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Numerical calculation of bending fatigue life of thin-rim spur gears J. Kramberger * , M. SSraml, I. Potr cc, J. Fla ssker 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 toexamine the bending fatigue life of thin-rim spur gears of truck gearboxes. The gear service life is divided into theinitiation phase of the damage accumulation and the crack growth, respectively. The analysis of thin-rim gear fatiguelife has been performed using the finite element method and the boundary element method. The continuum mechanicsbased 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-elasticfracture mechanics. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Mechanical elements; Numerical modelling; Thin-rim gears; Bending fatigue; Crack initiation; Fatigue crack growth; Lifeprediction 1. Introduction The fatigue process of mechanical elements is a material characteristic and depends upon cyclic plas-ticity, 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). Bendingfatigue 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 gearmaterial. However, bending fatigue can be generally divided into two main phases: (i) initiation of micro-cracks 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 givedetailed information of the bending fatigue life of gears, especially thin-rim gears. In gear strength cal-culations, 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)00024-9Engineering Fracture Mechanics 71 (2004) 647–656www.elsevier.com/locate/engfracmech force varies during meshing of gears. Some researchers have investigated the effects of the rim thickness onstresses [3,8,9], but only a few have performed a fatigue crack growth analysis for thin-rim gears [5,6]. Nomenclature b exponent of material strength c fatigue ductility exponent d o outside diameter d h hub diameter h whole depth m modulus n exponent of Paris equation n 0 material hardening exponent t rim thickness w tooth and rim width C parameter of Paris equation C load loading factor C size size factor C surf surface factor E Young Õ s elastic modulus F applied normal force at the highest point of a single tooth contact K 0 material strength coefficient K I mode I stress intensity factor D K stress intensity factor range N 0 f number of loading cycles for crack initiation R stress (loading) ratio T time e 0 f fatigue ductility coefficient D e deformation range D e p plastic deformation range m Poisson ratio number r 1 maximum principal stress r 0 f fatigue strength coefficient D r stress range D r FL fatigue limit of polished laboratory specimen D r FLr real fatigue limit u tangential angle at tooth root x addendum modification factorBEM boundary element methodDBEM dual boundary element methodFEM finite element methodHCF high cycle fatigueLCF low cycle fatigueLEFM linear elastic fracture mechanicsSIF stress intensity factorP transition point 648 J. Kramberger et al. / Engineering Fracture Mechanics 71 (2004) 647–656 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 bedeveloped.The purpose of present study is to elaborate a new model for prediction of bending fatigue life of thin-rim spur gears of the truck gearboxes, which consists of fatigue crack initiation and fatigue crack propa-gation phase, respectively. The proposed model of crack initiation (phase I) is based on the continuummechanics assumptions, considering the adequate fatigue parameters of the materials. Stress–strain cal-culations of thin-rim gear are performed in the framework of the finite element method (FEM), and theseresults are further used for fatigue analysis [7,10]. Concerning the numerical model of the selected me-chanical component, the material is assumed to be homogeneous and isotropic, i.e. without imperfectionsor 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 phaseof proposed model.Once a fatigue crack has started, the crack propagation phase should be studied (phase II). This includesdetermining 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-rimgears. Gear tooth crack propagation is simulated using computer program BEASY [2], which is based onthe 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 elementmodel 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 differentslots in the model. The loaded gear tooth is subjected to a normal force F ¼ 1737 N/mm applied at thehighest 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 inthe critical section of the gear tooth root. Five cases are considered with the same basic tooth geometry anddifferent rim thickness t (solid, 2 : 5 m , 2 m , 1 : 5 m and 1 m , 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. J. Kramberger et al. / Engineering Fracture Mechanics 71 (2004) 647–656 649 sides of tooth root, the most critical for further crack propagation is the tensile side of tooth root. The stressdistribution in the tooth fillet for tensile side is given in Fig. 2, where the maximum principal stresses r 1 (tensile stresses) are plotted. Position in the root area is described by angle u , which is defined as the anglebetween the symmetry line of the tooth and the tangent to the fillet curve, as shown in Fig. 2. The mag-nitude of the stresses decreases as the rim thickness decreases and then increases for the thinnest thickness(1 m ). 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 followingrelation: D r ¼ K 0 ð D e p Þ n 0 ; ð 1 Þ where n 0 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 deforma-tions, stresses and the number of loading cycles [11,12]. A frequently used procedure for calculating numberof cycles needed for fatigue crack to occur is based on strain–life relationship ( e – N ), and includes material Fig. 2. Maximum principal stress distribution in the tooth fillet.Table 1 r 1 stresses in material pointRim thickness t Max. principal stresses in the materialpoint at tooth root r 1 [MPa]Location of max. stresses u ( ° )1 m 918 651 : 5 m 813 272 m 835 282 : 5 m 900 28Solid 1120 31650 J. Kramberger et al. / Engineering Fracture Mechanics 71 (2004) 647–656 parameters––fatigue strength coefficient ( r 0 f ) and fatigue ductility coefficient ( e 0 f ) 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 numberof loading cycles. Material curve can be fully characterised by previously described material parameters r 0 f , b , e 0 f , c (Fig. 4) as is shown in Eq. (2). This curve is divided into an elastic component and a plasticcomponent, 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: D e 2 ¼ r a E þ D e p 2 ¼ r 0 f E ð 2 N 0 f Þ b þ e 0 f ð 2 N f Þ c : ð 2 Þ It is a fact that the e – N method is not ideal to analyse fatigue damage initiation on the micro-structurallevel, as the micro-crack initiation in crystal grains and dislocation theory are not taken into account. It hasbeen established in work of Suresh [11] and Bhattacharya et al. [13] that fatigue damage initiation isrepresented by the transition of a certain number of loading cycles when the first fatigue damage occurs, onthe basis of the assumed initial homogeneous state of material. Thus, the e – N procedure represents a veryuseful 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( C size ), the type of loading ( C load ) and effect of surface finish and treatment ( C sur ), which is taken into ac-count in bending fatigue analysis of thin-rim gears. Practically, all kinds of bending fatigue failures start atthe external surface of the component. It is well known, that fatigue properties are very sensitive to surfaceconditions. The surface finish correction factor C sur is presented in Fig. 5, in dependence on surfaceroughness R a and tensile strength of the material R m [7]. Using this assumption the real service life of gears Table 2Basic geometry and material parameters of the thin-rim gearProfile Involute Material 42CrMo4Number of teeth on gear 39 Modulus of elasticity E ¼ 2 : 06  10 5 MPaNormal pressure angle 24 ° Poisson Õ s ratio m ¼ 0 : 3Module m ¼ 4 : 5 mm Fatigue strength coefficient r 0 f ¼ 1820 MPaWhole depth h ¼ 10 : 41 mm Fatigue ductility coefficient e 0 f ¼ 0 : 65Addendum modification factor x ¼ 0 : 0593 Exponent of strength b ¼À 0 : 08Outside diameter d o ¼ 184 : 7 mm Fatigue ductility exponent c ¼À 0 : 76Tooth and rim width w ¼ 28 mm Hardening exponent n 0 ¼ 0 : 14Hub diameter d h ¼ 100 mm Strength coefficient K 0 ¼ 2259 MPaRim thickness t ¼ð 1 ; 1 : 5 ; 2 ; 2 : 5 Þ m Surface factor C surf ¼ 0 : 5 – 1 : 0Fig. 3. The loading cycle for fatigue analysis. J. Kramberger et al. / Engineering Fracture Mechanics 71 (2004) 647–656 651
