-Based Fatigue-Crack-Initiation Model

KWAI S. CHAN

This article presents results on the development of a microstructure-based fatigue-crack-initiationmodel which includes explicit crack-size and microstructure-scale parameters. The current status ofmicrostructure-based fatigue-crack-initiation models is briefly reviewed first. Tanaka and Mura’smodels[1,2] for crack initiation at slipbands and inclusions are then extended to include crack size andrelevant microstructural parameters in the response equations. The microstructure-based model forcrack initiation at slipbands is applied to predicting the crack size at initiation, small-crack behavior,and notch fatigue in structural alloys. The calculated results are compared against the experimentaldata for steels and Al-, Ti-, and Ni-based alloys from the literature to assess the range of predictabilityand accuracy of the fatigue-crack-initiation model. The applicability of the proposed model for treatingvariability in fatigue-crack-initiation life due to variations in the microstructure is discussed.

I. INTRODUCTION for the development of probabilistic life-prediction methods,since the tail ends of the defect distribution, which are oftenFATIGUE in engineering structures involves crack-initi-related to the microstructural-size distribution, mostly influ-ation and growth processes. Current life-predictionence fatigue life.approaches for military gas turbines address both crack initi-

Prior reviews established the important roles of micro-ation and propagation lives. Typically, a gas-turbine compo-structure in fatigue-crack initiation and growth in metals.nent is designed to have (1) a minimum low-cycle fatigueThe need to introduce a microstructural-size parameter in a(LCF) crack-initiation life exceeding the total specified ser-fatigue-crack-growth model was identified in a review articlevice life and (2) a crack-propagation life from an assumedby Bailon and Antolovich,[4] which compared fatigue-crack-initial inspection that is twice the number of service cyclesgrowth models against experimental data available at thatbetween inspections.[3] According to Cowles,[3] the currenttime. A microstructure-based fatigue-crack-growth modellife-prediction systems are expensive to establish and sub-for large cracks was developed by the author in an earlierstantiate, because large experimental databases are required.investigation.[5] Applications of the fatigue-crack-growthThe current life-prediction systems may be improved bymodel in a probabilistic life-prediction framework have alsodevelopment of better crack-initiation-life models, probabi-been demonstrated for steels and a Ti alloy.[6] Microstructure-listic treatment of the variability of crack-initiation life, andbased fatigue-crack-initiation models have been developedenhanced local notch analysis capability.[3] One of the limita-by Tanaka and Mura on the basis of the accumulation oftions of current crack-initiation-life models is that they aredislocation dipoles generated by irreversible slip on slip-incapable of predicting the crack size at fatigue-crack initia-bands.[1,2] These models treat slipband crack initiation intion. Consequently, an initial crack size must be assumed infavorably oriented surface grains, often known as stage Ithe prediction of crack-growth life.fatigue, as well as crack initiation at inclusions by slipRecent advances in materials processing have facilitatedimpingement and dislocation-dipole accumulation. With onethe design and manufacture of components with tailoredexception,[7] these microstructure-based crack-initiationmicrostructures at desired locations. For example, a coarse-models have rarely been applied to structural alloys. Recentgrained microstructure may be placed in regions where creepgeneralization of the dislocation-dipole model by Venkatara-resistance is desired, while a fine-grained microstructureman et al.[8,9] led to a stress–initiation life relation thatmay be introduced in regions where fatigue-crack-initiationappears to predict a grain-size dependence that is contradic-resistance is needed. This design approach requires precisetory to the models proposed by Tanaka and Mura.[1,2] Thus,control of the microstructure during processing as well asthere exists a need to evaluate the various microstructure/detailed knowledge of the microstructure/property relation-fatigue-life relationships against experimental data. In addi-ship, so that optimization of microstructures to achieve ation, there is also a need for a microstructure-based fatigue-balance of material properties can be executed in an efficientcrack-initiation model that explicitly incorporates crack size,and effective manner. In particular, the time to materialmicrostructural size, and other relevant material parameters.development can be significantly reduced if computer-based

This article presents results on an investigation whose aimmethods can be implemented to reduce the amount of itera-was to develop microstructure-based fatigue-crack-initiationtions required to develop the desired microstructures andmodels that include explicit crack-size and microstructure-properties for the intended applications. For computer-basedscale parameters. Since fatigue is a wide topic, the presentdesign of damage-tolerant materials and structures, micro-article focuses solely on the microstructural-size effects andstructure-based models of fatigue-crack initiation andthe crack size at fatigue-crack initiation at ambient tempera-growth are required. These models would also be essentialture under uniaxial cyclic loading. While they are important,the roles of temperature, frequency, multiaxial loading, andenvironment will not be addressed. In this article, the currentKWAI S. CHAN, Institute Scientist, is with the Southwest Researchstatus of microstructure-based fatigue-crack-initiation mod-Institute, San Antonio, TX 78238. Contact e-mail: [email protected]

Manuscript submitted November 9, 2001. els is briefly reviewed. Following Tanaka and Mura’s

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 34A, JANUARY 2003—43

approach,[1,2] fatigue-crack initiation is modeled by consider- was obtained by relating the slip offsets in the PSBs tothe applied-strain amplitude. Statistical considerations ofing the accumulation of dislocation dipoles generated on

slipbands during cyclic loading. Tanaka and Mura’s models plastic-strain accumulation in a localized slipband led to avalue of 0.5 for a.[9] In an earlier article,[19] Saxena andfor crack initiation at slipbands and inclusions[1,2] are

extended to include crack size and relevant microstructural Antolovich demonstrated that the parameter a in Cu-Alalloys varied from 0.5 to 1; this variation was related to theparameters in the response equations. The microstructure-

based model is then applied to treating fatigue-crack initia- stacking-fault energy and to the slip morphology. Thus, thereis experimental evidence and a general consensus that thetion and the growth of small cracks in structural alloys such

as steels and Al-, Ti-, TiAl-, and Ni-based alloys. Stress– value of a ranges from 0.5 to 1,[9] depending on the plastic-strain-localization process. Although it originated from con-initiation life (S-Ni) and fatigue-crack-growth (dc/dN )

curves are computed as functions of crack length and micro- sideration of PSB formation by irreversible random slip, Eq.[3] by itself does not provide any insight into the role ofstructural-scale sizes. These results are compared against

experimental data from the literature to assess the range of microstructure in the crack-initiation process.More recent fatigue-crack-initiation models incorporatepredictability and accuracy of the fatigue-crack-initiation

models, as well as the feasibility of their implementation microstructural parameters in a direct and explicit manner.In a series of three articles, Tanaka and Mura proposedinto the current damage-tolerance life-prediction framework.

Applicability of the models for treating the variability in dislocation models for treating fatigue-crack initiation atslipbands, inclusions, and notches.[1,2,20]fatigue-crack-initiation life due to variations in the micro-

structure will also be discussed. Tanaka and Mura envisioned that fatigue-crack initiationoccurs by the accumulation of dislocation dipoles duringstrain cycling. For crack initiation along a slipband, the

II. REVIEW OF EXISTING FATIGUE-CRACK- dislocation-dipole accumulation model is given by[1]

INITIATION MODELS

Current fatigue-crack-initiation models are formulated (Dt 2 2k)N 1/2i 5 F8mWs

pd G1/2

[4]based on either stress or strain. For uniaxial cyclic loading,fatigue-crack initiation in metals can be described in terms

where Dt is the shear-stress range, k is the friction stress ofof the stress-life (S-Nf) approach via the relation[10]

dislocation, m is the shear modulus, d is the grain size, andWs is the specific fracture energy per unit area along thesa 5 s 8f N a

f [1]slipband. For crack initiation at a grain boundary, Eq. [4]

where sa is the stress amplitude, s 8f is the fatigue-strength still applies, but the term Ws is replaced by the specificcoefficient, Nf is the number of cycles to failure, and a is fracture energy per unit area of grain boundary (WG). Simi-the fatigue-strength exponent. Failure is usually defined as larly, the criterion for crack initiation by inclusion fracturefatigue fracture of the specimens, but can also be defined due to impinging slip and dislocation pileup is given by[2]

as initiation of a fatigue crack to a certain length or depth.In the strain-life approach, the plastic strain range (D«p) is

(Dt 2 2k)N 1/2i 5 F4m (m 1 m 8)WI

pm 8j G1/2

[5]related to the cycles to failure according to the Coffin–Manson relation, given by[11,12]

where m 8 is the shear modulus of the inclusion, j is theinclusion size, and WI is the specific fracture energy per unitD«p

