James C. Newman, Jr.Mississippi State University,

Mississippi State, MS 39762

e-mail: j.c.newman.jr@ae.msstate.edu

Balkrishna S. AnnigeriPratt & Whitney,

East Hartford, CT 06118

e-mail: balkrishna.annigeri@pw.utc.com

Fatigue-Life Prediction MethodBased on Small-Crack Theory inan Engine MaterialPlasticity effects and crack-closure modeling of small fatigue cracks were used on a Ti-6Al-4V alloy to calculate fatigue lives under various constant-amplitude loading condi-tions (negative to positive stress ratios, R) on notched and un-notched specimens. Fatiguetest data came from a high-cycle-fatigue study by the U.S. Air Force and a metallic mate-rials properties handbook. A crack-closure model with a cyclic-plastic-zone-correctedeffective stress-intensity factor range and equivalent-initial-flaw-sizes (EIFS) were usedto calculate fatigue lives using only crack-growth-rate data. For un-notched specimens,EIFS values were 25-lm; while for notched specimens, the EIFS values ranged from 6 to12 lm for positive stress ratios and 25-lm for R¼�1 loading. Calculated fatigue livesunder a wide-range of constant-amplitude loading conditions agreed fairly well with thetest data from low- to high-cycle fatigue conditions. [DOI: 10.1115/1.4004261]

Keywords: fatigue, cracks, crack growth, crack closure, plasticity, titanium alloy

1 Introduction

The observation that small or short fatigue cracks can growmore rapidly than those predicted by linear-elastic fracturemechanics (LEFM) based on large-crack data, and grow at DKlevels well below the large-crack threshold, has attracted consid-erable attention [1–5]. Some consensus is emerging on crackdimensions, mechanisms, and possible methods to correlate and topredict small-crack behavior. A useful classification of smallcracks has been made by Ritchie and Lankford [6]. Naturally-occurring (three-dimensional) small cracks, often approachingmicrostructural dimensions, are largely affected by crack shape(surface or corner cracks), enhanced crack-tip plastic strains dueto micro-plasticity, local arrest at grain boundaries, and the lackof crack closure in the early stages of growth.

Research on small-crack behavior and improved analysis meth-ods have shown that fatigue is “crack propagation” from micro-structural discontinuities in a number of engineered materials,such as aluminum alloys, titanium alloys and steels [7–11]. Large-crack thresholds (DKth) on a wide class of materials may also beinadvertently too high and crack-growth rates too low in the near-threshold regime due to the load-shedding test method used togenerate these data [12–14]. New threshold test methods are beingdeveloped with compression precracking [15,16] to generatecrack-growth-rate data at very low initial stress-intensity factorswith minimal load-history effects. But small-crack data should begenerated on these materials to validate the fatigue-life predictionmethods based on crack growth from microstructural flaw sizes.

In the present paper, plasticity effects and crack-closure model-ing of small fatigue cracks were used on a Ti-6Al-4V titaniumalloy to calculate fatigue lives under various constant-amplitudeloading conditions (negative to positive stress ratios, R) onnotched and un-notched specimens. Fatigue test data came from ahigh-cycle-fatigue program by the U.S. Air Force [17–21] and theMetallic Materials Properties Development Standards (MMPDS)Handbook [22]. A crack-closure model with a cyclic-plastic-zone-corrected effective stress-intensity factor range and equivalent-ini-

tial-flaw-sizes (EIFS) were used to calculate fatigue lives usingonly crack-growth-rate data.

2 Material and Specimen Configurations

The titanium alloy considered herein is from the United StatesAir Force High-Cycle-Fatigue (HCF) program [17–21], that wasin the solution treated and over-aged (STOA) condition. The forg-ing, heat-treatment and aging process resulted in a microstructurewith an average grain size of 20 lm. The yield stress (rys) was931 MPa, the ultimate tensile strength (ru) was 979 MPa, and themodulus of elasticity (E) was 116 GPa. Additional fatigue datawas obtained from MMPDS [22] on an STOA titanium alloy ofslightly higher strength.

The large-crack DK-rate data for the titanium alloy wasobtained from C(T) specimens [13] using material obtained fromthe same batch of material as used in the HCF test program. Thefatigue specimens analyzed are shown in Fig. 1. They were: (a)uniform stress (KT¼ 1) un-notched specimen in a flat sheet or rodform, (b) circular-hole (KT¼ 3.0) specimen in a flat sheet, and (c)double-edge-notch tension (KT¼ 3.06) in plate form. All speci-mens were chemically polished to remove a small layer of dis-turbed material, which may have contained some machiningresidual stresses. Here the stress concentration factor, KT, isexpressed in terms of remote (gross) stress, S, instead of the net-section stress.

