Gear Crack Propagation Investigations

IntroductionA common design goaJ for gears in heli-

copter or turboprop power transmissions isreduced weight To,help meet this goal, somegear de igns use thin rims. Rims that are toothin. however, may lead to bending fatigueproblems and cracks. The most commonmethods of gear design and analysis arebased on standards published by theAmerican Gear Manufacturers Association,Included in the standards arerating formulasfor gear tooth bending to prevent crack initi-ation (Ref. I), These standards can. includethe effect of rim thicknes on tooth bendingfatigue (Ref. 2). The standards. however, donol indicate the crack propagation path or theremaining life once a crack has started,Fracture mechanics has developed into auseful di cipline for predicting strength andlife of cracked structures.

Ahmad and Lao (Ref. 3) applied fracturemechanics [Q gear teeth to illustrate the pro-cedure and estimate crack propagation direc-lion. Honda and Conway (Ref, 4) alsoapplied fracture mechanic to simulate toothcrack propagation. compute threshold loadsand calculate tooth life. Fla ker and Jezernik(Ref. 5) applied fracture mechanics to gearteeth to estimate stress intensity factors andgear life. Researchers at Tohoku Universityin Japan performed a series of analy es andexperiments to determine Ithe effect of re id-ual tress on crack initiation and propagation(Refs. 6, 7). Moo, Daniewicz et aI. (Ref. 8)developed a comprehensive, self-containedanalysis package to refine the spur gearbending fatigue theory using fracturemechanics. Lastly, Flasker and Pehaa (R~f.9) described their method for calculatingcrack propagation in gear teeth using fracturemechanics. Much of the work of the abovereference considered only an initial crack,and propagation paths were not considered.Many of '!he reference that did com idercrack propagation assumed the propagationoccurred in a straight path. In adduioa,experimental validation of the cited analyseswas sparse. Finally, no work using fracture18 GEAR TE.CHNOLOG'f

David G. L-ewic~kiand Roberto, BaUaliini

mechanics was performed for thin-rim gears.The objective ofthis study was to deter-

mine the effect of gear rim thickness allcrack: propagatien life, From M extensiverudy (Ref. 10'), linear elastic fracture

mechanics was used. ~o analyze gear toothbending fatigue in standard and tllin-rimmedgears, Finite element computer programwere u ed to' determine stress di tribution ,estimate stre intens:ity factors and modelcrack propagation, Various fatigue crackgrowth model were used to estimate crackpropagation life. Experimental tests wereperformed to validate predicted crack propa-gation results.

Fatigue 'Crack GrowthMany machine elements, such as gear

teeth, are cyclically loaded in application.The overall fatigue life of such componentsmaybe repre ented bythree di tinci. phases:I) crack initiation, 2) crack propagation !IJJld3) final failure. Once crack initiation hasoccurred, fracture mechanics may be used to.estimate crack propagation fatigue growthrate and time to final failure,

The mo I universally used method to cal-culate crack propagation crack growth waspostulated by Paris and Erdogall (Ref. I.I),Purely Mode I loaded specimens subjectedto cyclic load were considered. Unstablecrack growth such that the stress intensityfactor grew with increasing crack size wasalso considered. Parispostulated that therateof crack growth with respect to number ofstress cycles was a logarithmic relationshipwith the stress intensity factor range a

~ =C(.1Kf'dN

where da is the change in crack lengthfor dNnumber of stress cycles, .1K is the range ofthe Mode I stress intensity factor .1 a giventime and Cand 1'1 are material constant. Thematerial constants C and n must be deter-mined by some experimental. means,

Further research of fatigue crack growthhas shown three important factors 1I0[ con-sidered ill the Paris model. First wasthe

effect of load ratio R on crack growth IN =minimum cyclic load/maximum cyclic load),Second was the instability of crack: growthobserved when the tre intensity [actorrange approached the material's fracturetoughness index, KIC" Thlird was the pre neeofa stress intensity threshold factor M(rho