25 « 8f N b

f [2]area of inclusion. Other forms of the model for treatingcrack initiation at debonded inclusions and notches are alsowhere « 8f is the fatigue-ductility coefficient, and b is theavailable in the literature.[2] Several modifications of thefatigue-ductility exponent. Both phenomenological modelsbasic model have also been made by Mura and co-workers,(stress-life and strain-life equations) are well known, butwhich are described in a series of recent articles.[8,9,21–24]

neither contain parameters that describe the role of micro-The crack-initiation model by developed Venkataraman etstructure in the fatigue-crack-initiation process or theal.[8,9] incorporates a surface-energy term (gs) and a slip-fatigue life.irreversibility factor (e), which has a value between 0 and 1.Fatigue-crack initiation is often associated with the forma-The stress-life relation obtained by Venkataraman et al.[9] istion of persistent slipbands (PSBs).[13–17] These PSBs are

produced by random irreversible slip on surface grains that(Dt 2 2k)N a

i 5 0.37 1mdeh21 gs

md21/2

[6]leads to surface roughening, the formation of extrusions andintrusions, and eventually to fatigue-crack initiation withinthe PSBs. A crack-nucleation criterion based on the forma- which predicts a grain-size dependence that is contradictorytion of a critical notch depth by random slip in PSBs was to Eq. [4] and experimental data.proposed by Cheng and Laird.[18] This criterion, which is The scaling law for crack initiation proposed by Chan[25]

essentially identical to the Coffin–Manson relation,[11,12] has is an extension of the Coffin–Manson LCF equation tothe form given by[18]

include a microstructure-size parameter. The modifiedstrain-life equation gives a grain-size dependence of fatigue-Dgp

2N a

i 5 C8 [3] life initiation that is similar to that described in Eq. [4]. Themodel of Chang et al.[26] is intended to treat fatigue-crackinitiation at inclusions as the result of slip impingement andfor metals subjected to a plastic-shear-strain range (Dgp).

The number of fatigue cycles to crack initiation is Ni , and dislocation pileups. The form of the model by Chang etal.[26] is similar to Eq. [5]. The statistical model by Ihara andC8 is a constant. The value of the exponent a is 0.78, which

44—VOLUME 34A, JANUARY 2003 METALLURGICAL AND MATERIALS TRANSACTIONS A

Tanaka[27] considers the accumulation of stochastic damagewithin the dislocation-cell structure formed during thefatigue process.

Recently, Harvey et al.[28] proposed a fatigue-crack initia-tion based on the deepening of slip steps into a fatigue crackduring strain cycling. Slip-height displacements on the orderof nanometers were measured using atomic-force micros-copy. Fatigue-crack initiation was assumed to occur when thecumulative slip-height displacement exceeded a thresholdcrack-opening displacement. According to this approach, thenumber of cycles to fatigue-crack initiation is determinedby the large-crack fatigue-crack-growth threshold (DKth),according to[28]

Ni 5DK2

th

4sysEfD«phs[7]

Fig. 1—A schematic showing the accumulation of dislocation dipoles gen-erated by irreversible slip during fatigue loading. The coalescence of thewhere E is the Young’s modulus, sys is the yield stress, D«pdislocation dipoles was postulated by Tanaka and Mura[1] as the processis the plastic-strain range, f is the fraction of plastic-strainthat a fatigue crack initiates along a planar slipband.

range that produces the surface displacement, and hs is theslip spacing.

The review indicated that only a few existing fatigue-crack-initiation models contain microstructural-size parame- stored strain energy (DU ) of dislocations per unit thickness isters. None of the existing fatigue-crack-initiation models given by[1]

contain crack length in the formulation. As a result, noneof the crack-initiation models can be used to predict the size DU 5

p (1 2 v)(Dt 2 2k)2

m[8]

of a fatigue crack at initiation. For models based on therandom irreversible slip process,[18,19,28] there is little distinc- during stress cycling between t1 and t2, such that Dt 5 t1tion between the crack-initiation and crack-growth proc- 2 t2. The number of fatigue cycles to crack initiation wasesses, since both involve deepening of the surface-slip steps. then obtained by equating the stored energy in the dislocationFor these models, fatigue cracks initiate early at minute dipoles to the specific fracture energy (Ws) per unit area.sizes and propagate to greater lengths at subsequent fatigue The energy balance leads one to[1]

loading. The advance of crack-detection techniques such as2NiDU 5 4dWs [9]atomic-force microscopy makes it feasible to measure crack

sizes as small as 100 nm or less.[28] However, detection ofwhich givescracks in this range is difficult and may not be practical or

necessary. The crack size at initiation is uncertain in most(Dt 2 2k)2 Ni 5

4mWs

p (1 2 v)d[10]structural alloys, and the ability to predict its occurrence

has not been established. Thus, there exists a need for awhen Eq. [8] is substituted into Eq. [9]. Equation [10], firstquantitative definition of crack initiation as well as a method-derived by Tanaka and Mura, describes, correctly, the inverseology that can predict the cycles to initiation of a fatiguerelationship between fatigue-crack-initiation life and thecrack of a given size.grain size (d ). The crack size, however, is arbitrary and notpredicted by this formulation.

III. FORMULATION OF CRACK SIZE–BASED The crack size can be incorporated into Eq. [10] by consid-FATIGUE-CRACK-INITIATION MODELS ering the energy of the fatigue-crack formation. According

to Mura and Nakasone,[23] the Gibbs free-energy changeThe formulation of crack size–based fatigue-crack-initia-(DG) associated with the nucleation of a fatigue crack fromtion models is described in this section. The dislocation-a double pileup of dislocation dipoles is[23]

dipole model of Tanaka and Mura[1] will be extended totreat crack initiation along the slipband. Subsequently, the DG 5 2We 2 Wm 1 2cgs [11]approach will be utilized to derive the model for treating

where We is the elastic strain energy stored in dislocations,crack initiation at inclusions and notches. The essence ofWm is the mechanical energy released due to the openingthe fatigue-crack-initiation process by the dislocation-dipoleup of a crack, c is the half-length (or depth) of the incipientmechanism, operating in a surface grain, is depicted in Figurefatigue crack, and gs is the surface energy of the crack.1. During fatigue loading, irreversible slip occurs in a favor-A detailed description of the energetics of fatigue-crackably oriented surface grain, leading to dislocation motioninitiation, presented in the Appendix, indicates that the Gibbson a slip plane (I) and dislocation pileups at grain boundaries.free-energy change is a function of the fatigue cycle, asUpon unloading, dislocations with opposite signs are acti-given by[23]

vated on an adjacent plane (II), producing reverse slip andthe formation of vacancy and interstitial dislocation dipoles DG 5 2z1N 2 2 z2N 2 1 2z3Ngs [12]at the ends of the double pileup. Representing the dislocationpileups in terms of continuum (i.e., continuously distributed) where z1, z2, and z3 are given by Eqs. [A7], [A8], and [A9],

respectively. The first term (z1N 2) is the stored energy ofdislocations, Tanaka and Mura showed that the change in the


N 5 0, crack initiation can also be defined on the basis ofDG # 0, and the resulting fatigue cycle at crack initiationwould be twice as large as that predicted by the instabilitycriterion (DG/N 5 0). The instability criterion is preferredhere because it yields a more conservative result. It shouldalso be noted that most of the work done by the external(a)load is expected to spent by irreversible slip and is dissipatedas heat. Only a small portion of the external work is storedin the dislocation substructure.

The approach of Mura and Nakasone[23] is based on theassumption that all the dislocation dipoles in the slipbandscontribute to the formation of the fatigue crack. The numberof dislocation dipoles (neq) in a single slipband is given by[8]

neq 5 0.051 dbh2!Weq

m[13]

where h is the slipband width and Weq is the strain energystored in the dislocation dipoles of a single or an equivalentslipband. In general, not all dislocation dipoles contributeto the formation of the fatigue crack, at least not immediatelyat crack initiation.[8] Under this circumstance, it is moreappropriate to consider fatigue-crack initiation as occurringonly at a portion of the slipband, as shown in Figure 2. Thesurface energy required to create this crack would be 2cgs ,(b)and the strain energy within the incipient crack that would

Fig. 2—Energetics of fatigue crack formation: (a) the formation of a fatigue be released to form the crack is cWeq/d. The maximum incrack of length (2c) or crack depth (c) at the tip of a planar slipband, and the Gibbs free energy (Eq. [12]) can be postulated to occur(b) the free-energy change associated with the crack initiation process.

when the condition given by[9]

2cgs 5cWeq

d[14]

the dislocation dipoles, the second term (z2N 2) is the elastic-strain-energy release rate (Eq. [A3]) due to the opening of is satisfied, leading tothe incipient crack during the initiation process, and the third

Weq 5 2dgs [15]term (2z3Ngs) is the surface energy associated with the crack-formation process. Equation [12] reduces to the theory of at incipient crack initiation. The crack length (c) at crackbrittle fracture[69] or to the theory of elastic/plastic frac- initiation is given by the number of dislocations (nc) thatture[70,71] subject to small-scale yielding under monotonic contribute to crack formation, according to[9]

loading when the stored energy of dislocation (first term) isc 5 ncb [16]ignored and N is taken to be unity for monotonic loading.