3 ASTM Load-Reduction and Compression

Pre-Cracking Behavior

Currently, in North America, the threshold crack-growth re-gime is experimentally defined by using ASTM Standard E647,which has been shown in many cases to exhibit anomalies due tothe load-reduction (LR) test method. The test method has beenshown to induce remote closure, which prematurely slows downcrack growth and produces an abnormally high threshold. Thefatigue-crack growth rate properties in the threshold and near-threshold regimes for the titanium alloy were obtained from Ref.13, which used both the LR test method and an improved testmethod. The improved test method used “compression-compression” precracking, as developed by Suresh [15], Pippan etal. [16] and others [12–14], to provide fatigue-crack-growth rate

Contributed by the International Gas Turbine Institute (IGTI) Division of ASMEfor publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manu-script received April 26, 2011; final manuscript received May 4 2011; publishedonline December 28, 2011. Editor: Dilip R. Ballal.

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data under constant-amplitude loading in the near-threshold re-gime, with minimal load-history effects. Test data were obtainedfrom Ref. 13 over a wide range in stress ratios (R¼ 0.1 to 0.7) oncompact C(T) specimens for three different widths (25, 51 and76-mm) to help determine the DKeff-rate relation for large cracks.

The test data at R¼ 0.1 for the ASTM LR method are shown inFig. 2(a). These data show a “fanning out” of data at lower growthrates as a function of specimen width (w). These results were verysimilar to those presented by Garr and Hresko [23] on Inconel-718, which showed a width effect on threshold behavior using theASTM LR method. The 51-mm wide tests produced a lowerthreshold and faster rates at a given DK value than the 76-mmwide specimens. The solid curve is a predicted curve based on theR¼ 0.7 data (as the DKeff-rate curve) using the crack-closuremodel [24] with a constraint factor (a) of two. This curve alsoshows that the 51- and 76-mm specimens produced data far fromtheir expected trend (solid curve).

In contrast, data from the three specimen widths for R¼ 0.1loading using the compression precracking constant amplitude(CPCA) test method [12] show drastically different behavior, asshown in Fig. 2(b), with no “fanning” nor specimen width depend-ency, as was noted with the E647 LR method. Data for the threespecimen widths plotted directly on top of each other over thesame range in crack-growth rates examined. The DK-rate curve isclearly independent of specimen width and crack length, and therate is only as a function of the applied DK, a key assumption inthe fracture mechanics approach to life prediction. These resultsindicate that this titanium alloy is very sensitive to load reduction;and caution must be used whenever LR procedures are used.Again, the solid curve in Fig. 2(b) is the predicted behavior forR¼ 0.1 using the crack-closure model (a¼ 2) and the R¼ 0.7data from Ref. 13. In the low- to mid-rate regimes, the rates wereover predicted by about 25%, but slightly over predicted thethreshold at the ASTM defined threshold rate.

Newman’s crack-closure model [24] was used to correlate DK-rate data on the three width C(T) specimens to generate the effec-tive stress-intensity factor against rate relation. From past analyseson titanium alloys, a constraint factor (a) of two had been foundto correlate test data over a wide range in stress ratios in the mid-rate regimes. To convert from DK to DKeff, the Elber [25] relationwas used

DKeff ¼ 1� Ko=Kmaxð Þ= 1� Rð ÞDK ¼ UDK (1)

where Ko is the crack-opening stress-intensity factor and Kmax isthe maximum value. For low applied stresses and a constraint fac-tor (a) of two, the crack-opening ratio [26,27] is:

Ko=Lmax¼ 0:343þ 0:027Rþ 0:917R2 � 0:287R3 (2)

Equation (2) applies for any material that correlates crack-growth-rate data on a DKeff-basis with a constraint factor of two. Figure2(c) shows all test data using the CPCA test method [13]. Thedata correlated very well over a wide range in stress ratios and

Fig. 2 (a) Fatigue-crack-growth-rate data using the ASTM load-reduction test method. (b) Fatigue-crack-growth-rate data usingthe CPCA test method. (c) DKeff-rate data from the CPCA testmethod and small-crack estimates.

Fig. 1 Fatigue specimens analyzed

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specimen widths [28]. In these analyses, test data for R¼ 0.7 wasassumed to be closure-free and; thus DK¼DKeff. The solid curvewas a fit to the data shown by symbols. Because there were nodata available below a rate of 1e-10 m/cycle, several assumptionswere made. First, a (DKeff)th value of 2.5 MPaHm was selected(vertical dashed line). Second, a linear extrapolation was madefrom the lower test data (dashed line) and the lower solid curve isan estimate for small-crack behavior. Small cracks have beenshown to grow below the large-crack threshold on a variety ofmaterials. These three extrapolated curves will be used to see howthey influence fatigue-life calculations. For high rates, aconstraint-loss regime (plane-strain to plane-stress behavior) isexpected at a (DKeff)T value of about 38 MPaHm. The constraint-loss range was estimated and constraint change was assumed to belinear on log rate [26,27]. Nearly plane-strain (a¼ 2) conditionsapply for low rates and plane-stress (a¼ 1) conditions apply forhigh rates. The elastic fracture toughness, KIe, was 66 MPaHm.

