Wear 257 (2004) 1167–1175

A study of spur gear pitting formation and life prediction

K. Aslantas∗, S. Tasgetiren

Technical Education Faculty, Afyon Kocatepe University, Afyon, Turkey

Received 1 March 2004; received in revised form 9 August 2004; accepted 9 August 2004Available online 28 September 2004

Abstract

In this study, a numerical prediction on pitting formation is carried out in spur gear made from austempered ductile iron. General two-dimensional rolling sliding contact situations are considered for the development of the analytical model. The problem is assumed underlinear elastic fracture mechanics and the finite element method is used for numerical solutions. Mixed mode stress intensity factorsKI andKII for cyclic loading are evaluated and related to crack extension by a Paris-type equation. The maximum tangential stress criterion is usedto determine the crack-turn-angle during crack propagation under cyclic loading.

A series of experimental study is also carried out to determine the pitting formation life. Test specimens were first austenitized in salt bath at9 ricala©

K

1

tvciue

agecs[pd

t

ectedcon-ry.utedan-e-

thents.con-

pit-nicsgth,

r andrface

thethethe

atiguector

0d

00◦C for 90 min after which they were quenched in salt bath at 325 and 425◦C, for 60 min. A comparison is carried out between numend experimental results.2004 Elsevier B.V. All rights reserved.

eywords:Spur gear; Austempered ductile iron; Pitting; Fatigue; Finite element analysis

. Introduction

Rolling contact fatigue cracks are one of the most impor-ant problems for the gear industry. Under rolling contact,arious surface damages (pitting, spalling and cracking) oc-ur and also cracks develop in the machine units, thus lead-ng to loss of serviceability of the machine part. Pitting fail-res are formed in all rolling pairs such as gear, rail-wheeltc.

Austempered ductile iron has recently been developed forn increasing number of the engineering applications such asears, crankshaft, connecting rods and other heavy machin-ry and transportation equipment because they offer excellentombinations of high strength, ductility, toughness, fatiguetrength and wear resistance over other grades of cast irons1]. Mechanical properties of austempered ductile irons de-end on the heat treatment conditions, graphite size and theistribution of defects in the microstructure.

Modelling studies of crack propagation and spall forma-ion based on the fracture mechanics have been carried out

∗

by researchers in wear studies. Aslantas¸ and Tas¸getiren[2]studied the behaviour of a subsurface-edge crack subjto moving normal and tangential loads. The problem issidered under the linear elastic fracture mechanics theoKIandKII stress intensity factors at the crack tip are compfor different load positions and different load applicationgles. Tas¸getiren and Aslantas¸ [3] also studied the growth bhaviour of surface crack having three different angles tosurface for different load positions and friction coefficieThe Hertzian contact model was considered for loadingditions.

Chue and Chung[4] have analysed the mechanism ofting caused by rolling contact by using fracture mechaapproach. The authors considered the initial crack lencrack angle, contact force, friction, strain hardened layehydraulic pressure of trapped fluid acting on the crack suas variables of the analyses. Keer and Bryant[5] attemptedto simulate a crack propagation mechanism for pitting. Instudy, pitting formation life is calculated and the effect ofcontact friction, lubricant pressure and friction betweencrack faces are discussed. The authors determined the flife by increasing the crack’s length and stress intensity fa

Corresponding author. Tel.: +90 272 228 1235; fax: +90 272 228 1235.

E-mail address:aslantas@aku.edu.tr (K. Aslantas¸). using an analytical method.

043-1648/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2004.08.005

1168 K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175

Sheppard et al.[6] have considered an elastic half-planecontaining a surface-breaking crack normal to the free sur-face, subjected to loading by uniform tractions over a givenlength of its surface. Stress intensity factors are computedfor various crack lengths, friction coefficients and ratios ofapplied tractions. Hills and Comninou[7] studied an elastichalf-plane subjected to a uniform pressure over part of itssurface. A normal edge crack is assumed to lie at the edgeof the loaded region. The effect of Coulomb friction betweenthe crack faces is considered, and the various possibilities in-volving stick and slip regions along the crack are examined.Stress intensity factors are obtained.

