Another approach to examine the data is to plot the SIF history for each point
along the crack front. Figure 5.9 shows the SIF history for point 29 in Figure 5.8
(roughly the midpoint of the crack front). The magnitude of K;, Ky, and Kj; is plotted
as a function of time, or load step. The figure also includes K,,, which was calculated
using Newman’s crack closure equations described by Equations (4.1), (4.2), (4.4),
and (4.5).

10000
8000
6000

4000

2000 ~

17
Load Step

-4000
Figure 5.9: Typical SIF history for one load cycle for one point on crack front.

When the individual points in Figure 5.9 are connected with straight lines, the
plots represent the loading cycle on the tooth. The minimum load has been taken to be
zero. In actuality, the minimum load in the tooth root might be compressive. When a
tooth 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 load cycle is modeled as R equals zero, i.e. K pin = K pimin = K tmin = 9+

The difference in the single tooth and double tooth contact loads is evident in
the mode I SIFs. The plateaus in the curve correspond to the two contact stages. K; is
significantly larger during single tooth contact (load steps 5-11) compared to the
double tooth contact stages (load steps 1-4 and 12-15). The magnitudes of K; are
significantly greater than Ky. As a result, it will be assumed that mode II does not
contribute to the crack growth.

Based on gear theory, the curves in Figure 5.9 should be continuous 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



Figure 3.1: Coordinate system at a crack tip.

A significant feature of Equation (3.1) is that as r goes to zero, or as one
approaches the crack tip, this first term of the series solution approaches infinity.
However, the higher order terms of the series will remain finite. For this reason, a
large portion of LEFM focuses on this first term of the series expansion only. In
reality, the stresses do not approach infinity at the crack tip. There is a zone around
the tip where linear elastic conditions do not hold and plastic deformation takes place.
This zone is called the plastic zone and results in blunting of the sharp crack tip.
LEFM holds when the plastic zone is small in relation to the length scale of the crack.

The SIF is a convenient way to describe the stress and displacement
distributions near a flaw in linear elastic bodies. The SIF for any mode is a function of
geometry, crack length, and loading. The general equation for a SIF is

K = BoVna (3.2)
Bis a dimensionless factor that depends on geometry, 2a is the crack length, and ois
the far field stress. It can be seen from Equation (3.2) that the units of K are

stress *,/length .

For a crack to propagate, the energy supplied to the system must be greater
than or equal to the energy necessary for new surface formation. When supplying
energy to the system, the energy can primarily go into plastic deformation or new
surface formation. LEFM assumes that all of the energy supplied goes into forming
new surfaces. As a result, LEFM predicts the material at a crack tip will fail when the
mode I SIF, K;, reaches a critical intensity called the fracture toughness, Kic. Fracture
toughness is a material property and by definition is not dependent on geometry.
Therefore, the criterion for fracture, or crack propagation, under LEFM, in mode I, is
K, =Kjc. Standard tests can be performed to measure values of fracture toughness

[ASTM 1997]. The tests subject a standard specimen to pure mode I loading. The
crack growth direction under pure mode I loading is self-similar. In other words, the
crack tip in Figure 3.1 under only mode I Joading will extend along the x-axis.
However, it is rare that a crack is subjected to pure mode I loading. More
realistically, the loading will be a combination of all the modes. The mixed mode
loading affects the fracture criterion and crack trajectory. For example, Mode II

NASA/CR—2000-210062 22



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Table 5.1: Crack propagation data from trajectory prediction.

Propagation N Crack Front Depth’ Area
Step [cycles] __Length [in] _[inx 10°] _[in?x 10°]
0 N.A. 0.200 5.00 0.579
1 15,000 0.233 5.53 0.743
2 20,200 0.258 5.58 0.837
3 38,200 0.237 5.17 0.933
4 56,200 0.453 5.88 1.69
5 66,500 0.506 6.32 2.19
6 76,800 0.595 6.85 2.77
7 94,900 0.774 8.10 4.27
8 121,000 0.940 9.50 5.76
9 147,200 1.11 11.4 777
10 187,400 1.15 12.9 10.1
11 227,000 1.19 14.9 12.5
12 274,000 1.29 16.9 15.7
13 311,000 1.42 18.8 18.6
Nroral = 311,000

Initial notch

a) Tooth surface

{
|
Lo. a ial a,

b) Cross section of tooth at midpoint of crack
Figure 5.13: Initial crack; N =0 cycles.

” The approximate location along the tooth length of the initial crack’s midpoint is used to measure the
crack depth.

NASA/CR—2000-210062 65



Figure 6.9: Low magnification (31.4) view of tooth #5’s fracture surface on convex
side.

A third tooth (#9) was also observed with the SEM. All of the features
observed on teeth #5 and #11, with the exception of the ductile fracture area, were
observed on tooth #9. No additional features could be seen. It is concluded that the
observations made of teeth #5 and #11 are good representations of the crack patterns
on all of the fractured teeth.

Figure 6.10: Magnified view of ductile rupture at location C in Figure 6.9 (1670x).

NASA/CR—2000-210062 77



of the ellipse is the mean contact point, which determines the contact ellipse’s location
on the tooth surface. The orientations of the ellipse’s minor and major axes are
defined by the tooth surface’s geometry, curvature, and the alignment between the
gear and pinion. The length of the axes is a function of the load. It can be shown that
the ratio of the axes’ lengths is constant and is not a function of the load. The form of
the equations for the length of the ellipse’s semi-major and semi-minor axes, a and b,
respectively, is [Johnson 1985] [Timoshenko et al. 1970]:

3a %
a=s[7=| (2.1a)
%
o=e{ 7] ° (2.1b)

where f and g are functions defined by the geometry. The magnitude of force, P,
exerted on the tooth is proportional to the input torque level and gear geometry.

The meshing of the mating gear teeth is a continuous process. The position of
the area of contact and magnitude of the force exerted between the teeth varies with
time as the gear rotates. Figure 2.8 illustrates schematically the progression of the
contact area along a tooth of a left-handed spiral bevel pinion. In the schematic, the
continuous process has been discretized into a series of elliptical contact patches, or
load step increments. The darkened arrow demonstrates the direction the load moves.
The actual tooth contact pattern during operation is a function of the alignment of the
gear and pinion.

Heel

Single tooth
contact

Double tooth
contact

Figure 2.8: Schematic of tooth contact shape and direction during one load cycle of a
left-handed spiral bevel pinion tooth.

Overlap in tooth contact between adjacent teeth results in two modes of contact:
single tooth contact and double tooth contact. At the beginning of a meshing cycle for
one tooth, two teeth of the pinion are in contact with the gear. As the pinion rotates,
the adjoining tooth loses contact with the gear and only one pinion tooth receives all of
the force. As the pinion continues to rotate, the load moves further up the pinion
tooth, and the next pinion tooth comes into contact with the gear; the force on a pinion

NASA/CR—2000-210062 12



the rectangular region along a portion of the tooth length near the bottom right corner
and is labeled A in the figures.

Concave side Convex side

---- Fatigue: rubbed

—-—- Fatigue: partially rubbed

Tooth surface — Ductile rupture

Crack initiation
from EDM notch >.

‘. Ridge

ae 7
we J
oe f
“3 a

Figure 6.2: Sketch of crack propagation through typical pinion tooth’s cross section.

Figure 6.3: Low magnification (4.6x) view of tooth #5’s fracture surface on concave
side.

The left side of both Figures 6.3 and 6.4 is the toe end of the tooth length. This

is true for all of the concave side pictures in this chapter. On the other hand, the right
side of the photographs from the convex side is the toe end of the tooth length. All of

NASA/CR—2000-210062 72



must also be defined. A cyclic loading history like that shown in Figure 3.3 is
assumed. The minimum applied traction is assumed to be zero, and the maximum
applied traction is S,,,, =100ksi. A tensile traction is applied normal to the top edge.
All of the nodes along the bottom edge are restrained in the vertical direction, and the
far right corner node is restrained in the horizontal direction. If desired, at this stage
the finite element solver could be run to calculate displacements, strains, and stresses
in the uncracked geometry.

Figure 3.7: Two dimensional finite element model.

Next, a crack is introduced into the geometry model. With the change in
geometry, the model must be remeshed. However, the damage to the mesh model is
localized, and, therefore, only a small region around the crack must be remeshed. The
mesh around the crack tip is a rosette of eight triangular, six-noded quarter point
elements, Figure 3.8a. The remaining area is meshed with quadratic six-noded
elements. Figure 3.8b shows the initial edge crack and locally remeshed region. The
boundary conditions, material properties, and loads were defined earlier and do not
need to be redefined. At this point, displacements, strains, and stresses are solved for
in the cracked geometry.

A method, such as the displacement correlation technique, is used to compute
the stress intensity factors at the crack tip based on the relative displacements of the
crack faces. Once the SIFs are calculated, Paris’ model (Equation (3.6)) can be used
to calculate the amount of crack growth for one load cycle, da/dN. A method, e.g.
maximum principal stress (Equation (3.4)), is used to determine the direction of crack
growth from the calculated SIFs. In most cases, the amount of crack growth for one
load cycle will be on the order of 10° - 10% inches. Since this is significantly smaller
than the geometry features of the gear, it would be inefficient to update the geometry
model for every load cycle. Consequently, a number of load cycles is assumed, e.g.
N, =2,000 cycles. Finally, the crack in the geometry model is extended by an amount

NASA/CR—2000-210062 29



predicted by this moving load method differed from a failure in a tested pinion;
however, the method succeeded in predicting a fatigue life comparable to the
experimental data.

Other issues related to modeling crack growth in a gear were also investigated.
For example, the effect of shifting the load location along a tooth’s length on the crack
trajectories was confirmed. For a crack that has initiated in the tooth’s root, when the
load location is directly above the crack, the crack trajectory will remain very close to
the root. Additionally, the effect of compressive loads on fatigue crack growth rates in
AISI 9310 steel was examined. This examination is significant because a principal
focus of current gear design is to minimize a gear’s weight. Reducing the amount of
material in the gear may increase the magnitude of the compressive stresses in a gear
tooth’s root, which could influence crack growth rates. It was discovered that the
compressive portion of a load cycle did not significantly modify the rates when crack
closure was incorporated into Paris’ model to calculate fatigue crack growth rates. As
a result, the BEM/LEFM analyses of a spiral bevel pinion were carried out ignoring
the compressive portions of the loading history.

The predictions from the moving load crack propagation method were
compared to predictions when only HPSTC was considered. HPSTC is a more
simplified approach and has been commonly used in past research when numerically
analyzing crack propagation in gears. The HPSTC method utilized existing fatigue
crack growth theories since there was a single load location and proportional loading.
The analyses in this thesis with the two loading methods predicted different fatigue
lives and crack trajectories. The lack of experimental fatigue crack growth rate data
hindered an evaluation of the crack growth rates predicted by the two methods. The
moving load method’s crack trajectory predictions agreed more closely to the tested
pinion failures. Crack trajectories are of primary importance to predict the failure
mode.

The dearth of fatigue crack growth rate data and crack front shape information
from tooth failures in a tested spiral bevel pinion motivated SEM observations of the
fracture surfaces. A crack growth scenario was devised from the observations. In
addition, the observations suggested that the failure mechanism along the majority of
the surface was fatigue. This result supported the use of the numerical simulations to
predict fatigue crack growth trajectories in the gear.

As this thesis was a first attempt at predicting fatigue crack growth in spiral
bevel gears, certain limitations were encountered. The limitations can be summarized
as follows:

¢ A scarcity of experimental data prohibited validations of calculated crack growth
rates, fatigue life predictions, and crack front shape evolution.

¢ The effect of tooth deflections on the contact area between mating gear teeth was
not modeled. Capturing this effect will increase the accuracy of the model since
crack trajectories are strongly determined by the load locations.

¢ It is anticipated that the deflections of a cracked spiral bevel gear tooth will be
limited by the adjacent tooth picking up the load. The magnitude of this

NASA/CR—2000-210062 102



is significant in the context of the overall goal of this thesis, which is to study aspects
of gear geometry that affect damage tolerance.

It is assumed initially in this chapter that the stresses induced in a gear tooth
under positive (tensile) and negative (compressive) parts of the load cycle are
“proportional.” In other words, the shape of the stress intensity factor distribution
along the crack front is the same under both tensile and compressive loading. In two
dimensional analyses, this is not a concern because the crack only consists of a tip,
where the deformation can be tensile only or compressive only, not a combination of
the two. In three dimensions, however, the distribution of the loading (deformation)
along the crack front might be different in the compressive and tensile load cases. In
the end, whether the positive and negative parts of the load cycle are proportional is
not of major concern. As will be shown in the remaining sections, damage occurs
only during the tensile portion of the load cycle.

Elber [1971] observed that during unloading a crack actually closes prior to the
applied load being entirely removed. This phenomenon has been called fatigue crack
closure. Fatigue crack closure also explains why, for a given AK, fatigue tests show
the crack growth rate increasing as R increases. Figure 4.1 shows typical fatigue crack
growth rate data as a function of SIF range [Kurihara er al. 1986]. Kurihara er 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 two data points off Kurihara et al.’s plots for each value of
R. The horizontal scatter in the curves is a result of the different R-values. Note that
as R increases, the curves shift to the left, producing an increase in fatigue crack

growth rate for a given AK.

1.00E-02
05 4

1.00E-03 q
5 j
ES 0.5
4 -0.33
=I
E 1.00E-04 +
2
3 0: hy ? Ros
3 | //

1.00E-05 =

6.67
/:
1.00E-06 or T - Sa
1 10 100 1000

AK [MPa*m°*}

Figure 4.1: Fatigue crack growth rate data for pressure vessel steel at various R-values
(data taken from [Kurihara et al. 1986}).

NASA/CR—2000-210062 36



|.

Figure 5.11: Griffith crack problem: Straight, through thickness crack in an infinite
plate subjected to uniform tensile stresses [Griffith 1921].

4. Repeat step 3 for every load step of the cycle to get the final coordinates and angle
for the trajectory during one load cycle. The final crack trajectory is approximated
by a straight line from the initial crack tip location to the final crack growth
location. Based on simple geometry, the final length, da’, and final angle, 6,
after one load cycle are calculated in the following manner:

da,’ =i +a’ (5.5)

6, = ww (F) (5.6)
M
where l= da’ j-+,)c08(64-1.) (5.7)
=
y
Ai = Seda‘ sin'u-an) (5.8)

jal

Figure 5.12 illustrates this step schematically, assuming the load cycle has
been discretized into four steps, ie. M = 4. Note that the arc length, which is the
sum of the da’y.1,), is equal to the amount of growth predicted by the crack-
closure-modified Paris model, Equation (5.1). The arc length is given
mathematically by

M
da’ = ¥\da'(j-4,)) (5.9)
j=0
5. Repeat steps 3 and 4 for every point along the crack front.

NASA/CR—2000-210062 61



There are no closed form solutions to calculate SIF distributions along the
crack front for arbitrary three dimensional cracks. As a result, a conventional
approach to calculate the SIF distribution is to discretize the front into a series of two
dimensional crack tips. For example, the finite plate model presented in Section 3.2.2,
in reality, has a finite width. Therefore, the crack must have a finite width. The crack
front shape might be that shown in Figure 3.10. In this example, the crack width is
equal to the plate thickness.

Discretized three dimensional crack front Two dimensional crack tip
Figure 3.11: Discrete crack front points treated as two dimensional problems.

Next, the crack front is discretized, as shown by the lines intersecting the crack
front in Figure 3.11. Once the crack front is discretized, each point is treated as a two
dimensional problem. The two dimensional methods to calculate SIFs are applied at
each discrete point. The discrete point is propagated by an amount and at an angle
uniquely defined by the SIFs associated with that point. Once each discrete crack
front point is propagated individually, a least squares curve fit is performed through
the new discrete crack front points, Figure 3.12.

A potential difference in the three dimensional approach, as opposed to the two
dimensional method, is that singular crack front elements might not be used along the
crack front. Since the BEM is implemented in this thesis, the volume of the three
dimensional model is not meshed; only the surfaces are meshed. Therefore, elements
that represent the crack tip singularity are not available along the crack front. The
main drawback of this is that some SIF accuracy along the crack front is sacrificed.

NASA/CR—2000-210062 32



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 in displacements. Consequently, the predicted crack front shape
and trajectory will be inaccurate.

5.5 Simulation Results

Three dimensional fatigue crack trajectory simulations are performed using the
method described for non-proportional loading in Section 5.4.2. The full pinion
boundary element model is used and an initial crack is introduced into the model as
described in Section 5.3. In order to validate the predictions, the dimensions and
shape of the crack are similar to a starter notch that was used in a tested pinion. The
experimental results and data are reported in Chapter 6.

Thirteen crack propagation steps are carried out, which model the crack at
thirteen distinct stages of growth. In each propagation step, the SIFs are evaluated
using the displacement correlation method at the mesh nodes in the first row of
elements along the crack front. To reduce the computational time for each
propagation step, the second stage of double tooth contact, load steps twelve through
fifteen, are ignored. The method for propagation adopted here only allows crack
growth during loading. The final four load steps represent the unloading portion of the
load cycle and, therefore, do not contribute to crack growth in the simulations.

The Paris growth rate model modified 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). Kis set equal to three (plane strain). To calculate Smax »
which is a variable in the K,, calculation, f is set to one. The material properties listed
in Table 2.2 are used.

A least squares curve fit to the predicted discrete crack front points is used to
determine a smooth crack front curve. The approach is to fit a polynomial curve of
second or third order to groups of points. To allow unsymmetric crack front shapes,
the crack front points are divided into one, two, or three groups. A curve is then fit
independently through each group.

The tooth contact locations and magnitudes defined for the 100% design load
are used throughout the thirteen propagation steps. The loading simplifications for
these data were presented in Section 5.2.1.

Table 5.1 contains the crack geometry and growth rate data for each 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. At the end of propagation step thirteen, the crack has propagated
neither entirely through the tooth width nor through the rim. The analyses are stopped
at this step because the toe end of the crack has reached the top land. It was decided
that continuing the analyses would lead to no additional insights because, as reported
in Chapter 7, the prediction differed from the experimental results. Also, one can
imagine that if the simulations were continued, since the trajectory has turned on both
the heel and toe end, this middle portion of the tooth will break away from the gear.

NASA/CR—2000-210062 64



maximum deflection is unknown. As a result, the simulations did not model this
behavior.

« The method to predict crack trajectories under the moving load was time
consuming since every discrete load step had to be analyzed for a full pinion
model. For the method to be more practical to a gear designer, improvements in
the computational time and implementation would have to be made.

In summary, the achievements of this thesis brought the current gear design
approach closer to developing a numerical tool to evaluate safety aspects of gear
geometries. Critical areas that must be understood in greater detail prior to predicting
more accurate crack trajectories and growth rates in three dimensions were identified.
By incorporating non-proportional loading, complex three dimensional geometry, and
arbitrary three dimensional crack models, the work extended the current capabilities
and theories for predicting fatigue crack growth rates and crack trajectories.

8.2 Recommendations for Future Research

The accomplishments of this thesis produced many new questions and issues
related to simulating fatigue crack growth in spiral bevel gears. Future research
focused on the following areas will further assist the development of numerical design
tools to evaluate a gear design’s safety.

e A testing program centered on fatigue crack growth rates and trajectories from
non-proportional loads should be carried out. A more fundamental understanding
of crack behavior in this type of loading environment is needed to enhance the
accuracy of fatigue crack growth rate models. To gain insights into the
fundamental behavior, the tests do not have to be conducted with spiral bevel
gears. To judge the correctness of the predictions in this thesis, however, more
tests on spiral bevel gears that record crack growth rates and trajectories over a
time period are necessary. Such test data are essential to confirm the accuracy of
the simulations and to evaluate the proposed method for predicting crack growth
under the moving tooth load.

¢ Amore detailed understanding of the contact between a cracked gear tooth mating
with an uncracked gear tooth is required. The load redistribution effects in this
scenario could be studied either experimentally or numerically. Numerical studies
would require a fully three dimensional, LEFM, and contact analysis of the rolling
process between two mating gears. Once the redistribution effects are captured,
the findings could be applied to crack growth simulations. The expectation is that
the simulations would model more accurately the observed behavior in real
failures.

¢ The analyses in this thesis considered only loads normal to the tooth surface.
However, a gear tooth is subjected to a variety of loads in operation. For example,
the contact between the mating gears also produces frictional forces along with the
normal loads over the contact ellipse area. Gears in rotorcraft applications operate
at elevated temperature, and, therefore, thermal effects might be included. In
addition, the rotation of the gear produces centrifugal forces. Dynamic loads are

NASA/CR—2000-210062 103



Table 4.1: Slope and intercept of curves in Figures 4.4 and 4.5.

Test n Cc sep 05 \0
[(inlcycle)/(ksi*inTM” )"]
R=-1 (Air) 3.3 6.4e-12
R=0.05 (Air) 3.5 7.3e-11
R=0.5 (Air) 3.9 5.3e-11
R=-1 (Oil) 3.2 Lle-11
R=0.01 (Oil) 3.2 9.9e-11 :
R=0.5 (il) 3.8 7.9e-11

For a given R, the n values are similar between the two environments. The
effect 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 rates. However, the
environment effect will not be considered in this investigation.

Similar to the pressure vessel steel analyses, da/dN versus AKeg plots are
generated using Equations (4.2), (4.4), and (4.5). A value of « =1 best describes the
condition at the crack tip in the test specimen. Figures 4.6 and 4.7 illustrate the
various curves collapsing into an intrinsic fatigue crack growth rate curve.

1.00E-03 =
j n=3.63
| [ C¥=4.26E-10
LOOE-04 =
z
3 1.005-05
3
é —R=1
z =
S 1008-06 4 R005
3 “R05
| Linear Curve Fit
LOOE-07 +
j
L.00E-08 ;

1 10 100

AK og (ksi*in®*]
Figure 4.6: Intrinsic fatigue crack growth rate for AISI 9310 in room temperature air;
K=1.

Figures 4.4 through 4.7 demonstrate that Newman’s crack closure model
accounts for the scatter in fatigue crack growth rates at different stress ratios in AISI
9310 steel. Table 4.2 contains the slopes and vertical intercepts from the lines in the
figures. In addition, a linear least squares curve is fit through the data in Figures 4.6
and 4.7. The slope and vertical intercept from each curve fit are also included in the

NASA/CR—2000-210062 42



causes for the discrepancies between the predictions and the experimental results. The
fatigue crack growth rate model parameters, Section 7.3.1, crack closure model
parameters, Section 7.3.2, and the contact position and magnitude, Section 7.3.3, are

researched further.

7.3.1 Fatigue Crack Growth Rate Model Parameters
Limited fatigue crack growth rate data is available in the literature for AISI

9310 steel. The predictions in Section 5.5 used values for the crack growth rate model
parameters, n and C, taken from a curve fit to the intrinsic fatigue crack growth rate
data (no closure) for AISI 9310 tested in 250° oil. These values were 3.36 and
6.19e-20 (in/cycle)(psi*in®*)", respectively. The range of values for n and C from the
literature is reported in Table 7.1. Au ef al.’s data 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 7.1: Fatigue crack growth rate model parameters.

Source h j c oon daldN” Cycles Fatigue
[Gn/cycle)/(psi*inTM)]" | [in/cycle] /inch Life’

vo o84] a. | 163 1.08e-13 9.30e-6 | 107,527 | 675,838

his 2.56 2.72e-17 2.03e-6 | 492,611 | 3,096,201
[Proprietary 1998] 3.63 5.49e-21 154e-5 | 64,935 | 408,145
[Proprietors 1998] 3.36 6.19e-20 1.23e-5 | 81,301 | 511,000

The fatigue life estimates using each set of parameters in Table 7.1 assume that
the number of cycles that each source would predict for the gear’s fatigue life is
roughly proportional to the ratio of the oil test’s cycles/inch to each set’s cycles/inch.
Forman et al., Au et al., and the air test data each predict a fatigue life of 675,838
cycles (32% increase), 3,096,201 cycles (506% increase), and 408,135 cycles (20%
decrease), respectively.