Crack-growth analyses were performed using a multilineartable-lookup method. The DKeff-rate value used in subsequent lifeanalyses was the solid curve with circular symbols, as shown inFig. 2(c). These values are given in Table 1.

4 Crack-Closure Modeling

The crack-closure model [24,26] was used to calculate crack-opening stresses under constant-amplitude loading to show theinfluence of an initial defect size on the crack-closure behavior ofsmall cracks. Previous studies [8–11] have shown that small-crackeffects are more pronounced for negative stress ratios.

Some typical results of calculated crack-opening stress-inten-sity factor (Ko) normalized by the maximum applied stress-intensity factor (Kmax) as a function of surface crack half-length,c, is shown in Fig. 3. The crack-growth analysis was performedunder stress ratios (R) of �1, 0.1, 0.5 and 0.8 for a low appliedstress (HCF), solid curves, and a high applied stress (LCF), dashedcurves, with a constraint factor (a) of 2 (a¼ 3 for plane strain;a¼ 1 for plane stress). (An unresolved issue is that a small crackmay be acting under plane-stress conditions and not under thehigh constraint (a¼ 2) conditions assumed in the model andneeded for large-crack behavior.) The initial discontinuity(ai¼ ci¼ 12 lm) was assumed to be a void (hn¼ 6 lm) fully openon the first cycle. As the crack grows into the forward plastic-zoneregion, crack-opening stresses rapidly builds until the steady-statevalue for large-crack behavior is approached. The negative stressratio results show a significant crack-closure transient (due to theassumed void height instead of a tight crack), while the high stressratio results give crack-opening values as the minimum appliedstress-intensity factor (no crack-closure behavior). Herein, the ini-tial ai/ci ratio was assumed to be unity; and the initial void height,hn, was assumed to be one-half of ai (or ci).

The J-integral is one of the most commonly used parameters fornonlinear crack-growth analyses. El Haddad et al. [29] and Hudakand Chan [30] have made DJ estimates for small cracks. Becausecrack-closure effects may be one of the key elements in small-crack growth, DJ should be computed using only that portion ofthe load cycle during which the crack is fully open (or DJeff).Modifications to account for crack-closure effects are discussedlater. To develop a nonlinear crack-tip parameter for small cracks,it is convenient to define an equivalent plastic stress-intensity fac-tor KJ, in terms of the J-integral, as

K2J¼ JE= 1� g2

� �¼ rodE= 1� g2

� �(3)

where E is the modulus, g¼ 0 for plane stress, g¼ v (Poisson’s ra-tio) for plane strain, ro is the flow stress, and d is the crack-tip-opening displacement. As shown by Rice [31] from the Dugdalemodel, the J-integral is equal to rod. A common practice inelastic-plastic fracture mechanics has been to add a portion of theplastic-zone size (q) to the crack length to account for crack-tipyielding. An estimate for J was determined in Ref. 32 by firstdefining a plastic-zone-corrected stress-intensity factor as

Kp¼ Sffiffiffiffiffiffiffiffiffipdð Þ

pF d=w; d=r;:::ð Þ (4)

where d¼ cþ cq, F is the boundary-correction factor, w is speci-men width and r is hole radius. The term c was assumed to be aconstant and was evaluated by equating Kp to KJ for severalcracked bodies [32]. A value of 1/4 was found to give good agree-ment between Kp and KJ up to large values of applied stress toflow stress (S/ro) ratios and q/a up to 100.

Elber’s effective stress-intensity factor range [25] was based onlinear-elastic analyses. To account for plasticity, a portion of theDugdale cyclic-plastic-zone length (x) has been added to thecrack length. Thus, the cyclic-plastic-zone-corrected effectivestress-intensity factor range is:

DKp

� �eff¼ Smax � Soð Þ

ffiffiffiffiffiffiffiffiffipdð Þ

pF d=w;d=r:::ð Þ (5)

where Smax is the maximum stress, So is the crack-opening stress,d¼ cþ cx and F is the boundary-correction factor. For large-crack behavior, the cyclic-plastic-zone correction was found to beinsignificant. For small cracks emanating from a hole or notch, thecyclic-plastic-zone corrected stress-intensity factor was found tobe very significant for applied stress levels greater than about one-half of the flow stress of the material.

Table 1 Effective stress-intensity factor range against raterelation for small- and large-cracks in Ti-6Al-4V (STOA) alloy

DKeff, MPaHm dc/dN, m/cycle

2.0 2.0e-122.15 2.0e-112.5 1.3e-102.8 3.0e-107.5 1.0e-088.7 4.0e-0813.0 2.0e-0723.5 1.0e-0650.0 1.0e-05a¼ 2 � 1.0e-06a¼ 1 � 1.0e-05KIe¼ 66 MPaHm ro¼ 955 MPa

Fig. 3 Crack-closure behavior for small cracks under low- andhigh-stress levels

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5 Small- and Large-Crack Behavior

Small- and large-crack data was obtained on the titanium alloyin the USAF report [17–21] at room temperature and lab-air con-ditions at several stress ratios. The large-crack data was obtainedfrom standard compact C(T) specimens (6.35 mm thick; w¼ 51mm), and the scatter band for the R¼ 0.1 data [18] is shown inFig. 4(a). (All test data and curves are DK values unless noted asDKeff). The small-crack data (open circles) was obtained from cir-cular rods (5 mm diameter) using the replica method employingacetyl cellulose film [19]. Small cracks initiated as surface cracksgrowing at the free surface of the rods and they were assumed tobe semicircular surface cracks [19]. Various reports [17,33,34]also discuss the possibility that some of the surface cracks mayhave initiated as sub-surface embedded cracks. And the small-crack data showed a pronounced small-crack effect that will bediscussed later.

