This thesis concerns the formulation of continuum mechanical models and the development of numerical methods for frictional heating, wear and damage in tribological systems.

The models are formulated in the spirit of generalized standard materials, i.e. by defining observable and internal state variables and stating the constitutive relations by means of gradients of a free energy and a dissipation potential. Here two extensions of this framework is discussed: (i) the case when the free energy is non-smooth and (ii) when gradients of internal variables are included in the constitutive model. The latter extension is used to formulate a model of damage which is non-local in the sense that the evolution of damage is governed by a boundary-value problem instead of a local evolution law. This way the well-known problem with mesh-dependent solutions is removed. Another feature of the gradient model is that the boundary conditions can be used to include couplings between bulk damage and processes at the boundary. This feature is used to establish a coupling between bulk damage and wear.

The governing equations are stated as a variational problem which is discretized and reformulated as a system of semi-smooth equations. The numerical methods are based on a modified Newton method for semi-smooth equations. The Newton method is applied in two different ways, either directly to the whole system of equations or by splitting the system of equations into different parts which are treated in a Gauss-Seidel scheme. In this case the Newton method is used to solve the different parts in each Gauss-Seidel step. This approach is applied to three different problems: (i) thermoelastic wear problems including frictional heating and heat conduction through the contact interface (ii) the gradient damage problem and (iii) the coupled wear and damage problem. For the thermoelastic wear problem it is found that the decoupled approach is slightly more efficient than the the direct application of the Newton method. The numerical solution shows how the wear process is influenced by the change of geometry due to frictional heating. Furthermore, it is shown how fretting can be induced by a prescribed varying boundary temperature. For the plain gradient damage problem the direct application of the Newton method is more efficient. Finally, for the coupled wear and damage problem the two approaches are about equally efficient. Numerical solutions of this problem shows how the initiation of damage depends on contact geometry, friction and wear.