The free-energy change described by Eq. [12] is shown where b is the magnitude of the Burgers vector. The valueschematically as a function of fatigue cycle in Figure 2. of nc is obtained from Eq. [13] by replacing Weq by cWeq/d,Initially, the free-energy change is positive (DG . 0), indi- since only that amount of strain-energy density is required tocating that there is an energy barrier for crack initiation. The overcome the surface energy. This substitution leads one to[9]

free-energy change reaches a maximum where DG/N 50, which indicates that the dislocation-dipole structure and

nc 5 0.051 dbh2!cWeq

dm[17]the incipient crack are energetically in quasi equilibrium.

Beyond the maximum, DG/N , 0 and the dislocation-dipole structure is unstable with respect to crack formation. which can be combined with Eq. [16] to giveMura and Nakasone[23] postulated that the onset of crackinitiation occurs when the Gibbs free-energy change reaches c 5 0.0051d

h22

1gs

m2 [18]a maximum, as described by[23]

at fatigue-crack initiation. Imposing the condition that gs 5DGN

5 0 Ws leads one to

and the number of cycles to reach the maximum free-energy(Dt 2 2k)N 1/2

i 5 F 8m 2

lp (1 2 v)G1/2

1hd21c

d21/2

[19]change is taken to be the number of cycles to fatigue-crackinitiation. At DG/N 5 0, the free-energy change is stillpositive (DG . 0) and crack initiation is not spontaneous, as the criterion for the initiation of a fatigue crack of length

2c or depth c. For fatigue-crack initiation in polycrystallinebut must overcome the surface-energy barrier. The energybarrier can easily be provided by the stored strain energy materials, the applied-stress range (Ds ) can be related to

the shear-stress range (Dt ) on the slip plane through theof the dislocation dipoles, since the dipole structure is unsta-ble with respect to the crack configuration. Instead of DG/ Taylor factor (M ), leading to


(Ds 2 2Mk)N ai 5 F 8M2m 2

lp (1 2 v)G1/2

1hd21c

d21/2

[20]

when the exponent of Ni has been generalized to a, where0 , a # 1. An exponent not limited to 1/2 is necessary,because it provides better agreement with experimental data.Furthermore, experimental evidence and statistical simula-tion of fatigue-crack initiation by irreversible slip indicatedthat the value of a is not a constant, but depends on thestacking-fault energy[19] and on the degree of slip irre-versibility.[9,18,19]

Using a similar procedure, the governing equation forfatigue-crack initiation by slip impingement at an inclusioncan be derived. In this case, the surface energy for slipbandcracking in Eq. [18] is replaced by the corresponding surface

(a)energy for creating two fracture surfaces in an inclusion (gi).The resulting equation is then combined with Eq. [5] to give

(Ds 2 2Mk)N ai [21]

5 F8(m 1 m 8)lm 8

G1/2 F Mmh2

d(h 1 d )GFcjG

1/2

for fatigue-crack initiation at inclusions.

IV. STRESS–INITIATION LIFECALCULATIONS

The applications of the proposed model (Eq. [20]) forcomputing the stress–initiation life curves of a number ofstructural alloys are demonstrated in this section. Materialconstants in the model include the shear modulus (m ), Pois-

(b)son’s ratio (n ), and grain size (d ). The term 2Mk representsthe fatigue limit below which fatigue-crack initiation does Fig. 3—Fatigue-life curves for PM Astroloy: (a) plot of Ds -Dse used to

determine the material constants in the fatigue crack initiation model, and (b)not occur. These parameters and the fatigue limit were evalu-computed stress (S)–initiation life (Ni) compared with experimental data.[29]ated from available experimental data in the literature. The

Taylor factor for the optimally oriented grain is taken to be2, and the parameter l has a universal value of 0.005 basedon the investigation by Venkataraman et al.[9] The half-crack initiate a crack length of one-half of the grain size are com-

paratively short. Near the fatigue limit, most of the fatiguelength or crack depth (ci) at final failure is taken to be about1000 mm, unless an experimental value is available. The life is spent in the initiation of a grain-sized crack.

The number of cycles to crack initiation is available forslipband width (h) is generally not available, and it is usedas a fitting parameter in the stress-life calculations. a number of steels. Applications of the proposed fatigue-

crack-initiation model to these steels are presented in FigureTo determine the material constants, the fatigue limit wasfirst evaluated from the experimental data. A log-log plot 5 for 4340[31] and three other low-carbon (LC) steels.[32] The

finite grain size for the 4340 steel is about 1 mm,[4] whileof Ds - Dse vs Nf was then constructed to obtain the fatigue-strength exponent from the slope of the linear plot. This is it is 7.8, 20.5, and 55 mm for LC steels A through C,

respectively. In all cases, the experimental data at failureillustrated in Figure 3(a) for the PM Astroloy alloy.[29] Afterspecifying a final crack depth, the slipband width was evalu- were used to determine the model constants. Using this set

of material constants, the fatigue-crack-initiation the modelated by fitting the crack-initiation model to the experimentaldata. Once h was determined, S-Ni curves were calculated was used to compute the S-Ni response for other crack sizes.

Figure 5(a) shows that the agreement between the calculatedfor various crack depths. The calculated curves of PMAstroloy for four different ci values are shown in Figure 3(b). curve and initiation data for 38 mm is good at high stresses.

It is not as good, but reasonable, near the fatigue limit. ThereA summary of the material constants for PM Astroloy[29] ispresented in Table I, together with those for other are no experimental data available for crack sizes less than

38 mm. Good agreement is also obtained for LC steels Aalloys.[30–39]

Figures 4(a) and (b) compare the calculated S-Ni curves through C, as shown in Figures 5(b) through (d), respectively.In all three cases, the crack lengths at crack initiation wereagainst experimental data for PM Astroloy[29] and wrought

IN100[30] at 600 8C to 650 8C. The experimental data corres- about 20 to 90 mm.The effect of grain size on fatigue-crack initiation is bestponded to specimen fracture, which was simulated by crack

lengths of about 800 to 1000 mm. For both alloys, the cycles illustrated by the results shown in Figure 6(a) for Ti-6Al-4V[33] and in Figure 6(b) for lamellar TiAl alloys.[34,35,36]to crack initiation at a given stress range are reduced with

decreasing crack lengths. At high stresses, the cycles to The Ti-6Al-4V alloy was heat treated to become a bimodal


Table I. Summary of Material Parameters for Various Structural Alloys

Fatigue LimitGrain Size Dsc 5 2 Mk, Poisson’s Shear Modulus Fatigue Strength Slipband Width

Materials T, 8C d, mm R MPa Ratio m, MPa Exponent, a h, mm

PM Astroloy[29] 25 20 '0 500 0.3 8.01 3 104 0.54 0.15650 20 '0 580 0.3 6.72 3 104 0.44 0.05

IN100[30] 600 20 * 600 0.3 6.72 3 104 0.50 0.034340 steels[31] 25 1 21 600 0.3 7.76 3 104 0.50 1.5 3 1023

LC steels A[32] 25 7.8 21 476 0.3 7.76 3 104 0.783 0.3LC steels B[32] 25 20.5 21 340 0.3 7.76 3 104 0.616 0.4LC steels C[32] 25 55 21 300 0.3 7.76 3 104 0.664 3Ti-6Al-4V[33] 25 6 0.1 335 0.3 4.4 3 104 0.5 0.03Lamellar TiAl[34,35,36] 25, 650 150 0.1 310 0.3 4.4 3 104 0.2 8 3 1023

X7075[37] 25 30 21 100 0.3 2.58 3 104 0.225 1.2 3 1022

220 21 100 0.3 2.58 3 104 0.225 1.2 3 1021

2024 Al[37] 25 50 21 110 0.3 2.58 3 104 0.308 9 3 1022

X2024 Al[37] 25 50 21 70 0.3 2.58 3 104 0.197 1.9 3 1022

ULTIMET[38] 25 80 0.05 437 0.3 8.9 3 104 0.351 7.5 3 1022

Cu[39] 25 25 21 75 0.3 4.84 3 104 0.63 5

*Strain-controlled tests at zero minimum strain.

decrease in fatigue life, even though there is considerablescatter near the fatigue limit. In contrast, the S-Ni curve forlamellar TiAl alloys is relatively flat. The fatigue strengthis not very sensitive to the number of cycles to crack initia-tion or the fatigue life. The calculated S-Ni curve for thegrain-sized crack (ci 5 75 mm) is similar to that for 500mm. Thus, fatigue failure in the lamellar alloys is imminentonce a grain-sized crack has been initiated.

Grain size affects the S-Ni curves of X-7075 Al[37] in adifferent way. As illustrated in Figure 7(a), an increase in thegrain size lowers the fatigue strength of X7075-Al relativelyuniformly over the 104 to 107 cycles regime, by shifting theentire S-Ni curve to a lower stress. This behavior is faithfullyreproduced by the crack-initiation model (Figure 7(a)) byallowing the fatigue limit to vary with the grain size. Nonuni-

(a) form slip has been shown to lower the fatigue strength in2024-Al[37] (Figure 7(b)). The crack-initiation model is capa-ble of simulating this aspect of the fatigue process throughthe slipband-width parameter. Localized slip results in asmaller slipband-width, while uniform slip would lead to alarge slipband width.