In Fig. 4(a), the solid lines with circular symbols are the DKeff-rate baseline relation determined from C(T) specimens in Fig.2(c) with data from Ref. 13. Again, the lower dashed lines withcircular symbols are the estimated behavior for small cracks. Thesolid and dashed lines are the predicted and estimated behaviorfor R¼ 0.1 loading, which agreed well for rates greater than about

1e-9 m/cycle. But the large-crack data from the USAF report[17,18] is approaching a higher threshold than the predicted curvebecause of load-history effects from load shedding, similar to thatshown in Fig. 2(a).

Small-crack data from the USAF report [17,19] for R¼ 0.5loading are shown as square open symbols in Fig. 4(b). Again, thesmall cracks were assumed to be semicircular surface cracks, butsome of these cracks may have initiated as sub-surface embeddedcracks. These small-crack data also show some pronounced small-crack effects for low values of DK. The DKeff-rate curve (solidand dashed lines with symbols) and the predicted behavior forR¼ 0.5 and �1 loading are shown as solid and dashed lines. AtR¼ 0.5 and DK values greater than about 3 MPaHm, the small-and large-surface-crack data agreed fairly well with the predictedlarge-crack results from C(T) specimens, except at the very highrates. Also, the triangular symbols are test data on middle-cracktension specimens on a mill-annealed titanium alloy [35]. Again,the test data agreed fairly well with the predicted behavior atR¼�1, except the mill-annealed alloy did not show a sharp tran-sition in the mid-region.

In an effort to try to explain the pronounced small-crack effectsshown in Figs. 4(a) and 4(b), some measured rates on small cracksin the round-bar specimens [19,33,34] are shown in Fig. 5. Herecrack-growth rate is plotted against the surface crack half-length,c. Five tests were conducted at a maximum stress level of 613MPa at R¼ 0.1, as shown by the solid curves with symbols. Thefatigue lives for these tests ranged from 1.6 to 3� 106 cycles withan average of 2.1� 106 cycles. The measured results show anextremely high initial rate followed by a rapid drop and a slowrise at longer crack lengths. These small-crack effects could notbe explained from crack-closure transients nor plasticity effects atR¼ 0.1.

For the KT¼ 1 specimens, a large amount of scatter wasobserved at this particular stress level [16,19,20]. Fatigue livesranged from 140,000 to about 5� 106 cycles. The reason for thelarge amount of scatter was also not known.

FASTRAN [26] with a 25-lm (initial semicircular surfacecrack) and the DKeff-rate curve (Table 1), predicted about 150,000cycles to failure. (Reason for selecting the 25-lm flaw will be dis-cussed later.) This life; however, agreed fairly well with the lowerbound of the fatigue tests. The solid curve in Fig. 5 is the pre-dicted results from the life-prediction code. It shows a slight dropin the initial stages and followed by a slow rise at larger cracklengths, similar to the test data. But the analyses still predictedslower rates than measured from the small-crack tests. (FAS-TRAN Version 5.33 was used herein.)

Since various reports [16,19,33,34] have discussed the possibil-ity of sub-surface crack initiation sites, the extremely high rates atvery small crack lengths may be explained by Fig. 6. The crack ata sub-surface initiation site will grow as a fish-eye crack undervacuum. When the near circular crack penetrates the free surface,the observed crack length could be very small, but the stress-intensity factor would be very large due to the vertex at the crackfront [34]. This could explain the very rapid crack-growth rates atvery small crack lengths.

The sub-surface initiation sites could also help explain the largedifferences in the fatigue life scatter observed at this particularstress level. Fatigue cracks grow much slower in vacuum thanlab-air, so several million cycles could elapse before the crackpenetrates the free surface. This could also explain why the USAFreport stated that the “crack propagation life was very small com-pared to the total fatigue life.”

Figure 7 shows the DK-rate results on the small-crack datafrom the round-bar specimens and large-crack data on C(T) speci-mens [18–20]. Square symbols are C(T) tests at R¼ 0.1 constant-amplitude loading, which agree very well with the FASTRAN cal-culations for large cracks. The solid circular symbols are C(T)tests conducted using the ASTM load-shedding method, whichproduced an elevated threshold and slower rates in the near-threshold regime. FASTRAN calculations produced faster rates

Fig. 4 (a) Small- and large-crack-growth-rate data with LEFMand closure-based relations at R 5 0.1. (b) Small- and large-crack-growth-rate data with LEFM and closure-based relationsat R 5 0.5 and 21.

