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free xed rollers Figure 5.5: Simplified model 3: ignore A semi-elliptical crack is introduced The crack is 0.64 inches long and model. applied to the middle tooth both shafts and half of gear rim. in the root 0.14 inches over the middle third of the deep. of the tooth middle tooth A simplified length in each load is and tooth height. The shape of the traction patch is rectangular, and the traction across the patch is constant. The shape and location of the traction patch is different from those described in Section 5.2.1. However, the difference is not important because the intent is to analyze differences in SIFs between models after changing keeping all the rest of the model parameters constant. To achieve all the models, the mesh in the region of the crack and load patch The model's rate context, rate SIFs increase on average SIFs for simplification changes is proportional magnitude of the by 7%, one, two, of this magnitude to KI raised exponent 8%, and three, and 11% is approximately changes in the SIFs have dramatic effects concluded from this study that the full with Recall (Equation 3.4. variable consistency is identical. respectively. are significant. to a power one respect between to the In a fatigue that (3.6)). For Consequently, the crack AISI and growth 9310, seemingly full growth the small on the crack growth rate predictions. It is model should be used for all trajectory predictions. To verify that only explicitly modeling three teeth yields accurate results, a nine teeth model is analyzed. If the SIFs between the three teeth and nine teeth models are similar, then it can be concluded that not all of the nineteen teeth of the pinion need to be modeled. An edge crack is introduced in the three and nine teeth models, in the middle of the tooth length, in the root of the concave side of the middle tooth. The crack shape is semi-elliptical, and is 0.125 inches long and 0.05 inches deep. An effort is made to keep the meshes between the two models identical. The difference in SIF distribution under load steps 1, 5, and 8 is investigated. As shown in Figure 5.6, the percent difference in Kl between the two models three load cases is below 5%. The_.a_bsolute magnitude of KII for both models load cases appears as is significantly a large percent smaller than difference KI. Consequently, between the models. a small variation Instead of for all and all in Kn percent differences, Figure 5.7 shows the absolute Kn values for all the loads and models. It is evident from the figures that the three teeth and nine teeth models yield similar results, NASA/CR--2000-210062 53 leading to the prediction conclusion that the three teeth model is sufficient for the trajectory analyses. i [ 8 4 i 6 _ •6 2 _ @ -2 Load 8 -'-- 1 A---_ _- _- Crack front position (Orientation: heel to toe) Load 5 ! -6 _ i -8 4 -10 Figure 5.6: Percent difference in K_ between three teeth and nine teeth models for load cases one, five, and eight. Crack front position one corresponds to the heel end of the crack front. 2000 10.] Load 1 ------*.--3 Teeth --- all 9Teeth /9 L°ad5 _b Load 5 0 T o=_ _ k e.,. 6 7 8 9 10 11 12 14 13 \ IJ/_lSS/_9 /w jr, jr.- Load Crack front position Orientation heel to toe) _2 -2000 •_-looo _ ( " " : -3000 -40OO Figure 5.7: Ku distribution 5.3 Initial SIF History To simulate static BEM analyses NASA/CR--2000-210062 for three teeth and nine teeth five, and eight. Under the moving are Moving Each 54 for load cases one, Load load during one load cycle performed. models analysis on a pinion's represents oqe tooth, fifteen of the fifteen discrete Section time steps as the contact area moves up the pinion tooth, as discussed in 5.2.1. Recall these contact ellipses are defined for a full design load input torque of 3120 semi-elliptical in-lb. The full pinion boundary edge crack is introduced into concave The The side. dimensions approximately Each crack is located are 0.125 inches approximately long by 0.050 normal to the surface. load step produces a unique SIF distribution changes between load Figure in the middle inches deep. SIF distribution steps because 5.8 shows the crack the load position mode front. I SIF distribution single twelve tooth contact load steps. The second stage of double tooth contact, through fifteen, are omitted from the figure to simplify it. Modes The The remaining seven curves are the bottom most of the seven curves the result from for each single step number locations tooth creates load step the SIF curves a greater in the tooth the last single contact increases, of the contact location SIFs, load eleven, SIFs under corresponds patches tooth are progressing moment arm. under first