Au ef al.’s combination of n and C predicts the smallest growth rate and
therefore the longest fatigue life. The benefits and conservatism of considering crack
closure in the predictions is demonstrated by comparing the predictions using the
intrinsic parameters to the predictions with Au er al.’s parameters. Figure 7.6 contains
the fatigue crack growth rate curves for the various sets of n and C. The curves are
generated from the data in Table 7.1.

'° Calculations are based on an assumed value for AK = 18,000 psi*in®?.

' 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



al. approximated the geometry of gears in contact as a pair of disk rollers compressed
together. The linear elastic stresses in the disks could be written in closed form. The
SIFs were calculated using the closed form expressions. Bingyuan et al.’s primary
focus was to calculate surface fatigue life and compute crack growth rates. No
trajectory predictions were made.

The majority of the aforementioned research on spiral bevel gears is unrelated
to failure, but rather associated with design and efficiency; methods have been
developed to create numerical models of spiral bevel gears and predict contact areas.
Crack trajectories have been predicted in gears with simpler geometry that can be
represented by two dimensional models. This thesis is a natural extension of the
research to date. The next step is to computationally model fatigue crack trajectories
in spiral bevel gears.

1.3 Overview of Chapters

This thesis is divided into eight chapters. The first and last chapters are
overview and summary. The remaining chapters each build upon one another and
propose, apply, and evaluate methods for predicting fatigue crack growth in spiral
bevel gears.

Chapter Two contains background information on gears, with particular
attention to spiral bevel gears. The objective is to define vocabulary and concepts
related to spiral bevel gears that will be used throughout the thesis. In addition, the
work of the thesis is further motivated by examples of gear failures and the current
design objectives for gears.

A focus of this thesis is to demonstrate that computational fracture mechanics
can be used to analyze complex gear geometries under realistic loading conditions.
LEFM and fatigue theories that are utilized to accomplish this task are presented in
Chapter Three. Methods that are currently implemented in two and three dimensions
to compute crack trajectories are demonstrated through examples.

Chapter Four explores the significance of compression loading on calculated
crack growth rates. The magnitude of compressive stresses in a gear’s tooth root is a
function of the rim thickness. If fatigue crack growth rates are highly sensitive to this
compression, then growth rates may warrant more attention in designing gears. The
concept of fatigue crack closure is used to investigate fatigue crack propagation rates
in AISI 9310, a common gear steel. First, the concept of fatigue crack closure is
discussed. A material-independent method is presented for obtaining fatigue crack
growth rate data that do not vary with stress ratio. The method is demonstrated using
data at various stress ratios for pressure vessel steel. Next, the concepts are applied to
AISI 9310 steel data to obtain an intrinsic fatigue crack growth model. This model is
used to investigate the effect of low stress ratios on fatigue crack growth in AISI 9310.

Chapter Five is an initial investigation into predicting three dimensional
fatigue crack trajectories in a spiral bevel pinion under a moving load. First, a
boundary element model of a pinion is developed. A method to represent the moving
contact area on a gear tooth is discussed. Next, studies are conducted to determine the
smallest model that still accurately represents the operating conditions of the pinion.
Once the model is defined, a crack is introduced into the model, and the initial stress

NASA/CR—2000-210062 5



TABLE OF CONTENTS

CHAPTER ONE: INTRODUCTION. .........:ccccssssssessessessesseeseesesseeseeseeseeseesteaseseeseenes 1
1.1 Background .

1.2. Numerical Analyses of Gears
1.3. Overview of Chapters

CHAPTER TWO: GEAR GEOMETRY AND LOADING
21 Untroduction ......ceceesseseeseeseeeeeeeeseenes
2.2 Basics of Spiral Bevel Gear Geometry.
2.3. Teeth Contact and Loading of a Gear Tootl
2.4 Gear Materials wo... cesseessesseeseees
2.5 Motivation to Model Gear Failure:

2.5.1 Gear Failures ........s.0000

2.5.2 OH-58 Spiral Bevel Gear Design Objectives
2.6 Chapter Summary

CHAPTER THREE: COMPUTATIONAL FRACTURE MECHANICS.
3.1 Introduction
3.2 Fracture Mechanics and Fatigue
3.2.1 Fatigue ae
3.2.2 Example: Two dimensional, mode I dominant fatigue crack growth
simulation with static, proportional loading... 27
3.2.3. Example: Three dimensional, mode I dominant fatigue crack growth
simulation with static, proportional loading
3.3 Fracture Mechanics Software a
3.4 Chapter SUMMALY .......sessessecssssesseeseesseseeseeneenseneeneeneeneenecaeennenensenenenneneennenaes

CHAPTER FOUR: FATIGUE CRACK GROWTH RATEG............ccccseeseeses 35
41 Introduction
42 Fatigue Crack Closure Concept.
4.3. Application of Newman’s Model to AISI 9310 Steel
4.4 — Sensitivity of Growth Rate to Low R
46 Chapter Summary

CHAPTER FIVE: PREDICTING FATIGUE CRACK GROWTH
TRAJECTORIES IN THREE DIMENSIONS UNDER MOVING, NON.
PROPORTIONAL LOAD:
5.1 Introduction
5.2. BEM Model
5.2.1 Loading Simplifications
5.2.2 Influence of Model Size on SIF Accuracy . ae
5.3 Initial SIF History Under Moving Load .......ccccesecssssssssesesseeseessseeeseessseeneseenes

NASA/CR—2000-210062 v



‘max

%%

= (0.415« -0.071K?

A, =1-A,-A,—-A,

A; = 2A, + A, -1
Kis a constraint factor taking on a lower bound of | for plane stress conditions and an
upper bound value of 3 to simulate plane strain conditions. The flow stress, oo, is the
average between the uniaxial yield stress and the uniaxial ultimate tensile strength of
the material.

Because Newman’s model for K,, is a function of material constants (09), R,
and k, it is applicable for any fatigue crack where LEFM holds and the loading
conditions and material properties are known. Figure 4.3 is an example of how the
curves in Figure 4.1 collapse into an intrinsic curve when crack closure is taken into
account. Equations (4.4) and (4.5) are used to calculate AK,,. U is calculated using
Equation (4.2).°

1,00B-02 5
] 2*da/dN (R =0) :
j ~~
j ¥ 7 daldN (R=0)
| : —+ R=-.
1.00E-03 = é
_ q 1 —* R=-3
zg I / 0.5*da/dN (R=0) 4, pao
g 7a / 3 R=-
= 1.00E-04 + } eR=05
z F ~eR=0.33
aS | —+R=0
1.00E-05 R05
—R=0.67
+ R=0.8
1.00E-06 + — neal
1 10 100 1000

0.5.

AK og [MPa*mTM]
Figure 4.3: Intrinsic fatigue crack growth rate data for pressure vessel steel using
Newman’s equations for AK,g; K =1 (using data taken from [Kurihara er a/. 1986]).

The crack tip condition in the fatigue test specimen Kurihara ef al. used, a thin
plate with a center crack, is best described by plane stress. Therefore, a value of
k =1 was selected for the preliminary graphs. x was then increased, and the amount

5 Note that Newman claims Equation (4.4b) is valid for negative R-values greater than or equal to -1.
However, Kurihara et a/.’s data extends to -5.0. Equation (4.4b) was used for the cases when R = -5.0,
-3.0, and -2.0. Figure 4.3 illustrates, at least for this case, the equation can also hold for these low R-
values.

NASA/CR—2000-210062 39



Qualitative information can be taken from the pinion test to validate the
predictions. For example, all five failures were teeth fractures. There was no
evidence of a rim type of failure. A macroscopic view of the crack trajectory can be
determined from the tested pinion. Figure 6.1 shows photographs of a typical tooth
failure in the tested pinion. The rectangular EDM notch on the concave side of the
tooth near the root is observable in all of the pictures. The crack trajectory, which is
assumed to initiate from the EDM notch, is deeper into the rim in the middle of the
tooth length than on the toe end. In addition, a ridge is observed where the crack
trajectory turned toward the root of the convex side from the initial path into the rim.
All five of the fractured teeth had remaining portions of the tooth left intact at the heel
end. Furthermore, four of the notches were small enough such that they do not appear
to have grown during the 4.9 million load cycles.

Crack growth rate data and crack front shape information during propagation
could both be used to validate the numerical analyses. Since this data was
unavailable, the tooth fractures were observed with a SEM. The SEM observations
are summarized in the next section.

6.3 Fractography

6.3.1 Overview

Three teeth that had fractured from the tested OH-58 pinion were examined
using a SEM. Tooth #11 contained a “short” notch (~4 mm long x ~1mm deep), and
was the initial flaw from which the numerical simulations were based. Tooth #5
contained a longer and deeper starter notch (~18 mm x ~1.7 mm deep). Pictures of
these teeth’s fracture surfaces are contained in this section. The starter notch of the
third tooth examined was similar to tooth #11. No distinguishing features were found
on the tooth that were not seen on the other teeth.

There were several goals of the examinations. The primary objectives were to
determine how much of the crack growth occurred due to fatigue, where the
mechanism of failure might have changed, and, if it did change, to what type of
fracture mechanism. Additionally, information on the rate of crack growth and the
crack front shape during propagation was desired. All of this information could verify
how well the numerical simulations predicted the actual crack evolution.

The sketch in Figure 6.2 shows a typical view of how the crack propagated
through a pinion tooth’s cross section. The crack initiated from the EDM notch on the
side of the tooth with the applied load. This side is the loaded or concave side. The
opposite side of the tooth, where the crack ends, is the load free or convex side. The
region where the crack deviates from the original path, into the rim, towards the fillet
on the convex side will be referred to as the ridge.

Figures 6.3 and 6.4 are of tooth #5 and #11, respectively. The fractured teeth
are lying on their side with the tooth surface at the bottom of the figure and the ridge
at the top. As illustrated in Figure 6.2, the crack growth direction was from the bottom
of the photograph toward the ridge at the top of the photograph. The EDM notch is

NASA/CR—2000-210062 11


CHAPTER SEVEN:
DISCUSSION AND SENSITIVITY STUDIES

7.1 Introduction

In Chapter 5, fatigue crack growth in the OH-58 spiral bevel pinion under
moving, non-proportional loads was predicted. Chapter 6 presented experimental
fatigue crack data from a tested OH-58 spiral bevel pinion. The present chapter
compares these two sets of results to evaluate the success of the predictions and
investigates the sensitivity of the prediction results to variations in the methods
assumptions.

In Section 7.2, the crack growth simulations are compared to the experimental
results of a tested pinion; the fatigue lives and crack trajectories are evaluated to
determine the accuracy of the prediction method. Sensitivity studies are conducted in
Section 7.3 to explore variations in tooth contact position and magnitude and the
sensitivity of the crack-closure-based fatigue crack growth rate models to variations in
the model parameters.

Crack growth predictions from the moving load analyses (Section 5.5) are
compared to crack growth results from analyses that consider only highest point of
single tooth contact (HPSTC) loading in Section 7.4. HPSTC loading has been
commonly adopted in past research because it is a more simplified approach than the
moving load. When using HPSTC loading, existing fatigue crack growth theories can
be implemented since there is a single load location and proportional loading. The
two loading methods’ results are compared to evaluate the need for the moving, non-
proportional load method; the least computationally intensive model and method
which produces reasonable crack growth results is the most practical for a gear
designer.

7.2 Comparisons of Crack Growth Results

Figure 7.1 shows the predicted and experimental trajectories on the tooth
surface and through the cross section of a pinion tooth. The predicted results are the
same as those reported in Section 5.5. The experimental trajectories are approximated
from measurements and photographs of the failure surfaces of tooth #11. As
discussed in Chapter 6, the failure associated with tooth #11 is representative of all the
failures in the tested pinion. In addition, the size of the initial flaw in the predictions
was taken from the dimensions of the EDM notch in tooth #11.

A ridge is not observed in the predicted crack path through the cross section
because the simulations were stopped prematurely. The simulations were halted
because the trajectory along the tooth surface on the toe end varied significantly from
the experiment. As a result, it was concluded that the simulations were not completely
accurate and further propagation of the crack would lead to no additional insights. An
explanation for this discrepancy might be that the simulation did not properly account
for changes in the load shape and location as the crack grew. This changing load
scenario will be investigated further in Section 7.3.3.

NASA/CR—2000-210062 81



NASA/CR—2000-210062 ARL-CR-451

U.S. ARMY

RESEARCH LABORATORY

Simulating Fatigue Crack Growth
in Spiral Bevel Gears

Lisa E. Spievak, Paul A. Wawrzynek, and Anthony R. Ingraffea
Cornell University, Ithaca, New York

DISTRIBUTION STATEMENT A
Approved for Public Release ? 0000 8 { ) 02 7
Distribution Unlimited



#0005

Figure B.2: Typical picture of flat, polished area on tooth #11 (400x). Photograph was
taken near location B in Figure 6.4.

Figure B.3: Machining grooves and pits on tooth surface (280x).

NASA/CR—2000-210062 107



Acknowledgments

The research contained in this thesis was conducted under grant NAG3-1993 between Cornell University and
NASA Glenn Research Center. I wish to thank Dr. David Lewicki and Dr. Robert Handschuh of the U.S.
Army Research Laboratory at NASA Glenn Research Center. Much of this thesis’ work is a direct
result of their advice and expertise. Lehigh University professor Dr. Eric Kaufmann’s time and
technical knowledge were instrumental with the scanning electron microscope observations
contained in this thesis. In addition, Dr. Richard N. White at Cornell University
volunteered his time and skills to photograph the tested spiral bevel pinion.

Many of his photographs are contained in this volume.

Available from
NASA Center for Aerospace Information National Technical Information Service
7121 Standard Drive 5285 Port Royal Road
Hanover, MD 21076 Springfield, VA 22100
Price Code: A06 Price Code: A06



APPENDIX C

20000 >
— Model -*Load1
---- Modell Load 5
15000 - “Load 11
< 10000 +
&
&
&
&
“” 5000
0 :2 = 7
oo u 16 21
Crack front position
(Orientation: toe to heel)
-5000

Figure C.1: SIFs from toe-shifted load steps one, five, and eleven.

20000 +
— Model —* Load 1
---- Modell ~—+Load5
15000 - —- Load 11

10000 =

5000 =

Crack front position
(Orientation: toe to heel)

-5000
Figure C.2: SIFs from heel-shifted load steps one, five, and eleven.

NASA/CR—2000-210062 108



Similarly, equations for Ky and Ky; can be written as a function of u;, the displacement
due to in plane shear, and w-, the displacement due to out of plane shear, respectively.
It is important to note that as ; approaches zero, the accuracy of the SIFs will decrease
when using the displacement correlation method if the crack front elements are not
capable of representing the singularities at the crack tip.

Crack growth rates are calculated from the SIF information and experimentally
determined fatigue crack growth model parameters. The SIF information is also used
to calculate the angle of propagation, e.g. Equation (3.4).

3.2.2 Example: Two dimensional, mode I dominant fatigue crack growth

simulation with static, proportional loading
The purpose of this example is to demonstrate how fatigue crack predictions

can be performed on a simple two dimensional model. The model assumptions are:

1. The location of applied load is not changing. This will be referred to as
static loading.

2. The loading is proportional.

3. The crack growth can primarily be attributed to mode I opening. In
other words, K, >> K,,. This will be referred to as mode I dominant.

4. Crack closure effects will be ignored.
5. LEFM holds.

The method to predict crack trajectories in two dimensions is incremental. A
series of finite element analyses are run which incrementally increase the crack length
by a significant amount in relation to the model’s geometry. For a given increase in
crack length, the number of cycles to achieve that amount of growth can be calculated.
For a given propagation step i, there are N; load cycles associated with it.

The amount of crack growth for one cycle is calculated as a function of the
maximum stress in the load cycle. Because it is assumed the loading is proportional, it
is straightforward to calculate the direction the crack will grow during the cycle using
the maximum principal stress theory. However, there is no proposed method to
calculate the final amount and direction of crack growth during one load cycle if the
ratio K,,/K; varies during the cycle, i.e. non-proportional loading.

NASA/CR—2000-210062 27



Using load step eleven’s SIF results from the initial crack (Figure 5.8), which
are the largest mode I SIFs during the load cycle, the plastic zone size along the crack
front ranges from 1.44x10TM inches to 1.07x10° inches. These dimensions are only
0.29% and 2.14%, respectively, of the initial crack depth. It is concluded, therefore,
that the plane strain assumption along the entire crack front most accurately represents
the conditions in the real gear.

Crack Growth Rate Sensitivity to B

Smax, the far field applied stress, is a function of B , c (half of the crack length),
and K;. B is a dimensionless quantity that considers geometry effects when relating K;
to the applied stress. Values of B from handbook solutions can vary from one half to
two [Murakami 1987]. Since the gear geometry is complex and unlike any handbook
solution, a value of B =1 was selected. An alternate approach could have been to use

a known factor for a similar, simplified geometry. This alternate approach will now
be investigated and growth rates between the two methods will be compared.

I.

Figure 7.7: Finite thickness plate with a semi-elliptical surface crack subjected to
mode I uniform stress.

The initial crack in the gear is approximated by a finite thickness plate with a
semi-elliptical surface flaw subjected to mode I uniform tensile stress, o, Figure 7.7.
The magnitude of K; varies along the crack front. Kjmax and Kimin occur at the surface
and midpoint of the crack front, respectively. They are given by Broek [1986] as:

12 12
Kina. = 7 GVA and K jin 22 leo (7.2)
c

NASA/CR—2000-2 10062 88



Section 7.3.3. The goal is to predict what operating conditions would be necessary to
obtain the fracture path observed in the tested pinion.

For the constant torque level of 3120 in-lb, approximately five hundred
thousand cycles were predicted to propagate the crack through the tooth thickness.
This value is the same order of magnitude as that which occurred in the tested pinion.
The number of cycles to produce the tooth fractures in the test is smaller than 4.9
million. The sensitivity of the fatigue life prediction to the values chosen for the
model parameters is studied in Section 7.3.1.

Alban’s condition number four for “classic tooth-bending fatigue” scenario
(Section 2.5) is not captured in the numerical work. The magnitude of the applied
loads during the cycle was kept constant during the crack propagation analyses, Figure
7.2a. This type of loading scenario is considered load control and results in the SIFs
increasing continuously as the crack grows, Figure 7.2b. Fatigue crack growth will
occur at an increasing rate until the SIFs satisfy a fracture criterion, such as K; = Kjc.

Crack growth simulations under load control will predict a shorter number of cycles to
failure than observed. This is because, in reality, when a cracked and uncracked tooth
mesh, the cracked tooth will deflect a limited amount before it’s adjacent tooth picks
up a portion of the load [Alban 1985]. The displacement of the crack faces reaches a
maximum and will be roughly equal for every remaining load cycle. As a result, the
rate of increase in the SIFs will decrease, and reach roughly 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,

a) | Time p) | Time

°) | Time
Figure 7.2: Schematic of load cycles: a) Load versus time, b) K; versus time, load
control, and c) K; versus time, displacement control.

NASA/CR—2000-210062 83



the gear is left- or right-handed, respectively. So that the teeth can fit together, or
mesh, a spiral bevel gear and pinion will always have opposite hands. The thickness
and height of a spiral bevel gear tooth varies along the cone. The larger end of the
tooth is the heel, and the smaller the toe. The curvature of the tooth creates concave
and convex tooth surfaces on opposite sides of the tooth, Figure 2.3.

Heel

Convex side

Concave side

Tooth root

Figure 2.3: Schematic of a single spiral bevel gear tooth.

The tooth profile, as shown in Figure 2.4, is one side of the cross section of a
gear tooth. The fillet curve is at the bottom of the tooth profile where it joins the space
between the teeth. The region of the tooth near the fillet is the bottom land, and the
area near the top of the profile is the top land.

Top Land

Tooth Profile

Fillet Curve

Bottom Land
Figure 2.4: Schematic of cross section of a gear tooth.

The advantage of the spiral bevel gear’s curved teeth is to allow for more than
one tooth to be in contact at a time. This makes it significantly stronger than a straight
bevel gear of equal size. Consequently, spiral bevel gears are commonly found in
high speed and high force applications. One such application, which is the focus of
this thesis, is in helicopter transmission systems. The mating spiral bevel gears in the

NASA/CR—2000-210062 9



In the BEM model, the shape of a contact ellipse is approximated by straight
lines connecting the axes’ end points. The straight line approximation is valid because
Saint Venant’s principle holds; as long as the total applied forces and resulting
moments are kept constant, the elliptical shape of the traction can be approximated by
a patch with straight sides without altering the stress distribution along the crack front.
Frictional forces are neglected, and, consequently, the traction is constant over the
patch. Each patch has a unique magnitude of traction.

The four figures in Figure 5.2 demonstrate how the traction patches are built
into the model geometry. The purpose of the models is to calculate SIFs from all
fifteen static load cases. The combination of all fifteen SIF distributions represents
one load cycle on the tooth. Figure 5.2 shows how a single BEM model can
incorporate multiple load cases. Not all of the contact ellipses can be modeled in one
BEM model because there is overlap between the ellipses. The multiple load case
feature minimizes the computational time. For example, the boundary element model
for load cases one, five, eight, and thirteen is virtually identical. The only difference
between them is the boundary conditions. Hence, with the multiple load case feature,
the two most computationally expensive steps of the boundary element solver, setting
up the boundary integral equations and factoring the stiffness matrix, occur only once.
The different boundary conditions are then applied individually, and the corresponding
equations are solved for the unknown displacements and tractions for each load case.

5.2.2 Influence of Model Size on SIF Accuracy

The fewer the number of elements, or unknowns, in a boundary element
model, the less computationally intensive the model is. Minimizing the number of
elements can primarily be accomplished by 1) using a coarser mesh with larger
elements or 2) by modeling less of the geometry of the solid. A disadvantage of the
first option is the accuracy of the solution is sacrificed. The elements used in all of the
studies in this thesis are linear. Therefore, only linear variations in displacement
across an element can be represented. Likewise, the geometry is approximated by a
series of linear segments. Because the geometry of the pinion is complex with
significant amounts of curvature, larger elements do not represent the geometry
adequately. As a result, this option is disregarded, and the second option, simplifying
the model, is considered.

Simplifying the model also has drawbacks. The smaller the portion of the
pinion modeled, the less accurate the representation of the boundary conditions. Three
simplified models are investigated. The first simplification, Figure 5.3, is to ignore the
long shaft in the full model. The new faces that are created 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 pinion in half, Figure 5.5. The boundary conditions for this
model are the same as the second simplification, with the addition of roller boundary
conditions (displacement in the direction of the local normal set to zero) applied to the

NASA/CR—2000-210062 SI



Load Step 1
0.4 + = 150% Load
-s 125% Load
03+ — 100% Load
~ 0.2
x
< o14
0 Bae,
0 5 0 20 25
0.1 -! Crack front position
. (Orientation: heel to toe)
02+
a)
0.5
Load Step 11
0.4 5 + 150% Load
| = 125% Load
03- — 100% Load
~ 0.2 -
x
< ony
j
o/__
0 5 10 1s 20 25
-0.1 - Crack front position
(Orientation: heel to toe)
-0.2 -
b)

Figure 7.12: Ky/K; distribution for load step one (a) and load step eleven (b).

NASA/CR—2000-210062 95



the photographs of the fracture surfaces designate the tooth number, the level of
magnification, and a length scale.

Arm

26.8 kV

Figure 6.4: Low magnification (5.3x) view of tooth #11’s fracture surface on concave
side.

