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International Journal for Computational Methods in Engineering Science and Mechanics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ucme20

Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion a

a

a

Vineet Agarwal , Pramod R. Zagade , Danish Khan & B P Gautham

a

a

TRDDC Process Engineering Innovation Labs, Tata Consultancy Services , Pune , India , 411013 Accepted author version posted online: 06 Feb 2014.Published online: 06 Feb 2014.

To cite this article: International Journal for Computational Methods in Engineering Science and Mechanics (2014): Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion, International Journal for Computational Methods in Engineering Science and Mechanics To link to this article: http://dx.doi.org/10.1080/15502287.2014.882434

Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to this version also.

PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

ACCEPTED MANUSCRIPT Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion

Vineet Agarwal, Pramod R. Zagade, Danish Khan and BP Gautham1

TRDDC Process Engineering Innovation Labs, Tata Consultancy Services, Pune, India, 411013

Downloaded by [B P Gautham] at 07:35 06 February 2014

1Corresponding

author email: [email protected]

Abstract A computational model for studying the fatigue crack propagation characteristics in a gear tooth root in the presence of inclusion is presented. A step-by-step crack growth scheme is implemented to predict the crack path using the finite element method and linear elastic fracture mechanics. Paris law approach is used to model fatigue crack propagation. Effect of size and location of hard circular inclusion on growth of a surface initiated crack and service life in a gear tooth is studied.

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1. Introduction Service life of a gear is determined from its performance under fatigue loading. Failure of gear tooth can occur due to surface pitting on the flanks or cracks generated in the root region due to repeated loading. Estimation of service life of gear is very important from the perspective of a gear designer. Detailed

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understanding of the development and propagation of fatigue crack in the root region under repeated bending load is crucial in design phase. Various flaws such as nonmetallic inclusions are inevitable in the steel gears. Inclusions are known to have significant effect on the fatigue and fracture properties of the material. It may depend on the nature of inclusions, their size, shape and location. It can affect the crack propagation path and service life of the gear. If the inclusions are close to surface it can affect the crack initiation location as well. It is important to understand the role of inclusion in the service life of a gear. Aim of the present work is to build a computational model to study the effect of size and location of inclusions on the fatigue properties of a gear tooth. It may enable determination of tolerable inclusion parameters that can be specified to the supplier of steel and provide guidelines in making decisions on subsequent manufacturing operations in order to achieve improved performance under fatigue loading. There are two phases in service life of gear 1) crack initiation phase and 2) crack propagation phase [1]. Generally a significant part of service life is attributed to crack initiation phase in case of high cycle fatigue. Typically, crack starts from the surface of the component subjected to bending fatigue. If the inclusion is located close to surface, crack can initiate at the inclusion interface in the subsurface level. After development of significantly large micro-crack in the gear material, crack propagation characteristics determine remaining life. The size, location and type of the inclusion, affect the crack path. It is observed in

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ACCEPTED MANUSCRIPT the experiments by researchers [2] that crack tends to shift towards soft inclusions if sufficiently sized inclusions are in the vicinity of the crack path. Hard inclusions, which are tightly bonded to the metal matrix such as carbide in steel gears, tend to deflect cracks away from the inclusions. In the present work computational model is developed for the prediction of crack propagation path in steel gear when subjected to high cycle fatigue. Algorithm is implemented in the commercial software Ansys to study effect of tightly bonded hard inclusion on the crack path and service life. The model is used to study fatigue crack behavior

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in a steel gear. Several researchers have studied fatigue crack behavior in steels. In earlier studies numerical methods were applied for predicting service life of the gears [3]. Linear Elastic Fracture Mechanics (LEFM) approach is used to model gear tooth root fatigue behavior and prediction of the service life of the gears [1]. In this approach crack initiation point is determined by performing FE analysis. Point of maximum stress is assumed to be the crack initiation point. Crack is introduced in the structure and further step by step growth is simulated by local re-meshing of the model. Paris law [4] is used for prediction of the service life. Lei et. al., [5] have developed a computational model to simulate the crack paths in the vicinity of different types of inclusions using Boundary Element Method (BEM). Nisitani et. al., [6] simulated the crack propagation in a plate under tensile stress, when circular inclusions are present. In case of gears, generally, load acting at highest point of single tooth contact (HPSTC) is considered for determining the stress range. Effect of moving tooth load is also studied using BEM [7] and LEFM [8]. Numerous experimental investigations have been done to understand fatigue crack behavior in the presence of inclusion. Prasannavenkatesan et.al., [9] have carried out experimental investigation to characterize the driving force for fatigue crack nucleation at subsurface primary inclusions in carburized and shot peened gear steels. Toyoda et. al., [10] studied inclusions causing fatigue crack in surface treated gears.

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ACCEPTED MANUSCRIPT In the present work a computational model is developed using finite element framework in Ansys. Initially 2D stress analysis is used for locating crack initiation point. LEFM is the used to determine stress intensity factor of the loaded structure. Depending on direction for crack progress, as predicted using LEFM, step by step crack growth is modeled. Paris law is used for the prediction of number of cycles for crack increment [1]. For the current study hard circular inclusions, tightly bonded to the metal matrix are considered. The circular shaped inclusion is modeled using appropriately fine mesh and by specifying higher stiffness than

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rest of the structure. For the purpose of performing parametric studies crack initiation location is fixed on the surface based on the initial analysis by assuming a small flaw on the gear surface, at that location. Effect of size and location of inclusion on fatigue crack path and life is studied.

