30120130405011

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME
88
CRACK PATH PREDICTION OF GEAR TOOTH WITH DIFFERENT
PRESSURE ANGLES -NUMERICAL STUDY
BASIM M.FADHIL
Petroleum Engineering Department, Faculty of Engineering,
Koya University, Erbil, Kurdistan Region, Iraq,
ABSTRACT
A finite element study was conducted to investigate the influence of the gear pressure angle
associated with rim thickness on gear tooth crack initiation and propagation besides the fatigue life.
Three values of pressure angles (15o
, 20o
, and 22o
) are taken in account associated with three values
of rim thickness. A finite element programs FRANC2D and ABAQUS were used to simulate gear
tooth initiation and propagation. The analysis used principles of linear elastic mechanics. FRANC
program had a unique feature to automated crack propagation using automated re-meshing scheme.
The computed stress intensity factors used for determine crack propagation direction. With a simple
Paris equation, fatigue life, has been calculated. The results show that the gear pressure angle
associated with rim thickness has a significant effect on the crack initiation position and crack
propagation path in addition of fatigue life.
Keywords: pressure angle, crack propagation, gear tooth, finite element.
1. INTRODUCTION
Gears form the man mechanical elements in power transmission and are frequently
responsible for gear box failure. They are designed in general according to standard such as.
Generally the tooth failure can takes place under cycling loading that may cause bending fatigue.
Reducing the mechanical elements weight designers especially those that are used in aircrafts and
helicopters may form a significant goal for designers, and one of these elements are gears, so with
appropriate design of gear may help to meet this goal. So some gear designs use thin rim, but with
too thin rim may lead bending fatigue. The gear life depends mainly on appropriate design to pervert
bending fatigue [1, 2].
A computational model has been used for determination of service life of gears with regard to
bending fatigue in a gear tooth root, shows that gear tooth fillet radius affects the polymer gear
performance severely [3]. a study was conducted to follow the crack propagation in the tooth foot of
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 5, September - October (2013), pp. 88-102
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89
a spur gear by using Linear Elastic Fracture Mechanics (LEFM) and the Finite Element Method
(FEM)[4].Lewicki et al. [5, 6] studied, numerically and experimentally, how to validate predicted
results by considering the gear body rim thickness and gear speed effects on the crack propagation
angle when the crack occurs in a gear tooth foot.Sfakiotakis et al. Sfakiotakis et al. [7], Goldez et al.
[8], simulate, in quasi-static behavior, the stiffness of a toothed wheel couple, where one of the gears
contains a crack. They analyze the evolution of the stress intensity factors on the profile of the
pinion, based on the contact point position in the toothed couple. Kramberger et al. [9], predict the
gear service life in fatigue, in the presence of an initial crack in the tooth foot. The FEM has been
used to simulate the crack propagation based on LEFM, and in the correlation displacement method
to determine the relation between intensity factor and length of the crack.
The objective of this study was to determine the effect of gear pressure angle associated with
rim thickness on the crack initiation location and crack propagation path with different load values
and different rim thicknesses.
2. BASICS OF FRACTURE MECHANICS
Consider three kinds of loading on a cracked plate (Fig. 1). For mode I, the loads are
subjected perpendicular to the crack plane and try to open the crack. Mode II indicates to in-plane
shear loading or sliding. Mode III refers to out-of-plane loading or tearing. Linear elastic fracture
mechanics, as the name means, is depends on a linear elastic material with no plastic deformation.
Fig .1. Three kinds of loading on a cracked body (a) Mode I. (b) Mode II (c) Mode III.
Williams (1957) [10] showed that the stress portioning and displacement area in front of a
crack tip in an isotropic linear elastic material can be written as
ߪ௜௝ ൌ
ଵ
√ଶగ௥
ൣ‫ܭ‬ூ݂ூ௃
ூ ሺߠሻ ൅ ‫ܭ‬ூூ݂ூ௃
ூூሺߠሻ ൅ ‫ܭ‬ூூூ݂௜௝
ூூூ
ሺߠሻ൧ ሺ1ሻ
‫ݑ‬௜ ൌ
1
ߤ
ට
‫ݎ‬
2ߨ
ሾ‫ܭ‬ூ݃௜
ூሺߠሻ ൅ ‫ܭ‬ூூ݃௜
ூூሺߠሻ ൅ ‫ܭ‬ூூூ݃௜
ூூூ
ሺߠሻሿ ሺ2ሻ
where σij are the elements of the stress tensor, ui are the displacements and θ are location
coordinates (Fig. 2), KI, KII, and KIII are the stress intensity factors for mode I, mode II, and mode III,
respectively, µis the shear modulus, and fij
I
, fijII, fij
III
, gi
I
, gi
II,
and gi
III
are known functions.

