Gradient Theory of Damage Coupled to Frictional Contact and Wear, and Its Numerical Treatment

In this paper finite element approaches for fretting fatigue are proposed on the basis of a non-local model of continuum damage coupled to friction and wear. The model is formulated in the frame-work of a standard material. In a previous paper this was done in the spirit of Maugin, where an extra entropy flux is introduced in the second law in order to include the gradient of the internal variable in a proper manner. In this paper we follow instead the ideas of Fremond and others, where this extra entropy flux is no longer needed, but instead new non-classical balance laws associated to damage, friction and wear, respectively, are derived from the principle of virtual power. The standard material is then defined as usual by state laws based on free energies and complementary laws based on dissipation potentials. In particular, we pick free energies and dissipation potentials that correspond to a non-local continuum damage model coupled to friction and wear. In addition, the boundary conditions at the contact interface creates a coupling between damage and wear. This is a key feature of our model, which makes it very useful in studies of fretting fatigue. By starting from a variational formulation of the governing equations, two different finite element algorithms are implemented. Both algorithms are based on a Newton method for semi-smooth equations. In the first algorithm the Newton method is applied to the entire system of equations, while in the second algorithm the system of equations is split into two different parts such that an elastic wear problem is solved for fixed damage followed by the solution of the damage evolution problem for the updated displacements and contact forces in an iterative process. The latter algorithm can be viewed as a Gauss-Seidel scheme. The numerical performance of the algorithms is investigated for three two-dimensional examples of increasing complexity. Based on the numerical solutions, the behavior of the model is also discussed. For instance, it is shown numerically how the initiation of damage depends on the contact geometry, the coefficient of friction and the evolution of wear.