2. Fatigue crack modeling 2.1 Fatigue crack initiation High cycle fatigue failure occurs in two main phases, crack initiation phase and crack propagation phase. Further crack initiation can be divided into two stages: 1) micro-crack nucleation and 2) short crack growth. Crack propagation phase consists of 1) long crack growth and 2) occurrence of final failure. In high cycle fatigue (HCF), crack initiation period accounts for most of the service life. Please refer Figure 1. Total service life N (number of stress cycles) is divided into Ni (number of stress cycles required for the fatigue crack initiation) and Np (number of stress cycles required for a crack to propagate till critical crack length i.e. when the final failure is expected) [1].

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ACCEPTED MANUSCRIPT Fatigue life of gears in HCF can be predicted using Basquin type equation [1], (∆σ)ki Ni = Ci

(1)

where, Δσ is the applied stress range, ki and Ci are the material constants. In Figure 1, if crack initiation curve is correlated with Wholer curve then, crack initiation life can be expressed in terms of ΔσFL and NFL,

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Ni = NFL (

∆σFL ki ) ∆σ

(2)

The exponent ki can be determined as (refer Figure 1),

ki =

log(4NFL ) log(σU ⁄∆σFL )

(3)

where, σU is the ultimate strength. Experimental results have shown good correlation with this equation [11]. The material parameter, ΔσFL required while predicting crack initiation life Ni is determined using appropriate standard test specimen. In the present work more emphasis is given on studying the effect of inclusion on the crack propagation path and life. Crack initiation location is kept same for the parametric studies by enforcing a condition of slightly flawed gear surface to ensure initial stress concentration and crack initiation. Prediction of crack initiation period is calculated using the parameters mentioned in the previous studies [1]. 2.2 Fatigue Crack propagation Linear elastic fracture mechanics is applied for the prediction of crack growth. Fatigue crack growth rate, da/dN, is a function of stress intensity range K = Kmax- Kmin. Paris law [1] can be used for predicting crack

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ACCEPTED MANUSCRIPT growth rate as given below, da = C [K(a)]m dN

(4)

where, C and m are the experimentally determined material parameters. Number of cycles required for propagation of crack, from initial length, a0 to critical crack length, ac can be found using following equation,

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Np

ac

1 da ∫ dN = ∫ C [K(a)]m 0

(5)

a0

In the current work Finite Element Method along with Linear Elastic Fracture Mechanics is used for simulation of the fatigue crack growth [1]. Quarter point elements are used at the crack tip. Stress intensity factors in mode I and II fracture in plane strain conditions are determined using Quarter Point Displacement Technique (QPDT) as,

KI =

2G π . √ . [vd − vb ] (3 − 4𝑣) + 1 2L

(6a)

K II =

2G π . √ . [ud − ub ] (3 − 4𝑣) + 1 2L

(6b)

where, G is the shear modulus of the material, ν is the Poisson’s ratio for the material, ui and vi are displacements of nodes on the element face, L is the finite element length on crack face, The combined Stress Intensity Factor (K) in the mixed mode is then defined as,

K = √(K 2I + K 2II ). (1 − 𝑣 2 )

6

(7)

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ACCEPTED MANUSCRIPT Number of cycles required for propagation of crack is predicted using Paris law. Then crack front is advanced by specified length. The criterion based on stress intensity factors is used for determining direction for crack advancement [12]. The crack extension angle θ0 is determined as,

θ0 = cos

−1

3. K 2II + √K 4I + 8K 2I K 2II ( ) K 2I + 9K 2II

(8)

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The gear model is reconstructed with the extended crack and is meshed again for the next analysis. The procedure is repeated until the stress intensity factor reaches the critical value K Ic, when the complete fracture is expected. Above mentioned procedure is implemented in Ansys13 through APDL macro. The model is validated with results from literature. Glodez et. al., [1] have reported comparison of simulation results with experimental results for service life of the gear under various loading conditions. Using the reported gear geometry, material and fatigue parameters, simulation for prediction of crack path and crack propagation life is carried out using current model. Table 1 shows the comparison of predicted crack propagation life in both the cases.

Predicted crack path is compared with simulated crack path in the literature [1] and is found to be in good agreement. Error in the predicted results may be due to different mesh and crack increment lengths used in both the cases. Total service life of the gear as predicted by current model is plotted and compared with experimental results adapted from literature (Please refer Figure 2). The predicted service life contour plot lies well within the experimental results scatter.

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ACCEPTED MANUSCRIPT 3. Crack propagation in presence of inclusions In steel gears, various types of non-metallic inclusions such as carbides, oxides or sulfides may be present. These inclusions can be very hard particles, difficult to break during subsequent processes. It may be tightly bonded to metal matrix. During certain manufacturing processes inclusions may get partially or completely debonded from the metal matrix. Typical sizes of non-metallic inclusions that may have effect on