90
For the investigation, the analysis was reduced to a two-dimensional problem and supposed only
mode I and mode II loading. From Eq. (1), the stress ahead of the crack tip can be explained by the
stress intensity factor. The stress intensity factor is related to load and geometry. A number of
methods can be used to estimate the stress intensity factor such as Green’s functions, weight
functions, boundary integral equations, finite element method (FEM), or experimental techniques.
For other than simple geometry and loading, closed-form solutions for the stress intensity factor are
not available and methods such as FEM or experiments are used. With the growing capacities of
computers today FEM techniques have become extremely popular. Also from Eq. (1), the stress
distribution near the crack shows a1/√r singularity. By using the FEM technique with traditional
finite elements, a big number of elements close to the crack tip are need for high accuracy [11].
Work by Henshell and Shaw (1975) [12] and Barsoum (1976) [13] overcame this deficiency. A six-
node triangular element have been used, besides the mid-side nodes on sides adjacent to the crack tip
moved from the mid-position to one-quarter of the length (Fig. 3). It has been shown by these
investigations that this kind of mesh modeled the inverse square-root singularity of stress flied near a
crack tip. The output of the finite element method is determined nodal displacements for which nodal
forces, stresses, and strains can be calculated. For fracture mechanics, stress intensity factors are of
essential significant and can be estimated as well depends on the forces and nodal displacements.
Numerous methods to determine stress intensity factors have been established based on the nodal
values.
Fig.2. Axes of coordinate ahead of
Crack tip.
Fig.3. Isoparametric quarter-node, six-node
triangular, elements used for the zone near a
Crack tip.
One common method to determine stress intensity factors is called the displacement
correlation method. By related the displacement relationship of Eq. (2) with the is placement
relationship of the finite element analysis using quarter node elements, it can be indicated [14] that
the stress intensity factors as a function of the nodal displacements are
‫ܭ‬ூ ൌ
ߤ
݇ ൅ 1
ඨ
2ߨ
‫ܮ‬
ሾ4ሺ‫ݒ‬௕ െ ‫ݒ‬ௗ ሻ ൅ ‫ݒ‬௘ െ ‫ݒ‬௖ ሿ ሺ3ሻ

91
‫ܭ‬ூூ ൌ
ߤ
݇ ൅ 1
ඨ
2ߨ
‫ܮ‬
ሾ4ሺ‫ݒ‬௕ െ ‫ݒ‬ௗ ሻ ൅ ‫ݑ‬௘ െ ‫ݑ‬௖ ሿ ሺ4ሻ
ߤ ൌ െ
‫ܧ‬
2ሺ1 ൅ ‫ݒ‬ሻ
ሺ5ሻ
‫ܭ‬ ൌ ൝
3 െ 4‫ݒ‬ for plane strain
3 െ ‫ݒ‬
1 ൅ ‫ݒ‬
for plane stress
ሺ6ሻ
Where E is the modulus of elasticity, v is Poisson’s ratio, L is the element length, and ui and
vi are nodal displacements in the x and y directions, respectively (Fig. 3).Once the mode I and II
stress intensity factors are known, the predicted crack propagation angle can be estimated under
mixed mode loading. The technique of Erdogan and Sih [15] was states that the crack extension
begins at the crack tip and moves (grows) in the radial direction in the plane normal to the direction
of the maximum tangential tensile stress. Mathematically, the predicted crack propagation angle can
be written as
ߠ௠ ൌ 2‫݊ܽݐ‬ିଵ
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬‫ܭ‬ூ
‫ܭ‬ூூ
േ ඨቀ
‫ܭ‬ூ
‫ܭ‬ூூ
ቁ
ଶ
4
൅ 8
‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
ሺ7ሻ
The predicted crack propagation angle is defined relative to the coordinate system shown in
Fig. 2 and setting θ = θm. In Fig. 2, θ is shown in the positive sense.
3. SURFACE LIFE OF GEAR TOOTH
3.1 CRACK NUCLEATION LENGTH
The life of gear tooth can be divided into crack nucleation period Ni and into crack
propagation period Np [16].
N=Ni+Np (8)
Where Ni represents the number of cycles required for microcrack initiation till reach the
ath.while Np represents the numbers of cycles required for crack propagation from the initial to
critical length where tooth fracture takes place.
Kitagawa-Takahashi plot of applied stress range required for crack growth is a suitable
representative for the fatigue crack growth, (Fig.4).

