Contact Problem with Coulomb Friction

In the previous two chapters we have introduced problems with obstacles and the problem of linear elasticity. The goal of this chapter is to combine those two and to introduce probably the most important ”nonsmooth“ problem of mechanics: the problem of an elastic body in contact with a rigid obstacle. A simple way how to define this problem is to restrict the normal displacement of certain boundary points in the elasticity problem by means of the unilateral contact constraints known from Chapter 9. This leads to a formulation known as contact problem without friction; in most cases, this formulation does not fully reflect the physical reality. To achieve this, one has to take into account the friction between the body and the obstacle—this friction restricts the tangential component of the displacement on the ”contact boundary“. There are several models of contact problems with friction (cf., e.g., Klarbring, 1986; Lemaitre and Chaboche, 1994). The most realistic one is probably the model of Coulomb friction. In the following sections we will introduce this model, derive its reciprocal (dual) formulation which leads to a quasi-variational inequality and apply the nonsmooth Newton’s method of Chapter 3 to the numerical solution of a suitable discrete approximation. We further show that the discretized problem can be formulated as a linear complementarity problem which, again, can be solved by the nonsmooth Newton’s method. Finally, we formulate a design problem; the control (design) variable is the coefficient of friction (a property of the material) and the goal to maximize the tangential adhesion between the body and the obstacle. This design problem is an MPEC with an LCP as equilibrium constraint.