6.3.2 Results

The morphologies of teeth #5 and #11 are very similar. Lines emanating
radially from the starter notch are visible at low magnifications (4.6x in Figure 6.3 and
5.3x in Figure 6.4). These lines are indicative of fatigue crack growth.

On tooth #11, some fatigue striations can be seen near the EDM notch. Region
A in Figure 6.5a is the EDM notch. Figure 6.5b is a magnified view of the striated
region.® As expected, the striations are roughly parallel to the edge of the EDM notch.
The crack growth direction was perpendicular to the striations and, as mentioned
above, from the bottom of the figure towards the top. In general, however, most of the
surface near the notch was flat with no significant features or texture. This flat surface
leads to the conclusion that significant rubbing took place. The rubbing “polished” the
surface and removed all features that would have indicated the mode of fracture, e.g.
fatigue striations, dimples, etc. The rubbed surface was visible over approximately
80% of the surface when moving away from the notch toward the ridge of the fracture.
Point B in Figure 6.3 is the approximate location where Figure 6.6 was taken. Figure
6.6 is an example of the typical surface appearance in the rubbed region.” The

8 Figure B.1 in Appendix B also contains magnified views of the striated region.
° Figure B.2 in Appendix B is another picture of the typical surface appearance in the rubbed region.
Point B in Figure 6.4 is the approximate location where Figure B.2 was taken.

NASA/CR—2000-210062 73



Least squares
curve fit

New discrete crack
front points

Figure 3.12: Least squares curve fit through new discrete crack front points.

3.3 Fracture Mechanics Software

A suite of fracture mechanics software developed by the Cornell Fracture
Group is used in this thesis [FRANC3D 1999a, 1999b]. The codes were developed to
handle the complexities of three dimensional crack trajectory predictions. OSM is
used to define a three dimensional solid geometry model of an object. The program is
based on defining the surfaces of the model explicitly in Cartesian space. The
boundary of a solid is generated by adjacent surfaces, or faces. Each face of the
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 of the solid. The local coordinate system might also be of
significance when defining boundary conditions.

The geometry model is then read into FRANC3D. With FRANC3D, a user
can create a finite element or boundary element mesh based on the geometry model.
Displacement or force/traction boundary conditions must be defined for all the faces
of the solid. The conditions must be specified in all three Cartesian directions with
respect to either the local or the global coordinate system. Material properties are also
assigned to regions of the model using FRANC3D.

Cracks are added to the solid by explicitly defining the vertices, edges, and
faces that model the cracks. A crack has two distinct faces that must be meshed
identically.

As mentioned in Section 3.2.3, a crack front must be discretized prior to
calculating SIFs and to propagating the crack. Within FRANC3D, there are three
options to discretize the crack front. The discrete points can be defined by the mesh
nodes, the midpoints of the elements sides along the crack, or at a user defined number
of equally spaced points along the crack front. The built in feature in FRANC3D to
calculate SIFs uses the displacement correlation technique. The most accurate results
are obtained when a row of four sided elements is used along the crack front. This
will give a set of equally spaced points behind the crack front where the SIFs can be

NASA/CR—2000-210062 33



conclusions would still be valid. This is a simplification to the loading cycle and
method.

Table 4.4: Crack growth rate calculations for a wide range of R-values taking into
account crack closure effects. The percent change in da/dN is due to AR = 0.005.

R da/dN % Change
(inlcycle x daldN
10”)
0.705 2.434 0.153 =
0.700 2.467 0.160 4.446
0.505 3.542 0.549 --
0.500 3.565 0.561 2.135
0.255 4.388 1.136 -
0.250 4.399 1.146 0.873
0.005 4.714 1.1449 --
0.000 4.716 1.451 0.130
-0.495 4.886 1.637 --
-0.500 4.888 1.639 0.120
-0.955 5.058 1.841 =
-1.000 5.060 1.844 0.116
1.995 5.402 2.303 =
-2.000 5.404 2.306 0.108
-2.995 5.746 2.841 =
-3.000 5.748 2.844 0.102

daldN 3.00E-07 >

2.50E-07 +

1.00E-07

5.00E-08 +

rf 0-00E+00— T .R

-3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Figure 4.8: da/dN as a function of AK gy.

NASA/CR—2000-210062 45



Schijve 1988

Schijve 1996

Sfakiotakis et al. 1997

Sih 1974

Singh et al. 1998

Swift 1984

Timoshenko et al. 1970

Townsend et al. 1991

Westergaard 1939

Williams 1957

NASA/CR—2000-210062

Schijve, J., 1988, “Fatigue Crack Closure: Observations and Technical
Significance,” Mechanics of Fatigue Crack Closure, ASTM STP 982,
J.C. Newman, Jr. and W. Elber, Eds., American Society for Testing
and Materials, pp. 5-34.

Schijve, J., 1996, “Predictions on Fatigue Life and Crack Growth as an
Engineering Problem. A State of the Art Survey,” Fatigue 96:
Proceedings of the Sixth International Fatigue Congress Volume II,
Editors G. Luterjering and H. Nowak, Elsevier Science Inc., pp. 1149-
1164.

Sfakiotakis, V.G., Katsareas, D.E., and Anifantis, N.K., 1997,
“Boundary Element Analysis of Gear Teeth Fracture,” Engineering
Analysis with Boundary Elements, Volume 20, pp. 169-175.

Sih, G.C., 1974, “Strain-Energy-Density Factor Applied to Mixed

* Mode Crack Problems,” International Journal of Fracture, Volume

10, pp. 305-321.

Singh, R., Carter, B.J., Wawrzynek, P.A., and Ingraffea, A.R., 1998,
"Universal Crack Closure Integral for SIF Estimation," Engineering
Fracture Mechanics, Volume 60, pp. 133-146.

Swift, T., 1984, “Fracture Analysis of Stiffened Structure,” Damage
Tolerance of Metallic Structures: Analysis Methods and Application,
ASTM STP 842, J.B. Chang and J.L. Rudd, Eds., American Society for
Testing and Materials, pp. 69-107.

Timoshenko, S.P., and Goodier, J.N., 1970, Theory of Elasticity, Third
Edition, McGraw-Hill, Inc.

Townsend, D.P., and Bamberger, E.N., 1991, “Surface Fatigue Life of
MSONiL and AISI 9310 Gears and Rolling-Contact Bars,” Journal of
Propulsion and Power, Volume 7, pp. 642-649.

Westergaard, H.M., 1939, “Bearing Pressures and Cracks,” Journal of
Applied Mechanics, Volume 6, pp. 49-53.

Williams, M.L., 1957, “On the Stress Distribution at the Base of a
Stationary Crack,” Journal of Applied Mechanics, Volume 24, pp.
109-114.

116



The location and size of the fifteen discrete contact patches were provided by
NASA/GRC. The data were determined numerically by a procedure similar to that
described by Litvin et al. [1991]. The mean point of contact between the gears is
taken as the center of the ellipse. Hertzian contact theory along with the applied
torque level is used to determine the width of the ellipse. The patches were calculated
for operating conditions of 300 horsepower, 6060 rotations per minutes, and 3120 in-
Ib torque. These conditions are approximating the 100% design load condition, which
is defined as 3099 in-Ib torque.

if
Figure 5.2: Contact ellipses defined at the geometry level in the numerical models.

NASA/CR—2000-210062 50



4.5 Chapter Summary

Highlights of this chapter can be summarized as follows:

© The crack closure concept and Newman’s model were presented. It was shown
that the model predicts that fatigue damage occurs only during the portion of the
load cycle when the crack faces are not in contact.

e Newman’s crack closure model was applied to empirical data for crack growth
rates of a pressure vessel steel. It was shown that crack closure explains well the
apparent dependence of crack growth rates on R. In fact, the material has an
intrinsic crack growth rate. R is a parameter that determines during what portion
of the load range the crack faces are not in contact. This range is called the
effective stress intensity factor range.

e Newman’s model was applied to AISI 9310 steel, a typical steel used for gears.
There was much less crack growth data available for this steel as compared to the
pressure vessel steel. Nevertheless, it was shown that the crack closure model
works for this small data set.

e It was demonstrated that, in the regime of negative R-values, the model predicts
that the crack growth rates as a function of the effective stress intensity factors are
only a weak function of the magnitude of R.

e The observation made in this chapter that crack growth rates are not highly
sensitive to R in the negative R-regime will be used in Chapter 5 when modeling
the load history.

NASA/CR—2000-210062 46



striations were present on the surface, the fact that no other obvious signs of failure
mode were observed, lead to the conclusion that the crack propagated by fatigue in
this region under an applied load range which was inadequate to produce striations.
The lack of rubbing also suggests that the fracture surfaces were created in the later
stages of crack growth.

The combination type of surface was also found over approximately 90% of
the surface on the load free side of the tooth. Figure 6.8, of a partially rubbed surface,
was taken from location A in Figure 6.9. Region B in Figure 6.9 is the tooth surface
on the load free side. Therefore, point A is approximately 0.75 mm from where the
crack ended on the tooth surface. In addition, there are fatigue striations evident in
Figure 6.8. Because this figure is from the convex side of the tooth, the crack growth
direction was from the top of the figure to the bottom. This combination of evidence
leads to the conclusion that the crack continued to grow in fatigue mode along the
convex side of the tooth.

A light band can be seen in Region C of Figure 6.9. The darkened region
separating region B and C is assumed to be oxidation of the fracture surface. Recall
that the fatigue striations in Figure 6.8 are from location A. The surface in Region C
shows obvious signs of ductile rupture, Figure 6.10. This observation is encouraging
because it demonstrates that the material is capable of failing by ductile fracture, and
the areas where this type of fracture occurred should be obvious and visible under the
SEM. This result also leads to the conclusion that the primary mode of crack growth
on the concave and convex sides of the tooth was fatigue.

20.0 wy 1m #0005

Figure 6.8: Picture of partially rubbed surface with fatigue striations on load free side
(825x). Photograph was taken at location A in Figure 6.9.

NASA/CR—2000-210062 76



Air test

14

LD Oil test
Bi
2 Forman et al.
085
aS

0.6 -

04+

02 Auetal.

0 = _
tC) 20000 40000 60000 80000 100000 120000

N [eycles}

Figure 7.6: Fatigue crack growth rate curves for the sets of model parameters in
Table 7.1.

An exact evaluation of which set of material constants is most accurate is
nearly impossible since many parameters of the pinion test are unknown. The
calculations presented in this section demonstrate a trend that the fatigue life
calculations will be more accurate when material constants from intrinsic fatigue crack
growth rate curves are used. :

7.3.2. Crack Closure Model Parameters

For the simulation results presented in Section 5.5, values for the crack closure
model parameters, kand B, were assumed in order to calculate the fatigue crack
growth rates. This section investigates the validity and sensitivity of the results to the
assumed values.

Crack Growth Rate Sensitivity to k

kK incorporates three dimensional effects into the crack growth rate
calculations. Newman specifies that x varies between one and three for plane stress to
plane strain, respectively. For the predictions reported in Section 5.5, K was equal to
three. However, for extremely shallow cracks or portions of the crack front near the
free surface, a value of kK equal to one might represent the crack conditions more
accurately.

One method to evaluate the crack tip conditions is to compare the size of the
crack tip plastic zone to the crack’s geometry. The plastic zone is larger in plane
stress than in plane strain. An approximation of the plastic zone size, rp, is

r at Ky (7.1)

’ 3n\o

y

NASA/CR—2000-210062 87



of correlation between the curves was visually inspected. As x increased, the curves
became more scattered, validating the choice of k =1.
The equation of a line in Figure 4.3 is given by:

to $2] n’ log(AK, )+1og(C’) (4.6)

In Figures 4.1 and 4.3, the slope for a given curve (R-value) is uniquely defined by the
data points. According to the crack growth models, all of the curves should have the
same slope. Ideally, this would be the case for the plots in Figure 4.3. The relatively
small scatter in the magnitude of the slopes at different R-values is attributed to scatter
in the experimental results.

Figure 4.3 includes the intrinsic curve predicted by the intrinsic R=0 data.
This curve falls roughly in the middle of the predicted curves. To give an idea of the
scatter in the curves, the figure also includes lines corresponding to one half and two
times the crack growth rate for R=0. All of the predicted intrinsic curves fall into
this envelope. As a result, it is concluded that the AK equations produce good
correlation.

These results with 500 MPa pressure vessel steel demonstrate that an intrinsic
fatigue crack growth rate curve can be obtained using Newman’s 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 Model to AISI 9310 Steel

An open literature search for fatigue crack growth 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 R-values and frequencies for carburized and noncarburized AISI 9310 steel.
Because they were investigating the correlation between fatigue 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 AK, there is very little scatter in the curves. This
suggests that the crack growth rate is not sensitive to R or that the applied loads were
high enough such that K,, <K,,i,- Since the objective of this study is to correlate

fatigue crack growth rate data at different R-values, including the negative R regime,
Au et al.’s data is inadequate.

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



Glodez et al. 1998

Griffith 1921

Handschuh er al. 1991

Hourlier et al. 1985

Irving et al. 1999

Irwin 1957

Jeong et al. 1997

Johnson 1985

Kurihara et al. 1986

Laird 1967

Lewicki et al. 1993

NASA/CR—2000-210062

Glodez, S., Pehan, S., and Flasker, J., 1998, “Experimental Results of
the Fatigue Crack Growth in a Gear Tooth Root,” International
Journal of Fatigue, Volume 20, pp. 669-675.

Griffith, A.A., 1921, “The Phenomena of Rupture and Flow in Solids,”
Philosophical Transactions of the Royal Society of London, Volume
A221, pp. 163-198.

Handschuh, R.F., and Litvin, F.L., 1991, “A Method for Determining
Spiral-Bevel Gear Tooth Geometry for Finite Element Analysis,”
NASA Technical Paper 3096/AVSCOM Technical Report 91-C-020.

Hourlier, F., d’Hondt, H., Truchon, M., and Pineau, A., 1985, “Fatigue
Crack Path Behavior Under Polymodal Fatigue,” Multiaxial Fatigue,
ASTM STP 853, K.J. Miller and M.W. Brown, Eds., American Society
for Testing and Materials, Philadelphia, pp. 228-248.

Irving, P.E., and Buller, R.G., 1999, “Prediction of Fatigue Life Under
Helicopter Loading Spectra for Safe Life and Damage Tolerance
Design,” Fatigue and Fracture Mechanics: Twenty Ninth Volume,
ASTM STP 1332, T.L. Panontin and $.D. Sheppard, Eds., American
Society for Testing and Materials, West Conshohocken, PA, pp. 727-
742.

Irwin, G.R., 1957, “Analysis of Stresses and Strains Near the End of a
Crack Traversing a Plate,” Journal of Applied Mechanics, Volume 24,
pp. 361-364.

Jeong, D.Y., Tang, Y.H., and Orringer, O., 1997, “Damage Tolerance
Analysis of Detail Fractures in Rail,” Theoretical and Applied
Mechanics, Volume 28, pp. 109-115.

Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press,
Great Britain.

Kurihara, M., Katoh, A., and Kawahara, M., 1986, “Analysis on
Fatigue Crack Growth Rates Under a Wide Range of Stress Ratios,”
Journal of Pressure Vessel Technology, Volume 108, pp. 209-213.

Laird, C., 1967, “Mechanisms and Theories of Fatigue,” Fatigue and
Microstructure,” pp. 149-203.

Lewicki, D.G., Handschuh, R.F., Henry, Z.S., and Litvin, F.L., 1993,

“Low-Noise, High-Strength, Spiral-Bevel Gears for Helicopter
Transmissions,” NASA TM-106080.

112



NASA/CR—2000-210062

U.S. ARMY

RESEARCH LABORATORY

Simulating Fatigue Crack Growth
in Spiral Bevel Gears

Lisa E. Spievak, Paul A. Wawrzynek, and Anthony R. Ingraffea
Cornell University, Ithaca, New York

Prepared under Grant NAG3-1993

National Aeronautics and
Space Administration

Glenn Research Center

ARL-CR-451

May 2000


~4000

APPENDIX A

Crack font position
(Orientation: heel to toe)

Load |
—= Load 2
—+ Load 3
+ Load 4
Load 5
—> Load 6
—+ Load 7
— Load 8
— Load 9
—+ Load 10
—& Load 11

Figure A.1: Mode II SIF distribution for load steps one through eleven.

Km [psi*ir? |

-4000

Crack font position
(Orientation: heel to toe)

—-Load 1
—* Load 2
+ Load 3
> Load 4
Load 5
—e Load 6
+ Load 7
— Load 8
— Load 9
—> Load 10
-& Load 11

Figure A.2: Mode III SIF distribution for load steps one through eleven.

NASA/CR—-2000-210062

105



5.4.3 Approximations of Method

The accuracy of the proposed method is limited by the accuracy of the BEM
results. In addition, several approximations and simplifications 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 away from the crack front such that the deformation due to local contact
stresses does not contribute to the crack tip field, St. Venant’s principle holds. In this
case, the shape of the traction area is inconsequential and the straight line
approximation is legitimate. However, if the crack trajectory is significantly 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 are kept constant throughout the crack propagation. In reality, the
change in flexibility of the tooth could change the contact areas and in tum affect the
SIF distribution in the later stages of propagation. This aspect will be investigated in
Chapter 7. However, if the same contact areas are used during the entire crack
propagation simulation, the fatigue life predictions will most likely be conservative.
This is because the SIFs continuously increase as the crack advances when the loading
scenario is kept constant and the crack length is increasing.

It is assumed that the crack front conditions are characterized by plane strain
along the entire front. This is consistent with the maximum principal stress theory
equations implemented to calculate the kink angle. However, shallow cracks and
portions of edge cracks near the free surface are usually characterized by plane stress,
not plane strain. Nevertheless, crack growth rates will be larger in plane strain
conditions than plane stress. This assumption errs on the conservative side and will
predict shorter fatigue lives. The size of the plastic zone in the gear is investigated
more in Section 7.3.2.

An additional approximation is introduced when incrementally calculating the
amount of crack growth. The method assumes that crack growth only occurs during
the tensile portion of the load cycle. However, it can not be experimentally shown that
this is true. In fact, it is generally accepted that crack growth occurs during the
opening and closing portion of the load cycle [Laird 1967]. The tensile portion creates
plastic deformation at the crack tip and causes it to blunt. During unloading, the
plastic deformation creates a wedging action at the crack tip that acts to advance the
crack.

The reasoning behind Equation (5.3) governing the crack growth rate is that, if
the loading were to become proportional, the amount of crack extension predicted
during one loading cycle would be equal to that predicted by the crack-closure-
modified Paris’ model. An additional assumption is that the method assumes mode I
dominant fatigue crack growth. If the ratio of Ky to K;, and likewise the ratio of Kj to
K;, were to become large enough, the mode II (or mode III) loading could contribute to
the crack growth. In this case, AK, should be a function of K;, Ky, and/or Ky. The
models proposed for crack growth rates do not incorporate the mode II and III effects.

For each propagation step, a value for N is chosen that is large enough to
increase a majority of the crack front a significant distance in relation to the model’s
geometry. It is assumed during the N loading cycles that the variations in the

NASA/CR—2000-210062 63



c) Close up view of teeth
Figure 5.1: Typical boundary element model of OH-58 pinion.

In operation, the input torque is applied at the end of the pinion’s long shaft.
The small shaft sits on roller bearings. When the torque is applied, the gear rotates
and the teeth of the pinion successively contact the gear’s teeth. When contact occurs,
load is transferred across the teeth. The boundary conditions shown in Figure 5.la
model these operating conditions. This model will be referred to as the full model.
The face patches at the end of the long shaft are fixed in all directions. The
displacements on the surfaces of the smaller shaft are restrained in the local normal
direction. Though not explicitly shown in Figure 5.1, contact areas are modeled as
distinct face patches on the middle tooth. Traction normal to the patch is defined
which equals the load that is transferred across the contacting teeth for a given input
torque and rotation angle. More detail on how these contact patches are defined is
given in the next section.

5.2.1 Loading Simplifications

The meshing of the gear and pinion is a continuous process. The magnitude of
force between the gear teeth varies during the meshing as adjacent teeth come into and
out of contact. Figure 2.8 is a schematic of the continuous process that 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: four double tooth contact patches, seven single tooth contact patches, followed
by four more double tooth contact patches. Each load step is a unique face patch in
the boundary element model. The load steps will be referred to as numbers one
through fifteen, corresponding to the patches from the gear root to the top land,
respectively. This is consistent with the progression of contact area along a pinion
tooth from the root toward the top. One progression through the fifteen load steps is
one load cycle on the tooth. One rotation of the gear results in one load cycle on each
tooth.

NASA/CR—2000-210062 49



A simplified loading method that assumes a cyclic load at the HPSTC on the
pinion tooth during meshing was investigated. The failure mode predicted by this
method was the same as the moving load predictions. However, the crack trajectory
and fatigue life calculations varied between the two methods. The HPSTC method is
advantageous because it significantly reduces the computational time. However, upon
comparison of the results from the two methods to experimental results, it is
concluded that the moving load method’s trajectories are more accurate.

In summary, insights into the intricacies of modeling fatigue crack growth in
three dimensions were gained. Preliminary steps toward accurately modeling crack
growth in complicated three dimensional objects such as spiral bevel gears were
completed successfully. To improve the accuracy of the simulations, the change in
contact between spiral bevel gear teeth during operation as a crack evolves is needed.

NASA/CR-—2000-210062 100



loading will turn, or kink, the crack away from self-similar crack propagation. There
are several proposed methods to predict the direction of crack growth under mode I
and II loading. The most widely accepted methods are the maximum principal stress
theory [Erdogan ef al. 1963], the maximum energy release rate theory [Nuismer 1975],
and the minimum strain energy density theory [Sih 1974]. Due to ease of
implementation and demonstrated accuracy, the maximum principal stress theory will
be used in this thesis. The method is based on two assumptions. First, the crack will
propagate radially from the crack tip. The second is that the crack will propagate in a
direction that is perpendicular to the maximum tangential stress. In other words, the
crack will kink at an angle 6,, where Og is a maximum. For mode I and II loading,
assuming plane strain conditions, Ogg is

1 7] 20 3 .
Og = eke cos” aka sno] (3.3)

The direction of crack growth can also be shown to correspond to the principal stress
direction. Setting the partial derivative of Ogg with respect to @ equal to zero, the angle
@,, will be that which satisfies the equation
K, sin0 +K,, (3cos@ -1)=0 (3.4)
From Equation (3.4), it is seen that if Kj equals zero, i.e. pure mode I loading,
then the crack will propagate at an angle equal to zero. Figure 3.2 illustrates
schematically the angle of crack trajectory, 6,,, with respect to the crack front
coordinate system.

Self-similar crack propagation
K,>0; Ky, =0; 6, =0

Mixed mode crack trajecto
K,>0; K;, #0; 6,40 ~

m

Figure 3.2: Angle of crack trajectory with respect to crack tip.

3.2.1 Fatigue

Cracks have been known to grow when the mode I SIF is less than Kjc. In
these instances, the flaw has been subjected to cyclic loading. Cyclic loading can
produce fatigue crack growth at loads significantly smaller than the fracture toughness
of the material. Figure 3.3 illustrates how cyclic loading is characterized by the tensile
load range, AS, and the load ratio, R. R is defined as the ratio of minimum stress, Simin,

NASA/CR—2000-210062 23



is given by McFadden [1985]. These examples demonstrate that gear failures do
occur in service.

Gear experts are researching ways to make gears quieter and lighter through
changes in the geometry. However, at the same time there is a tradeoff between
weight, noise, and reliability. Geometry changes could have negative effects on the
strength and crack trajectory characteristics of the gear. A design tool to predict the
performance of proposed gear designs and changes, such as discussed by Lewicki
[1995], would be extremely useful. Savage er al. [1992] used an optimization
procedure to design spiral bevel gears using gear tooth bending strength and contact
parameters as constraints. Including effects of geometry changes on the strength and
failure modes could contribute greatly to his procedures.

