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table. These curve numerical analyses. sensitivity of fatigue fit values Crack crack will be used closure growth in the concepts rates crack growth rate will now be extended models for the to investigate to low R-values. 1.00E-03 q -i 1.00E-04 17=3.36 _ C *=7.44E-10 1.00E-05 --R=-I _ 1.00E-06 R=0.01 [ ---*- R=0.5 _Linear Curve Fit 1.00E-07 I 1.00E-08 , , , I i d , i , 10 , i 100 AKef: [ksi*in °'5] Figure 4.7: Intrinsic Table fatigue 4.2: Intrinsic Test crack growth and non-intrinsic C n rate for AISI 9310 fatigue [(in/cycle)/(ksi,ino.5).] crack growth C* in 250 ° oil; I¢ = 1. rate parameters. [(in/cycle)/(ksi,inO :).] R = -1 (Air) 3.3 6.4e-12 6.30e-10 R = 0.05 (Air) 3.5 7.3e-11 8.80e-10 R = 0.5 (Air) 3.9 5.3e-11 1.98e-10 R --I (Oil) 3.2 1.1e-11 8.52e-10 R = 0.01 (Oil) 3.2 9.9e-11 1.09e-9 R=0.5 (Oil) 3.8 7.9e-11 2.87e-10 Curve Fit Air 3.6 NA 6 4.26e- Curve Fit Oil 3.4 NA 1 7.44e-10 6 Not Applicable NASA/CR--2000-210062 43 10 the 4.4 Sensitivity Table Equations assumed. Table of Growth 4.3 contains (4.1), (4.2), results (4.4), 4.3: Calculations to Low from calculations and (4.5). values R-values using for Kma,, _', and Smax/c7o are for a constant U K,,,_x; SIF 0.705 1.000 0.100 10.000 Kmin 7.050 0.700 1.000 0.100 10.000 7.000 3.000 0.822 2.467 0.505 1.000 0.100 10.000 5.050 4.950 0.716 3.542 0.500 1.000 0.100 10.000 5.000 5.000 0.713 AK 2.950 0.825 AKe£f 2.434 3.565 4.388 0.255 1.000 0.100 10.000 2.550 7.450 0.589 0.250 1.000 0.100 10.000 2.500 7.500 0.587 4.399 0.005 1.000 0.100 10.000 0.050 9.950 0.474 4.714 0.000 1.000 0.100 10.000 0.000 10.000 0.472 4.716 -0.495 1.000 0.100 10.000 -4.950 t 4.950 0.327 4.886 10.000 10.000 -5.00 15.000 0.326 4.888 -9.950 19.950 0.254 5.058 20.000 0.253 5.060 1.000 0.100 1.000 0.100 1.000 0.100 10.000 -10.000 -1.995 1.000 0.100 10.000 -19.950 29.950 0.180 5.402 -2.000 1.000 0.100 10.000 -20.000 30.000 0.180 5.404 -2.995 1.000 0.100 10.000 -29.950 39.950 0.144 5.746 -3.000 1.000 0.100 10.000 -30.000 40.000 0.144 5.748 in Table The crack growth rate is calculated in Table 4.4 based on the effective SIF data 4.3. C and n are assumed to be 7.44e-10 (in/cycle)/(ksi*in°'5) '' and 3.4, respectively. for the AISI These values are taken from the curve 9310 steel tests conducted in heated plotted in Figure 4.8. The curve in Figure regime is less sensitive R equal context, 4.8 shows to variations that the fit to the intrinsic growth oil. da/dN as a function crack in R compared growth rate to the positive rate data of R is in the negative R regime. R Between to zero and -3.00, the crack growth rate varies by a factor of 1.96. In a fatigue a difference of this order of magnitude is acceptable. As a result, one can conclude that when not change modeling significantly parameter geometry to characterize to be damage geometry affect R. It has been shown fatigue crack for R < 0. damage tolerant, evolution a primary that when taken as R = 0 A/CR--2000-210062 under the growth, AK,._, or likewise Therefore, crack significant change in the crack growth rates utilized in the numerical analyses discussed NAS of AKeff at different Constant s,,°xloo -0.500 be R to find AKe_ over a range of R-values units are ksi*in °5 R does Rate the magnitude Kop or da/dN, of R is not a useful in gears. In the context of designing gear concern need not be how aspects of gear closure is taken for negative in Chapters assumption that, 44 if into account R-values. 5 and 7. R < 0, the there is not a This result will be The load ratio will general results and conclusions method. Table account would still 4.4: Crack crack be valid. growth closure This rate calculations effects. The 0.705 2.434 0.700 0.505 0.500 0.255 0.250 0.005 0.000 2.467 3.542 3.565 4.388 4.399 4.714 4.716 -0.495 -0.500 4.886 4.888 -0.955 -1.000 5.058 5.O6O -1.995 -2.000 -2.995 -3.000 5.402 5.404 5.746 5.748 is a simplification for a wide )ercent change daJdN (incycle 10 .7) 0.153 0.160 0.549 range to the of R-values cycle taking in da/dN is due to _ % Change x daldN and into = 0.005. 