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U are S Figure 3.12: 3.3 Fracture squares Mechanics A suite Group Least is used curve fit through new discrete crack front points. Software of fracture in this thesis mechanics software [FRANC3D 1999a, developed 1999b]. by the Cornell The codes were Fracture developed to handle the complexities of three used to define a three dimensional dimensional crack trajectory predictions. OSM solid geometry model of an object. The program based of the on defining boundary the of a solid surfaces is generated by model adjacent explicitly in Cartesian surfaces, or faces. is is space. The face of the Each boundary element model has a three dimensional local coordinate system associated with it. In order to define a closed solid, all of the local face normals must point away from the interior significance The can create the solid. The when defining boundary geometry model is then a finite Displacement of the solid. respect of element the local coordinate system conditions. read into FRANC3D. or boundary or force/traction The conditions to either local element mesh With based might coordinate system. be FRANC3D, Material model. for all the directions properties faces with are also assigned faces to regions of the model using FRANC3D. Cracks are added to the solid by explicitly defining the vertices, that model the cracks. A crack has two distinct faces that must of a user on the geometry boundary conditions must be defined must be specified in all three Cartesian or the global also identically. As mentioned calculating SIFs in Section and options to discretize nodes, the midpoints 3.2.3, to propagating the the crack front. of the elements a crack front crack. Within will give when a row a set of equally NASA/CR--2000-210062 of four spaced be discretized FRANC3D, prior there are to three The discrete points can be defined by the mesh sides along the crack, or at a user defined number of equally spaced points along the crack front. calculate SIFs uses the displacement correlation are obtained must edges, and be meshed sided points elements behind 33 The built technique. is used the crack in feature in FRANC3D to The most accurate results along front the crack where front. the SIFs This can be evaluated. Additionally, to improvethe performanceof the crack front elements,the ratio of the elements'width to lengthshouldbecloseto one[FRANC3D 1999c]. When a crack is propagated,the geometrymodel chan_es. However, the geometrychangesonly nearthe crack. Therefore,only the meshmodelnearthe new crack is damagedandrequiresremeshing, The remainderof the geometryandmesh model is left unchanged.This is a distinct advantageof FRANC3D. The programBES is usedto solvefor the displacementsandstressesusing the boundaryelement technique. FRANC3D is used as a post-processorto view the deformedshape,stresscontours,andextractnodalinformation. FRANC3D usesthe samefunctionalform to interpolatethe geometryandfield variablevariationsoveran element.The form is given by the associatedelementtype. In all of the models,only isoparametricthree- and four-noded elementsare used. Quadraticelementsare available;however,basedon the work in [FRANC3D 1999c], the gain in accuracydoesnotjustify the significantincreasein computationaltime. 3.4 Chapter Summary This chapter covered growth numerically. angles are calculated calculate correlation and three and fatigue from The maximum dimensional simulations, and BES SIFs. examples Some that will be used |ia Chapter crack of LEFM importance is how pertinent crack principal growth stress to modeling rates theory trajectories under mixed mode loading. In addition, method was introduced as a technique to evaluate SIFs. numerical studies theories Of primary in those chapters in a spiral NASA/CR--2000-210062 of the in the simulations 2 and this chapter trajectories demonstrated features were will be Utilized cover bevel issues how the software covered. in the work related gear. 34 to predicting The are applied FRANC3D, background Of chapters three will be used to the displacement Two dimensional theories programs crack and trajectory in OSM, provided 4, 5, and 6. The dimensional fatigue CHAPTER FOUR: FATIGUE CRACK GROWTH 4.1 RATES Introduction The goal of this chapter the fatigue crack growth is to determine rate in a common the context of gears because the magnitude root is a function of the rim thickness. sensitive to compression, designing gears. then crack On the other hand, 4.2, the concept shows that crack closure the factors that control growth rates may stress crack closure ratios affect This is of interest in in a gear's tooth rates are highly warrant stresses than the loading cycle portion of the cycle. of fatigue