92
Fig.4. Kitagawa-Takahashi plot
In the region of constant value of threshold stress intensity range Kth, linear elastic fracture
mechanics (LEFM) can be used to analysis the fatigue crack growth. The threshold crack length ath,
below which LEFM is not valid, may be estimated approximately as [8]
ܽ௧௛ ൎ
1
ߨ
൬
∆‫ܭ‬௧௛
∆ߪி௅
൰
ଶ
ሺ9ሻ
Where σFL is the fatigue limit, see Fig. 1. The threshold crack length ath thus defines the
transition point between short and long cracks, i.e. the transition point between the initiation and
propagation periods in engineering applications. However, a wider range of values have been
selected for ath in the literature, usually between 0.05 and 1 mm for steels where high strength steels
have the smallest values [9].
3.2 FATIGUE CRACK GROWTH
Most of the life of the component is spent while the crack length is relatively small. In
addition, the crack growth rate increases with increased applied stress. The application of LEFM to
fatigue is based upon the assumption that the fatigue crack growth rate, da/dN, is a function of the
stress intensity range K=Kmax_Kmin where a is the crack length and N is the number of load cycles.
In this study the simple Paris equation is used to describe of the crack growth rate [17]
݀ܽ
݀ܰ
ൌ ‫ܥ‬ሾ∆‫ܭ‬ሺܽሻሿ௠
ሺ10ሻ
Where C and m are material parameters. In respect to the crack propagation period Np
according to Eq. (9), and with integration of Eq. (10), one can obtain the number of loading cycles
Np to tooth fracture
න ݀ܰ ൌ
1
‫ܥ‬
ே೛
଴
න
݀ܽ
ሾ∆‫ܭ‬ሺܽሻሿ௠
ሺ11ሻ
௔೎
௔೟೓

93
Eq. (11) indicates that the required number of loading cycles Np for a crack to propagate
from the initial length ath to the critical crack length ac can be explicitly determined, if C, m and
K(a) are known. C and m are material parameters and can be obtained experimentally, usually by
means of a three-point bending test according to the standard procedure ASTM E 399-80 [18]. For
simple cases the dependence between the stress intensity factor and the crack length K=f (a) can be
determined using the methodology given in the literature [17, 18].
For more complicated geometry and loading cases it is necessary to use alternative methods.
In this work the finite element package FRANC2D [19] has been used for simulation of the fatigue
crack growth. A unique feature of FRANC2D is the automatic crack propagation capability.
4. PRACTICAL MODEL
The crack growth was accomplished on the gear wheel with basic data given in Table 1. The
gear is made of high-strength alloy steel 14CiNiMo13- with Young’s modulus E = 2.07×105 MPa,
Poison’s ratio υ = 0.3.Table 2 shows the material parameters for crack propagation.
Table 1. Basic data of treated spur gear pair [20]
Magnitude Value
Number of teeth for pinion Z1=28
Number of teeth for wheel Z2=28
Module m=3.175
Flank angle of tool α=20o
Table 2. Material parameters for crack propagation [20]
Magnitude Value
Threshold stress intensity range =122Nmm-3/2
Fracture toughness KIc=2954 Nmm-3/2
Material parameter of Paris equation C=3.128E-13
Material exponent of Paris equation M=2.954
Fatigue limit σFL=450MPa
5. NUMERICAL MODEL
According to the gear parameter in the table 1 ,and via the AutoCAD code a complete two-
dimensional gear was created .in order to obtain the correct boundary conditions ,just three tooth are
included in the model with three different values of pressure angles(15o
,20o
,and 22o
)and three
different rim thickness (0.3h,0.5h and 1h),where (h) is the tooth hight.Boundary conditions of the left
and right edge portions are kept fixed Figs(). The three gear tooth coordinates used as input data to
create finite element mesh with CASCA program .FRANC then used this mesh and accomplished
crack propagation simulations.
FRANC (FRacture Analysis Code) computer program described by Wawrzynek (1991) [21]
was used in this study. FRANC is a general purpose finite element code for the static analysis of
cracked structures.
FRANC is designed for two-dimensional problems and is capable of analyzing plane strain,
plane stress, or axi-symmetric problems. Figs.(5) illustrate