The streintensit:y tbre hold factor is thehighest sire s inten ity factor in which nocrack growth would occur. The Colliprie t

crack growth model (Ref.12) account forthese effects where

1 M(l ).+(::. r,wf.1(!~~~~·)1)(2)tlK'h

] II addressing appli cati ons '1.0 gears, Inoueet al. (Ref 7) describe fatigue crack growth

of gear bending fatigue tests. Here, crackgrowth equations were derived as a functionof crack depth through a. gear tooth. Theexpression derived for crack growth rateda/dN. as a function of sires inlensity rangeLlK was

where the parameters K,co' IX, LtK:co .1K'H TIand A were aJI estimated as a function oftooth hardness (Ref. 7),

Crack PflopaglltioD Simulation,,1) Theanalysu of the current sludy used the

FRANC (FRacture ANalysis Code) comput-erprogram described by Wawrzynek (Ref]3), FRANC isa general purpose finite ele-ment code forlihe static analysis of crackedsmicnrres. FRANC is designed Ior two-dimen ional problems and i capable of ana-lyzing plane strain, plane tress or axi-sym-metric problem .

Among the variety of capabilities, aunique feature of FRANC i the ability to

model a crack in a structure. fRANC use amelhod cal1ed "delete and fill" to accom-pli h this ..To illustrate. the user would firstdefine an initial crack by identifyingtbenode of the crack mouth and coordinates ofthe crack lip. FRANC will then delete theelements in the vicinity of the crack tip.FRANC will next insert a rosette of quarter-point. six-node lrlangular elements aroundthe crack tip to model the inver e quare-root stre s singuLarily (Ref. 14. IS). Finally.FRANC will filii the remaining area. betweenthe roselle and original mesb with conven-tional! ix-node triangular elements. The usercan then nm the unite element, equationelver to determine nodal displaeemems,

force, 'stresses and strains.

A further uaiq ue feature of FRANC is !heautomatic crack prepagation capability,After an initial crack i 'inserted in a me h.FRANC model a propagated crack as anumber of lJaighl line egment. For each, egment, FRANC models the crack. tip u inga 1'0 ene of quarter-point elem nt . FRANCthen solves, the fini.te element equations. cal-culates the stre inrensity factors and calcu-Iatesthe crack propagation angle. After llIecrack propagation angle is determined.FRANC then places the new crack tip atthecalculated angle and at a user-defined crackincrement length. The model i thenreme hed u ing the "delete and fill" methodde cribed above. The procedure is repealedII pecific number of times as pecified bythe user. 10 the current mely, the tresintensity factors were determined from thecalculated nodal. di placements using jhedisplacement correlation metbod I(Ref. 16).

TIle method of rdogan and Sih (Ref. 1.7)wa used in the current tudy to determinethe crack propagation angle.

Once lhe stres inten ity factors aredetennined for each segment, the predictednumber of crack: propagation, cycles can beestimated using, the fatigue crack growthmodels. Regardless of th model used •. theera k growlll rates daMN. were of the form

~",g(M() (4)dN

where g(.M() is given by Eq. ] for the Pari .relationship. 1&;1. 2 for the CoHiprie t rela-tionsilip or 13q. 3 f01' Inoue's method. Thepredicted number of crack: propagationcycles for me i!.1J crack segment, N" was esti-mated by

Fig. m - lnlte 'element model of gean used In.cmck propagation studies" solid model, mR '" 3.3.

r Test Gears

Oil Seal Gu Row 'l,r

III,I I

I rI I

Viewing Port Drive Shafl

Tesl Gear,Coyer '

Shaft OilSeal

TestLubricant ~Inlet ...»"

,L Test Lubricant Dullel

TemperatureMeasurement Location

.Fig,2: -!\SA Lewis Spur Gear Fatigue Rig.

(b) C-Q4-0231l1

Fig. 3 'ean used Ito delennine' I/:ltcct of rim lhi.ckncss(8) mil = 3,,3,. (b,) mil = '0.3.

N,OVEMBERIOEC ,!.IBER '187 19

Fig. 4 - Specialized cra.ck propagation gages for gear tooth crack growtb measurements. (a), increasein gage electrical res.lst:mce as 'the number of broken strands increases. (b) Installatlen 'of crack propa-g ulon gllge. on test gea:J.

UMBER OF BROKEN STRANDS

(b) C-94-05937

~r----------------------------------.