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the fatigue life need to be determined. Murakami [14] have reported practical observations of metal fatigue tests in the presence of inclusions. Shapes of inclusions can affect the fatigue life significantly. Inclusions generally found in the metals can have arbitrary shapes. It can be approximated to shapes such as circular or elliptical with varying dimensions for the ease of analysis. Inclusions nearer to surface have greater impact on the fatigue life of gears. It can shift the crack initiation location to the subsurface level. Larger inclusions in original crack path may have larger impact on the crack propagation life. Hard circular inclusions bonded tightly to the gear metal matrix are considered for current simulation studies. Inclusions are modeled as rigid, hard particles by providing sufficiently higher stiffness to the elements inside inclusion boundary, than rest of the elements. As explained earlier, crack propagation model is implemented to study the crack propagation behavior in a gear tooth. In order to construct quarter point elements around crack tip near the inclusion without interfering inclusion mesh, smaller crack increments are used. This was implemented to avoid meshing problems and situation of occurring dissimilar materials inside quarter point elements row at the crack tip. Model is used for the prediction of crack path in presence of inclusion. Figure 3 shows typical behavior of a crack in the vicinity of a hard inclusion as predicted by the current model. After the crack initiation as the crack progresses, it can be seen that it gets deflected away from the inclusion which is near the original

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ACCEPTED MANUSCRIPT crack path. Near the inclusion, crack propagates close to the interface for certain distance and then gets deflected into the base material. It was observed that, crack does not propagate to debond the inclusion from the metal or to enter the inclusion. Observed crack path is similar to the crack paths reported in the literature [5] for the similar cases. Number of cycles for the propagation of crack is observed to increase in case of the tightly

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bonded hard inclusion. This is due to change in the stress intensity factor near the inclusion. Figure 4 shows stress intensity factor variation when crack is near the inclusion. As stress intensity factor reduces significantly near the inclusion, which slows down the growth of crack and results in increase in crack propagation cycle count. 4. Effect of inclusion parameters on crack growth The model is used to study effect of various inclusion parameters on crack path and service life of a gear tooth. Gear tooth of module 3.6 mm with pressure angle 200 was selected for the study. Load is assumed to be applied at highest point of single tooth contact (HPSTC) leading to a maximum tensile stress of 648 MPa in the inclusion free gear. Gear material parameters for alloy steel, 42CrMo4 are used, with Young’s modulus E = 2.1 x 1011 Pa, and Poisson’s ratio ν = 0.3. Other parameters, ultimate tensile strength σU = 1100 MPa, fatigue limit σFL = 550 MPa, number of cycles at the knee of the Wohler Curve NFL = 3 x 106 and the fracture toughness KIc = 2620Mpa√mm are taken from the literature [1]. The crack propagation life of the gear without inclusion was determined to be 3.85 x 105 cycles. Studies are carried out by varying following inclusion parameters and results are compared with results of inclusion free gear.

Inclusion size

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Distance of inclusion from the crack path

Location of inclusion along crack path and inclusion size

Figure 5 illustrates various terms used in defining parameters related to the location of inclusion used in the study. Suitably sized inclusion is assumed to be present near the gear surface and crack path so as to affect original crack of the inclusion free gear. Effect of size of the inclusion (diameter di) is studied.

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Distance from the crack path (dc) is varied to study effect of proximity of inclusion to the original crack path. Effect of inclusion size and distance of inclusion from the surface (dg) along the original crack path is studied. 4.1 Effect of inclusion size In order to study the effect of size of inclusion on crack growth, the diameter of a hard circular inclusion is varied. Position of the inclusion is fixed with its center at 160 µm from the gear surface (i.e. dc=160 µm) and 20 µm from the original crack path (i.e. dg=20 µm). Diameter of the inclusion di is varied in such a way that, it creates situations of a) non-interfering inclusion, b) inclusion just touching the crack path, c) inclusion slightly interfering crack path and d) inclusion completely interfering with the crack path. Effect of these cases on the crack path and crack growth is studied. Figure 6 shows variation of crack propagation cycles with increasing size of the inclusion. When the inclusion size is small and it is not interfering with original crack path, effect on crack propagation life is negligible. As the size of inclusion increases and it starts interfering with the crack path and crack growth is observed to be slower in the vicinity of inclusion. Crack path tends to deflect away and amount of deflection increases with the increase in inclusion size. Cycles required for crack propagation increase with the increasing inclusion diameter.

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ACCEPTED MANUSCRIPT 4.2 Effect of distance of inclusion from crack path Change in the distance of inclusion from original crack path affects the crack behavior. If the inclusion is sufficiently away from the original crack path, effect is negligible. In order to study this behavior, inclusion of 100 µm diameter is modeled. Distance from original crack path (dc) is varied, keeping distance from surface (dg) constant. Figure 7 shows variation in the crack propagation life with the change in distance

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from original crack path. Inclusion in the crack path affects crack propagation cycle count significantly. As explained earlier, change in the crack path and drop in stress intensity factor is observed near the inclusion. Amount of deflection in the crack path is higher and crack has to travel more distance near the inclusion. This effect is greater in the cases where inclusion is closer to the crack path. Effect on the crack growth behavior reduces with the increasing distance of inclusion from the crack path. 4.3 Effect of location of inclusion along crack path and size of inclusion Effect of change in the size as well as location of inclusion along the crack path is studied. Distance from the gear tooth surface, dg is varied along crack path by keeping distance from the crack path, dc constant. Simulations are carried out for 3 different inclusion sizes, to create situations of inclusion a) touching the original crack path, b) slightly interfering original crack path and c) completely interfering the original crack path. Please refer Figure 8 for the results. Crack path is always observed getting deflected away near the inclusion along with the increase in crack propagation cycle count. This effect is pronounced in presence of larger inclusion in the path of original crack. For the inclusions placed nearer the gear surface, crack encounters the inclusion earlier. Stress intensity factor is lower and it slows down the crack growth significantly. It results in the significant increase in cycle count. As the inclusions are moved in the interior of the gear, effect on the cycle count is reduced.