Application of the crack-initiation model to ULTIMET,[38]

a Co-based alloy, is presented in Figure 8(a). The grain sizeof the alloy was 80 mm, and the size of just-initiated slipbandcracks was on the order of 100 to 200 mm. The calculatedS-Ni curves are in agreement with the experimental data(Figure 8(a)). Comparison of the model calculation andexperimental data of Cu[39] is presented in Figure 8(b). Themodel simulates the initiation of grain-sized microcracksaccurately, but the computed curve for specimen failure isnot as good. In particular, the model underpredicted thefatigue limit.

(b)

Fig. 4—Comparisons of computed and measured S-Ni curves for Ni alloysat 600 8C to 650 8C: (a) PM Astroloy[29] and (b) IN100.[30]

V. CRACK-SIZE PREDICTIONS

One of the important features of the proposed model isthe ability to predict the crack size at fatigue-crack initia-microstructure of primary a grains and lamellar a 1 b

colonies with an average grain size of 6 mm. The S-Ni curve tion. The crack length at initiation can be computed basedon two methods, utilizing either Eq. [18] or [20]. The useis typical, exhibiting an increase in fatigue strength with a


(a) (b)

(c) (d )

Fig. 5—Measured S-Ni data compared against model calculations for steels at ambient temperature: (a) 4340 steel,[31] (b) LC steel A,[32] (c) LC steel B,[32]

and (d ) LC steel C.[32]

of Eq. [20] for computing the crack size is demonstrated required is 1 3 105 J/m2, as shown in Figure 9. Thus, it isin Figures 3 through 8 for selected alloys. The validity of imperative that the proper surface energy, one that includesEq. [18] for computing the fatigue-crack size at initiation the plastic dissipation in the slipband, be used in the crack-is evaluated in this section. The evaluation involved a size calculation when Eq. [18] is used. The surface energyreview of the experimental data for fatigue-crack size at for cleavage fracture in metals would underpredict the crackinitiation. A summary of the experimental data[31,32,38,40–53]

size at initiation considerably.for the crack size at initiation reported in the literature is The fatigue-crack size detected at initiation for otherpresented in Table II. alloys is compared against the model calculations. From

Figure 9 shows the measured values of 4340 steels[31]Figure 10, it appears that experimental data form roughly

and LC steels[32] as a function of grain size. Despite the two scatter bands, corresponding to two sets of h and gslarge scatter, the crack size is seen to increase with the values. These correlations might be fortuitous, since it isgrain size, according to a power law with an exponent of obvious from Eq. [18] that the crack size depends on h,about 2. The theoretical crack size was computed based on gs , and the shear modulus. The results underscore the needEq. [18] using a slipband with of 0.4 mm and a surface

for more accurate values for h and gs if Eq. [18] is usedenergy of 2 J/m2. This particular h value was obtained fromto predict the fatigue-crack size at initiation.the S-Ni calculations described earlier, while the gs value

The values of h are expected to range from the slip-of 2 J/m2 is typical for cleavage fracture in metals. At aplane spacing to the grain size. The former correspondsgiven grain size, the computed crack sizes are five ordersto localized slip on a single slip plane, while the latterof magnitude lower than those observed experimentally,corresponds to uniform slip distributed evenly within theeven though the correct grain-size dependence (slope ofentire grain. Computed h values for various alloys are corre-the linear plot) was obtained. This discrepancy is attributedlated against the grain size in Figure 11, which shows thatto the use of a surface energy that does not reflect thethe slipband widths lie between the two bounds. For aplastic energy dissipated during irreversible slip along thegiven alloy, the h value appears to increase slightly withslipband. To obtain good agreement between model calcu-

lations and measured crack sizes, the surface energy increasing grain size.


(a) (a)

(b)(b)

Fig. 6—Comparisons of computed and measured S-Ni curves for Ti-basedFig. 7—Calculated and observed S-Ni responses in Al alloys:[37] (a) grainalloys: (a) Ti-6Al-4V[33] and (b) lamellar TiAl alloys.[34,35,36]

size effect in X-7075 Al and (b) slip morphology effect in 2024 Al.

alloy. The computed crack-growth rates as a function ofVI. SMALL-CRACK BEHAVIORthe applied stress for small cracks are presented in Figure

The crack-size-based fatigue-crack-initiation model 12, which shows that dc/dN is zero at a stress below theallows one to investigate the growth behavior of just-initiated fatigue limit, but increases rapidly with increasing stressessmall cracks. This can be accomplished by differentiating above the fatigue limit. The crack-growth rate at a constantEq. [20] with respect to the fatigue cycle, and the resulting stress and R ratio is, however, a constant, independentfatigue-crack-growth rate (dc/dN ) expression is given by of the crack length or the stress-intensity-factor range. A

comparison of the computed da/dN based on the crack-initiation model against experimental data of smalldc

dN5 2a 3Flp (1 2 v)

8M2 GFDs 2 2Mkm G2

1d 3

h22c2a2141

2a

cracks[54] and large cracks[55] is shown in Figure 13 forTi-6Al-4V. Small cracks in Ti-6Al-4V were observed topropagate at stress-intensity ranges below those of the[22]large-crack threshold, but they exhibited the same growthrates as those of the large crack at DK . DKth when thewhich is reduced tolength of the small cracks became longer. In contrast, thecomputed crack-growth rate based on the crack-initiationdc

dN5 Flp (1 2 v)

8M2 GFDs 2 2Mkm G2

1d 3

h22 [23]model is a constant, independent of DK. Furthermore, thegrowth rate is in agreement with experimental data when

and the crack-growth rate is independent of crack length the crack size is greater than the grain size, which is aboutwhen a 5 1/2. In contrast, dc/dN increases with crack length 6 mm.when a . 1/2, but dc/dN decreases with crack length whena , 1/2.

VI. CRACK INITIATION AT A STRESSThe fatigue-crack-growth response of small cracks in Ti-GRADIENT6Al-4V[54] was computed using Eq. [23] using the materials

constants obtained from S-Ni calculations and shown in The utilization of the crack-size-based initiation modelfor treating fatigue-crack initiation at a stress gradient isTable I. Equation [23] was used because a 5 1/2 for this


as the maximum length of a fatigue crack that can be initiatedin the notch field. Figure 15 shows a comparison of Eq. [26a]against experimental data of several structural alloys.[58–67] Asummary of the notch data is presented in Table III. Thereis reasonable agreement between the experimental data andmodel calculations, which is consistent with the previousfinding by Lukas and Klesnil.[68] It is also important to notethat the stresses acting on a small crack emanating from anotch change from the notch-stress field to the bulk-stressfield as the length of the small crack increases. The transitionpoint has been calculated as

l*r

5 (k2t 2 1)21 [26b]

which is based on an analysis by Dowling[31] and is shownas the dotted line in Figure 15. Fatigue is limited by crackinitiation at stresses below the boundary, but is also limited

(a) by crack growth at stresses above the boundary. From Figure15, it is apparent that the proposed fatigue-crack initiationis most pertinent for fatigue of notches with a kt value ofless than 3 or 4, depending on the nominal stress.

For a crack of length c, where 0 , c # l*, initiated withinthe notch-stress field, the average stress acting on the crackis given by

Ds(c) 51c #

c

0

Ds (x)dx 51c #

c

0

ktDS 1 rr 1 4x2

1/2

dx [27]

which can be integrated to yield

Ds(c) 5ktDSr

2c F11 14cr 2

1/2

21G [28]

and further simplified to

(b)Ds(c) 5 ktDSF1 2

crG [29]

Fig. 8—Measured and calculated S-Ni curves for a Co alloy (ULTIMET)and copper: (a) Co alloy, ULTIMET;[38] and (b) Cu.[39]

when the approximation

illustrated in this section for notch fatigue. Figure 14 shows 11 14cr 2

1/2

' 1 112 14c

r 2 218 14c

r 22

[30]a notch of a root radius (r ) subjected to a remotely appliedstress range (DS). The distance directly ahead of the tip of is invoked on the basis of the Binomial theorem. Substitutionthe notch root is designated as x, while the distance normal to of Eq. [29] into [20] leads one tothe notch is designated as y. The elastic-stress-concentrationfactor of the notch is kt . According to Neuber,[56] the distribu- FktDS11 2

cr2 2 DseGN a

i

[31]tion of the local stress (syy) of an element (x) located directlyahead of the notch tip ( y 5 0) is given by[56,57]

5 F 8M2m 2

lp (1 2 v)G1/2

1hd21c

d21/2

Ds 5 Dsyy 5 ktDSF rr 1 4xG

1/2

[24]

for crack initiation in a notch field, where Dse 5 2Mk iswhose distribution is depicted by the solid line in Figure 14.the fatigue limit.According to the proposed fatigue-crack-initiation model,