Magalhaes et al.[8] studied surface and subsurface crackinitiation in austempered ductile iron discs. The authors usedseveral auxiliary surface analysis techniques. Magalhaes andSeabra[9] analysed wear and scuffing properties of gearsmade from austempered ductile iron. In the study, FZG testswere performed for five gear sets and three different austem-pering temperatures were used.

In this study, a numerical model for the prediction of pit-ting formation life is carried out in spur gear made fromaustempered ductile iron. For this purpose, a subsurface crackis considered first. Then the crack growth is modelled by us-ing the linear elastic fracture mechanics principles. The ana-lytical model developed is considered as in the general two-d des l-u ation.T minet yclicl ut toc prop-e

2

2

on-t ntactr earsa smis-s f theg veryu th ane rm

d f thec

c

w -t

Fig. 1. Equivalent radius for gear and pinion.

determined using Eq.(2) in whichEp andEg are the elasticitymodulus of the pinion and the gear, respectively andRp andRg are radius of the pinion and the gear.

E∗ = 2EpEg

Eg(1 − ν2p) + Ep(1 − ν2

g)R∗ = RpRg

Rp + Rg(2)

The pinion used in the study is made of surface hardened (58HRc) AISI 8620 steel whose elasticity modulus is 207 GPa.The gear material is the investigated austempered ductile iron.The elasticity modulus and Poisson’s ratio of the studied ADIare obtained experimentally as 170 GPa and 0.25, and doesnot change with applied heat treatments. The geometricalparameters of the gear and the pinion are given inTable 3.The radius of the equivalent cylinders are obtained as 8.75and 21 mm for pinion and gear, respectively. As a result, theequivalent elasticity modulusE∗ and the equivalent radiusR∗are obtained as 201.8 GPa and 6.17 mm.

The normal force is a function of the maximum Hertzcontact pressureP0:

P0 = 2P

πc(3)

Distribution of the normal contact pressure in the contactarea can be written as:

p

TC

CSMPSMCNMCATS

imensional rolling sliding contact situations. Mixed motress intensity factorsKI andKII for cyclic loading are evaated and related to crack extension by a Paris-type equhe maximum tangential stress criterion is used to deter

he crack-turn-angle during crack propagation under coading. Several experimental studies are also carried oharacterize the material and to determine the materialrties to be used in the numerical model.

. Numerical procedures

.1. Contact analysis

When two solids in relative motion are brought into cact, the resulting normal and tangential forces at the coegion are transmitted from a surface to its pair. The gre one of the examples of solid bodies used for the tranion of forces. For the purpose of easier consideration oear contact parameters in numerical computations, it isseful to replace the complicated gear pair geometry wiquivalent model of two cylinders[10]. Equivalent cylindeodel for gear pair is shown inFig. 1.According to the Hertz elastic theory[11], when two cylin-

rical elements move on each other, the half-length oontact area can be determined from:

2 = 4PR∗

πE∗ (1)

hereP is the applied normal force,E∗ the equivalent elasicity modulus andR∗ is the equivalent radius.E∗ andR∗ are

(x) = P0(c2 − x2)1/2

c(4)

able 1hemical composition (wt.%) of the gear material

3.40i 2.69n 0.19

0.020.01

g 0.044r 0.04i 0.73o 0.23u 0.87l 0.015i 0.004n 0.007

K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175 1169

Fig. 2. (a) The location of the nodes used to calculate the stress intensity factors; (b) six-node quarter point isoparametric elements at the crack tip.

2.2. Calculation of the stress intensity factors

A mixed mode (mode I and II) stress intensity factor so-lution is developed for the general crack geometry using thefinite element methodology. In this study, the displacementcorrelation method is used for calculation of the stress inten-sity factors. The method is one of the most popular methodsto calculate stress intensity factors by numerical techniques.After the finite element or boundary element solutions forcracked structure are obtained, nodal displacement values ofnodes 2, 3, 4 and 5 (Fig. 2) are determined. These displace-ments are utilized for the calculation of the stress intensityfactors.