by the seven double load steps H and HI tooth contact. the single tooth contact load steps. to load five. The topmost curve is contact is roughly shift up. for the load steps followed are plotted similarly in Appendix A. The four bottom curves in Figure 5.8 are the SIFs The and magnitude load steps, tooth contact length. is oriented from four double the crack eleven the initial 5.1. A on the of the tooth The along Figure tooth varies SIFs step to step. element model is used, the root of the middle step. The equivalent. However, This is explained up the pinion As a result, total applied force as the by the fact that tooth. The the displacements, load the change in and likewise root will thus increase. 18000 16000 Load I 14000 Load 2 Load 3 12000 Load 4 :_ 10000 Load 5 Load 6 _. _2 --+- Load 7 8ooo Load 8 6000 Load 9 Load 10 4000 Load 11 2000 1 6 11 16 21 26 31 36 41 46 51 56 61 Crack front position (Orientation: heel to toe) Figure NAS 5.8: Mode A/CR--2000-210062 I SIF distribution for load steps 55 one through eleven. Another the crack along (roughly approach to examine the data is to plot the SIF history for each point front. Figure 5.9 shows the SIF history for point 29 in Figure 5.8 the midpoint of the crack front). The magnitude of K1, KH, and Km is plotted as a function of time, or load step. The figure also includes Kop, which was calculated using Newman's crack closure equations described by Equations (4.1), (4.2), (4.4), and (4.5). 10000 i 6000 "-" 4000 f- \ ! 0__ i_6_ i -2000 \ _j 17 _ _K Load Step II -4000 Figure 5.9: Typical SIF history for one load cycle for one point on crack front. plots When the individual points in Figure 5.9 are connected with straight lines, the rei_resent the loading cycle onthe tooth. The minimum load has been taken to be zero. tooth In actuality, the minimum load in the tooth root might be compressive. When a is loaded, compressive stresses could result in the root of the convex side. Depending on the magnitude of these stresses, they may extend into the concave root of the adjacent tooth. However, Chapter 4 demonstrated that the crack growth rates do not vary significantly for negative R-values when crack closure is taken into account. Therefore, the 10_id cycle The the mode difference I SIFs. significantly The larger is modeled in the single plateaus during asReci'uaiszero, tooth in the curve single tooth ke. and double correspond contact (load K ln,in = g llmi n = K lllmin = O. tooth contact loads to the two contact steps 5-11) is evident stages. compared Kt is to the double tooth contact stages (load steps 1-4 and 12-15). The magnitudes of/(1 significantly greater than Kin. As a result, it will be assumed that mode HI does contribute to the crack growth. Based on gear theory, the curves in Figure 5.9 should be continuous in are not and smooth. The continuous curves would most likely show a large increase in slope as the loading progresses from double tooth contact to single tooth contact. One can NASA/CR 2000-210062 56 imagine that as the number of discrete load steps increases, the curves in Figure will become smoother. However, due to transmission error and noise, it is known 5.9 that the curves in reality that the fifteen load steps are sufficient are neither The moving continuous nor smooth. to approximate load on the pinion's tooth Therefore, the true loading cycle. proportional Since loading, the ratio propagation during grow this type of loading under which is changing results changes. kink gear tooth, A method is required the ratio the method not be used. in a constant in the spiral-bevel the load cycle conditions. is non-proportional; changes during the load cycle, Figure 5.10. Consequently, three dimensional crack described in Section 3.2.3 can assumed it is assumed to propagate a That method angle for the the predicted to determine and is proposed of Kit to KI how a crack in Section load angle of would 5.4. 0.2 i i i i _ t t J i i p 13 14 15 Load Step -0.4 -0.2 "_ 3 -0.6 -0.8 -1 -1.2 - Figure The 5.10: Typical KH to K_ ratio also growth. Mode I dominant fatigue et al. [ 1996] studied mixed mode They selected various crack the test lengths indicates which geometries I dominant, II dominant, characterized mode respectively. as balanced I dominant assumed balanced orientations crack that the fatigue NASA/CR--2000-210062 mode loading from to achieve analyses, they selected five different geometries 1.812, and 16.725. The ratios covered crack mode moving load. mode is driving the crack crack growth is associated with smaller ratios. Qian I and II crack growth in four point bend specimens. specimen and Kit to KI ratio under FEM different analyses Ktl to KI ratios. with KidKt values growth