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Section 7.3.3. The obtain the fracture For the goal is to predict path observed constant what operating in the tested torque level of conditions number million. of cycles The model in-lb, sensitivity of the the tooth approximately fatigue fractures life 2.5) is not captured loads during 7.2a. This the cycle type increasing prediction continuously at an increasing Crack growth failure during the crack is considered crack grows, under tooth will deflect of the load [Alban will predict a limited 1985]. The when amount displacement fatigue" magnitude criterion, a shorter a cracked it's 4.9 for the scenario analyses, crack Figure in the SIFs growth will such as K 1 = K_c. number of cycles to and uncracked tooth adjacent picks of the crack maximum and will be roughly equal for every remaining load rate of increase in the SIFs will decrease, and reach roughly than of the applied and results Fatigue before hundred chosen propagation 7.2b. a fracture in reality, values load control Figure load control This is because, to the tooth thickness. in the tested pinion. tooth-bending The five test is smaller to the work. rate until the SIFs satisfy simulations the cracked scenario as the than observed. up a portion numerical was kept constant of loading occur in the through occurred in the parameters is studied in Section 7.3.1. Alban's condition number four for "classic (Section mesh, to produce be necessary pinion. 3120 thousand cycles were predicted to propagate the crack This value is the same order of magnitude as that which The would tooth faces reaches cycle. As a result, the a constant maximum every cycle. An idealization of this is shown schematically in Figure 7.2c. Propagating the crack under these conditions is considered displacement control. When the maximum SIF ceases to increase, the fatigue crack growth rate is relatively constant, and the number of cycles to failure increases. P_ K / Tithe a) b) K Time c) Figure 7.2: Schematic control, NASA/CR--2000-210062 of load cycles: a) Load versus time, b) Kt versus and c) KI versus time, displacement control. 83 a time, load To demonstratethatthe simulationsarecapableof predictingthe turning that is necessaryto predict the ridge in the fracturedtooth, a large crack shapeis assumed and inserted into the full pinion BEM model. The crack front coordinatesare determinedby the location of the ridge in a fractured tooth from the experiment. Figure7.3 is a photographof the fracturesurfacewith the approximatelocationof the assumedcrack front designatedby the dashedline. Basedon the SEM observations, the assumedcrack has propagatedalong the root from the initial notch to the toe surface. Figure 7.4 is a picture of the BEM geometrymodel that illustrates the assumedinitial crack trajectory on the tooth surface. Sincethe correct contact areas for a tooth that is flawed to this large of an extentareunknown,the HPSTCload step from the movingload analyses(load stepeleven)is used. Figure7.3:Assumedlocationof crackfront (ridge). Figure7.4:Tooth surfaceshowingassumedshapeof largecrack (dashedline). A few cyclesof crack growth are carried out using the methoddescribedin Section 3.2.3. Thismethod assumesmode I dominant fatigue crack growth with static,proportionalloading. The cyclesof crack growth arenecessaryto demonstrate the direction the crack front progressesfrom its assumedlocation at the beginningof the ridge formati0nl Figure 7.5 showsthe trajectorythroughthe thicknessof the tooth at approximatelythe middle of the tooth length. The initial trajectoryinto the rim is assumedto be fiat, andthe curving at the endof the crack length showsthe formation of the ridge. This demonstrationof the ridge formation basedon an assumedcrack NASA/CRm2000-210062 84 front from the SEM observationsupportsthe crack growth scenariodevelopedin Section6.3.2. \ \ , \ ! Initial crack \ \ l°I°n Figure 7.5: Crack One could the classical tooth trajectory argue failure through that the tested conditions did not fracture tooth spiral described from the gear; thickness bevel pinion in Section a portion for assumed failure 2.5 since of the tooth large crack. did not exactly the entire at the heel length remained meet of the intact. This suggests that the loading might have been biased toward the toe end of the tooth. Numerical analyses with shifted load locations are presented in Section 7.3.3 that give insight into the sensitivity The predictions of the crack in Section trajectories to loading 5.5 did not consider location. changes in the original contact locations during propagation. The increasing tooth deflections as the crack grows might cause the original contact locations to shift and the distribution of load and the size of the contact fracture mechanics ellipses to change. simulation of the necessary to capture the scope of this work. A three dimensional, contact rolling process between two the load redistribution effects This type of analysis is not in It is impossible to determine the exact amount of rubbing between the crack based on the BEM analyses. In Section 6.3.2 it was concluded that the surfaces faces with greater since these