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and crack growth at lower DK values than the load-shedding testdata. Further study is required to determine whether the sub-surface initiation sites, modified stress-intensity factor solutions,and vacuum crack growth are the reasons for the unusual small-crack effects observed in these tests.

6 Fatigue-Life Calculations

Small-crack theory and equivalent initial flaw sizes (EIFS) havebeen used to calculate the fatigue lives on three types of fatiguetests. Small-crack theory is the use of measured small-crack dataand the nonlinear crack-closure model to calculate or predict thefatigue life of smooth and notched specimens or components.Herein, crack growth in the a- and c-directions was assumed to beequal (i.e., DKeff-da/dN and DKeff-dc/dN relations were thesame). The fatigue test data were obtained from either the USAFHCF report [17–21] or the Metallic Materials Properties Develop-ment Standards Handbook [22]. All tests were subjected toconstant-amplitude loading over a wide range of stress levels andstress ratios. First, smooth (KT¼ 1) flat sheets or round-bar speci-mens were analyzed. Second, fatigue tests on circular hole(KT¼ 3) flat sheet specimens were compared with the calculatedresults. And last, double-edge-notch (KT¼ 3.06) specimens wereanalyzed.

6.1 Smooth (KT 5 1) Specimens. Figure 8 shows fatiguetests conducted on round bars (D¼ 4 to 5 mm) that had been pol-ished and these data were obtained from the USAF report [17,19].The maximum stress level is plotted against the fatigue life. Fa-tigue tests were conducted by three different organizations (opensymbols) and the solid symbols are fatigue lives from cyclicstress-strain tests [16]. The diamond symbol and scatter bandshows the fatigue tests [19,33,34] that were used to measuresmall-crack growth rates, as shown in Figs. 5 and 7. If small-cracktests and measurements had been made on some of the shorterfatigue-life tests, which may have resulted in surface initiation

sites, then a much different small-crack effect may have been seenfrom these tests.

In the FASTRAN analyses, a 25-lm semicircular surface crackwas assumed as the initial flaw in a square bar (see dashed lineinsert in Fig. 6), since the stress-intensity factor solution for a sur-face crack in a round bar was not available in the life-predictioncode [26]. (Since the majority of the fatigue life is spent as amicro-structurally small crack, the difference between a square-or round-bar would not be significant.) The calculated fatiguelives (solid curve) agreed very well with the test data at the higherstress levels, but under estimated the endurance limit. At veryhigh applied stress levels, the life-prediction model predicts fail-ure when the applied stress is nearly equal to the flow stress (ro)of the material

Using the linear extrapolated curve in Fig. 2(c) and the 25-lmflaw, the calculated fatigue lives fell very short for low appliedstress levels, but matched the test data for applied stresses greaterthan about 600 MPa. The separation point between the solid anddashed curves is at a rate of about 1e-10 m/cycle. But using aneffective threshold, (DKeff)th, of 2.5 MPaHm matched the testdata quite well with a 25-lm flaw. However, the solid curve inFig. 2(c) will be used for further fatigue-life calculations becausethe long-life tests are suspected to have been initiated as sub-surface flaws, as previously stated.

Fig. 5 Measured and calculated small-crack growth in round bar under constant-amplitude loading

Fig. 6 Assumed sub-surface initiation site (fish-eye) withextremely high rates observed at free-surface crack penetratinglocation

Fig. 7 Measured and calculated small- and large-crack-growthrates at R 5 0.1

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Figure 9 shows fatigue tests conducted on flat rectangularsheets (w¼ 25.4 mm; B¼ 1.6 or 3.2 mm) that had been polishedand cleaned. These test data were obtained from Ref. 22. Again,the maximum applied stress, Smax, is plotted against the fatiguelife. Tests were conducted at R¼�1, 0 or 0.05 and 0.54 over awide range in remote applied stress levels.

In the FASTRAN analyses, a corner crack in a sheet wasassumed as the initial flaw. A trial-and-error method was used tofind an initial quarter-circular crack that would result in reasona-ble life calculations for the R¼�1 test conditions. Again, a 25-lm flaw produced reasonable calculated fatigue lives for most ofthe stress levels (lower solid curve), and matched the endurancelimit for the R¼�1 tests quite well. Using the same initial flawsize for the R¼ 0 and 0.54 loading, produced reasonable lives atthe higher stress levels and matched the endurance limit for thehigh R tests, but under estimated the endurance limit for the R¼ 0tests, similar to the results in Fig. 8. The upper failure stress for allR ratios was very near to the flow stress of the material.

