The ALE-formulation of bodies in rolling contact Theoretical foundations and finite element approach

Numerical methods for the analysis of rolling contact problems are of high importance for the optimization of the components, car tires for example. Today, finite element methods for the computation of stationary rolling contact have been developed for industrial use. However, from a review of the scientific literature it has been concluded that the theoretical foundations of the relative kinematic description, i.e. an arbitrary Lagrangian Eulerian (ALE) formulation, seems to be not well understood in detail. This presentation is aimed to close this gap. Starting from the basic kinematic description of the ALE-formulation of rolling the weak form of the equations of motion are developed. By this careful analysis an additional flux contribution is derived which has not been recognized in the scientific literature so far. The contact problem is described for the treatment of two deformable bodies in rolling contact within the framework of well established methods for the numerical treatment of contact in the Lagrangian picture in general. Essential differences between the Lagrangian description and the ALE-description of rolling are shown up and discussed in detail. For a comprehensive presentation the finite element discretization is restricted to the case of a deformable wheel rolling on a rigid plane surface. However, the sketched algorithmic approach is valid in general. The efficiency of the numerical algorithms developed so far are discussed by the analysis of a simple three-dimensional example. Critical remarks and an outlook to further research in this field concludes this presentation.