mechanisms I and II effects, mode II dominant, Using these ratios as guidelines, the mode ! and !I effects during the earlier growth during the later stages crack growth is driven by mode 57 of the cycle. I. that considered From the of 0, 0.262, 0.701, of pure mode I, and highly mode gear situation can be stages of the cycle to However, it will be 5.4 Method for Three Dimensional Under Non-Proportional Loading As shown in Section non-proportional 5.3, a crack loading. trajectories in three proportional fatigue As a result, dimensions crack Fatigue in a spiral not in Section Section 5.4.1 Literature of growth rates to two dimensional and research related predicting trajectories determines failure is subjected literature crack the approximations of under (pure mode angles to of the method. and work fatigue analyses. in importance life. Schijve fatigue trajectories The numerical [1996] gives life and crack the majority is related to is also an overview growth. non-proportional There loading because mode I and II loading. mode I and n tests, I) and maximum experimental the high stage They and compare shear results. loading is no mention scenarios. frequency stress give experimental the maximum (pure However, mode their mode. Bower et al. [1994] They incrementally advanced of contact. If the SIFs largely of methods Crack in the context of gears because the trajectory will be catastrophic. The number of cycles to mixed predictions for mixed moving contact load. each to A method to predict loads is proposed growth, work are of primary importance whether the failure mode proportional non- applicable on a gear's results in very short times from crack initiation to tooth or rim failure. Bold et al. [1992] is the most extensive report covering fatigue crack steels to growth review that were to non-proportional fatigue crack The limited amount of numerical to predicting crack is of secondary tooth tooth Review crack confined A Predictions to predict of relevant work. under non-proportional summarizes In the literature related of the work is experimental. predicting pinion only a few methods Section 5.4.1 is a summary fatigue crack trajectories 5.4.3 Growth methods adequate. revealed the gear model; three dimensional 5.4.2. bevel conventional are growth Crack met results tangential II) theory work growth from in non- stress theory for predicting kink contains no theoretical considered brittle fracture under a the load and evaluated the SIFs at their fracture criterion, then the crack was propagated based on the mode I and II SIFs for that load position using the maximum principal stress criterion. Their approach incorporates non-proportional loading in an incremental manner; however, the work is limited to brittle fracture, does not include fatigue, and does not include three dimensional Hourlier et at.'s [1985] focus was effects. to determine which of three theories predicted trajectories closest worked in terms of kl, which to experimental data for non-proportional loading. They is the mode I stress intensity factor for a small advance of the crack three at an angle a maximum, fatigue most match 2) direction growth mechanism where rate da/dN. method. experimental primarily forms as functions maximum da/dN. NASA/CR--2000-210062 theories investigated Akl is a maximum, The rate is calculated and is a function inaccurate this thesis 0. The results. Hourtier because it requires and of time and work the moving the angle J: :. in which was kl is of maximum I dominant found method is not practical to find 58 work da/dN one to express 0 in order 3) direction a mode Their the maximum et al.'s 1) direction assuming of ki,,_.¥(0) and Ak1(0). In general, were that growth 1) was the found to best for the purposes of load and kl in closed corresponding to the Three dimensional wheel position model is analyzed over finite element a railroad track for consecutive analyses containing stages have been a crack of wheel performed [Olzak position to simulate et al. 1993]. the The rail and the SIFs are calculated for each stage. However, Olzak et al. did not propagate was to determine what happens to the crack displacement the crack. Their primary goal and contact shape when the load bevel is directly likely never over the crack. be directly The most numerically over the crack significant modeled and first found. Next calculated. rate of/_ angle, from it is assumed and Panasyuk therefore, they the crack that their set up closed form also does not directly constant kink angle similar 5.4.2 into for the entire incremental Proposed take approach account the [1995]. crack under To calculate from model which is updated geometry was and However, They a moving the kink 0(2.)] are the