The crack face displacements in both opening and slidingmodes are related to the stress intensity factors for mode Iand mode II. The opening modeKI and the shear modeKIIare calculated by[12]:

KI = G

κ + 1

√2π

L[4(v2 − v4) + (v5 − v3)] (5)

ce crac

KII = G

κ + 1

√2π

L[4(u2 − u4) + (u5 − u3)] (6)

κ =∣∣∣∣∣3 − 4ν, for plane strain

3−ν1+ν

, for plane stress(7)

whereL is the element length,ν the Poisson’s ratio, anduiandvi are the nodal displacements in thex andy directions,respectively. In the present study the angle of crack growthassociated with each load position and crack length is foundfrom the maximum tangential stress theory using the currentload steps SIF as:

θ = 2 tan−1

1

4

KI

KII±

√(KI

KII

)2

+ 8

(8)

2.3. Finite element model of the gear

The finite element method has been widely used to solveproblems in linear elastic fracture mechanics. The main diffi-

Fig. 3. (a) Finite element mesh; (b) the subsurfa

k in gear tooth and surface loads effect on tooth surface.

1170 K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175

Fig. 4. Microstructure of the ductile iron material: (a) as-cast; (b) austempered at 325◦C for 60 min.

culty in these calculations stems from the occurrence of infi-nite stresses at the crack tip. An eight-node isoparametric ele-ment is applied in a plane strain configuration that representsthe contact between the gear and pinion. In the mesh aroundthe subsurface crack, shown inFig. 3a, six-node isoparamet-ric elements with mid-side nodes adjacent to the crack tip areused.

The gear is modelled as a half cylinder. The displace-ments at the bottom are fixed in thex andz directions. Theeffect of the pinion is shown as elliptic loading accordingto the Hertz theory. The position of the load is shown with“d” which is the distance between the centre of the crackand the maximum load in the elliptic variation. InFig. 3b,the depth of the subsurface crack is shown with “h”. Thedepth is determined by stress analysis for uncracked finiteelement model. Von-Mises yield criteria is used for the stressanalysis.

The simulations were performed with the FRANC2D[13]finite element code. Among the variety of capabilities, aunique feature of FRANC2D is the ability to model a crackin the structure; 1080 elements and 3311 nodes were used forthe entire mesh.

3. Experimental study

3

phitec po-s -t tm ear

TH

S

ABA

samples for pitting testing were machined from ‘Y’ blocksof thickness 50 mm. In addition, metallographic sampleswere prepared according to the standard procedures. Thegear material has 95% nodularity and the nodule count is200 nod/mm2.

Three different samples are studied in the experiments.One as-cast, one (A) austenitized at 900◦C for 90 min andquenched in salt bath at 325◦C for 60 min. Austenitizationof the last specimen (B) is the same as the second specimenbut this is quenched at 425◦C for the same duration. Themechanical properties of the specimens are given inTable 2.The values of theTable 2are the mean of three samplesresults.

3.2. Pitting tests of gears

Pitting tests of the gear pairs are carried out on a FZG(Forschungsstelle fur Zahnrader und Getriebebau) test ma-chine which has different gear centre distances. The FZG testmachine is back-to-back spur gear equipment with a closedpower circuit[14,15]. Test pinion and test gear are mountedon two parallel shafts which connect the drive gear stage withthe same gear ratio (Fig. 5). While not in motion, the trans-mission can be blocked by a locking pin. By twisting of theload clutch using weights on the load lever, the desired testt ringc

obilGg en inT op-

TG

n

NPTMPDW

.1. Material

The material used in the present study is a nodular graast iron alloyed with Ni and Mo. The chemical comition of the material is given inTable 1. The microstrucure of the as-cast material is shown inFig. 4. The as-casaterial has 80% pearlitic and 20% ferritic structure. G

able 2eat treatment and mechanical properties of the gear material

ample Austemperingtemperature (◦C)

HardnessHRC

σy (MPa) σUTS (MPa) δ (%)

325 42 804 1228 4.3425 37 565 862 9.2

s-cast 23 493 780 7.1

orque is applied, which is indicated at the torque measulutch.