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5. Summary LEFM based FE model is used to predict crack path and fatigue life of a gear tooth. The simulation model results were found to be in good agreement with the experimental results taken from the literature. The model is used to study effect of tightly bonded hard inclusion on the crack path and crack propagation life

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for surface originated cracks. Various cases were simulated to study effect of variation in size and location of the inclusion with respect to crack in the inclusion free gear. It was observed that for the hard inclusions near to the original crack paths, crack propagation tends to be slower. The effects are pronounced with size of inclusion and proximity of the inclusion with original crack path. Further investigation can be carried out to study effect of cracks originating from subsurface level in presence of inclusion. Besides crack behavior in presence of multiple inclusions can be studied.

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ACCEPTED MANUSCRIPT References [1] Glodez S., Abersek B., Flasker J. and Kramberger J, Evaluation of service life in regard to bending fatigue in a gear tooth root, Key Engineering Materials, Vols. 251-252 (2003), 297-302. [2] Jajam K., Tippur H., Role of inclusion stiffness and interfacial strength on dynamic matrix crack growth: An experimental study, International Journal of Solids and Structures, 49(2012), 1127-1146.

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[3] Pehan S., Hellen T. K. , Flasker J., Applying numerical methods for determining the service life of gears, Fatigue and Fracture of Engineering Materials and Structures, Vol. 18(1995), No. 9, 971979. [4] Pugnoa N., Ciavarellab M., Cornettia P., Carpinteri A., A generalized Paris’ law for fatigue crack growth, Journal of Mechanics and Physics of Solids, 54(2006), 1333-1349. [5] Lei J., Wang Y.S., Gross D., Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM, Engineering Analyses with Boundary Elements, 2005, Vol 29(8), 802-813. [6] Nisitani H., Saimoto A., Effect of a circular inclusion on crack propagation path in a plate under tension, Key Engineering Materials, 1998, Vol 145-149, 61-70. [7] Lewicki D.G., et.al., Consideration of moving tooth load in gear crack propagation predictions, Journal of Mechanical Design, ASME, March 2001-123, 118-124. [8] Podurug S., Jelaska D., Glodez S., Influence of different load models on gear crack path shapes and fatigue lives, Fatigue & Fracture of Engineering Materials & Structures, 2008, 31, 327-339. [9] Prasannavenkatesan R., et. al., 3D modeling of subsurface fatigue crack nucleation potency of primary inclusions in heat treated and shot peened martensitic gear steels, Int. J. Fatigue, 2009, Vol. 31(7), 1179-1189. [10] Toyoda T., Kanazawa T., Matsumoto K., A study of inclusions causing fatigue cracks in steels for

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ACCEPTED MANUSCRIPT carburized and shot-peened gears, JSAE Review, Vol. 11, 1190, No. 1, 50-54. [11] Jelaska D., 2000, Proc. Int. Conf. Life Assessment and Management for Structural Components, Kiev, 239-246. [12] Cotterell B., and Rice J. R., Slightly curved or kinked cracks, International Journal of Fracture, 1980, Vol. 16, 155–169. [13] Niemann G., Winter H., Maschinenelemente - Band II, Springer Verlag, 1983.

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[14] Murakami Y., Metal Fatigue: Effects of small defects and non-metallic inclusions, Elsevier, first edition, 2002.

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Table 1: Crack propagation life of gear

Sr.

Maximum

Crack propagation life

Crack propagation life

no.

tensile stress

as predicted by current

as from [1]

(Mpa)

model

1

527

8.75 x 105

9.4 x 105

6.9

2

659

3.28 x 105

3.77 x 105

12.9

3

790

1.63 x 105

1.77 x 105

7.9

4

922

8.09 x 104

9.32 x 104

13.1

15

% Error

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ACCEPTED MANUSCRIPT σ σu Wohler Curve Crack initiation

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ΔσFL 1/4

Ni

NFL

NP

N

Figure 1: Service life in fatigue [1]

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Figure 2: Comparison with experimental results [1]

Hard inclusion

Figure 3: Crack path near hard inclusion

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Stres Intensity Factor ( Mpa.mm0.5)

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700 600 500 SIF with inclusion SIF without inclusion

400 300 200 100 0 0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

Crack length (m)

Figure 4: Stress Intensity Factor (SIF) near hard inclusion

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distance from gear surface (dg)

distance from crack path (dc) crack path without inclusion

Figure 5: Size and location of inclusion in gear tooth root

6.5E+05 Crack propagation cycles (Np)

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inclusion, diameter (di)

6.0E+05 5.5E+05 with inclusion without inclusion

5.0E+05 4.5E+05 4.0E+05 3.5E+05 0

50 100 Size of inclusion (µm)

150

200

Figure 6: Effect of inclusion size on crack growth life

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Crack propaagation cycles (Np)

4.90E+05 With Inclusion Without inclusion

4.70E+05 4.50E+05 4.30E+05 4.10E+05 3.90E+05 3.70E+05 3.50E+05 0

100 200 300 400 Distance of inclusion centre from original crack path (mm)

500

Figure 7: Effect of distance from original crack path

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Crack propagation cycles (Np)

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5.0E+05

40 micron

4.8E+05

80 micron 120 micron

4.6E+05

without inclusion

4.4E+05 4.2E+05 4.0E+05

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3.8E+05 3.6E+05 200

300

400 500 600 700 Distance from surface along crack path (µm)

800

Figure 8: Effect of inclusion size and location along crack path

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International Journal for Computational Methods in Engineering Science and Mechanics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ucme20

Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion a

a

a

Vineet Agarwal , Pramod R. Zagade , Danish Khan & B P Gautham

a

a

TRDDC Process Engineering Innovation Labs, Tata Consultancy Services , Pune , India , 411013 Accepted author version posted online: 06 Feb 2014.Published online: 06 Feb 2014.