amounts surfaces away. kinematics varying The amounts of rubbing were older, of the of rubbing were formed the features geometry observed and on the the varying degrees of rubbing to time, contact between the fracture surfaces_ _¢ that does not allow the ridge's forces between the crack faces contact between is necessary 7.3 fully. mechanics, and mating gears is the crack to determine Sensitivity These faces in the earlier stages of crack growth; of the surfaces had more time to rub loading fracture it could loading is another surfaces. be attributed might deflect the true cause loading than for the attribute to the magnitude of the tooth in a manner fracture surfaces to rub, but does near the notch. A three dimensional with accurate explanation Rather conditions create large contact analysis modeling on the tooth surface of the rubbing. Studies studies are performed to gain insight into the crack growth rates and predicted crack trajectories to growth load magnitude, and load location. They are also conducted NASA/CR--2000-210062 85 sensitivity of predicted rate model assumptions, to investigate possible causes for the discrepancies fatigue crack parameters, researched 7.3.1 growth between rate Section further. 7.3.2, and steel. The predictions parameters, in Section closure) 6.19e-20 (in/cycle)/(psi*in°5) for AISI literature is reported Parameters data is available tested 7.1. The range data closure Section fatigue These are for AISI rate model crack growth values of values The model 7.3.3, for the crack growth in 250 ° oil. Au et al.'s results. crack in the literature fit to the intrinsic ", respectively. in Table 7.3.1, and magnitude, 5.5 used values a curve 9310 and the experimental Section position Rate Model growth rate n and C, taken from data (no parameters, the contact Fatigue Crack Growth Limited fatigue crack 9310 the predictions model were rate 3.36 and for n and C from the are not from intrinsic fatigue crack growth rate curves. Their data are from fatigue crack growth tests with R = 0.05. The other three sets of model parameters have been normalized to an intrinsic fatigue crack growth rate curve. Table Source Forman _ 7.1: Fatigue n et al. 2.56 3.63 [Proprietary 1998] Oil test [Proprietary 1998] life estimates cycles et al., Au (32% et al., and 9.30e-6 107,527 675,838 2.72e-17 2.03e-6 492,611 3,096,201 5.49e-21 1.54e-5 64,935 408,145 6.19e-20 1.23e-5 81,301 511,000 using each increase), the air test 3,096,201 decrease), respectively. Au et al.'s combination therefore the longest closure in the intrinsic parameters the fatigue generated fatigue predictions crack life. cycles of n and The set of parameters growth rate curves data (506% in Table 7.1 assume that predict for the gear's fatigue life is cycles/inch to each set's cycles/inch. each predict increase), C predicts benefits is demonstrated to the predictions from the data in Table Fatigue Life 1 1.08e-13 the number of cycles that each source would roughly proportional to the ratio of the oil test's Forman rate model parameters. da/dN l° Cycles /inch I [in/cycle] ....... _3.36 The fatigue growth [(in/cycle)/(psi*in°5)]" 1.63 [19841 Au et al. [1981]11 Air test crack C the a fatigue and 408,135 smallest and conservatism by comparing with Au et al.'s for the various the life of 675,838 cycles growth rate of considering predictions parameters. sets of n and Figure C. The using (20% and crack the 7.6 contains curves are 7.1. _0Calculations are based on an assumed value for AK = 18,000 psi*in °'5. I_ This data was taken from a fit to fatigue crack growth rate data for non-carburized AISI 9310 tested in wet air, at a loading frequency of 1.0 Hz, and R = 0.05. The parameters are not from intrinsic fatigue crack growth rate data. It is should be noted that when parameters from the air test at R = 0.05 are used the calculated growth rate is 2.02e-6 in/cycle. NASA/CR--2000-210062 86 1.8 I 1.6 Air test 1.4 1.2 1 Oil test __ Forman et al. ._ o.8 0.6 0.4 0.2 Au et al. 0 , . 0 1 20000 40000 60000 80000 100000 120000 N [cycles] Figure An nearly 7.6: Fatigue exact growth 7.3.2 model since many rate curves Crack Model parameters, fl, were rand Sensitivi Rate t¢ incorporates calculations. Newman accurately. One tip plastic stress than of in plane _c equal to evaluate zone assumed in order the validity dimensional test trend from are unknown. is The that the fatigue life intrinsic fatigue crack strain. 