Three different tests are performed using lubricant (Mear SP 150 with kinematics viscosityvk = 150 mm2/s). Theeometrical parameters used in the experiments are givable 3. The gear pair has been subjected to 250 Nm

able 3eometrical parameters of gears

Test gear Pinio

umber of teeth 41 17itch diameter (mm) 123 51ooth depth (mm) 5.41odule 3ressure angle (◦) 20istance of centres (mm) 87idth (mm) 19

K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175 1171

Fig. 5. The FZG test machine used in the pitting tests.

erational torque and the number of revolutions of pinionwas 1500 rev/min. Test gear tooth surfaces have been ob-served in per 5× 105 revolution. When pits of the size about0.5 mm have been observed on any tooth flank, the test runwas stopped. The pitting size values obtained experimentsare the mean of five gear teeth samples results. Pitting sizesare measured approximately using microscope (50×) and theexact pitting sizes are measured using scanning electron mi-croscope after the experiments.

3.3. Fatigue crack growth testing

Crack propagation testing is necessary to obtain the pa-rameters of Paris–Erdogan equation. These are namely the“C” and “m” values used in Eq.(9). For this purpose fa-tigue tests were conducted for the M(T) specimens witha thickness of 4 mm, according to ASTM E647 standard.The notch preparation is made by electrical-discharge ma-chining. The crack propagation experiments are carried outwith a unaxial tension–compression test machine (Instron8501). All experiments are carried out at room temperature.The loading frequency is selected as 15 Hz. The load ratio(R) is 0.1. The specimens are clamped by hydraulic grips.The crack length is observed using a travelling microscope(30×) during the crack propagation. The fatigue parametersoE threes

TG hrees

S )

ABA

4. Results and discussion

4.1. Experimental results

In the results of experimental studies, it is observed thatpitting failures occur roughly between the initial point of sin-gle tooth pair contact and pitch line. This was an expectedphenomenon. Because, this location of the tooth is subjectedto highest load due to the single tooth pair contact. The pittingfailure size observed on the tooth surface changes between150 and 500�m at the pitch line. Maximum pitting size atthe tooth surface was taken into consideration to stop the ex-periment. Maximum pitting sizes on the tooth surface and theaverage number of load cycle are given inTable 4.

Gear materials are generally subjected to heat treatmentto prevent surface fatigue failures such as pitting or spalling.Spherical ductile iron alloyed with Ni and Cu has a good hard-enability property when appropriate parameters are selectedfor austempering process. Pitting formation life increaseswhen the austempering temperature decreases. Particularly,the effect of austempering process on the pitting formationlife can be seen when comparing the gear austempered at325◦C with as-cast gear. Therefore, the highest hardness ofthe 325◦C was responsible for longest pitting formation life.This feature can be the major advantage for gear applications.

4

aga-t ad-

TF

S

ABA

btained from experimental analyses are given inTable 5.ach value given in table is the mean value of at leastamples.

able 4ear pitting test results for maximum pitting size (mean value of t

amples)

ample Average pitting formation life(cycles× 106)

Pitting size (maximum(�m)

4.700 4003.500 460

s-cast 1.850 500

.2. Subsurface crack initiation and propagation

There are two stages, i.e., crack nucleation and propion, in machine components subject to rolling contact lo

able 5atigue parameters obtained from fatigue crack growth experiments

ample C (mm/cycle (MPa√

m)) m

1.06× 10−12 5.13.79× 10−13 6.5

s-cast 2.07× 10−14 7.01

1172 K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175

Fig. 6. SEM micrograph of pitting failure on gear tooth surface (austemperedat 325◦C for 60 min).

ing. Nodule is the crucial factor for crack initiation in austem-pered ductile iron materials. Dommarca et al.[16] stated thatthe crack nucleation resistance of austempered ductile ironis lower than that of steel. Because austempered ductile ironmaterials behave as a metal matrix composite, cyclic load-ing causes to weaken the interface bond between the matrixand the nodule. Therefore, microcracks emanate from thegraphite nodules and grow to link with other graphites.

