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5.4.3 Approximations The accuracy results. In of Method of the proposed addition, several method is limited approximations and by the accuracy simplifications of the BEM in the loading conditions and crack growth rate models could affect the accuracy. The shape of the elliptical contact patches is estimated by straight lines. When the applied traction is far enough stresses away from the crack front such does not contribute case, the shape approximation of that the deformation to the crack tip field, the traction is legitimate. area However, is St. Venant's due to local principle inconsequential and if the crack trajectory contact holds. the In this straight is significantly line close to a contact patch, then this assumption is no longer valid. As a result, the accuracy of the SIFs could be comprised. Furthermore, the size and location of each of the fifteen contact areas change are kept in flexibility SIF distribution Chapter 7. propagation constant throughout of the tooth could change in the later stages However, if the simulation, the the crack the contact of propagation. same contact fatigue life propagation. This areas along the entire equations that front. This implemented portions of edge not plane conditions during will most the likely front conditions is consistent to calculate cracks the near the free with kink surface the maximum when are usually the loading by plane principal However, An additional characterized strain stress shallow theory qracks by plane lives. The size approximation of crack growth. The method of the plastic zone and stress, is introduced when assumes that crack in the gear is investigated incrementally growth calculating only occurs the tensile portion of the load cycle. However, it can not be experimentally this is true. In fact, it is generally accepted that crack growth occurs opening crack be conservative. are characterized angle. entire in strain. Nevertheless, crack growth rates will be larger in plane strain than plane stress. This assumption errs on the conservative side and will predict shorter fatigue more in Section 7.3.2. amount the crack the will be investigated This is because the SIFs continuously increase as the crack advances scenario is kept constant and the crack length is increasing. It is assumed the areas and in turn affect aspect are used predictions In reality, and closing plastic deformation plastic crack. deformation portion creates The reasoning the loading of the load cycle at the crack were a wedging behind action Equation to become [Laird 1967]. tip and causes at the crack (5.3) governing proportional, the The tensile it to blunt. During during shown that during the portion creates unloading, tip that acts to advance the crack growth amount of crack the the the rate is that, extension if predicted during one loading cycle would be equal to that predicted by the crack-closuremodified Paris' model. An additional assumption is that the method assumes mode dominant fatigue crack growth. Kt, were to become the crack growth. models proposed For each increase geometry. If the ratio of Kit to Kt, and likewise large enough, the mode II (or mode IID loading In this case, AKeyy should be a function for crack growth rates do not incorporate propagation a majority of the crack It is assumed NASA/CR--2000-210062 step, a value the N loading 63 distance cycles I of Km to could contribute to of KI, Km and/or Kin. The the mode II and HI effects. for N is chosen front a significant during the ratio that is large in relation that the enough to to the model's variations in the displacements and stresses are negligible. In reality, the crack is growing slowly during the cycles and therefore the displacements and stresses at the crack front will vary as the crack advances. If the value of N is too large, then the model will not pick up subtle changes and trajectory in displacements. 5.5 Simulation Three method shape dimensional for fatigue crack non-proportional of the crack distinct the are similar along propagation fifteen, growth stages the simulations are performed 5.4.2. to a starter step, method the second Paris that was used in Chapter 6. are carried out, correlation front. To stage of double using The full at the step, mesh the tooth in a tested which propagation reduce do not contribute growth rate model to crack modified the pinion model the nodes load the SIFs The crack growth at are evaluated in the computational contact, pinion. first time steps row for twelve of each through The method for propagation adopted here only allows The final four load steps represent the unloading portion and, therefore, The shape crack is introduced into the model as the predictions, the dimensions and notch In each crack are ignored. during loading. load cycle front in Section of growth. displacement elements crack loading experimental results and data are reported Thirteen crack propagation steps using predicted trajectory element model is used and an initial in Section 5.3. In order to validate thirteen the Results described boundary described Consequently, will be inaccurate. crack of the in the simulations. to incorporate crack closure is used in this study. The model parameters are held constant during the propagation steps. Values for C and n are taken from a curve fit to AISI 9310 steel crack growth rate data from 250 ° oil (Section 4.4). t¢ is set equal which is a variable in the Kop calculation, in Table 2.2 are used. to three (plane 13is set to one. strain). To calculate The material properties determine A least squares curve fit to the predicted discrete crack front points a smooth crack front curve. The approach is to fit a polynomial second or third the crack front order to groups of points. points are divided into