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Numerical calculation of bending fatigue life of thin-rim spur gears
Engineering Fracture Mechanics 71 (2004) 647–656 www.elsevier.com/locate/engfracmech
Numerical calculation of bending fatigue life of thin-rim spur gears raml, I. Potrc, J. Flasker J. Kramberger *, M. S Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Received 30 October 2002; received in revised form 4 November 2002; accepted 11 November 2002
Abstract Mechanical elements subjected to cyclic loading have to be designed against fatigue. The aim of this paper is to examine the bending fatigue life of thin-rim spur gears of truck gearboxes. The gear service life is divided into the initiation phase of the damage accumulation and the crack growth, respectively. The analysis of thin-rim gear fatigue life has been performed using the finite element method and the boundary element method. The continuum mechanics based approach is used for the prediction of the fatigue process initiation phase, where the basic fatigue parameters of the materials are taken into account. The remaining life of gear with an initial crack is evaluated using the linear-elastic fracture mechanics. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Mechanical elements; Numerical modelling; Thin-rim gears; Bending fatigue; Crack initiation; Fatigue crack growth; Life prediction
1. Introduction The fatigue process of mechanical elements is a material characteristic and depends upon cyclic plasticity, local deformation, dislocation motion, formation of micro- and macro-cracks and their propagation, etc. Gears fail by contact fatigue failure (pitting) as by bending fatigue failure (tooth breakage). Bending fatigue is of a great importance in engineering applications of gears, where specific variable loads appear. This operational loads result in stresses, which can be equal to or lower than the yield stress of a gear material. However, bending fatigue can be generally divided into two main phases: (i) initiation of microcracks and (ii) propagation of cracks. The most common methods of gear design are based on conventional standard procedures like DIN, AGMA and ISO. Although the standards for calculation are the most up-to date methods, they do not give detailed information of the bending fatigue life of gears, especially thin-rim gears. In gear strength calculations, the gear tooth can be considered as a cantilever subjected to a pulsating force, where the applied
*
Corresponding author. Tel.: +386-2-2207721; fax: +386-2-2207729. E-mail address: [email protected] (J. Kramberger).
0013-7944/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00024-9
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Nomenclature b c do dh h m n n0 t w C Cload Csize Csurf E F K0 KI DK Nf0 R T e0f De Dep m r1 r0f Dr DrFL DrFLr u x BEM DBEM FEM HCF LCF LEFM SIF P
exponent of material strength fatigue ductility exponent outside diameter hub diameter whole depth modulus exponent of Paris equation material hardening exponent rim thickness tooth and rim width parameter of Paris equation loading factor size factor surface factor YoungÕs elastic modulus applied normal force at the highest point of a single tooth contact material strength coefficient mode I stress intensity factor stress intensity factor range number of loading cycles for crack initiation stress (loading) ratio time fatigue ductility coefficient deformation range plastic deformation range Poisson ratio number maximum principal stress fatigue strength coefficient stress range fatigue limit of polished laboratory specimen real fatigue limit tangential angle at tooth root addendum modification factor boundary element method dual boundary element method finite element method high cycle fatigue low cycle fatigue linear elastic fracture mechanics stress intensity factor transition point
force varies during meshing of gears. Some researchers have investigated the effects of the rim thickness on stresses [3,8,9], but only a few have performed a fatigue crack growth analysis for thin-rim gears [5,6].
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Prediction of bending fatigue life of thin-rim gears is very important from a practical point of view. Regarding the accurate prediction of crack initiation and propagation, an advanced model needs to be developed. The purpose of present study is to elaborate a new model for prediction of bending fatigue life of thinrim spur gears of the truck gearboxes, which consists of fatigue crack initiation and fatigue crack propagation phase, respectively. The proposed model of crack initiation (phase I) is based on the continuum mechanics assumptions, considering the adequate fatigue parameters of the materials. Stress–strain calculations of thin-rim gear are performed in the framework of the finite element method (FEM), and these results are further used for fatigue analysis [7,10]. Concerning the numerical model of the selected mechanical component, the material is assumed to be homogeneous and isotropic, i.e. without imperfections or damages at the beginning of the fatigue analysis. Finally, permanent damage due to accumulation of plastic deformation in the material under repeated loads is taken into account in the fatigue initiation phase of proposed model. Once a fatigue crack has started, the crack propagation phase should be studied (phase II). This includes determining of crack propagation trajectory (which can be through the tooth root or through the gear rim) and remaining service life to final failure. This kind of failure mode is essential for the design of thin-rim gears. Gear tooth crack propagation is simulated using computer program BEASY [2], which is based on the boundary element method (BEM).