To cite this article: International Journal for Computational Methods in Engineering Science and Mechanics (2014): Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion, International Journal for Computational Methods in Engineering Science and Mechanics To link to this article: http://dx.doi.org/10.1080/15502287.2014.882434

Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to this version also.

PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

ACCEPTED MANUSCRIPT Fatigue Crack Propagation in Gear Tooth in Presence of Inclusion

Vineet Agarwal, Pramod R. Zagade, Danish Khan and BP Gautham1

TRDDC Process Engineering Innovation Labs, Tata Consultancy Services, Pune, India, 411013

Downloaded by [B P Gautham] at 07:35 06 February 2014

1Corresponding

author email: [email protected]

Abstract A computational model for studying the fatigue crack propagation characteristics in a gear tooth root in the presence of inclusion is presented. A step-by-step crack growth scheme is implemented to predict the crack path using the finite element method and linear elastic fracture mechanics. Paris law approach is used to model fatigue crack propagation. Effect of size and location of hard circular inclusion on growth of a surface initiated crack and service life in a gear tooth is studied.

1

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1. Introduction Service life of a gear is determined from its performance under fatigue loading. Failure of gear tooth can occur due to surface pitting on the flanks or cracks generated in the root region due to repeated loading. Estimation of service life of gear is very important from the perspective of a gear designer. Detailed

Downloaded by [B P Gautham] at 07:35 06 February 2014

understanding of the development and propagation of fatigue crack in the root region under repeated bending load is crucial in design phase. Various flaws such as nonmetallic inclusions are inevitable in the steel gears. Inclusions are known to have significant effect on the fatigue and fracture properties of the material. It may depend on the nature of inclusions, their size, shape and location. It can affect the crack propagation path and service life of the gear. If the inclusions are close to surface it can affect the crack initiation location as well. It is important to understand the role of inclusion in the service life of a gear. Aim of the present work is to build a computational model to study the effect of size and location of inclusions on the fatigue properties of a gear tooth. It may enable determination of tolerable inclusion parameters that can be specified to the supplier of steel and provide guidelines in making decisions on subsequent manufacturing operations in order to achieve improved performance under fatigue loading. There are two phases in service life of gear 1) crack initiation phase and 2) crack propagation phase [1]. Generally a significant part of service life is attributed to crack initiation phase in case of high cycle fatigue. Typically, crack starts from the surface of the component subjected to bending fatigue. If the inclusion is located close to surface, crack can initiate at the inclusion interface in the subsurface level. After development of significantly large micro-crack in the gear material, crack propagation characteristics determine remaining life. The size, location and type of the inclusion, affect the crack path. It is observed in

2

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ACCEPTED MANUSCRIPT the experiments by researchers [2] that crack tends to shift towards soft inclusions if sufficiently sized inclusions are in the vicinity of the crack path. Hard inclusions, which are tightly bonded to the metal matrix such as carbide in steel gears, tend to deflect cracks away from the inclusions. In the present work computational model is developed for the prediction of crack propagation path in steel gear when subjected to high cycle fatigue. Algorithm is implemented in the commercial software Ansys to study effect of tightly bonded hard inclusion on the crack path and service life. The model is used to study fatigue crack behavior

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in a steel gear. Several researchers have studied fatigue crack behavior in steels. In earlier studies numerical methods were applied for predicting service life of the gears [3]. Linear Elastic Fracture Mechanics (LEFM) approach is used to model gear tooth root fatigue behavior and prediction of the service life of the gears [1]. In this approach crack initiation point is determined by performing FE analysis. Point of maximum stress is assumed to be the crack initiation point. Crack is introduced in the structure and further step by step growth is simulated by local re-meshing of the model. Paris law [4] is used for prediction of the service life. Lei et. al., [5] have developed a computational model to simulate the crack paths in the vicinity of different types of inclusions using Boundary Element Method (BEM). Nisitani et. al., [6] simulated the crack propagation in a plate under tensile stress, when circular inclusions are present. In case of gears, generally, load acting at highest point of single tooth contact (HPSTC) is considered for determining the stress range. Effect of moving tooth load is also studied using BEM [7] and LEFM [8]. Numerous experimental investigations have been done to understand fatigue crack behavior in the presence of inclusion. Prasannavenkatesan et.al., [9] have carried out experimental investigation to characterize the driving force for fatigue crack nucleation at subsurface primary inclusions in carburized and shot peened gear steels. Toyoda et. al., [10] studied inclusions causing fatigue crack in surface treated gears.

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ACCEPTED MANUSCRIPT In the present work a computational model is developed using finite element framework in Ansys. Initially 2D stress analysis is used for locating crack initiation point. LEFM is the used to determine stress intensity factor of the loaded structure. Depending on direction for crack progress, as predicted using LEFM, step by step crack growth is modeled. Paris law is used for the prediction of number of cycles for crack increment [1]. For the current study hard circular inclusions, tightly bonded to the metal matrix are considered. The circular shaped inclusion is modeled using appropriately fine mesh and by specifying higher stiffness than

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rest of the structure. For the purpose of performing parametric studies crack initiation location is fixed on the surface based on the initial analysis by assuming a small flaw on the gear surface, at that location. Effect of size and location of inclusion on fatigue crack path and life is studied.