The direction of microcrack changes depending on thepresence of nodules near the crack tip. For this reason, thepitting failures have irregular form. The typically smoothspall surface observed in bearing steel such as SAE 52100[16] contrasts to the irregular one observed in austemperedductile irons. Dommarca et al. have also observed that spallfailures occur as V-shape pointing opposite to rolling direc-tion in SAE 52100. It can be seen fromFigs. 5 and 6that thefracture surfaces of the pitting failures have irregular mor-phology in the materials studied in the present analysis. Inaddition,Figs. 6 and 7also show the subsurface crack propa-gation due to cyclic contact loadings in black zone indicatedwith arrows.

F t gearm

4.3. Numerical results

Variation of theKI andKII at the crack tips are analysedfor different load positions. Friction is not considered becausethe movement at the pitch line is only rolling. Rebbechi etal. [17] measured the friction coefficient for this contact as0.04–0.06 which may be regarded frictionless. Fifteen dif-ferent positions fromd = −0.35 to +0.35 mm are selectedfor the load (seeFig. 3b). The mode I and II stress intensityfactors are computed by Eqs.(5)–(7). Fig. 8gives the resultsof these analyses.KII have both negative and positive valuesand this affects both the propagation direction and the prop-agation rate importantly according to maximum tangentialstress criterion. Presence of positiveKII values deflects thecrack away from free surface. NegativeKII values cause thecrack to deviate toward the surface. Maximum and minimumvalues ofKII occur atd = −0.15 and +0.15 mm load posi-tions, respectively. Therefore, the subsurface crack deflectsfrom free surface atd = −0.15 mm and deviates towards thefree surface atd= +0.15 mm. MinimumKI value occurs atd= 0 and values ofKI are negative for all load positions.

4.4. Simulation of the crack propagation and pittingformation

frac-t s inf e be-t edb

w ter-m nderc t thecgT

�

w� evm vei rowtha

eacht eachl ntlK thecn ingt

ig. 7. SEM micrograph of pitting failure on gear tooth surface (as-casaterial).

Pitting formation at the surface can be analysed usingure/fatigue mechanics. In the majority of investigationracture mechanics, it is assumed that the dependencween log (da/dN) and log�K is linear and can be describy the equation derived by Paris and Erdogan[18]:

da

dN= C[�K]m (9)

hereC andm are the material parameters that are deined experimentally. For a growing subsurface crack u

onstant amplitude compressive loads, the conditions arack tip are defined by the current value of�K. For crackrowth due to mixed mode cyclic loading,�K is defined byanaka[19] as:

K4eff = �K4

I + 8�K4II (10)

here�KI = KI,max − KI,min and�KII = KII ,max − KII ,min.KI is taken to be zero if bothKI,maxandKI,min have negativ

alues.�KI = KI,max if KI,min < 0 andKI,max > 0 [20]. In theajority of the studies about subsurface cracks, negatiKI

s not considered because it has no effect on the crack gndKI,min is assumed zero[21,22].

In order to calculate the time elapsed for the crack to ro the surface, following procedures are practiced foroading cycle: FirstlyKII values are determined for differeoad positions, then the load positions occurringKII ,max and

II ,min are determined for initial crack length. Secondly,rack length is increased step by step by a “da” increment andewKII ,maxandKII ,min values are determined by consider

he load position (Fig. 9).

K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175 1173

Fig. 8. Variation ofKI andKII at the left and right crack tips with respect to load position.