2. Fatigue crack initiation model A model for crack initiation under bending fatigue in a thin-rim gear is first presented. The finite element model for calculating stress–strain field at tooth root for the fatigue initiation phase is presented in Fig. 1. Several models of gears with various values of rim thickness have been modelled by incorporating different slots in the model. The loaded gear tooth is subjected to a normal force F ¼ 1737 N/mm applied at the highest point of single tooth contact, as is shown in Fig. 1b. Plain strain conditions are considered. The general procedure in the proposed bending fatigue model is to first compute stress concentration in the critical section of the gear tooth root. Five cases are considered with the same basic tooth geometry and different rim thickness t (solid, 2:5m, 2m, 1:5m and 1m, where m denotes module as standard measure of spur-gear size). Although the crack initiation due to maximum stress concentrations can appear on both
Fig. 1. FEM model of gear: (a) mesh, (b) mesh refinement at loaded tooth root.
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Fig. 2. Maximum principal stress distribution in the tooth fillet.
sides of tooth root, the most critical for further crack propagation is the tensile side of tooth root. The stress distribution in the tooth fillet for tensile side is given in Fig. 2, where the maximum principal stresses r1 (tensile stresses) are plotted. Position in the root area is described by angle u, which is defined as the angle between the symmetry line of the tooth and the tangent to the fillet curve, as shown in Fig. 2. The magnitude of the stresses decreases as the rim thickness decreases and then increases for the thinnest thickness (1m). It is observed that location of maximum stresses moves to the root area as the rim thickness decreases (see Fig. 2 and Table 1). Material model is assumed as linear elastic, while cycle stress–strain curve is defined using following relation: n0
Dr ¼ K 0 ðDep Þ ;
ð1Þ
where n0 is the cyclic strain-hardening exponent and K 0 is the cyclic strength coefficient (see Table 2). An assumption for fatigue analysis is a loading cycle (see Fig. 3) of gear meshing, presumed as pulsating (R ¼ 0). Methods for fatigue analysis are most frequently based on Coffin–Manson relation between deformations, stresses and the number of loading cycles [11,12]. A frequently used procedure for calculating number of cycles needed for fatigue crack to occur is based on strain–life relationship (e–N ), and includes material Table 1 r1 stresses in material point Rim thickness t
Max. principal stresses in the material point at tooth root r1 [MPa]
Location of max. stresses u (°)
1m 1:5m 2m 2:5m Solid
918 813 835 900 1120
65 27 28 28 31
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Table 2 Basic geometry and material parameters of the thin-rim gear Profile Number of teeth on gear Normal pressure angle Module Whole depth Addendum modification factor Outside diameter Tooth and rim width Hub diameter Rim thickness
Involute 39 24° m ¼ 4:5 mm h ¼ 10:41 mm x ¼ 0:0593 do ¼ 184:7 mm w ¼ 28 mm dh ¼ 100 mm t ¼ ð1; 1:5; 2; 2:5Þm
Material Modulus of elasticity PoissonÕs ratio Fatigue strength coefficient Fatigue ductility coefficient Exponent of strength Fatigue ductility exponent Hardening exponent Strength coefficient Surface factor
42CrMo4 E ¼ 2:06 105 MPa m ¼ 0:3 r0f ¼ 1820 MPa e0f ¼ 0:65 b ¼ 0:08 c ¼ 0:76 n0 ¼ 0:14 K 0 ¼ 2259 MPa Csurf ¼ 0:5–1:0
Fig. 3. The loading cycle for fatigue analysis.