2. Fatigue crack modeling 2.1 Fatigue crack initiation High cycle fatigue failure occurs in two main phases, crack initiation phase and crack propagation phase. Further crack initiation can be divided into two stages: 1) micro-crack nucleation and 2) short crack growth. Crack propagation phase consists of 1) long crack growth and 2) occurrence of final failure. In high cycle fatigue (HCF), crack initiation period accounts for most of the service life. Please refer Figure 1. Total service life N (number of stress cycles) is divided into Ni (number of stress cycles required for the fatigue crack initiation) and Np (number of stress cycles required for a crack to propagate till critical crack length i.e. when the final failure is expected) [1].

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ACCEPTED MANUSCRIPT Fatigue life of gears in HCF can be predicted using Basquin type equation [1], (∆σ)ki Ni = Ci

(1)

where, Δσ is the applied stress range, ki and Ci are the material constants. In Figure 1, if crack initiation curve is correlated with Wholer curve then, crack initiation life can be expressed in terms of ΔσFL and NFL,

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Ni = NFL (

∆σFL ki ) ∆σ

(2)

The exponent ki can be determined as (refer Figure 1),

ki =

log(4NFL ) log(σU ⁄∆σFL )

(3)

where, σU is the ultimate strength. Experimental results have shown good correlation with this equation [11]. The material parameter, ΔσFL required while predicting crack initiation life Ni is determined using appropriate standard test specimen. In the present work more emphasis is given on studying the effect of inclusion on the crack propagation path and life. Crack initiation location is kept same for the parametric studies by enforcing a condition of slightly flawed gear surface to ensure initial stress concentration and crack initiation. Prediction of crack initiation period is calculated using the parameters mentioned in the previous studies [1]. 2.2 Fatigue Crack propagation Linear elastic fracture mechanics is applied for the prediction of crack growth. Fatigue crack growth rate, da/dN, is a function of stress intensity range K = Kmax- Kmin. Paris law [1] can be used for predicting crack

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ACCEPTED MANUSCRIPT growth rate as given below, da = C [K(a)]m dN

(4)

where, C and m are the experimentally determined material parameters. Number of cycles required for propagation of crack, from initial length, a0 to critical crack length, ac can be found using following equation,

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Np

ac

1 da ∫ dN = ∫ C [K(a)]m 0

(5)

a0

In the current work Finite Element Method along with Linear Elastic Fracture Mechanics is used for simulation of the fatigue crack growth [1]. Quarter point elements are used at the crack tip. Stress intensity factors in mode I and II fracture in plane strain conditions are determined using Quarter Point Displacement Technique (QPDT) as,

KI =

2G π . √ . [vd − vb ] (3 − 4𝑣) + 1 2L

(6a)

K II =

2G π . √ . [ud − ub ] (3 − 4𝑣) + 1 2L

(6b)

where, G is the shear modulus of the material, ν is the Poisson’s ratio for the material, ui and vi are displacements of nodes on the element face, L is the finite element length on crack face, The combined Stress Intensity Factor (K) in the mixed mode is then defined as,

K = √(K 2I + K 2II ). (1 − 𝑣 2 )

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(7)

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ACCEPTED MANUSCRIPT Number of cycles required for propagation of crack is predicted using Paris law. Then crack front is advanced by specified length. The criterion based on stress intensity factors is used for determining direction for crack advancement [12]. The crack extension angle θ0 is determined as,

θ0 = cos

−1

3. K 2II + √K 4I + 8K 2I K 2II ( ) K 2I + 9K 2II

(8)

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The gear model is reconstructed with the extended crack and is meshed again for the next analysis. The procedure is repeated until the stress intensity factor reaches the critical value K Ic, when the complete fracture is expected. Above mentioned procedure is implemented in Ansys13 through APDL macro. The model is validated with results from literature. Glodez et. al., [1] have reported comparison of simulation results with experimental results for service life of the gear under various loading conditions. Using the reported gear geometry, material and fatigue parameters, simulation for prediction of crack path and crack propagation life is carried out using current model. Table 1 shows the comparison of predicted crack propagation life in both the cases.

Predicted crack path is compared with simulated crack path in the literature [1] and is found to be in good agreement. Error in the predicted results may be due to different mesh and crack increment lengths used in both the cases. Total service life of the gear as predicted by current model is plotted and compared with experimental results adapted from literature (Please refer Figure 2). The predicted service life contour plot lies well within the experimental results scatter.

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ACCEPTED MANUSCRIPT 3. Crack propagation in presence of inclusions In steel gears, various types of non-metallic inclusions such as carbides, oxides or sulfides may be present. These inclusions can be very hard particles, difficult to break during subsequent processes. It may be tightly bonded to metal matrix. During certain manufacturing processes inclusions may get partially or completely debonded from the metal matrix. Typical sizes of non-metallic inclusions that may have effect on