10

1

0.0 1.5 2.00.5 1.0

CRACK LENGTH, mm

Fig . .s - Crack propagation rutigugrowlb for 'Iest 1" ma = 0.3.20 GEAR TECHNOLOGV

(5)

where Qj was the crack lengUil of the i,), seg-

ment, Dr_I wasthe crack length of the (i-II'lnsegment, Nj.( was the Dumber of cycles of

the (i-I )111 egment and g( &9 was the aver-age crack growth rate of the i,},and (i-I),hsegments. Note that G1 was the initial crack

length. and NI '" 0 and i varied from 2lO thetotal number of segments.

Gear Finite Element Modeling

Basic gear tooth geometry data was inputto a tooth coordinate generation computer

program. The tooth coordinate generatorprogram used the method of Hefeng et aI.

(Ref. ,18) to determine the tooth coordinates.

The output was tooth coordinate and rimcoordinate data which defined a single toothector of a gear. This oulput w u ed by a

commercially available pre- and post-pro-I I cessing finite element analysis software

package (Ref. 19). This package created thefinite element me h of rue complete gear.

FRANC then used this mesh and performedcrack propagation simulations.

Fig. J shows a sample finite element

mesh of an uncracked gear. The tooth geom-etry used modeled that of the te t gears of theNASA Lewis Spur Gear Fatigue Rig(de cribed in the following ection), Theanalysis used 8-node, place lIe ,quadrilat-eral finite elements. The mesh was refilled inthe region of the loaded tooth for improvedaccuracy. The model of Fig. I had 2,353 ele-ments and 7.295 nodes. Material propertiesused were that of AJSI 93 W steel, The toothload was placedat the highest point of single

tooth contact. For boundary conditions, fourhub node were fixed, In addition, gears with

various rimthicknes e were modeled. Theparameter describing the rim thicknes wasthe backup ratio mlJ. where

m --LB- h, (6)

where b was the rim thiekne s, and h wa thetooth whole depth. Gears with various back-up rados were modeled by inco!pOmlingslots in the model. All cases used the samefinite element mesh for the loaded tooth.

Test. Facility

Crack propagation experiments were per-formed ill the NASA Lewis SpUI: GearFatigue Rig (Fig. 2). The test stand operated

all a torque regenerative principle in wltich

'torque was circulated in a loop of test gearsand slave gears. Oil pre u:re was supplied toload vanes in one slave gear which displacedthe gear with respect to its shaft. This pro-duceda torque all the test gears, slave gearsand connecting han proportional to theamount of applied oil pre SUE. A [9 KW (25hp). variable-speed motor provided peed tothe drive shaft using a belt and pulley. Thelubricant used for the gears, bearings andloading systems was a synthetic paraffinicoil. The test gear lubricant was filteredthrough a 5-microll fibergla s filter.

Test GearsThe test gears were 28-tootn, g-pitch, 20"

pre ure angle external pur gears with II facewidth of 6.35 mm (0.25"). The teeth hadinvolute profiles wi,lh linear tip lrelief startingat. 'the highe t point. of ingle tooth contactand ending at the tooth tip at an amount ofO.oJ3 mm (0.0005"). AI] test gears used inthe experiments were fabricated andmachined from a single batch of materialThe te t gear material wa vacuum-melted,consumable-electrode AISI 93IO steel. Thegears were case carburizedaad ground. Theteeth were hardened to a case hardness of Rc6] and a core hardnes of Rc 38. The effec-live case depth (depth at a hardness of Rc 50)was 0.81 rom (0.032"). Two different te t

gear designs were considered. The first wasa thick-rirnmedgear with .3 backup ratio ofrnB = 3.3 (Fig.. 3a). The second was a thin-rimmed gear whicb incorporated slots (Fig.3b). The backup ratio of the thin-rimmedgear was rnB = 0.3.