parameters––fatigue strength coefficient (r0f ) and fatigue ductility coefficient (e0f ) related to fatigue process, exponent of strength (b) and fatigue ductility exponent (c), etc. [7,11]. The usual way of presenting fatigue test results is to plot elastic and plastic strain range against number of loading cycles. Material curve can be fully characterised by previously described material parameters r0f , b, e0f , c (Fig. 4) as is shown in Eq. (2). This curve is divided into an elastic component and a plastic component, respectively. The transition point P (Fig. 4) defines the difference between high cycle fatigue (HCF) versus low cycle fatigue (LCF). This type of behaviour is known as the Coffin–Manson relation, described by the following equation: De ra Dep r0f ¼ þ ¼ ð2Nf0 Þb þ e0f ð2Nf Þc : 2 E 2 E
ð2Þ
It is a fact that the e–N method is not ideal to analyse fatigue damage initiation on the micro-structural level, as the micro-crack initiation in crystal grains and dislocation theory are not taken into account. It has been established in work of Suresh [11] and Bhattacharya et al. [13] that fatigue damage initiation is represented by the transition of a certain number of loading cycles when the first fatigue damage occurs, on the basis of the assumed initial homogeneous state of material. Thus, the e–N procedure represents a very useful method to determine, where fatigue damage initiation, in time domain, is most probable [11,12]. Basic parameters influencing fatigue life of thin-rim gears are: particularly machine component size (Csize ), the type of loading (Cload ) and effect of surface finish and treatment (Csur ), which is taken into account in bending fatigue analysis of thin-rim gears. Practically, all kinds of bending fatigue failures start at the external surface of the component. It is well known, that fatigue properties are very sensitive to surface conditions. The surface finish correction factor Csur is presented in Fig. 5, in dependence on surface roughness Ra and tensile strength of the material Rm [7]. Using this assumption the real service life of gears
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Fig. 4. Strain–life (e–N ) method for fatigue crack initiation.
Fig. 5. Surface finish correction factor Csur .
may be reduced in regard to the appropriate value of Csur , which is in present study considered through the following equation: DrFLr ¼ DrFL Csur ;
ð3Þ
where DrFLr is the real fatigue limit and DrFL is the fatigue limit of polished laboratory specimen. Table 3 shows the results of fatigue analysis as number of cycles for fatigue crack initiation at tooth root of thin-rim gear. Fatigue analyses have been carried out for the most loaded material point at the tooth root, which was the position of the greatest tensile stress (see Fig. 2). The thin-rim gear design (different values of the rim thickness), as well as the influence of surface finish and treatment, has the most influence upon the fatigue crack initiation at tooth root. Cases of design and
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Table 3 Number of cycles for fatigue crack initiation Rim thickness t [mm]
1m 1:5m 2m 2:5m Solid
Ni ––Number of loading cycles for fatigue crack initiation Surface finish and surface treatment Poor machined and no treatment
Averaged machined and no treatment
Good machined and no treatment
Polished and no treatment
Good machined and shot peened
Polished and shot peened
Good machined and nitrided
Polished and nitrided
1.231 105 5.809 105 2.118 105 1.231 105 2.079 104
2.940 105 1.549 106 5.284 105 2.940 105 4.068 104
7.429 105 4.334 106 1.388 106 7.429 105 8.605 104
1.866 107 >108 3.950 107 1.866 107 1.343 106
8.301 106 >108 1.701 107 8.301 106 6.681 105
8.441 107 >108 >108 8.441 107 6.195 106
>108 >108 >108 >108 9.276 107
>108 >108 >108 >108 >108
surface treatment of the truck gears are most common. The suitable choice for a given case can be established based on the present analyses (see Table 3).
3. Fatigue crack growth In order to perform fatigue crack growth analysis, a BEM model has been created. The latest development known as the dual boundary element method (DBEM) crack modelling strategy, which is incorporated in the BEASY analysis programme [2], has been used to perform the crack growth analysis. For crack growth simulation, an incremental type analysis is used, where knowledge of both the direction and size of the crack increment extension are necessary. The incremental direction at the crack tip for next extension is determined by linear fracture mechanics criteria, involving stress intensity factors (SIF) as the prime parameters [1]. The boundary element model for crack growth analysis is shown in Fig. 6. The initial crack has been placed perpendicularly to the surface at the previously determined location for the initiation in the tensile area of the tooth root (see Section 2). The length of the initial crack is set equal to 0.2 mm. This value is estimated as a crack size, which identifies the transition from nucleation to propagation phase. For
Fig. 6. Boundary element model of gear with initial crack in the tooth fillet.
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Fig. 7. Mode I stress intensity factors versus crack length.
incremental simulation, the automatic crack propagation option of BEASY [2] was used. A uniform crack increment size of 0.2 mm is prescribed. The J -integral method to calculate crack tip stress intensity factors and maximum principal stress theory as a crack growth direction criterion are used. Fig. 7 shows the mode I stress intensity factors as a function of the crack length. The mode I stress intensity factors gradually increase with increasing crack length. The values of mode I stress intensity factors have been determined to be significantly larger than those of mode II stress intensity factors. Fig. 8 shows the calculated crack propagation paths for different values of the rim thickness. The results suggest how the gear will behave once a crack has initiated in the fillet region. For cases solid, 2:5m, 2m and 1:5m, crack propagates only through the tooth. For rim thickness 1m, the crack trajectory is through the rim. This kind of failure mode (tooth breakage or rim breakage) can be an important design information.