2.5.2 OH-58 Spiral Bevel Gear Design Objectives

In rotorcraft applications, a primary source of vibration of the gear box is
produced by the spiral bevel gears [Coy et al. 1987] [Lewicki et a/. 1993]. In turn, the
vibration of the gear box accounts for the majority of the interior cabin noise. As a
result, recent design has focused on modifying the gear’s geometry to reduce the
vibration and noise. In addition, due to the application of the gear, a continuous
design objective is to make the gear lighter and more reliable.

Adjusting the geometry of the gear, however, may jeopardize the gear’s
strength characteristics. Lewicki et al. [1997a] showed that the failure mode in spur
gears is closely related to the gear’s rim thickness. It was demonstrated that if an
initial flaw exists in the root of a tooth, the crack would propagate either through the
rim or through the tooth for a thin rimmed and thick rimmed gear respectively. As a
result, a tool to evaluate the strength and fatigue life characteristics of proposed gear
designs would be useful.

Albrecht [1988] demonstrated that AGMA standards to determine gear stresses
and life were insufficient. He also showed the advantages of a numerical simulation
method, such as the FEM, over the currently accepted AGMA standards at that time.
The work of this thesis is an extension of the numerical approaches to determine gear
stresses and life.

2.6 Chapter Summary

This chapter covered basic terminology and geometry aspects of gears.
Concepts related to spiral bevel gears were the primary focus. In addition, methods to
visualize and model the contact between mating spiral bevel gears were presented.
Characteristics of a common gear steel, AISI 9310, were summarized. These
materials properties will be used in the numerical simulations. Finally, some
examples of gear failures and gear design objectives were discussed to motivate the
significance of modeling gear failures numerically.

NASA/CR—2000-210062 19



produced from the contact between the mating gears in conjunction with the high
loading frequency. Several researchers have also modeled the residual stresses in
a gear, which result from the difference in hardness between the case and core, in |
numerical analyses by thermal loads. Parametric studies to determine the
significance of all these loading variables on predicted fatigue crack growth in a
spiral bevel gear is imperative to the development of numerical gear design tools.

e Further work to enhance the speed of the numerical analyses and increase the SIF
accuracy would be beneficial. One method to improve the accuracy is to use the
FEM. However, meshing three dimensional volumes with cracks introduces
additional difficulties. Research is currently underway, and should be continued,
to overcome these meshing difficulties. The computational time will continue to
decrease as computer technology rapidly advances. The overall objective of
developing a practical and accurate numerical design tool for any type of gear is
foreseeable in the near future.

NASA/CR—2000-210062 104



length and orientation through five propagation steps. For the assumed material
properties and loading in this example, the calculated number of load cycles to grow
the crack from the initial length in Figure 3.8 to that in Figure 3.9 is 4,900 cycles.

3.2.3 Example: Three dimensional, mode I dominant fatigue crack growth
simulation with static, proportional loading

The assumptions of the two dimensional example will apply to this three
dimensional example. In three dimensions, the procedure to predict fatigue crack
trajectories is very similar to that in two dimensions. As in two dimensions, the
geometry model must be defined, the mesh created, and the model attributes assigned.
The main complexity with three dimensional crack growth simulations is that there is
not a single crack tip, but rather a three dimensional crack front. For a given three
dimensional crack, there is no longer a single value for the SIF in each mode, but
rather a SIF distribution along the crack front for each mode. In addition, the crack
length might also vary along the crack front.

In this thesis, all of the three dimensional models are boundary element
models. In the boundary element method, the primary variables are load and
displacement. Strains and stresses are secondary variables. The BEM is based on an
integral equation formulation. An advantage of the method is that the number of
unknowns in the equations is proportional to the surface discretization. This is in
contrast to the FEM where the number of unknowns is proportional to the volume
discretization. In computational fracture mechanics when predicting crack trajectories
and remeshing are necessary, an advantage of the BEM is that only the surfaces near
the crack need to be remeshed, as opposed to the entire volume which must be
remeshed when using the FEM. Volume meshing with cracks can be rather difficult;
whereas, surface meshes are straightforward with and without cracks.

crack face

crack front

crack face

Hl
i
i
i
i
1
1

Figure 3.10: Schematic of three dimensional crack front.

NASA/CR—2000-210062 31



Miller et al. 1999

Murakami 1987

Newman 1976

Newman 1981

Newman 1984

Nuismer 1975

Olzak et al. 1993

Owen et al. 1983

Panasyuk et al. 1995

Paris et al. 1961

NASA/CR—2000-210062

Miller, M., Rufin, A.D., Westre, W.N., and Samavedam, G., 1999,
“High-Speed Civil Transport Hybrid Laminate Sandwich Fuselage
Panel Test,” Fatigue and Fracture Mechanics: Twenty Ninth Volume,
ASTM STP 1332, T.L. Panontin and S.D. Sheppard, Eds., American
Society for Testing and Materials, West Conshohocken, PA, pp. 713-
726.

Murakami, Y., (Eds.), 1987, Stress Intensity Factors Handbook,
Pergamon Press.

Newman, J.C., Jr., 1976, “A Finite-Element Analysis of Fatigue Crack
Closure,” Mechanics of Crack Growth, ASTM STP 590, American
Society for Testing and Materials, pp. 281-301.

Newman, J.C., Jr., 1981, “A Crack-Closure Model for Predicting
Fatigue Crack Growth Under Aircraft Spectrum Loading,” Methods
and Models for Predicting Fatigue Crack Growth under Random
Loading, ASTM STP 748, J.B..Chang and C.M. Hudson, Eds.,
American Society for Testing and Materials, pp. 53-84.

Newman, J.C., Jr., 1984, “A Crack Opening Stress Equation for
Fatigue Crack Growth,” International Journal of Fracture, Volume
24, pp. R131-R135.

Nuismer, R.J., 1975, “An Energy Release Rate Criterion for Mixed
Mode Fracture,” /nternational Journal of Fracture, Volume 11, pp.
245-250.

Olzak, M., Stupnicki, R., and Wojcik, R., 1993, “Numerical Analysis
of 3D Cracks Propagating in the Rail-Wheel Contact Zone,” Rail
Quality and Maintenance for Modern Railway Operations, Kluwer
Academic Publishers, pp. 385-395.

Owen, D.R.J., and Fawkes, AJ., 1983, Engineering Fracture
Mechanics: Numerical Methods and Applications, Pineridge Press
Limited.

Panasyuk, V.V., Datsyshyn O.P., and Marchenko, H.P., 1995, “The
Crack Propagation Theory Under Rolling Contact,” Engineering
Fracture Mechanics, Volume 52, pp. 179-191.

Paris, P.C., Gomes, M.P., and Anderson, W.E., 1961, “A Rational
Analytical Theory of Fatigue,” The Trend in Engineering, Volume 13,
pp. 9-14.

114



discrete time steps as the contact area moves up the pinion tooth, as discussed in
Section 5.2.1. Recall these contact ellipses are defined for a full design load input
torque of 3120 in-Ib. The full pinion boundary element model is used, Figure 5.1. A
semi-elliptical edge crack is introduced into the root of the middle tooth on the
concave side. The crack is located approximately in the middle of the tooth length.
The dimensions are 0.125 inches long by 0.050 inches deep. The crack is oriented
approximately normal to the surface.

Each load step produces a unique SIF distribution along the crack front. The
SIF distribution changes between load steps because the load position and magnitude
varies from step to step. Figure 5.8 shows the mode I SIF distribution for the first
eleven load steps, the initial four double tooth contact load steps followed by the seven
single tooth contact load steps. The second stage of double tooth contact, load steps
twelve through fifteen, are omitted from the figure to simplify it. Modes II and IIT
SIFs are plotted similarly in Appendix A.

The four bottom curves in Figure 5.8 are the SIFs under double tooth contact.
The remaining seven curves are the SIFs under the single tooth contact load steps.
The bottom most of the seven curves corresponds to load five. The topmost curve is
the result from load eleven, the last single tooth contact step. The total applied force
for each single tooth contact load step is roughly equivalent. However, as the load
step number increases, the SIF curves shift up. This is explained by the fact that the
locations of the contact patches are progressing up the pinion tooth. The change in
location creates a greater moment arm. As a result, the displacements, and likewise
SIFs, in the tooth root will thus increase.

18000 5
|

—*Load 1
—® Load 2
—*- Load 3
Load 4
—*- Load 5
—*Load 6
—+ Load 7
— Load 8
—Load 9
—*Load 10
—# Load II

05)

K, [psi*in'

0 r 7 —T T r
1 6 1 16 21 26 31 36 41 46 51 56 61
Crack front position
(Orientation: heel to toe)

Figure 5.8: Mode I SIF distribution for load steps one through eleven.

NASA/CR—2000-210062 55



El-Zafrany 1993

Elber 1971

Erdogan et al. 1963

Flasker et al. 1993

Fleck er al. 1988

Forman er al. 1984

Forsyth 1962

FRANC3D 1999a

FRANC3D 1999b

FRANC3D 1999c

Fu et al. 1995

NASA/CR—2000-210062

El-Zafrany, A., 1993, Techniques of the Boundary Element Method,
Ellis Horwood Limited.

Elber, W., 1971, “The Significance of Fatigue Crack Closure,”
Damage Tolerance in Aircraft Structures, ASTM STP 486, American
Society for Testing and Materials, pp. 230-242.

Erdogan, F., and Sih, G.C., 1963, “On the Crack Extension in Plates
Under Plane Loading and Transverse Shear,” Journal of Basic
Engineering, Volume 85, pp. 519-527.

Flasker, J., and Pehan, S., 1993, “Crack Propagation in Tooth Root
with Variable Loading,” Communications in Numerical Methods in
Engineering, Volume 9, pp. 103-110.

Fleck, N.A., and Newman, J.C., Jr., 1988, “Analysis of Crack Closure
Under Plane Strain Conditions,” Mechanics of Fatigue Crack Closure,
ASTM STP 982, J.C. Newman, Jr. and W. Elber, Eds., American
Society for Testing and Materials, pp. 319-341.

Forman, R.G., and Hu, T., 1984, “Application of Fracture Mechanics
on the Space Shuttle,” Damage Tolerance of Metallic Structures:
Analysis Methods and Applications, ASTM STP 842, J.B. Chang and
J.L. Rudd, Eds., American Society for Testing and Materials, pp. 108-
133.

Forsyth, P.J.E, 1962, “A Two Stage Process of Fatigue Crack
Growth,” Proceedings of the Crack Propagation Symposium, Volume
1, pp. 76-94.

FRANC3D V1.14 Concepts & Users Guide, 1999, developed by the
Cornell Fracture Group, Cornell University, Ithaca, NY.

FRANC3D V1.14 Menu & Dialog Reference, 1999, developed by the
Cornell Fracture Group, Cornell University, Ithaca, NY.

FRANC3D VI1.14 Validation & Verification Reference, 1999,
developed by the Cornell Fracture Group, Cornell University, Ithaca,

NY.

Fu, M., and Shang, B., 1995, “Stress Analysis of the Precision Forging
Die for a Bevel Gear and its Optimal Design Using the Boundary-
Element Method,” Journal of Materials Processing Technology,
Volume 53, pp. 511-520.

lll



polishing might have resulted from rubbing of the crack faces while the gear was in
operation or rubbing against a part of the gearbox after fracturing away from the
pinion. The extent and uniformity of the flat, polished surfaces support the former

hypothesis.

188m

#90683

Figure 6.5: Fatigue striations near EDM notch on tooth #11 at 30x (a) and at 307x (b).

A transition from the flat, polished area to one with some texture combined
with flattened areas was observed further from the notch near the ridge (point C in
Figures 6.3 and 6.4). In Figures 6.3 and 6.4, this combination, or partially rubbed,
type of surface was found along the transition line from the darker region (flat,
polished area) to the lighter region of the upper Jeft corer. Recall the light region in
both figures is near the toe end of the tooth. Figure 6.7, taken from region C in Figure
6.3, shows clearly the features of the partially rubbed surface. The appearance of the
raised areas is as if they have been flattened, while the lower lying regions have
morphology indicative of fatigue. However, no well developed fatigue striations are

observed.

NASA/CR—2000-210062 74



In regime III, the crack growth is unstable. A crack can grow in fatigue only
when K, < K;c. Asa result, regime III is bounded on the right by AKic.

da\h
aN
I Il

|

Paris Regime

|
>
AK. AKic log(AK, )

Figure 3.4: Typical shape of a fatigue crack growth rate plot.

Crack growth in regime II creates striations on the fracture surface in certain
materials under appropriate loading conditions. It has been shown that the spacing
between striations is roughly equal to the macroscopic crack growth rate da/dN
[Forsyth 1962]. In general, ductile alloys, e.g. aluminum alloys, form the most well
developed striations. The material of interest in this thesis, AISI 9310 steel, is capable
of forming striations [Bhattacharyya et al. 1979] [Au et al. 1981] [McElvily et al.
1996]. Au er al. successfully correlated fatigue crack growth rates to fatigue striations
in AISI 9310 steel.

Paris first proposed C as a material property. However, experimental research
has found that C varies as a function of the stress ratio. The crack growth rate
increases as the stress ratio increases. Fatigue crack growth data in regime II from
tests conducted at different stress ratios, plots as shown in the left graph of Figure 3.5.
The spread in the curves is explained by fatigue crack closure [Elber 1971]. In
general, it has been found that a crack will prematurely close prior to the tensile load
being entirely removed. The level of stress at which this premature closing occurs is
Sop (or, due to similitude, K,,). Incorporating fatigue crack closure phenomenon into
Paris’ model should collapse the curves into a single line (the right graph of Figure
3.5). This is accomplished by plotting on the abscissa AKyg (AK gy = Kinay — K gy),

rather than AK. This single curve is referred to as the “intrinsic” fatigue crack growth
rate. More details of fatigue crack closure will be discussed in Chapter 4.

NASA/CR—2000-210062 25



30000

25000 ~
20000
Se ; - 150% Load
Z 15000 + -= 125% Load
= — 100% Load
x
10000
Load Step 1
5000 oy
0 T T —— 1
0 5 10 15 20 25
Crack front position
a) (Orientation: heel to toe)
30000 5
25000 -
20000
“15000 -
x
&
“= 10000 Load Step 11
x “+= 150% Load
5000 -= 125% Load
— 100% Load
0 ~ : 7
5 10 15 20 25
-5000 Crack front position
b) (Orientation: heel to toe)

Figure 7.10: K; distribution for load step one (a) and load step eleven (b).

NASA/CR—2000-210062

93



intensity factor history under the moving load is calculated. A method to predict
fatigue crack trajectories under the moving load is proposed. The method is then
applied to predict fatigue crack growth trajectories and rates in a spiral bevel pinion.
Fatigue crack growth results from a spiral bevel pinion in operation are
necessary to validate the predictions. The sponsor of the research efforts of this thesis,
NASA-Glenn Research Center (NASA/GRC), provided a pinion that was tested in
their gear test fixture. Notches were fabricated into several of the teeth’s roots prior to
beginning the test. The test data and crack growth results are presented in Chapter 6.
In addition, in an effort to obtain crack front shape and crack growth rate information,
the fracture surfaces are observed with a scanning electron microscope, and the results

are given in the chapter.
The crack trajectory and fatigue life results from the simulation and the tested

pinion are compared in Chapter Seven. To gain insight into the discrepancies between
the prediction and test, the influence of model parameter assumptions and loading
simplifications on crack trajectories and calculated fatigue crack growth rates are
studied. Next, the necessity of the moving, non-proportional load crack growth
method is evaluated by comparing the results to predictions that assume proportional
loading.

Finally, Chapter Eight summarizes the accomplishments of the work in the
previous chapters. Implications of the research conducted and suggestions for future
work are given.

NASA/CR—2000-210062 6



#0001

Figure 6.6: Typical picture of flat, polished area on tooth #5 (410x). Photograph was
taken near location B in Figure 6.3.

#0002

Figure 6.7: Typical picture of partially rubbed surface (695x). Photograph was taken
from location C in Figure 6.3.

The lighter region in the upper left comers of Figures 6.3 and 6.4 (point D)

shows little to no signs of rubbing. The surface also shows no obvious signs of
fracture mode, e.g. intergranular fracture, ductile rupture, dimpling. Although no

NASA/CR—2000-210062 75



—_——————

—

ae
_

Experimental
‘ Predicted

a) Tooth surface

b) Cross section at location of midpoint of initial crack
Figure 7.1: Predicted and experimental crack trajectories.

Qualitative comparisons of the analyses’ results to the tested pinion results can
be made. The numerical analyses predict tooth failure, which is concordant with the
experiment. In addition, in both the fractography study and numerical analyses, the
crack propagates more rapidly toward the toe than the heel. In both the test and
simulation, a portion of the tooth at the heel end remains intact. Additionally, the
simulations predict the crack propagating along a steeper trajectory into the gear rim in
the middle of the tooth length than on the toe end of the length. This behavior is also
observed in the tested pinion.

The final predicted trajectory through the thickness of the tooth agrees very
well with the initial path in the experiment. It is assumed that this path could lead to
the formation of a ridge if the simulations were continued. The entire predicted crack
trajectory, however, does not completely match the tested pinion. The simulations
predict the toe end of the crack turning up the tooth height at a steep angle. This
behavior is not seen in the tested pinion. One reason for the discrepancy could be that
the loading conditions for the simulations were not identical to the test. The gear was
tested at increasing torque levels over the 4.9 million cycles. The simulations,
however, were performed under a constant torque level. The increase in torque should
affect the tooth contact, which, in turn, will influence the crack trajectory. The
influences of the torque level and contact location on crack trajectories are explored in

NASA/CR—2000-210062 82



Be m=" a .
Figure 1.2: Spiral bevel gear tooth failure [Alban 1985].

In general, gears in rotorcraft applications are designed for infinite life; the
gears are designed to prevent any type of failure from occurring. Developing a
damage tolerant design approach could reduce cost and increase effectiveness of the
gear. Lewicki et al.’s work on determining the effect of gear rim thickness on crack
trajectories is a good example of how damage tolerance can be applied to gears.
Knowing how the gear’s geometry affects the failure mode allows a designer to select
a geometry such that, if a crack were to develop, the failure mode would be benign.
Other examples of damage tolerant design can be found in aircraft structures [Swift
1984] [Rudd 1984] [Miller et a/. 1999], helicopter rotor heads [Irving et al. 1999], and
train rails [Jeong er al. 1997].

Damage tolerance involves designing under the assumption that flaws exist in
the structure [Rudd 1984]. The initial design then focuses on making the structure
sufficiently tolerant to the flaws such that the structural integrity is not lost. Damage
tolerant design allows for multiple load paths to prevent the structure from failing
within a specified time after one element fails. In this regard, gears would be designed
for the benign failure mode, tooth failure, as opposed to rim failure, which could be
catastrophic.

Current American Gear Manufacturers Association (AGMA) standards use
tables and indices to approximate the strength characteristics of gears [AGMA 1996].
The finite element method (FEM) and boundary element method (BEM) are becoming
more useful and common approaches to study gear designs. A primary reason for this
is the tremendous increase in computing power. Section 1.2 summarizes recent
research related to modeling gears numerically.

Limited work has focused on predicting crack trajectories in spiral bevel gears.
This is most likely because a spiral bevel gear’s geometry is complex and requires a
three dimensional representation. Structures with uncomplicated geometries, such as
spur gears, can be modeled in two dimensions. Modeling an object in three
dimensions requires a crack to also be modeled in three dimensions. Three
dimensional crack representations introduce unique challenges that do not arise when
modeling in two dimensions.

A three dimensional crack model consists of a continuous crack front. When a
simpler geometry allows for a two dimensional simplification, a crack front is now

NASA/CR—2000-210062 2



Bingyuan et al. 1991

Blarasin et al. 1997

Blom et al. 1985

Bold et al. 1992

Bower et al. 1994

Broek 1986

Chan et al. 1970

Couchon et al. 1993

Coy et al. 1985

Coy et al. 1987

Dike 1978

Dudley 1996

NASA/CR—2000-210062

Bingyuan, J., Xitong, Z., and Lei, W., 1991, “Estimate by Fracture
Mechanics Theory for Rolling Contact Fatigue Life and Strength of
Case-Hardened Gear Materials with Computer,” Engineering Fracture
Mechanics, Volume 39, pp. 867-874.

Blarasin, A., Guagliano, M., and Vergani, L., 1997, “Fatigue Crack
Growth Predictions in Specimens Similar to Spur Gear Teeth,”
Fatigue and Fracture of Engineering Materials and Structures,
Volume 20, pp. 1171-1182.

Blom, A.F., and Holm, D.K., 1985, “An Experimental and Numerical
Study of Crack Closure,” Engineering Fracture Mechanics, Volume
22, pp. 997-1011.

Bold, P.E., Brown, M.W., and Allen, R.J., 1992, “A Review of Fatigue
Crack Growth in Steels Under Mixed Mode I and II Loading,” Fatigue
and Fracture of Engineering Materials and Structures, Volume 15, pp.
965-977.

Bower, A.F., and Fleck, N.A., 1994, “Brittle Fracture Under A Sliding
Line Contact,” Journal of the Mechanics and Physics of Solids,

Volume 42, pp. 1375-1396.
Broek, D., 1986, Elementary Engineering Fracture Mechanics, Fourth
revised edition, Kluwer Academic Publishers.

Chan, S.K., Tuba, LS., and Wilson, W.K., 1970, “On the Finite
Element Method in Linear Fracture Mechanics,” Engineering Fracture
Mechanics, Volume 2, pp. 1-17.

Couchon, D.C., Barnes, G.K., Cedoz, R.W., 1993, Shot-Peened Gear
Failures Due to Operation in a Misaligned Condition, AIAA 93-2147.

Coy, J.J., Townsend, D.P., and Zaretsky, E.V., 1985, “Gearing,”
NASA Reference Publication 1152/AVSCOM Technical Report 84-C-

15.

Coy, J.J., Handschuh, R.F., Lewicki, D.G., Hugg, R.G., Krejsa, E.A.,
and Karchmer, A.M., 1987, “Identification and Proposed Control of
Helicopter Transmission Noise at the Source,” USAAVSCOM-TR-87-

C-2.

Dike, G., 1978, Geometry and Stresses of Straight Bevel Gears,
Transactions of Machine Elements Division, Lund Technical
University, Lund, Sweden.

Dudley, D.W., 1996, “Fatigue and Life Predictions of Gears,” ASM
Handbook Volume 19: Fatigue and Fracture, ASM International, pp.
345-354.

110



Tooth root concave side

Figure 6.12: Sketch of crack propagation scenario on loaded side devised from
fracture surfaces.

6.4 Chapter Summary

This chapter was devoted to presenting data from an OH-58 spiral bevel pinion
test. The test was conducted by NASA/GRC. EDM notches were introduced into the
root of nine of the pinion teeth to serve as starter cracks for fatigue crack growth.
Limited observations of the crack growth during the test were made, and, as a result,
the fracture surfaces were observed with a SEM.

Overall, the microscopy identified fatigue crack growth regions and regions of
ductile rupture successfully. In addition, the crack face morphology showed
significant signs of rubbing, which had “polished” the surface. This polishing
removed any discernable fracture surface features on the majority of the surfaces. The
signs of fatigue on the loaded and load free sides of the fracture surface indicated the
majority of the crack growth was attributed to fatigue. At the ridge near the toe end,
the surface showed little to no signs of rubbing. This observation suggested that the
surface was created in the latter stages of crack growth. It was inferred from the
jagged and torn appearance of the fracture surface near the heel that this region was
the last remaining ligament of the tooth after rupture occurred in the root of the convex
side.