growth rate is at that growth and an elastic solve angle, K,(2.), for N cycles the process is half plane, analytically and, for Kt, Kit, applied to gear model because in closed form. The method non-proportional load cycle. is developed al. of K = F[Kt(_.), and 0. Once again, their method can not be directly neither the traction nor the geometry can be expressed will most used, and growth rates were of the contact are expressed propagates equations et edge _, are calculated, in the numerical the load are not applicable. the load to the crack. that the crack gear, Panasyuk to an extremum et al. assumed could by a two dimensional 0, Kt, and KH at these repeated. done principal stress theory was The translation and location that correspond Finally, and was propagated of 2., the distance the values of the spiral and their findings work contact load. The maximum calculated by Paris' model. as functions In the case loading their method and assumes is extended a and a in the next section. Method Compared to a two dimensional static problem, the problem continuous in time and in a third space dimension. Methods have been at hand presented previous chapters and sections discretizations, two dimensional With applied. to discretize both crack propagation of these dimensions. theories can be is in the In summary, the proposed method discretizes the continuous loading in time into a series of elliptical contact patches, or load increments. Two dimensional fatigue crack propagation points theories are from the three proposed account method then used dimensional to predict time varying to propagate crack fatigue front. crack incrementally The remainder trajectories a series of this section in three dimensions of discrete outlines taking a into SIFs. Method 1. Discretize 2. Calculate by the displacement correlation FRANC3D/BES, the mode I, II, and III discrete tooth point contact along path into 15 load the ( j = 0 - M ). In general, front single are taken point front the nodes as the discrete i along NASA/CR--2000-210062 crack the crack points. steps (Section 5.2.1). method, using a feature built SIFs (Kti(j_, Ku'_j;, Kmi_j)), where (i=l-num__points) and of the first row of mesh Figure 59 nodes 5.9 is a typical front for the entire loading j cycle. is the behind load in to i is a case the crack plot of the SIFs for a . The goal of this step is to calculate • extension, da'oq ' j), during extension during a load incrementally during place the when Koj. a load This in only implies that a given from j'l j-1 to j. I when will that propagation between take the of crack grows i only takes steps is the opening place for the crack at point load than only amount 0 C/q,J) is the angle is _eater KI'j i the i to j. SIF growth point It is assumed In addition, mode and 0, step from cycle. change (KI' j -Kli(j-l))> point, a load step for positive, SIF at that during the loading portion of the cycle. To calculate most accurately the total amount of crack growth over one cycle, crack closure is taken into account. The amount of extension during one load cycle is predicted by a modified Paris' dai=C(AKej) where using kKej required. (4.4) and (4.5). to calculate Figure adjusted crack closure, (5.1) Figure U is given by Equation 5.9 shows that the loading U, Smax, the far field 5.11 to incorporate " = K/,,,a., " - Kop i = U'K, im,,. Equations by R=0. In order model, shows the Griffith stress crack (4.2). in a Griffith geometry Kop' is found is characterized crack [Griffith problem, 1921]. is The gear's geometry is obviously different= from a Griffith crack problem. Therefore, an equivalent S,,a.,' must be calculated for the gear. First, K/,,,ax is found in step 2. Sma.,.i is then found by solving Equation (3.2) for S,,,,,.,i: S,,_,.i = _,,_,.i _ Kti,,o., • Lastly, it is assumed (5.2) • that, at a given point, the amount of extension between load steps is proportional to the ratio of the change in mode I SIF to the effective SIF. The amount of crack growth for each load increment is given by: i do i {j-l,j) _" i K t (j)- K I 0.01 inches. model accomplish crack cycles (5.10) such that da_,j the FRANC3D the the crack da ii,,,Ii = N * daT i 7. of because to update that a series Because amount step is necessary is too to assume the geometry. a significant crack squares A single that has curve all load steps are analyzed curve may be fit into a user-defined curves are fit to each set. to locally remesh the model by an fit is performed polynomial may be divided grown number After the crack prior running the with the new crack. at step 2. Y X l :1,2) i (2,3) h i Ii Figure 5.12: Schematic NASA/CR--2000-210062 of crack extension for one point load cycle. 62 along the crack front after one