94
Fig.5. Teeth with pressure angle
equals to 15o
and the rim
thickness equal to 0.3h.
equal to 15o
and the rim
thickness equal to 0.5h
Fig.7. Teeth with pressure
angle equal to 15o
and the rim
thickness equal to 1h
. Fig.8. Teeth with pressure angle
equal to 20o
and the rim thickness
equal to 0.3h.
equal to 20o
and the rim
angle equal to 20o
and the rim
thickness equal to 1h
.
equal to 22o
and the rim thickness
equal to 0.3h.
angle equal to 22o
and the rim
thickness equal to 0.5h
angle equal to 22o
and the rim
thickness equal to 1h.
the nine finite element models with three different pack ratio (0.3h,0.5h and 1h) and three
different pressure angles (15o
,20o
and 22o
) respectively ,also with boundary conditions on the left and
right edges. in order to model the crack in the structure ,FRANC uses a technique called(delete and
fill)to perform this .to illustrate the rule ,first adopt a finite element mesh for an uncracked model
(Fig.14a) ,then the user would define the position of the initial crack by specifying the node of the
mouth where the maximum principal stress and the coordinate of the crack tip (crack
size)(Fig.14b).then the program will delete the elements near the crack tip (Fig.14c).By identifying
the number of the element, the program will insert a rosette of quarter –point ,six-node triangular
element around the crack tip (Fig.14d).eventually the program (FRANC)will fill the remaining area
by six triangular elements(Fig.14d).

95
Fig.14.Scheme of crack modeling of computer.(a)Uncracked mesh.(b)User defined FRANC of the
initial crack.(c)deletion of element near crack tip .(d)Rosette of triangular elements.(e)Final mesh of
cracked surface.
FRANC can then calculate stress intensity factors using the displacement correlation method
and the predicted crack propagation angle using the maximum tangential stress theory.
Another feature of FRANC is the automatic crack growth capability. After an initial crack is
inserted in a mesh, FRANC models a propagated crack as a number of straight line segments. For
each segment, FRANC models the crack tip using a rosette of quarter point elements. FRANC then
solves the finite element equations, calculates the stress intensity factors, and calculates the crack
propagation angle. FRANC then places the new crack tip at the calculated angle and at a user-
defined crack increment length. The model is then re-meshed using the “delete and fill” method
described above. The procedure is repeated a specific number of times as specified by the user. It
should be noted that the local x-y coordinate system of Figs. 2 and 3 moves with the crack tip as
crack propagation is numerically simulated. The analysis used 8-node and plane stress, quadrilateral
finite elements.
In this study the tooth load was placed at the location of the highest point of single tooth
contact on the cracked tooth with three different values 600N/mm, 800N/mm and 1000N/mm
(Fig.15).
F
Fig.15 Load acting at the highest point of the single contact
6. RESULTS AND DISCUSSION
6.1 CRACK INITIATION LOCATION AND CRACK PROPAGATION PATH
The initial crack has been assumed to be perpendicular to the tooth surface (or
perpendicular to the surface) and corresponds to the threshold crack length ath with 0.02mm.due to
crack increment size to be specified in advance, crack increment length is taken to be 0.1 mm to the
critical length of crack.

96
6.1.1 GEAR WITH PRESSURE ANGLE EQUAL TO15O
The crack initiation location depends on the maximum principal stress position, with rim
thickness equal to 0.3h the maximum principal stress was at the minimum area which between the
teeth where the crack initiate and propagate toward the rim edge cause the gear fracture (Fig.16)
Fig.16.Crack initiation position
for gear with pressure angle
equal to15o
and rim thickness
equal to 0.3h.
Fig.17. Crack initiation position
equal to15o
and rim thickness
equal to 0. 5h.
Fig.18.Crack initiation
position for gear with
pressure angle equal to15o
and rim thickness equal to
1h.
While with rim thickness was equal to 0,5h and 1h the crack initiated at the left tooth flank
and propagated toward the right flank (figs.17 and 18).
6.1.2 GEAR WITH PRESSURE ANGLE EQUAL TO 20O
With rim thickness 0.3h and 0.5h the crack initiate at area between teeth and propagate
toward the rim edge, as shown in figs.19and 20. At rim thickness equal to 1h, the crack initiate at the
root fillet where the maximum principal stress (tension) and propagate toward the right flak (Fig.21).
Fig.19. Crack initiation position for
gear with pressure angle equal 20o
and rim thickness equal to 0.3h.
for gear with pressure angle equal
to20o
0.5h.
equal to20o
and rim thickness
equal to 1h.
6.1.3 GEAR WITH PRESSURE ANGLE EQUAL TO22O
Crack initiation location and crack propagation path for gear with pressure angle 22o
is the
same for the previous case Fig.22, 23, and24.