Due to the dearth of well-developed fatigue striations on the fracture surfaces,
no observations were made on the crack growth rates. In addition, the large amounts
of rubbing removed all indications of crack front shape during propagation.
Nevertheless, a scenario of crack propagation was devised. The next chapter
compares these test results to the simulations results from Chapter 5.

NASA/CR—2000-210062 79



APPENDIX B

#ABOL

Figure B.1: Fatigue striations in Figure 6.5 at 360x (a) and 1,000x (b).

An additional point of interest is captured in Figure B.3. The figure is of the
tooth surface. The horizontal lines are the grooves from the machining of the gear. In
addition, two pits are observed on the surface. These pits could have resulted from
several variables such as particles that were caught on the surface during meshing of
the teeth, manufacturing flaws, surface wear, etc. The cuts and pits give an indication

of the initial flaw size from which fatigue cracks may originate.

NASA/CR—2000-210062

106



Table 7.2: Crack trajectories from contact locations shifted along tooth length.
Contact location Schematic of resulting crack
trajectory on surface in root

oe

Heel-shifted

Load Magnitude

The tested spiral bevel pinion was run at varying levels of input torque detailed
in Table 6.1. However, the simulation results reported in Section 5.5 assumed contact
areas and load magnitudes produced by 3120 in-lb torque (100% design load). The
goal of the current study is to identify the influences of the increased torque levels on
crack trajectories.

The SIF distributions and trajectories under larger torque levels of 3874 in-lb
and 4649 in-Ib (125% and 150% design load, respectively) are explored. From
Hertzian contact theory, it is known that the lengths of the contact ellipses’ axes are
proportional to the cube root of the applied load (Equation (2.1)). Consequently, the
lengths of the major and minor axes increase by 7.72% (125% design load torque) and
14.47% (150% design load torque) under the larger loads. It is assumed that the mean
contact points (center of the ellipses) are the same as the points defined for the 100%
design load. Similar to the shifted load analyses, the crack from propagation step five
in the moving load simulations (Section 5.5) is selected to analyze under the larger
torque levels. Figure 7.9 shows the locations of the crack and of the contact ellipses
defined for 125% design load.

NASA/CR—2000-210062 91



imagine that as the number of discrete load steps increases, the curves in Figure 5.9
will become smoother. However, due to transmission error and noise, it is known that
the curves in reality are neither continuous nor smooth. Therefore, it is assumed that
the fifteen load steps are sufficient to approximate the true loading conditions.

The moving load on the pinion’s tooth is non-proportional; the ratio of Ky to Ky
changes during the load cycle, Figure 5.10. Consequently, the method to propagate a
three dimensional crack described in Section 3.2.3 can not be used. That method
assumed proportional loading, which results in a constant kink angle for the load
cycle. Since the ratio is changing in the spiral-bevel gear tooth, the predicted angle of
propagation during the load cycle changes. A method to determine how a crack would
grow under this type of loading is required and is proposed in Section 5.4.

15
Load Step

Ki lKy

Figure 5.10: Typical Kj to K; ratio under moving load.

The Kj to K; ratio also indicates which loading mode is driving the crack
growth. Mode I dominant fatigue crack growth is associated with smaller ratios. Qian
et al. [1996] studied mixed mode I and II crack growth in four point bend specimens.
They selected the test specimen geometries from FEM analyses that considered
various crack lengths and orientations to achieve different Kj; to K; ratios. From the
analyses, they selected five different geometries with Ky/K; values of 0, 0.262, 0.701,
1.812, and 16.725. The ratios covered crack growth mechanisms of pure mode I,
mode I dominant, balanced mode I and II effects, mode II dominant, and highly mode
II dominant, respectively. Using these ratios as guidelines, the gear situation can be
characterized as balanced mode I and II effects during the earlier stages of the cycle to
mode I dominant crack growth during the later stages of the cycle. However, it will be
assumed that the fatigue crack growth is driven by mode I.

NASA/CR—2000-210062 S7



fixed

rollers
Figure 5.5: Simplified model 3: ignore both shafts and half of gear rim.

A semi-elliptical crack is introduced in the root of the middle tooth in each
model. The crack is 0.64 inches long and 0.14 inches deep. A simplified load is
applied to the middle tooth over the middle third of the tooth length 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 one variable and
keeping all the rest of the model parameters constant. To achieve consistency between
all the models, the mesh in the region of the crack and load patch is identical.

The SIFs increase on average by 7%, 8%, and 11% with respect to the full
model’s SIFs for simplification one, two, and three, respectively. In a fatigue growth
rate context, changes of this magnitude are significant. Recall that the crack growth
rate is proportional to K; raised to a power (Equation (3.6)). For AISI 9310, the
magnitude of the exponent is approximately 3.4. Consequently, seemingly small
changes in the SIFs have dramatic effects on the crack growth rate predictions. It is
concluded from this study that the full 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 K; between the two models for all
three load cases is below 5%. The absolute magnitude of K;, for both models and all
load cases is significantly smaller than K;. Consequently, a small variation in Ky
appears as a large percent difference between the models. Instead of percent
differences, Figure 5.7 shows the absolute K,; 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



Applied Load

=P

Maximum Maximum

tensile compressive
stress \ 4“ stress
Ap

Plieg Loag

a)

Maximum Maximum
tensile compressive
stress stress

b) a

Figure 2.10: Stresses in cantilever beam (a) are analogous to gear tooth root (b).

The compressive stresses are noteworthy because Lewicki et al. [1997b]
showed that the magnitude of the compressive stress increases as a gear’s rim thickness
decreases. The compressive stress could affect the crack propagation trajectories and
crack growth rates. However, it is demonstrated in Chapter 4 that low stress ratios, i.e.
large compressive stresses compared to tensile stresses, do not have a significant
influence on crack predictions.

Up to this point, only frictional loads and traction normal to the tooth’s surface
have been discussed. The normal loads are the only loading conditions to be
considered in this thesis. However, additional sources do produce forces on the gear.
Some of these additional loads include dynamic effects, centrifugal forces, and
residual stresses due to the case hardening of the gear. In addition, since a lubricant is
always used when gears are in operation, lubricant could get inside a crack and create
hydraulic pressure.

NASA/CR—2000-210062 15



4.4 Sensitivity of Growth Rate to Low R

Table 4.3 contains results from calculations of AK 4g at different R-values using
Equations (4.1), (4.2), (4.4), and (4.5). Constant values for Kmax» K; and Smax/Oo are
assumed.

Table 4.3: Calculations to find AK. over a range of R-values for a constant Kyax; SIF
units are ksi*in®>.

R K_| Smax/Oo | Kmax Kmin AK U_| Aker
0.705 1.000 0.100 | 10.000 | 7.050 2.950 _| 0.825 2.434
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 3.565
0.255 | 1.000 | 0.100 | 10.000 | 2.550 | 7.450 | 0.589 | 4.388
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 | 14.950 | 0.327 4.886
-0.500 | 1.000 0.100 | 10.000 | -5.00 | 15.000 | 0.326 4.888
-0.955_| 1.000 0.100 | 10.000 | -9.950 | 19.950 | 0.254 5.058
-1.000 | 1.000 0.100 | 10.000 | -10.000 | 20.000 | 0.253 5.060
-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

The crack growth rate is calculated in Table 4.4 based on the effective SIF data
in Table 4.3. C and n are assumed to be 7.44e-10 (in/cycle)/(ksi*in”*)" and 3.4,
respectively. These values are taken from the curve fit to the intrinsic growth rate data
for the AISI 9310 steel tests conducted in heated oil. da/dN as a function of R is
plotted in Figure 4.8.

The curve in Figure 4.8 shows that the crack growth rate in the negative R
regime is less sensitive to variations in R compared to the positive R regime. Between
R equal to zero and —3.00, the crack growth rate varies by a factor of 1.96. Ina fatigue
context, a difference of this order of magnitude is acceptable. As a result, one can
conclude that when modeling fatigue crack growth, AK,, or likewise Kop or da/dN,
does not change significantly for R <0. Therefore, the magnitude of R is not a useful
parameter to characterize damage evolution in gears. In the context of designing gear
geometry to be damage tolerant, a primary concern need not be how aspects of gear

geometry affect R.

It has been shown that when crack closure is taken into account there is not a
significant change in the crack growth rates for negative R-values. This result will be
utilized in the numerical analyses discussed in Chapters 5 and 7. The load ratio will
be taken as R=O under the assumption that, if R<0O, the general results and

NASA/CR—2000-210062 44



model for this thesis. The remainder of the pinion solid model was built from a
drawing of the pinion. The basic shape of the shafts and gear rim were modeled.
Some subtle details of the pinion drawing were ignored in cases where the geometry
would complicate the geometry model and have negligible effects on the computed
SIFs. The surfaces of the solid model were meshed using three- and four-noded
elements. Figure 5.1 contains three views of a typical boundary element model (recall
that the meshes shown in the figures are surface meshes). The volume of the gear is
not meshed. The conical shape of the gear rim and the cylindrical shape of the shafts
are seen best in Figure 5.1b. 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.

om
>, Fixed displacement
4 boundary. conditions

Long shaft

SS====4)

Gear rim

b) Section A-A from (a): profile of shaft

NASA/CR—2000-210062 48



Figure 6.11 summarizes the surface appearance on the loaded and load free
sides of the fractured pinion teeth. A scenario of crack growth progression is
developed based on these observations. The fatigue crack growth initiates from the
EDM notch. The growth continues into the rim and at a larger rate towards the toe
than the heel since it is assumed that the rubbed areas are the older surfaces. Once the
crack reaches the ridge, the crack continues to grow toward the toe end. Figure 6.12 is
a sketch of this scenario on the loaded side. The numbers in the sketch correspond to
the progression of the crack front. When the crack reaches the tooth surface at the toe
end, the extent of crack growth has dramatically changed the stress distribution in the
remaining ligament. Consequently, the crack front turns toward the fillet on the
convex side, and progresses by fatigue along the convex side. When the crack front
becomes sufficiently close to the root of the convex side, ductile rupture occurs in the
remaining ligament. After this, any additional load on the tooth causes the torsional
tearing of the ligament on the heel end. Figure 6.2 sketches the crack growth through
the tooth width. This sketch is applicable to cross-sections from the toe end to
approximately the middle of the tooth length.

No rubbing

Partially rubbed

Heel

Toe Rubbed
[EDM Noth |
Tooth root concave side
Ridge
Heel Partially rubbed Toe

Ductile rupture
Tooth root convex side

Figure 6.11: Sketch of loaded and load free sides of a pinion tooth’s fracture surface
appearance along the length. Orientation is consistent with SEM pictures.

NASA/CR—2000-210062 3


transmission system convert the power from the horizontal engine shaft to the vertical
shaft of the main rotor. Gears in this application typically operate at rotational speeds
of 6000 rpm and transmit on the order of 300 hp of power.

Many parallel axis gears, such as spur gears, have involute tooth profiles. As
sketched in Figure 2.5, the involute curve can be visualized by unwrapping thread
from a spool while keeping the thread taut. The path traced by the end of the string is
an involute curve. The spool is the evolute curve. All involute gear geometries are
generated from circle evolute curves. The involute curve then becomes a spur gear
tooth’s profile. A closed form solution for the coordinates along the curve exists for
this type of geometry. As a result, the tooth’s surface coordinates can be calculated
with relative ease. However, the geometry of a spiral bevel tooth is much more
complex, and there is no closed form solution to describe the surface coordinates.
Handschuh et al. [1991] developed a program to numerically calculate the surface
coordinates of a spiral bevel gear tooth. The program models the kinetics of the
cutting process in creating the gear, along with the basic gear geometry. The program
calculates the coordinates of a spiral bevel gear tooth in three dimensions for use as
input to a finite element model. The numerical models in this thesis were all created
using the tooth geometry coordinates as defined by Handschuh et al.’ s program.

Involute Curve

Evolute Curve:
Base Circle

Figure 2.5: Generation of an involute curve.

NASA/CR—2000-210062 10



new faces. The boundary conditions for each model are chosen because they most
closely match those of the full model (Figure 5.1a).

Figure 5.3: Simplified model 1: ignore long shaft.

In each of the simplified models, the flexibility of the pinion changes. When
an identical crack is introduced into all of the models, the SIFs might vary from model
to model. To determine whether a simplified model is valid, the SIFs from the
simplified models are compared to the full model’s SIFs for identical cracks. It is
assumed that the full model most accurately represents the operating conditions and
loading paths.

fixed

Figure 5.4: Simplified model 2: ignore both shafts.

NASA/CR—2000-2 10062 52



5.4. Method for Three Dimensional Fatigue Crack Growth Predictions Under Non-
Proportional Loading
5.4.1 Literature Review
5.4.2. Proposed Method.
5.4.3. Approximations of Method
5.5 Simulation Results
5.6 | Chapter Summary.

CHAPTER SIX: EXPERIMENTAL RESULTS........:ccscccsssecsseessseseeseeeneenneenennies 69
6.1 Introduction... se
6.2 Test Results
6.3. Fractography..
6.3.1 Overview
6.3.2 Results
6.4 Chapter Summary.

CHAPTER SEVEN: DISCUSSION AND SENSITIVITY STUDIE:
TAL Introduction ......ceeseesesseseeeeeerereeeeteees
7.2. Comparisons of Crack Growth Results

7.3 Sensitivity Studies 00... cesses
731 Fatigue Crack Growth Rate Model Parameter:

7.3.2 Crack Closure Model Parameters
7.3.3, Loading Assumptions
7.4 Highest Point of Single Tooth Contact (HPSTC) Analysi:
7.5 Chapter Summary

CHAPTER EIGHT: CONCLUDING REMARKS
8.1 Accomplishments and Significance of Thesi
8.2 Recommendations for Future Research

APPENDIX A ooeecscscssessssessssessesesseseesescseseeseseeseseseeaeenenesneaneneneqneaeanenesneneseesnasssnansacens 104
APPENDIX B oue.ceecescsessesssssssssssesssscsessescsssscsesneasseesesncscenensenenensesesneseenenesnenesseneenenee 106
APPENDIX Co ooneeeesesscsesssssssssessssssssssscssesesesnenssesnceneasecacescseseenesessennensaneneeneneerensans 108
REFERENCES ..0.0...ccscscssssssssssssessesessssesesscscseescsesseneesenceseseeseseeseneaneaeeneneenserencaneneneaee 110

NASA/CR—2000-210062 vi


number of cycles to failure are unknown. Some of the crack growth may have
occurred during the first million cycles at 2479 in-lb torque, or there may have been
no crack growth until the last 1.9 million cycles at 4649 in-Ib torque. In addition,
since no observations of crack growth during the 4.9 million cycles were made, the
predicted fatigue life curve (Figure 5.15) can not be validated. The only quantitative
information is an upper bound of 4.9 million cycles on the total number of cycles to
failure.

Figure 6.1: Typical tooth failure in tested pinion.

NASA/CR—2000-210062 70



to maximum stress, Sinax, Which, due to similitude, is equal to the ratio of minimum
mode I SIF, Kymin, to maximum mode I SIF, Kinay-

pw Sun — Kn (5)

Sina K max
Cyclic load histories can also be classified as proportional or non-proportional. When
the ratio of Ky to K; is constant during the loading cycle, the loading is proportional.

Non-proportional is the case when this ratio varies with time.

Stress or SIF
S max + ® mae

S min» K min

Time

Figure 3.3: Cyclic load cycle.

There are three regimes of fatigue crack growth as demonstrated in Figure 3.4.
Regime I is related to crack initiation and small crack effects. As noted on the plot,
there is a threshold value, AK,;,, below which fatigue crack growth will not occur. For
AISI 9310 steel, values for AK,, are reported to range from approximately 3.5 ksi*in®>
~ 12 ksi*in®> [Binder et a/. 1980], [Forman er al. 1984], [Proprietary source 1998].
As the stress ratio goes from positive to negative, the threshold value increases.

Regime II is commonly referred to as the Paris regime. The work of this thesis
will only focus on crack growth in regime II. Crack initiation, small crack effects, and
unstable crack growth (regime III) will be ignored. A seminal development in
predicting fatigue crack growth was from [Paris et al. 1961] and [Paris et al. 1963].
They discovered that a crack grows in fatigue at a rate that is a function of AK;. They
proposed that the nature of the curve in regime II could be described by:

& =C(AK,)' (3.6)
where N is the number of cycles, and C and n were proposed as material constants.
Equation (3.6) is commonly referred to as the Paris model. When the crack growth
rate in regime II is plotted on a log-log scale as a function of AK, the slope of the
curve is n. If the curve is extrapolated to the vertical axis, the intercept is C.

NASA/CR—2000-210062 24



0.2 +
0.18 +
0.16

a

0.14

in

2
=

2

= Depth
0.08 = Area

Depth [in] or Area [i

0.06 -
0.04 -
0.02

0 50000 100000 150000 200000 250000 300000 350000
N [cycles]
Figure 5.15: Predicted crack depth and crack area versus number of cycles.

The accuracy of the crack trajectory and fatigue life predictions will be
evaluated in Chapter 7 by comparing the simulation results to experimental data. The
experimental data is obtained from a spiral bevel pinion test that was carried out by
NASA/GRC.

5.6 Chapter 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 tooth
was found. It was determined that the loading on the tooth was non-proportional. As
a result, a method was developed to propagate the three dimensional crack under the
non-proportional load. A crack trajectory prediction in the OH-58 spiral bevel pinion
was performed using this method. In Chapter 7, the simulation results will be
compared to experiment results; Chapter 6 presents the experimental data. In addition,
in Chapter 7 parameters of the crack propagation method will be investigated to
determine the sensitivity of the growth rates and trajectories to variations in the
method and crack growth rate model variables.

NASA/CR—2000-210062 67



3. The goal of this step is to calculate for a given point i the amount of crack

extension, da‘g.1, ), during a load step from j-1 to j. 6'(1, ) is the angle for the
extension during a load step from j-1 to j. It is assumed that the crack grows
incrementally during a load cycle. In addition, propagation at point i only takes
place when the change in mode I SIF between load steps is positive,
(K,'; —K,'(j-)) > 0, and only when K7' is greater than the opening SIF at that
point, Kop. This implies that growth will only take place 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 predicted by a modified Paris’ model, adjusted to incorporate crack closure,
is

da'=C(AK qj)" 6.1)
where AKeg = = Ki\nax — Kop = U*Ki'max. U is given by Equation (4.2). Kop) is found
using Equations (4.4) and (4.5). Figure 5.9 shows that the loading is characterized
by R=0.

In order to calculate U, Sma, the far field stress in a Griffith crack problem, is
required. Figure 5.11 shows the Griffith crack geometry [Griffith 1921]. The
gear’s geometry is obviously different from a Griffith crack problem. Therefore,
an equivalent S,,q.\ must be calculated for the gear. First, Ki max is found in step 2.
Smax' is then found by solving Equation (3.2) for Smnax'

i _ Ky max
Sav =O mae = 6.2)
Baa
Lastly, it is assumed 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:
a ane
Kr 7K uy dai (5.3)
AK vy
The angle of crack growth associate with each load increment is found from the
maximum principal stress theory using the current load steps SIFs as:

da'(j-1.;) =

(5.4)

NASA/CR—2000-210062 60



The focus of this thesis is on tooth bending fatigue failure because this is one
of the most common failures. In general, tooth bending fatigue crack growth can lead
to two types of failures. In rotorcraft applications, the type of failure could be either
benign or catastrophic. Crack propagation that leads to the loss of one or more
individual teeth will most likely be a benign type of failure. The remaining gear teeth
will still be able to sustain load, and the failure should be detected due to excessive
vibration and noise. On the other hand, a crack that propagates into and through the
tim of the gear leaves the gear inoperable. The gear will no longer be able to carry
any load, and will most likely lead to loss of aircraft and life.

Alban [1985, 1986] proposes a “classic tooth-bending fatigue” scenario. He
suggests five conditions that characterize the “classic” failure:

1. The origin of the fracture is on the concave side in the root.
. The origin is midway between the heel and the toe.

3. The crack propagates first slowly toward the zero-stress point in the root.
As the crack grows, the location of the zero-stress point moves toward a
point under the root of the convex side. The crack then progresses outward
through the remaining ligament toward the convex side’s root.

4. As the crack propagates, the tooth deflection increases only up to a point
when the deflection is large enough that the load is picked up
simultaneously by the next tooth. Since the load on the first tooth is
relieved, the rate of increase in the crack growth rate decreases.

5. No material flaws are present.

Alban presents results from a photoelastic study of mating spur gear teeth. The
study demonstrates the shift in the zero-stress point. The zero-stress point is where the
tensile stresses in the root of loaded side of the tooth shift to compressive stresses on
the load free side. Figure 2.11 shows stress contours for two mating spur gear teeth.
In the bottom gear, one of the teeth is cracked and another tooth has already fractured
off. The teeth of the top gear are not flawed. By comparing contours between the
mating cracked and uncracked teeth, it is easy to pick out the zero-stress location shift
toward the root of the load free side. The shift of the zero-stress location demonstrates
the changing stress state in the tooth. This changing stress state drives the crack to
turn. The point in the two dimensional cross section where the crack turns is actually
a ridge when the third spatial dimension, the length of the tooth, is considered. This
classic tooth failure scenario will be used as a guideline when evaluating the
prediction and experimental results in the following chapters.

NASA/CR—2000-210062 17



evaluated. Additionally, to improve the performance of the crack front elements, the
ratio of the elements’ width to length should be close to one [FRANC3D 1999c].

When a crack is propagated, the geometry model changes. However, the
geometry changes only near the crack. Therefore, only the mesh model near the new
crack is damaged and requires remeshing. The remainder of the geometry and mesh
model is left unchanged. This is a distinct advantage of FRANC3D.

The program BES is used to solve for the displacements and stresses using the
boundary element technique. FRANC3D is used as a post-processor to view the
deformed shape, stress contours, and extract nodal information.

FRANC3D uses the same functional form to interpolate the geometry and field
variable variations over an element. The form is given by the associated element type.
In all of the models, only isoparametric three- and four-noded elements are used.
Quadratic elements are available; however, based on the work in [FRANC3D 1999c],
the gain in accuracy does not justify the significant increase in computational time.

3.4 Chapter Summary

This chapter covered theories of LEFM and fatigue pertinent to modeling crack
growth numerically. Of primary importance is how crack growth rates and trajectory
angles are calculated from SIFs. The maximum principal stress theory will be used to
calculate trajectories under mixed mode loading. In addition, the displacement
correlation method was introduced as a technique to evaluate SIFs. Two dimensional
and three dimensional examples demonstrated how the theories are applied in
numerical simulations. Some features of the software programs FRANC3D, OSM,
and BES that will be used in the simulations were covered. The background provided
in Chapter 2 and this chapter will be utilized in the work of Chapters 4, 5, and 6. The
studies in those chapters cover issues related to predicting three dimensional fatigue

crack trajectories in a spiral bevel gear.

NASA/CR—2000-210062 34



leading to the conclusion that the three teeth model is sufficient for the trajectory
prediction analyses.

6 Load 8

z eee
=e 2
g Nee
5 0 =)
€ 123 4 5 15 16 17 18 19
a 2) Crack front position
* Load 5 (Orientation: heel to toe)
44
6
8
-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
Load 1
—+—3 Teeth
1000 -a- 9 Teeth Toad 5
Load 8
0 <i a 1

18 19
Crack front position
(Orientation: heel to toe)

14 15,

10 12. 13

il

K y [psi*in?}

8
3
8

-3000 +

-4000
Figure 5.7: Kj, distribution for three teeth and nine teeth models for load cases one,
five, and eight.