Fig. 8. Simulated crack propagation paths for various rim thickness (2:5m, 2m, 1:5m and 1m).
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Table 4 Load spectrum of first gears of truck gearbox [4] Load case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Relative loading Repetition in 1000 km
1.07 1
1.00 13
0.92 202
0.84 487
0.76 160
0.69 65
0.62 81
0.54 62
0.47 86
0.39 108
0.31 114
0.23 105
0.15 94
0.08 69
Fig. 9. Predicted crack propagation life of gear tooth.
In the framework of the LEFM, the propagation rate of cracks can be described by the ParisÕ equation da n ¼ CðDKÞ ; dN
ð4Þ
where DK is the stress intensity factor range, C and n are experimentally determined parameters. The cyclic load used for determination of remaining life has been based on the measurements on loaded vehicles [4] (Table 4). The value of loading and the corresponding number of its repetitions on the gear for characteristic driving conditions are given for an interval of 1000 km. The relative load is normalised by the load F ¼ 1737 N/mm, which has been used for the loading cycle at fatigue crack initiation analysis (see Fig. 3). Two couples of material constants have been used in ParisÕ law (C ¼ 1:575 1018 mm/cycle/ p different n (MPap mm) , n ¼ 4:689 for carburised layer with thickness of 0.6 mm, and C ¼ 4:202 1017 mm/cycle/ (MPa mm)n and n ¼ 4:144 for core, respectively [4]). Previously calculated stress intensity factor ranges, shown in Fig. 7, have been used for prediction of remaining life of the cracked tooth. The predicted number of crack propagation cycles is given in Fig. 9. It is evident that the largest life occurs in the case of rim thickness equal 1:5m since it has the lowest stress intensity factors.
4. Conclusions This paper presents a numerical model of bending fatigue life for thin-rim gears of truck gearboxes. The results include damage initiation as well as crack propagation in the most critical region for bending
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fatigue. On the basis of presented analyses, the number of loading cycles required for the damage initiation and number of loading cycles for fatigue crack propagation have been predicted. Critical region for crack initiation has been determined through a stress–strain analysis in the framework of the FEM. The fatigue initiation has been based on maximum principal stress r1 distribution. The tensile side of the loaded tooth, which is further important for crack growth phase, is focused. It is observed that the critical location for crack initiation moves toward the tooth root area with decreasing rim thickness. Providing e–N fatigue initiation analysis for different kinds of surface finish and treatment and different types of thin-rim gear design, the corresponding number of loading cycles for initial fatigue damage is determined. Then, a numerical analysis to investigate the effect of rim thickness on gear tooth crack propagation path and remaining life has been performed. The results obtained show the importance of rim thickness on the kind of expected failure mode. For a rim thickness greater than 1:5m, the predicted crack would propagate rather through the tooth and not through the rim of the gear. This kind of damage is less critical for the presented application of the truck gearbox. The proposed model enables to determine the whole service life under given loading cycles, adequate fatigue material parameters and predicted crack growth directions. The estimated bending fatigue life of the gear can deviate from real service life because some effects like residual stresses and possible causes of retardation of crack propagation (crack closure) have not been taken into account. In order to validate the present numerical results, an adequate experimental work should be performed. More reliable prediction could be expected by evaluating effects of gear geometry (thickness and position of web, etc.), and this requires further research work.