97
Fig.22. Crack initiation
position for gear with pressure
angle equal to22o
and rim
for gear with pressure angle equal
to22o
0.5h.
Fig.24. Crack initiation
position for gear with
pressure angle equal to 22o
and rim thickness equal to 1h.
The crack initiation location and propagation path does not differ with the load value
(600N/mm, 800N/mm and 1000N/mm) for all cases.
With rim thickness equal to 0.5h, the predicted crack propagation path was unstable for gears
with pressure angle 20o
and 22o
.
6.2 STRESS INTENSITY FACTOR VERSUS CRACK LENGTH
6.2.1 RIM THICKNESS =0.3h
Figs. 25-27 illustrate the stress intensity factor values that numerically computed versus the
crack length for rim thickness equal to 0.3h with pressure angles 15o
, 20o
and 22o
and with different
load values (600N,800N and 1000N).gear with pressure angle 22o
has the less values of KI.
Fig.26. The stress intensity factor vs.
crack length for rim thickness equal
to 0.3h with pressure angles 15o,
20o,
and 22o
at load equal to 800N/mm.
Fig.27. The stress intensity
factor vs. crack length for rim
thickness equal to 0.3h with
pressure angles 15o,
20o
,and 22o
Fig.25. The stress intensity factor
vs. crack length for rim thickness
equal to 0.3h with pressure angles
15o,
20o
,and 22o
at load equal to
600N/mm.
6.2.2 RIM THICKNESS =0.5h
Due to the unstable crack for gears with pressure angle 20o
and 22o
, the stress intensity factor
does not meet the fracture toughness (KIc=2954N mm-3/2
).while the gear with pressure angle =15o
the
stress intensity factor reach the fracture toughness and the crack is stable(Figs.28-30) .
0
500
1000
1500
2000
2500
3000
0 1 2 3
Crack length[ mm]
KI[MPamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
500
1000
1500
2000
2500
3000
3500
0 1 2 3
Crack Length [mm]
KI[MPamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
500
1000
1500
2000
0 1 2 3
Crack Length[mm]
KI[Mpamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22

98
Fig.28. The stress intensity factor vs.
crack length for rim thickness equal
to 0.5h with pressure angles
15o,
20o
,and 22o
at load equal to
600N/mm.
15o,
20o
,and 22o
at load equal to
800N/mm.
15o
, 20o
, and 22o
at load equal to
1000N/mm.
6.2.3 RIM THICKNESS =1h
With rim thickness equal to1h, the crack was stable and the stress intensity factor reach the
fracture toughness for the three gear types. Gear with pressure angle 22o
has the lower value of KI
regardless of load value (Figs.31-33).
thickness equal to 1h with
pressure angles 15o
, 20o
, and
22o
thickness equal to 1h with
pressure angles 15o
, 20o
, and 22o
equal to 1h with pressure angles
15o
, 20o
, and 22o
at load equal to
1000N/mm.
6.3 NUMBER OF CYCLES
6.3.1 RIM THICKNESS =0.h
Fig.34. show the number of cycles that numerically computed by FRANC versus pressure
angle with different values of loads. Obviously the difference of number of cycles for the three
pressure angles is significant. With increasing the load a noticeable decreasing in the number of
loading cycle appear.
0
1000
2000
3000
4000
0 2 4 6
Crack Length[ mm]
KI[MPamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
1000
2000
3000
4000
5000
0 1 2 3 4 5
Crack length [mm]
KI[MPamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
1000
2000
3000
4000
0 2 4 6
Crack length[mm]
KI[Mpamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
750
1500
2250
3000
3750
0 2 4 6
Crack Length[mm]
KI[MPamm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6
Crack Length[ mm]
KI[Mpa.mm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6
Crack length[mm]
KI[Mpa.mm0.5
]
Pr.angle=15
Pr.angle=20
Pr.angle=22