5.3 Initial SIF History Under Moving Load
To simulate the moving load during one load cycle on a pinion’s tooth, fifteen
static BEM analyses are performed. Each analysis represents one of the fifteen

NASA/CR—2000-210062 54





where ¢ ~ FS]

Ble
From Equation (7.2), the maximum and minimum expressions for B are
Bou = and By = 2 | 73)
4 @ Ve

Based on the initial crack geometry, Byax and B,,i, are 0.783 and 0.701, respectively.
K,» and da are recalculated using these values for B, the SIFs from the initial notch
analyses (Figure 5.8), and the same model parameters as were used in Section 5.5.

2.55
Ky Bras)
eda (Bnav)
2 Ky Brin)
eda (Brin)

Percent Change

0 — . a
1 6 11 16 21 26 31 36 4l 46 St 36 61
Crack front position
(Orientation: heel to toe)

Figure 7.8: Change in K,, and da as functions of Bnay and Byin with respect to original
calculations with B =0.

Figure 7.8 shows the percent change of K,, and da with respect to the original
calculations. The data show that the largest percent difference is 2.2%. As the crack
grows the ratio of a to c will become smaller since the tooth length is longer than the
tooth width. This will increase @ and, therefore, increase Byqy and decrease Bin.
However, as the crack grows, this closed form solution for K; in the gear is no longer
valid since the crack’s geometry changes dramatically. Therefore, no further
conclusions can be stated on the effect of 8 on crack growth calculations.

7.3.3 Loading Assumptions

The intent of this study is to determine how the crack trajectory changes under
different contact conditions. One motivation for this is that the simulation 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



1.00E-03

1,00E-04

1.00E-05

1.00E-06

da [dN {in/cyle]

1.00E-07

1.00E-08 + - woe
1 10 100

AK [ksi*in®*]
Figure 4.4: Fatigue crack growth rate data for AISI 9310 steel in room temperature air.

1.00E-03

z
B
=
g
g —R=1
+ R=0.01
+ R=0.5
. ——— _
1 10 100

AK [ksi*in®>}
Figure 4.5: Fatigue crack growth rate data for AISI 9310 steel in 250° oil.

NASA/CR—2000-210062 41



To demonstrate that the simulations are capable of predicting the turning that is
necessary to predict the ridge in the fractured tooth, a large crack shape is assumed
and inserted into the full pinion BEM model. The crack front coordinates are
determined by the location of the ridge in a fractured tooth from the experiment.
Figure 7.3 is a photograph of the fracture surface with the approximate location of the
assumed crack front designated by the dashed line. Based on the SEM observations,
the assumed crack has propagated along the root from the initial notch to the toe
surface. Figure 7.4 is a picture of the BEM geometry model that illustrates the
assumed initial crack trajectory on the tooth surface. Since the correct contact areas
for a tooth that is flawed to this large of an extent are unknown, the HPSTC load step
from the moving load analyses (load step eleven) is used.

Figure 7.4: Tooth surface showing assumed shape of large crack (dashed line).

A few cycles of crack growth are carried out using the method described in
Section 3.2.3. This method assumes mode I dominant fatigue crack growth with
static, proportional loading. The cycles of crack growth are necessary to demonstrate
the direction the crack front progresses from its assumed location at the beginning of
the ridge formation. Figure 7.5 shows the trajectory through the thickness of the tooth
at approximately the middle of the tooth length. The initial trajectory into the rim is
assumed to be flat, and the curving at the end of the crack length shows the formation
of the ridge. This demonstration of the ridge formation based on an assumed crack

NASA/CR—2000-210062 84



Three dimensional finite element analyses have been performed to simulate the
wheel position over a railroad track containing a crack [Olzak et al. 1993]. The rail
model is analyzed for consecutive stages of wheel position and the SIFs are calculated
for each stage. However, Olzak er al. did not propagate the crack. Their primary goal
was to determine what happens to the crack displacement and contact shape when the
load is directly over the crack. In the case of the spiral bevel gear, the load will most
likely never be directly over the crack and their findings are not applicable.

The most significant work was done by Panasyuk et al. [1995]. They
numerically modeled and propagated a two dimensional edge crack under a moving
contact load. The maximum principal stress theory was used, and growth rates were
calculated by Paris’ model. The translation and location of the contact are expressed
as functions of A, the distance from the load to the crack. To calculate the kink angle,
first the values of A that correspond to an extremum of K = F[Kj(A), Ki(A), @A)] are
found. Next 6, K;, and K;; at these 4 are calculated, from which the growth rate is
calculated. Finally, it is assumed that the crack propagates for N cycles at that growth
rate and angle, and the crack in the numerical model is updated and the process is
repeated. Panasyuk ef a/. assumed that their geometry was an elastic half plane, and,
therefore, they could set up closed form equations and solve analytically for K;, Ki,
and @. Once again, their method can not be directly applied to gear model because
neither the traction nor the geometry can be expressed in closed form. The method
also does not directly take into account the non-proportional loading and assumes a
constant kink angle for the entire load cycle. However, their method is extended and a
similar incremental approach is developed in the next section.

5.4.2 Proposed Method

Compared to a two dimensional static problem, the problem at hand is
continuous in time and in a third space dimension. Methods have been presented in
previous chapters and sections to discretize both of these dimensions. With the
discretizations, two dimensional crack propagation theories can be applied. 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 theories are then used to propagate incrementally a series of discrete
points from the three dimensional crack front. The remainder of this section outlines a
proposed method to predict fatigue crack trajectories in three dimensions taking into
account time varying SIFs.

Method

1. Discretize tooth contact path into 15 load steps (Section 5.2.1).

2. Calculate by the displacement correlation method, using a feature built in to
FRANC3D/BES, the mode I, II, and III SIFs (Kj, Ku'y. Kui'y)), where i is a
discrete point along the crack front (i=l-num_points) and j is the load case
( j =0-M ). In general, the nodes of the first row of mesh nodes behind the crack
front are taken as the discrete points. Figure 5.9 is a typical plot of the SIFs for a
single point i along the crack front for the entire loading cycle.

NASA/CR—2000-210062 59


front from the SEM observation supports the crack growth scenario developed in
Section 6.3.2.

Initial crack
logation

Figure 7.5: Crack trajectory through tooth thickness for assumed large crack.

One could argue that the tested spiral bevel pinion failure did not exactly meet
the classical failure conditions described in Section 2.5 since the entire length of the
tooth did not fracture from the gear; a portion of the tooth at the heel remained 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 of the crack trajectories to loading location.

The predictions in Section 5.5 did not consider 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 ellipses to change. A three dimensional, contact mechanics, and
fracture mechanics simulation of the rolling process between two mating gears is
necessary to capture the load redistribution effects fully. This type of analysis is not in
the scope of this work.

It is impossible to determine the exact amount of rubbing between the crack
faces based on the BEM analyses. In Section 6.3.2 it was concluded that the surfaces
with greater amounts of rubbing were formed in the earlier stages of crack growth;
since these surfaces were older, the features of the surfaces had more time to rub
away. The kinematics of the geometry and loading is another explanation for the
varying amounts of rubbing observed on the fracture surfaces. Rather than attribute
the varying degrees of rubbing to time, it could be attributed to the magnitude of
contact between the fracture surfaces. The loading might deflect the tooth in a manner
that does not allow the ridge’s fracture surfaces to rub, but does create large contact
forces between the crack faces near the notch. A three dimensional analysis modeling
contact between the crack faces with accurate loading conditions on the tooth surface
is necessary to determine the true cause of the rubbing.

7.3 Sensitivity Studies

These studies are performed to gain insight into the sensitivity of predicted
crack growth rates and predicted crack trajectories to growth rate model assumptions,
load magnitude, and load location. They are also conducted to investigate possible

NASA/CR—2000-210062 85



Create
geometry model

Define
finite element model

Define attributes
«Material properties

*Fixities
*Loads
i=l
Initiate crack
nd Remesh
Propagate crack Solve equations
Propagation step: i=i+1
Load cycles: Nyoia) = ZN; Post-process
*Compute SIFs
l¢—_____| *Compute da;
*Compute N;,
*Compute angle

Figure 3.6: Flow chart of process to predict fatigue crack trajectory.

As outlined in Figure 3.6, the process begins with a geometry model. The
geometry model is then discretized into a finite element mesh. Figure 3.7 shows the
finite element mesh for an arbitrary geometry model that will be used for
demonstrative purposes. This particular initial mesh consists entirely of quadratic
eight-noded elements.

Model attributes must be defined next. The material properties are specified
within the finite element program as a Young’s Modulus of 29,000 ksi and Poisson’s
ratio of 0.25. The thickness of the model is taken to be 1 inch. Boundary conditions

NASA/CR—2000-210062 28



[1997] used the FEM to look at two and three dimensional spur gear models. Residual
stresses due to case hardening were modeled as nodal thermal loads. Two different
sized models were analyzed: one tooth including the arc length of the gear rim directly
below the tooth and three teeth with the corresponding gear rim arc length. To
determine the new crack front, they used a criterion such that the SIFs along the new
front should be constant. Paris’ model was used to calculate the fatigue lives based on
the SIFs near the midpoint of the crack front. A constant load location with constant
magnitude and simple spur gear geometry allowed Pehan er al. to consider only crack
opening (mode I) effects. Their method for determining the new crack fronts is
computationally intensive and limited since three dimensional effects are not
accommodated.

Lewicki et al. [1998] performed three dimensional crack propagation studies
using the FEM and BEM to investigate fracture characteristics of a split tooth gear
configuration. The geometry of the split tooth configuration is similar to a spur gear.
The analyses used single load locations and explored propagation paths for various
crack locations. The strong point of this work is that three dimensional simulations of
crack trajectories were performed in addition to calculating fatigue crack growth rates.

Very little work, in addition to Lewicki er al.’s research, has used the BEM to
analyze gears. Sfakiotakis [1997] performed two dimensional BEM analyses of gear
teeth considering mechanical and thermal loads. Rather than perform trajectory
predictions, they calculated SIFs for different size initial cracks with various loading
conditions and crack locations. Fatigue loading was not considered. Fu et al. [1995]
also used the BEM for stress analysis related to optimizing the forging die of spiral
bevel gears.

The progression of research related to computer analysis of gears has led to the
investigation of crack growth in spiral bevel gears. FEM models of spiral bevel gears
can be created from Handschuh er al.’s [1991] computer program that models the
cutting process of spiral bevel gears to determine tooth surface coordinates in three
dimensions. Litvin er al. [1996] utilized this program, in conjunction with tooth
contact analysis [Litvin et al. 1991], to determine how bearing (contact between
mating gear teeth) changes with different spiral bevel gear tooth surface designs.
Transmission error curves were generated that gave an indication of the efficiency of
the gear.

Along with Litvin er al.’s work [1991], tooth contact analysis of mating gears
has been explored by Bibel et al. [1995 and 1996], Savage et al. [1989], and Bingyuan
et al. [1991]. Bibel et al. successfully modeled multi-tooth spiral bevel gears with
deformable contact using the FEM. They conducted a stress analysis of mating spiral
bevel gears and analytically modeled, using gap elements from general purpose finite
element codes, the rolling contact between the gear teeth. Bibel et al.’s work can be
used to investigate how changes in gear geometry affect tooth deflections. Variations
in tooth deflections can alter the contact zone between gear teeth. Savage et al.
developed analytical methods to predict, using tooth contact analysis, the shift in
contact ellipses due to elastic deflections of a spiral bevel gear’s shafts and bearings
under loads. Savage et al. and Bibel et al.’s work was related to spiral bevel gears,
however, they did not incorporate fracture mechanics. On the other hand, Bingyuan et

NASA/CR—2000-210062 4



table. These curve fit values will be used in the crack growth rate models for the
numerical analyses. Crack closure concepts will now be extended to investigate the
sensitivity of fatigue crack growth rates to low R-values.

1.00B-03
n=3.36
1.00E-04 C*=7.44E-10
i
z
3 1,008.05 =
g i
3
é —R=1
z =
© 1008-06 R001
3 + R=0.5
Linear Curve Fit
1,00E-07 4
1.00E-08 + — mer 1
1 10 100

AK og [ksi*in®*]
Figure 4.7: Intrinsic fatigue crack growth rate for AISI 9310 in 250° oil; x =1.

Table 4.2: Intrinsic and non-intrinsic fatigue crack growth rate parameters.

Test n c exe 0.5) c* ye 0.5)

[(inlcycle)(ksi*inTM )"] [(inlcycle)I(ksi*inTM )]
R=-1 (Air) 33 6.4e-12 6.30e-10
R =0.05 (Air) 3.5 7.3e-11 8.80e-10
R=05 (Air) 3.9 5.3e-11 1.98e-10
R=-1 (il) 3.2 1.le-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® 4.26e-10
Curve FitOil 3.4 NA! 7.A4e-10

© Not Applicable

NASA/CR—2000-210062 43



represented by a single point, the crack tip. Ata crack tip there are only two modes of
displacement; in three dimensional models, however, there is a distribution of three
modes of displacement along the crack front. Propagating a crack in two dimensions
is completely defined by a single angle and extension length. On the other hand, along
the crack front there is a distribution of angles and lengths.

Codes developed by the Cornell Fracture Group at Cornell University, such as
Object Solid Modeler (OSM) and FRacture ANalysis Code — 3D (FRANC3D), have
been developed to handle three dimensional fracture problems. FRANC3D explicitly
models cracks and predicts crack trajectories under static loads. The crack growth
models are based on accepted fatigue crack growth and linear elastic fracture
mechanics (LEFM) mixed mode theories.

Because gears operate at high loading frequencies, the actual time from crack
initiation to failure is limited. As a result, crack trajectories and preventing
catastrophic failure modes are the primary concern in gear design. Crack growth rates
are not as important. The goal of this research is to investigate issues related to
predicting three dimensional fatigue crack growth in spiral bevel gears. A simulation
that allows for arbitrarily shaped curved crack fronts and crack trajectories will be
most accurate. In addition, the loading on a tooth as a function of time, position, and
magnitude should be considered.

1.2 Numerical Analyses of Gears

Computational fracture mechanics applied to gear design is a relatively novel
research area. As a result, the majority of work has been limited to two dimensional
analyses. In three dimensions, very little work has predicted crack trajectories in
gears. This section summarizes some pertinent developments in applying numerical
methods and fracture mechanics to gear design.

The complexity of two dimensional gear analyses has evolved. Albrecht
[1988] used the FEM to investigate gear tooth stresses, gear resonance, and
transmission noise. Individual gear teeth were modeled in two dimensions and the
increase in accuracy when using the FEM over AGMA standard indices for
calculating gear tooth root stresses was demonstrated. Blarasin et a/. [1997] used the
FEM and weight function technique to evaluate stress intensity factors (SIFs) in
specimens similar to spur gear teeth. Cracks with varying depths were introduced in
two dimensional models and a constant single point load was applied. The SIFs were
determined as a function of crack depth. Fatigue lives were calculated, but predictions
of the crack trajectory were never performed. Flasker er al. [1993] used two
dimensional FEM to analyze fatigue crack growth in a gear of a car gearbox. The
analyses considered highest point of single tooth contact (HPSTC), but variable
loading at that point. Residual stresses from the case and core were simulated with
thermal loading. Based on a given load history, the crack was incrementally
propagated. Lewicki et al. [1997a, 1997b] combined FEM and LEFM to investigate
crack trajectories in thin rimmed spur gears. The work successfully matched crack
trajectory predictions to experiments.

Limited three dimensional crack analyses of gears have been achieved. The
work most often concerns simple geometries and loading conditions. Pehan et al.

NASA/CR—2000-210062 3



Crack closure can be attributed to a number of factors. During the opening
portion of a load cycle, the material at the crack tip plastically deforms. As the cycles
repeat, a wake of plastic deformation remains as the crack propagates through the
body. The plastic deformation wake results in a mismatch between the crack faces.
Although not considered here, crack closure can also occur due to differences in the
surface roughness of the crack faces, due to mixed mode loading, or oxidation of the
crack surfaces.

Elber modified Paris’ model to account for crack closure. The modification
allows crack propagation to occur only while the crack tip is open. He introduced S.,
as the stress level where the crack first opens during the tensile part of the load cycle.
His equation for the crack propagation rate is:

da
— =C(AK,,)" =C(UAK)" 41
Se = C(AK yg)" =CUAK) (4.1)
where U, the effective stress range ratio, is defined as
Snax ~Sop__1=°%6,
Us = (4.2)
Sina ~ Sirin 1-R
Figure 4.2 illustrates the relationships among various K values. Sy» (Kop) is difficult to
measure experimentally. In addition, the value varies with loading conditions. As a
result, Elber developed an empirical relationship between U and R. From this

relationship, S,, (Kop) could be backed out.

Kp
Kyax
AK,
(an f o
Koh \ ff :
fl i /;
f \ +t \ > Time
\ } \ f|
nO ff NN JI
Kins mt Ih \ a7 |!
‘ / / \\ /
\ )
i\ /|f \ asi
Kn ‘ ; . ;
\ | \ |
\ . 7
\_/ \3/
Runring / =

Figure 4.2: Constant AK for different stress ratios.

When da/dN is plotted as a function of AK,g, the scattered curves (due to
different R-values) collapse into a single, “intrinsic” crack growth rate curve. In
crack-closure-based fatigue models, da/dN is a function of AK.y. This implies that
crack growth occurs only while the crack tip is open. If Knox is kept constant between
various tests with different R-values, then K,,i, must change. If it can be shown that

NASA/CR—2000-210062 37



2.4 Gear Materials

As discussed in Section 2.2, spiral bevel gears are commonly used in helicopter
transmission systems. In this application, the gear’s material impacts the life and
performance of the gear. Most often a high hardenable iron or steel alloy is used. The
traditional material for the OH-58 spiral bevel gear is AISI 9310 steel (AMS 6265 or
AMS 6260). Some other aircraft quality gear steels are VASCO X-2, CBS 600, CBS
1000, Super Nitroalloy, and EX-53. The choice of material is dependent on operating
variables such as temperature, loads, lubricant, and cost. The material characteristics
most important for gears are surface fatigue life, hardenability, fracture toughness, and
yield strength. Table 2.1 shows the chemical composition of AISI 9310 [AMS 1996].
Table 2.2 contains relevant material properties.

Table 2.1: Chemical composition of AISI 9310 by weight percent [AMS 1996].
C |Mn/| P Ss Si_| Cu | Ni | Cr | B_ | Mo| Fe

Minimum | 0.07 | 0.40 | -- -- |0.15} -- | 3.00} 1.00| -- | 0.08 | 95.30
Maximum | 0.13 | 0.70 | 0.015 | 0.015 | 0.35 | 0.35 | 3.50 | 1.40 | 0.001 | 0.15 [93.39

Most gears are case hardened. Case hardening increases the wear life of the gear.
In general, the gears are vacuum carburized to an effective case depth! of 0.032 in —
0.040 in (0.813 mm — 1.016 mm). The case hardness specification is 60 - 63 Rockwell
C (RC), and the core hardness is 31 - 41 RC [AGMA 1983].

Table 2.2: Material properties of AISI 9310.

Tensile Strength” 185 x 10° psi

Yield Strength" 160 x 10° psi

Young’s Modulus 30 x 10° psi

Poisson’s Ratio 0.3

Fracture Toughness* 85 ksi*in”TM
a4 ASTM No. 5 or finer

Average Grain Size’ (< 0.00244 in)

2.5 Motivation to Model Gear Failures
Gear failures can be categorized into several failure modes. Tooth bending,

pitting, spalling, and thermal fatigue can all be placed in the category of fatigue
failures. Examples of impact type of failures are tooth shear, tooth chipping, and case
crushing. Wear and stress rupture are two additional modes of failure. According to
[Dudley 1986], the three most common failures are tooth bending fatigue, tooth
bending impact, and abrasive tooth wear. He gives examples of a variety of failures
from tooth bending fatigue to spalling to rolling contact fatigue in both spur and spiral
bevel gears.

' The effective case depth is defined as the depth to reach 50 RC.
' [Coy et al. 1995]

? [Townsend er al. 1991]

> [AMS 1996]

NASA/CR—2000-210062 16



Tooth
fractures

Figure 2.6: OH-58 spiral bevel pinion with two fractured teeth.

A spiral bevel gear set is used in the main rotor transmission of the U.S.
Army’s OH-58 Kiowa Helicopter. An OH-58 spiral bevel pinion that exhibited tooth
fracture during an experiment is shown in Figure 2.6.

The geometry of the OH-58 gear set will be used throughout this thesis. In the
set, a 19 tooth spiral bevel pinion meshes with a 71 tooth spiral bevel gear. The
pinion’s shafts are supported by ball bearings. The input torque is applied at the end
of the pinion’s large shaft. The approximate dimensions of a pinion tooth are given
schematically in Figure 2.7.

0.227"
Figure 2.7: Approximate dimensions of OH-58 spiral bevel pinion tooth.

2.3 Teeth Contact and Loading of a Gear Tooth

According to the theory of gears, there is a point of contact between a spiral
bevel gear and pinion at any instant in time where their surfaces share a common
normal vector. In reality, the tooth surfaces deform elastically under the contact. The
deformation spreads the point of contact over a larger area. The larger area has
traditionally been approximated using Hertzian contact theory. This contact is
conventionally idealized to spread over an elliptical area [Johnson 1985]. The center

NASA/CR—2000-210062 1



AGMA

BEM

EDM

FEM

FRANC3D

HPSTC

LEFM

NASA/GRC

OSM

RC

SEM

SIF

LIST OF ABBREVIATIONS

American Gear Manufacturers Association
Boundary element method
Electro-discharge machined

Finite element method

FRacture ANalysis Code — 3D

Highest point of single tooth contact

Linear elastic fracture mechanics

National Aeronautics and Space Administration - Glenn Research

Center

Object Solid Modeler
Rockwell C

Scanning electron microscope

Stress intensity factor

NASA/CR—2000-210062 vii



|

AGMA 1983

AGMA 1996

Alban 1985

Alban 1986

Albrecht 1988

AMS 1996

ASTM 1997

Au etal. 1981

REFERENCES

AGMA 246.02A, 1983, “Practice for Carburized Aerospace Gearing,”
American Gear Manufacturers Association, Alexandria, VA.

ANSI/AGMA 2005-C96, 1996, “Design Manual for Bevel Gears,”
American Gear Manufacturers Association, Alexandria, VA.

Alban, L.E., 1985, Systematic Analysis of Gear Failures, American
Society for Metals.

Alban, L.E., 1986, “Failures of Gears,” Metals Handbook, 9" Edition.,
Volume 11, Failure Analysis and Prevention, ASM International, pp.
586-601.

Albrecht, C., 1988, “Transmission Design Using Finite Element
Method Analysis Techniques,” Journal of the American Helicopter
Society, Volume 33, no. 2, pp. 3-14.

AMS 6265J, 1996, Aerospace Material Specification, SAE
International, Pennsylvania.

ASTM E 399-90, 1997, “Standard Test Method for Plane-Strain
Fracture Toughness of Metallic Materials,” Annual Book of ASTM
Standards, Metals Test Methods and Analytical Procedures, Volume
03.01, pp. 408-438.

Au, J.J., and Ke, J.S., 1981, “Correlation Between Fatigue Crack
Growth Rate and Fatigue Striation Spacing in AISI 9310 (AMS 6265)
Steel,” Fractography and Materials Science, ASTM STP 733, L.N.
Gilbertson and R.D. Zipp, Eds., American Society for Testing and
Materials, pp. 202-221.

Bhattacharyya et al. 1979 Bhattacharyya, S., Johnson, V.E., Agarwal, S., and Howes, M.A.H.,

Bibel et al. 1995

Bibel et al. 1996

Binder et al. 1980

Bingyuan et al. 1991

NASA/CR—2000-210062

Eds., 1979, LITRI Fracture Handbook, Failure Analysis of Metallic
Materials by Scanning Electron Microscopy, The Materials Research
Division IT Research Institute, Chicago, Illinois.