References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13]
Aliabadi MH, Rooke DP. Numerical fracture mechanics. Dordrecht: Kluwer Academic Publishers; 1992. BEASY User Guide, vol. 2, Computational mechanics, 1998. Chong TH, Suzuki T, Aida T, Kubo A, Fujio H. Tooth fillet stresses of gear with thin rim. Bull JSME 1983;26(220):1799–806. Flasker J, Pehan S. Crack propagation in tooth root with variable loading. Commun Numer Methods Engng 1993;9(2):103–10. raml M, Potrc I, Flasker J. Numerical calculation of bending fatigue life of thin rim spur gears. In: 2nd Kramberger J, S International Conference on Fracture and Damage Mechanics, Milano, Italy. Advances in fracture and damage mechanics II. 2001. p. 113–8. Lewicki DG, Ballarini R. Rim thickness effects on gear crack propagation life. Int J Fract 1997;87:59–86. MSC/Corporation: MSC/FATIGUE Quick Start Guide Version 8, The MacNeal-Schwendler Corporation, Los Angeles, 1999. Oda S, Nagamura K, Aoki K. Stress analysis of thin rim spur gears. Bull JSME 1981;24(193):1273–80. Opalic M, Kolacek Z. Tooth root stress analysis of thin rim gear, DMMI Õ99. J Mech Engng 1999:173–9 (Special edition). raml M. Modelling of contact fatigue of mechanical elements. Dissertation, Abstract in English, Maribor, Slovenia, 2001. S Suresh S. Fatigue of materials. 2nd ed. Cambridge: Cambridge University Press; 1998. Fatemi A, Yang L. Cumulative fatigue damage and life prediction theories: a survey of the state of the art for homogeneous materials. Int J Fatigue 1998;20(1):9–34. Bhattacharya B, Ellingwood B. Continuum damage mechanics analysis of fatigue crack initiation. Int J Fatigue 1998;20(9):631–9.
Numerical calculation of bending fatigue life of thin-rim spur gears raml, I. Potrc, J. Flasker J. Kramberger *, M. S Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia Received 30 October 2002; received in revised form 4 November 2002; accepted 11 November 2002
Abstract Mechanical elements subjected to cyclic loading have to be designed against fatigue. The aim of this paper is to examine the bending fatigue life of thin-rim spur gears of truck gearboxes. The gear service life is divided into the initiation phase of the damage accumulation and the crack growth, respectively. The analysis of thin-rim gear fatigue life has been performed using the finite element method and the boundary element method. The continuum mechanics based approach is used for the prediction of the fatigue process initiation phase, where the basic fatigue parameters of the materials are taken into account. The remaining life of gear with an initial crack is evaluated using the linear-elastic fracture mechanics. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Mechanical elements; Numerical modelling; Thin-rim gears; Bending fatigue; Crack initiation; Fatigue crack growth; Life prediction
1. Introduction The fatigue process of mechanical elements is a material characteristic and depends upon cyclic plasticity, local deformation, dislocation motion, formation of micro- and macro-cracks and their propagation, etc. Gears fail by contact fatigue failure (pitting) as by bending fatigue failure (tooth breakage). Bending fatigue is of a great importance in engineering applications of gears, where specific variable loads appear. This operational loads result in stresses, which can be equal to or lower than the yield stress of a gear material. However, bending fatigue can be generally divided into two main phases: (i) initiation of microcracks and (ii) propagation of cracks. The most common methods of gear design are based on conventional standard procedures like DIN, AGMA and ISO. Although the standards for calculation are the most up-to date methods, they do not give detailed information of the bending fatigue life of gears, especially thin-rim gears. In gear strength calculations, the gear tooth can be considered as a cantilever subjected to a pulsating force, where the applied
*
Corresponding author. Tel.: +386-2-2207721; fax: +386-2-2207729. E-mail address: [email protected] (J. Kramberger).
0013-7944/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00024-9
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Nomenclature b c do dh h m n n0 t w C Cload Csize Csurf E F K0 KI DK Nf0 R T e0f De Dep m r1 r0f Dr DrFL DrFLr u x BEM DBEM FEM HCF LCF LEFM SIF P
exponent of material strength fatigue ductility exponent outside diameter hub diameter whole depth modulus exponent of Paris equation material hardening exponent rim thickness tooth and rim width parameter of Paris equation loading factor size factor surface factor YoungÕs elastic modulus applied normal force at the highest point of a single tooth contact material strength coefficient mode I stress intensity factor stress intensity factor range number of loading cycles for crack initiation stress (loading) ratio time fatigue ductility coefficient deformation range plastic deformation range Poisson ratio number maximum principal stress fatigue strength coefficient stress range fatigue limit of polished laboratory specimen real fatigue limit tangential angle at tooth root addendum modification factor boundary element method dual boundary element method finite element method high cycle fatigue low cycle fatigue linear elastic fracture mechanics stress intensity factor transition point
force varies during meshing of gears. Some researchers have investigated the effects of the rim thickness on stresses [3,8,9], but only a few have performed a fatigue crack growth analysis for thin-rim gears [5,6].