one, independently through each group. The tooth contact locations are used throughout the thirteen these data were presented Table 5.1 contains To allow two, or three and magnitudes propagation unsymmetric defined steps. in Section 5.2.1. the crack geometry and crack groups. loading growth rate listed is used curve front A curve for the The S,,,_.,-;, 100% shapes, is then design for each fit load simplifications data to of for of the propagation steps. N is rounded to the nearest 100 cycles. Figures 5.13 and 5.14 show the initial and final crack trajectories along the tooth surface and the depth of the crack into the gear rim. neither entirely through at this step because that continuing At the end the tooth the analyses would lead step nor through thirteen, the crack the rim. The analyses has reached to no additional the top land. insights has propagated are stopped It was decided because, as reported differed from the experimental results. Also, were continued, since the trajectory has turned and toe end, this middle NASA/CR--2000-210062 width the toe end of the crack in Chapter 7, the prediction imagine that if the simulations the heel of propagation portion of the tooth 64 will break away from one can on both the gear. Table5.1: Crackpropagationdatafrom trajectoryprediction. Propagation N Crack Front Depth 7 Area Step 0 [cycles] 1 15.000 20200 2 a) Tooth N.A° Length [in] 0.200 0.233 [in x 10 "2] 5.00 5.53 [in 2 x 10 "2] 0.579 0.743 0.258 5.58 0.837 3 4 38200 0.237 5.17 0.933 56.200 0.453 5.88 1.69 5 66500 0.506 6.32 2.19 6 76.800 0.595 6.85 2.77 7 94.90O 121 300 0.774 8.10 4.27 8 9.50 11.4 5.76 7.77 9 147 200 0.940 1.11 10 ll 187,400 1.15 12.9 10.1 227,000 1.19 14.9 12.5 12 274,000 1.29 16.9 15.7 13 311,000 1.42 18.8 18.6 N_tut = 311,000 surface / / / / \ \ / / ! / / / / _, f '_------_ b) Cross Figure 7 The approximate crack depth. location NAS A/CR--2000-210062 5.13: Initial along the tooth length crack; /Initial section of tooth at midpoint of crack N = 0 cycles. of the initial crack's 65 / .... midpoint is used to measure the a) Tooth surface / / ./ / / / /' / i / ",, i ! t / , \ z b) Cross section initial crack Figure 5.14: In halfway Crack the prediction simulation, through the tooth by estimating the tooth the Width. width after thirteen propagation majority the of A rough inches steps; crack estimate as 0.227 of front number 511,000. of cycles This to failure number could beginning midpoint of cycles. progressed almost of the total llfe _of the gear is made at the from at N -- 311,000 has toe end growth rate from step thirteen. From these data, approximately necessary for the crack front to progress through the remaining the tooth the be non-conservative and using the 200,000 ligament. initial notch since a constant average cycles are Therefore, is estimated to be K is used to calculate the remaining life. On the other hand, the fatigue life prediction does not take into account the cycles leading up to crack initiation. It assumes that the crack begins propagating immediately that in a real gear a number propagation from the notch. Figure after the introduction Of the notch. It is most likely of cycles are attributed to initiation of the crack 5.15 is a plot of the calculated fatigue life. The location along the tooth length of the initial crack's midpoint is used to measure the crack depth, which is plotted against total number Of cycles. The crack face area as a function of total number crack of cycles is also plotted for comparison length. NAS A/CR--2000-210062 66 since the crack depth varies along the 0.2 0.18 0.16 _.'= 0.14 i O.l "-' 0.08 Area e_ 0.06 0.04 0.02 0 50000 100000 150000 200000 250000 300000 350000 number of cycles. N [cycles] Figure 5.15: The evaluated Predicted accuracy in Chapter experimental NASA/GRC. data 5.6 Chapter crack of the depth crack 7 by comparing is obtained and crack trajectory area versus and fatigue the simulation from a spiral bevel results pinion life predictions to experimental test that was will data. carried be The out by Summary In this chapter, a boundary element model of a spiral bevel pinion was presented. Different size models were investigated to determine the smallest model which achieved accurate SIF results. An initial crack was introduced into the model and the SIF history along the crack front under the moving load on the pinion was found. It was determined that the loading on the tooth was non-proportional. a result, a method non-proportional was was developed load. performed using compared to experiment in Chapter 7 parameters determine method the sensitivity and crack NAS A/CR--2000-210062 growth A crack this to propagate trajectory method. In results; Chapter of the crack of the growth rate model the three prediction Chapter rates variables. 67 in the OH-58 7, the 6 presents propagation and dimensional spiral simulation the experimental method will trajectories crack under bevel results tooth As the pinion will be data. In addition, be investigated to to variations in the CHAPTER EXPERIMENTAL SIX: RESULTS 6.1 Introduction Experimental made in Chapter results 5. can be used Recordings to evaluate of crack length number of cycles are necessary to confirm crack front shape during propagation would There is a limited amount the numerical results. validate provided limited. and accuracy depth of the predictions as a function of total the fatigue life prediction. Knowing also assist in verifying the predictions. of useful experimental data Test data from an OH-58 the for the OH-58 pinion to spiral bevel pinion are by NASA/GRC. The crack growth observations made during the test are As a result, the fracture surfaces of the tested pinion are observed under a scanning electron fracto_aphy growth the microscope results scenario (SEM). are summarized during The test in Section the test is formulated data 6.2 and from from NASA/GRC 6.3, and respectively. the A crack the SEM observations. 