provides fatigue negative of compressive stresses If fatigue crack growth if the compressive crack growth rate predictions greatly, simplified by ignoring the compressive In Section how highly gear steel, AISI 9310. more attention in do not alter the fatigue for a gear tooth is discussed. This a convenient framework within which crack growth. A material-independent can be section to understand method is presented for obtaining fatigue crack growth rate data that does not vary with stress ratio, R. The crack closure approach is extended beyond aluminum alloys, considered by Elber Section growth stress [1971] and Newman 4.3 applies model. Section range, magnitude and [1981] the concepts and discussed to AISI 9310 4.4 demonstrates likewise the crack in Section data to obtain that in the range growth rate, 4.2, to steels. an intrinsic of negative is not highly Next, fatigue crack R, the effective sensitive to the of R. 4.2 Fatigue Crack Closure Concept Due to the cyclical loading on a gear's occur. The load range, AS, or stress ratio, R, characterizes cyclic loading. stress, Smi,,, to maximum minimum mode They geometry. increased et al. [1997b] also found during The majority growth rates. commonly S,_a_, which, is equal to the ratio I SlY, Km_., (Equation of R in spur gears decreases, R becomes of the crack of the load cycle closure when negative approach the crack R-values as low as is a function more of the gear due is that damage faces flaw to failure is usually sensitive to the magnitude interest in the context of gears. The of 3.5). to the only are not in contact. of the literature's discussion of crack closure covers its effect Since gears have such high load frequencies, crack growth of secondary might factor range, AK, along with the load R is defined as the ratio of minimum found that spur gear teeth can have interpretation the portion crack propagation due to similitude, mode that the magnitude As the rim thickness bending of the gear rim. A general occurs intensity Recall, I SlY, Kmin, to maximum Lewicki -3.0. stress, tooth, fatigue time from on crack rates are detectable insignificant. However, if the crack growth rate is highly of the compressive portion of the load cycle, then crack growth rates may warrant more attention. On the other hand, if, for negative values of R, the crack growth rate is relatively insensitive to the magnitude of R, then the effect of geometry on R need not be the primary NASA/CR--2000-210062 concern 35 in gear design. This demonstration is siguificant in the contextof the o,rerallgoal of this thesis,which is to study aspects of geargeometrythataffect damagetolerance. It is assumedinitially in this chapterthat the stressesinducedin a gear tooth under positive (tensile) and negative (compressive)parts of the load cycle are "proportional." In other words, the shapeof the stressintensity factor distribution alongthe crack front is the sameunderboth tensile andcompressiveloading. In two dimensionalanalyses,this is not a concernbecausethe crack only consistsof a tip, wherethe deformationcanbe tensileonly or compressiveonly, not a combinationof the two. In three dimensions,however,the distribution of the loading (deformation) alongthe crack front might be different in the compressiveandtensile load cases. In the end, whetherthe positive andnegativeparts of the load cycle are proportional is not of major concern. As will be shown in the remaining sections,damageoccurs only during the tensileportionof the load cycle. Elber [ 1971] observedthatduringunloadinga crack actuallyclosesprior to the appliedload beingentirely removed. This phenomenonhasbeencalled fatiguecrack closure. Fatiguecrack closurealso explainswhy, for a given AK, fatigue testsshow the crack_owth rate increasingasR increases. Figure 4. ! shows typical fatigue crack growth rate data as a function of SIF range [Kurihara et al. 1986]. Kurihara et al. conducted fatigue tests with 500 MPa class C-Mn steel, which is used in pressure vessels. The tests covered a wide range of stress ratios from -5.0 to 0.8. Figure 4.1 was obtained by selecting R. The horizontal scatter as R increases, growth the two data points off Kurihara et al.'s plots for each value of in the curves is a result of the different R-values. Note that curves rate for a given shift to the left, producing an increase in fatigue crack AK. 1.00E-02 -0.5 ,,-1 _;_ 1.00E-03 _ 0.5 3_.2 1.00E-04 47 R=-5 1.00E-05 0.67 .... 1.00E-06 I r ...... i 10 t-r1-- , ...... 100 I000 AK [MPa*m °5] Figure 4.i: Fatigue NASA/CR--2000-210062 crack growth rate data for pressure vessel steel at various (data taken from [Kurihara et al. 1986]). 