It was believed that tooth bending fatiguecracks would be difficult to initiate based onthe load capacity of the te I rig. Due 10 this,notches were fabricated in tlJe fillet region(loaded side) on one tooth of each of the te t

gears 10. promote crack initiation. The notch-es were fabricated using electrical dischargemachining (EDM) willi a 0.10 mID (0.004")diameter wire electrode. The nominal notchdimensions were 0.20 mm (0'()()8") in lengthand O.[3 mm (0.1005") in width along thefull face width of the tooth. The notche werelocated at the same locatIon for both te t

gears ..This location was at a radius of 40.49mm (1.594") on the fillet. which was theposition of the greate t [ensile stress for thesolid gear (mB "'. 3.30). The notches produceda Ires concentration factor of approximate-Iy three as determined using a finite dementanaiysi .

InstrumentatlonThe standard te t rig instrumentation

monitored test gear speed, oil load pres ure,test gear and slave gear oil. pressure and oiltemperatures. Also, overall test stand vibra-tion was monitored using an accelerometermounted on the top housing. Inaddnion tothe standard facility vibration sensor, anadvanced vibration processing diagnosticsystem was installed ill the test stand to helpassist in crack detection. Crack propagationgages were used in the experiments to deter-mine fatigue crack growth. Special gageswere fabricated for installation in the toothfillet region of the test gears. The gages hadten circular strand with an inner radius of1.52 mm (0•.060") and an outer radius of 3.05mm (0.120") (Fig. 4). The strand weredesigned to break as the crack propagatedthrough them. which in tum increased theelectrical resistance of the gage (Fig. 4a) ..Fig.4b shows the installation of a gage in the fil-let region of a notched tooth. A gage wasinstalled on each side of the tooth flank foreach gear instrumented with crack gages. Theelectrical resistance of the crack gages wasmonitored along with the load cycle count toestimate cycles as a function of crack length.The information from the rotating crackgages wastransferred through brush-type sliprings. Also, an infrared tach sensor was usedto measure number of load cycles.

MeasuJred Gear Fatigue 'Crack GrowtbThethin-rimmed gear was u ed ill Test 1.

The test wa run at 89 Nm (786inllb) torqueand 10,000 rpm speed for 6.5 hrs .• at whichtime rim fracture occurred. Fig .. 5 plots thenumber of load cycles as a function of themeasured crack:lengtll. The crack gagere ults indicated the crack growth was non-ull.iform throughout the tooth face width. Acrack: started on the rear flank of the tooth atthe tip of the notch and reached an initial sizeof 0.46 mm (Om 8") at l ,060,000 cycles.The crack: continued to propagate throughthe rear Hank, but did not reach the frontflank until approximately 2,.680,000 cycles.At 2,910,000 cycles, the crack reached 11 sizeof 0.64 mm ~O.025")on the front flank, butcompleted propagation through the rear gageby this lime. Even though the crack initiationtime was not uniform throughout the toothface width, the crack propagatiourate wasuniform. This wa indicated by the similari-ty in slopes of the curves in Fig. 5 for Gages1 aad 2.

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ITflW'CIRCLE. 110

NOV.EW8ERIOEceWBER 1U7 21

0.20

0.15~

""d>- ().10UZ9

~ 0.05

0.000.0

Gage I. Front Tooth Flank

Gage 2, Rear Tooth Flank

0.5 1.0 1.5

CRACK LENGTH. mm

Fig. 6 - Cl'ackpropag!!tion, fatigue growth fOI" Test 2, rna ::: 3.3.

e 50--...'"~Ii 40

~~G 30

""Z~ 20""""~Iii 10UJc0 (a)::i: 0

0.0 0.5 1.0 1.5 2.0 2.5CRACK LENGTH. mm

I. Pans equauoe, n = 2.204. C = 1.460. 10-' mmlcyd(Mpa-/m)'2. Puis equation, n = 2.954,. C = 8.433 1H)"' m_n1Jcycl(MPtr/m'iJ. Pan, equation, n = 2.555. C = 2.512 • W·' mm,"-yd(MPa:or.;;r4. Pari, equation, n = 2.420. C = 3.988 • 1()"' mmJoyd(MPtr/m'i5. Colhpriest equation6. Inoue equation 6

