Comparison of non-elliptic contact models: Towards fast and accurate modelling of wheel–rail contact

Highlights

• Non-elliptic contact models based on virtual penetration are evaluated and compared to each other for the application to wheel–rail contact.

• The accuracy of the virtual-penetration-based models is problem dependent which results in less accurate patch estimation in certain contact cases.

• The pressure distribution estimation by these models deviates from the results of more accurate, but computationally expensive, CONTACT code.

• The effect of spin variation within the contact patch on creep forces in non-elliptic contact cases is shown.

Abstract

The demand to investigate and predict the surface deterioration phenomena in the wheel–rail interface necessitates fast and accurate contact modelling. During the past 20 years, there have been attempts to determine more realistic contact patch and stress distributions using fast simplified methods. The main aim of the present work is to compare some of these state-of-the-art, non-elliptic contact models available in the literature. This is considered as the first step to develop a fast and accurate non-elliptic contact model that can be used on-line with vehicle dynamics analysis. Three contact models, namely STRIPES, Kik-Piotrowski and Linder are implemented and compared in terms of contact patch prediction, as well as contact pressure and traction distributions. The evaluation of these models using CONTACT software indicate the need for improvement of contact patch and pressure estimation in certain contact cases.

Introduction

Accurate determination of the wheel–rail contact patch and calculation of stress distributions within it has gained importance due to the increasing need for the prediction of wear and rolling contact fatigue. At the same time, on-line application of the contact model in multi-body-systems (MBS) codes imposes restrictions on computational cost of the model. Thus, there is always a trade-off between accuracy and computational efficiency of the methods used to solve the wheel–rail contact problem. Generally, in today's commercial MBS codes [1], the contact patch is considered to be elliptic based on the Hertz theory. Some models approximate a non-elliptic contact patch by one or more ellipses (equivalent elastic, multi-Hertzian). The tangential stresses and creep forces are then calculated using the FASTSIM algorithm [2]. The accuracy of the obtained contact forces may be considered acceptable for the dynamic analysis of the vehicle–track interaction. However, the non-physical characterisation of the contact patch and inaccurate estimation of the stresses in certain cases leads to unrealistic wear and fatigue predictions. Therefore, there is a need for a non-elliptic contact model that is almost as fast.

Non-elliptic contact models can be divided into two categories: the models based on numerical methods such as the finite element method (FEM) or the boundary element method (BEM), and the models which predict the contact patch analytically. In this paper, these two categories are named as advanced numerical methods and fast non-elliptic methods, respectively.

In advanced numerical methods the contact conditions are satisfied in discrete points of the bodies in contact. The bodies should therefore be discretized into elements. The contact solution is highly dependent on the element size. To obtain smoother and more accurate results a finer mesh is needed which requires more computational effort.

FEM: The most rigorous method to use for solving contact problems is FEM. This is because it can include different sources of non-linearity originating from material property or contact geometry. However, due to numerical issues, the contact conditions and the friction law should be regularised. The regularisation should be treated carefully in order to achieve reliable results. Moreover, the need for a very fine mesh in the contact zone makes it too expensive from a computational point of view. For a summary on wheel–rail contact modelling using FEM see [3].

BEM: Solving rolling contact problems using BEM seems to be more common than using FEM. BEM-based methods generally utilize half-space theory. This requires the contact patch dimensions to be much smaller than the dimensions of the bodies in contact. Furthermore, it implies a flat (planar) contact patch. Using half-space theory, the deformations can be related to the stresses through influence functions. The deformation in one point is calculated by superposing the effect of tractions applied over the contact patch. The contact equations may be solved using different mathematical techniques. The CONTACT code [4] based on Kalker's complete theory of three-dimensional rolling [2] utilizes the variational method while Björklund and Andersson's method [5], ContLab, uses matrix inversion.

Since the real contact patch is not known in advance and the surface deformation–traction relationship is dependent on an integral over the contact area, one should start with a so-called potential contact area and do iterations to arrive at the real contact patch and contact pressure distribution. But such an iterative procedure slows down the models based on BEM. Knothe and LeThe [6] also proposed a model based on BEM for contact between bodies of revolution. This model seems to be faster since the patch is divided into strips rather than rectangular elements and the contact pressure distribution in the rolling direction is, in advance, assumed to be elliptic.

The computational expenses associated with advanced numerical methods motivate attempts to look for faster non-elliptic contact models. As mentioned in the preceding section, the iterative procedure to determine the contact patch is time consuming, hence, for a fast non-elliptic method the contact patch should be determined analytically. Several non-elliptic methods are introduced in the literature which are based on a so-called virtual penetration concept. A survey of such models is given in [7]. Apart from these models, the normal contact may be treated using Winkler elastic foundation in which the elastic deformation at each point is only dependent on the pressure at that point. However, it is believed that such a simplification leads to poor accuracy [2]. The Winkler foundation method may be improved by considering the effect of neighbouring points. Telliskivi [8] proposed a so-called semi-Winkler contact model in which the neighbouring points are connected by linear spring elements with the stiffness determined by experimental work or FEM analysis.

In this paper, the contact models based on the virtual penetration concept are investigated. Different formulations of the concept are discussed in the next section. The models based on these formulations are then compared and evaluated for wheel–rail contact in wheelset central and offset positions.