6.2 Test Results A pinion under contains (EDM) that was a separate tested research the loading by NASA/GRC project data from into the root of a tooth's is used the test. concave in their spiral for A notch side. was torque detailed had fractured in Table from for an additional 6.1. cycles at increasing of the 4.9 million cycles, 6.1: Pinion levels five of teeth # of EDM [cycles] 1 million [rpm] 6060 Torque [in-lb] 1550 l million 6060 2324 1 1 million 6060 3099 1 1 million 6060 3874 1 2 million 6060 4649 1 1 million 4848 2479 9 1 million 6060 3099 9 l million 6060 1.9 million 6O6O 3874 4649 9 9 the latter 4.9 million Therefore, NASA/CR--2000-210062 more The test data. Speed During cycles As a result, eight roots of the pinion. Time growth. machined up to 4649 in-lb torque at the end of until the completion of the six million 4.9 million At the completion 6.1 the pinion. Table crack Table electro-discharge cycles; at which time, there was no observable crack growth. notches of varying sizes were fabricated into individual tooth then ran continuously gear test fixture The gear was run for six million beginning at 1550 in-lb torque and progressing the six million cycles. The test was not stopped pinion bevel comparison/validation. cycles, the sequence the test was never of events 69 for the tooth Notches stopped fractures to observe the and the exact number of cycles occurred during no crack growth to failure the first until are million the last unknown. cycles Some at 2479 1.9 million of the crack in-lb torque, cycles at 4649 or there in-lb since no observations of crack growth during the 4.9 million predicted fatigue life curve (Figure 5.15) can not be validated. information failure. is an upper bound of 4.9 million cycles growth on the total number b) NASA/CR--2000-210062 6.1: Typical tooth 7O failure in tested have have been In addition, cycles were made, the The only quantitative a) Figure may torque. may pinion. of cycles to Qualitative information can be taken from the pinion test to validate the predictions. For example, all five failures were teeth fractures. There was no evidenceof a rim type of failure. A macroscopicview of the crack trajectorycan be determinedfrom the testedpinion. Figure 6.1 showsphotographsof a typical tooth failure in the testedpinion. The rectangularEDM notch on the concaveside of the tooth nearthe root is observablein all of the pictures. The crack trajectory,which is assumedto initiate from the EDM notch, is deeperinto the rim in the middle of the tooth length than on the toe end. In addition, a ridge is observedwhere the crack trajectoryturnedtowardthe root of the convexsidefrom the initial path into the rim. All five of the fracturedteethhadremainingportionsof the tooth left intact at the heel end. Furthermore,four of the notchesweresmallenoughsuchthatthey do not appear to havegrownduring the 4.9 million load cycles. Crack growth rate data andcrack front shapeinformation during propagation could both be used to validate the numerical analyses. Since this data was unavailable,the tooth fractureswere observedwith a SEM. The SEM observations aresummarizedin the next section. 6.3 Fractography 6.3.1 Overview Three teeth that using a SEM. Tooth was the initial flaw had fractured examined starter notch are contained was similar determine were how mechanism fracture several much of failure mechanism. goals The of the might crack have opposite in Figure tooth's side of the tooth cross Figures are lying growth due if it did on the a typical The load. were rate objectives to fatigue, change, side with the tooth found crack view of how initiated from ends, is the load path, #5 and #11, surface where growth and the crack of the verify propagated the EDM notch on the or concave side. The free or convex side. The respectively. The of the figure at the top. As illustrated in Figure 6.2, the crack growth direction of the photograph toward the ridge at the top of the photograph. 71 could to the type into the rim, towards at the bottom were to what of crack This side is the loaded the crack 6.3 and 6.4 are of tooth NASA/CR--2000-210062 examined deep), and Tooth #5 was desired. All of this information predicted the actual crack evolution. section. where occurred the crack deviates from the original side will be referredto as the ridge. on their were features The primary and, information 6.2 shows with the applied side of the tooth, region where on the convex pinion teeth. changed, Additionally, sketch a pinion OH-58 (-18 mm x -1.7 mm deep). Pictures of in this section. The starter notch of the of the examinations. crack front shape during propagation how well the numerical simulations through tested to tooth #i 1. No distinguishing on the tooth that were not seen on the other There the #11 contained a "short" notch (-4 mm long x -lmm from which the numerical simulations were based. contained a longer and deeper these teeth's fracture surfaces third tooth from the fillet fractured teeth and the ridge was from the bottom The EDM notch is the rectangular and is labeled region along a portion A in the figures. Concave of the tooth side length Convex near the bottom right comer side Fatigue: .... I .... -- Tooth surface Fatigue: Ductile partially rubbed rupture rubbed cross section. Crack initiation from EDM Figure notch--... 6.2: Sketch The Ridge of crack propagation left side of both Figures is true for all of the concave side of the photographs NASA/CR--2000-210062 from through _ typical pinion tooth's 6.3 and 6.4 is the toe end of the tooth side pictures the convex in this chapter. On the other side is the toe end of the tooth 72 length. hand, length. This the right All of