The existence of a 25-lm initial flaw early in the fatigue life onthe KT¼ 1 specimens may be unreasonable, since the averagegrain size is only about 20-lm [16]. It may be that the estimatedDKeff-rate curve in Fig. 2(c) is still not correct, the state-of-stressfor small cracks is more like plane stress (a¼ 1 to 1.1) instead ofplane strain (a¼ 2), and/or the influence of the microstructuralgrain orientation on crack-closure behavior is needed. However,the answers to these questions may have to wait for more small-

crack data, since the small-crack tests in the USAF report [16,19]had some concerns about sub-surface initiation sites and improperstress-intensity factor solutions for a free surface penetrating crack(see Fig. 6). In addition, crack-closure measurements on smallcracks at both positive and negative stress ratios are needed andmore elastic-plastic stress analyses on the early stages of small-crack growth in the proper microstructure could lead to a betterunderstanding.

6.2 Circular Hole (KT 5 3) Specimens. Figure 10 shows fa-tigue tests conducted on flat rectangular sheets (w¼ 25.4 mm;B¼ 1.6 or 3.2 mm) containing a central circular hole (D¼ 1.6mm) that had been polished and cleaned. Again, these test datawere obtained from Ref 22. Here the maximum applied grossstress, Smax, is plotted against the fatigue life, instead of the net-section stress. Tests were conducted at R¼�1, 0 and 0.54 over awide range in remote applied stress levels.

In the FASTRAN analyses, a single surface crack at the centerof the circular hole was assumed as the initial flaw. Again, a trial-and-error method was used to find an initial crack size that wouldresult in reasonable life calculations for R¼�1 test conditions. A6-lm semicircular flaw produced reasonable calculated fatiguelives (solid curve) for all of the stress levels, and even matchedthe endurance limit very well. Using the same initial flaw size forthe R¼ 0 and 0.54 loading, reasonable fatigue lives were

Fig. 8 Stress-life and calculated behavior for round bar with KT 5 1

Fig. 9 Stress-life and calculated behavior for flat sheet with KT 5 1

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calculated at all stress levels with a slight over estimations on thelong lives for the high-R tests.

The smaller initial flaw size for the circular-hole specimens arereasonable, since a much smaller volume of material is being sub-jected to the high stress level around the hole. For the un-notched(KT¼ 1) specimens, a much larger volume of material is subjectedto the applied stress and the presence of a larger flaw may be theinitiation site. Thus, the un-notched specimens may be more likelyto have sub-surface initiation sites than the circular-holespecimens.

6.3 Double-Edge Notch (KT 5 3.06) Specimens. Figures11(a) and 11(b) show fatigue tests conducted on double-edge-notched specimens with a KT¼ 3.06 (based on gross appliedstress) that were obtained from the USAF report [16,21]. Themaximum gross stress level, Smax, is plotted against the fatiguelife for tests conducted at a stress ratio of R¼�1, 0.1, 0.5 and0.8.

In the FASTRAN analyses, a surface crack located at the centerof one semicircular edge notch was considered with a total widthof 2w [see Fig. 1(c)], since the double-edge-notch configuration isnot in the current life-prediction code. Thus, the adjacent edgenotch was not considered in the stress-intensity factor solution.This assumption is satisfactory because most of the fatigue life isconsumed in the small-crack regime. Of course, the stress-concentration factor accounted for the presence of the doublenotches and the stress-intensity factor solution for a surface crackis influenced by the local stress-concentration factor.

In Figs. 11(a) and 11(b), the fatigue-life calculations have beenmade for semicircular surface cracks of 12- and 25-lm. The 25-lm flaw fit the R¼�1 test data fairly well. Whereas, the 12-lmflaws fit the test results for the R¼ 0 and 0.5 loading conditions.And the 12- and 25-lm flaw bounded the test results for R¼ 0.8loading.

7 Concluding Remarks

Small-crack theory and equivalent-initial-flaw-sizes (EIFS)were used to calculate fatigue lives (stress-life) for notched andun-notched specimens made of a Ti-6Al-4V (STOA) alloy andtested at room temperature and laboratory-air conditions. Smoothspecimens (flat sheet and round bars), flat sheet with circular holesand double-edge-notched plates were analyzed. The followingconclusions were found:

(1) For large cracks, the load-reduction test method caused ele-vated thresholds and slower crack-growth rates than thecompression precracking constant-amplitude (CPCA) testmethod.

(2) Plasticity effects on the effective stress-intensity factorrange were small, even for very high applied stress levels,but the crack-closure transients appeared to be the dominatemechanism for rapid small-crack growth.

(3) Using FASTRAN (constraint factor, a¼ 2) and small-cracktheory, smooth (KT¼ 1) fatigue specimens produced anEIFS (surface or corner cracks) of 25-lm for moderate tohigh applied stress levels at R¼�1 to 0.54, but under esti-mated the endurance limits at R¼ 0 or 0.1. Sub-surfacecrack initiation sites may have been the reason for thehigher endurance limits.

