Keywords: gears, bending fatigue, service life, fatigue crack initiation, fatigue crack propagation, computational simulations.

A computational model for determination the service life of gears in regard to bending fatigue in a tooth root is presented. Gears are very specific machine parts subjected to fatigue loads (surface contact fatigue and bending fatigue). The bending fatigue process leading to tooth breakage is divided into crack initiation () and crack propagation () period, where the complete service life is defined as .

The model for the fatigue crack initiation presented in this paper is based on the continuum mechanics approach, where it is assumed that material is homogeneous and isotropic. The strain-life method ( ) is used to determine the number of stress cycles required for the fatigue crack initiation, where it is assumed that the crack is initiated at the point of the highest stress in the material. The total cyclic strain range comprises two components (elastic and plastic cyclic strain range and ) [1,2]. The number of stress cycles required for the fatigue crack initiation for the applied stress range is solved iterative and determined numerically using finite element method (FEM), and the appropriate material parameters like are Young's modulus , cyclic strain hardening exponent , fatigue strength coefficient , fatigue ductility coefficient , exponent of strength and the fatigue ductility exponent . The computational analyses also consider the influence of surface finish factor on fatigue strength, which is strongly related to tensile strength of the material and surface roughness of gear flanks .

Gear tooth crack propagation was simulated using a FEM method based computer program which uses principles of linear elastic fracture mechanics (LEFM). The initial crack was placed at the previously determined location in the fillet area of gear tooth. It has been assumed that the initial crack corresponds to the threshold crack length and that condition for validity of LEFM are fulfilled. The functional relationship between the stress intensity factor and crack length , which is needed for determination of the required number of loading cycles for a crack propagation from the initial to the critical length, is obtained using displacement correlation method (DCM). For prediction of the crack extension angle the maximum tangential stress (MTS) criterion is used. The simple Paris equation is then used for the further prediction of the fatigue crack growth.

The presented model is used for the computational determination of the complete service life of real spur gear, which is made of high strength alloy steel 42CrMo4 (through-hardened) with Young's modulus MPa and Poison's ratio . Using the material parameters , MPa, , and , the stress cycles, , have been determined using the program MSC/FATIGUE [3]. The FEM-programme FRANC2D [4] has been used for the numerical simulation of the fatigue crack growth, where the initial crack length was equal to mm. The fracture toughness MPa , and the material parameters mm/cycl/(MPa ) and have been determined previously by the three-point bending samples according to ASTM E 399-80 standard.

The results of the computational analyses show that the ratio among the periods of initiation and end of propagation (i.e. final breakage) depends on the stress level. At low stress level almost all service life is spent in crack initiation, but at high stress levels the significant part of the life is spent in the crack propagation. It is very important cognition by determination the service life of real gears in the praxis, because majority of them really operate with loading conditions close to the fatigue limit. The computational results correspond well with the available experimental data for the same material and same thermal treatment as used in this study.

1

Suresh S., "Fatigue of Material", Cambridge University Press (Second Edition), Cambridge, 1998.

2

Dieter G.E., "Mechanical Metallurgy", McGraw-Hill Book Company, Metric Edition, 1988.

3

MSC/FATIGUE, "Quick Start Guide", McNeal-Schwendler Corporation, Los Angeles, 1999.

4

FRANC2D, "User's Guide", Version 2.7, Cornell University, 1998.

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