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the fatigue life need to be determined. Murakami [14] have reported practical observations of metal fatigue tests in the presence of inclusions. Shapes of inclusions can affect the fatigue life significantly. Inclusions generally found in the metals can have arbitrary shapes. It can be approximated to shapes such as circular or elliptical with varying dimensions for the ease of analysis. Inclusions nearer to surface have greater impact on the fatigue life of gears. It can shift the crack initiation location to the subsurface level. Larger inclusions in original crack path may have larger impact on the crack propagation life. Hard circular inclusions bonded tightly to the gear metal matrix are considered for current simulation studies. Inclusions are modeled as rigid, hard particles by providing sufficiently higher stiffness to the elements inside inclusion boundary, than rest of the elements. As explained earlier, crack propagation model is implemented to study the crack propagation behavior in a gear tooth. In order to construct quarter point elements around crack tip near the inclusion without interfering inclusion mesh, smaller crack increments are used. This was implemented to avoid meshing problems and situation of occurring dissimilar materials inside quarter point elements row at the crack tip. Model is used for the prediction of crack path in presence of inclusion. Figure 3 shows typical behavior of a crack in the vicinity of a hard inclusion as predicted by the current model. After the crack initiation as the crack progresses, it can be seen that it gets deflected away from the inclusion which is near the original

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ACCEPTED MANUSCRIPT crack path. Near the inclusion, crack propagates close to the interface for certain distance and then gets deflected into the base material. It was observed that, crack does not propagate to debond the inclusion from the metal or to enter the inclusion. Observed crack path is similar to the crack paths reported in the literature [5] for the similar cases. Number of cycles for the propagation of crack is observed to increase in case of the tightly

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bonded hard inclusion. This is due to change in the stress intensity factor near the inclusion. Figure 4 shows stress intensity factor variation when crack is near the inclusion. As stress intensity factor reduces significantly near the inclusion, which slows down the growth of crack and results in increase in crack propagation cycle count. 4. Effect of inclusion parameters on crack growth The model is used to study effect of various inclusion parameters on crack path and service life of a gear tooth. Gear tooth of module 3.6 mm with pressure angle 200 was selected for the study. Load is assumed to be applied at highest point of single tooth contact (HPSTC) leading to a maximum tensile stress of 648 MPa in the inclusion free gear. Gear material parameters for alloy steel, 42CrMo4 are used, with Young’s modulus E = 2.1 x 1011 Pa, and Poisson’s ratio ν = 0.3. Other parameters, ultimate tensile strength σU = 1100 MPa, fatigue limit σFL = 550 MPa, number of cycles at the knee of the Wohler Curve NFL = 3 x 106 and the fracture toughness KIc = 2620Mpa√mm are taken from the literature [1]. The crack propagation life of the gear without inclusion was determined to be 3.85 x 105 cycles. Studies are carried out by varying following inclusion parameters and results are compared with results of inclusion free gear.

Inclusion size

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Distance of inclusion from the crack path

Location of inclusion along crack path and inclusion size

Figure 5 illustrates various terms used in defining parameters related to the location of inclusion used in the study. Suitably sized inclusion is assumed to be present near the gear surface and crack path so as to affect original crack of the inclusion free gear. Effect of size of the inclusion (diameter di) is studied.

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Distance from the crack path (dc) is varied to study effect of proximity of inclusion to the original crack path. Effect of inclusion size and distance of inclusion from the surface (dg) along the original crack path is studied. 4.1 Effect of inclusion size In order to study the effect of size of inclusion on crack growth, the diameter of a hard circular inclusion is varied. Position of the inclusion is fixed with its center at 160 µm from the gear surface (i.e. dc=160 µm) and 20 µm from the original crack path (i.e. dg=20 µm). Diameter of the inclusion di is varied in such a way that, it creates situations of a) non-interfering inclusion, b) inclusion just touching the crack path, c) inclusion slightly interfering crack path and d) inclusion completely interfering with the crack path. Effect of these cases on the crack path and crack growth is studied. Figure 6 shows variation of crack propagation cycles with increasing size of the inclusion. When the inclusion size is small and it is not interfering with original crack path, effect on crack propagation life is negligible. As the size of inclusion increases and it starts interfering with the crack path and crack growth is observed to be slower in the vicinity of inclusion. Crack path tends to deflect away and amount of deflection increases with the increase in inclusion size. Cycles required for crack propagation increase with the increasing inclusion diameter.

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ACCEPTED MANUSCRIPT 4.2 Effect of distance of inclusion from crack path Change in the distance of inclusion from original crack path affects the crack behavior. If the inclusion is sufficiently away from the original crack path, effect is negligible. In order to study this behavior, inclusion of 100 µm diameter is modeled. Distance from original crack path (dc) is varied, keeping distance from surface (dg) constant. Figure 7 shows variation in the crack propagation life with the change in distance

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from original crack path. Inclusion in the crack path affects crack propagation cycle count significantly. As explained earlier, change in the crack path and drop in stress intensity factor is observed near the inclusion. Amount of deflection in the crack path is higher and crack has to travel more distance near the inclusion. This effect is greater in the cases where inclusion is closer to the crack path. Effect on the crack growth behavior reduces with the increasing distance of inclusion from the crack path. 4.3 Effect of location of inclusion along crack path and size of inclusion Effect of change in the size as well as location of inclusion along the crack path is studied. Distance from the gear tooth surface, dg is varied along crack path by keeping distance from the crack path, dc constant. Simulations are carried out for 3 different inclusion sizes, to create situations of inclusion a) touching the original crack path, b) slightly interfering original crack path and c) completely interfering the original crack path. Please refer Figure 8 for the results. Crack path is always observed getting deflected away near the inclusion along with the increase in crack propagation cycle count. This effect is pronounced in presence of larger inclusion in the path of original crack. For the inclusions placed nearer the gear surface, crack encounters the inclusion earlier. Stress intensity factor is lower and it slows down the crack growth significantly. It results in the significant increase in cycle count. As the inclusions are moved in the interior of the gear, effect on the cycle count is reduced.

