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, 20

o, and 22

o) 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

© IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com

IJMET

© I A E M E



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

�� √�� 1�

�� 1�� 2� �� 2�

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 fijI, 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

�� 1 2�! �4�#$ % #& � � #' % #( � �3�



91

�� 1 2�! �4�#$ % #& � � �' % �( � �4�

� � % *2�1 � #� �5�

� � , 3 % 4# for plane strain3 % #1 � # for plane stress 8 �6�

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

�: � 2;<=>�?@@@@A�� B C��D

�

4 � 8FGGGGH �7�

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]

<JK L 1� M∆�JK∆�OPQ� �9�

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]

S<ST � U�∆��<��: �10�

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

W ST � 1UXYZ W S<�∆��<��: �11�

[\[]^



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,20

o,and 22

o)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 15oand the rim

thickness equal to 0.3h.


equal to 15oand the rim

thickness equal to 0.5h

Fig.7. Teeth with pressure

angle equal to 15oand the rim

thickness equal to 1h

. Fig.8. Teeth with pressure angle

equal to 20oand the rim thickness

equal to 0.3h.


equal to 20oand the rim




thickness equal to 1h

.


equal to 22oand the rim thickness

equal to 0.3h.



thickness equal to 0.5h



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,20

o and 22

o) 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 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 and rim thickness equal to

0.5h.




equal to 1h.

6.1.3 GEAR WITH PRESSURE ANGLE EQUAL TO22O

Crack initiation location and crack propagation path for gear with pressure angle 22

o 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 and rim thickness equal to

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 22

o.

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, 20

o and 22

o 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

at load equal to 1000N/mm.

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 20

o and 22

o, 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

[MP

am

m0

.5]

Pr.angle=15

Pr.angle=20

Pr.angle=22

0

5001000

1500

2000

25003000

3500

0 1 2 3

Crack Length [mm]

KI

[ M

Pa

mm

0.5

]

Pr.angle=15

Pr.angle=20

Pr.angle=220

500

1000

1500

2000

0 1 2 3

Crack Length[ mm]

KI

[Mp

a m

m0

.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,20

o,and 22

o at load equal to

600N/mm.




15o,20

o,and 22

o at load equal to

800N/mm.




15o, 20

o, and 22

o 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 at load equal to 600N/mm.



thickness equal to 1h with

pressure angles 15o, 20o, and 22o

at load equal to 800N/mm.



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

[MP

a m

m0

.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

[MP

am

m0

.5]

Pr.angle=15

Pr.angle=20

Pr.angle=22

0

1000

2000

3000

4000

0 2 4 6

Crack length [mm]

KI

[Mp

am

m0

.5]

Pr.angle=15

Pr.angle=20

Pr.angle=22

0

750

1500

2250

3000

3750

0 2 4 6

Crack Length[ mm]

KI

[MP

am

m0

.5]

Pr.angle=15

Pr.angle=20

Pr.angle=220

500

1000

15002000

2500

3000

3500

0 2 4 6

Crack Length[ mm]

KI

[Mp

a.m

m0

.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[

Mp

a.m

m0

.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 20

oand

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:o

f cycle

s

600N

800N

1000N

0

20000

40000

60000

80000

100000

15 20 22


No: of cycle

s 600 N

800 N

1000 N

0

20000

40000

60000

80000

100000

15 20 22


No:o

f cycle

s

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 22

o 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 15

o conducted by

ABAQUS.

Fig.39. Crack initiation location and crack propagation for pressure angle equal to 20o conducted by

ABAQUS.

Crack

initiation

Crack

initiation



101

Fig.40. Crack initiation location and crack propagation for pressure angle equal to 22

o 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 20oand

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] Srđan 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