Fig. 10 Stress-life and calculated behavior for flat sheet with KT 5 3

Fig. 11 (a) Stress-life and calculated behavior for double-edge-notch specimens with KT 5 3.06 at R 5 21 and 0.1. (b) Stress-life and calculated behavior for double-edge-notch specimenswith KT 5 3.06 at R 5 0.5 and 0.8.

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(4) Using FASTRAN (constraint factor, a¼ 2) and small-cracktheory, notched (KT¼ 3 or 3.06) fatigue specimens pro-duced an EIFS (surface cracks) from 6 to 12-lm for posi-tive stress ratios (R¼ 0 and 0.54), and 25-lm for negativestress ratio (R¼�1) loading.

Acknowledgment

Authors thank the U. S. Air Force Research Laboratories(AFRL) for the fatigue test data on the titanium alloy; and Pratt &Whitney for financial support of this research effort.

Nomenclaturea¼ crack depth measured in thickness directionai¼ initial crack depthB¼ thicknessc¼ crack half-length measure in width directionci¼ initial crack half-lengthD¼ specimen, circular-hole or notch diameterE¼ modulus of elasticity

hn¼ void (or crack) half-heightJ¼ path-independent integral around crack tip

KIe¼ elastic fracture toughnessKJ¼ stress-intensity factor computed from J-integralKo¼ crack-opening stress-intensity factor

Kmax¼ maximum stress-intensity factorKp¼ plastic-zone corrected stress-intensity factorKT¼ elastic stress-concentration factorNf¼ cycles to failureR¼ stress ratio (Smin/Smax)S¼ applied stress

Smax¼ maximum applied stressSmin¼ minimum applied stress

So¼ crack-opening stressw¼ specimen widtha¼ constraint factor

DJeff¼ effective J-integralDK¼ stress-intensity factor range

DKeff¼ effective stress-intensity factor range(DKeff)T¼ effective stress-intensity factor range at transition(DKeff)th¼ effective stress-intensity factor range threshold(DKp)eff¼ cyclic-plastic-zone corrected effective stress-intensity

factor rangeDKth¼ threshold stress-intensity factor range

q¼ plastic-zone sizero¼ flow stress (average yield and ultimate)rys¼ yield (0.2% offset) stressru¼ ultimate tensile strengthx¼ cyclic plastic-zone size

References[1] Pearson, S., 1975, “Initiation of Fatigue Cracks in Commercial Aluminum

Alloys and the Subsequent Propagation of Very Short Cracks,” Eng. Fract.Mech., 7, pp. 235–247.

[2] Kitagawa, H., and Takahashi, S., 1976, “Applicability of Fracture Mechanics toVery Small Cracks or the Behaviour in the Early Stages,” Proc. Second Interna-tional Conference on Mechanical Behavior of Materials, Boston, MA, pp.627–631.

[3] Zocher, H., ed., 1983, “Behaviour of Short Cracks in Airframe Components,”AGARD Conf. Proc. No. 328.

[4] Ritchie, R. O., and Lankford, J., eds., 1986, Small Fatigue Cracks, The Metal-lurgical Society, Inc., Warrendale, PA.

[5] Miller, K. J., and de los Rios, E. R., eds., 1986, “The Behaviour of Short Fa-tigue Cracks,” European Group on Fracture, Publication No. 1.

[6] Ritchie, R. O., and Lankford, J., 1986, “Overview of the Small Crack Problem,”Small Fatigue Cracks, R. O. Ritchie and J. Lankford, eds., The MetallurgicalSociety, Inc., Warrendale, PA, pp. 1–5.

[7] Newman, J. C., Jr., 1983, “A Nonlinear Fracture Mechanics Approach to theGrowth of Small Cracks,” Behaviour of Short Cracks in Airframe Components,Zocher, H., ed., AGARD Conf. Proc. 328, pp. 6.1–6.26.

[8] Newman, J. C., Jr., Swain, M. H., and Phillips, E. P., 1986, “An Assessment ofthe Small-Crack Effect for 2024-T3 Aluminum Alloy,” Small Fatigue Cracks,The Metallurgical Society, Inc., Warrendale, PA, pp. 427–452.

[9] Newman, J. C., Jr., and Edwards, P. R., 1988, “Short-Crack Growth Behaviourin an Aluminum Alloy - an AGARD Cooperative Test Programme,” AGARDConf. Proc. R-732, pp. 1–96.

[10] Newman, J. C., Jr., Phillips, E. P., and Swain, M. H., 1997, “Fatigue-Life Pre-diction Methodology using Small-Crack Theory,” NASA Paper No. TM-110307.

[11] Newman, J. C. Jr., Phillips, E. P., and Everett, R. A., Jr., 1999, “Fatigue Analy-ses Under Constant- and Variable-Amplitude Loading using Small-Crack The-ory,” NASA Paper No. TM-209329.

[12] Newman, J. C., Jr., Schneider, J., Daniel, A., and McKnight, D., 2005,“Compression Pre-Cracking to Generate Near Threshold Fatigue-Crack-Growth Rates in Two Aluminum Alloys,” Int. J. Fatigue, 27, pp.1432–1440.