4.446 0.561 1.136 1.146 1.1449 1.451 1.637 2.135 1.639 1.841 1.844 2.303 2.306 0.120 2.841 2.844 loading 0.873 0.130 0.116 0.108 0.102 daMN 3.00E-07 2.50E-07 2.00E-07 1.00E-07 5.00E-08 0.50 NASA/CR--2000-210062 45 1.00 4.5 Chapter Summary Highlights • The crack that the model closure load cycle • can be summarized concept predicts when and Newman's that fatigue the crack faces crack closure model of a pressure vessel Newman's rates apparent intrinsic of the dependence crack load effective • of this chapter growth range stress Newman's There was was It was growth faces It was rates not presented. during It was the portion shown of the to empirical that crack on R. data closure In fact, explains the that determines in contact. for crack well material during This the has what range growth an portion is called the range. demonstrated growth it was that, in the regime rates as a function to R in the negative R-regime that crack will be used 46 the crack R-values, of the effective the load history. NASA/CR--2000-210062 shown of negative only a weak function of the magnitude of R. The observation made in this chapter that sensitive only model was applied to AISI 9310 steel, a typical steel used for gears. much less crack growth data available for this steel as compared to the that the crack • shown are pressure vessel steel. Nevertheless, works for this small data set. • applied R is a parameter factor were occurs are not in contact. steel. crack intensity model damage of crack rate. the as follows: the model stress growth closure intensity rates in Chapter are 5 when model predicts factors not are highly modeling CHAPTER FIVE: PREDICTING FATIGUE CRACK TRAJECTORIES MOVING, 5.1 GROWTH IN THREE DIMENSIONS UNDER NON-PROPORTIONAL LOADS Introduction Chapter growth 5 covers trajectories numerical in three chapter is to model crack growth Section 1.2, most previous work assumed one fixed load location. operation, fixed spiral location bevel loading, A boundary 5.2. The Section modeling dimensions related bevel to predicting pinion. The fatigue goal crack of this under realistic operating conditions. As covered in in the area of predicting crack trajectories in gears The location was usually the HPSTC. However, in gears are subjected therefore, issues in a spiral to a load moving could lead to incorrect element model tooth coordinates in three dimensions. three dimensional The trajectories. of the OH-58 spiral bevel pinion is presented in and a dimensioned drawing of the pinion were provided by NASA/GRC, along with the coordinates for discrete elliptical contact areas along a spiral bevel gear tooth. OSM/FRANC3D is used to create the model from these data. Studies are conducted to determine the smallest model that still achieves accurate SIF results. Once this model is defined, initial analyses for the discrete moving load cases are conducted. The SIF history load is presented in Section 5.3. Section 5.4 develops a method for an initial crack subjected to predict three dimensional to the fatigue crack growth trajectories under a moving load. The method increments a set of discrete points along the crack front for each step in the load cycle to find the total amount of extension and final angle of growth after fifteen load steps (1 load cycle). The propagation path for each point is then approximated with a straight cycles are specified, and the crack front is advanced an amount extension for one cycle times the number of assumed cycles, line. A number equal and to the at the of crack angle calculated for one cycle. Next, a curve is fit through the new crack tip locations to define the new crack front. The FRANC3D geometry model is updated, and the process is repeated. Finally, in Section implemented to predict 5.5 the proposed fatigue crack moving growth load crack trajectories propagation in the OH-58 method spiral is bevel pinion. 