J. Kramberger et al. / Engineering Fracture Mechanics 71 (2004) 647–656
649
Prediction of bending fatigue life of thin-rim gears is very important from a practical point of view. Regarding the accurate prediction of crack initiation and propagation, an advanced model needs to be developed. The purpose of present study is to elaborate a new model for prediction of bending fatigue life of thinrim spur gears of the truck gearboxes, which consists of fatigue crack initiation and fatigue crack propagation phase, respectively. The proposed model of crack initiation (phase I) is based on the continuum mechanics assumptions, considering the adequate fatigue parameters of the materials. Stress–strain calculations of thin-rim gear are performed in the framework of the finite element method (FEM), and these results are further used for fatigue analysis [7,10]. Concerning the numerical model of the selected mechanical component, the material is assumed to be homogeneous and isotropic, i.e. without imperfections or damages at the beginning of the fatigue analysis. Finally, permanent damage due to accumulation of plastic deformation in the material under repeated loads is taken into account in the fatigue initiation phase of proposed model. Once a fatigue crack has started, the crack propagation phase should be studied (phase II). This includes determining of crack propagation trajectory (which can be through the tooth root or through the gear rim) and remaining service life to final failure. This kind of failure mode is essential for the design of thin-rim gears. Gear tooth crack propagation is simulated using computer program BEASY [2], which is based on the boundary element method (BEM).
2. Fatigue crack initiation model A model for crack initiation under bending fatigue in a thin-rim gear is first presented. The finite element model for calculating stress–strain field at tooth root for the fatigue initiation phase is presented in Fig. 1. Several models of gears with various values of rim thickness have been modelled by incorporating different slots in the model. The loaded gear tooth is subjected to a normal force F ¼ 1737 N/mm applied at the highest point of single tooth contact, as is shown in Fig. 1b. Plain strain conditions are considered. The general procedure in the proposed bending fatigue model is to first compute stress concentration in the critical section of the gear tooth root. Five cases are considered with the same basic tooth geometry and different rim thickness t (solid, 2:5m, 2m, 1:5m and 1m, where m denotes module as standard measure of spur-gear size). Although the crack initiation due to maximum stress concentrations can appear on both
Fig. 1. FEM model of gear: (a) mesh, (b) mesh refinement at loaded tooth root.
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Fig. 2. Maximum principal stress distribution in the tooth fillet.
sides of tooth root, the most critical for further crack propagation is the tensile side of tooth root. The stress distribution in the tooth fillet for tensile side is given in Fig. 2, where the maximum principal stresses r1 (tensile stresses) are plotted. Position in the root area is described by angle u, which is defined as the angle between the symmetry line of the tooth and the tangent to the fillet curve, as shown in Fig. 2. The magnitude of the stresses decreases as the rim thickness decreases and then increases for the thinnest thickness (1m). It is observed that location of maximum stresses moves to the root area as the rim thickness decreases (see Fig. 2 and Table 1). Material model is assumed as linear elastic, while cycle stress–strain curve is defined using following relation: n0
Dr ¼ K 0 ðDep Þ ;
ð1Þ
where n0 is the cyclic strain-hardening exponent and K 0 is the cyclic strength coefficient (see Table 2). An assumption for fatigue analysis is a loading cycle (see Fig. 3) of gear meshing, presumed as pulsating (R ¼ 0). Methods for fatigue analysis are most frequently based on Coffin–Manson relation between deformations, stresses and the number of loading cycles [11,12]. A frequently used procedure for calculating number of cycles needed for fatigue crack to occur is based on strain–life relationship (e–N ), and includes material Table 1 r1 stresses in material point Rim thickness t
Max. principal stresses in the material point at tooth root r1 [MPa]
Location of max. stresses u (°)
1m 1:5m 2m 2:5m Solid
918 813 835 900 1120
65 27 28 28 31
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Table 2 Basic geometry and material parameters of the thin-rim gear Profile Number of teeth on gear Normal pressure angle Module Whole depth Addendum modification factor Outside diameter Tooth and rim width Hub diameter Rim thickness
Involute 39 24° m ¼ 4:5 mm h ¼ 10:41 mm x ¼ 0:0593 do ¼ 184:7 mm w ¼ 28 mm dh ¼ 100 mm t ¼ ð1; 1:5; 2; 2:5Þm
Material Modulus of elasticity PoissonÕs ratio Fatigue strength coefficient Fatigue ductility coefficient Exponent of strength Fatigue ductility exponent Hardening exponent Strength coefficient Surface factor
42CrMo4 E ¼ 2:06 105 MPa m ¼ 0:3 r0f ¼ 1820 MPa e0f ¼ 0:65 b ¼ 0:08 c ¼ 0:76 n0 ¼ 0:14 K 0 ¼ 2259 MPa Csurf ¼ 0:5–1:0
Fig. 3. The loading cycle for fatigue analysis.