[13] Ruschau, J. J., and Newman, J. C., Jr., 2008, “Compression Precracking to Gen-erate Near Threshold Fatigue-Crack-Growth Rates in an Aluminum and Tita-nium Alloy,” J. ASTM International, 5(7), pp. 1–11.

[14] Yamada, Y., and Newman, J. C., Jr., 2009, “Crack Closure Under High Load-Ratio Conditions for Inconel-718 Near Threshold Conditions,” Eng. Fract.Mech., 76, pp. 209–220.

[15] Suresh, S., 1985, “Crack Initiation in Cyclic Compression and Its Application,”Eng. Fract. Mech., 21, pp. 453–463.

[16] Pippan, R., Plochl, L., Klanner, F., and Stuwe, H. P., 1994, “The Use of FatigueSpecimens Precracked in Compression for Measuring Threshold Values andCrack Growth,” J. Test. Eval., 22, p. 98.

[17] Gallagher, J. P., 2001, “Improved High-Cycle Fatigue (HCF] Life Prediction,”AFRL-ML-WP-TR-2001-4159.

[18] deLaneuville, R. E., and Sheldon, J. W., 2001, “Crack Growth,” AFRL-ML-WP-TR-2001-4159, Appendix 3B, pp. 1–12.

[19] Lenets, Y. N., 2001, “Nucleation and Propagation of Small Cracks,” AFRL-ML-WP-TR-2001-4159, Appendix 3C, pp. 1–14.

[20] Bellows, R., 2001, “Effect of Specimen Geometry and Frequency on HCFSmooth Specimen Behavior,” AFRL-ML-WP-TR-2001-4159, Appendix 3F,pp. 1–4.

[21] deLaneuville, R. E., and Sheldon, J. W., 2001, “Notch Fatigue,” AFRL-ML-WP-TR-2001-4159, Appendix 3J, pp. 1–6.

[22] Metallic Materials Properties Development and Standardization, U.S. Depart-ment of Transportation, Federal Aviation Administration, 2003, Washington,D.C., Paper No. DOT/FAA/AR-MMPDS-01.

[23] Garr, K. R., and Hresko, G. C., 2000, “A Size Effect on the Fatigue CrackGrowth Rate Threshold of Alloy 718,” ASTM Spec. Tech. Publ., 1372, pp.155–174.

[24] Newman, J. C., Jr., 1981, “A Crack Closure Model for Predicting Fatigue CrackGrowth Under Aircraft Spectrum Loading,” ASTM Spec. Tech. Publ., 748, pp.53–84.

[25] Elber, W., 1971, “The Significance of Fatigue Crack Closure,” ASTM Spec.Tech. Publ. 486, pp. 230–242.

[26] Newman, J. C., Jr., 1992, “FASTRAN-II - A Fatigue Crack Growth StructuralAnalysis Program,” Paper No. NASA TM 104159.

[27] Newman, J. C., Jr., 1992, “Effects of Constraint on Crack Growth Under Air-craft Spectrum Loading,” Fatigue of Aircraft Materials, A. Beukers ed., DelftUniversity Press, The Netherlands, pp. 83–109.

[28] Newman, J. C. Jr., Ruschau, J. J., and Hill, M. R., “Improved Test Method forVery Low Fatigue-Crack-Growth-Rate Data,” Fatigue Fracture Eng. Mater.Struct. (to be published).

[29] El Haddad, M. H., Dowling, N. E., Topper, T. H., and Smith, K. N., 1980, “J-Integral Application for Short Fatigue Cracks at Notches,” Int. J. Fracture, 16,pp. 15–30.

[30] Hudak, S. J., Jr., and Chan, K. S., 1986, “In Search of a Driving Force to Char-acterize the Kinetics of Small Crack Growth,” Small Fatigue Cracks, R. O.Ritchie and J. Lankford, eds., Metallurgical Society, Warrendale, PA, pp.379–405.

[31] Rice, J. R., 1968, “A Path Independent Integral and the Approximate Analysisof Strain Concentration by Notches and Cracks,” J. Appl. Mech., 35, pp.379–386.

[32] Newman, J. C., Jr., 1992, “Fracture Mechanics Parameters for Small FatigueCracks,” ASTM Spec. Tech. Publ. 1149, pp. 6–28.

[33] Lenets, Y. N., Bellows, R., and Merrick, H., 2000, “Propagation Behavior ofNaturally Initiated Cracks in Round Bars of Ti-6Al-4V,” Proc. 5th NationalTurbine High Cycle Fatigue Conference, HCF’00.

[34] Lenets, Y. N., Nelson, R., and Merrick, H., 2002,” Growth of Small Cracks atDifferent Stress Ratios,” Proc. 7th National Turbine Engine High Cycle FatigueConference, HCF’02.

[35] Lanciotti, A., and Galatolo, R., “Short Crack Observations in Ti-6Al-4V underConstant Amplitude Loading,” AGARD Conf. Proc., R-732, pp. 10.1–10.7.

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