99
Fig.34. Computed number of cycles vs. pressure angle with different values of loads at rim thickness
equal to 0.3h.
equal to 0.5h.
6.3.2 RIM THICKNESS = 0.5h
Due to the instability of crack with this rim thickness for gears with pressure angles 20o
and
22o
, the number of cycles does not reflect the accurate number of cycles, nevertheless, the gear with
pressure angle 22o
has the highest number of loading cycles(Fig.35).
6.3.3 RIM THICKNESS =1h
Fig.36. illustrate that the gear with pressure angle 22o with the three different loading values,
has the highest number of loading cycles.
equal to 1h.
0
2000
4000
6000
8000
10000
12000
15 20 22
Pressure angle[degree]
No:ofcycles
600N
800N
1000N
0
20000
40000
60000
80000
100000
15 20 22
No:ofcycles
600 N
800 N
1000 N
0
20000
40000
60000
80000
100000
15 20 22
No:ofcycles
600 N
800 N
1000 N

100
6. 4 COMPARISON OF CRACK INITIATION LOCATION AND CRACK PATH
PREDICTION TO EXPERIMENTS
Fig.37a and b. shows the experimental results taken from [effect of rim thickness.].
Reasonable agreement between numerical (present study) and experimental result. But for thin rim
thickness, the numerical determined crack initiation location and path significantly differs from
experimental results due to a fabricated notch in tooth fillet region of test gear to promote crack
initiation.
(a) (b)
Fig.37. experimental sample of crack propagation path. (a) Tooth fracture at rim thickness equal to
1h and more, (b) rim fracture at rim thickness equal to o.3h.
In order to confirm the result in this study numerically, three finite elements models has been
performed by ABAQUS package related for teeth with pressure angles 15o
, 20o, and 22o
with rim
thickness equal to 0.5h where it is considered as critical rim thickness where the crack being
instable.Fig.38-40.show a good agreement with the present results especially for the crack initiation
location.
Fig.38. Crack initiation location and crack propagation for pressure angle equal to 15o
conducted by
ABAQUS.
conducted by
ABAQUS.
Crack
initiation
Crack
initiation

101
conducted by
ABAQUS.
7. CONCLUSIONS
A numerical study was performed to investigate the effect of the pressure angle associated
with different rim thickness on crack initiation location and crack propagation path of gear tooth.
Gear tooth crack initiation and propagation was simulated using finite element program based
computer which used principles of linear elastic mechanics .stress intensity factors were computed
and used to determine crack propagation direction beside the fatigue life. Comparison with previous
experimental study has been done to validate the predicted results. The following conclusions were
made:
1) The pressure angle plays an important role for specifying the crack initiation location regardless of
the rim thickness.
2) For rim thickness equal to 0.5h, an instability takes place for gears with pressure angles 20o
and
22o.
3) For rim thickness equal to 1h, tooth with pressure angle 22o
has the lower stress intensity factors
comparing to others for the same values of crack length.
4) The pressure angle has a significant effect for increasing the fatigue life, where increasing the
pressure angle a noticeable increase of fatigue life will takes place.
REFERENCES
[1] J. Kramberger *, M. Sˇraml, S. Glodezˇ, J. Flasˇker, I. Potrc” Computational model for the
analysis of bending fatigue in gears”,computer and structure,82(2004),pp.2261-2269.
[2] Sr an Podrug,Srečko Glodež, Damir Jelaska” Numerical Modelling of Crack Growth in a Gear
Tooth Root” Journal of Mechanical Engineering, 57(2011)7-8, 579-586.
[3] S. Senthilvelan, R. Gnanamoorthy,”Effect of gear tooth fillet radius on the performance of
Injection molded Nylon 6/6 gears”, Materials and Design 27 (2006) 632–639.
[4] S. Zouari • M. Maatar • T. Fakhfakh M.Haddar” Following Spur Gear Crack Propagation in the
Tooth Foot by Finite Element Method” J Fail. Anal. and Preven. (2010) 10:531–539.
[5] David G. Lewicki, Roberto Ballarini,” Effect of Rim Thickness on Gear Crack Propagation
Path”, Army Research Laboratory Technical Memorandum 107229 Technical Report ARL–
TR–1110, (1996).
[6] Lewicki, D.: Effect of Speed (Centrifugal Load) on Gear Crack Propagation. U.S. Army
Research Laboratory, Glenn Research Center, Cleveland, OH (2001)
[7] Sfakiotakis, V.G., et al.: Finite Element Modeling of Spur Gearing Fractures. Machine Design
Laboratory, Mechanical &Aeronautics Engineering Department, University of Patras,
Patras, Greece (2001).
[8] Goldez, S., et al.: A Computational Model for Determination of Service Life of Gears. Faculty
of Mechanical Engineering, University of Maribor, Maribor, Slovenia (2002).
Crack
initiation