As seen from theFig. 9, positiveKII values decrease andnegativeKII values increase when the crack length increases.NegativeKII values increase as four times relative to the initialcrack length when the right crack tip reaches the surface. Inaddition, the load position which max and minKII valuesoccur do not depend on the crack length. As a result of theseconsiderations,Keff value in Eq.(10) is revised as:

�K4eff = 8�K4

II (11)

�Keff values are calculated for each crack length increasedby “da” for every step and these are used in the Paris–Erdoganequation. The increment “da” is used as 0.5 of initial cracklength. Therefore, Eq.(9) is rearranged as:

da

dN= C[�Keff]

m (12)

The local finite element mesh around the initial crack isshown inFig. 10(step 1). During finite element analysis, theright crack tip is extended in the direction of the maximum

F itioni

tangential stress. The incremental procedure is repeated untilthe right crack tip reaches the contact surface. InFig. 10, steps4–11 illustrate the crack propagation towards the surface.Step 15 depicts the crack propagation from the left crack tipafter the right tip has reached the surface. Numerical analyseshave shown that stress intensity factor at the right crack tip ismuch higher than the left crack tip. Therefore, the right cracktip will first start to propagate[10].

4.5. Comparison of the numerical and experimentalresults

The predicted pitting size by the finite element analysis isabout 450�m. The pitting size observed on the tooth surfacesafter the FZG test changes between 400 and 500�m. Pittingsizes and number of cycle for maximum pitting are given inTable 6for comparison. For the determination of the depthof the failure, the teeth are cut to obtain the cross-section.Fig. 11shows SEM photographs of the pitting failure fromthe cross-section. It is concluded that the maximum depth ofthe failures reach up to 180�m. A finite element study carriedout by Aslantas¸ et al. [23] has shown this value as 150�m.This also gives crack nucleation depth of Hertz contact theory[11]. The difference is caused due to existence of nodules.The predicted subsurface crack propagation path and the testr ckp veryg

TC

G

34A

ig. 9. Variation ofKII with respect to the crack length and the load posn as-cast gear material for right crack tip.

esults are shown inFig. 11. As a result, the predicted craath and pitting size agree with experimental results in aood manner.

able 6omparison of experimental and numerical results

ear Experiment Numerical

Pitting size(�m)

Number of cycles(cycle× 106)

Pitting size(�m)

Number of cycles(cycle× 106)

25◦C 400 4.700 ∼450 5.16825◦C 460 3.500 3.238s-cast 500 1.850 1.672

1174 K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175

Fig. 10. Fatigue crack growth of subsurface crack and variation of�KII values at crack tips.

Fig. 11. SEM photograph of the cross-section, austempered at 325◦C (a), as-cast (b) and numerically obtained pitting form of the tooth surface.

5. Conclusions

The present study considers the pitting failure model basedon the finite element analysis and linear elastic fracture me-chanics. Also the effect of austempering process on the pit-ting formation in spur gear made of ductile iron is analysed.A series of experimental study is carried out to determine the

pitting formation life on the tooth surface. Experimental andnumerical results from obtained analyses are compared.

In the experimental analysis, the pitting failure size ob-served on the tooth surface changes between average 150and 500�m at the pitch line. Pitting formation life increaseswhen the austempering temperature decreases. So the pittingresistance of the ductile iron gears are affected by austem-

K. Aslantas, S. Ta¸sgetiren / Wear 257 (2004) 1167–1175 1175

pering temperature and time. In addition, fracture surfaces ofthe pitting failures have irregular morphology and the shapesof pitting do not depend on rolling direction.

Numerical analysis used in the paper is appropriate fordetermining the pitting resistance of the ductile iron gears.Finite element results provide a good prediction tool bothfor the formation life and pitting shape in a reasonable accu-racy with the necessary material properties such as elasticitymodulus, Poisson’s ratio and the constant of Paris–Erdoganequation which can be obtained by standard experimentalprocedures if not available in the literature.

It can be concluded that, for the gears made from austem-pered ductile iron, the pitting failure time can be predictedwithout testing of gears. When some standard experimentalvalues are known, the finite element method and fracture me-chanics analyses can be utilized for the life prediction of thegears.

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