0.2

0.1 -

0.5 1.0 1.5 2.0 2.5

CRACK LENGTH. mm

F,ig.,7 - Compari on, of predicted crack propagatton cycles using Paris, Collipriest and Inoue eqna-lions. Mod~lfor mn 0; 0.3. (a) Mode 1 stress intensity factor; (b) Life comparison.22 GIE. .. R TECHNOLOGY

The thick-rimmed gear was used in Te t

2. This gear was run at 136 Nm (1200 in/lb)torque and 10,000 rpm speed for ]5 minutes,at which time tooth fracture occurred. Fig. 6gives the processed crack propagation results

for Test 2. Note that the crack. initiation andcrack propagation was fairly uniform

throughout the tooth face width for this test.Comparison of Predicted and

Measured Crack GrowthThe FRANC computer program Was used

to simulate crack propagation and 'calculateMode I stress intensity factors as III functionof crack length. The predicted stress inten i-ty factors were then used with three different

fatigue crack growth models (Paris.Coll:ipriest and Inoue) to estimate crackpropagation.

A comparison of predicted crack propa-

gation cycles using the Paris .•Collipriest andInoue methods is shown in Fig. 7. For this,the thin-rimmed model (mo = 0'.3) was used

to simulate the test gear of Fig. 3b. An initialcrack of 0.64 mm (0.025") was placed in the

tooth fillet at the location of the maximumtensile stress. Crack propagation was thensimulated, and the Mode [ stress intensityfactor as a function of crack length is giveniII Fig, 7a. From tbis, six differeat fatiguegrowth cases were considered. The first fourcases used. the Paris equation and materialconstants of AISI 9310 specimens fromexperiments of Au and Ke (Ref. 20'). The

fifth Case used the Collipriest equation andA~S[ 9310 material constants from Forman

and Hu (Ref. 21). The load ratio used was.R= 2.6,. as determined from the fillite element

analysis, The sixth case used Inoue'smethod and the material constarus of theSCM 4]5 material (SCM 415 is a high-

strength Japanese steel with properties simi-lar to A]S[ 9310). The predicted number of

cycles per crack length varied significantlyamong the cases studied (Fig. 7b). Note thatthe cycles were defined as the number of

crack propagation cycles after an initialcrack of 0..64 rnm (0..0.25").

Predicted crack growth for the rno = 0.3and 3.3 gears was compared to the measu.redcrack growth from the experiments. Again,

the six different prediction schemes men-tioned above were used. The predicted num-ber of crack propagation cycles using thesixth schemes were, for the most pan,extremely low compared to the measurednumber of cycles from the experiments. To

account for this, the concept of fatigue crackclosure was investigated. Elber (Ref. 22)performed crack experiments on aluminumalloys and deduced (hat residual compres-sive stresses existed near the crack tip regionbecause of pLasticdeformation. These resid-ual stresses reduced the effective stressintensity Factor range (and thus, increasedcrack propagation life) and provided a betterfit to experimental data than other empiricalexpressions, Elber proposed. an. effectivestress intensity range ratio U such tll.at

where t1KdJ was the effective stress intensityfactor range. Elber then used the effectivestress intensity factor range in the Parisfatigue crack growth model. ill addition,Elber defined U through experimental stud-ies as a linear function of the load ratio R.

The concept of fatigue crack closure wasapplied to the current gear crack experi-ments and predictions. A study was conduct-ed to estimate the effective stress intensityfactor range ratio for the experiments. Thepredicted number of crack propagationcycles using the same six schemes wereplotted versus crack length at a variety ofarbitrarily chosen U ratios. For the Parisequation and material. constants 11 := 2.954and C '" 8.433 X 10'9 rrunlcyC!(MPadm)D,good correlation between predicted crackcycles andtheexperiments occurred when:1) U '" 0.4 for R '" 2.6, and 2) U '" 0.8 for R'" 0.1. Assuming a linear relation between U

and R produced

:= 0.82 + O.16{R)

Fig. g shows a sample comparison of pre-dicted and measured crack growth. when thefatigue crack closure concept was used. Thecycles were defined as the number of crackpropagation cycles after an initial crack of0.64 rom (0.025")., ~t should be noted thatgood correlation was also achieved whenthe Collipriest equation was used with cer-tain U values ..This produced a relationshipsimilar to Bq, 8, but with different coeffi-cients (Ref. W).