parameters––fatigue strength coefficient (r0f ) and fatigue ductility coefficient (e0f ) related to fatigue process, exponent of strength (b) and fatigue ductility exponent (c), etc. [7,11]. The usual way of presenting fatigue test results is to plot elastic and plastic strain range against number of loading cycles. Material curve can be fully characterised by previously described material parameters r0f , b, e0f , c (Fig. 4) as is shown in Eq. (2). This curve is divided into an elastic component and a plastic component, respectively. The transition point P (Fig. 4) defines the difference between high cycle fatigue (HCF) versus low cycle fatigue (LCF). This type of behaviour is known as the Coffin–Manson relation, described by the following equation: De ra Dep r0f ¼ þ ¼ ð2Nf0 Þb þ e0f ð2Nf Þc : 2 E 2 E
ð2Þ
It is a fact that the e–N method is not ideal to analyse fatigue damage initiation on the micro-structural level, as the micro-crack initiation in crystal grains and dislocation theory are not taken into account. It has been established in work of Suresh [11] and Bhattacharya et al. [13] that fatigue damage initiation is represented by the transition of a certain number of loading cycles when the first fatigue damage occurs, on the basis of the assumed initial homogeneous state of material. Thus, the e–N procedure represents a very useful method to determine, where fatigue damage initiation, in time domain, is most probable [11,12]. Basic parameters influencing fatigue life of thin-rim gears are: particularly machine component size (Csize ), the type of loading (Cload ) and effect of surface finish and treatment (Csur ), which is taken into account in bending fatigue analysis of thin-rim gears. Practically, all kinds of bending fatigue failures start at the external surface of the component. It is well known, that fatigue properties are very sensitive to surface conditions. The surface finish correction factor Csur is presented in Fig. 5, in dependence on surface roughness Ra and tensile strength of the material Rm [7]. Using this assumption the real service life of gears
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Fig. 4. Strain–life (e–N ) method for fatigue crack initiation.
Fig. 5. Surface finish correction factor Csur .
may be reduced in regard to the appropriate value of Csur , which is in present study considered through the following equation: DrFLr ¼ DrFL Csur ;
ð3Þ
where DrFLr is the real fatigue limit and DrFL is the fatigue limit of polished laboratory specimen. Table 3 shows the results of fatigue analysis as number of cycles for fatigue crack initiation at tooth root of thin-rim gear. Fatigue analyses have been carried out for the most loaded material point at the tooth root, which was the position of the greatest tensile stress (see Fig. 2). The thin-rim gear design (different values of the rim thickness), as well as the influence of surface finish and treatment, has the most influence upon the fatigue crack initiation at tooth root. Cases of design and
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Table 3 Number of cycles for fatigue crack initiation Rim thickness t [mm]
1m 1:5m 2m 2:5m Solid
Ni ––Number of loading cycles for fatigue crack initiation Surface finish and surface treatment Poor machined and no treatment
Averaged machined and no treatment
Good machined and no treatment
Polished and no treatment
Good machined and shot peened
Polished and shot peened
Good machined and nitrided
Polished and nitrided
1.231 105 5.809 105 2.118 105 1.231 105 2.079 104
2.940 105 1.549 106 5.284 105 2.940 105 4.068 104
7.429 105 4.334 106 1.388 106 7.429 105 8.605 104
1.866 107 >108 3.950 107 1.866 107 1.343 106
8.301 106 >108 1.701 107 8.301 106 6.681 105
8.441 107 >108 >108 8.441 107 6.195 106
>108 >108 >108 >108 9.276 107
>108 >108 >108 >108 >108
surface treatment of the truck gears are most common. The suitable choice for a given case can be established based on the present analyses (see Table 3).
3. Fatigue crack growth In order to perform fatigue crack growth analysis, a BEM model has been created. The latest development known as the dual boundary element method (DBEM) crack modelling strategy, which is incorporated in the BEASY analysis programme [2], has been used to perform the crack growth analysis. For crack growth simulation, an incremental type analysis is used, where knowledge of both the direction and size of the crack increment extension are necessary. The incremental direction at the crack tip for next extension is determined by linear fracture mechanics criteria, involving stress intensity factors (SIF) as the prime parameters [1]. The boundary element model for crack growth analysis is shown in Fig. 6. The initial crack has been placed perpendicularly to the surface at the previously determined location for the initiation in the tensile area of the tooth root (see Section 2). The length of the initial crack is set equal to 0.2 mm. This value is estimated as a crack size, which identifies the transition from nucleation to propagation phase. For
Fig. 6. Boundary element model of gear with initial crack in the tooth fillet.