5.2 BEM Model A boundary OSM/FRANC3D. fillet curve from a program All points were element model of the OH-58 spiral bevel pinion was built The Cartesian coordinates for a tooth surface, tooth profile, provided that models on the generated by NASA/GRC. The the gear cutting tooth surface are points during the manufacturing process [Litvin 1991]. the tooth geometry program was to generate element analysis. This program's NASA/CR--2000-210062 data were process output was 47 along generated with of tangency with and automatically the gear kinematics. to the cutter surface A primary motivation for developing data for a three dimensional finite adapted to develop a boundary element model for drawing Some would this thesis. of the The pinion. remainder The basic of the shape pinion of the solid shafts model and gear was built rim were from a modeled. subtle details of the pinion drawing were ignored in cases where the geometry complicate the geometry model and have negligible effects on the computed SIFs. The elements. surfaces Figure that the meshes not meshed. of the solid 5.1 contains shown The model three views in the figures conical shape were meshed of a typical are surface using three- boundary meshes). The and four-noded element model volume of the gear is of the gear rim and the cylindrical shape (recall of the shafts are seen best in Figure 5.lb. As seen in Figures 5.1a and 5.1c, three of the nineteen teeth of the pinion are modeled explicitly. Section 5.2.2 discusses studies to verify the accuracy of the three teeth model. _, Fixed /_,_?_.__y_ "'_ "I_A a) Overall view "Short shaft boundary Oe_' " of full model _//_- Tooth Gear rim b) Section A-A from (a): profile NAS A/CR---2000-210062 displacement of shaft 48 conditions 31t c) Close up view of teeth Figure 5.1 : Typical In operation, The small shaft the input model The these face operating patches boundary torque sits on roller and the teeth of the pinion load is transferred across / is applied bearings. When successively the teeth. conditions. at the element end model face which equals torque given patches at the end of the model long the load and rotation tooth. that is transferred angle. More of the pinion's are detail fixed across shaft. rotates contact occurs, in Figure 5. l a to as the full in all model. directions. The are restrained in the local normal 5.1, contact areas are modeled as Traction normal to the the contacting on how long the gear When shown will be referred shaft pinion. is applied, contact the gear's teeth. The boundary conditions This on the middle of OH-58 the torque displacements on the surfaces of the smaller shaft direction. Though not explicitly shown in Figure distinct / these teeth contact patch is defined for a given patches are input defined is in the next section. 5.2.1 Loading Simplifications The meshing of the gear and pinion force between the gear teeth varies out of contact. Figure 2.8 is a continuous during the meshing is a schematic process. as adjacent of the continuous The magnitude teeth process come that of into and has been discretized into fifteen load steps. In order to perform numerical crack propagation studies of the pinion, the continuous contact between the teeth is discretized into fifteen contact patches, or load steps: by four four double more tooth double contact tooth the boundary element model. through fifteen, corresponding respectively. tooth from patches, contact seven patches. single Each The load steps will to the Patches from This is consistent with the progression the root toward the top. One progression one load cycle tooth. NASA/CR--2000-210062 on the tooth. One rotation tooth load patches, face followed patch be referred to as numbers the gear root to the top in one land, of contact area along a pinion through the fifteen load steps is of the gear results 49 contact step is a unique in one load cycle on each The location NASA/GRC. The described by Litvin taken as the center torque level is used for operating lb torque. is defined and size data of the fifteen were determined et al. [1991]. of the ellipse. to determine conditions discrete contact numerically patches were by a procedure provided similar to that The mean point of contact between the gears is Hertzian contact theory along with the applied the width of the ellipse. of 300 horsepower, 6060 These conditions are approximating as 3099 in-lb torque. the rotations 100% The patches were per minutes, design calculated and load condition, 3120 / / 5.2: Contact