Fig. 9 displays the effect of rim thicknesson predicted Mode I stress intensity factorsand predicted crack propagation cycles. Thestress intensity factors were determinedfrom FRANC using the appropriate finite

0.2

<IlU.I...JU>-U

~0.1 -

~::2

0.00.0 0.5

(7) CRACK LENGllI. mm

1.0 1.5 2.0 2.5

Fig. 8:- Comparison of predicted crack propagation cycles to experiments. Paris fatigue crack g,-owthmodel, II = .2.954. C = 8.433 '" 10'9 mm!cycl(MPa..[m)", R '" -2.6, V '" 0.4, used for predlctlens. Test I. GageI, front 'Dank, for expernnents,

25 (a)

20 -

15

10

5

DO

6

5 -

4

3 -

2 -

0'0'

.1118=0'.5

2 3 4

CRACK LENGTH. mm

(8)

CRACK LENGTH, mm2 3 4

s 6

5 6

Fi.g.9 -!Effect of backup rauo on tress intensity factors and crack propagation cycles. (a) Mode' 1 stressintensity factors. (b) Crack propagation cycles, Paris fatigue crack growth model, n '" 2.954, C '" 8,433 X

10,9 mmicycl(MPa..Jm)n, U = 0..82 + 0.16 R.

element models. The Paris equation wasused along with the effective stress intensi-ty range ratios of Eq. 8. The initial cracks ofthe various models were placed at the loca-tion of the maximum tensile stress in thetooth fillet. The stress intensity factors werelowest for the rns := 0.5 case. This gave the

highest predicted number of cycles for 1II.ecases studied. The cycles were all defined asthe number of crack propagationcycles afteran initial crack of 0.28 rom (0.011 "), Thestress intensity factors were highest for thernB ;;;;:0.3 case. However, the predicted lifefor this was somewhere between the ca e of

NQVEM6ERIOECE·1oI6ER '~1I7 23

ms = 0.5 and l.0 due to the fatigue crack c1o-su.re effect. The cases of mB = 3.3 and 1.0gave nearly the same predicted life.

ConclusionsBased on these analytical and experimen-

tal studies, the following conclusions weremade: 1) Good correlation between predict-ed and measured gear crack. growth wasachieved when the predictions used the Pariscrack growth equation and the concept offatigue crack closure, 2) For thin rims, adecrease in rim thickness caused an increasein both the stress intensity factorand thecompressi ve cyclic stress in the gear toothfillet region, The increase in stress intensityfactor promoted crack growth, while theincrease in cyclic compressive stress tend-ed to retard crack growth and increase thenumber of propagation cycles to failure, 0

Acknowledgments: The authors wish tothank Dr. Paul A. Wawrzynek of FractureAnalysis Consultants, Inc..for fruitful discus-sions andJor providing .theFRANC program.

Prepared for An Integrated Monitoring,Diagnostics & Prevention TechnologyShowcase, Society for Machinery FailurePrevention Technology, Mobile, AL, April22-26, 1996.

References:1. "Fundamental Rating Factors and

Calculation Methods for Involute Spur andHelical Gear Teeth." ANSIIAGMA 2001-B88, American Gear ManufacturersAssociation, Alexandria, VA, 1990.

2. Drago, R.. J. and R.. V. Lutthans,"Combined Effects of Rim Thickness andPitch Diameter of Spur Gear Tooth Stresses."Journal of the American Helicopter Society.Vol. 28, July, L983, pp. 13-19.

3. Ahmad, J. and F. T. Loo. "On the Useof Strain Energy Density Fracture Criterionin the Design of Gears Using Finite ElementMethod." ASME Paper No.. 77-DET~158.Presented at the Design TechnicalConference, Chicago, IL, Jun., 1977,

4, Honda, H. and J. C. Conway, "AnAnalysis by Finite Element Techniques of theEffects of a Crack in the Gear Tooth Filletand Its Applicability to Evaluating Strengthof the Flawed Gears." Bulletin of the JSME,Vol. 22, No. 174, Dec., 1979, pp. 1848-1855.