Bibel, G.D., Kumar, A., Reddy, S., and Handschuh, R., 1995, “Contact
Stress Analysis of Spiral Bevel Gears Using Finite Element Analysis,”
Journal of Mechanical Design, Volume 117, pp. 235-240.

Bibel, G.D., and Handschuh, R., 1996, “Meshing of a Spiral Bevel
Gearset with 3D Finite Element Analysis,’ NASA Technical
Memorandum 107336/ARL-TR-1224.

Binder, S., and Mack, J.C., 1980, “Experience with Advanced High
Performance Gear Steel,” ASME Paper No. 80-C2/DET-77.

Bingyuan, J., Xitong, Z., and Lei, W., 1991, “Estimate by Fracture
Mechanics Theory for Rolling Contact Fatigue Life and Strength of

109



/

a) Tooth surface

b) Cross section of tooth at midpoint of
initial crack
Figure 5.14: Crack prediction after thirteen propagation steps; N = 311,000 cycles.

In the simulation, the majority of the crack front has progressed almost
halfway through the tooth width. A rough estimate of the total life of the gear is made
by estimating the tooth width as 0.227 inches at the toe end and using the average
growth rate from step thirteen. From these data, approximately 200,000 cycles are
necessary for the crack front to progress through the remaining ligament. Therefore,
the number of cycles to failure beginning from the initial notch is estimated to be
511,000. This number could be non-conservative since a constant 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 after the introduction of the notch. It is most likely
that in a real gear a number of cycles are attributed to initiation of the crack
propagation from the notch.

Figure 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 of cycles is also plotted for comparison since the crack depth varies along the
crack length.

NASA/CR—2000-210062



CHAPTER THREE:
COMPUTATIONAL FRACTURE MECHANICS

3.1 Introduction

This chapter discusses areas of computational fracture mechanics relevant to
the work of this thesis. The areas of focus are LEFM, fatigue, and the BEM. The
BEM is used in a fashion similar to the more common FEM. The primary difference
between the methods in three dimensional elasticity problems is that with the BEM
only the surfaces, or boundaries, are meshed, as opposed to the volume that is meshed
in the FEM. In computational LEFM, the displacement and/or stress results from a
numerical analysis are used to calculate the SIFs. The SIFs are in turn used to predict
how and where a crack may grow.

The analyses of this work are conducted using a suite of computational fracture
mechanics programs developed by the Cornell Fracture Group. OSM is used to create
a geometry model of the OH-58 spiral bevel pinion. FRANC3D is used as a pre- and
post-processor to the boundary element solver program, BES. FRANC3D has built in
features to compute SIFs using the displacement correlation technique.

3.2 Fracture Mechanics and Fatigue

Westergaard [1939], Irwin [1957], and Williams [1957] were the first to write
closed form solutions for the stress distribution near a flaw. Their solutions were
limited to very specific geometries and loading conditions. Their results, in the form
of a series solution, showed that the stress a distance r from a crack tip varied as ri,
It can be shown that, under linear elastic conditions, the first term of the series solution
for the stress near a flaw in any body, under mode I, or opening, loading is given by:

oM= K, fi) .1)

i V2nr
where r and @ are polar coordinates as defined in Figure 3.1, fi; is a function of 6 that is
dependent on the mode of loading, and K; is the mode I stress intensity factor. The
sub- and super-scripts (J) denote mode I loading. Similarly, two other modes of
loading can be defined as in-plane shear, mode II, and out-of-plane shear, mode III.
The stress solutions for mode II and III loading are identical in form to Equation (3.1),
but with all of the sub- and super-scripts J replaced with JI or III.

NASA/CR—2000-210062 21



CHAPTER TWO:
GEAR GEOMETRY AND MODELING

2.1 Introduction

Chapter Two covers the basic terms and geometry aspects of a spiral bevel gear.
This terminology and background is essential to motivate the numerical simulations of
this thesis. A gear’s design and geometry can be quite complex; however, only the
fundamentals are explained in this chapter.

2.2 Basics of Spiral Bevel Gear Geometry

Gears are used in machinery to transmit motion. Gears operate in pairs. The
two mating gears have similar shapes. The smaller of the mating gears is called the
pinion, and the larger the gear. Motion is transferred from one gear to another by
successively engaging teeth.

There are various types of gears. The shape of the teeth and the angle at which
the mating gears are mounted are a few of the distinguishing characteristics between
the gear types. Gears with intersecting shafts are called bevel gears. The most
common angle to mount bevel gears is 6 = 90°, although any intersecting angle could
be used. A bevel gear’s form is conical. For comparison, as illustrated in Figure 2.1,
spur gears are cylindrical, and the shafts of the gears are parallel. The geometry of a
spur gear can be almost fully illustrated in two dimensions. However, the conical
shape of a bevel gear requires a three dimensional illustration. This two and three
dimensional difference is where the complexity of the work contained in this thesis
lies.

Pinion

Axes of gears run
parallel to each other

®

Gear

a) Spur gears operate with parallel axes

NASA/CR—2000-210062 7



Intrinsic
Rt ‘curve
=>
t t
log(AK’;) log (AK eg)

Figure 3.5: Schematic of fatigue crack growth rate data in Paris regime at different
stress ratios collapsing into a single “intrinsic” curve.

Using Paris’ model, the amount of crack growth per cycle for a given cracked
object and load history can be predicted from the SIFs. In computational fracture
mechanics, the FEM or BEM is used to calculate the SIFs. Several ways to calculate
SIFs using numerical methods include the displacement correlation method [Chan er
al. 1970], stiffness derivative [Parks 1974], J-integral [Rice 1968], and the universal
crack closure integral [Singh ef al. 1998]. The displacement correlation technique is
used in this work because it relies only on displacement information on the boundary
near the crack tip and because the method is computationally efficient. The numerical
analyses of the spiral bevel pinion are conducted using the boundary element method,
which solves for displacement information only on the boundaries. The displacement
correlation method is computationally efficient since only a single numerical analysis
is adequate to calculate the SIFs, unlike some of the other techniques that require two.
Additionally, the mode I, II, and I SIFs are all calculated by the same method.

The displacement correlation method utilizes the fact that the displacements
near a crack tip are proportional to the SIFs. Under pure mode I loading, the opening
displacement, uy, is given by [Owen et al. 1983]

u, =X I) aca nsin 2 )-sin 30 (3.7)
* 4u \2n 2 2
where K= 3-v for plane stress
l+v

kK =3-4v for plane strain

[is the shear modulus of the material, v is Poisson’s ratio, and @ is the angle between
the location of the displacement and the normal to the crack tip. Equation (3.7) can be
rearranged to solve for K; = f (uy). Along the crack front @ = 180°. Knowing the

material properties (E (elastic modulus) and v), and the crack opening displacement
Uy, at a given distance r from the crack front, K; can be calculated.

u,E an
= 3.8
x 8d-v?) Vr G8)

NASA/CR—2000-210062 26



between the moving load method and a simplified loading method. Again, the
assumption is that the moving load method is most accurate. The simplified method
assumes a cyclic load at the HPSTC on the pinion tooth. The HPSTC is taken as
contact ellipse number eleven from the discretized moving load; load step eleven is the
final step of single tooth contact in the discretized load data. The magnitude of the
load is defined as 100% design load. The model parameters and material properties
from the moving load analyses are used in the HPSTC predictions.

The initial crack location and geometry are the same as those from the moving
load analyses. The method to propagate the crack under the HPSTC is described in
Section 3.2.3. The method assumes proportional loading. It is assumed that the
HPSTC produces Kina: and that R is zero. The direction of growth is determined by
the maximum principal stress theory using the ratio of Ky to K; from the HPSTC
loading. The extensions for the discrete crack front points are calculated with Paris’
model modified to account for crack closure. Figure 7.14 is a comparison of the crack
trajectories from the moving and HPSTC load methods. Roughly 190,000 cycles have
occurred. The cross section view is taken at the approximate location along the tooth
length of the initial crack front’s midpoint.

EEE

we
| Fixed (Experimental

a) Tooth surface

b) Cross section of tooth at midpoint of
initial crack

Figure 7.14: Comparison of crack trajectories from moving load and HPSTC load
(fixed location) methods after ~190,000 cycles.

NASA/CR—2000-210062 97



length at the toe (Figure 6.1b). The trajectory in the simulation turned, out of the root,
up the tooth height and eventually reached the top land (Figure 5.14a). It is
hypothesized that the differences may be attributed to inconsistencies between the
contact conditions (loading conditions).

The inconsistencies could result from misalignment during the test or
inaccurate representation in the simulation of the actual contact areas in the test.
Glodez et al.’s [1998] experimental work with spur gears supports this hypothesis.
They considered two load cases: i) loading along the entire length of the tooth, and ii)
loading along one half of the length. With load case ii) the crack in the 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 goal of the remainder of this section is to investigate whether shifted loads
have the same influence on crack trajectories in spiral bevel gears as Glodez et al.

showed in spur gears.

Load Location

Two shifted load scenarios are investigated. For both scenarios, the cracked
BEM model from propagation step number five is analyzed under the shifted contact.
This model was chosen because the crack trajectory began turning sharply from this
step onward in the preliminary analyses. The contact areas are shifted approximately
+0.3 inches along the tooth length.

The crack trajectory for the shifted contact areas is calculated using the non-
proportional load method described in Section 5.4.2. The load cycle is approximated
by the discrete load steps one, five, and eleven. Table 7.2 sketches the predicted
trajectories on the tooth surface for the original and two shifted analyses. The mode I
and II SIFs from the shifted loading scenarios are given in Appendix C.

When the contact is central, the crack tums up the tooth height on both ends.
The trajectory “wraps around” the contact location. However, when the contact is
shifted toward the heel (toe), the tendency for the crack to kink up on the heel end (toe
end) is suppressed. This is most clearly seen when comparing the central and toe
contact location trajectories. This result is consistent with Glodez ef al.’s
observations. As a result, it is assumed that, if the fatigue crack growth simulations
were carried out further with the shifted contact locations, a flatter trajectory that
maintains a path very near the root under the contact location will result.

The discrepancy of the toe end trajectory between the test and simulation is
explained by the fact that, in the test, the contact was closer to the toe end. The shifted
contact could have resulted from increasing deflections of the tooth. As the crack
grew, the tooth’s stiffness decreased, and the load could have been redistributed along
the tooth length. The subtleties of the redistribution and its effect on crack trajectories
can only be modeled accurately with a three dimensional contact analysis between the
mating gears in conjunction with a fracture mechanics simulation.

NASA/CR—2000-210062 90



Pinion

B

b) Bevel gears operate with intersecting axes
Figure 2.1: Schematics of spur (a) and bevel (b) gears.

The cone defined by the angle between a bevel gear’s axis and the line of
tangency with the mating gear is called the pitch cone. In Figure 2.1b, @ and
define the pitch cones. The gear ratio is the ratio of the angular frequencies of the
mating gears, @/@, which also equals the ratio of sin(@) to sin(@;), or, due to
geometry, the ratio of the number of gear teeth to the number of pinion teeth.

a) Straight bevel gear b) Spiral bevel gear
Figure 2.2: Bevel gear drawings [Coy et al. 1988].

Two common bevel gears are the straight bevel gear and the spiral bevel gear.
The main difference between these two gears is the shape of their teeth. The teeth of
the straight bevel gear are straight, and the teeth of the spiral bevel gear are curved.
Figure 2.2 illustrates this difference. When looking along the axis of a spiral bevel
gear, the teeth will either curve counterclockwise or clockwise, depending on whether

NASA/CR—2000-210062 8



@ 25 Load Step 1

[degrees] + 150% Load
- -& 125% Load
15 — 100% Load
104
54
o—- —— a : —
5 25
~ Crack front position
-10 (Orientation: heel to toe)
“15 -
-20
25
-30
-35 J
a) 35
0 _ _
0 5 10 15 20 25
54 Crack front position
(Orientation: heel to toe)
-10
“15
-20 «|
25 Load Step 11
—* 150% Load
-30 ~s- 125% Load
— 100% Load
-35
-40 4
0 45 -
by Mesrees]

Figure 7.13: Kink angle distribution based on the maximum principal stress theory for
load step one (a) and load step eleven (b).

7.4 Highest Point of Single Tooth Contact (HPSTC) Analysis
Comparison studies to determine the smallest model that accurately represents

the gear’s operating conditions were performed when developing the BEM model.

These results were reported in Section 5.2. Similar comparisons are now made

NASA/CR—2000-210062 96



CHAPTER EIGHT:
CONCLUDING REMARKS

8.1 Accomplishments and Significance of Thesis

This thesis investigated computationally modeling fatigue crack growth in
spiral bevel gears. Predicting crack growth is significant in the context of gear design
because a crack’s trajectory determines whether the failure will be benign or
catastrophic. Having the capability to predict crack growth in gears allows a designer
to prevent catastrophic failures. Prior to this thesis, numerical methods had been
limited to modeling cracking in gears with simpler geometry, such as spur gears. Spur
gear geometry permits the use of two dimensional analyses. However, spiral bevel
gears require a three dimensional model of the geometry and cracks. Three
dimensional models are much more complicated to create, require greater computing
power because of the significant increase in degrees of freedom, and no closed form
solutions exist to predict the growth of arbitrary three dimensional cracks. Prior to this
thesis, few predictions of crack growth in spiral bevel gears had been performed.
Accurately modeling three dimensional fatigue crack trajectories in a spiral bevel
pinion required expanding the state-of-the-art capabilities and theories for predicting
fatigue crack growth rates and crack trajectories.

The geometry of a spiral bevel pinion from the transmission system of the U.S.
Army’s OH-58 Kiowa Helicopter was used for demonstrative purposes. A BEM
model of the pinion was developed using a computer program developed by
NASA/GRC that calculates the surface coordinates of a spiral bevel gear tooth. Their
tooth contact analysis program was also used to determine the location, orientation,
and magnitude of contact between the pinion and its mating gear. The contact was
represented by discrete traction patches on the gear tooth.

LEFM theories were combined with the BEM to accomplish the crack growth
predictions. The simulations were based on computational fracture mechanics
software developed by the Cornell Fracture Group, which allow for arbitrarily shaped,
three dimensional curved crack fronts and crack trajectories. The crack trajectories
were determined by a Paris model, modified to incorporate crack closure, to calculate
fatigue crack growth rates in conjunction with the maximum principal stress theory to
calculate kink angles.

In operation, the load on a gear tooth varies over time in location and
magnitude. This moving load effect was incorporated into the propagation method.
Only loads normal to a gear tooth’s surface were considered. It was discovered that
the moving normal load produces a non-proportional load history in the tooth root.
Proposed prediction methods for fatigue crack growth under non-proportional loads in
the literature were determined to be insufficient for the spiral bevel gear model. As a
result, a method to predict three dimensional fatigue crack growth under non-
proportional loading was developed. The method incrementally advanced the crack
front for a series of discrete load steps throughout one load cycle. A number of load
cycles were then specified, and the crack was advanced an amount based on the
number of specified load cycles and the calculated trajectory from the single load
cycle; the process was then repeated. Some aspects of the final crack trajectory

NASA/CR—2000-210062 101



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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Simulating Fatigue Crack Growth in Spiral Bevel Gears
WU-581-30-13-00

6. AUTHOR(S) NAG3-1993
1L162211A47A
Lisa E. Spievak, Paul A. Wawrzynek, and Anthony R. Ingraffea
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
REPORT NUMBER

Cornell University
Cornel] Fracture Group

641 Frank Rhodes Hall B-12287
Ithaca, New York 14853

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING
U.S. Amy Research Laboratory AGENCY REPORT NUMBER
Cleveland, Ohio 44135-3191
and c NASA CR—2000-210062
NASA Glenn Research Center RL-C)

Cleveland. Ohio 44135-3191 A R451

17, SUPPLEMENTARY NOTES

Project Manager, David G. Lewicki, Structures and Acoustics Division, NASA Glenn Research Center,
organization code 5950, (216) 433-3970.

12a, DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited
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This publication is available from the NASA Center for AeroSpace Information. (301) 621-0390,

13. ABSTRACT (Maximum 200 words)

The majority of helicopter transmission systems utilize spiral bevel gears to convert the horizontal power from the engine
into vertical power for the rotor. Due to the cyclical loading on a gear’s tooth, fatigue crack propagation can occur. In
rotorcraft applications, a crack’s trajectory determines whether the gear failure will be benign or catastrophic for the
aircraft. As a result, the capability to predict crack growth in gears is significant. A spiral bevel gear’s complex shape
requires a three dimensional model of the geometry and cracks. The boundary element method in conjunction with linear
elastic fracture mechanics theories is used to predict arbitrarily shaped three dimensional fatigue crack trajectories in a
spiral bevel pinion under moving load conditions. The predictions are validated by comparison to experimental results.
The sensitivity of the predictions to variations in loading conditions and crack growth rate model parameters is explored.
Critical areas that must be understood in greater detail prior to predicting more accurate crack trajectories and crack
growth rates in three dimensions are identified.

14. SUBJECT TERMS 15. NUMBER OF PAGES
Gears; Spiral-bevel gears; Crack propagation; Fracture mechanics; 121
Boundary element method 16. PRICE CODE
A06
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF REPORT OF THIS PAGE OF ABSTRACT
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6. Determine the number of cycles, N, necessary to achieve a significant amount of
crack growth in relation to the pinion’s geometry. This step is necessary because
the amount of crack growth over one load cycle is too small to update the
geometry model. Therefore, it is necessary to assume that a series of load cycles
has occurred prior to changing the geometry. Because each point along the crack
front has a unique growth rate associated with it, the crack front will not grow
uniformly. Each point will grow by an amount dajinai -

a fygy = N * day’ (5.10)
In general, N is chosen such that dayinai > 0.01 inches.

7. Update the FRANC3D geometry model with the new crack that has grown by an
amount of dajing’. To accomplish this, a least squares curve fit is performed
through the new discrete crack front points. A single polynomial curve may be fit
through all of the points, or the points may be divided into a user-defined number
of sets and individual polynomial curves are fit to each set. After the crack
geometry is updated, it is necessary to locally remesh the model prior running the
BEM solver. Once again, all load steps are analyzed with the new crack.

8. Repeat process beginning at step 2.

Ya

Crack tip

* 7

Figure 5.12: Schematic of crack extension for one point along the crack front after one
load cycle.

NASA/CR—2000-210062 62



CHAPTER SIX:
EXPERIMENTAL RESULTS

6.1 Introduction

Experimental results can be used to evaluate the accuracy of the predictions
made in Chapter 5. Recordings of crack length and depth as a function of total
number of cycles are necessary to confirm the fatigue life prediction. Knowing the
crack front shape during propagation would also assist in verifying the predictions.

There is a limited amount of useful experimental data for the OH-58 pinion to
validate the numerical results. Test data from an OH-58 spiral bevel pinion are
provided by NASA/GRC. The crack growth observations made during the test are
limited. As a result, the fracture surfaces of the tested pinion are observed under a
scanning electron microscope (SEM). The test data from NASA/GRC and the
fractography results are summarized in Section 6.2 and 6.3, respectively. A crack
growth scenario during the test is formulated from the SEM observations.

6.2 Test Results

A pinion that was tested by NASA/GRC in their spiral bevel gear test fixture
under a separate research project is used for comparison/validation. Table 6.1
contains the loading data from the test. A notch was electro-discharge machined
(EDM) into the root of a tooth’s concave side. The gear was run for six million cycles
beginning at 1550 in-lb torque and progressing up to 4649 in-lb torque at the end of
the six million cycles. The test was not stopped until the completion of the six million
cycles; at which time, there was no observable crack growth. As a result, eight more
notches of varying sizes were fabricated into individual tooth roots of the pinion. The
pinion then ran continuously for an additional 4.9 million cycles at increasing levels of
torque detailed in Table 6.1. At the completion of the 4.9 million cycles, five teeth
had fractured from the pinion.

Table 6.1: Pinion test data.

Time Speed Torque # of EDM
[cycles] [rpm] [in-Ib] Notches
1 million 6060 1550 1
1 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
1 million 6060 3874 9

1.9 million 6060 4649 9

During the latter 4.9 million cycles, the test was never stopped to observe the
crack growth. Therefore, the sequence of events for the tooth fractures and the exact

NASA/CR—2000-210062 69



tooth is once again reduced due to the double tooth contact. The contact area will
differ for single tooth and double tooth contact. The change in area of the contact is
schematically illustrated in Figures 2.8 and 2.9.

Time Step Tooth 1 Tooth 2

ie)

Figure 2.9: Schematic of load progression on adjacent pinion teeth.

NASA/CR—2000-210062 13



The midpoint of the crack front is deeper in the HPSTC analyses after 190,000
thousand cycles. From Figure 7.14b, it appears that the moving load analysis
trajectory will produce rim failure. Figure 5.14, however, shows that the crack turns
when the predictions are continued. Therefore, both the static and moving load
method predict tooth failure. The slope of the trajectory into the rim in the moving
load prediction matches more closely the observed trajectories in the tested pinion.
This comparison is purely qualitative.

Several obvious differences between the trajectories predicted by the two
methods can be observed. As seen in Figure 7.14a, the HPSTC method predicts a
larger kink at the heel end; the moving load method predicts a larger kink at the toe
end. Considering the location of the HPSTC load, this result is consistent with the
shifted load analyses of Section 7.3.3.

One may conclude from Figure 7.14b that the HPSTC method predicts a larger
crack face area since the cross section view of the crack is deeper, yet the lengths of
the cracks on the tooth surface are roughly equal. Figure 7.15, in general, supports
this conclusion.

0.25 ]

0.2

Crack Area [in"]

Ss
a

+ Moving Load
© HPSTC

°

0.05

O+ —— T
0 50000 100000 150000 200000 250000 300000 350000
N [cycles]

Figure 7.15: Crack area versus number of load cycles for HPSTC and moving load
prediction methods.

In summary, the HPSTC analyses predict the same failure mode as the moving
load analyses. The crack trajectory and fatigue life calculations vary between the two
methods. Since no experimental fatigue life data exists, the accuracy of one methods
fatigue life prediction over the other methods can not be evaluated. The moving load

NASA/CR—2000-210062 98



Zero-stress point

Tensile stress

Compressive stress

Fractured
tooth

Crack

Figure 2.11: Photoelastic results from mating spur gear teeth (stress contour
photograph from [Alban 1985]).

2.5.1 Gear Failures
Gears in rotorcraft applications are currently designed for infinite life.

Therefore, gear failures are not common. However, failures do occur primarily as a
result from manufacturing flaws, metallurgical flaws, and misalignment.

Dudley [1996] gives an overview of the various factors affecting a gear’s life.
Some of the more common metallurgical flaws listed are case depth too thin or too
thick, grinding burns on the case, core hardness too low, inhomogeneities in the
material microstructure, composition of the steel not within specification limits, and
quenching cracks. In addition, examples of surface durability problems, such as
pitting, are presented. A pitting flaw could develop into a starter crack for a fatigue
failure.

Pepi [1996] examined a failed spiral bevel gear in an Army cargo helicopter.
A grinding burn was determined as the origin of the fatigue crack. In addition, it was
learned that the carburized case was deeper than acceptable limits in the area of the
crack origin, which contributed to crack growth. Roth er al. [1992] determined a
microstructure inhomogeneity, introduced during the remelting process, to be the
cause of a fatigue crack in a carburized AISI 9310 spiral bevel gear. Both of these
failures could be classified as manufacturing flaws.