102
[9] Kramberger, J., et al.: Computational Model for the Analysis of Bending Fatigue in Gears.
Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia (2004).
[10] Williams, M.L., “On the Stress Distribution at the Base of a Stationary Crack,” Journal of
Applied Mechanics, (1957), Vol. 24, No. 1, Mar.,pp. 109–114.
[11] Chan, S.K., Tuba, I.S., and Wilson, W.K., “On the Finite Element Method in Linear Fracture
Mechanics,” Engineering Fracture MechanicsVol. 2, No. 1-A, ,( 1970), pp. 1–17.
[12] Henshell, R.D., and Shaw, K.G., “Crack Tip Finite Elements Are Unnecessary,” International
Journal for Numerical Methods in Engineering,( 1975), Vol. 9, pp. 495–507.
[13] Barsoum, R.S., “On the Use of soparametric Finite Elements in Linear Fracture Mechanics,”
International Journal for NumericalMethods in Engineering, Vol. 10, (1976),No. 1, pp. 25–37.
[14] Tracey, D.M., “Discussion of ‘On the Use of Isoparametric Finite Elements in Linear Fracture
Mechanics’ by R.S. Barsoum,” International Journal for Numerical Methods in Engineering,
Vol. 11,( 1977) 1977,, pp. 401–402.
[15] Erdogan, F., and Sih, G.C., “On the Crack Extension in Plates Under Plane Loading and
Transverse Shear,” Journal of Basic Engineering, Vol. 85,( 1963), pp. 519–527.
[16] Shang DG, Yao WX, Wang DJ. A new approach to the determination of fatigue crack initiation
size. Int J Fatigue, 20,(1998),pp.683–687.
[17] Ewalds HL, Wanhill RJ. Fracture mechanics. London:Edward Arnold Publication; 1989.
[18] ASTM E 399-80, American standard, standard test method for plane-strain fracture toughness
of metallic materials.
[19] FRANC2D, User_s Guide, Version 2.7, Cornell University, 1998.
[20] Podrug, S., Jelaska D., Glodež, S. Influence of different load models on gear crack path shapes
and fatigue lives. Fatigue and Fracture of Engineering Materials and Structures, vol. 31, (2008),
pp. 327-339.
[21] Wawrzynek, P.A., “Discrete Modeling of Crack Propagation:Theoretical Aspects and
Implementation Issues in Two and Three Dimensions,” Ph.D. Dissertation, Cornell University.
(1991).
[22] Prabhat Kumar Sinha, Mohdkaleem, Ivan Sunit Rout And Raisul Islam, “Analysis & Modelling
Of Thermal Mechanical Fatigue Crack Propagation of Turbine Blade Structure” International
Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 3, 2013, pp. 155 -
176, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359, Published by IAEME
[23] Akash.D.A, Anand.A, G.V.Gnanendra Reddy And Sudev.L.J, “Determination Of Stress
Intensity Factor For A Crack Emanating From A Hole In A Pressurized Cylinder Using
Displacement Extrapolation Method” International Journal of Mechanical Engineering &
Technology (IJMET), Volume 4, Issue 2, 2013, pp. 373 - 382, ISSN Print: 0976 – 6340, ISSN
Online: 0976 – 6359, Published by IAEME
[24] Manjeet Singh, Dr. Satyendra Singh “Estimation of Stress Intensity Factor of A Central
Cracked Plate” International Journal of Mechanical Engineering & Technology (IJMET),
Volume 3, Issue 2, 2012, pp. 310 - 316, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359,
Published by IAEME
[25] Dr J M Prajapati And P A Vaghela “Factor Affecting The Bending Stress At Critical Section of
Asymmetric Spur Gear” International Journal of Mechanical Engineering & Technology
(IJMET), Volume 4, Issue 4, 2013, pp. 266 - 273, ISSN Print: 0976 – 6340, ISSN Online:
0976 – 6359, Published by IAEME

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