5. Flasker, J. and A. Jezernik. "TheComparative Analysis of Crack Propagationin the Gear Tooth." Proceedings of the Int.24 GEAR TECHNOLOGY

Corif. of Application of Fracture Mechanicsto Materials and Structures, Freiburg,Germany, June, 1983. pp. 971-982.

6. Kala, M. et al, "Strength Evaluation ofCarburized Gear Teeth Based on FractureMechanics." Proceedings of theKSMElJSME Joint Conference "Fractureand Strength '90," Seoul, Korea, 1990, PI',248-253.

7. Inoue, K. et al. "Fracture MechanicsBased Evaluation of Strength. of CarburizedGear Teeth." Proceedings of the JSMEInternational Conference on Motion andPower Transmissions. Hiroshima, Japan,Nov" 1991, pp. 80[-806,

8. Daniewicz, S, R. et al, ''The StressIntensity Factor and Stiffness for a CrackedSpur Gear Tooth," Journal of MechanicalDesign, Vol. 116, No.3, Sept., 1994,

9. Flasker, 1. and S. Pehan, "CrackPropagation in Tooth Root with VariableLoading .." Communications in NumericalMethods in Engineering, Vol.. 9, No.2, Feb.,1993,. pp. 103-110.

10. Lewicki, D. G. "Crack PropagationStudies to Determine Benign or CatastrophicFailure Modes for Aerospace Thin-RimGears.." Ph.D. Dissertation, Case WesternReserve University, May, 1995.

11. Paris, P. C. and E Erdogen. "A CriticalAnalysis of Crack Propagation Laws .."Journal of Basic Engineering, Vol. 85, L963,pp. 528-534.

12. Collipriest, l E. "AnExperimentalist's View of the Surface FlawProblem." The Surface Crack: PhysicalProblems and Computation Solutions,American Society of Mechanical Engineers,1972, pp. 43-61.

13. Wawrzynek, P. A. "Discrete Modelingof Crack Propagation: Theoretical. Aspectsand Implementation Issues in.Two and ThreeDimensions." Ph.D. Dissertation, CornellUniversity, 199L

14. Henshell, R. D. and K. G. Shaw,"Crack Tip Finite Elements AreUnnecessary." International Journal forNumerical Methods in Engineering, VoL 9,1975, pp. 495-507.

15. Barsoum, R. S. "On the Use ofIsoparametric Finite Elements in LinearFracture Mechanics." Journal for NumericalMethods in Engineering, Vol. 10, No. I,1976, pp. 25-37.

16. Tracey, D. M. "Discussion of 'On theUse of Isopara-metric Finite Elements in

Linear Fracture Mechanics' by R. S.Barsoum." International Journal forNumerical Methods in Engineering, Vol. 11,1977, pp. 401-402.

n Erdogan, F. and G. C. Sih, "On theCrack. Extension in Plates Under PlaneLoading and Transverse Shear." Journal ofBasic Engineering, Vol 85, 1963, pp.519-527.

18. Hefeng, B. etal. "Computer Modelingof Rack-Generated Spur Gears." Mechanismand Machine Theory, Vol. 20, No.4, 1985,pp.351-360.

19. P31PATRAN User Manual. POAEngineering. Costa. Mesa, CA, 1993.

20. Au, 1. J and J. S. Ke. "CorrelationBetween Fatigue Crack Growth Rate andFatigue Striation Spacing in AISI 9310Steel." F,:actography and Materials Science,ASTM STP 733, 1981, pp. 202-221.

21. Forman, R.. G. and T. Hu."Application of Fracture Mechanics on theSpace Shuttle." Damage Tolerance ofMetallic Structures, ASTM STP 842, 1984.pp. 108-133.

22. Elber, W. "The Significance ofFatigue Crack Closure." Damage Tolerancein Aircraft Structures, ASTM STP 486, 197..pp. 230-242.

David 'G..Lewickiis with the Vehicle Technology Center; U.s. ArmyResearch Laboratory. Lewis Research Center;Cleveland, OH.

Roberto 'Bal:lariniis in theDept. of Civil Engineering and Mechanical& Aerospace Engineering,Case Westen! ReserveUniversity, Cleveland, OR.

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