Numerical methods for determining stress intensity factors vs crack depth in gear tooth roots

Abstract

The calculation of the strength of gears is usually carried out by standard procedures. The calculation of the service life of a gear with a crack in a tooth root, however, is possible only by numerical methods. The first step in determining the service life in such a case is the evaluation of the stress intensity factor as a function of crack shape and depth. Two-dimensional analysis is appropriate for this since it is fast and efficient. Here, the direction of crack propagation from the tip of the initial crack is determined using a special numerical algorithm, whereby the direction of maximum strain energy release rate G is sought. The procedure is repeated incrementally. In order to study the influence of realistic applied loads at the point of contact, the gear has to be treated three-dimensionally. Here, the propagation of each point along the crack tip profile is also assumed to be in the direction of the maximum strain energy release rate. The crack depth is determined in such a way that the stress intensity factor on the crack tip profile is constant. The result of such numerical calculations gives a diagram of the stress intensity factor as a function of crack depth. With known gear material properties it is then possible to calculate the service life of the gear by numerically integrating the Paris equation. This article describes two- and three-dimensional methods for monitoring the crack propagation for a particular gear geometry, including the effects of varying through-thickness load behaviour.