36 R-values portion Crack closure can be attributed to a number of factors. During the opening of a load cycle, the material at the crack tip plastically deforms. As the cycles repeat, body. a wake of plastic deformation The plastic deformation wake Although not considered surface roughness crack surfaces. allows here, crack closure of the crack faces, due Elber modified crack propagation as the stress His equation Paris' model to occur U, the effective to mixed to account only while stress range ratio, S,,a,. - Sop Figure 4.2 illustrates the relationships measure experimentally. In addition, Elber developed due mode loading, for crack closure. the crack is defined = to differences in the or oxidation of the tip is open. the tensile The modification He introduced an Sop (Kop) could empirical be backed Sop part of the load cycle. (4.1) as 1- s _/s.... S,,a ,. -S,,i, ' relationship, also occur = C (AK _ )" = C (U2d_)" U - result, can level where the crack first opens during for the crack propagation rate is: da -_ where remains as the crack propagates through the results in a mismatch between the crack faces. (4.2) 1-R among various K values. Sop (Kop) is difficult to the value varies with loading conditions. As a relationship between U and R. From this out. K Figure 4.2: Constant ._6Xdf for different stress ratios. When da/dN is plotted as a function of ,SJfeff, the scattered different R-values) collapse into a single, "intrinsic" crack growth curves (due rate curve. to In crack-closure-based fatigue models, da/dN is a function of 2d_e_. This implies that crack growth occurs only while the crack tip is open. If K,,a., is kept constant between various tests with different R-values, then K,,i,, must change. If it can be shown that NAS A/CR--2000-210062 37 AKe/t- remains load cycle negative will nearly when constant R-value cases be investigated constant could does not be treated in Section R-values, to crack in the same 4.4. decreases. performed a series negative contribute Figure 4.2 manner. illustrates then the portion growth. The how of the Therefore, sensitivity all of AK,2. AK,,f¢ could remain as Kmm Elber aluminum alloy. developed the empirical Elber's range for various K,,,i _ <_Kop The U(R) of R-values stress of experimental ratio range was relationship U(R) = 0.5 + 0.4R relationship for which -0.1 when is valid he had fatigue only tests with sheets < R <__0.7. of 2024-T3 From the tests, alloy over - 0.1 _ 0 (4.4a) for - 1 < R < 0 (4.4b) _qlax when K op >_Kmi,,. The coefficients Ao NASA/CR--2000-210062 Ao - A3 are: = (0.825 - 0.34t¢ + 0.05K "2 38 )F ( mo]l': COS L ( 2Oo)1 (4.5) A_ = (0.415_¢ - 0.0711¢2) S"'a" O"0 A 2 = 1 - A 0 - A1 - A3 A 3 = 2A 0 + AI - 1 t¢ is a constraint factor taking on a lower upper bound value of 3 to simulate average between the uniaxial yield the material. Because Newman's curves in Figure Equations Equation (4.2). 5 (4.4) 1.00E-02 of 1 for plane strain conditions. and the uniaxial for Kop is a function for any fatigue crack properties are known. 4.1 collapse account. plane stress model and n', it is applicable conditions and material bound (4.5) are used curve conditions constants LEFM holds and 4.3 is an example when to calculate crack AKop. closure (o'0), R, the loading of how the is taken U is calculated into using -J 2*da/dN (R---O) "_... da/dN (R=0) ._ 1 _' and an The flow stress, or0, is the ultimate tensile strength of of material where Figure into an intrinsic and stress 1.00E-03 !o, i " --_ R=-5 • • 05*d /dN (R O) "-*- R=-2 _ R=-3 _R=-I ",I-- R=-0.5 1.00E-04 -" 1.00E-05 _, -4- R=-0.33 1 -'+- R=0 _ _R=0.67 --*- R=0.8 l 1.00E-06 ! 1 ....... 10 100 1000 M_'e_q[MPa*m°'5] Figure 4.3: Intrinsic Newman's equations The crack plate with _¢ = 1 was fatigue tip condition a center selected crack growth for AKe/y; _" = 1 (using in the fatigue rate data for pressure data taken from test specimen vessel [Kurihara Kurihara steel using et al. 1986]). et al. used, a thin crack, is best described by plane stress. Therefore, a value of for the preliminary graphs. _ was then increased, and the amount s Note that Newman claims Equation (4.4b) is valid for negative R-values greater than or equal to -1. However, Kurihara et al.'s data extends to -5.0. Equation (4.4b) was used for the cases when R = -5.0, -3.0, and -2.0. values. Figure 4.3 illustrates, NAS A/CR--2000-210062 at least for this case, the equation 39 can also hold for these low R- of correlation became more between scattered, The equation the curves validating was visually inspected. the choice of Ic -- 1. of a line in Figure 4.3 is given In Figures 4.1 and 4.3, the slope for a given data points. According to the crack growth As _c increased, the curves by: curve (R-value) is uniquely defined by the models, all of the curves should have the same slope. Ideally, this would be