Figure 16(a) shows the calculated S-Ni curves for initiatingcrack initiation is feasible only when Ds . Dse . There, thea crack depth of 200, 500, and 1000 mm from a notch withmaximum length of the fatigue crack that can be initiateda kt value of of 4. The calculations are based on Eq. [31]in a notch field is obtained from the equality given byusing material parameters for Ti-6Al-4V. Figure 16(a) indi-cates that at a given stress, the cycles to initiate a crack

ktDSF rr 1 l*G

1/2

5 Dse [25] increase with increasing crack length. At a large crack length(i.e., 1000 mm), the calculated S-Ni curve approaches that

which can be rearranged to yield of the smooth specimen. It occurs when the condition

kt11 2cr2 5 1 [32]

l*r

514F1ktDS

Dse2

2

2 1G [26a]


Tab

leII

.Su

mm

ary

ofM

icro

stru

ctur

e,G

rain

Size

,F

atig

ueC

rack

Init

iati

onSi

te,

Init

ial

Cra

ckL

engt

h,M

easu

rem

ent

Met

hod,

and

Fat

igue

Con

diti

ons

for

Var

ious

Allo

ys

Initi

alC

rack

Stre

ssFa

tigue

Gra

inSi

ze,

dL

engt

h,M

easu

rem

ent

Am

plitu

de,

Lim

it,N

i,N

f,M

ater

ial

Mic

rost

ruct

ure

(mm

)In

itiat

ion

Site

2ci,

mm

Met

hod

MPa

MPa

Cyc

les

Cyc

les

Ref

eren

ce

Pure

Cu

sing

leph

ase

10sl

ipba

nds

2O

M—

—1

310

52

310

6T

hom

pson

etal

.[13,

42]

sing

leph

ase

350

grai

nbo

unda

ries

50to

70—

——

53

103

33

104

Usa

mi[4

0]

(gb)

slip

band

150

Pure

Al

sing

leph

ase

100

gb12

.7O

M—

—10

310

43

310

5Sm

ith[4

2]

Fe-3

pct

Sisi

ngle

phas

e13

0gb

/slip

band

s20

to70

OM

331

270

13

105

63

105

Tana

kaet

al.[4

3]

sing

leph

ase

600

gb/s

lipba

nds

100

to14

0O

M24

022

56

310

52

310

6

LC

stee

ls(A

)fe

rrite

1pe

arlit

e7.

8sl

ipba

nds

90N

i5

Nf

2N

p26

523

23

310

51.

43

106

Tair

aet

al.[3

2]

3.1

245

232

1.1

310

65

310

6

LC

stee

ls(B

)fe

rrite

1pe

arlit

e20

.5sl

ipba

nds

118

245

187

2.2

310

56.

63

105

LC

stee

ls(C

)fe

rrite

1pe

arlit

e55

slip

band

s37

724

515

81.

33

105

3.9

310

5

104

184

158

9.0

310

53.

53

106

Car

bon

stee

lsfe

rrite

1m

aste

nsite

40sl

ipba

nds

in12

to25

0O

M27

927

91

310

7—

Kun

ioan

dY

amad

a[44]

ferr

ite43

20st

eels

—5.

0to

23.0

gb10

.0to

50.0

——

——

—M

cGre

evy

and

Soci

e[4

5]

HSL

Ast

eels

ferr

ite1

fine

Nb

7sl

ipba

nds

0.2

SEM

——

93

103

—B

hat

etal

.[53]

prec

ipita

tes

Ti-

6Al-

4Vbi

-mod

al6

prim

ary

alph

a15

SEM

720

680

——

Pete

rset

al.[4

6]

Ti-

6Al-

4Vbi

-mod

al10

alph

a10

to45

SEM

750

680

13

103

1.3

310

4D

emul

sant

and

Men

dez[4

7]

glob

ular

(equ

iaxe

d5

alph

a18

to62

SEM

750

680

13

103

9.1

310

3

alph

a)gl

obul

ar(e

long

ated

38el

onga

ted

alph

a40

to70

SEM

750

680

13

103

5.1

310

3

alph

a)L

amel

lar

TiA

la

21

g15

0sl

ipba

nds

14to

230

SEM

340

310

—6.

93

104

Cha

nan

dSh

ih[3

4]

Ti-

8.6A

leq

uiax

edgr

ains

20sl

ipba

nds

5to

45O

M65

0—

3.5

310

3—

Wag

ner

etal

.[41]

100

slip

band

s10

to80

OM

650

—1.

73

103

—A

stro

loy

g1

g8

30po

res

240

SEM

860

——

—D

avid

son

and

Cha

n[48,

49,5

0]

slip

band

s50

to70

—IN

718

g1

g8

1d

5.0

to50

carb

ides

18SE

M32

0—

——

Wei

sset

al.[5

1]

IN10

0g

1g

860

carb

ides

10to

20SE

M96

—8.

43

104

83

104

Wei

etal

.[30]

PML

Cg

1g

830

slip

band

s25

SEM

——

——

Jabl

onsk

i[52]

Ast

rolo

yIn

clus

ions

75to

150

(75

to15

0m

mA

l 2O

1)in

clus

ions

75to

150

(75

to15

0m

mSi

O2)

Mo/

Mo

allo

yssi

ngle

phas

e60

—11

SEM

450

420

13

107

—W

eiss

etal

.[51]

100

—29

385

360

4.8

310

7—

30—

2037

535

02

310

7—

Co

supe

rallo

ysi

ngle

phas

e80

(50

to25

0)sl

ipba

nds

100

to20

0SE

M58

645

0—

13

106

Jian

get

al.[3

8]


Fig. 11—Correlation of deduced values of the slipband width with grainsize for various structural alloys.

Fig. 9—A comparison of measured and computed crack sizes at crackinitiation for steels. The crack lengths computed based on the surface energy(gs 5 2 J/m2) is several orders of magnitude smaller than the experimentalvalues. A large surface energy on the order 1 3 105 J/m2 is required togive the experimentally observed crack lengths. The finding suggests thatthe plastic energy dissipated by irreversible slip during the crack initiationprocess must be taken into account in the surface energy term.

Fig. 12—Fatigue crack growth rates of just initiated small cracks as afunction of the applied stress ranges for Ti-6Al-4V.

relates explicitly the stress range and fatigue cycle to thecrack-size and microstructural-size parameters. The energet-ics of fatigue-crack formation have been clarified and firmlyestablished on the basis of the Orowan modification of the

Fig. 10—Experimental data of crack lengths at initiation for various struc- Griffith theory of brittle fracture. Such a relation is importanttural alloys compared against model calculations. for providing a scientific basis for predicting the crack size

at fatigue-crack initiation within a surface grain as well asat a notch root. Furthermore, the proposed model allows oneto predict the crack-extension rate for an incipient smallis satisfied. Thus, it is important to apply Eq. [31] judiciously

to the length scale where fatigue-crack initiation dominates. crack at initiation. The crack-size-based model, therefore,provides a link in the gap between the crack-initiation andFor kt 5 4, fatigue-crack initiation dominates at c/r , 0.067

(Figure 15), corresponding to c ' 100 to 200 mm. Thus, crack-propagation approaches that has, heretofore, beenmissing.the calculated S-Ni curve shown in Figure 16 is valid for c

5 200 mm. Calculated S-Ni curves for large crack depths One of the important findings of the survey performed aspart of this investigation is that the crack size at initiation,(.200 mm) might be inaccurate, since crack growth has not

been considered in these calculations. A comparison of the reported in the literature, is usually on the order of the grainsize or greater. Cracks of smaller size have been reported.calculated and measured S-Ni curves for smooth[33] and

notched[59,60] specimens of Ti-6Al-4V is presented in Fig- Even in these cases, the observed crack sizes are consider-ably larger than those predicted on the basis of Eq. [18] andure 16(b).a surface energy of 2 J/m2, which is a reasonable value forthe energy to create two new surfaces in metals. The large

VII. DISCUSSION discrepancy between the computed and observed crack sizesshown in Figures 9 and 10 necessitates a broader view ofA unique contribution of this article is the development

of a microstructure-based fatigue-crack-initiation model that what contributes to the “surface-energy” term in Eq. [18].