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5. Summary LEFM based FE model is used to predict crack path and fatigue life of a gear tooth. The simulation model results were found to be in good agreement with the experimental results taken from the literature. The model is used to study effect of tightly bonded hard inclusion on the crack path and crack propagation life

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for surface originated cracks. Various cases were simulated to study effect of variation in size and location of the inclusion with respect to crack in the inclusion free gear. It was observed that for the hard inclusions near to the original crack paths, crack propagation tends to be slower. The effects are pronounced with size of inclusion and proximity of the inclusion with original crack path. Further investigation can be carried out to study effect of cracks originating from subsurface level in presence of inclusion. Besides crack behavior in presence of multiple inclusions can be studied.

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ACCEPTED MANUSCRIPT References [1] Glodez S., Abersek B., Flasker J. and Kramberger J, Evaluation of service life in regard to bending fatigue in a gear tooth root, Key Engineering Materials, Vols. 251-252 (2003), 297-302. [2] Jajam K., Tippur H., Role of inclusion stiffness and interfacial strength on dynamic matrix crack growth: An experimental study, International Journal of Solids and Structures, 49(2012), 1127-1146.

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[3] Pehan S., Hellen T. K. , Flasker J., Applying numerical methods for determining the service life of gears, Fatigue and Fracture of Engineering Materials and Structures, Vol. 18(1995), No. 9, 971979. [4] Pugnoa N., Ciavarellab M., Cornettia P., Carpinteri A., A generalized Paris’ law for fatigue crack growth, Journal of Mechanics and Physics of Solids, 54(2006), 1333-1349. [5] Lei J., Wang Y.S., Gross D., Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM, Engineering Analyses with Boundary Elements, 2005, Vol 29(8), 802-813. [6] Nisitani H., Saimoto A., Effect of a circular inclusion on crack propagation path in a plate under tension, Key Engineering Materials, 1998, Vol 145-149, 61-70. [7] Lewicki D.G., et.al., Consideration of moving tooth load in gear crack propagation predictions, Journal of Mechanical Design, ASME, March 2001-123, 118-124. [8] Podurug S., Jelaska D., Glodez S., Influence of different load models on gear crack path shapes and fatigue lives, Fatigue & Fracture of Engineering Materials & Structures, 2008, 31, 327-339. [9] Prasannavenkatesan R., et. al., 3D modeling of subsurface fatigue crack nucleation potency of primary inclusions in heat treated and shot peened martensitic gear steels, Int. J. Fatigue, 2009, Vol. 31(7), 1179-1189. [10] Toyoda T., Kanazawa T., Matsumoto K., A study of inclusions causing fatigue cracks in steels for

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ACCEPTED MANUSCRIPT carburized and shot-peened gears, JSAE Review, Vol. 11, 1190, No. 1, 50-54. [11] Jelaska D., 2000, Proc. Int. Conf. Life Assessment and Management for Structural Components, Kiev, 239-246. [12] Cotterell B., and Rice J. R., Slightly curved or kinked cracks, International Journal of Fracture, 1980, Vol. 16, 155–169. [13] Niemann G., Winter H., Maschinenelemente - Band II, Springer Verlag, 1983.

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[14] Murakami Y., Metal Fatigue: Effects of small defects and non-metallic inclusions, Elsevier, first edition, 2002.

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Table 1: Crack propagation life of gear

Sr.

Maximum

Crack propagation life

Crack propagation life

no.

tensile stress

as predicted by current

as from [1]

(Mpa)

model

1

527

8.75 x 105

9.4 x 105

6.9

2

659

3.28 x 105

3.77 x 105

12.9

3

790

1.63 x 105

1.77 x 105

7.9

4

922

8.09 x 104

9.32 x 104

13.1

15

% Error

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ACCEPTED MANUSCRIPT σ σu Wohler Curve Crack initiation

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ΔσFL 1/4

Ni

NFL

NP

N

Figure 1: Service life in fatigue [1]

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Figure 2: Comparison with experimental results [1]

Hard inclusion

Figure 3: Crack path near hard inclusion

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Stres Intensity Factor ( Mpa.mm0.5)

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700 600 500 SIF with inclusion SIF without inclusion

400 300 200 100 0 0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

Crack length (m)

Figure 4: Stress Intensity Factor (SIF) near hard inclusion

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distance from gear surface (dg)

distance from crack path (dc) crack path without inclusion

Figure 5: Size and location of inclusion in gear tooth root

6.5E+05 Crack propagation cycles (Np)

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inclusion, diameter (di)

6.0E+05 5.5E+05 with inclusion without inclusion

5.0E+05 4.5E+05 4.0E+05 3.5E+05 0

50 100 Size of inclusion (µm)

150

200

Figure 6: Effect of inclusion size on crack growth life

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Crack propaagation cycles (Np)

4.90E+05 With Inclusion Without inclusion

4.70E+05 4.50E+05 4.30E+05 4.10E+05 3.90E+05 3.70E+05 3.50E+05 0

100 200 300 400 Distance of inclusion centre from original crack path (mm)

500

Figure 7: Effect of distance from original crack path

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Crack propagation cycles (Np)

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5.0E+05

40 micron

4.8E+05

80 micron 120 micron

4.6E+05

without inclusion

4.4E+05 4.2E+05 4.0E+05

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3.8E+05 3.6E+05 200

300

400 500 600 700 Distance from surface along crack path (µm)

800

Figure 8: Effect of inclusion size and location along crack path

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