5.5, values for the crack to calculate the and sensitivity effects that tc varies shallow between closure fatigue crack of the results the crack into represent tip conditions geometry. An approximation The to the 87 the 5.5, crack zone rate stress _¢ was equal front near conditions is to compare zone growth for plane of the crack plastic of the plastic 2 crack in Section or portions might the one and three reported cracks to one to the crack's I(K/] NASA/CR--2000-210062 in Section For the predictions for extremely method crack pinion accurate O, to r¢ three a value presented investigates specifies respectively. However, surface, the demonstrate a material constants is most in Parameters results Crack free of constants parameters are used. Closure Growth of material parameters For the simulation strain, set presented in this section will be more accurate when section three. rate curves for the sets of model Table 7.1. of which growth rates. This assumed values. plane growth evaluation impossible calculations calculations crack the is larger size, rp, is to to the more size of the in plane Using load stepeleven'sSIF resultsfrom the initial crack (Figure5.8), which arethe largestmodeI SIFsduring the load cycle, the plasticzonesizealongthe crack front rangesfrom 1.44x104 inches to 1.07x10-3 inches. Thesedimensionsare only 0.29% and2.14%,respectively,of the initial crack depth. It is concluded,therefore, that the planestrainassumptionalongthe entirecrack front mostaccuratelyrepresents the conditionsin the real gear. Crack Growth Rate SensitiviO, to/3 S,,,.,, the far field applied stress, is a function and KI. /3 is a dimensionless quantity to the applied of [3 from stress. Values that considers two [Murakami 1987]. solution, of /3 = 1 was selected. a value a known/3 factor be investigated Since the gear for a similar, and growth handbook geometry geometry solutions is complex An alternate simplified rates of ,6, c (half between / T° when relating/(1 can vary from one half and unlike could This alternate the two methods length), effects approach geometry. of the crack to any handbook have been approach to use will now will be compared. / | vl 2c / lo Figure 7.7: Finite The initial semi-elliptical The magnitude and midpoint thickness crack surface plate with a semi-elliptical mode I uniform stress. in the gear flaw of K1 varies of the crack subjected along front, is approximated to mode the crack by a finite I uniform front. respectively. They surface tensile crack to thickness plate stress, o', Figure Kl,,,_x and Ki,,,i,, occur are given subjected by Broek with a 7.7. at the surface [1986] as: /----- = 1-1217,f _ KI"'_" NASA/CR--2000-210062 and _) 88 Kl,,i,, - l'12Jao" O _c _ (7.2) where From Equation (7.2), the maximum and minimum expressions = 1.12 and ft,,i,, =I.12 Based on the initial crack geometry, Kop and da are recalculated analyses (Figure using for fl are /_- (7.3) tim,,, and flmi,, are 0.783 these values 5.8), and the same model and 0.701, for fl, the SIFs parameters from as were used respectively. the initial in Section notch 5.5. 2.5 --_ K oe(fl ,,,_., ) 2 "_Kop(fl,,,,,,) ! ---a--da(fl,,,,,,) --n- da (fl ,,,a_.) [ l _1.5' e- _ 1 0.5 0 --T I 6 I P 11 16 P _ 21 26 I I i 31 36 41 _ 46 51 , i 56 61 Crack front position (Orientation: heel to toe) Figure 7.8: Change Figure calculations. grows tooth 7.8 shows the percent The data show change since conclusions This will as the crack the increase grows, crack's can be stated this closed on the effect percent smaller tp and, geometry Loading Assumptions The intent of this study different contact conditions. of Kop and da with respect that the largest the ratio of a to c will become width. However, valid in Kop and da as functions of flma.,-and tim,,, with respect calculations with fl = 0. since therefore, form changes difference the tooth increase solution As the crack is longer fl,,,a.,, and growth than decrease for KI in the gear dramatically. of fl on crack to the original is 2.2%. length to original Therefore, the flmi,. is no longer no further calculations. 7.3.3 is to determine how the crack trajectory One motivation for this is that the changes simulation under and experiment's crack trajectories on the toe end do not match. The tested pinion's crack mouth remained relatively flat along the root until it reached the end of the tooth NASA/CR--2000-210062 89 length atthe toe (Figure6.lb). The trajectoryin the simulationturned,out of the root, up the tooth height and eventually reached the top land (Figure 5.14a). It is hypothesizedthat the differencesmay be attributedto inconsistenciesbetweenthe contactconditions(loadingconditions). The inconsistenciescould result from misalig-nmentduring the test or inaccuraterepresentationin the simulation of the actual contact areasin the test. Glodez et al.'s [1998] experimental work with spur gears supports this hypothesis. They considered loading two load cases: along one half i) loading of the length. along With the entire load case length of the tooth, ii) the crack in the and ii) unloaded portion of the tooth length turned out of the root and grew up the tooth height. On the other hand, the crack in load case i) remained flat along the entire length of the tooth root. The have the showed Load goal of the remainder same influence of this section on crack trajectories in spiral scenarios are investigated. whether bevel gears shifted as Glodez step onward +0.3 inches shifted load The in the preliminary analyses. along the tooth length. crack proportional the trajectory load discrete load five is analyzed trajectory began The for the shifted method described steps one, contact contact in Section five, For both and areas eleven. are shifted approximately is calculated The load Table the cracked under the shifted contact. turning sharply from this areas 5.4.2. scenarios, cycle using When the contact The trajectory shifted toward "wraps the heel is