Fig. 15—Calculated crack depth at initiation normalized by the notch radiuscompared against experimental data for several structural alloys.[58–69]

Table III. Summary of Notch Stress Concentration Factor(kt), Notch Radius (r ), and Crack Depth (c) for

Selected Alloys

Notch Radius, Crack Depth,Material kt mm mm

Ti-6Al-4V(STOA)[58] 2.25 1.27 16 to 370Ti-6Al-4V (MA)[59,60] 4.1 1.59 150 to 2540

12.6 0.25 950 to 5080Ti-6Al-4V (BA)[59,60] 4.1 1.59 265 to 5080

Fig. 13—A comparison of the calculated fatigue crack growth response ofTi-10V-2Fe-3Al[61] 12.6 0.25 635 to 2800just initiated small cracks against experimental data of small and large2024-T4[62] 2 2 10 to 20cracks in Ti-6Al-4V.[54,55]

2124-T4[62] 2 2 64 to 5007075-T6[63,64] 2 2 76Mild steels[65,66] 8 0.1 550

12 0.254 800LC steels[67] 8.48 0.16 167 to 854

the experimentally observed fatigue-crack sizes at initiationare comparable to the experimental values of the plasticwork dissipated during fatigue-crack propagation. Previousstudies by Fine, Davidson, and co-workers[72–76] indicatedthat the plastic work dissipated during fatigue-crack growthwas in the range of 104 to 106 J/m2 for various alloys includ-ing steels, Al alloys, and Ti alloys. During fatigue-crackinitiation, most of the energy provided by the applied loadis expended in irreversible slip and dissipated as heat, whilethat stored in the dislocation dipoles is comparably small.Thus, the surface-energy term in Eq. [18] should be identifiedwith the plastic work dissipated by irreversible slip during

Fig. 14—A schematic depicts the formation of a fatigue crack within the the crack-initiation process and not with the energy to createstress field of a notch subjected to a remotely applied stress range of DS. free surfaces.[77] In the event that fatigue-crack initiation

indeed occurs at the true surface energy, the crack size atinitiation would be so small (,0.1 mm) that it is certainlybelow the detection limit of most conventional crack-lengthThe concept of a surface energy for fatigue-crack forma-

tion is a direct extension of Orowan’s modification of the measurement methods. Subsequent growth of these fatiguesto a detectable size would invariably involve a surface energyGriffith energy for brittle fracture.[69] As in quasi-static frac-

ture,[70,71] the surface energy dissipated by fatigue is consid- on the order of the 104 to 106 J/m2 reported for the growthof fatigue cracks in a number of structural alloys.[72–76] Thus,erably larger than that required to create two fresh surfaces.

As a result, it is not surprising that the apparent surface there is both theoretical and experimental evidence to sup-port the notion that the gs term in Eq. [18] is dominatedenergy required to nucleate a fatigue crack of a size commen-

surate to the experimental data is much larger than the typical by the plastic work spent in irreversible slip during crackformation. Most, if not all, of this plastic work is likely tovalue of 2 J/m2 for the surface energy. It is also encouraging

that the apparent surface energy inferred from Eq. [18] and be dissipated as heat.


Three length-scale parameters are included in the crack-initiation model, which are the slipband width, grain ormicrostructural size, and the crack size. A simple and yetrigorous way for extending the microstructure-based crack-initiation model to a probabilistic design and life-predictionframework is to describe the distributions of the three length-scale parameters (h, d, and c) in terms of the means (h, d,and c) and randomness (Xh , Xd , and Xc).[5] The latter aredimensionless parameters that describe the randomness orstatistical distribution of the parameter of interest. Uponproper substitution, Eq. [20] can be expressed as

(Ds 2 2Mk)N ai 5 F 8M2m 2

lp (1 2 v)G1/2

1hd21c

d21/2

[33](a)

1Xh

Xd21Xc

Xd2

1/2

to provide a probabilistic relation between fatigue-crack-initiation life and variations in microstructural-length scales.If desired, the fatigue limit (2Mk) and the Taylor factor canbe described via the probabilistic treatment. Furthermore,the same approach can be applied to Eqs. [21] and [29] toobtain the pertinent equations for crack initiation at inclu-sions and notches, respectively.

The proposed crack-initiation model has been evaluatedagainst experimental data in the literature. Since not allthe microstructural-length parameters in the crack-initiationmodel were measured, the values of the parameters such asslipband width and the crack size at initiation or fracturecould only be estimated or, in some cases, fitted to the(b)experimental data of crack-initiation life. Even though it is

Fig. 16—S-Ni curves for notch fatigue: (a) computed S-Ni curves for initia- not a rigorous proof of the crack-initiation model, thetion to several crack sizes at a notch tip compared to that for smooth bardeduced slipband width and crack size at initiation or fracturefatigue, and (b) computed S-Ni curves compared against experimental dataare reasonable and consistent with experimental data whenfor Ti-6Al-4V.[33,59]

they are available in the literature.According to the crack-initiation models as given in Eqs.

[20], [21], and [29], the number of cycles to crack initiationThe difficulty with identifying the proper value for theincreases with increasing crack length and decreasing grainsurface-energy term can be avoided if Eq. [20] is used forsize. Both findings are supported by extensive experimentalrelating the cycles to initiation at a given crack length orobservations reported in the literature.[31,32,37] In addition,depth. This approach also offers the advantage that it canthe number of cycles to initiation increases with increasingbe readily adopted into a life-prediction scheme for treatingslipband width. The slipband width parameter can be inter-both fatigue-crack initiation and growth, as demonstratedpreted as a measure of the inhomogeneity of slip within theearlier by Dowling[31] and Socie et al.[78] Instead of experi-grain. A small slipband width accompanies inhomogeneous,mental estimates of crack length, as used by Dowling,[31]

localized slip, and it leads to easy crack initiation and athe crack-size-based crack-initiation model, if verified bylower crack-initiation life. In contrast, homogeneous slipfurther experimental means, allows one to compute the num-results in a large slipband width and a higher crack-initiationber of cycles to initiation for the desired crack length, aslife. The detrimental effect of localized slip and the beneficialwell as the uncertainty in fatigue-crack-initiation life due toeffect of uniform slip on fatigue-crack initiation are wellvariations in microstructural features or defect sizes.known and have been demonstrated by Starke and Lutjer-The formulation of the crack-initiation model is based oning.[37] From Figure 11, it is evident that the slipband widththe accumulation of dislocation dipoles on planar slip bands.is related to the grain size. The origin and nature of theThis type of crack-initiation process is most prevalent inrelationship between the slipband width and grain size isplanar-slip materials, where the formation of dislocationnot known. Undoubtedly, the widening of the slipband mustcells and patterning is difficult. For cell-forming materials,be related to the work-hardening behavior of the materialthe proposed crack-initiation model might be applicable oncewithin the slipband. Work hardening of the slipband is notPSBs are formed. Under this circumstance, the ladder-likeaddressed in the current model, but it can be incorporateddislocation structure within the PSBs is similar to the disloca-using crystal-plasticity theories. In the current investigation,tion pileup structure in planar-slip materials. Coalescencethe slipband width is used, essentially, as a fitting parameter.of edge-dislocation dipoles in the PSBs has also been pro-Experimental determination of this material parameter isposed as the mechanism of void formation in cell-forming

materials.[79,80] also desirable. It is also of interest to note that the slipband


1. The crack size at initiation can be incorporated into afatigue-crack-initiation model by considering the fractureenergy dissipated by the irreversible slip process duringfatigue. A set of crack-size-based fatigue-crack-initiationmodels has been developed for treating crack initiationat slipbands, inclusions, and notches.

2. The number of cycles to initiation increases with increas-ing crack sizes and slipband widths, but decreases withincreasing grain sizes, inclusion sizes, and notch-stress-concentration factors (decreasing notch radius).

3. The crack sizes at initiation, calculated based on thesurface energy, are several orders of magnitude smallerthan the experimental values. For realistic crack sizes,the surface-energy term must be interpreted as the plasticenergy dissipated by irreversible slip during the fatigue-crack-initiation process.

Fig. 17—Crack length vs fatigue cycle curves for three grain sizes computed 4. The crack-initiation model leads to a crack-growth-rateusing the proposed crack initiation model compared with the current equation for stage I fatigue that shows different character-approach for assigning a specified crack length after the low-cycle fatigue

istics than those of stage II large cracks.life has been exhausted. The calculations were performed for crack sizes5. Crack initiation at blunt notches (kt # 4) is treated by thebelow the crack detection limit of current nondestructive evaluation

methods. proposed model. The applicability of the crack-initiationmodel to sharp notches (kt . 4) might be limited, becausefatigue-crack growth is the controlling failure process in

values obtained in this study are consistent with the widths sharp notches.of PSBs or extrusions reported in the literature.[6,15–17,81] 6. Capabilities of the crack-size-based initiation model

For lifing of components, it is a common practice to include computations of the stress–initiation life curves,compute a crack-initiation life based on low-cycle-fatigue fatigue-crack growth curve for just-initiated cracks,data generated by testing smooth specimens and a crack- crack-length vs fatigue-cycle curve for crack sizes belowgrowth life based on fatigue-crack-growth data obtained the current crack-detection limit, and the variability offrom large-crack fracture-mechanics specimens.[3] The these properties due to microstructure variations.crack-initiation life is defined in terms of a specified cracksize that corresponds to the resolution of the crack-detection APPENDIXmethod. For a complex geometry, the crack-initiation life The energetics of fatigue fracturecorresponds to the cycles to initiate a surface crack of 750

According to Rice,[83,84] irreversible slip in metals can bemm in length.[3,82] The current approach is illustrated as therelated to internal dislocation arrangements and describeddashed line in Figure 17. The proposed crack-initiationin terms of changes in the internal variables. These internalmodel provides a framework for providing several pieces ofvariables could conceptually be considered as state variablesinformation that cannot be obtained by the current approach,that provide a full thermodynamic characterization of con-which include: (1) computing the initiation portion of thestrained equilibrium states corresponding to a given internalcrack length vs fatigue-cycle curve, (2) delineating the crack-dislocation arrangement, applied stress, and temperature. Forinitiation and the crack-growth regimes, as well as theircrack initiation from a pileup of dislocation dipoles, thetransition point, and (3) calculating the variation of the crack-internal variable would be the elastic-strain energy storedinitiation response due to microstructure variability. This isin the dislocations. The free-energy change associated withdemonstrated in Figure 17, which shows the calculated crackthe coalescence of dislocation dipoles into a fatigue cracklength as a function of fatigue cycles in the crack-initiationis the difference in the energy (2cgs) expended in formingregime for Ti-6Al-4V with three grain sizes. In Figure 17,a crack and the elastic-strain energy released by the disloca-an existing crack of 750 mm in length is shown as the dottedtions and crack opening, resulting in[22,23]line. The current approach of assigning a crack length of

750 mm after the low-cycle-fatigue life has been exhausted DG 5 2We 2 Wm 1 2cgs [A1]is represented by a step function and is shown by the dashed

which is also shown as Eq. [11] in the text. The We termline in Figure 17. In comparison, the computed crack length–corresponds to the stored-energy, which is comprised of thecycles curves are continuous, sensitive to the grain size, andself-energy and the interaction energy of the dislocationdifferent from that depicted by the step function at smalldipoles. The value of We is given bycrack sizes that are below the current crack-detection limit.