Albrecht [1988] gives an example of a series of failures in the Boeing Chinook
helicopter, which were caused by gear resonance with insufficient damping. Couchon
et al. [1993] gives an example of a gear failure resulting from excessive misalignment.
The excessive misalignment was due to a failed bearing that supported the pinion.
The misalignment led to a fatigue crack on the loaded side of the tooth. An analysis of
an input spiral bevel pinion fatigue crack failure in a Royal Australian Navy helicopter

NASA/CR—2000-210062 18



CHAPTER FIVE:
PREDICTING FATIGUE CRACK GROWTH
TRAJECTORIES IN THREE DIMENSIONS UNDER
MOVING, NON-PROPORTIONAL LOADS

5.1 Introduction

Chapter 5 covers numerical modeling issues related to predicting fatigue crack
growth trajectories in three dimensions in a spiral bevel pinion. The goal of this
chapter is to model crack growth under realistic operating conditions. As covered in
Section 1.2, most previous work in the area of predicting crack trajectories in gears
assumed one fixed load location. The location was usually the HPSTC. However, in
operation, spiral bevel gears are subjected to a load moving in three dimensions. The
fixed location loading, therefore, could lead to incorrect three dimensional trajectories.

A boundary element model of the OH-58 spiral bevel pinion is presented in
Section 5.2. The tooth coordinates 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 load cases are conducted. The SIF history for an initial crack subjected to the
moving load is presented in Section 5.3.

Section 5.4 develops a method to predict three dimensional 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 line. A number of
cycles are specified, and the crack front is advanced an amount equal to the crack
extension for one cycle times the number of assumed cycles, and at the 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 5.5 the proposed moving load crack propagation method is
implemented to predict fatigue crack growth trajectories in the OH-58 spiral bevel
pinion.

5.2 BEM Model

A boundary element model of the OH-58 spiral bevel pinion was built with
OSM/FRANC3D. The Cartesian coordinates for a tooth surface, tooth profile, and
fillet curve were provided by NASA/GRC. The data were generated automatically
from a program that models the gear cutting process along with the gear kinematics.
All points on the generated tooth surface are points of tangency to the cutter surface
during the manufacturing process [Litvin 1991]. A primary motivation for developing
the tooth geometry program was to generate data for a three dimensional finite
element analysis. This program’s output was adapted to develop a boundary element

NASA/CR—2000-210062 47



Load Step |
—= 150% Load

= 125% Load
— 100% Load ,“/ *

_— 5004

“

z

2 0 —
S 20 25
i

Crack front position
(Orientation: heel to toe)

-500

Load Step 11
+ 150% Load
-# 125% Load
— 100% Load
_— 6000 4
= 5000 -
Z
= 4000 4
< 3000 4
2000 +
1000
0 - - + :
0 5 10 15 20 25
Crack front position
b) (Orientation: heel to toe)

Figure 7.11: Kj distribution for load step one (a) and load step eleven (b).

NASA/CR—2000-210062 94



da, =(da/dN), *N;, at an angle @,, with respect to the self-similar crack trajectory
(Figure 3.2). Again, the model must be remeshed locally, and the process is repeated.

Le L i
a) Initial crack and quarter point element b) Finite element mesh after adding initial

rosette crack
Figure 3.8: Initial edge crack in model.

Figure 3.9: Predicted crack trajectory for model in Figure 3.8.

Figure 3.9 is a picture of the predicted crack trajectory for the finite element
model in Figure 3.8. The crack has been incrementally advanced from the initial

NASA/CR—2000-210062 30



In Figure 2.9, tooth 1 and 2 are two adjacent teeth of a spiral bevel pinion. The
ellipses represent “snap shot” areas of contact between a gear and a pinion’s tooth.
The darkened ellipse is the area that is currently in contact with the gear at a particular
instant in time. Similar to Figure 2.8, the larger ellipses represent single tooth contact,
and the smaller are areas of double tooth contact. The first row in Figure 2.9 begins
with tooth 1 at the last moment of single tooth contact. After a discrete time step, the
load on tooth 1 has progressed up the tooth and tooth 2 has come into contact near the
root, as depicted in row two. In the final row, or time step, tooth 1 loses contact and
tooth 2 advances into the stage of single tooth contact.

It is seen in Figures 2.8 and 2.9 that the contact area between mating spiral
bevel gear teeth moves in three spatial dimensions during one load cycle. Most of the
previous research into numerically calculating crack trajectories in gears has been
performed on spur gears with two dimensional analyses and has not incorporated the
moving load discussed above. Instead, a single load location on the spur gear tooth
that produces the maximum stresses in the tooth root during the load cycle has been
used to analyze the gear. This load position corresponds to the highest point of single
tooth contact (HPSTC). Contact between spur gear teeth only moves in two
directions, and, therefore, this simplification to investigate a spur gear under a fixed
load at the HPSTC has proven successful [Lewicki 1995] [Lewicki er al. 1997a].
However, since the contact area between mating spiral bevel gear teeth moves in three
dimensions, the crack front trajectories could be significantly influenced by this three
dimensional effect. As a result, trajectories under the moving load should be predicted
first and compared to trajectories considering only a fixed loading location at HPSTC.
This approach is detailed in Chapters 5 and 7.

It has been implicitly assumed in the above discussion that the traction, or
force over the contact area, is normal to the surface. Dike [1978] points out that this
assumption is valid if there are no frictional forces in the contact area. He also states
that is the case with gears since a lubricant is always used. The lubricant will make
the magnitude of the frictional forces small compared to the normal forces. This
assumption will be utilized in the numerical simulations.

In the same paper, Dike also asserts that there are two main areas in a gear
tooth where the bending stresses may cause damage. The first is the location of
maximum tensile stresses at the fillet of the tooth on the same side as the load. The
second is at the fillet of the tooth on the side opposite the load, where the maximum
compressive stresses occur.

This can be visualized by drawing an analogy between a cantilever beam and a
gear tooth, Figure 2.10. Basic beam theory predicts that the maximum tensile stress
occurs at the beam/wall connection on the outer most fibers on the same side as the
applied load. The maximum compressive stress occurs at the same vertical location,
on the side opposite the load. Similarly, as a gear tooth is loaded, it creates tensile
stresses in the tooth root of the loaded side. In the root of the side opposite the load,
there are compressive stresses. These compressive stresses might also extend into the
fillet and root of the next tooth.

NASA/CR—2000-210062 14



5.4 Method for Three Dimensional Fatigue Crack Growth Predictions
Under Non-Proportional Loading

As shown in Section 5.3, a crack in a spiral bevel pinion tooth is subjected to
non-proportional loading. As a result, conventional methods to predict crack growth
trajectories in three dimensions are not adequate. A literature review of non-
proportional fatigue crack growth revealed only a few methods that were applicable to
the gear model; Section 5.4.1 is a summary of relevant work. A method to predict
three dimensional fatigue crack trajectories under non-proportional loads is proposed
in Section 5.4.2. Section 5.4.3 summarizes the approximations of the method.

5.4.1 Literature Review

In the literature related to non-proportional fatigue crack growth, the majority
of the work is experimental. The limited amount of numerical work is related to
predicting crack growth rates and fatigue life. The numerical work is also largely
confined to two dimensional analyses. Schijve [1996] gives an overview of methods
and research related to predicting fatigue life and crack growth. There is no mention
of predicting crack trajectories in non-proportional loading scenarios. Crack
trajectories are of primary importance in the context of gears because the trajectory
determines whether the failure mode will be catastrophic. The number of cycles to
failure is of secondary importance because the high loading frequency on a gear’s
tooth 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 growth in
steels under mixed mode I and II loading. They give experimental results from non-
proportional mode I and II tests, and compare the maximum tangential stress theory
(pure mode I) and maximum shear stress (pure mode II) theory for predicting kink
angles to experimental results. However, their work contains no theoretical
predictions for mixed mode. Bower ef al. [1994] considered brittle fracture under a
moving contact load. They incrementally advanced the load and evaluated the SIFs at
each stage of contact. If the SIFs met 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 effects.

Hourlier er al.’s [1985] focus was to determine which of three theories
predicted trajectories closest to experimental data for non-proportional loading. They
worked in terms of k;, which is the mode I stress intensity factor for a small advance of
the crack at an angle @. The three theories investigated were 1) direction in which k; is
a maximum, 2) direction where Ak; is a maximum, and 3) direction of maximum
fatigue growth rate da/dN. The rate is calculated assuming a mode I dominant growth
mechanism and is a function of kimax(@) and Ak,(@). Their work found that 1) was the
most inaccurate method. In general, the maximum da/dN method was found to best
match experimental results. Hourlier et al.’s work is not practical for the purposes of
this thesis primarily because it requires one to express the moving load and k; in closed
forms as functions of time and @ in order to find the angle corresponding to the
maximum da/dN.

NASA/CR—2000-210062 58



Lewicki 1995

Lewicki et al. 1997a

Lewicki et al. 1997b

Lewicki et al. 1998

Litvin et al. 1991

Litvin 1994

Litvin er al. 1996

McClung et al. 1989a

McClung et al. 1989b

McEvily et al. 1996

McFadden 1985

NASA/CR—2000-210062

Lewicki, D.G., 1995, Crack Propagation Studies to Determine Benign
or Catastrophic Failure Modes for Aerospace Thin-Rim Gears, Ph.D.
Dissertation, Case Western Reserve University.

Lewicki, D.G., and Ballarini, R., 1997, “Effect of Rim Thickness on
Gear Crack Propagation Path,” Journal of Mechanical Design,
Volume 119, pp. 88-95.

Lewicki, D.G., and Ballarini, R., 1997, “Rim Thickness Effects on
Gear Crack Propagation Life,” International Journal of Fracture,
Volume 87, pp. 59-86.

Lewicki, D.G., Sane, A.D., Drago, R.J., and Wawrzynek, P.A., 1998,
“Three-Dimensional Gear Crack Propagation Studies,” NASA/TM-
1988-208827/ARL-TR-1833.

Litvin, F.L., and Zhang, Y., 1991, “Local Synthesis and Tooth Contact
Analysis of Face-Milled Spiral Bevel Gears,” NASA Contractor
Report 4342/AVSCOM Technical Report 90-C-028.

Litvin, F.L., 1994, Gear Geometry and Applied Theory, PTR Prentice
Hall, New Jersey.

Litvin, F.L., Wang, A., and Handschuh, R.F., 1996, “Computerized
Design and Analysis of Face-Milled, Uniform Tooth Height Spiral
Bevel Gear Drives,” NASA Technical Memorandum 107348/ARL-
TR-1251.

McClung, R.C., and Sehitoglu, H., 1989, “On the Finite Element
Analysis of Fatigue Crack Closure — 1. Basic Modeling Issues,”
Engineering Fracture Mechanics, Volume 33, pp. 237-252.

McClung, R.C., and Sehitoglu, H., 1989, “On the Finite Element
Analysis of Fatigue Crack Closure — 2. Numerical Results,”
Engineering Fracture Mechanics, Volume 33, pp. 253-272.

McEvily, A.J., Pohl, K., and Mayr, P., 1996, “Comparison of the
Fractographic Features of a Carburized Steel Fractured Under
Monotonic or Cyclic Loading,” Materials Characterization, Volume
36, pp. 153-157.

McFadden, P.D., 1985, “Analysis of the Vibration of the Input Bevel
Pinion in RAN Wessex Helicopter Main Rotor Gearbox WAK 143
Prior to Failure,” Aeronautical Research Report, Number AR-004-
049.

113



Paris et al. 1963

Parks 1974

Pehan et al. 1997

Pepi 1996

Proprietary Source 1998

Qian et al. 1996

Rice 1968

Roth et al. 1992

Rudd 1984

Savage et al. 1989

Savage et al. 1992

NASA/CR—2000-210062

Paris, P.C., and Erdogan, F., 1963, “A Critical Analysis of Crack
Propagation Laws,” Journal of Basic Engineering, Volume 85, pp.
528-534.

Parks, D.M., 1974, “A Stiffness Derivative Finite Element Technique
for Determination of Crack Tip Stress Intensity Factors,” International
Journal of Fracture, Volume 10, pp. 487-502.

Pehan, S., Hellen, T., Flasker, J., and Glodez, S., 1997, “Numerical
Methods for Determining Stress Intensity Factors vs. Crack Depth in
Gear Tooth Roots,” Jnternational Journal of Fatigue, Volume 19, pp.
677-685.

Pepi, M., 1996, “Metallurgical Examination of a Failed Spiral Bevel
Gear,” Technology Showcase: Integrated Monitoring, Diagnostics,
and Failure Prevention. Proceedings of a Joint Conference, Mobile,
Alabama, April 22-26.

Qian, J., and Fatemi, A., 1996, “Fatigue Crack Growth Under Mixed-
Mode I and II Loading,” Fatigue and Fracture of Engineering
Materials and Structures, Volume 19, pp. 1277-1284.

Rice, J.R., 1968, “A Path Independent Integral and the Approximate
Analysis of Strain Concentration by Notches and Cracks,” Journal of
Applied Mechanics, Volume 35, pp. 379-386.

Roth, M., and Sander, K., 1992, “Fatigue Failure of a Carburized Steel
Gear From a Helicopter,” Handbook of Case Histories in Failure
Analysis, Volume 1, ASM International, pp. 228-230.

Rudd, J.L., 1984, “Air Force Damage Tolerance Design Philosophy,”
Damage Tolerance of Metallic Structures: Analysis Methods and
Applications, ASTM STP 842, J.B. Chang and J.L. Rudd, Eds.,
American Society for Testing and Materials, pp. 134-141.

Savage, M., Altidis, P.C., Lewicki, D.G., Coy, J.J., and Litvin, F.L.,
1989, “Tooth Contact Shift in Loaded Spiral Bevel Gears,” NASA
Technical Memorandum 101438/AVSCOM Technical Report 87-C-
36.

Savage, M., Prasanna, M.G., and Coe, H.H., 1992, “Maximum Life
Spiral Bevel Reduction Design,” NASA Technical Memorandum
105790/AVSCOM Technical Report 92-C-004.



CHAPTER ONE:
INTRODUCTION

1.1 Background

A desirable objective in the design of aircraft components is to minimize the
weight. A lighter aircraft operates more efficiently. A helicopter’s transmission
system is one example where design is focused on weight minimization. A
transmission system utilizes various types of gears, such as spur gears and spiral bevel
gears. Because spur gear geometry is relatively simple, optimizing the design of these
gears using numerical methods has been researched significantly. However, the
geometry of spiral bevel gears is much more complex, and less research has focused
on using numerical methods to evaluate their design and safety.

One obvious method to minimize the weight of a gear is to reduce the amount
of material. However, removing material can sacrifice the strength of the gear. In
addition, fatigue cracks in gears are a design concern because of the cyclical loading
on a gear tooth. Research shows that the size of a spur gear’s rim with respect to its
tooth height determines the crack trajectories [Lewicki et al. 1997a, 1997b]. This
knowledge is critical because it allows the designer to predict failure modes based on
geometry.

Two common failure modes of a gear are rim fracture and tooth fracture. Rim
fracture, shown in Figure 1.1 [Albrecht 1988], can be catastrophic and lead to the loss
of the aircraft and lives. On the other hand, Figure 1.2 is an example of a tooth
fracture [Alban 1985]. Tooth fracture is the benign failure mode because it is most
often detected prior to catastrophic failure. Knowing how crack trajectories are
affected by design changes is important with respect to these two failure modes.

(PHOTOS

1 PHOTO 8 ga ae a
Figure 1.1: Spiral bevel gear rim failure [Albrecht1988].

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Figure 7.9: Geometry model with crack showing contact areas one and eleven defined
for 125% design torque.

The mode I and II SIFs from load steps one and eleven are presented in Figures
7.10 and 7.11, respectively. The mode I SIFs do not increase linearly with the larger
loads; the 125% load has a larger effect than the 150% load. The smaller spread
between the curves produced by load step one at the toe end is most likely explained
by the fact that the load is not over this portion of the crack. In contrast, the SIFs
increase uniformly along the entire crack front for load step eleven; the major axis of
contact ellipse eleven is larger than the length of the crack mouth, and the ellipse is
located directly above the crack. On the other hand, for load step one the influence of
the increased load on the mode II SIFs is opposite. There is a larger spread in the
curves over the portion of the crack with no load above it (toe end).

The ratio Ky/K; is important because it determines the crack trajectory angle
and the amount of rubbing between the crack faces. The larger the ratio is, the larger
the kink angle will be and the greater the amount of rubbing. Figure 7.12 contains
these ratios produced by the two load locations and all three load magnitudes. The
curves demonstrate that the ratio of Kj to K; increases as the magnitude of load and
size of the contact area increases. This implies that Kj; is more sensitive to the
changes in the torque level than K;. This result supports the fractography
observations. A large percentage of the fracture surface displayed signs of significant
amounts of rubbing between the crack faces. The SIF ratios from the initial crack
propagation analyses were not necessarily large enough to support the extent of
rubbing observed. However, it appears that the increased torque levels will increase
the amount of rubbing between the crack faces.

The kink angles calculated by the maximum principal stress theory for the
various load locations and magnitudes are given in Figure 7.13. The largest absolute
change in angle is 9.7° and 6.4° for load step one and eleven, respectively.

NASA/CR—2000-210062 92


CHAPTER FOUR:
FATIGUE CRACK GROWTH RATES

4.1 Introduction

The goal of this chapter is to determine how highly negative stress ratios affect
the fatigue crack growth rate in a common gear steel, AISI 9310. This is of interest in
the context of gears because the magnitude of compressive stresses in a gear’s tooth
root is a function of the rim thickness. If fatigue crack growth rates are highly
sensitive to compression, then crack growth rates may warrant more attention in
designing gears. On the other hand, if the compressive stresses do not alter the fatigue
crack growth rate predictions greatly, than the loading cycle for a gear tooth can be
simplified by ignoring the compressive portion of the cycle.

In Section 4.2, the concept of fatigue crack closure is discussed. This section
shows that crack closure provides a convenient framework within which to understand
the factors that control fatigue crack growth. A material-independent 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 [1971] and Newman [1981] and discussed in Section 4.2, to steels. Next,
Section 4.3 applies the concepts to AISI 9310 data to obtain an intrinsic fatigue crack
growth model. Section 4.4 demonstrates that in the range of negative R, the effective
stress range, and likewise the crack growth rate, is not highly sensitive to the
magnitude of R.

4.2 Fatigue Crack Closure Concept

Due to the cyclical loading on a gear’s tooth, fatigue crack propagation might
occur. The load range, AS, or stress intensity factor range, AK, along with the load
ratio, R, characterizes cyclic loading. Recall, R is defined as the ratio of minimum
stress, Sin, to maximum stress, Siar, Which, due to similitude, is equal to the ratio of
minimum mode I SIF, Kin, to maximum mode I SIF, Kyay (Equation 3.5).

Lewicki er al. [1997b] found that spur gear teeth can have R-values as low as
-3.0. They also found that the magnitude of R in spur gears is a function of the gear
geometry. As the rim thickness decreases, R becomes more negative due to the
increased bending of the gear rim.

A general interpretation of the crack closure approach is that damage only
occurs during the portion of the load cycle when the crack faces are not in contact.
The majority of the literature’s discussion of crack closure covers its effect on crack
growth rates. Since gears have such high load frequencies, crack growth rates are
commonly of secondary interest in the context of gears. The time from detectable
flaw to failure is usually insignificant. However, if the crack growth rate is highly
sensitive to the magnitude 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 concern in gear design. This demonstration

NASA/CR—2000-210062 35



predictions match the experimental trajectory into the rim and through the cross
section of the tooth better than the HPSTC prediction. Since the trajectory into or
through the rim is what determines tooth failure or rim failure, it is concluded that the
moving load method is necessary to capture that result most accurately. All of the
trajectories on the tooth surface at the heel end, however, are in reasonable agreement.
Nonetheless, a distinct advantage of the HPSTC method is the significant decrease in
computational time to perform the crack propagation predictions since only one load
case needs to be analyzed.

7.5 Chapter Summary

The results from a fatigue crack growth simulation in a spiral bevel pinion
were compared to crack growth observations in a tested pinion. The comparisons are
summarized as follows:

The simulations predicted a reasonable fatigue life with respect to the test data.
The original trajectory predictions failed to capture detailed aspects of the
observed fracture surfaces in the test. It was determined that the simulated loading
on the tooth probably modeled the tooth contact in the test incorrectly. The tooth
contact information used in the predictions assumed perfect alignment between the
pinion and the gear and that the gears were not flawed. Some explanations for the
differences in contact between the test and theory were determined to be:
1. Change in contact location in the test as the crack grew and the tooth
became more flexible.
2. Differences in the magnitude of loading.
3. Crack growth under load control (simulation) versus displacement control
(test).
4. Misalignment between the gear and pinion in the test.

e Additional simulations demonstrated the capability to predict the crack trajectory
observed in the test. A large initial crack, which was assumed to approximate the
location of the crack front just prior to the formation of the ridge, was used and the
crack was propagated through a series of steps.

Sensitivity studies were conducted to determine how changes in some of the
crack growth method’s assumptions would modify the predictions. The studies
determined that:

e The fatigue crack growth rate model parameters used in the initial prediction
yielded conservative results.
The crack front condition is best described as plane strain.

e A reasonable approximation of the dimensionless quantity B, which incorporates
geometry effects when calculating SIFs, is a value of 1.

e The trajectory observed in the tested pinion would result from a contact biased
toward the toe end.

e The increased torque levels might explain the significant amounts of rubbing seen
on the fracture surfaces of the tested pinion.

NASA/CR—2000-210062 99



AK. remains nearly constant for various negative R-values, then the portion of the
load cycle when K,,;, SK, does not contribute to crack growth. Therefore, all

min >
negative R-value cases could be treated in the same manner. The sensitivity of AK ey
will be investigated in Section 4.4. Figure 4.2 illustrates how AK. could remain
constant as K,,,;, decreases.

Elber performed a series of experimental fatigue tests with sheets of 2024-T3
aluminum alloy. The stress ratio range was —0.1<R<0.7. From the tests, he
developed the empirical relationship

U(R)=0.5+0.4R when -0.1SR<0.7 (4.3)

Elber’s U(R) relationship is valid only for 2024-T3 aluminum alloy over the
range of R-values for which he had experimental data. His work inspired many to
develop empirical relationships between U and R for a variety of materials and ranges
of R. Schijve [1988] summarizes several of these empirical relationships for different
alloys and ranges of R. However, it is expensive to develop this relationship
empirically every time one wants to model crack closure in a new material. This led
to attempts to numerically model crack closure [Newman 1976, 1981], [Fleck er al.
1988], [McClung er al. 1989], and [Blom et a/. 1985]. Through the thirty plus years
of research related to crack closure, it has been found that the amount of crack closure
is dependent on many variables. Specimen size, specimen geometry, crack length,
applied stress state, and prior loading conditions all affect the magnitude of Spy.
Newman’s work attempts to incorporate all of these factors.

Newman developed and applied a hybrid analytical/numerical crack closure

model that simulated plane strain and plane stress conditions. He successfully
matched crack growth rates under constant-amplitude loading from his analytical
model to experimental data. The material he focused on initially was 2219-T851
aluminum alloy. The model has since been applied to a variety of metals. Newman’s
model is the most comprehensive and has been successfully validated against
experiments. As a result, his model will be utilized in this thesis.
All variables in Equation (4.2) are defined immediately from the loading conditions
with the exception of S,». To find an expression for Sop/Smax, Newman [1984] fit
equations to his numerical results for 2043-T3 aluminum alloy over a large range of R-
values and load levels. He worked in terms of applied loads, but due to similitude, S$
in his expressions can be replaced with K, giving:

K

rae =A, +A,R+A,R? + AR? for R20 (4.4a)
K.,

zk =A, +AR for -1<R<0 (4.4b)

‘max

when K,, 2 K,,i,- The coefficients Ag — A; are:

Ve
Ay = (0.825 - 0.34% + 0.054? oof Se | (45)

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