suppressed. is central, the crack 7.2 sketches the This maintains is most clearly further with the explained contact discrepancy by the fact that, could have the tooth's the tooth of the length. result that, toe end when mode on both I ends. contact the contact trajectory from decreased, subtleties increasing locations, location was closer and the load could of the redistribution 90 central a flatter and toe trajectory that will result. between deflections the with Glodez et al.'s crack growth simulations the test and simulation to the toe end. of the have tooth. been and its effect can only be modeled accurately with a three dimensional contact mating gears in conjunction with a fracture mechanics simulation. NASA/CR--2000-210062 height comparing is consistent if the fatigue in the test, the contact resulted stiffness The seen shifted a path very near the root under The grew, out up the tooth predicted The around" the contact location. However, when the contact is (toe), the tendency for the crack to kink up on the heel end (toe contact location trajectories. Th!s observations. As a result, it is assumed carried turns the non- is approximated trajectories on the tooth surface for the original and two shifted analyses. and II SIFs from the shifted loading scenarios are given in Appendix C. were et al. Location BEM model from propagation step number This model was chosen because the crack end) loads in spur gears. Two by is to investigate The shifted As the redistributed on crack analysis is crack along trajectories between the Table7.2: Cracktrajectories Contact from contact locations location shifted along tooth length. Schematic of resulting crack trajectory on surface in root Heel-shifted = Central Toe-shifted Load Magnitude The tested spiral in Table 6.1. However, areas and load bevel pinion was run at varying levels of input torque the simulation results reported in Section 5.5 assumed magnitudes produced by 3120 in-lb goal of the current study is to identify the influences crack trajectories. The SIF distributions and trajectories under and 4649 Hertzian in-lb (125% and contact theory, _t is known proportional to the cube lengths of the major 14.47% contact (150% design points (center design load. in the moving Similar load 150% and minor A/CR--2000-210062 that the lengths axes increase design torque respectively) levels are of the contact (Equation by 7.72% (2.1)). (125% to the shifted simulations load analyses, (Section 5.5) the locations 91 the crack is selected of the crack and of 3874 explored. ellipses' on in-lb From axes are the load torque) and that the mean for the 100% propagation to analyze The levels Consequently, design from load). torque load torque) under the larger loads. It is assumed of the ellipses) are the same as the points defined torque levels. Figure 7.9 shows defined for 125% design load. NAS larger load, load (100% of the increased design root of the applied torque detailed contact under of the contact step five the larger ellipses f Load • Load l l ? / Figure 7.9: Geometry The mode model I and II SIFs from 7.10 and 7.11, respectively. loads; the 125% load has between the curves by the fact increase ellipse directly the increased curves over The the kink load in the torque A large the various change front likely explained In contrast, for load step eleven; There faces. The the the major is a larger the amount load locations larger SIFs axis of all three in the trajectory angle Figure load is of spread the ratio of rubbing. and between K_. This of the fracture the crack faces. not necessarily it appears that the crack calculated and magnitudes by result is, the larger 7.12 contains magnitudes. surface The supports displayed SIT ratios large enough the increased The the fractography signs from of significant the initial crack to support the extent of torque levels will increase faces. the maximum are given principal in Figure is 9.7 ° and 6.4 ° for load step one and eleven, NASA/CR--2000-210062 is most of the crack. is opposite. the crack than percentage angles load locations in angle one at the toe end in Figures with the larger smaller spread with no load above it (toe end). because it determines the crack greater level between of rubbing kink step II SIFs by the two propagation analyses were rubbing observed. However, The are presented defined that the ratio of K_I to KI increases as the magnitude of load and area increases. This implies that KH is more sensitive to the of rubbing the amount one and eleven this portion crack between be and produced observations. amounts the entire of rubbing will one and eleven is larger than the length of the crack mouth, and the ellipse the crack. On the other hand, for load step one the influence on the mode curves demonstrate size of the contact changes by load is not over the portion of the crack ratio KI1/K_ is important angle ratios along eleven above and the amount these load load steps areas The mode I SIFs do not increase linearly a larger effect than the 150% load. The produced that the uniformly contact located with crack showing contact for 125% design torque. 92 7.13. stress The respectively. theory largest for the absolute