NASA/CR--2000-210062 ellipses defined at the geometry 50 level in the numerical in- which / Figure by models. In the BEM model, the shapeof a contactellipse is approximatedby straight lines connectingthe axes'endpoints. The straightline approximationis valid because Saint Venant's principle holds; as long as the total applied forces and resulting momentsarekept constant,the elliptical shapeof the tractioncan beapproximatedby a patchwith straightsideswithout alteringthe stressdistributionalongthe crack front. Frictional forces are neglected,and, consequently,the traction is constantover the patch. Eachpatchhasa uniquemagnitudeof traction. The four figures in Figure 5.2 demonstratehow the traction patchesare built into the model geometry. The purposeof the models is to calculateSIFs from all fifteen static load cases. The combinationof all fifteen SIF distributionsrepresents one load cycle on the tooth. Figure 5.2 shows how a single BEM model can incorporatemultiple load cases. Not all of the contactellipsescanbe modeledin one BEM model becausethere is overlap betweenthe ellipses. The multiple load case featureminimizesthe computationaltime. For example,the boundaryelementmodel for load casesone, five, eight, andthirteen is virtually identical. The only difference betweenthem is the boundaryconditions. Hence,with the multiple load casefeature, the two mostcomputationallyexpensivestepsof the boundaryelementsolver, setting up the boundaryintegralequationsandfactoringthe stiffnessmatrix,occur only once. The differentboundaryconditionsarethenappliedindividually, andthe corresponding equationsaresolvedfor theunknowndisplacementsandtractionsfor eachload case. 5.2.2 Influence of Model Size on SIF Accuracy The fewer the number of elements, or unknowns, model, the less elements elements intensive studies is the accuracy in this thesis of the solution are linear. model is. amounts of curvature, As a result, this option is considered. Simplifying the model also pinion modeled, the less accurate simplified models are investigated. shaft in the full model. is sacrificed. Therefore, an element can be represented. of linear segments. Because significant adequately. the model, long the Minimizing element the number can primarily be accomplished by 1) using a coarser mesh with or 2) by modeling less of the geometry of the solid. A disadvantage first option across series computationally in a boundary only The elements linear in all of the in displacement Likewise, the geometry is approximated the geometry of the pinion is complex by a with larger elements do not represent the geometry is disregarded, and the second option, simplifying has drawbacks. The smaller the representation of the boundary The first simplification, Figure The used variations of larger of the new faces that are created the portion of the conditions. Three 5.3, is to ignore the when the long shaft is disregarded are restrained in all directions. Secondly, the smaller shaft is removed, Figure 5.4. The boundary conditions on the heel end are the same as simplification one, and the new faces on the toe end are set to traction free. The final simplification is to cut the rim of the model are the same conditions in half, as the second (displacement NASA/CR--2000-210062 pinion Figure simplification, in the direction 5.5. The boundary with the addition of the local normal 51 conditions of roller set to zero) for this boundary applied to the new faces. closely The match boundary those conditions of the full model for each (Figure model are chosen because they most 5.1 a). mll___ fixed J Figure In each an identical to model. simplified assumed loading 5.3: Simplified of the simplified models, model 1" ignore the flexibility long shaft. of the pinion changes. When crack is introduced into all of the models, the SIFs might vary from model To determine whether a simplified model is valid, the SIFs from the models that the are compared full model most to the full model's accurately SIFs represents the for identical cracks. It is operating conditions and paths. free Figure NASA/CR--2000-210062 5.4: Simplified fixed model 52 2: ignore both shafts.