Keywords: contact mechanics, rolling, friction, wear, discretisation techniques, iterative solvers.

In order to study the dynamic behaviour of railway vehicles, and to numerically assess wear and rolling contact fatigue (RCF), one must solve the frictional rolling contact problem. Drawbacks of the variational contact model "CONTACT" by Kalker [1] are its first order accuracy and its consequent high computation times. In this paper we present two important leads in pursuit of second order accurate rolling contact models.

"CONTACT" uses a half-space approach. The first central equation is the definition of the (micro-) slip s, in terms of the prescribed rigid slip w and (spatial derivative of) displacement differences u. The second important equation describes u as the integral over the contact area of surface tractions p multiplied by influence functions A.

Our first observation is that the rolling contact problem is dominated by the equation for the slip, a (hyperbolic) transport phenomenon. In the case of infinite friction coefficient, u can be solved entirely from the equation for the slip, after which the traction-displacements relation may be used to recover the corresponding p. In this view, the original discretisation used in "CONTACT" can be interpreted as using the implicit Euler rule for the discretisation of the slip. This new view opens the way for alternatives, such as using the trapezoidal rule which is second order accurate.

Our second new observation is that the two equations may be combined and reorganised. It leads to a formulation employing an elliptical operator, a symmetric positive definite system matrix A, instead of a matrix A-A' representing a space derivative with less favourable properties. A further attractive feature of this approach is that it simplifies the treatment of the boundaries. It removes the need to know u at the leading edge boundary and allows incorporation of p=0 immediately. Second order accurate discretisation of the reorganised form seems possible but has not yet been formulated.

As a result of the latter insight we have found a new iteration scheme that is called SteadyGS. Experiments with this approach demonstrate that it increases performance by a factor two or more, at the same time the robustness of the model is improved as well.

1

J. Kalker, "Three-Dimensional Elastic Bodies in Rolling Contact", Solid Mechanics and its Applications, Kluwer Academic Publishers, 1990.

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