The enhanced ability of being able to compute the cracksize at initiation for a crack size below the crack-detection We 5

j1bd 2 (Dt 2 2k)2 N 2

4p 2A[A2]

limit is a unique attribute of the proposed crack-initiationmodel. with

A 5 mb/2p (1 2 v)VIII. CONCLUSIONS where j1 is a numerical constant. The Wm term is Irwin’s

strain-energy release rate due to the crack opening[71] andThe conclusions reached in this investigation are asfollows. is given by[22,23]


andWm 5 #11 2 v

m 2(K2I 1 K2

II)dc [A3]

Ni 52z3 gs

z1 1 z2[A13]

where KI and KII are the Mode I and II stress-intensityfactors, respectively. Integrating [A3] lead one to which is twice as large as that shown in Eq. [A10].

Wm 5(1 2 v)(Dt 2 2k)2 Dt 2d 2b2N 2

8p m A2 [A4]ACKNOWLEDGMENTS

This research was performed while the author was anas the elastic-strain-energy release rate associated with theERLE Visiting Scientist at Air Force Research Laboratoryopening up of the incipient fatigue crack. Furthermore, the(AFRL). The work was supported by AFRL through Con-incipient crack length can be related to the dislocation pileup,tract No. F33615-99-C-5803, Dr. James M. Larsen, Techni-as given by[23]

cal Manager. Technical assistance by Mr. J. Lawson, M.Dent, and J. Jira, AFRL, Dr. V. Nagarajan of Systran Federalc 5

bd(Dt 2 2k)NpA

[A5]Corporation, and clerical assistance by Ms. L. Salas, South-west Research Institute, is acknowledged.

which is combined with Eqs. [A1], [A2], and [A4] to give

DG 5 2z1N 2 2 z2N 2 1 2z3Ngs [A6] REFERENCES

1. K. Tanaka and T. Mura: ASME J. Appl. Mech., 1981, vol. 48, pp.where97-103.

2. K. Tanaka and T. Mura: Metall. Trans. A, 1982, vol. 13A, pp. 117-23.z1 5

j1bd 2 (Dt 2 2k)2

4p 2A[A7] 3. B.A. Cowles: Mater. Sci. Eng., 1988, vol. A103, pp. 63-69.

4. J.-P. Bailon and S.D. Antolovich: in Fatigue Mechanisms: Advancesin Quantitative Measurements of Physical Damage, ASTM STP 811,J. Lankford, D.L. Davidson, W.L. Morris, and R.P. Wei, eds., ASTM,z2 5

(1 2 v)(Dt 2 2k)2 Dt 2d 2b2

8pmA2 [A8]Philadelphia, PA, 1983, pp. 313-49.

5. K.S. Chan: Metall. Trans. A, 1993, vol. 24A, pp. 2473-86.6. K.S. Chan and T.-Y. Torng: ASME Trans., J. Eng. Mater. Technol.,and

1996, vol. 118, pp. 379-86.7. R. Tyron and T.A. Cruse: ASME J. Eng. Mater. Technol., 1997, vol.

z3 5bd (Dt 2 2k)

pA[A9] 119, pp. 65-70.

8. G. Venkataraman, Y.W. Chung, and T. Mura: Acta Metall. Mater.,1991, vol. 39, pp. 2621-29.The partial derivation of DG with respect to N gives 9. G. Venkataraman, Y.W. Chung, and T. Mura: Acta Metall. Mater.,1991, vol. 39, pp. 2631-38.

10. M.R. Mitchell: in Fatigue and Microstructure, M. Meshii, ed., ASM,DGN

5 22z1N 2 2z2N 1 2z3 gsMetals Park, OH, 1978, pp. 385-437.

11. L.F. Coffin, Jr.: Trans. ASME, 1954, vol. 76, pp. 931-50.and setting DG/N 5 0 leads one to 12. S.S. Manson and M.H. Hirschberg: in Fatigue: An Inter-Disciplinary

Approach, Syracuse University, Syracuse, NY, 1964, pp. 133-78.13. N. Thompson, N.J. Wadsworth, and N. Louat: Phil. Mag., 1956, vol.

Ni 5z3 gs

z1 1 z2[A10] 1, pp. 113-26.

14. J.C. Grosskreutz: in Metal Fatigue Damage—Mechanism, Detection,Avoidance, and Repair, ASTM STP 495, S.S. Manson, ed., ASTM,

when DG/N 5 0 is taken as the condition for crack initia- Philadelphia, PA, 1971, pp. 5-60.tion. The crack size at initiation is then obtained by substitut- 15. C. Laird: in Fatigue and Microstructure, M. Meshii, ed., ASM, Metals

Park, OH, 1978, pp. 149-203.ing Eq. [A10] into [A5], leading to16. M.E. Fine and R.O. Ritchie: in Fatigue and Microstructure, M. Meshii,

ed., ASM, Metals Park, OH, 1978, pp. 245-78.ci 5

bd(Dt 2 2k)pA 1 z3

z1 1 z22 gs [A11] 17. L. Remy: in Fatigue 84, C.J. Beevers, ed., EMAS, Warley, UnitedKingdom, 1984, vol. I, pp. 15-30.

18. A.S. Cheng and C. Laird: Fat. Fract. Eng. Mater. Struct., 1981, vol.for initiation in which all the dislocation dipoles in the pileup 4, pp. 343-53.

19. A. Saxena and S.D. Antolovich: Metall. Trans. A, 1975, vol. 6A, pp.contribute to the formation of the crack. A simpler crack-1809-28.size expression has been obtained and is shown in Eq. [18]

20. K. Tanaka and T. Mura: Mech. Mater., 1981, vol. 1, pp. 63-73.when only a portion of the dislocation pileup contributes to 21. M.R. Lin, M.E. Fine, and T. Mura: Acta Metall., 1986, vol. 34, pp.crack initiation. For both cases, ci is proportional to the 619-28.

22. G. Venkataraman, Y.-W. Chung, Y. Nakasone, and T. Mura: Actasurface energy, gs . The surface energy is considered to con-Metall. Mater., 1990, vol. 38, pp. 31-40.sist of an elastic component and a plastic component. The

23. T. Mura and Y. Nakasone: J. Appl. Mech., 1990, vol. 57, pp. 1-6.latter reflects the plastic work spent by the irreversible slip 24. T. Mura: Mater. Sci. Eng., 1994, vol. A176, pp. 61-70.that accompanies the crack formation and the opening up 25. K.S. Chan: Scripta Metall. Mater., 1995, vol. 32 (2), pp. 235-40.of the crack surfaces. 26. R. Chang, W.L. Morris, and O. Buck: Scripta Metall., 1979, vol. 13,

pp. 191-94.Instead of using DG/N 5 0 as the crack-initiation crite-27. C. Ihara and T. Tanaka: Fat. Fract. Eng. Mater. Struct., 2000, vol.rion, the onset of crack initiation can be considered to occur

23, pp. 375-80.at DG # 0. Using Eq. [A6], this crack-initiation criterion 28. S.E. Harvey, P.G. Marsh, and W.W. Gerberich: Acta Metall. Mater.,leads one to 1994, vol. 42, pp. 3493-3502.

29. Isomoto and N. Stoloff : Mater. Sci., 1990, vol. A124, pp. 171-81.DG 5 z1N 2 z2N 1 2z3 gs 5 0 [A12] 30. W. Wei, H. Floge, and E.E. Affeldt: Scripta Metall. Mater., 1991, vol.


57. M.M. Hammouda, R.A. Smith, and K.J. Miller: Fat. Eng. Mater.25, pp. 1757-61.Struct., 1979, vol. 2, pp. 139-54.31. N. Dowling: Fat. Eng. Mater. Struct., 1979, vol. 2, pp. 129-38.