the case for the plots in Figure 4.3. The small scatter in the magnitude of the slopes at different R-values is attributed in the experimental results. Figure 4.3 includes the intrinsic curve predicted by the This curve falls roughly in the middle of the predicted curves. scatter in the curves, the figure also includes lines corresponding times the crack growth this envelope. correlation. As These results crack growth fatigue rate for a result, R = 0. All of the predicted it is concluded that with 500 MPa pressure vessel rate curve can be obtained the using R = 0 data. To give an idea of the to one half and two intrinsic Mfeff steel intrinsic relatively to scatter curves equations produce demonstrate Newman's fall into good that an intrinsic material-independent model to account for crack closure. It is also shown that a possibility exists to extend the model beyond the range of R _>-1. Consequently, in Section 4.3 the model will be applied to AISI 9310 steel to determine how negative R-values influence crack propagation rates. 4.3 Application of Newman's An open literature search Model to AISI for fatigue crack 9310 growth Steel rate data for AISI 9310 steel at various R-values revealed little published information. A report by Au et al. [1981] contains the most information. Au et al. performed tests in different environments at various Because R-values they and were frequencies for carburized investigating and the correlation noncarburized between fatigue AISI 9310 steel. striations and crack growth rates, only two tests were performed on noncarburized steel in the same environment and at the same load frequency but at different R-values. The load levels used in the tests were not reported. When their measured fatigue crack growth rates at R -- 0.05 and 0.5 are plotted against Mr, there is very little scatter in the curves. This suggests that the crack high enough such that growth rate Kop <_K,,,,. fatigue crack growth rate data Au et al.'s data is inadequate. is not sensitive Since at different the to R or that the applied objective R-values, of this including study loads were is to correlate the negative R regime, Additional fatigue test data for AISI 9310 was provided by a helicopter manufacturer on the condition that the data's source not be identified. Data points are extracted from the fatigue crack growth rate curves obtained from tests in two different environments. Figure 4.4 shows growth rates for AISI 9310 steel in room temperature air for R =-1, 0.05, and 0.5. The curves in Figure 4.5 are obtained by extracting data points from fatigue crack growth rate tests in 250 ° oil for R = -1, 0.01, and 0.5. Table 4.1 summarizes the slopes and intercepts for the various curves. NASA/CR--2000-210062 40 7 1.00E-03 1.00E-04 1.00E-05 1.00E-06 1.00E-07 1.00E-08 1 10 100 AK [ksi*in °5] Figure 4.4: Fatigue 1.OOE-03 crack growth rate data for AISI 9310 steel in room temperature : 1.00E-04 'U ._ 1.00E-05 1.00E-06 1.00E-07 1.00E-08 1 10 100 AK [ksi*in °'5] Figure NAS 4.5: Fatigue A/CR--2000-210062 crack growth rate data 41 for AISI 9310 steel in 250 ° oil. air. Table 4.1: Slope and intercept Test R=-I effect of curves C n in Figures 4.4 and 4.5. [(in/cycle)/(ksi* in°5) n] 6.4e-12 (Air) 3.3 R = 0.05 (Air) R = 0.5 (Air) 3.5 3.9 7.3e- 11 5.3e-11 R = -1 R = 0.01 (Oil) (Oil) 3.2 3.2 1.1e-11 9.9e-11 R = 0.5 (Oil) 3.8 7.9e- 11 For a given R, the n values are similar between the two environments. The of the environment can be see in the variations of C. C is consistently larger in the heated oil environment. A larger C will result in faster growth environment effect will not be considered in this investigation. Similar generated condition various to the pressure vessel steel using Equations (4.2), (4.4), at the crack tip in the test curves collapsing analyses, and (4.5). specimen. into an intrinsic fatigue da/dN A value Figures crack rates. versus However, the 2ug,,ff plots are of tc = 1 best describes 4.6 and 4.7 illustrate growth the the rate curve. 1.00E-03 C *--4.26E1.00E-04 ! i y# + i •_ 1.00E-06 10 n =3.63 _ --R=-I i _ -'*R=0.05 R---0.5 .] -- Linear Curve Fit / ! 1.00E-07 _ii 1.00E-08 i+ / + ' ...... _ 1 ' .... I0 100 AKej 1 [ksi*in°Sl Figure 4.6: Intrinsic accounts 9310 crack Figures 4.4 through 4.7 for the scatter in fatigue steel. figures. and 4.7. fatigue Table In addition, The slope NASA]CR--2000-210062 4.2 contains the growth rate for AISI k'=l. 9310 in room demonstrate that Newman's crack growth rates at different slopes and vertical intercepts temperature crack closure stress ratios from the lines air, model in AISI in the a linear least squares curve is fit through the data in Figures 4.6 and vertical intercept from each curve fit are also included in the 42