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Fig. 7. Mode I stress intensity factors versus crack length.
incremental simulation, the automatic crack propagation option of BEASY [2] was used. A uniform crack increment size of 0.2 mm is prescribed. The J -integral method to calculate crack tip stress intensity factors and maximum principal stress theory as a crack growth direction criterion are used. Fig. 7 shows the mode I stress intensity factors as a function of the crack length. The mode I stress intensity factors gradually increase with increasing crack length. The values of mode I stress intensity factors have been determined to be significantly larger than those of mode II stress intensity factors. Fig. 8 shows the calculated crack propagation paths for different values of the rim thickness. The results suggest how the gear will behave once a crack has initiated in the fillet region. For cases solid, 2:5m, 2m and 1:5m, crack propagates only through the tooth. For rim thickness 1m, the crack trajectory is through the rim. This kind of failure mode (tooth breakage or rim breakage) can be an important design information.
Fig. 8. Simulated crack propagation paths for various rim thickness (2:5m, 2m, 1:5m and 1m).
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655
Table 4 Load spectrum of first gears of truck gearbox [4] Load case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Relative loading Repetition in 1000 km
1.07 1
1.00 13
0.92 202
0.84 487
0.76 160
0.69 65
0.62 81
0.54 62
0.47 86
0.39 108
0.31 114
0.23 105
0.15 94
0.08 69
Fig. 9. Predicted crack propagation life of gear tooth.
In the framework of the LEFM, the propagation rate of cracks can be described by the ParisÕ equation da n ¼ CðDKÞ ; dN
ð4Þ
where DK is the stress intensity factor range, C and n are experimentally determined parameters. The cyclic load used for determination of remaining life has been based on the measurements on loaded vehicles [4] (Table 4). The value of loading and the corresponding number of its repetitions on the gear for characteristic driving conditions are given for an interval of 1000 km. The relative load is normalised by the load F ¼ 1737 N/mm, which has been used for the loading cycle at fatigue crack initiation analysis (see Fig. 3). Two couples of material constants have been used in ParisÕ law (C ¼ 1:575 1018 mm/cycle/ p different n (MPap mm) , n ¼ 4:689 for carburised layer with thickness of 0.6 mm, and C ¼ 4:202 1017 mm/cycle/ (MPa mm)n and n ¼ 4:144 for core, respectively [4]). Previously calculated stress intensity factor ranges, shown in Fig. 7, have been used for prediction of remaining life of the cracked tooth. The predicted number of crack propagation cycles is given in Fig. 9. It is evident that the largest life occurs in the case of rim thickness equal 1:5m since it has the lowest stress intensity factors.
4. Conclusions This paper presents a numerical model of bending fatigue life for thin-rim gears of truck gearboxes. The results include damage initiation as well as crack propagation in the most critical region for bending
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fatigue. On the basis of presented analyses, the number of loading cycles required for the damage initiation and number of loading cycles for fatigue crack propagation have been predicted. Critical region for crack initiation has been determined through a stress–strain analysis in the framework of the FEM. The fatigue initiation has been based on maximum principal stress r1 distribution. The tensile side of the loaded tooth, which is further important for crack growth phase, is focused. It is observed that the critical location for crack initiation moves toward the tooth root area with decreasing rim thickness. Providing e–N fatigue initiation analysis for different kinds of surface finish and treatment and different types of thin-rim gear design, the corresponding number of loading cycles for initial fatigue damage is determined. Then, a numerical analysis to investigate the effect of rim thickness on gear tooth crack propagation path and remaining life has been performed. The results obtained show the importance of rim thickness on the kind of expected failure mode. For a rim thickness greater than 1:5m, the predicted crack would propagate rather through the tooth and not through the rim of the gear. This kind of damage is less critical for the presented application of the truck gearbox. The proposed model enables to determine the whole service life under given loading cycles, adequate fatigue material parameters and predicted crack growth directions. The estimated bending fatigue life of the gear can deviate from real service life because some effects like residual stresses and possible causes of retardation of crack propagation (crack closure) have not been taken into account. In order to validate the present numerical results, an adequate experimental work should be performed. More reliable prediction could be expected by evaluating effects of gear geometry (thickness and position of web, etc.), and this requires further research work.
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