58. M.A. Moshier, T. Nicholas, and B.M. Hillberry: Fatigue and Fracture32. S. Taira, K. Tanaka, and M. Hoshina: Fatigue Mechanisms, ASTMMechanics, ASTM STP1417, W.G. Reuter and R.S. Piascik, eds.,STP 675, ASTM, Philadelphia, PA, 1979, pp. 135-73.ASTM, West Conshohocken, PA, 2002, vol. 33, in press.33. R.S. Bellows: Improved High Cycle Fatigue Life Prediction, Final

59. G.R. Yoder, L.A. Cooley, and T.W. Crooker: Fracture Mechanics:Report No. F33615-96-C-5269, University of Dayton Research Insti-16th Symposium, ASTM STP868, M.F. Kanninen and A.T. Hopper,tute, Dayton, OH, 2001.eds., ASTM, Philadelphia, PA, 1985, pp. 392-405.34. K.S. Chan and D.S. Shih: Metall. Mater. Trans. A, 1997, vol. 28A,

60. G.R. Yoder, L.A. Cooley, and T.W. Crooker: Proc. 23rd Structures,pp. 79-90.Structural Dynamics and Materials Conf., CP823, Part 1, American35. K.S. Chan and D.S. Shih: Metall. Mater. Trans. A, 1998, vol. 29A,Institute of Aeronautics and Astronautics, New York, NY, 1982, pp.pp. 73-87.132-36.36. J.M. Larsen: Air Force Research Laboratory, Wright-Patterson Air

61. G.R. Yoder, L.A. Cooley, and R.R. Boyer: in Microstructure, FractureForce Base, Dayton, OH, unpublished research, 2001.Toughness and Fatigue Crack Growth Rate in Titanium Alloys, A.K.37. E.A. Starke, Jr. and G. Lutjering: in Fatigue and Microstructure, M.Chakrabarti and J.C. Chesnutt, eds., TMS, Warrendale, PA, 1987, pp.Meshii, ed., ASM, Metals Park, OH, 1978, pp. 205-43.209-29.38. L. Jiang, C.R. Brooks, P.K. Liaw, and D.L. Klarstrom: in Superalloys

62. S.S. Manson: Exper. Mech., 1965, vol. 5, pp. 193-226.2000, T.M. Pollock, R.D. Kissinger, R.R. Bowan, K.A. Green, M.63. J.C. Grosskreutz and G.G. Shaw: Technical Report No. 66-96, AirMcLean, S. Olson, and J.J. Schirra, eds., TMS, Warrendale, PA, 2000,

Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton,pp. 583-91.OH, May 1966.39. W. Hessler, H. Mullner, B. Weiss, and R. Sticker: Met. Sci., 1981,

64. S.S. Manson and M.H. Hirschberg: Technical Note D-3146, NASA,vol. 15, pp. 235-40.Cleveland, OH, June 1967.40. S. Usami: in Small Fatigue Crack, R.O. Ritchie and J. Lankford, eds.,

65. N.E. Frost and D.S. Dugdale: J. Mech. Phys. Solids, 1957, vol. 5, pp.TMS, Warrendale, PA, 1986, pp. 559-83.182-92.41. L. Wagner, J.K. Gregory, A. Gysler, and G. Lutjering: in Small Fatigue

66. M.H. El Haddad, T.H. Hopper, and K.N. Smith: Eng. Fract. Mech.,Cracks, R.O. Ritchie and J. Lankford, eds., TMS, Warrendale, PA,1979, vol. 11, pp. 573-84.1986, pp. 117-28.

67. K. Tanaka and Y. Nakai: Fat. Eng. Mater. Struct., 1983, vol. 6, pp.42. G.C. Smith: Proc. R. Soc., 1957, vol. A242, pp. 189-97.315-27.43. K. Tanaka, M. Hojo, and Y. Nakai: in Fatigue Mechanisms: Advances

68. P. Lukas and M. Klesnil: Mater. Sci. Eng., 1978, vol. 34, pp.in Quantitative Measurement of Physical Damage, ASTM STP811,61-68.J. Lankford, D.L. Davidson, W.L. Morris, and R.P. Wei, eds., ASTM,

69. A.A. Griffith: Phil. Trans. R. Soc. A, 1921, vol. 22A, pp. 163-98.Philadelphia, PA, 1983, pp. 207-32.70. E. Orowan: Rep. Progr. Phys., 1948, vol. XII, p. 185.44. T. Kunio and K. Yamada: in Fatigue Mechanisms, ASTM STP675,71. G.R. Irwin: Fracturing of Metals, ASM, Cleveland, OH, 1948, pp.J.T. Fong, ed., ASTM, Philadelphia, PA, 1979, pp. 342-70.

147-66.45. T.E. McGreevy and D.F. Socie: Fat. Fract. Eng. Mater. Struct., 1999, 72. P.K. Liaw, M.E. Fine, and D.L. Davidson: Fat. Eng. Mater. Struct.,

vol. 22. 1980, vol. 3, pp. 59-74.46. M. Peters, A. Gysler, and G. Luetjering: in Titanium ’80 Science and 73. P.K. Liaw, S.I. Kwun, and M.E. Fine: Metall. Trans. A, 1981, vol.

Technology, H. Kimura and O. Izumi, eds., AIME, Warrendale, PA, 12A, pp. 49-55.1980, vol. 3, pp. 1777-85. 74. M.E. Fine and D.L. Davidson: in Fatigue Mechanisms, J. Lankford

47. X. Demulsant and J. Mendez: Fat. Fract. Eng. Mater. Struct., 1995, et al., eds., ASTM STP811, Philadelphia, PA, 1983, pp. 350-70.vol. 18, pp. 1483-97. 75. D.L. Davidson and J. Lankford: in Environment-Sensitive Fracture of

48. D.L. Davidson and K.S. Chan: Acta Metall., 1989, vol. 37 (4), pp. Engineering Materials, Z.A. Foroulis, ed., TMS-AIME, Warrendale,1089-97. PA, 1977, pp. 581-94.

49. D.L. Davidson and S.J. Hudak, Jr.: Metall. Mater. Trans. A, 1995, vol. 76. D.L. Davidson: Morris E. Fine Symp., P.K. Liaw, J.R. Weertman,26A, pp. 2247-57. H.L. Marcus, and J.S. Santner, eds., TMS, Warrendale, PA, 1991, pp.

50. D.L. Davidson: Fat. Fract. Eng. Mater. Struct., 2000, vol. 23, pp. 355-62.445-52. 77. S.R. Bodner, D.L. Davidson, and R.J. Lanford: Eng. Fract. Mech.,

51. B. Weiss, R. Stickler, and A. Fathulla: in Short Fatigue Cracks, 1983, vol. 17, pp. 189-91.R.O. Ritchie and J. Lankford, eds., TMS, Warrendale, PA, 1986, pp. 78. D.F. Socie, N.E. Dowling, and P. Kurath: Fracture Mechanics: 15th471-97. Symp., ASTM STP833, R.I. Sanford, ed., ASTM, Philadelphia, PA,

52. D.A. Jablonski: Mater. Sci. Eng., 1981, vol. 48, pp. 189-98. 1984, pp. 284-99.53. S.P. Bhat, R.S. Cline, and Y.-W. Chung: M.E. Fine Symp., P.K. Liaw, 79. J.G. Antonopoulus, L.M. Brown, and A.T. Winter: Phil. Mag., 1976,

J.R. Weertman, H.L. Marcus, and J.S. Santner, eds., TMS, Warrendale, vol. 34, pp. 549-63.PA, 1991, pp. 49-59. 80. U. Essman and H. Mughrabi: Phil. Mag., 1979, vol. 40, pp. 731-56.

54. Y.N. Lenets: Improved High Cycle Fatigue Life Prediction, Final 81. P.J.E. Forsyth: Proj. R. Soc., 1957, vol. A242, pp. 198-202.Report No. F33615-96-C-5269, University of Dayton Research Insti- 82. Y. Dai, N. Marchand, and M. Hongoh: Can. Aero. Space J., 1993,tute, Dayton, OH, 2001, Appendix 3C. vol. 39, pp. 35-44.

55. R.E. deLaneuville and J.W. Sheldon: Improved High Cycle Fatigue 83. J. Kestin and J.R. Rice: Proc. Critical Review of ThermodynamicsLife Prediction, Final Report No. F33615-96-C-5269, University of Symposium, E.B. Stuart, G.-O. Benjamin, and A.J. Brainard, eds.,Dayton Research Institute, Dayton, OH, 2001, Appendix 3B. University of Pittsburgh School of Engineering Publication, Pittsburgh,

PA, 1970, pp. 275-98.56. H. Neuber: Theory of Notch Stress, translated by F.A. Raven, Edwards,Ann Arbor, MI, 1946. 84. J.R. Rice: J. Appl. Mech., 1970, vol. 37, pp. 728-37.


Documents

A Microstructure-Based Fatigue-Crack-Initiation Model.pdf